From fef1cf6c802595da3c60d3ebbb2f4627833a574c Mon Sep 17 00:00:00 2001
From: dhruvbalwada <dhruvbalwada@users.noreply.github.com>
Date: Sun, 28 Jan 2024 14:15:37 +0000
Subject: [PATCH] deploy: d20706857d8a0ce64dc951969b03910fb220ef91

---
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new file mode 100644
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diff --git a/_sources/bibliography.md b/_sources/bibliography.md
new file mode 100644
index 00000000..d977a9bd
--- /dev/null
+++ b/_sources/bibliography.md
@@ -0,0 +1,4 @@
+# References
+
+```{bibliography}
+```
diff --git a/_sources/intro.md b/_sources/intro.md
new file mode 100644
index 00000000..23a5fef7
--- /dev/null
+++ b/_sources/intro.md
@@ -0,0 +1,7 @@
+# Introduction
+
+The climate system is composed of many interacting components (e.g., ocean, atmosphere, ice) and described by complex nonlinear equations. To simulate, understand and predict climate, these equations are solved numerically under a number of simplifications, therefore leading to errors. The errors are the result of numerics used to solve the equations and the lack of appropriate representations of processes occurring below the resolution of the climate model grid (i.e., subgrid processes).
+
+The goal of this book is to conceptualize the problems associated with climate models within a simple and computationally accessible framework. We will introduce the readers to climate modeling by using a simple tool, the {cite}`Lorenz1995` (L96) two-timescale model. We discuss the numerical aspects of the L96 model, the approximate representation of subgrid processes (known as parameterizations or closures), and simple data assimilation problems (a data-model fusion method). We will then use the L96 results to demonstrate how to learn subgrid parameterizations from data with machine learning, and then test the parameterizations offline (apriori) and online (aposteriori), with a focus on the interpretability of the results.
+
+The book was created by and as part of [M<sup>2</sup>LInES](https://m2lines.github.io/), an international collaboration supported by [Schmidt Futures](https://www.schmidtfutures.com/), to improve climate models with scientific machine learning. The goal for this book was for our team to work together and learn from each other; in particular, to get up to speed on the key scientific aspects of our collaboration (parameterizations, machine learning, data assimilation, uncertainty quantification) and to develop new ideas. Ultimately, we are happy to share these resources with the scientific community, to introduce our research ideas and foster the use of machine learning techniques for tackling climate science problems.
diff --git a/_sources/notebooks/DA_demo_L96.ipynb b/_sources/notebooks/DA_demo_L96.ipynb
new file mode 100644
index 00000000..1b8f2da4
--- /dev/null
+++ b/_sources/notebooks/DA_demo_L96.ipynb
@@ -0,0 +1,951 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a013986b",
+   "metadata": {},
+   "source": [
+    "# Data Assimilation demo in the Lorenz 96 (L96) two time-scale model\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c33e3435",
+   "metadata": {},
+   "source": [
+    "## What is DA? Why do we do it?\n",
+    "\n",
+    "Data assimilation is an approach for combining data (e.g. observations) with prior knowledge (e.g. from a physical model) to obtain an estimate of the true state of a system. We use data assimilation to generate accurate initial conditions for weather and climate predictions, and to produce reanalysis products (state estimates) of the atmosphere-ocean-sea ice-land system. \n",
+    "\n",
+    "Conceptually, data assimilation follows the Bayes' Rule:\n",
+    "\n",
+    "\\begin{align}\n",
+    "\\ P(X|Obs)=\\frac{P(Obs|X)P(X)}{P(Obs)}\n",
+    "\\end{align}\n",
+    "\n",
+    "where $X$ is the state of the dynamic system, $P(X|Obs)$ is the posterior distribution, or the analysis in DA terms, and $P(X)$ is the prior distribution, or the forecast in DA terms. $P(Obs|X)$ is the observation distribution given the prior. Simply speaking, we want to know the best estimate of the current state $X$ of the system given the available observations. Below is a simple schematic representation of one DA step.\n",
+    "\n",
+    "```{figure} figs/DA_step.png\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "208a8f14",
+   "metadata": {},
+   "source": [
+    "## Design of the data assimilation experiments in L96\n",
+    "\n",
+    "We use the 2-scale Lorenz-96 model as our truth, or our representation of the \"real world\". We then use the 1-scale L96 model as our DA model, or our representation of the \"GCM\". Detailed description of this setup can be find here (https://m2lines.github.io/L96_demo/notebooks/gcm-analogue.html). This is what we call an idealized biased-model framework in DA research, in the sense that our DA model is biased compared to the truth, similar to real-world situations where our GCMs are biased against the natural weather or climate systems. \n",
+    "\n",
+    "We will follow the follwing steps when doing DA in this idealized biased L96 framework:\n",
+    "1. Specify a biased “GCM” model for performing DA, and set the parameters for the \"GCM\", observations, and DA method in `Config`.\n",
+    "    * Parameters to experiment in `Config`: forcing `F`, seasonal cycle, experiment length, \n",
+    "2. Generate a “truth” signal from the 2-scale L96 model\n",
+    "3. Collect sparse and noisy observations, sampled from “truth”\n",
+    "    * Parameters to experiment in `Config`: observation density in space, frequency in time, observational error\n",
+    "4. Perform DA using a method of choice\n",
+    "    * Parameters to experiment in `Config`: DA method (EnKF, 3DVAR, Hybrid EnKF), DA frequency, ensemble size (for EnKF), localization, inflation method and factor\n",
+    "    * Optional: parameter estimation to recover the true forcing\n",
+    "5. Assess DA performance\n",
+    "    * RMSE of DA experiment relative to truth\n",
+    "6. Analyze DA increments to assess GCM model error (optional)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "66584b6f",
+   "metadata": {},
+   "source": [
+    "### 1. Define our \"GCM\" and DA parameters to use throughout notebook"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9f81b262-6d04-44d8-9a61-ada0ea0c9acb",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import os\n",
+    "import time\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "from L96_model import L96, RK4, L96_2t_xdot_ydot, L96_eq1_xdot, L96s\n",
+    "import DA_methods\n",
+    "\n",
+    "rng = np.random.default_rng()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ebf223e6-ed8a-425a-b808-55b0cc59c69c",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def GCM(X_init, F, dt, nt, param=[0]):\n",
+    "    \"\"\"A toy `General Circulation Model` (GCM) that uses the single time-scale Lorenz 1996 model\n",
+    "    with a parameterised coupling term to represent the interaction between the observed coarse\n",
+    "    scale processes `X` and unobserved fine scale processes `Y` of the two time-scale model.\n",
+    "\n",
+    "    Args:\n",
+    "        X_init: Initial conditions of X\n",
+    "        F: Forcing term\n",
+    "        dt: Sampling frequency of the model\n",
+    "        nt: Number of timesteps for which to run the model\n",
+    "        param: Weights to give to the coupling term\n",
+    "\n",
+    "    Returns:\n",
+    "        Model output for all variables of X at each timestep along with the corresponding time units\n",
+    "    \"\"\"\n",
+    "    time, hist, X = (\n",
+    "        dt * np.arange(nt + 1),\n",
+    "        np.zeros((nt + 1, len(X_init))) * np.nan,\n",
+    "        X_init.copy(),\n",
+    "    )\n",
+    "    hist[0] = X\n",
+    "\n",
+    "    for n in range(nt):\n",
+    "        X = X + dt * (L96_eq1_xdot(X, F) - np.polyval(param, X))\n",
+    "        hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "\n",
+    "    return hist, time"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3d2efdbe",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class Config:\n",
+    "    # fmt: off\n",
+    "    K = 40                              # Dimension of L96 `X` variables\n",
+    "    J = 10                              # Dimension of L96 `Y` variables\n",
+    "    obs_freq = 10                       # Observation frequency (number of sampling intervals (si) per observation)\n",
+    "    F_truth = 10                        # F for truth signal\n",
+    "    F_fcst = 10                         # F for forecast (DA) model\n",
+    "    GCM_param = np.array([0, 0, 0, 0])  # Polynomial coefficicents for GCM parameterization\n",
+    "    ns_da = 2000                        # Number of time samples for DA\n",
+    "    ns = 2000                           # Number of time samples for truth signal\n",
+    "    ns_spinup = 200                     # Number of time samples for spin up\n",
+    "    dt = 0.005                          # Model timestep\n",
+    "    si = 0.005                          # Truth sampling interval\n",
+    "    seasonal = False                    # Option for adding a seasonal cycle to the forcing in the L96 truth model\n",
+    "    B_loc = 5                           # Spatial localization radius for DA in terms of model grid points\n",
+    "    DA = \"EnKF\"                         # DA method\n",
+    "    nens = 500                          # Number of ensemble members for DA\n",
+    "    inflate_opt = \"relaxation\"          # Method for DA model covariance inflation\n",
+    "    inflate_factor = 0.65                # Inflation factor\n",
+    "    hybrid_factor = 0.1                 # Weighting factor for hybrid EnKF method (weight of static B matrix)\n",
+    "    param_DA = False                    # Switch to turn on parameter estimation with DA\n",
+    "    param_sd = [0.01, 0.02, 0.1, 0.5]   # Polynomal parameter standard deviation\n",
+    "    param_inflate = \"multiplicative\"    # Method for parameter variance inflation\n",
+    "    param_inf_factor = 0.02             # Parameter inflation factor\n",
+    "    obs_density = 0.2                   # Fraction of spatial gridpoints where observations are collected\n",
+    "    DA_freq = 10                        # Assimilation frequency (number of sampling intervals (si) per assimilation step)\n",
+    "    obs_sigma = 0.5                     # Observational error standard deviation\n",
+    "    save_fig = False                    # Switch to save figure file\n",
+    "    save_data = False                   # Switch to save\n",
+    "    # fmt: on"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6d3052af",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 2. Generate `truth` run from two time-scale L96 model\n",
+    "\n",
+    "We initialise the L96 two time-scale model using a set of random normally distributed values for $X$, and zeros for $Y$, and run it forward for a period `ns_spinup` to allow the model to spinup. The $X$ and $Y$ components of the spinup are then used as initial conditions for another forward run for a time period `ns` to represent the unobserved `truth` field from which our observations will be derived.\n",
+    "\n",
+    "_Note that in reality we do not observe the fine scale components ($Y$s) and so this is what we aim to represent in the parameterisation in the `GCM()` function._"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e393d1bd-5838-402d-a381-a617c987d1f8",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Set up the \"truth\" 2-scale L96 model\n",
+    "M_truth = L96(Config.K, Config.J, F=Config.F_truth, dt=Config.dt)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a4696c6b-cfbd-4623-9952-bcf7a3e75783",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Generate the initial conditions of $X$ and $Y$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "18d8e02a-2dec-4724-9175-f5809414533e",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "M_truth.set_state(rng.standard_normal((Config.K)), 0 * M_truth.j)\n",
+    "\n",
+    "# The model runs for `ns_spinup` timesteps to spin-up\n",
+    "X_spinup, Y_spinup, t_spinup = M_truth.run(Config.si, Config.si * Config.ns_spinup)\n",
+    "\n",
+    "# The initial conditions for the first forecast (prior to DA)\n",
+    "# are the last time sample after spinup\n",
+    "X_init = X_spinup[-1, :]\n",
+    "Y_init = Y_spinup[-1, :]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a68c402e-e083-4cfe-b678-d4d8cd5e098f",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Using the generated initial conditions above, run **L96** to generate the `truth`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ae0f3b33-7cf7-47ae-9460-f735f8d46627",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "M_truth.set_state(X_init, Y_init)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4916b273-2823-4c6c-b3e4-fd1f3b5ce805",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "If given in `Config`, give `F` a **seasonal cycle** in the truth model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fceac2dd-543e-4688-99ce-a7eb2bcc4e41",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "if Config.seasonal:\n",
+    "    ann_period = 2000\n",
+    "    mon_period = 100\n",
+    "    mon_per_ann = ann_period / mon_period\n",
+    "    X_truth, Y_truth, t_truth = M_truth.run(Config.si, Config.si * mon_period)\n",
+    "    for i in range(1, int(Config.ns / mon_period)):\n",
+    "        M_truth.set_state(X_truth[-1, ...], Y_truth[-1, ...])\n",
+    "        M_truth.set_param(F=Config.F_truth + 2 * np.sin(2 * np.pi * i / mon_per_ann))\n",
+    "        X_step, Y_step, t_step = M_truth.run(Config.si, Config.si * mon_period)\n",
+    "        X_truth = np.concatenate((X_truth, X_step[1:None, ...]))\n",
+    "        Y_truth = np.concatenate((Y_truth, Y_step[1:None, ...]))\n",
+    "        t_truth = np.concatenate((t_truth, t_truth[-1] + t_step[1:None]))\n",
+    "else:\n",
+    "    X_truth, Y_truth, t_truth = M_truth.run(Config.si, Config.si * Config.ns)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c90c0fef-53be-4d0b-9105-9601862f6690",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Now, we generate climatological background (temporal) covariance `B_clim2` for the 2-scale L96 model."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "66ac4faf-ff3f-4152-bfb9-fcdb1f2c2234",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "B_clim2 = np.cov(X_truth.T)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "61df79fe-b945-43be-b08a-b63b93bca400",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Plotting values of $X$ and $Y$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1e66be99-a6d1-4134-92a4-e1a2ab8a6916",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(12, 10), dpi=150)\n",
+    "plt.subplot(221)\n",
+    "\n",
+    "# Snapshot of X[k]\n",
+    "plt.plot(M_truth.k, X_truth[-1, :], label=\"X\")\n",
+    "plt.plot(M_truth.j / M_truth.J, Y_truth[-1, :], label=\"Y\")\n",
+    "plt.legend()\n",
+    "plt.xlabel(\"k\")\n",
+    "plt.title(\"$X,Y$ @ $t=N\\Delta t$\")\n",
+    "plt.plot(M_truth.k, X_truth[0, :], \"k:\")\n",
+    "plt.plot(M_truth.j / M_truth.J, Y_truth[0, :], \"k:\")\n",
+    "plt.subplot(222)\n",
+    "\n",
+    "# Sample time-series X[0](t), Y[0](t)\n",
+    "plt.plot(t_truth, X_truth[:, 0], label=\"X\")\n",
+    "plt.plot(t_truth, Y_truth[:, 0], label=\"Y\")\n",
+    "plt.xlabel(\"t\")\n",
+    "plt.title(\"$X[0,t]$, $Y[0,t]$\")\n",
+    "plt.subplot(223)\n",
+    "\n",
+    "# Full model history of X\n",
+    "plt.contourf(M_truth.k, t_truth, X_truth)\n",
+    "plt.colorbar(orientation=\"horizontal\")\n",
+    "plt.xlabel(\"k\")\n",
+    "plt.ylabel(\"t\")\n",
+    "plt.title(\"$X(k,t)$\")\n",
+    "plt.subplot(224)\n",
+    "\n",
+    "# Full model history of Y\n",
+    "plt.contourf(M_truth.j / M_truth.J, t_truth, Y_truth)\n",
+    "plt.colorbar(orientation=\"horizontal\")\n",
+    "plt.xlabel(\"k\")\n",
+    "plt.ylabel(\"t\")\n",
+    "plt.title(\"$Y(k,t)$\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8a17c582-090e-48bf-b26a-adbd66fdbefb",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Generating climatological background covariance `B_clim1` for 1-scale L96 model. This climatological covariance will be used by the 3-dimensional variational (3DVar) DA method, or by the hybrid 3DVar-EnKF method. Since we cannot observe every state variable at every possible location, when assimilting sparse observations into the model, we need to use each observation to update multiple model locations, usually around the observed location. The covariance are thus needed to project the model-observation misfits between different model locations in certain DA methods such as 3DVar and EnKF."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "dd8b5cc7",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "M_1s = L96s(Config.K, F=Config.F_truth, dt=Config.dt, method=RK4)\n",
+    "M_1s.set_state(X_init)\n",
+    "X1_truth, t1_truth = M_1s.run(Config.si * Config.ns)\n",
+    "B_clim1 = np.cov(X1_truth.T)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d3f14f72",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 3. Generate synthetic observations\n",
+    "\n",
+    "Here we generate a set of sparse observations by sampling from the `X_truth` run and adding some random Gaussian noise. The observations are stored as individual samples, each containing the time, location, and value."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e01dbe50-741b-4ce5-88ac-4f583e19d11c",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Sample the \"truth\" to generate observations at certain times (t_obs) and locations (l_obs)\n",
+    "t_obs = np.tile(\n",
+    "    np.arange(Config.obs_freq, Config.ns_da + Config.obs_freq, Config.obs_freq),\n",
+    "    [int(Config.K * Config.obs_density), 1],\n",
+    ").T\n",
+    "\n",
+    "l_obs = np.zeros(t_obs.shape, dtype=\"int\")\n",
+    "for i in range(l_obs.shape[0]):\n",
+    "    l_obs[i, :] = rng.choice(\n",
+    "        Config.K, int(Config.K * Config.obs_density), replace=False\n",
+    "    )\n",
+    "\n",
+    "X_obs = X_truth[t_obs, l_obs] + Config.obs_sigma * rng.standard_normal(l_obs.shape)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "89ee8b4d-0539-4621-8d45-30769452a155",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The covariance matrix `R` for the observations (used to express the uncertainty in the observations during DA) is given as a diagonal matrix with entries defined by the square of the \"observational error standard deviation\". $R$ is a diagonal matrix because we assume that the errors of different observations are independent. The function `find_obs` is used here to find the locations and values of a set of observations given the observed times. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "abc69e52",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Calculated observation covariance matrix, assuming independent observations\n",
+    "R = Config.obs_sigma**2 * np.eye(int(Config.K * Config.obs_density))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "34000d8e-bb3c-4ca7-b760-e186ec7c2f80",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=[10, 6], dpi=150)\n",
+    "plt.plot(range(1000, 1500), X_truth[1000:1500, 0], label=\"truth\")\n",
+    "plt.scatter(\n",
+    "    t_obs[100:150, 0],\n",
+    "    DA_methods.find_obs(0, X_obs, t_obs, l_obs, [t_obs[100, 0], t_obs[150, 0]]),\n",
+    "    color=\"k\",\n",
+    "    label=\"obs\",\n",
+    ")\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c72bc6d5",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "#### Check the Model Background Covariances for the variable `X`\n",
+    "\n",
+    "The climatological covariances of both the \"truth\" (2-scale L96) and \"GCM\" (1-scale L96) are computed a-priori from a long run. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c053ebb0-0e36-41cc-bca5-141ddd15a165",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "B_corr1 = np.zeros(B_clim1.shape)\n",
+    "B_corr2 = np.zeros(B_clim2.shape)\n",
+    "for i in range(B_clim1.shape[0]):\n",
+    "    for j in range(B_clim1.shape[1]):\n",
+    "        B_corr1[i, j] = B_clim1[i, j] / np.sqrt(B_clim1[i, i] * B_clim1[j, j])\n",
+    "        B_corr2[i, j] = B_clim2[i, j] / np.sqrt(B_clim2[i, i] * B_clim2[j, j])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "577e9b3d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(16, 6), dpi=150)\n",
+    "plt.subplot(121)\n",
+    "plt.contourf(B_corr1, cmap=\"bwr\", extend=\"both\", levels=np.linspace(-0.95, 0.95, 20))\n",
+    "plt.colorbar()\n",
+    "plt.title(\"Background correlation matrix: 1-scale L96\")\n",
+    "plt.subplot(122)\n",
+    "plt.contourf(B_corr2, cmap=\"bwr\", extend=\"both\", levels=np.linspace(-0.95, 0.95, 20))\n",
+    "plt.colorbar()\n",
+    "plt.title(\"Background correlation matrix: 2-scale L96\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "760a249a",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 4. Run Data Assimilation\n",
+    "\n",
+    "During each DA cycle, we produce an updated state estimate for all $K$ grid points, and all `n` number of ensemble members.\n",
+    "\n",
+    "The inputs are:\n",
+    "\n",
+    "- prior estimate (the forecast at time t for all K grid points, and all n number of ensemble members)\n",
+    "- observations (N observations)\n",
+    "- operator matrix (N x K)\n",
+    "- covariance over obs (N x N)\n",
+    "- covariance over model (K x K)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b7c5c9f4-0ae8-4d07-a458-6d2b1ec4af33",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "start_time = time.time()\n",
+    "\n",
+    "# Set up array to store DA increments\n",
+    "X_inc = np.zeros((int(Config.ns_da / Config.DA_freq), Config.K, Config.nens))\n",
+    "if Config.DA == \"3DVar\":\n",
+    "    X_inc = np.squeeze(X_inc)\n",
+    "\n",
+    "t_DA = np.zeros(int(Config.ns_da / Config.DA_freq))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8d49ce4c-2a11-4207-9bd1-05b16607b2fd",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Initialize ensemble with perturbations\n",
+    "i_t = 0\n",
+    "ensX = X_init[None, :, None] + rng.standard_normal((1, Config.K, Config.nens))\n",
+    "X_post = ensX[0, ...]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "236c576b-be9f-4353-9172-455853f08d2e",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Localize the covariance matrix in advance to avoid the calculation at each DA step\n",
+    "# The localization weight matrix is also saved for EnKF\n",
+    "\n",
+    "B_loc, W_clim = DA_methods.localize_covariance(B_clim1, loc=Config.B_loc)\n",
+    "\n",
+    "if Config.param_DA:\n",
+    "    mean_param = np.zeros((int(Config.ns_da / Config.DA_freq), len(Config.GCM_param)))\n",
+    "    spread_param = np.zeros((int(Config.ns_da / Config.DA_freq), len(Config.GCM_param)))\n",
+    "    param_scale = Config.param_sd\n",
+    "    W = np.ones((Config.K + len(Config.GCM_param), Config.K + len(Config.GCM_param)))\n",
+    "    W[0 : Config.K, 0 : Config.K] = W_clim\n",
+    "else:\n",
+    "    W = W_clim\n",
+    "    param_scale = np.zeros(Config.GCM_param.shape)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "50e95487-435f-4bd4-bfab-6e0cb6893b30",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "ens_param = np.zeros((len(Config.GCM_param), Config.nens))\n",
+    "for i in range(len(Config.GCM_param)):\n",
+    "    ens_param[i, :] = Config.GCM_param[i] + rng.normal(\n",
+    "        scale=param_scale[i], size=Config.nens\n",
+    "    )"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "46ef8b1f",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "for cycle in np.arange(0, Config.ns_da / Config.DA_freq, dtype=int):\n",
+    "    # Set up array to store model forecast for each DA cycle\n",
+    "    ensX_fcst = np.zeros((Config.DA_freq + 1, Config.K, Config.nens))\n",
+    "\n",
+    "    # Model forecast for next DA cycle\n",
+    "    for n in range(Config.nens):\n",
+    "        ensX_fcst[..., n] = GCM(\n",
+    "            X_post[0 : Config.K, n],\n",
+    "            Config.F_fcst,\n",
+    "            Config.dt,\n",
+    "            Config.DA_freq,\n",
+    "            ens_param[:, n],\n",
+    "        )[0]\n",
+    "\n",
+    "    i_t = i_t + Config.DA_freq\n",
+    "\n",
+    "    # Get prior/background from the forecast\n",
+    "    X_prior = ensX_fcst[-1, ...]\n",
+    "\n",
+    "    # Call DA\n",
+    "    t_DA[cycle] = t_truth[i_t]\n",
+    "    if Config.DA == \"EnKF\":\n",
+    "        H = DA_methods.observation_operator(Config.K, l_obs, t_obs, i_t)\n",
+    "\n",
+    "        # Augment state vector with parameters when doing parameter estimation\n",
+    "        if Config.param_DA:\n",
+    "            H = np.concatenate(\n",
+    "                (H, np.zeros((H.shape[0], len(Config.GCM_param)))), axis=-1\n",
+    "            )\n",
+    "            X_prior = np.concatenate((X_prior, ens_param))\n",
+    "        B_ens = np.cov(X_prior)\n",
+    "        B_ens_loc = B_ens * W\n",
+    "        X_post = DA_methods.EnKF(X_prior, X_obs[t_obs == i_t], H, R, B_ens_loc)\n",
+    "\n",
+    "        X_post[0 : Config.K, :] = DA_methods.ens_inflate(\n",
+    "            X_post[0 : Config.K, :],\n",
+    "            X_prior[0 : Config.K, :],\n",
+    "            Config.inflate_opt,\n",
+    "            Config.inflate_factor,\n",
+    "        )\n",
+    "\n",
+    "        if Config.param_DA:\n",
+    "            X_post[-len(Config.GCM_param) :, :] = DA_methods.ens_inflate(\n",
+    "                X_post[-len(Config.GCM_param) :, :],\n",
+    "                X_prior[-len(Config.GCM_param) :, :],\n",
+    "                Config.param_inflate,\n",
+    "                Config.param_inf_factor,\n",
+    "            )\n",
+    "            ens_param = X_post[-len(Config.GCM_param) :, :]\n",
+    "\n",
+    "    elif Config.DA == \"HyEnKF\":\n",
+    "        H = DA_methods.observation_operator(Config.K, l_obs, t_obs, i_t)\n",
+    "        B_ens = (\n",
+    "            np.cov(X_prior) * (1 - Config.hybrid_factor)\n",
+    "            + B_clim1 * Config.hybrid_factor\n",
+    "        )\n",
+    "        B_ens_loc = B_ens * W\n",
+    "        X_post = DA_methods.EnKF(X_prior, X_obs[t_obs == i_t], H, R, B_ens_loc)\n",
+    "        X_post = DA_methods.ens_inflate(\n",
+    "            X_post, X_prior, Config.inflate_opt, Config.inflate_factor\n",
+    "        )\n",
+    "\n",
+    "    elif Config.DA == \"3DVar\":\n",
+    "        X_prior = np.squeeze(X_prior)\n",
+    "        H = DA_methods.observation_operator(Config.K, l_obs, t_obs, i_t)\n",
+    "        X_post = DA_methods.Lin3dvar(X_prior, X_obs[t_obs == i_t], H, R, B_loc, 3)\n",
+    "\n",
+    "    elif Config.DA == \"Replace\":\n",
+    "        X_post = X_prior\n",
+    "        X_post[l_obs[t_obs == i_t]] = X_obs[t_obs == i_t]\n",
+    "\n",
+    "    elif Config.DA == \"None\":\n",
+    "        X_post = X_prior\n",
+    "\n",
+    "    if not Config.DA == \"None\":\n",
+    "        # Get current increments\n",
+    "        X_inc[cycle, ...] = (\n",
+    "            np.squeeze(X_post[0 : Config.K, ...]) - X_prior[0 : Config.K, ...]\n",
+    "        )\n",
+    "\n",
+    "        # Get posterior info about the estimated parameters\n",
+    "        if Config.param_DA:\n",
+    "            mean_param[cycle, :] = ens_param.mean(axis=-1)\n",
+    "            spread_param[cycle, :] = ens_param.std(axis=-1)\n",
+    "\n",
+    "    # Reset initial conditions for next DA cycle\n",
+    "    ensX_fcst[-1, :, :] = X_post[0 : Config.K, :]\n",
+    "    ensX = np.concatenate((ensX, ensX_fcst[1:None, ...]))\n",
+    "\n",
+    "print(f\"Time to complete DA: {round(time.time() - start_time, 2)} (s)\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5d243a69",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 5. Post Processing and Visualization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f3c37a85-433c-4d2e-a49c-1daf4f6a9ca8",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "mean_X = np.mean(ensX, axis=-1)\n",
+    "clim = np.max(np.abs(mean_X - X_truth[0 : (Config.ns_da + 1), :]))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3f6f05d0",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "fig, axes = plt.subplots(2, 3, figsize=(15, 10))\n",
+    "ch = axes[0, 0].contourf(\n",
+    "    M_truth.k,\n",
+    "    t_truth[0 : (Config.ns_da + 1)],\n",
+    "    mean_X - X_truth[0 : (Config.ns_da + 1), :],\n",
+    "    cmap=\"bwr\",\n",
+    "    vmin=-clim,\n",
+    "    vmax=clim,\n",
+    "    extend=\"neither\",\n",
+    ")\n",
+    "\n",
+    "plt.colorbar(ch, ax=axes[0, 0], orientation=\"horizontal\")\n",
+    "axes[0, 0].set_xlabel(\"s\")\n",
+    "axes[0, 0].set_ylabel(\"t\", rotation=0)\n",
+    "axes[0, 0].set_title(\"X - X_truth\")\n",
+    "\n",
+    "axes[0, 1].plot(\n",
+    "    t_truth[0 : (Config.ns_da + 1)],\n",
+    "    np.sqrt(((mean_X - X_truth[0 : (Config.ns_da + 1), :]) ** 2).mean(axis=-1)),\n",
+    "    label=\"RMSE\",\n",
+    ")\n",
+    "axes[0, 1].plot(\n",
+    "    t_truth[0 : (Config.ns_da + 1)],\n",
+    "    np.mean(np.std(ensX, axis=-1), axis=-1),\n",
+    "    label=\"Spread\",\n",
+    ")\n",
+    "axes[0, 1].plot(\n",
+    "    t_truth[0 : (Config.ns_da + 1)],\n",
+    "    Config.obs_sigma * np.ones((Config.ns_da + 1)),\n",
+    "    label=\"Obs error\",\n",
+    ")\n",
+    "axes[0, 1].legend()\n",
+    "axes[0, 1].set_xlabel(\"time\")\n",
+    "axes[0, 1].set_title(\"RMSE (X - X_truth)\")\n",
+    "axes[0, 1].grid(which=\"both\", linestyle=\"--\")\n",
+    "\n",
+    "axes[0, 2].plot(\n",
+    "    M_truth.k,\n",
+    "    np.sqrt(((mean_X - X_truth[0 : (Config.ns_da + 1), :]) ** 2).mean(axis=0)),\n",
+    "    label=\"RMSE\",\n",
+    ")\n",
+    "X_inc_ave = X_inc / Config.DA_freq / Config.si\n",
+    "axes[0, 2].plot(M_truth.k, X_inc_ave.mean(axis=(0, -1)), label=\"Inc\")\n",
+    "axes[0, 2].plot(\n",
+    "    M_truth.k,\n",
+    "    DA_methods.running_average(X_inc_ave.mean(axis=(0, -1)), 7),\n",
+    "    label=\"Inc Ave\",\n",
+    ")\n",
+    "axes[0, 2].plot(\n",
+    "    M_truth.k,\n",
+    "    np.ones(M_truth.k.shape) * (Config.F_fcst - Config.F_truth),\n",
+    "    label=\"F_bias\",\n",
+    ")\n",
+    "axes[0, 2].plot(\n",
+    "    M_truth.k,\n",
+    "    np.ones(M_truth.k.shape) * (X_inc / Config.DA_freq / Config.si).mean(),\n",
+    "    \"k:\",\n",
+    "    label=\"Ave Inc\",\n",
+    ")\n",
+    "axes[0, 2].legend()\n",
+    "axes[0, 2].set_xlabel(\"s\")\n",
+    "axes[0, 2].set_title(\"Increments\")\n",
+    "axes[0, 2].grid(which=\"both\", linestyle=\"--\")\n",
+    "\n",
+    "plot_start, plot_end = 1000, 1500\n",
+    "plot_start_DA, plot_end_DA = int(plot_start / Config.DA_freq), int(\n",
+    "    plot_end / Config.DA_freq\n",
+    ")\n",
+    "plot_x = 0\n",
+    "\n",
+    "axes[1, 0].plot(\n",
+    "    t_truth[plot_start:plot_end], X_truth[plot_start:plot_end, plot_x], label=\"truth\"\n",
+    ")\n",
+    "axes[1, 0].plot(\n",
+    "    t_truth[plot_start:plot_end], mean_X[plot_start:plot_end, plot_x], label=\"forecast\"\n",
+    ")\n",
+    "axes[1, 0].scatter(\n",
+    "    t_DA[plot_start_DA - 1 : plot_end_DA - 1],\n",
+    "    DA_methods.find_obs(plot_x, X_obs, t_obs, l_obs, [plot_start, plot_end]),\n",
+    "    label=\"obs\",\n",
+    ")\n",
+    "axes[1, 0].grid(which=\"both\", linestyle=\"--\")\n",
+    "axes[1, 0].set_xlabel(\"time\")\n",
+    "axes[1, 0].set_title(\"k=\" + str(plot_x + 1) + \" truth and forecast\")\n",
+    "axes[1, 0].legend()\n",
+    "\n",
+    "if Config.param_DA:\n",
+    "    for i, c in zip(np.arange(len(Config.GCM_param), 0, -1), [\"r\", \"b\", \"g\", \"k\"]):\n",
+    "        axes[1, 1].plot(\n",
+    "            t_DA,\n",
+    "            DA_methods.running_average(mean_param[:, i - 1], 100),\n",
+    "            c + \"-\",\n",
+    "            label=\"C{} {:3f}\".format(\n",
+    "                i - 1, mean_param[int(len(t_DA) / 2) :, i - 1].mean()\n",
+    "            ),\n",
+    "        )\n",
+    "        axes[1, 1].plot(\n",
+    "            t_DA,\n",
+    "            DA_methods.running_average(\n",
+    "                mean_param[:, i - 1] + spread_param[:, i - 1], 100\n",
+    "            ),\n",
+    "            c + \":\",\n",
+    "            label=f\"SD {spread_param[int(len(t_DA) / 2):, i - 1].mean():3f}\",\n",
+    "        )\n",
+    "        axes[1, 1].plot(\n",
+    "            t_DA,\n",
+    "            DA_methods.running_average(\n",
+    "                mean_param[:, i - 1] - spread_param[:, i - 1], 100\n",
+    "            ),\n",
+    "            c + \":\",\n",
+    "        )\n",
+    "    axes[1, 1].legend()\n",
+    "    axes[1, 1].grid(which=\"both\", linestyle=\"--\")\n",
+    "\n",
+    "else:\n",
+    "    axes[1, 1].axis(\"off\")\n",
+    "\n",
+    "axes[1, 2].text(\n",
+    "    0.1,\n",
+    "    0.2,\n",
+    "    (\n",
+    "        f\"RMSE={np.sqrt(((mean_X - X_truth[0 : (Config.ns_da + 1), :]) ** 2).mean()):3f}\\n\"\n",
+    "        f\"Spread={np.mean(np.std(ensX, axis=-1)):3f}\\n\"\n",
+    "        f\"GCM param={Config.GCM_param}\\n\"\n",
+    "        f\"DA={Config.DA}, {Config.nens}\\n\"\n",
+    "        f\"DA_freq={Config.DA_freq}\\n\"\n",
+    "        f\"B_loc={Config.B_loc}\\n\"\n",
+    "        f\"inflation={Config.inflate_opt}, {Config.inflate_factor}\\n\"\n",
+    "        f\"obs_density={Config.obs_density}\\n\"\n",
+    "        f\"obs_sigma={Config.obs_sigma}\\n\"\n",
+    "        f\"obs_freq={Config.obs_freq}\\n\"\n",
+    "    ),\n",
+    "    fontsize=15,\n",
+    ")\n",
+    "axes[1, 2].axis(\"off\")\n",
+    "\n",
+    "if Config.save_fig:\n",
+    "    output_dir = os.path.join(\".\", \"DA_data\")\n",
+    "    os.makedirs(output_dir, exist_ok=True)\n",
+    "    exp_number = np.random.randint(1, 10000)\n",
+    "    with open(os.path.join(output_dir, f\"config_{exp_number}.txt\"), \"w\") as f:\n",
+    "        f.write(\n",
+    "            output_dir\n",
+    "            + str({k: v for k, v in Config.__dict__.items() if not k.startswith(\"_\")})\n",
+    "        )\n",
+    "    plt.savefig(os.path.join(output_dir, f\"fig_{exp_number}.png\"))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "63475b82-5e45-4a2c-93f5-f0e089320c7b",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Save DA output for further analysis"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "65993585",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "if Config.save_data:\n",
+    "    output_filename = (\n",
+    "        f\"K_{Config.K}_\"\n",
+    "        f\"J_{Config.J}_\"\n",
+    "        f\"obs_freq_{Config.obs_freq}_\"\n",
+    "        f\"F_truth_{Config.F_truth}_\"\n",
+    "        f\"F_fcst_{Config.F_fcst}_\"\n",
+    "        f\"ns_da_{Config.ns_da}_\"\n",
+    "        f\"ns_{Config.ns}_\"\n",
+    "        f\"ns_spinup_{Config.ns_spinup}_\"\n",
+    "        f\"dt_{Config.dt}_\"\n",
+    "        f\"si_{Config.si}_\"\n",
+    "        f\"B_loc_{Config.B_loc}_\"\n",
+    "        f\"DA_{Config.DA}_\"\n",
+    "        f\"nens_{Config.nens}_\"\n",
+    "        f\"inflate_opt_{Config.inflate_opt}_\"\n",
+    "        f\"inflate_factor_{Config.inflate_factor}_\"\n",
+    "        f\"hybrid_factor_{Config.hybrid_factor}_\"\n",
+    "        f\"obs_density_{Config.obs_density}_\"\n",
+    "        f\"DA_freq_{Config.DA_freq}_\"\n",
+    "        f\"obs_sigma_{Config.obs_sigma}.npz\"\n",
+    "    )\n",
+    "    output_dir = os.path.join(\".\", \"DA_data\")\n",
+    "    os.makedirs(output_dir, exist_ok=True)\n",
+    "    np.savez(\n",
+    "        os.path.join(output_dir, output_filename),\n",
+    "        mean_X=mean_X,\n",
+    "        X_truth=X_truth,\n",
+    "        X_inc_ave=X_inc_ave,\n",
+    "    )"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/GP_lorenz96.ipynb b/_sources/notebooks/GP_lorenz96.ipynb
new file mode 100644
index 00000000..7cd27ed8
--- /dev/null
+++ b/_sources/notebooks/GP_lorenz96.ipynb
@@ -0,0 +1,926 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Gaussian Processes"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "import tensorflow as tf\n",
+    "import gpflow\n",
+    "from L96_model import L96, L96_eq1_xdot, integrate_L96_2t\n",
+    "from L96_model import EulerFwd, RK2, RK4\n",
+    "\n",
+    "time_method = RK4"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Recap: Lorenz 96"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "```{figure} https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376@1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png\n",
+    ":width: 400px\n",
+    ":name: l96-equation-figure\n",
+    "\n",
+    "*Visualisation of a two-scale Lorenz '96 system with J = 8 and K = 6. Global-scale variables ($X_k$) are updated based on neighbouring variables and on the local-scale variables ($Y_{j,k}$) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from [Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.](https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436)*.\n",
+    "\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "{cite}`Lorenz1995` describes a \"two time-scale\" model in two equations (2 and 3) which are:\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}\n",
+    "\\\\\n",
+    "\\frac{d}{dt} Y_{j,k}\n",
+    "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - X_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
+    "\\end{align}\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Recap: Gaussian Processes (GPs)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Exact inference\n",
+    "\n",
+    "Consider the following case where we have a dataset $\\mathcal{D}$ comprising $n$ pairs of observations $\\mathcal{D} = \\{(x_1,y_1),(x_2,y_2),...,(x_n,y_n)\\}$, where $x_i$ could e.g., be the coarse-grid term at time $i$ and $y_i$ the corresponding sub-grid term. In the standard supervised learning problem, we wish to recover the function $f$ which maps a given input $x_i$ to a given output $y_i$, where we assume that the outputs have been corrupted by some random noise $\\varepsilon_i$:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "y_i = f(x_i) + \\varepsilon_i,\n",
+    "\\end{equation}\n",
+    "or in vector format:\n",
+    "\\begin{equation}\n",
+    "\\textbf{y} = f(\\textbf{x}) + \\pmb{\\varepsilon}. \n",
+    "\\end{equation}\n",
+    "\n",
+    "Then after learning the function $f$, we wish to make reliable predictions at un-observed test input locations $\\textbf{x}_* \\in \\mathbb{R}^{n_*}$, such that:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\textbf{f}_* = f(\\textbf{x}_*). \n",
+    "\\end{equation}\n",
+    "\n",
+    "In the GP framework, we assume that the noise term $\\pmb{\\varepsilon}$ represents independent and identically distributed zero-mean Gaussian noise, with variance $\\sigma^2$, such that $\\varepsilon_i \\sim \\mathcal{N}(0,\\sigma^2)$, and that the function $f$ itself is a GP:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "f(\\pmb{\\phi}) \\sim \\mathcal{GP}(m(\\pmb{\\phi}),k(\\pmb{\\phi},\\pmb{\\phi})),\n",
+    "\\end{equation}\n",
+    "\n",
+    "where $m(\\pmb{\\phi})$ and $k(\\pmb{\\phi},\\pmb{\\phi})$ are the mean and covariance (or kernel) of the GP respectively, and $\\pmb{\\phi}$ represents arbitrary function inputs - either  $\\textbf{x}$ or $\\textbf{x}_*$. Through the assumption that $f$ is a GP, we effectively place prior probabilities over all conceviable functions $f_i$, where $i = \\{1,2,...,\\infty\\}$, but assign higher probabilities to specific types of functions (e.g., linear, periodic, anisotropic, non-stationary etc) through our choice of $m(\\pmb{\\phi})$ and $k(\\pmb{\\phi},\\pmb{\\phi})$. After which, we use the observations $\\textbf{y}$ to update these prior probabilities in the form of a posterior distribution over functions. Let's see how we do this in practice:\n",
+    "\n",
+    "Continuing with the notation that $\\textbf{x} = \\{x_1,x_2,...,x_n\\}$ represents the set of training inputs, and $\\textbf{x}_* = \\{x_{*_1},x_{*_2},...,x_{*_{n_*}}\\}$ the set of test inputs, we wish to train a model which is able to make accurate predictions at the test locations: $\\textbf{f}_* = \\{f(x_{*_1}),f(x_{*_2}),...,f(x_{*_{n_*}})\\}$. We start by defining a series of covariance functions associated with the training and test inputs:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\begin{aligned}\n",
+    "\\textbf{K} &= k(\\textbf{x},\\textbf{x}) + \\sigma^2\\pmb{I} \\\\\n",
+    "\\textbf{k}_{*} &= k(\\textbf{x},\\textbf{x}_*) \\\\\n",
+    "\\pmb{k}_* &= k(\\textbf{x}_*,\\textbf{x}_*).\n",
+    "\\end{aligned}\n",
+    "\\end{equation}\n",
+    "\n",
+    "The covariance function of the GP prior is a particularly important concept, as this is what allows us to model the different kinds of functions described above, so consideration should go into choosing the most appropriate function (there are data driven ways to make this choice. We could e.g., use cross-validation, or more commonly in Bayesian statistics; compare models via the *log marginal likelihood* - GPflow is setup to do the latter). Now, because a GP can ultimately be described as a collection of random Gaussian variables, under our GP prior (defined by $m(\\pmb{\\phi})$ and $k(\\pmb{\\phi},\\pmb{\\phi})$), the function values $\\textbf{f}_*$ and the training observations $\\textbf{y}$ are considered to form a joint Gaussian distribution: \n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\begin{bmatrix}\n",
+    "\\textbf{f}_* \\\\\n",
+    "\\textbf{y}\n",
+    "\\end{bmatrix} \\sim \\mathcal{N}\\Bigg(\n",
+    "\\begin{bmatrix}\n",
+    "m(\\textbf{x}_*) \\\\\n",
+    "m(\\textbf{x})\n",
+    "\\end{bmatrix},\n",
+    "\\begin{bmatrix}\n",
+    "\\pmb{k}_* & \\textbf{k}_*^\\text{T} \\\\\n",
+    "\\textbf{k}_* & \\textbf{K}\n",
+    "\\end{bmatrix} \\Bigg),\n",
+    "\\end{equation}\n",
+    "\n",
+    "where we see that the marginal distributions (under the prior) of both $\\textbf{f}_*$ and $\\textbf{y}$ are given by:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\textbf{f}_* \\sim \\mathcal{N}\\big(m(\\textbf{x}_*),\\pmb{k}_*\\big) \\quad , \\quad \\textbf{y} \\sim \\mathcal{N}\\big(m(\\textbf{x}),\\textbf{K}\\big).\n",
+    "\\label{marginalsGP}\n",
+    "\\end{equation}\n",
+    "\n",
+    "In reality however, we are interested in the conditional distribution of the function values, given the training data - $p(\\textbf{f}_* | \\textbf{y},\\textbf{x},\\textbf{x}_*)$, which is also Gaussian such that $\\textbf{f}_* | \\textbf{y},\\textbf{x},\\textbf{x}_* \\sim \\mathcal{N}(\\bar{\\textbf{f}}_*,\\sigma^2_{\\textbf{f}_*})$. The mean and covariance of this distribution then tells us everything we need about the posterior distribution of the function values, and can be derived analytically through a bit of linear algebra (see {cite}`Rasmussen_Williams_2005` or {cite}`Bishop2006` for details). Here is the final solution: \n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\begin{aligned}\n",
+    "\\bar{\\textbf{f}}_* &= m(\\textbf{x}_*) + \\textbf{k}_*^\\text{T}\\textbf{K}^{-1}\\big(\\textbf{y} - m(\\textbf{x})\\big) \\\\\n",
+    "\\sigma^2_{\\textbf{f}_*} &= \\pmb{k}_* - \\textbf{k}_*^\\text{T}\\textbf{K}^{-1}\\textbf{k}_*.\n",
+    "\\end{aligned}\n",
+    "\\end{equation} "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Variational inference \n",
+    "\n",
+    "The analytical solution in the above description relies on the assumption that the data are Gaussian distributed, which makes the maths straightfoward because the product of 2 Gaussians (the prior and likelihood) results in another Gaussian (the posterior). In the case where the data are not Gaussian distributed, the resultant posterior will no longer have a closed-form solution, so instead we need to approximate the posterior as a multivarite Gaussian (or other) using variational techniques. To do this, GPflow implements the approach outlined in [Opper and Archambeau, (2009)](http://www0.cs.ucl.ac.uk/staff/c.archambeau/publ/neco_mo09_web.pdf), and we give a brief summary here to follow on from the section above.\n",
+    "\n",
+    "Recall the posterior predictive distribution from before: $p(\\textbf{f}_* | \\textbf{y})$, where we have removed the dependence on the inputs $\\textbf{x}$ and $\\textbf{x}_*$ to keep the notation uncluttered. In the following discussion we assume that this distribution is no longer Gaussian, because e.g., the observations $\\textbf{y}$ are no longer Gaussian, and hence we no longer have a Gaussian likelihood. Therefore instead, we wish to approximate $p(\\textbf{f}_* | \\textbf{y})$ by a multivariate Gaussian $q(\\textbf{f}_*) \\sim \\mathcal{N}(\\textbf{m},\\textbf{S})$. This is achieved by choosing the values of $\\textbf{m}$ and $\\textbf{S}$ which minimise the *variational free energy* cost function: \n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\mathcal{J}\\big(q(\\textbf{f}_*)\\big) = - \\ln p(\\textbf{y}) + \\mathcal{K}\\mathcal{L}[q(\\textbf{f}_*) \\vert\\vert p(\\textbf{f}_* | \\textbf{y})], \n",
+    "\\end{equation}\n",
+    "\n",
+    "where\n",
+    "\n",
+    "\\begin{equation}\n",
+    "p(\\textbf{y}) = \\int p(\\textbf{y} | \\textbf{f}_*)p(\\textbf{f}_*) \\ \\ d\\textbf{f}_*\n",
+    "\\end{equation}\n",
+    "\n",
+    "is the marginal likelihood (the probability of the observations under the model prior; commonly referred to as the *evidence*), and \n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\mathcal{K}\\mathcal{L}[q(\\textbf{f}_*) \\vert\\vert p(\\textbf{f}_* | \\textbf{y})] = \\int q(\\textbf{f}_*)\\ln \\dfrac{q(\\textbf{f}_*)}{p(\\textbf{f}_* | \\textbf{y})} \\ \\ d\\textbf{f}_*\n",
+    "\\end{equation}\n",
+    "\n",
+    "is the Kullback-Leibler divergence which compares the similarity of two probability distributions (in this case, $q(\\textbf{f}_*)$ and $p(\\textbf{f}_* | \\textbf{y}$)).\n",
+    "\n",
+    "The variational free energy $\\mathcal{J}\\big(q(\\textbf{f}_*)\\big)$ represents an upper bound on the negative (log) evidence $-\\ln p(\\textbf{y})$, and is the quantity that GPflow minimises in the variational setting. It is however common practice to track the maximisation of the negative variational free energy, which then represents a lower bound on the (log) evidence, known as the *Evidence Lower BOund* (ELBO):\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\ln p(\\textbf{y}) = \\text{ELBO} - \\mathcal{K}\\mathcal{L}[q(\\textbf{f}_*) \\vert\\vert p(\\textbf{f}_* | \\textbf{y})] \\geq \\text{ELBO}.\n",
+    "\\end{equation}\n",
+    "\n",
+    "Once the values of $\\textbf{m}$ and $\\textbf{S}$ have been optimised, the Gaussian approximation for the mean and covariance of the posterior can be computed as:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\begin{aligned}\n",
+    "\\bar{\\textbf{f}}_* &= m(\\textbf{x}_*) + \\textbf{k}_*^\\text{T}\\textbf{K}^{-1}\\big(\\textbf{m} - m(\\textbf{x})\\big) \\\\\n",
+    "\\sigma^2_{\\textbf{f}_*} &= \\pmb{k}_* + \\textbf{k}_*^\\text{T}\\textbf{K}^{-1}\\big(\\textbf{S} - \\textbf{K}\\big)\\textbf{K}^{-1}\\textbf{k}_*.\n",
+    "\\end{aligned}\n",
+    "\\end{equation} \n",
+    "\n",
+    "Note that in practice there is a fair bit of maths involved in computing $\\mathcal{J}\\big(q(\\textbf{f}_*)\\big)$ which we do not show here, however the reader is referred to the [GPflow docs](https://gpflow.github.io/GPflow/2.4.0/notebooks/theory/vgp_notes.html) and the [Opper and Archambeau, (2009)](http://www0.cs.ucl.ac.uk/staff/c.archambeau/publ/neco_mo09_web.pdf) paper for more information. See also this nice [Jupyter notebook](https://github.com/sotakao/moap-bml-workshop/blob/main/2_sparse_gp.ipynb) workthrough by So Takao from the University College London Centre for AI, which includes the derivations of the KL divergence term in the Appendix."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Sparse GPs\n",
+    "\n",
+    "Due to the need to invert the kernel matrix $\\textbf{K}$ (which has size $n \\times n$) in both the exact and variational settings, GP problems scale with complexity $\\mathcal{O}(n^3)$. Therefore for data sets with more than a few thousand training points, the standard GP implementations above become problematic. A solution to this is to consider sparse GPs which rely on a set of *inducing variables* to learn the function $f$, as outlined in [Titsias, (2009)](https://proceedings.mlr.press/v5/titsias09a/titsias09a.pdf). Note that GPflow has 2 commonly used classes for implementation of the inducing points method, namely 'SGPR' and 'SVGP'. Although at first it perhaps may seem that 'SVGP' is for variational inference and 'SGPR' is for exact inference, this is not actually the case. Sparse learning requires variational inference in either case, it is just that 'SVGP' is used for non-Gaussian data (as described above), while 'SGPR' is used for Gaussian data. Let's introduce the inducing points method now:\n",
+    "\n",
+    "Consider the set of points $\\textbf{z} = \\{z_1,z_2,...,z_m\\}$ where $m << n$, which could perhaps correspond to a subset of the training inputs, or a hold-out set etc, however the important point is that they exist in the same space as the inputs. These points are known as the *inducing points*, and subsequently their corresponding latent function values $\\textbf{u} = \\{f(z_1),f(z_2),...,f(z_m)\\}$ are known as the *inducing variables*. If we then assume that the posterior predictive function values at the input locations $\\textbf{x}$ and $\\textbf{x}_*$ (given by $\\textbf{f}$ and $\\textbf{f}_*$ respectively) are conditionally independent given $\\textbf{u}$, we can describe the posterior $p(\\textbf{f}_* | \\textbf{y})$ as a function of $\\textbf{u}$ as follows:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "p(\\textbf{f}_* | \\textbf{y}) = \\int p(\\textbf{f}_* | \\textbf{u})p(\\textbf{u} | \\textbf{y}) \\ \\ d\\textbf{u}.\n",
+    "\\end{equation}\n",
+    "\n",
+    "We can subsequently derive an analytical solution for $p(\\textbf{f}_* | \\textbf{u})$ above by recalling in the first section on exact inference how $\\textbf{y}$ and $\\textbf{f}_*$ formed a joint Gaussian under the GP prior. The same principles apply here to $\\textbf{u}$ and $\\textbf{f}_*$, hence the mean and covariance of $p(\\textbf{f}_* | \\textbf{u})$ follows the same form as those defined by $\\bar{\\textbf{f}}_*$ and $\\sigma_{\\textbf{f}_*}$ previously, except this time we replace all occurrences of $\\textbf{x}$ with $\\textbf{z}$, as well as re-defining the covariance matrices:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\begin{aligned}\n",
+    "\\textbf{K} = k(\\textbf{z},\\textbf{z}) \\\\\n",
+    "\\textbf{k}_* = k(\\textbf{z},\\textbf{x}_*) \\\\\n",
+    "\\pmb{k}_* = k(\\textbf{z},\\textbf{z}).\n",
+    "\\end{aligned}\n",
+    "\\end{equation}\n",
+    "\n",
+    "The second distribution above, $p(\\textbf{u} | \\textbf{y})$, does not have a closed-form solution however, so we need to rely on variational inference to approximate it as a multivariate Gaussian $q(\\textbf{u}) \\sim \\mathcal{N}(\\textbf{m},\\textbf{S})$. Similar to what we saw in the variational inference discussion above, both $\\textbf{m}$ and $\\textbf{S}$ (as well as $\\textbf{z}$) can be optimised by minimising the variational free energy cost function $\\mathcal{J}\\big(q(\\textbf{f}_*)\\big)$. After which, we can produce an analytical solution for the approximate posterior mean and covariance as we have already seen, by once again replacing all occurrences of $\\textbf{x}$ with $\\textbf{z}$, as well as re-defining the covariance matrices. This results in much cheaper inference than before as the covariance matrix $\\textbf{K}$ is now of size $m \\times m$, hence the computational cost of the matrix inverse is now $\\mathcal{O}(m^3)$ as opposed to $\\mathcal{O}(n^3)$. \n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# - a GCM class without any parameterization\n",
+    "class GCM_no_param:\n",
+    "    def __init__(self, F, time_stepping=time_method):\n",
+    "        self.F = F\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, param):\n",
+    "        return L96_eq1_xdot(X, self.F)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        # X0 - initial conditions, dt - time increment, nt - number of forward steps to take\n",
+    "        # param - parameters of our closure\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time\n",
+    "\n",
+    "\n",
+    "# - a GCM class including a linear parameterization in rhs of equation for tendency\n",
+    "\n",
+    "\n",
+    "class GCM:\n",
+    "    def __init__(self, F, parameterization, time_stepping=time_method):\n",
+    "        self.F = F\n",
+    "        self.parameterization = parameterization\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, param):\n",
+    "        return L96_eq1_xdot(X, self.F) - self.parameterization(param, X)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        # X0 - initial conditions, dt - time increment, nt - number of forward steps to take\n",
+    "        # param - parameters of our closure\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time\n",
+    "\n",
+    "\n",
+    "# - a GCM class including a GP parameterization in rhs of equation for tendency\n",
+    "\n",
+    "\n",
+    "class GCM_GP:\n",
+    "    def __init__(self, F, GP, time_stepping=time_method, stochastic=True, local=True):\n",
+    "        self.F = F\n",
+    "        self.GP = GP\n",
+    "        self.time_stepping = time_stepping\n",
+    "        self.stochastic = stochastic\n",
+    "        self.local = local\n",
+    "\n",
+    "    def rhs(self, X, param):\n",
+    "        if self.local:  # i.e. outputs are shape n*K\n",
+    "            X_ten = tf.convert_to_tensor(np.atleast_2d(X).T, dtype=tf.float64)\n",
+    "        else:  # i.e. outputs are shape n x K\n",
+    "            X_augment = np.hstack((np.atleast_2d(X).T, np.atleast_2d(np.arange(K)).T))\n",
+    "            X_ten = tf.convert_to_tensor(X_augment, dtype=tf.float64)\n",
+    "        if self.stochastic:\n",
+    "            f_mean = self.GP.predict_f_samples(\n",
+    "                X_ten, 1\n",
+    "            )  # produce 1 random sample from the posterior over functions\n",
+    "        else:\n",
+    "            f_mean, f_variance = self.GP.predict_f(X_ten)\n",
+    "        return L96_eq1_xdot(X, self.F) + np.squeeze(f_mean)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        # X0 - initial conditions, dt - time increment, nt - number of forward steps to take\n",
+    "        # param - parameters of our closure\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time\n",
+    "\n",
+    "\n",
+    "# Define the function which will be used to track the optimisation of the models. Much of the code in the 'GP'\n",
+    "# function is borrowed from the example on the GPflow website:\n",
+    "# https://gpflow.github.io/GPflow/2.4.0/notebooks/advanced/gps_for_big_data.html\n",
+    "\n",
+    "\n",
+    "def trainGP(\n",
+    "    model,\n",
+    "    training_data=None,\n",
+    "    validation_data=None,\n",
+    "    batch_size=None,\n",
+    "    iterations=1000,\n",
+    "    verbose=0,\n",
+    "):\n",
+    "    tf.random.set_seed(766)\n",
+    "\n",
+    "    n = len(training_data[0])\n",
+    "    if batch_size is None:\n",
+    "        batch_size = n\n",
+    "    train_dataset = (\n",
+    "        tf.data.Dataset.from_tensor_slices(training_data).repeat().shuffle(1000)\n",
+    "    )\n",
+    "    train_iter = iter(train_dataset.batch(batch_size))\n",
+    "    training_loss = model.training_loss_closure(train_iter, compile=True)\n",
+    "\n",
+    "    optimizer = tf.optimizers.Adam()\n",
+    "\n",
+    "    @tf.function\n",
+    "    def optimization_step(trainable_variables):\n",
+    "        optimizer.minimize(training_loss, trainable_variables)\n",
+    "\n",
+    "    elbo_log = []\n",
+    "    iters = []\n",
+    "    pred_error = []\n",
+    "    for step in range(iterations):\n",
+    "        optimization_step(model.trainable_variables)\n",
+    "        if step % 10 == 0:\n",
+    "            loss = -training_loss().numpy()\n",
+    "            prediction, uncertainty = model.predict_f(validation_data[0])\n",
+    "            pred_error.append(np.mean((prediction - validation_data[1]) ** 2))\n",
+    "            elbo_log.append(loss)\n",
+    "            iters.append(step)\n",
+    "        if verbose > 0:\n",
+    "            if step < 1000:\n",
+    "                if step % 100 == 0:\n",
+    "                    print(f\"ELBO at step {step}: {loss:.2f}\")\n",
+    "            else:\n",
+    "                if step % 1000 == 0:\n",
+    "                    print(f\"ELBO at step {step}: {loss:.2f}\")\n",
+    "    return model, iters, elbo_log, pred_error\n",
+    "\n",
+    "\n",
+    "def plot_loss(x, train, valid, labels=[\"ELBO\", \"MSE (Validation)\"]):\n",
+    "    fig, ax = plt.subplots(1, 2, figsize=(17, 5), dpi=150)\n",
+    "    plots = [train, valid]\n",
+    "    for c in range(2):\n",
+    "        ax[c].spines[\"top\"].set_color(None)\n",
+    "        ax[c].spines[\"right\"].set_color(None)\n",
+    "        ax[c].plot(x, plots[c], lw=2)\n",
+    "        ax[c].set_xlabel(\"iteration\", fontsize=12)\n",
+    "        ax[c].set_ylabel(labels[c], fontsize=12)\n",
+    "        ax[c].grid(True, alpha=0.3)\n",
+    "    plt.subplots_adjust(wspace=0.1)\n",
+    "    plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Learn the sub-grid tendencies offline"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "time_steps = 20000\n",
+    "Forcing, dt, T = 18, 0.01, 0.01 * time_steps\n",
+    "\n",
+    "# Create a \"real world\" with K=8 and J=32\n",
+    "W = L96(8, 32, F=Forcing)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Get training and validation data for the GP model\n",
+    "val_size = 4000\n",
+    "\n",
+    "# - Run the true state and output subgrid tendencies (the effect of Y on X is xytrue):\n",
+    "Xtrue, _, _, xytrue = W.run(dt, T, store=True, return_coupling=True)\n",
+    "gcm_no_param = GCM_no_param(Forcing)\n",
+    "\n",
+    "# train:\n",
+    "X_train = Xtrue[:-val_size, :]\n",
+    "Y_train = xytrue[:-val_size, :]\n",
+    "\n",
+    "# test:\n",
+    "X_valid = Xtrue[-val_size:, :]\n",
+    "Y_valid = xytrue[-val_size:, :]\n",
+    "\n",
+    "n, K = Y_train.shape\n",
+    "ns = len(X_valid)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Local mappings with a linear kernel\n",
+    "\n",
+    "The linear GP kernel is equivalent to:\n",
+    "\n",
+    "$k(\\textbf{x},\\textbf{x}') = \\theta^2 \\big(\\textbf{x}\\textbf{x}'^\\text{T}\\big)$, where $\\theta$ is a tunable hyperparameter.\n",
+    "\n",
+    "First we will build a GP with a linear kernel while assuming the outputs (i.e., each of the Ks) are independent.  The extension to learning non-linear functions is trivial. We simply define a new kernel, e.g., gpflow.kernels.Matern32(). Incorporating non-local information (i.e., from other ks (we have 8 ks in total)) is a \n",
+    "little more tricky, but we will tackle this later."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "np.random.seed(2)\n",
+    "\n",
+    "x_train = tf.convert_to_tensor(X_train.reshape(n * K, 1, order=\"F\"), dtype=tf.float64)\n",
+    "y_train = tf.convert_to_tensor(Y_train.reshape(n * K, 1, order=\"F\"), dtype=tf.float64)\n",
+    "x_valid = tf.convert_to_tensor(X_valid.reshape(ns * K, 1, order=\"F\"), dtype=tf.float64)\n",
+    "y_valid = Y_valid.reshape(ns * K, 1, order=\"F\")\n",
+    "\n",
+    "M = 50  # number of inducing variables\n",
+    "Z = tf.convert_to_tensor(\n",
+    "    np.atleast_2d(np.linspace(np.min(x_train), np.max(x_train), M)).T, dtype=tf.float64\n",
+    ")\n",
+    "\n",
+    "kernel = gpflow.kernels.Linear()\n",
+    "lik = gpflow.likelihoods.Gaussian()\n",
+    "gp_linear = gpflow.models.SVGP(kernel=kernel, likelihood=lik, inducing_variable=Z)\n",
+    "gp_linear.likelihood.variance.assign(np.var(y_train))\n",
+    "gp_linear, iters, elbo, pred_error = trainGP(\n",
+    "    gp_linear,\n",
+    "    training_data=(x_train, y_train),\n",
+    "    validation_data=(x_valid, y_valid),\n",
+    "    iterations=5000,\n",
+    "    batch_size=100,\n",
+    "    verbose=0,\n",
+    ")\n",
+    "plot_loss(iters, elbo, pred_error)\n",
+    "gp_linear"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "prediction_linGP, uncertainty_linGP = gp_linear.predict_f(x_valid)\n",
+    "prediction_linGP = np.squeeze(prediction_linGP.numpy())\n",
+    "uncertainty_linGP = np.squeeze(uncertainty_linGP.numpy())\n",
+    "\n",
+    "n_std = 3\n",
+    "start = 31000\n",
+    "end = start + 1000\n",
+    "prediction_linGPplot = prediction_linGP[start:end]\n",
+    "uncertainty_linGPplot = n_std * np.sqrt(uncertainty_linGP[start:end])\n",
+    "\n",
+    "fig, ax = plt.subplots(1, 2, figsize=(17, 5), dpi=150)\n",
+    "ax[0].spines[\"top\"].set_color(None)\n",
+    "ax[0].spines[\"right\"].set_color(None)\n",
+    "ax[0].plot(prediction_linGPplot, label=\"Predicted values\", color=\"teal\")\n",
+    "ax[0].fill_between(\n",
+    "    np.arange(1000),\n",
+    "    prediction_linGPplot + uncertainty_linGPplot,\n",
+    "    prediction_linGPplot - uncertainty_linGPplot,\n",
+    "    alpha=0.2,\n",
+    "    color=\"teal\",\n",
+    ")\n",
+    "ax[0].plot(y_valid[start:end], label=\"True values\", color=\"k\")\n",
+    "ax[0].grid(True, alpha=0.3)\n",
+    "ax[0].legend(fontsize=7)\n",
+    "\n",
+    "a, b = zip(*sorted(zip(X_valid.reshape(ns * K, order=\"F\"), prediction_linGP)))\n",
+    "a, c = zip(\n",
+    "    *sorted(zip(X_valid.reshape(ns * K, order=\"F\"), n_std * np.sqrt(uncertainty_linGP)))\n",
+    ")\n",
+    "a = np.array(a)\n",
+    "b = np.array(b)\n",
+    "c = np.array(c)\n",
+    "\n",
+    "ax[1].spines[\"top\"].set_color(None)\n",
+    "ax[1].spines[\"right\"].set_color(None)\n",
+    "ax[1].scatter(\n",
+    "    X_valid.reshape(ns * K, order=\"F\"),\n",
+    "    Y_valid.reshape(ns * K, order=\"F\"),\n",
+    "    s=1,\n",
+    "    color=\"k\",\n",
+    ")\n",
+    "ax[1].plot(X_valid.reshape(ns * K, order=\"F\"), prediction_linGP, color=\"teal\", lw=2)\n",
+    "ax[1].fill_between(a, b - c, b + c, color=\"teal\", alpha=0.2)\n",
+    "ax[1].grid(True, alpha=0.3)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Local mappings with a non-linear kernel\n",
+    "\n",
+    "Here we are going to opt for a combination of 2 kernels: The Rational Quadratic kernel:\n",
+    "\n",
+    "$k_1(\\textbf{x},\\textbf{x}') = \\theta_1^2 \\bigg(1 + \\dfrac{\\Vert \\textbf{x} - \\textbf{x}' \\Vert^2}{2\\theta_2^2\\theta_3}\\bigg)^{-\\theta_3}$\n",
+    "\n",
+    "and the linear kernel:\n",
+    "\n",
+    "$k_2(\\textbf{x},\\textbf{x}') = \\theta_4^2 \\big(\\textbf{x}\\textbf{x}^\\text{T}\\big)$\n",
+    "\n",
+    "such that our kernel becomes $k(\\textbf{x},\\textbf{x}') = k_1(\\textbf{x},\\textbf{x}') + k_2(\\textbf{x},\\textbf{x}')$\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "np.random.seed(2)\n",
+    "\n",
+    "x_train = tf.convert_to_tensor(X_train.reshape(n * K, 1, order=\"F\"), dtype=tf.float64)\n",
+    "y_train = tf.convert_to_tensor(Y_train.reshape(n * K, 1, order=\"F\"), dtype=tf.float64)\n",
+    "x_valid = tf.convert_to_tensor(X_valid.reshape(ns * K, 1, order=\"F\"), dtype=tf.float64)\n",
+    "y_valid = Y_valid.reshape(ns * K, 1, order=\"F\")\n",
+    "\n",
+    "M = 50  # number of inducing variables\n",
+    "Z = tf.convert_to_tensor(\n",
+    "    np.atleast_2d(np.linspace(np.min(x_train), np.max(x_train), M)).T, dtype=tf.float64\n",
+    ")\n",
+    "\n",
+    "kernel = gpflow.kernels.RationalQuadratic() + gpflow.kernels.Linear()\n",
+    "lik = gpflow.likelihoods.Gaussian()\n",
+    "gp_nonlinear = gpflow.models.SVGP(kernel=kernel, likelihood=lik, inducing_variable=Z)\n",
+    "gp_nonlinear.likelihood.variance.assign(np.var(y_train))\n",
+    "gp_nonlinear, iters, elbo, pred_error = trainGP(\n",
+    "    gp_nonlinear,\n",
+    "    training_data=(x_train, y_train),\n",
+    "    validation_data=(x_valid, y_valid),\n",
+    "    iterations=10000,\n",
+    "    batch_size=100,\n",
+    "    verbose=0,\n",
+    ")\n",
+    "plot_loss(iters, elbo, pred_error)\n",
+    "gp_nonlinear"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "prediction_nonlinGP, uncertainty_nonlinGP = gp_nonlinear.predict_f(x_valid)\n",
+    "prediction_nonlinGP = np.squeeze(prediction_nonlinGP.numpy())\n",
+    "uncertainty_nonlinGP = np.squeeze(uncertainty_nonlinGP.numpy())\n",
+    "\n",
+    "n_std = 3\n",
+    "start = 31000\n",
+    "end = start + 1000\n",
+    "prediction_nonlinGPplot = prediction_nonlinGP[start:end]\n",
+    "uncertainty_nonlinGPplot = n_std * np.sqrt(uncertainty_nonlinGP[start:end])\n",
+    "\n",
+    "fig, ax = plt.subplots(1, 2, figsize=(17, 5), dpi=150)\n",
+    "ax[0].spines[\"top\"].set_color(None)\n",
+    "ax[0].spines[\"right\"].set_color(None)\n",
+    "ax[0].plot(prediction_nonlinGPplot, label=\"Predicted values\", color=\"teal\")\n",
+    "ax[0].fill_between(\n",
+    "    np.arange(1000),\n",
+    "    prediction_nonlinGPplot + uncertainty_nonlinGPplot,\n",
+    "    prediction_nonlinGPplot - uncertainty_nonlinGPplot,\n",
+    "    alpha=0.2,\n",
+    "    color=\"teal\",\n",
+    ")\n",
+    "ax[0].plot(y_valid[start:end], label=\"True values\", color=\"k\")\n",
+    "ax[0].grid(True, alpha=0.3)\n",
+    "ax[0].legend(fontsize=7)\n",
+    "\n",
+    "a, b = zip(*sorted(zip(X_valid.reshape(ns * K, order=\"F\"), prediction_nonlinGP)))\n",
+    "a, c = zip(\n",
+    "    *sorted(\n",
+    "        zip(X_valid.reshape(ns * K, order=\"F\"), n_std * np.sqrt(uncertainty_nonlinGP))\n",
+    "    )\n",
+    ")\n",
+    "a = np.array(a)\n",
+    "b = np.array(b)\n",
+    "c = np.array(c)\n",
+    "\n",
+    "ax[1].spines[\"top\"].set_color(None)\n",
+    "ax[1].spines[\"right\"].set_color(None)\n",
+    "ax[1].scatter(\n",
+    "    X_valid.reshape(ns * K, order=\"F\"),\n",
+    "    Y_valid.reshape(ns * K, order=\"F\"),\n",
+    "    s=1,\n",
+    "    color=\"k\",\n",
+    ")\n",
+    "ax[1].plot(a, b, color=\"teal\", lw=2)\n",
+    "ax[1].fill_between(a, b - c, b + c, color=\"teal\", alpha=0.2)\n",
+    "ax[1].grid(True, alpha=0.3)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Implement the linear GP parameterisation in an online setting"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "T_test = 10\n",
+    "\n",
+    "X_full, _, _, _ = W.run(dt, T_test, return_coupling=True)  # Full model\n",
+    "\n",
+    "init_cond = Xtrue[-1, :]\n",
+    "\n",
+    "gcm_gp_det = GCM_GP(Forcing, gp_nonlinear, stochastic=False)\n",
+    "Xgp_det, t = gcm_gp_det(init_cond, dt, int(T_test / dt), gp_nonlinear)\n",
+    "\n",
+    "gcm_gp_sto = GCM_GP(Forcing, gp_nonlinear, stochastic=True)\n",
+    "Xgp_sto, t = gcm_gp_sto(init_cond, dt, int(T_test / dt), gp_nonlinear)\n",
+    "\n",
+    "gcm_no_param = GCM_no_param(Forcing)\n",
+    "X_no_param, t = gcm_no_param(init_cond, dt, int(T_test / dt))\n",
+    "\n",
+    "naive_parameterization = lambda param, X: np.polyval(param, X)\n",
+    "gcm = GCM(Forcing, naive_parameterization)\n",
+    "X_param, t = gcm(init_cond, dt, int(T / dt), param=[0.85439536, 1.75218026])\n",
+    "\n",
+    "time_i = 200\n",
+    "fig, ax = plt.subplots(1, figsize=(8, 5), dpi=150)\n",
+    "ax.spines[\"top\"].set_color(None)\n",
+    "ax.spines[\"right\"].set_color(None)\n",
+    "ax.plot(t[:time_i], X_full[:time_i, 4], label=\"Full L96\", color=\"k\")\n",
+    "ax.plot(\n",
+    "    t[:time_i],\n",
+    "    Xgp_det[:time_i, 4],\n",
+    "    label=\"GP (nonlinear kernel det.)\",\n",
+    "    color=\"darkorange\",\n",
+    ")\n",
+    "ax.plot(\n",
+    "    t[:time_i], Xgp_sto[:time_i, 4], label=\"GP (nonlinear kernel stoch.)\", color=\"teal\"\n",
+    ")\n",
+    "ax.plot(t[:time_i], X_no_param[:time_i, 4], label=\"no param\", color=\"darkviolet\")\n",
+    "ax.plot(t[:time_i], X_param[:time_i, 4], label=\"polynomial\", color=\"crimson\")\n",
+    "ax.grid(True, alpha=0.3)\n",
+    "plt.legend(frameon=False, loc=\"upper left\", fontsize=7)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Use multi-task GPs to exploit non-local features\n",
+    "\n",
+    "In the GP literature, it is common to see the multi-task learning problem framed the same set of inputs are used to learn each of the functional mappings to the various outputs:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\textbf{y}_k = f_k(\\textbf{x}) + \\pmb{\\varepsilon}_k,\n",
+    "\\end{equation}\n",
+    "\n",
+    "however it is also possible to define a unique set of inputs which map to the corresponding $k$, i.e:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\textbf{y}_k = f_k(\\textbf{x}_k) + \\pmb{\\varepsilon}_k.\n",
+    "\\end{equation}\n",
+    "\n",
+    "This is more applicable to our case as we have 8 coarse grid terms, each of which map to their own respective sub-grid terms. We then want to model for any potential correlation between the outputs $\\textbf{y}_k$, and hence the various functions $f_k(\\textbf{x}_k)$. One way to achieve this is through a concept known as *Linear models for coregionalisation* (see e.g., [Álvarez et al, 2012](https://arxiv.org/pdf/1106.6251.pdf)), where we assume that the function $f(\\textbf{x})$ is drawn from the linear combination of some latent function $g(\\textbf{x})$, such that:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "f(\\textbf{x}) = \\textbf{W}g(\\textbf{x}),\n",
+    "\\end{equation}\n",
+    "\n",
+    "where $f(\\textbf{x}) \\in \\mathbb{R}^K$, $g(\\textbf{x}) \\in \\mathbb{R}^L$, and $\\textbf{W} \\in \\mathbb{R}^{K \\times L}$, where $L \\leq K$. The principle is that the outputs of $g(\\textbf{x})$ are uncorrelated, but through combining them with $\\textbf{W}$, they become correlated. In GPflow, the only possibility for incorporating the unique input-output task pairing is via a simpler approach known as the *Intrinsic model for Coregionalisation* (see the [docs](https://gpflow.github.io/GPflow/2.4.0/notebooks/advanced/coregionalisation.html)). In this approach we assume the covariance of the latent function values between two tasks is given by a linear combination of the input kernel:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\text{cov}\\big(f_i(\\textbf{x}),f_j(\\textbf{x}')\\big) = k(\\textbf{x},\\textbf{x}')\\cdot \\text{B}_{ij},\n",
+    "\\end{equation}\n",
+    "\n",
+    "where $\\textbf{B} = \\textbf{W}\\textbf{W}^\\text{T} + \\text{diag}(\\kappa)$, and $\\textbf{W} \\in \\mathbb{R}^{K \\times L}$. The values of $\\textbf{W}$ and $\\kappa$ are optimised along with the kernel hyperparameters with the ELBO.\n",
+    "\n",
+    "We will use the same nonlinear kernel (Rational Quadratic + Linear) as before."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from gpflow.kernels import Coregion\n",
+    "\n",
+    "np.random.seed(2)\n",
+    "\n",
+    "heterosked = False  # separate noise variance for each task\n",
+    "\n",
+    "# in this case we use the idea of 'coregionalisation' to model the dependence between outputs\n",
+    "task_id_train = np.atleast_2d(np.kron(np.arange(K), np.ones(n))).T\n",
+    "task_id_test = np.atleast_2d(np.kron(np.arange(K), np.ones(ns))).T\n",
+    "\n",
+    "# augment the inputs with an index for which output each row belongs to\n",
+    "x_train = tf.convert_to_tensor(\n",
+    "    np.hstack((X_train.reshape(n * K, 1, order=\"F\"), task_id_train)), dtype=tf.float64\n",
+    ")\n",
+    "if heterosked:\n",
+    "    y_train = tf.convert_to_tensor(\n",
+    "        np.hstack((Y_train.reshape(n * K, 1, order=\"F\"), task_id_train)),\n",
+    "        dtype=tf.float64,\n",
+    "    )\n",
+    "    lik = gpflow.likelihoods.SwitchedLikelihood(\n",
+    "        [gpflow.likelihoods.Gaussian() for k in range(K)]\n",
+    "    )\n",
+    "else:\n",
+    "    y_train = tf.convert_to_tensor(\n",
+    "        Y_train.reshape(n * K, 1, order=\"F\"), dtype=tf.float64\n",
+    "    )\n",
+    "    lik = gpflow.likelihoods.Gaussian()\n",
+    "x_valid = tf.convert_to_tensor(\n",
+    "    np.hstack((X_valid.reshape(ns * K, 1, order=\"F\"), task_id_test)), dtype=tf.float64\n",
+    ")\n",
+    "y_valid = Y_valid.reshape(ns * K, 1, order=\"F\")\n",
+    "\n",
+    "M = 56\n",
+    "Z = np.array(\n",
+    "    [\n",
+    "        np.linspace(np.min(X_train[:, k]), np.max(X_train[:, k]), M // K)\n",
+    "        for k in range(K)\n",
+    "    ]\n",
+    ").T.reshape((M // K) * K, 1, order=\"F\")\n",
+    "Z = tf.convert_to_tensor(\n",
+    "    np.hstack((Z, np.atleast_2d(np.kron(np.arange(K), np.ones(M // K))).T)),\n",
+    "    dtype=tf.float64,\n",
+    ")\n",
+    "\n",
+    "rank = K  # for larger data sets it may be quicker to set a rank lower than full-rank.\n",
+    "coreg = Coregion(\n",
+    "    output_dim=K, rank=rank, active_dims=[x_train.shape[1] - 1]\n",
+    ")  # the last column of the input data\n",
+    "# contains the indices for each task\n",
+    "\n",
+    "dims = np.arange(x_train.shape[1] - 1)\n",
+    "k1 = gpflow.kernels.Linear(active_dims=dims)\n",
+    "k2 = gpflow.kernels.RationalQuadratic(active_dims=dims)\n",
+    "kernel = (k1 + k2) * coreg\n",
+    "\n",
+    "gp_nonlocal = gpflow.models.SVGP(kernel=kernel, likelihood=lik, inducing_variable=Z)\n",
+    "gp_nonlocal, iters, elbo, pred_error = trainGP(\n",
+    "    gp_nonlocal,\n",
+    "    training_data=(x_train, y_train),\n",
+    "    validation_data=(x_valid, y_valid),\n",
+    "    iterations=5000,\n",
+    "    batch_size=100,\n",
+    "    verbose=0,\n",
+    ")\n",
+    "plot_loss(iters, elbo, pred_error)\n",
+    "gp_nonlocal  # print the model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "prediction_nonlocGP, uncertainty_nonlocGP = gp_nonlocal.predict_f(x_valid)\n",
+    "prediction_nonlocGP = np.squeeze(prediction_nonlocGP.numpy())\n",
+    "uncertainty_nonlocGP = np.squeeze(uncertainty_nonlocGP.numpy())\n",
+    "\n",
+    "n_std = 3\n",
+    "start = 31000\n",
+    "end = start + 1000\n",
+    "prediction_nonlocGPplot = prediction_nonlocGP[start:end]\n",
+    "uncertainty_nonlocGPplot = n_std * np.sqrt(uncertainty_nonlocGP[start:end])\n",
+    "\n",
+    "fig, ax = plt.subplots(1, 2, figsize=(17, 5), dpi=150)\n",
+    "ax[0].spines[\"top\"].set_color(None)\n",
+    "ax[0].spines[\"right\"].set_color(None)\n",
+    "ax[0].plot(prediction_nonlocGPplot, label=\"Predicted values\", color=\"teal\")\n",
+    "ax[0].fill_between(\n",
+    "    np.arange(1000),\n",
+    "    prediction_nonlocGPplot + uncertainty_nonlocGPplot,\n",
+    "    prediction_nonlocGPplot - uncertainty_nonlocGPplot,\n",
+    "    alpha=0.2,\n",
+    "    color=\"teal\",\n",
+    ")\n",
+    "ax[0].plot(y_valid[start:end], label=\"True values\", color=\"k\")\n",
+    "ax[0].grid(True, alpha=0.3)\n",
+    "ax[0].legend(fontsize=7)\n",
+    "\n",
+    "a, b = zip(*sorted(zip(X_valid.reshape(ns * K, order=\"F\"), prediction_nonlocGP)))\n",
+    "a, c = zip(\n",
+    "    *sorted(\n",
+    "        zip(X_valid.reshape(ns * K, order=\"F\"), n_std * np.sqrt(uncertainty_nonlocGP))\n",
+    "    )\n",
+    ")\n",
+    "a = np.array(a)\n",
+    "b = np.array(b)\n",
+    "c = np.array(c)\n",
+    "\n",
+    "ax[1].spines[\"top\"].set_color(None)\n",
+    "ax[1].spines[\"right\"].set_color(None)\n",
+    "ax[1].scatter(\n",
+    "    X_valid.reshape(ns * K, order=\"F\"),\n",
+    "    Y_valid.reshape(ns * K, order=\"F\"),\n",
+    "    s=1,\n",
+    "    color=\"k\",\n",
+    ")\n",
+    "ax[1].plot(a, b, color=\"teal\", lw=2)\n",
+    "ax[1].fill_between(a, b - c, b + c, color=\"teal\", alpha=0.2)\n",
+    "ax[1].grid(True, alpha=0.3)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "T_test = 10\n",
+    "\n",
+    "X_full, _, _, _ = W.run(dt, T_test, return_coupling=True)  # Full model\n",
+    "\n",
+    "init_cond = Xtrue[-1, :]\n",
+    "\n",
+    "gcm_gp_nonlinear = GCM_GP(Forcing, gp_nonlinear, stochastic=True)\n",
+    "Xgp_nonlinear, t = gcm_gp_nonlinear(init_cond, dt, int(T_test / dt), gp_nonlinear)\n",
+    "\n",
+    "gcm_gp_nonlocal = GCM_GP(Forcing, gp_nonlocal, stochastic=True, local=False)\n",
+    "Xgp_nonlocal, t = gcm_gp_nonlocal(init_cond, dt, int(T_test / dt), gp_nonlocal)\n",
+    "\n",
+    "gcm_gp_nonlocal_det = GCM_GP(Forcing, gp_nonlocal, stochastic=False, local=False)\n",
+    "Xgp_nonlocal_det, t = gcm_gp_nonlocal_det(init_cond, dt, int(T_test / dt), gp_nonlocal)\n",
+    "\n",
+    "time_i = 200\n",
+    "fig, ax = plt.subplots(1, figsize=(8, 5), dpi=150)\n",
+    "ax.spines[\"top\"].set_color(None)\n",
+    "ax.spines[\"right\"].set_color(None)\n",
+    "ax.plot(t[:time_i], X_full[:time_i, 4], label=\"Full L96\", color=\"k\")\n",
+    "ax.plot(\n",
+    "    t[:time_i],\n",
+    "    Xgp_nonlinear[:time_i, 4],\n",
+    "    label=\"GP (nonlinear local kernel)\",\n",
+    "    color=\"teal\",\n",
+    ")\n",
+    "ax.plot(\n",
+    "    t[:time_i],\n",
+    "    Xgp_nonlocal[:time_i, 4],\n",
+    "    label=\"GP (nonlinear nonlocal kernel sto)\",\n",
+    "    color=\"crimson\",\n",
+    ")\n",
+    "ax.plot(\n",
+    "    t[:time_i],\n",
+    "    Xgp_nonlocal_det[:time_i, 4],\n",
+    "    label=\"GP (nonlinear nonlocal kernel det)\",\n",
+    "    color=\"darkviolet\",\n",
+    ")\n",
+    "ax.grid(True, alpha=0.3)\n",
+    "plt.legend(frameon=False, fontsize=7)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/Improving_Neural_networks.ipynb b/_sources/notebooks/Improving_Neural_networks.ipynb
new file mode 100644
index 00000000..cc8b238f
--- /dev/null
+++ b/_sources/notebooks/Improving_Neural_networks.ipynb
@@ -0,0 +1,773 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# Improving Performance of Neural Networks\n",
+    "\n",
+    "In this notebook, we will briefly introduce several techniques that are generally used to improve the accuracy of neural networks."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "import time\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import torch\n",
+    "from torch import nn, optim\n",
+    "from torch.utils.data import TensorDataset, DataLoader\n",
+    "\n",
+    "from L96_model import L96"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Ensuring reproducibility\n",
+    "np.random.seed(14)\n",
+    "torch.manual_seed(14);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Setting up the Dataset, Model, and the Training Code\n",
+    "\n",
+    "First, we setup all the necessary code that's required to build the dataset, create our neural network and train the network on the dataset.\n",
+    "\n",
+    "```{note}\n",
+    "The dataset, the neural network, and the training and evaluation functions that we use in this notebook are same as the one defined in [Introduction to Machine Learning and Neural Networks](https://m2lines.github.io/L96_demo/notebooks/intro_ML_and_NNs.html) and [Using Neural Networks for L96 Parameterization](https://m2lines.github.io/L96_demo/notebooks/L96_offline_NN.html).\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "### Create the Dataset"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": [
+     "hide-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# Generating the Ground Truth\n",
+    "# ---------------------------\n",
+    "\n",
+    "time_steps = 2000\n",
+    "forcing, dt, T = 18, 0.01, 0.01 * time_steps\n",
+    "\n",
+    "W = L96(8, 32, F=forcing)\n",
+    "\n",
+    "X_true, _, _, xy_true = W.run(dt, T, store=True, return_coupling=True)\n",
+    "X_true, xy_true = X_true.astype(np.float32), xy_true.astype(np.float32)\n",
+    "\n",
+    "\n",
+    "# Splitting the into Training and Test Dataset\n",
+    "# --------------------------------------------\n",
+    "\n",
+    "val_size = 400\n",
+    "\n",
+    "# Training Data\n",
+    "X_true_train = X_true[:-val_size, :]\n",
+    "subgrid_tend_train = xy_true[:-val_size, :]\n",
+    "\n",
+    "# Test Data\n",
+    "X_true_test = X_true[-val_size:, :]\n",
+    "subgrid_tend_test = xy_true[-val_size:, :]\n",
+    "\n",
+    "\n",
+    "# Building the Dataset and the Dataloaders\n",
+    "# ----------------------------------------\n",
+    "\n",
+    "BATCH_SIZE = 1024\n",
+    "\n",
+    "# Training Data\n",
+    "nlocal_data = TensorDataset(\n",
+    "    torch.from_numpy(X_true_train),\n",
+    "    torch.from_numpy(subgrid_tend_train),\n",
+    ")\n",
+    "loader_train = DataLoader(dataset=nlocal_data, batch_size=BATCH_SIZE, shuffle=True)\n",
+    "\n",
+    "\n",
+    "# Test Data\n",
+    "nlocal_data_test = TensorDataset(\n",
+    "    torch.from_numpy(X_true_test), torch.from_numpy(subgrid_tend_test)\n",
+    ")\n",
+    "loader_test = DataLoader(dataset=nlocal_data_test, batch_size=BATCH_SIZE, shuffle=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "### Create the Neural Network\n",
+    "\n",
+    "We build a 3 layer neural network consisting of two hidden layers and an output layer. Each hidden layer is followed by the `ReLU` activation function."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": [
+     "hide-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "class NetANN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs\n",
+    "        self.linear2 = nn.Linear(16, 16)\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = self.relu(self.linear1(x))\n",
+    "        x = self.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Train and Evaluate the Network"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": [
+     "hide-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "def train_model(network, criterion, loader, optimizer):\n",
+    "    \"\"\"Train the network for one epoch\"\"\"\n",
+    "    network.train()\n",
+    "\n",
+    "    train_loss = 0\n",
+    "    for batch_x, batch_y in loader:\n",
+    "        # Get predictions\n",
+    "        if len(batch_x.shape) == 1:\n",
+    "            # This if block is needed to add a dummy dimension if our inputs are 1D\n",
+    "            # (where each number is a different sample)\n",
+    "            prediction = torch.squeeze(network(torch.unsqueeze(batch_x, 1)))\n",
+    "        else:\n",
+    "            prediction = network(batch_x)\n",
+    "\n",
+    "        # Compute the loss\n",
+    "        loss = criterion(prediction, batch_y)\n",
+    "        train_loss += loss.item()\n",
+    "\n",
+    "        # Clear the gradients\n",
+    "        optimizer.zero_grad()\n",
+    "\n",
+    "        # Backpropagation to compute the gradients and update the weights\n",
+    "        loss.backward()\n",
+    "        optimizer.step()\n",
+    "\n",
+    "    return train_loss / len(loader)\n",
+    "\n",
+    "\n",
+    "def test_model(network, criterion, loader):\n",
+    "    \"\"\"Test the network\"\"\"\n",
+    "    network.eval()  # Evaluation mode (important when having dropout layers)\n",
+    "\n",
+    "    test_loss = 0\n",
+    "    with torch.no_grad():\n",
+    "        for batch_x, batch_y in loader:\n",
+    "            # Get predictions\n",
+    "            if len(batch_x.shape) == 1:\n",
+    "                # This if block is needed to add a dummy dimension if our inputs are 1D\n",
+    "                # (where each number is a different sample)\n",
+    "                prediction = torch.squeeze(network(torch.unsqueeze(batch_x, 1)))\n",
+    "            else:\n",
+    "                prediction = network(batch_x)\n",
+    "\n",
+    "            # Compute the loss\n",
+    "            loss = criterion(prediction, batch_y)\n",
+    "            test_loss += loss.item()\n",
+    "\n",
+    "        # Get an average loss for the entire dataset\n",
+    "        test_loss /= len(loader)\n",
+    "\n",
+    "    return test_loss\n",
+    "\n",
+    "\n",
+    "def fit_model(network, criterion, optimizer, train_loader, val_loader, n_epochs):\n",
+    "    \"\"\"Train and validate the network\"\"\"\n",
+    "    train_losses, val_losses = [], []\n",
+    "    start_time = time.time()\n",
+    "    for epoch in range(1, n_epochs + 1):\n",
+    "        train_loss = train_model(network, criterion, train_loader, optimizer)\n",
+    "        val_loss = test_model(network, criterion, val_loader)\n",
+    "        train_losses.append(train_loss)\n",
+    "        val_losses.append(val_loss)\n",
+    "    end_time = time.time()\n",
+    "    print(f\"Training completed in {int(end_time - start_time)} seconds.\")\n",
+    "\n",
+    "    return train_losses, val_losses"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Regularization and Overfitting\n",
+    "\n",
+    "One of the most common issues that happen while training a neural network is when the model memorizes the training dataset. It causes the model to perform very accurately on the training set but shows very poor performance on the validation set. This phenomenon is termed overfitting. One of the ways to prevent overfitting is to add regularization to our model as described below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "```{figure} figs/overfitting.png\n",
+    ":name: overfitting-curve\n",
+    ":width: 700\n",
+    "\n",
+    "Curves showing different types of model fits. The image is taken from the Python Machine Learning book by Sebastian Raschka.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The curve on the far right of the plot above predicts perfectly on the given set, yet it's not the best choice. This is because if you were to gather some new data points, they most likely would not be on that curve. Instead, those new points would be closer to the curve in the middle graph since it generalizes better to the dataset."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "### Regularization Intuition\n",
+    "\n",
+    "Regularization can be thought of as **putting constraints on the model** to obtain better generalizability i.e. *avoiding remembering* the training data.\n",
+    "\n",
+    "One of the ways to achieve this can be by adding a term to the loss function such that:\n",
+    "> Loss = Training Loss + Regularization\n",
+    "\n",
+    "This puts a penalty for making the model more complex.\n",
+    "\n",
+    "Very braodly speaking (just to gain intuition) - if we want to reduce the training loss (reduce bias) we should try using a more complex model (if we have enough data) and if we want to reduce overfitting (reduce variace) we should simplify or constraint the model (increase regularization)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Regularization of Neural Networks"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Some of the ways to add regularization in neural networks are:\n",
+    "\n",
+    "- Dropout (added in the definition of the network). \n",
+    "- Early stopping\n",
+    "- Weight decay (added in the optimizer part - see `optim.Adam` in PyTorch)\n",
+    "- Data augmentation (usually for images)\n",
+    "\n",
+    "Below we show how some of these regularizations can be applied to the L96 parameterization problem. However, this particular problem is not very plagued by the issue of overfitting, and so the advantages of these processes are not clear in all their glory. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "#### Weight decay (L2 norm)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Weight decay is usually defined as a term that’s added directly to the update rule.\n",
+    "Namely, to update a certain weight $w$ in the $i+1$ iteration, we would use a modified rule:\n",
+    "\n",
+    "$w_{i+1} = w_{i} - \\gamma ( \\frac{\\partial L}{\\partial w} + A w_{i})$\n",
+    "\n",
+    "In practice, this is almost identical to L2 regularization, though there is some difference (e.g., see [here](https://bbabenko.github.io/weight-decay/))\n",
+    "\n",
+    "Weight decay is one of the parameters of the optimizer - see `torch.optim.SGD`"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "##### Using Weight Decay\n",
+    "\n",
+    "Now we try to train our `NetANN` model again but this time by adding a weight decay to it."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "nn_3l_decay = NetANN()\n",
+    "\n",
+    "n_epochs = 80\n",
+    "criterion = torch.nn.MSELoss()\n",
+    "optimizer_decay = optim.Adam(nn_3l_decay.parameters(), lr=0.003, weight_decay=0.1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "train_loss_decay, val_loss_decay = fit_model(\n",
+    "    nn_3l_decay, criterion, optimizer_decay, loader_train, loader_test, n_epochs\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Plotting the training and validation loss curves"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "\n",
+    "plt.plot(train_loss_decay, \"r\", label=\"Training loss, with weight decay\")\n",
+    "plt.plot(val_loss_decay, \"r--\", label=\"Validation loss, with weight decay\")\n",
+    "\n",
+    "plt.legend()\n",
+    "plt.yscale(\"log\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "#### Dropout"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Dropout means randomly deactivating or temporarily removing some units from a layer of the network while training, along with all its incoming and outgoing connections. See more details [here](http://jmlr.org/papers/v15/srivastava14a.html).\n",
+    "It is usually the most useful regularization that we can do in fully connected layers.\n",
+    "\n",
+    "In convolutional layers dropout makes less sense - see more discussion [here](https://www.kdnuggets.com/2018/09/dropout-convolutional-networks.html)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "```{figure} figs/Dropout_layer.png\n",
+    ":name: dropout-layer\n",
+    ":width: 700\n",
+    "\n",
+    "Dropout Neural Net Model. Left: A standard neural net with 2 hidden layers. Right: An example of a thinned net produced by applying dropout to the network on the left. Crossed units have been dropped. Image taken from [here](http://www.jmlr.org/papers/volume15/srivastava14a/srivastava14a.pdf).\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "In the network defined below, we add dropout to with a probability of 20% to each layer. This means that during each training step, random 20% of the units within each layer will be deactivated."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class NetANNDropout(nn.Module):\n",
+    "    def __init__(self, dropout_rate=0.2):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)\n",
+    "        self.linear2 = nn.Linear(16, 16)\n",
+    "        self.linear3 = nn.Linear(16, 8)\n",
+    "\n",
+    "        self.dropout = nn.Dropout(dropout_rate)  # Dropout regularization\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = self.relu(self.linear1(x))\n",
+    "        x = self.dropout(x)\n",
+    "        x = self.relu(self.linear2(x))\n",
+    "        x = self.dropout(x)\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Network with very high dropout\n",
+    "nn_3l_drop = NetANNDropout(dropout_rate=0.5)\n",
+    "\n",
+    "criterion = torch.nn.MSELoss()\n",
+    "optimizer_drop = optim.Adam(nn_3l_drop.parameters(), lr=0.003)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "train_loss_drop, val_loss_drop = fit_model(\n",
+    "    nn_3l_drop, criterion, optimizer_drop, loader_train, loader_test, n_epochs\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Plotting the training and validation loss curves."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(train_loss_drop, \"r\", label=\"Training loss, with dropout\")\n",
+    "plt.plot(val_loss_drop, \"r--\", label=\"Validation loss, with dropout\")\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In this particular problem drop out does not seem to provide much advantage. While the model with drop out never seems to reach a state of overfitting (validation loss>training loss), it also does not perform as well as the model without any dropout. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Choosing the Learning Rate\n",
+    "\n",
+    "While training a neural network **selecting a good learning rate (LR) is essential for both fast convergence and a lower error**. A high learning rate can cause the training loss to never converge while a too small learning rate will cause the model to converge extremely slowly."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Finding the Optimal Learning Rate\n",
+    "\n",
+    "To choose the optimal learning rate for our network, we can use an LR finding algorithm. The objective of a LR Finder is to find the highest LR which still minimises the loss and does not make the loss diverge/explode. This is done by first starting with an extremely small LR and then increasing the LR after each batch until the corresponding loss starts to explode. To read more about learning rate finders, read [this blog](https://towardsdatascience.com/speeding-up-neural-net-training-with-lr-finder-c3b401a116d0).\n",
+    "\n",
+    "For our use case, we use the LR finder from the `torch-lr-finder` package to find the best learning rate for our neural network."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from torch_lr_finder import LRFinder"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Define the model and the optimizer. The optimizer is **initialized with a very small learning rate**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "nn_3l_lr = NetANN()\n",
+    "optimizer = optim.Adam(nn_3l_lr.parameters(), lr=1e-7)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Now we setup the LR finder and make it run for 200 iterations during which the learning rate varies from 1e-7 to 100."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "lr_finder = LRFinder(nn_3l_lr, optimizer, criterion)\n",
+    "lr_finder.range_test(loader_train, end_lr=100, num_iter=200)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Now we plot the LR vs the loss curve to find the best learning rate."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Plot the lr vs the loss curve\n",
+    "lr_finder.plot()\n",
+    "\n",
+    "# Reset the model and optimizer to their initial state\n",
+    "lr_finder.reset()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "From the curve, we see that at the learning of approximately 0.01 we get the steepest gradient. So we choose 0.01 as the learning rate for our neural network."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "n_epochs = 20\n",
+    "optimizer = optim.Adam(nn_3l_lr.parameters(), lr=0.01)\n",
+    "\n",
+    "train_loss, val_loss = fit_model(\n",
+    "    nn_3l_lr, criterion, optimizer, loader_train, loader_test, n_epochs\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Plotting the training and validation loss curves."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(train_loss, \"b\", label=\"Training loss\")\n",
+    "plt.plot(val_loss, \"r\", label=\"Validation loss\")\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "From the loss curves we can see that **the loss has converged much faster** than before."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Recommended Reading"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Batch Normalization "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Normalize the activation values such that the hidden representation don’t vary drastically and also helps to get improvement in the training speed."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Cyclic learning rate"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The cyclic learning rate policy, introduced in [Cyclical Learning Rates for Training Neural Networks](https://arxiv.org/abs/1506.01186), cycles the learning rate between two boundaries with a constant frequency in a triangular fashion. To read more about the cyclic learning rates and the one cycle policy, read [here](https://sgugger.github.io/the-1cycle-policy.html).\n",
+    "\n",
+    "In PyTorch, cyclic learning rate can be used from `optim.lr_scheduler.CyclicLR`."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "```{figure} figs/cyclic_lr.png\n",
+    ":name: cyclic-lr\n",
+    ":width: 500\n",
+    "\n",
+    "Cyclical learning rates oscillate back and forth between two bounds when training, slowly increasing the learning rate after every batch update. Image taken from [here](https://pyimagesearch.com/2019/07/29/cyclical-learning-rates-with-keras-and-deep-learning/).\n",
+    "```"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/L96-two-scale-description.ipynb b/_sources/notebooks/L96-two-scale-description.ipynb
new file mode 100644
index 00000000..1b098cc6
--- /dev/null
+++ b/_sources/notebooks/L96-two-scale-description.ipynb
@@ -0,0 +1,348 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# The Lorenz-96 Two-Timescale System\n",
+    "\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "In 1996, Lorenz proposed a two-time scale dynamical system, referred to as the Lorenz-96 model (L96), whose behavior mimics the non-linear dynamics of the extratropical atmosphere with simplified representation of multiscale interactions and nonlinear advection {cite:p}`Lorenz1995`. The L96 model consists of two sets of equations coupling two sets of variables ($X_k$ and $Y_{j,k}$), which evolve over two (slow and fast) timescales and are arranged around a latitude circle as shown in Fig. 1. The equations comprising L96 are:\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k},\n",
+    "\\\\\n",
+    "\\frac{d}{dt} Y_{j,k}\n",
+    "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - Y_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
+    "\\end{align}\n",
+    "\n",
+    "where $X_k$, $k=1,\\ldots,K$, denotes $K$ slow (or low-frequency) variables, and $Y_{j,k}$ , $j=1,\\ldots,J$ denotes $J*K$ fast (or high-frequency) variables. \n",
+    "The slow equations are coupled to the fast equations via a coupling term, $\\sum_{j=0}^{J-1} Y_{j,k}$, which sums over the $J$ fast variables corresponding to a particular $k$. On the other hand, each fast equation is forced by a coupling term, $\\frac{hc}{b} X_k$, that depends on the slow variable corresponding to that particular $k$. \n",
+    "\n",
+    "The evolution of this two-time scale system depend on three key parameters: $b$, $c$ and $h$. Here $b$ determines the magnitude of the nonlinear interactions among the fast variables, $c$ controls how rapidly the fast variables fluctuate compared to the slow variables and, $h$ governs the strength of the coupling between the slow and fast variables. Moreover, the slow time-scale equation is forced by the parameter $F$, whose value determines the chaotic behaviour of the system. E.g. {cite:p}`Wilks2005`.\n",
+    "\n",
+    "The chaotic dynamical system L96 is very useful for testing different numerical methods in atmospheric modeling thanks to its transparency, low computational cost and simplicity compared to Global Climate Models (GCM). The interaction between variables of different scales makes the L96 model of particular interest when evaluating new parameterization methodologies. As such, it was used in assessing different techniques that were later incorporated into GCMs ({cite}`Crommelin2008`, {cite}`Dorrestijn2013`).\n",
+    "\n",
+    "The L96 model has been extensively used as a test bed in various studies including data assimilation approaches ({cite}`Law2016`, {cite}`Hatfield2017`), stochastic parameterization schemes ({cite}`Kwasniok2012`, {cite}`Arnold2013`, {cite}`Chorin2015`)  and Machine Learning-based parameterization methodologies ({cite}`Schneider2017`, {cite}`Dueben2018` , {cite}`Gagne2020`).\n",
+    "\n",
+    "\n",
+    "```{figure} https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376@1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png\n",
+    ":width: 400px\n",
+    ":name: l96-equation-figure\n",
+    "\n",
+    "*Visualisation of a two-scale Lorenz '96 system with J = 8 and K = 6. Global-scale variables ($X_k$) are updated based on neighbouring variables and on the local-scale variables ($Y_{j,k}$) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from [Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.](https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436)*.\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Code to numerically integrate the two time-scale L96 model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "from utils import display_source"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In the `L96_model` module we provide the function `L96_2t_xdot_ydot` (shown next) that returns the tendencies (sum of the RHS) for the X and Y equations, as well as the tendency corresponding to the coupling in the X equation, $-hc/b \\sum_{j=0}^{J-1} Y_{j,k}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from L96_model import L96_2t_xdot_ydot\n",
+    "\n",
+    "display_source(L96_2t_xdot_ydot)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "These tendencies describe the continuous evolution of the L96 model at a particular time. However, to obtain a discrete solution we must integrate numerically in time. In the `L96_model` module, we provide the function `integrate_L96_2t` (shown next) that uses fourth-order Runge-Kutta integration. Starting from the initial conditions `X0,Y0`, the function returns the trajectories of `X,Y` sampled at an interval `si`. There is a related function, `integrate_L96_2t_with_coupling`, that in addition to the trajectories, also returns the coupling term, $-hc/b \\sum_{j=0}^{J-1} Y_{j,k}$ at each point in the trajectory."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from L96_model import integrate_L96_2t\n",
+    "\n",
+    "display_source(integrate_L96_2t)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "An example of how to use `integrate_L96_2t` is shown below. Here we choose $K=36$ and $J=10$ (i.e. 36 values of $X$ and 10 values of $Y$ per value of $X$). Also, in accordance with previous studies, we set $h=1$, $c=10$, and $b=10$. The initial condition for $X$ is setup as $X(s)=s(s-1)(s+1)$, for $s=-1\\ldots 1$, and the $Y$ is initialized with zeros. The value of $F$ is set to $10$, which is sufficient to obtain chaotic behavior.\n",
+    "\n",
+    "*Note: if you increase $F$, you may need to reduce $dt$ for numerical stability.* "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "K = 36  # Number of globa-scale variables X\n",
+    "J = 10  # Number of local-scale Y variables per single global-scale X variable\n",
+    "nt = 1000  # Number of time steps\n",
+    "si = 0.005  # Sampling time interval\n",
+    "dt = 0.005  # Time step\n",
+    "F = 10.0  # Focring\n",
+    "h = 1.0  # Coupling coefficient\n",
+    "b = 10.0  # ratio of amplitudes\n",
+    "c = 10.0  # time-scale ratio"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def s(k, K):\n",
+    "    \"\"\"A non-dimension coordinate from -1..+1 corresponding to k=0..K\"\"\"\n",
+    "    return 2 * (0.5 + k) / K - 1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "k = np.arange(K)  # For coordinate in plots\n",
+    "j = np.arange(J * K)  # For coordinate in plots\n",
+    "\n",
+    "# Initial conditions\n",
+    "X_init = s(k, K) * (s(k, K) - 1) * (s(k, K) + 1)\n",
+    "Y_init = 0 * s(j, J * K) * (s(j, J * K) - 1) * (s(j, J * K) + 1)\n",
+    "\n",
+    "# \"Run\" model\n",
+    "X, Y, t = integrate_L96_2t(X_init, Y_init, si, nt, F, h, b, c, dt=dt)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "After running the model, we plot the results."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(12, 10), dpi=150)\n",
+    "plt.subplot(221)\n",
+    "\n",
+    "# Snapshot of X[k]\n",
+    "plt.plot(k, X[-1], label=\"$X_k(t=n_t dt)$\")\n",
+    "plt.plot(j / J, Y[-1], label=\"$Y_{j,k}(t=n_t dt)$\")\n",
+    "plt.plot(k, X_init, \"k:\", label=\"$X_k(t=0)$\")\n",
+    "plt.plot(j / J, Y_init, \"k:\", label=\"$Y_{j,k}(t=0)$\")\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.xlabel(\"k, k + j/J\")\n",
+    "plt.title(\"$X_k, Y_{j,k}$\")\n",
+    "plt.subplot(222)\n",
+    "\n",
+    "# Sample time-series X[0](t), Y[0](t)\n",
+    "plt.plot(t, X[:, 0], label=\"$X_0(t)$\")\n",
+    "plt.plot(t, Y[:, 0], label=\"$Y_{0,0}(t)$\")\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.xlabel(\"t\")\n",
+    "plt.subplot(223)\n",
+    "\n",
+    "# Full model history of X\n",
+    "plt.contourf(k, t, X)\n",
+    "plt.colorbar(orientation=\"horizontal\")\n",
+    "plt.xlabel(\"k\")\n",
+    "plt.ylabel(\"t\")\n",
+    "plt.title(\"$X_k(t)$\")\n",
+    "plt.subplot(224)\n",
+    "\n",
+    "# Full model history of Y\n",
+    "plt.contourf(j / J, t, Y)\n",
+    "plt.colorbar(orientation=\"horizontal\")\n",
+    "plt.xlabel(\"k + j/J\")\n",
+    "plt.ylabel(\"t\")\n",
+    "plt.title(\"$Y_{j,k}(t)$\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## A class for numerical integration of the two time-scale L96\n",
+    "\n",
+    "For convenience, we also provide a class named `L96` in `L96_model.py` which helps to reduce the amount of code when using the `L96_2t_xdot_ydot` function."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from L96_model import L96\n",
+    "\n",
+    "help(L96)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "For example, to setup the same model as above with $K=36$, $J=10$ and $F=10$, use"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create an instance of the L96 model, default parameters except those given\n",
+    "M = L96(36, 10, F=10, dt=0.005)\n",
+    "M"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note that parameters $F$, $h$, $b$, $c$ have defaults since in many of the applications in later notebooks these will not change. The initial conditions are automatically randomized to $X = b {\\cal N}$, $Y = {\\cal N}$ where ${\\cal N}$ is a random number drawn from a Guassian distribution. To set the same initial condition as used above, use"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Set the initial conditions (here X is the same cubic as above, Y=0)\n",
+    "M.set_state(s(M.k, M.K) * (s(M.k, M.K) - 1) * (s(M.k, M.K) + 1), 0 * M.j)\n",
+    "print(M)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "To sample the same tracjectories as above use the `.run()` method."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Run the model for 1000 sample intervals, or for time 1000*0.005 = 5\n",
+    "X2, Y2, t = M.run(0.005, 5)\n",
+    "\n",
+    "print(\"Mean absolute difference =\", np.abs(X2 - X).mean() + np.abs(Y2 - Y).mean())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Re-running the above cell gives the same answer each time because the same initial conditions are used for each invocation or `.run()`. To allow sequences of run, using the end state of one as the starting point for the next, add the parameter `store=True` which updates the initial condition stored in the object (`M`) with the final state of the run."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Run the model as above but use the end state as an initial condition for the next run.\n",
+    "# (Re-running this cell will give different answers after the first run)\n",
+    "X2, Y2, t = M.run(0.005, 5, store=True)\n",
+    "\n",
+    "print(\n",
+    "    \"Mean absolute difference 1st run=\", np.abs(X2 - X).mean() + np.abs(Y2 - Y).mean()\n",
+    ")\n",
+    "\n",
+    "X2, Y2, t = M.run(0.005, 5, store=True)\n",
+    "print(\n",
+    "    \"Mean absolute difference 2nd run=\", np.abs(X2 - X).mean() + np.abs(Y2 - Y).mean()\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The entire model creation and execution process demonstrated above can also be implemented in a one-liner version as shown below."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X3, Y3, t = (\n",
+    "    L96(36, 10, F=10, dt=0.005)\n",
+    "    .set_state(s(M.k, M.K) * (s(M.k, M.K) - 1) * (s(M.k, M.K) + 1), 0 * M.j)\n",
+    "    .run(0.005, 5)\n",
+    ")"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/L96_ResNet_RNN.ipynb b/_sources/notebooks/L96_ResNet_RNN.ipynb
new file mode 100644
index 00000000..388d5fe2
--- /dev/null
+++ b/_sources/notebooks/L96_ResNet_RNN.ipynb
@@ -0,0 +1,1015 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# ResNet and Recurrent Neural Networks"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "3Uai4mkDXGYx",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "from matplotlib.animation import FuncAnimation\n",
+    "from IPython.display import HTML\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "import math\n",
+    "import pandas as pd\n",
+    "\n",
+    "import torch\n",
+    "from torch.autograd import Variable\n",
+    "import torch.nn.functional as F\n",
+    "import torch.utils.data as Data\n",
+    "import torchvision\n",
+    "from torch import nn, optim\n",
+    "\n",
+    "# from torch_lr_finder import LRFinder\n",
+    "import torch.nn.functional as F\n",
+    "from torch import nn\n",
+    "from L96_model import L96, L96_eq1_xdot, integrate_L96_2t, EulerFwd, RK2, RK4"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "```{figure} https://www.asimovinstitute.org/wp-content/uploads/2019/04/NeuralNetworkZo19High.png\n",
+    ":width: 600px\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "Ud1XdNrKXNSx"
+   },
+   "source": [
+    "[Lorenz (1996)](https://www.ecmwf.int/en/elibrary/10829-predictability-problem-partly-solved) describes a \"two time-scale\" model in two equations (2 and 3) which are:\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}\n",
+    "\\\\\n",
+    "\\frac{d}{dt} Y_{j,k}\n",
+    "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - Y_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
+    "\\end{align}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "f82UE1jAYXYq"
+   },
+   "source": [
+    "```{figure} https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376@1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png\n",
+    ":width: 400px\n",
+    "\n",
+    "*Visualisation of a two-scale Lorenz '96 system with J = 8 and K = 6. Global-scale variables ($X_k$) are updated based on neighbouring variables and on the local-scale variables ($Y_{j,k}$) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from [Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.](https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436)*.\n",
+    "\n",
+    "```\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "74cig9YaV-Qq"
+   },
+   "source": [
+    "## Generating data from L96:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "cjRYqUxXkDwj",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "time_steps = 20000\n",
+    "Forcing, dt, T = 18, 0.01, 0.01 * time_steps\n",
+    "\n",
+    "# Create a \"real world\" with K=8 and J=32\n",
+    "W = L96(8, 32, F=Forcing)\n",
+    "\n",
+    "# Get training data for the neural network.\n",
+    "# - Run the true state and output subgrid tendencies (the effect of Y on X is xytrue):\n",
+    "X_true, y, t, xy_true = W.run(dt, T, return_coupling=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "4eBi3GvAkCzt",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "val_size = 4000  # number of time steps for validation\n",
+    "# train:\n",
+    "X_true_train = X_true[\n",
+    "    :-val_size, :\n",
+    "]  # Flatten because we first use single input as a sample\n",
+    "subgrid_tend_train = xy_true[:-val_size, :]\n",
+    "\n",
+    "# test:\n",
+    "X_true_test = X_true[-val_size:, :]\n",
+    "subgrid_tend_test = xy_true[-val_size:, :]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "0UhunGR0i__o",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Create non local training data\n",
+    "# Define a data loader (8 inputs, 8 outputs)\n",
+    "\n",
+    "# Define our X,Y pairs (state, subgrid tendency) for the linear regression local network.local_torch_dataset = Data.TensorDataset(\n",
+    "torch_dataset = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_train).double(),\n",
+    "    torch.from_numpy(subgrid_tend_train).double(),\n",
+    ")\n",
+    "\n",
+    "BATCH_SIZE = 1024  # Number of sample in each batch\n",
+    "\n",
+    "loader = Data.DataLoader(dataset=torch_dataset, batch_size=BATCH_SIZE, shuffle=True)\n",
+    "\n",
+    "# Define a test dataloader (8 inputs, 8 outputs)\n",
+    "\n",
+    "torch_dataset_test = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_test).double(), torch.from_numpy(subgrid_tend_test).double()\n",
+    ")\n",
+    "\n",
+    "loader_test = Data.DataLoader(\n",
+    "    dataset=torch_dataset_test, batch_size=BATCH_SIZE, shuffle=True\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "6vLWTQvBSYGE",
+    "user_expressions": []
+   },
+   "source": [
+    "\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}\n",
+    "\\\\\n",
+    "\\frac{d}{dt} Y_{j,k}\n",
+    "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - Y_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
+    "\\end{align}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "DPCFNuimR23o",
+    "user_expressions": []
+   },
+   "source": [
+    "Goal: Use $X_k$ to predict subgrid terms,  $- \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "2IuYac4orcTB",
+    "user_expressions": []
+   },
+   "source": [
+    "## Janni's fully connected, 3-layer Artificial Neural Network (ANN)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "9xUUyYZO621N",
+    "user_expressions": []
+   },
+   "source": [
+    "```{figure} https://miro.medium.com/max/720/1*VHOUViL8dHGfvxCsswPv-Q.png\n",
+    ":width: 400px\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "CRNfi_iTW01J",
+    "user_expressions": []
+   },
+   "source": [
+    "K=8 and J=32"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "N8CXFv5TfyKs",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "class Net_ANN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super(Net_ANN, self).__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs, 16 neurons for first hidden layer\n",
+    "        self.linear2 = nn.Linear(16, 16)  # 16 neurons for second hidden layer\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "    #         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = F.relu(self.linear1(x))\n",
+    "        x = F.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "Lv7KfTKqotrx",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def train_model(net, criterion, trainloader, optimizer):\n",
+    "    net.train()\n",
+    "    test_loss = 0\n",
+    "    for step, (batch_x, batch_y) in enumerate(trainloader):  # for each training step\n",
+    "        b_x = Variable(batch_x)  # Inputs\n",
+    "        b_y = Variable(batch_y)  # outputs\n",
+    "        if (\n",
+    "            len(b_x.shape) == 1\n",
+    "        ):  # If is needed to add a dummy dimension if our inputs are 1D (where each number is a different sample)\n",
+    "            prediction = torch.squeeze(\n",
+    "                net(torch.unsqueeze(b_x, 1))\n",
+    "            )  # input x and predict based on x\n",
+    "        else:\n",
+    "            prediction = net(b_x)\n",
+    "        loss = criterion(prediction, b_y)  # Calculating loss\n",
+    "        optimizer.zero_grad()  # clear gradients for next train\n",
+    "        loss.backward()  # backpropagation, compute gradients\n",
+    "        optimizer.step()  # apply gradients to update weights"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "XAJ8y0lsoyhQ",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def test_model(net, criterion, trainloader, optimizer, text=\"validation\"):\n",
+    "    net.eval()  # Evaluation mode (important when having dropout layers)\n",
+    "    test_loss = 0\n",
+    "    with torch.no_grad():\n",
+    "        for step, (batch_x, batch_y) in enumerate(\n",
+    "            trainloader\n",
+    "        ):  # for each training step\n",
+    "            b_x = Variable(batch_x)  # Inputs\n",
+    "            b_y = Variable(batch_y)  # outputs\n",
+    "            if (\n",
+    "                len(b_x.shape) == 1\n",
+    "            ):  # If is needed to add a dummy dimension if our inputs are 1D (where each number is a different sample)\n",
+    "                prediction = torch.squeeze(\n",
+    "                    net(torch.unsqueeze(b_x, 1))\n",
+    "                )  # input x and predict based on x\n",
+    "            else:\n",
+    "                prediction = net(b_x)\n",
+    "            loss = criterion(prediction, b_y)  # Calculating loss\n",
+    "            test_loss = test_loss + loss.data.numpy()  # Keep track of the loss\n",
+    "        test_loss /= len(trainloader)  # dividing by the number of batches\n",
+    "    #         print(len(trainloader))\n",
+    "    # disabling prints to only show graphs\n",
+    "    # print(text + ' loss:',test_loss)\n",
+    "    return test_loss"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "1bbQAWn2iI6T",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "criterion = torch.nn.MSELoss()  # MSE loss function"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "a7isegnHh0fa",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "nn_3l = Net_ANN().double()\n",
+    "# optimizer = optim.Adam(nn_3l.parameters(), lr=1e-7)\n",
+    "# lr_finder = LRFinder(nn_3l, optimizer, criterion)\n",
+    "# lr_finder.range_test(loader, end_lr=100, num_iter=200)\n",
+    "# lr_finder.plot() # to inspect the loss-learning rate graph\n",
+    "# lr_finder.reset() # to reset the model and optimizer to their initial state"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 959
+    },
+    "id": "o02WYIpQlw57",
+    "outputId": "aa5e5add-866c-4d9b-a34e-4110b24f20d8",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "n_epochs = 20  # Number of epocs\n",
+    "optimizer = optim.Adam(nn_3l.parameters(), lr=0.01)\n",
+    "validation_loss = list()\n",
+    "train_loss = list()\n",
+    "# time0 = time()\n",
+    "for epoch in range(1, n_epochs + 1):\n",
+    "    train_model(nn_3l, criterion, loader, optimizer)\n",
+    "    train_loss.append(test_model(nn_3l, criterion, loader, optimizer, \"train\"))\n",
+    "    validation_loss.append(test_model(nn_3l, criterion, loader_test, optimizer))\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(train_loss, \"b\", label=\"training loss\")\n",
+    "plt.plot(validation_loss, \"r\", label=\"validation loss\")\n",
+    "plt.ylabel(\"loss\")\n",
+    "plt.xlabel(\"epochs\")\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "final_losses = np.zeros(4)\n",
+    "final_losses[0] = validation_loss[-1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 265
+    },
+    "id": "CYoJoMUBp-eA",
+    "outputId": "81651f93-e26b-45b1-a0f3-480a4f7f1218",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "preds22 = nn_3l(torch.from_numpy(X_true_test[:, :]).double())\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(preds22.detach().numpy()[0:1000, 1], label=\"NN Predicted values\")\n",
+    "plt.plot(subgrid_tend_test[:1000, 1], label=\"True values\")\n",
+    "\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "sCwwt6bysNfl",
+    "user_expressions": []
+   },
+   "source": [
+    "3-layer ANN is (too) good!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "8Bn_9aswsS45",
+    "user_expressions": []
+   },
+   "source": [
+    "## ResNet\n",
+    "\n",
+    "ResNet is based on residual blocks, depicted below  \n",
+    "Residual blocks are based on skip connections  \n",
+    "Skip connections add output from one layer to that of a deeper layer"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "JgKr2RhCf2Eh",
+    "user_expressions": []
+   },
+   "source": [
+    "\n",
+    "```{figure} https://d1m75rqqgidzqn.cloudfront.net/wp-data/2020/10/22125000/Image-recreation-Sep-15-1-1-640x420.jpg\n",
+    ":width: 600px\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "bRAdV1k-sv00",
+    "user_expressions": []
+   },
+   "source": [
+    "ResNet is famously applied to *image recognition* using NN that are *very deep*  \n",
+    "\n",
+    "\n",
+    "> He, Kaiming, et al. “Deep Residual Learning for Image Recognition.” ArXiv.org, 2015, arxiv.org/abs/1512.03385. Accessed 5 Oct. 2021.  \n",
+    "\n",
+    "\n",
+    "Pretrained image recognition ResNet models are available with tens or >100 layers  \n",
+    "This is not our use case for Lorenz '96 \n",
+    "\n",
+    "\n",
+    "> However, adaptation for dynamic systems do exist, e.g. https://towardsdatascience.com/neural-odes-breakdown-of-another-deep-learning-breakthrough-3e78c7213795\\\\\n",
+    "\n",
+    "\n",
+    "Here, we'll look at skip connections and residual blocks, rather than ResNet   \n",
+    "First, need to 'break' Janni's model by adding too many layers"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "RHGr85pTuJT4",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "class Net_deepNN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super(Net_deepNN, self).__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs, 16 neurons for first hidden layer\n",
+    "        self.linear2 = nn.Linear(16, 16)  # 16 neurons for second hidden layer\n",
+    "        self.linear3 = nn.Linear(16, 16)\n",
+    "        self.linear4 = nn.Linear(16, 16)\n",
+    "        self.linear5 = nn.Linear(16, 16)\n",
+    "        self.linear6 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "    #         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = F.relu(self.linear1(x))\n",
+    "        x = F.relu(self.linear2(x))\n",
+    "        x = F.relu(self.linear3(x))\n",
+    "        x = F.relu(self.linear4(x))\n",
+    "        x = F.relu(self.linear5(x))\n",
+    "        x = self.linear6(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "APC8JO46ntQT",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "nn_deep = Net_deepNN().double()\n",
+    "# optimizer = optim.Adam(nn_deep.parameters(), lr=1e-7)\n",
+    "# lr_finder = LRFinder(nn_deep, optimizer, criterion)\n",
+    "# lr_finder.range_test(loader, end_lr=100, num_iter=200)\n",
+    "# lr_finder.plot() # to inspect the loss-learning rate graph\n",
+    "# lr_finder.reset() # to reset the model and optimizer to their initial state"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 1000
+    },
+    "id": "3TmNXy-fn8Qa",
+    "outputId": "41d461ba-00ef-4893-ec0f-dd3a12af87a9",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "n_epochs = 40  # Number of epocs\n",
+    "optimizer = optim.Adam(nn_deep.parameters(), lr=0.01)\n",
+    "validation_loss = list()\n",
+    "train_loss = list()\n",
+    "# time0 = time()\n",
+    "for epoch in range(1, n_epochs + 1):\n",
+    "    train_model(nn_deep, criterion, loader, optimizer)\n",
+    "    train_loss.append(test_model(nn_deep, criterion, loader, optimizer, \"train\"))\n",
+    "    validation_loss.append(test_model(nn_deep, criterion, loader_test, optimizer))\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(train_loss, \"b\", label=\"training loss\")\n",
+    "plt.plot(validation_loss, \"r\", label=\"validation loss\")\n",
+    "plt.ylabel(\"loss\")\n",
+    "plt.xlabel(\"epochs\")\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "final_losses[1] = validation_loss[-1]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "UwJ7ODPuu5ZD",
+    "user_expressions": []
+   },
+   "source": [
+    "Worse loss with twice the training epochs"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 265
+    },
+    "id": "_8ApZP9GvGu6",
+    "outputId": "a48e4b75-0eab-40df-e25a-224f97e93caf",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "preds_deep = nn_deep(torch.from_numpy(X_true_test[:, :]).double())\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(preds22.detach().numpy()[0:100, 1], label=\"3 layer Predictions\")\n",
+    "plt.plot(subgrid_tend_test[:100, 1], label=\"True values\")\n",
+    "plt.plot(preds_deep.detach().numpy()[0:100, 1], label=\"deepNN Predictions\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "Ma86AJSRnOox",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "class Net_deepNN_withSkips(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super(Net_deepNN_withSkips, self).__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs, 16 neurons for first hidden layer\n",
+    "        self.linear2 = nn.Linear(16, 16)  # 16 neurons for second hidden layer\n",
+    "        self.linear3 = nn.Linear(16, 16)\n",
+    "        self.linear4 = nn.Linear(16, 16)\n",
+    "        self.linear5 = nn.Linear(16, 16)\n",
+    "        self.linear6 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "    #         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x1 = F.relu(self.linear1(x))\n",
+    "        x2 = F.relu(self.linear2(x1))\n",
+    "        x3 = F.relu(self.linear3(x2)) + x1\n",
+    "        x4 = F.relu(self.linear4(x3))\n",
+    "        x5 = F.relu(self.linear5(x4)) + x3\n",
+    "        x6 = self.linear6(x5)\n",
+    "        return x6"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "2K62ND04zHi0",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "nn_deep_skips = Net_deepNN_withSkips().double()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 959
+    },
+    "id": "ziyE32LbzIBk",
+    "outputId": "48d05a40-f334-41a9-aaab-89541333bd8a",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "n_epochs = 20  # Number of epocs\n",
+    "optimizer = optim.Adam(nn_deep_skips.parameters(), lr=0.01)\n",
+    "validation_loss = list()\n",
+    "train_loss = list()\n",
+    "# time0 = time()\n",
+    "for epoch in range(1, n_epochs + 1):\n",
+    "    train_model(nn_deep_skips, criterion, loader, optimizer)\n",
+    "    train_loss.append(test_model(nn_deep_skips, criterion, loader, optimizer, \"train\"))\n",
+    "    validation_loss.append(test_model(nn_deep_skips, criterion, loader_test, optimizer))\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(train_loss, \"b\", label=\"training loss\")\n",
+    "plt.plot(validation_loss, \"r\", label=\"validation loss\")\n",
+    "plt.ylabel(\"loss\")\n",
+    "plt.xlabel(\"epochs\")\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "# final_losses=np.append(final_losses,validation_loss[-1])\n",
+    "final_losses[2] = validation_loss[-1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 265
+    },
+    "id": "aJKvu_mS2WBN",
+    "outputId": "447fa50c-c284-484f-cf21-6c0e03ed6e46",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "preds_deep_skips = nn_deep_skips(torch.from_numpy(X_true_test[:, :]).double())\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(preds22.detach().numpy()[0:100, 1], label=\"3 layer Predictions\")\n",
+    "plt.plot(subgrid_tend_test[:100, 1], label=\"True values\")\n",
+    "plt.plot(preds_deep.detach().numpy()[0:100, 1], label=\"deepNN Predictions\")\n",
+    "plt.plot(\n",
+    "    preds_deep_skips.detach().numpy()[0:100, 1], label=\"deepNN w/ skips Predictions\"\n",
+    ")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Adding skips led to two improvements  \n",
+    "* Improved loss, lower than that of deep network without skips and the original 3-layer NN\n",
+    "* Faster training"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "GuSgIFLHNz6l",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "class Net_3L_withSkip(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super(Net_3L_withSkip, self).__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs, 16 neurons for first hidden layer\n",
+    "        self.linear2 = nn.Linear(16, 16)  # 16 neurons for second hidden layer\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "    #         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = F.relu(self.linear1(x))\n",
+    "        x = F.relu(self.linear2(x)) + x\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "L9R5rAiVO8lo",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "nn_3l_skip = Net_3L_withSkip().double()\n",
+    "# optimizer = optim.Adam(nn_3l_res.parameters(), lr=1e-7)\n",
+    "# lr_finder = LRFinder(nn_3l_res, optimizer, criterion)\n",
+    "# lr_finder.range_test(loader, end_lr=100, num_iter=200)\n",
+    "# lr_finder.plot() # to inspect the loss-learning rate graph\n",
+    "# lr_finder.reset() # to reset the model and optimizer to their initial state"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 959
+    },
+    "id": "3_Pq6ntZPCdo",
+    "outputId": "db458eb8-8746-4466-e8b2-5bb2048d3cb1",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "n_epochs = 20  # Number of epocs\n",
+    "optimizer = optim.Adam(nn_3l_skip.parameters(), lr=0.01)\n",
+    "validation_loss = list()\n",
+    "train_loss = list()\n",
+    "# time0 = time()\n",
+    "for epoch in range(1, n_epochs + 1):\n",
+    "    train_model(nn_3l_skip, criterion, loader, optimizer)\n",
+    "    train_loss.append(test_model(nn_3l_skip, criterion, loader, optimizer, \"train\"))\n",
+    "    validation_loss.append(test_model(nn_3l_skip, criterion, loader_test, optimizer))\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(train_loss, \"b\", label=\"training loss\")\n",
+    "plt.plot(validation_loss, \"r\", label=\"validation loss\")\n",
+    "plt.ylabel(\"loss\")\n",
+    "plt.xlabel(\"epochs\")\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "final_losses[3] = validation_loss[-1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 265
+    },
+    "id": "Va1f2qqwPHg3",
+    "outputId": "699e3277-28a7-4559-a4e1-451e13d06426",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "preds_3l_skip = nn_3l_skip(torch.from_numpy(X_true_test[:, :]).double())\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(preds22.detach().numpy()[0:100, 1], label=\"3 layer Predictions\")\n",
+    "plt.plot(subgrid_tend_test[:100, 1], label=\"True values\")\n",
+    "plt.plot(preds_deep.detach().numpy()[0:100, 1], label=\"deepNN Predictions\")\n",
+    "plt.plot(\n",
+    "    preds_deep_skips.detach().numpy()[0:100, 1], label=\"deepNN w/ skips Predictions\"\n",
+    ")\n",
+    "plt.plot(preds_3l_skip.detach().numpy()[0:100, 1], label=\"3 layer w/ skip Predictions\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 175
+    },
+    "id": "FCGsrysp5_SU",
+    "outputId": "4b63680a-2fb8-416e-a4cc-1091b74c944a",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "model_names = [\"3 layer\", \"deep\", \"deep w/ skips\", \"3 layer w/ skip\"]\n",
+    "df = pd.DataFrame(final_losses, index=model_names, columns=[\"loss\"])\n",
+    "df[\"epochs\"] = [20, 40, 20, 20]\n",
+    "df"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Main points\n",
+    " - Skip connections are easy to program in pytorch  \n",
+    " - Deeper isn't always better  \n",
+    " - Residual structure can lower loss, and lead to faster training  \n",
+    " - Even shallow NN can benefit from using a residual block"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "id": "aOjZww_ObQzt",
+    "user_expressions": []
+   },
+   "source": [
+    "## Recurrent Neural Networks (RNN)\n",
+    "#What is RNN? >  Recurrent Neural Network.\n",
+    "The hidden state (hidden layer) is used in the next time step. Useful in operations which involve recurrence. \n",
+    "In the image below, $x_t$ is the vector of $X$ and $o_t$ is the vector of tendency terms. The network uses $x_t$ and hidden state $s_{t-1}$ as inputs to obtain the tendency term $o_t$. A normal fully connected ANN will only use $x_t$ to obtain $o_t$ but RNN uses $s_{t-1}$ in addition to $x_t$ to get the output, $o_t$. \n",
+    "\n",
+    "```{figure} https://upload.wikimedia.org/wikipedia/commons/0/05/RNN.png\n",
+    ":width: 600px\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "0P4byyMiEE-D",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "### Applying RNN to L96 model:\n",
+    "\n",
+    "# preparing data for training\n",
+    "\n",
+    "x1 = torch.Tensor(X_true)  # xk's w.r.t to time\n",
+    "x2 = torch.Tensor(xy_true)  # tendency terms\n",
+    "\n",
+    "\n",
+    "class L96_RNN(nn.Module):\n",
+    "    def __init__(self, in_n, hid, out_n):\n",
+    "        super(L96_RNN, self).__init__()\n",
+    "        self.h_size = hid\n",
+    "        self.linear1 = nn.Linear(\n",
+    "            in_n + hid, hid\n",
+    "        )  # note that X_k + hidden state is the input\n",
+    "        self.linear2 = nn.Linear(hid, out_n)\n",
+    "\n",
+    "    def forward(self, x, hidden):\n",
+    "        # forward code for RNN is different than ANN. Hidden state is also given as input to forward function\n",
+    "        x2 = self.linear1(torch.cat((x, hidden), 1))  # input layer to 1st layer\n",
+    "        x3 = torch.relu(\n",
+    "            x2\n",
+    "        )  # x2 is the hidden state which will be used as input in the next iteration\n",
+    "        x4 = self.linear2(x3)  # x4 is the output layer, tendency terms in L96 case.\n",
+    "\n",
+    "        return (\n",
+    "            x4,\n",
+    "            x3,\n",
+    "        )  # x4 is the output layer, and x3 is the hidden layer returned for using in the next iteration"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 612
+    },
+    "id": "LhxVdWq4EI6X",
+    "outputId": "4fc76fe7-1327-4dbc-812a-79a6f658360b",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "### deletes the model if it already exists\n",
+    "try:\n",
+    "    del model\n",
+    "except:\n",
+    "    pass\n",
+    "###\n",
+    "\n",
+    "\n",
+    "m = 1  # no. of time steps to use in RNN. untested, code might break if changed.\n",
+    "\n",
+    "in_n, hid, out_n = (\n",
+    "    8,\n",
+    "    8,\n",
+    "    8,\n",
+    ")  # X_k, k=1 to 8 and corresponding tendency terms. hid layer nodes can be chaned\n",
+    "\n",
+    "model = L96_RNN(in_n, hid, out_n)  # model initialize\n",
+    "\n",
+    "l_r = 1e-03  # learning rate\n",
+    "\n",
+    "optimizer = torch.optim.Adam(model.parameters(), lr=l_r)  # using Adam optimiser\n",
+    "loss_fn = torch.nn.MSELoss(reduction=\"mean\")  # loss function\n",
+    "\n",
+    "epochs = 20  # epochs\n",
+    "\n",
+    "loss_array = np.zeros([epochs, 2])  # to store loss for plotting purposes\n",
+    "\n",
+    "for p in range(epochs):\n",
+    "    for i in range(time_steps - 1):\n",
+    "        hidd = Variable(torch.randn((1, model.h_size)))\n",
+    "        inp = x1[i : i + m, :].reshape([1, 8])\n",
+    "        outp = x2[i : i + m, :].reshape([1, 8])\n",
+    "        pred, hidd = model.forward(inp, hidd)\n",
+    "        loss = loss_fn(pred, outp)\n",
+    "        # loss.backward()\n",
+    "        model.zero_grad()\n",
+    "        optimizer.zero_grad()\n",
+    "        loss.backward()\n",
+    "        # loss.backward(retain_graph=True)\n",
+    "        optimizer.step()\n",
+    "\n",
+    "    loss_array[p, 0] = p\n",
+    "    loss_array[p, 1] = loss.detach().numpy()\n",
+    "    # print('epoch no. >',p, 'loss >',loss.detach().numpy())\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(loss_array[:, 1])\n",
+    "plt.ylabel(\"loss\")\n",
+    "plt.xlabel(\"epochs\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "id": "iPhzyyWdEMYA",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# checking tendency term output from L96 RNN:\n",
+    "\n",
+    "hidd = Variable(\n",
+    "    torch.randn((1, model.h_size))\n",
+    ")  # hidden state needs to be initialized for the 1st iteration\n",
+    "\n",
+    "pred = torch.zeros([time_steps, 8])  # tendency prediction\n",
+    "\n",
+    "for j in range(time_steps - 1):\n",
+    "    inp = x1[j : j + m, :].reshape([1, 8])\n",
+    "    pred[j, :], hidd = model.forward(inp, hidd)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "colab": {
+     "base_uri": "https://localhost:8080/",
+     "height": 282
+    },
+    "id": "fJ3If9ZeFAr5",
+    "outputId": "46cf08c1-f80d-4647-c969-5a61385a37ae",
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "l = 1  # k value. l=k. l goes from 1 to 8 for this example\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, x2[:, l], \"-\", label=\"actual\")\n",
+    "plt.plot(t[0:-1], pred[:, l].detach().numpy(), \"-\", label=\"RNN\")\n",
+    "plt.xlim([0, 10])  # looking only till t=10\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.xlabel(\"t\")\n",
+    "plt.ylabel(\"tendency\")\n",
+    "plt.show();"
+   ]
+  }
+ ],
+ "metadata": {
+  "colab": {
+   "collapsed_sections": [],
+   "name": "ComparingSimpleANN.ipynb",
+   "provenance": []
+  },
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/L96_offline_training_NN.ipynb b/_sources/notebooks/L96_offline_training_NN.ipynb
new file mode 100644
index 00000000..759c6baf
--- /dev/null
+++ b/_sources/notebooks/L96_offline_training_NN.ipynb
@@ -0,0 +1,1120 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "c0d2675f-796a-4466-a76a-c9a354518d1b",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# Using Neural Networks for L96 Parameterization: Offline Training"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aa918476-598e-4ec0-9d9c-b2fd20d2e699",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Outline:\n",
+    "In this notebook, we'll apply the concepts learned in the [previous notebook](https://m2lines.github.io/L96_demo/notebooks/gradient_decent.html) to the parameterization task in L96 model, which was highlighted [here](https://m2lines.github.io/L96_demo/notebooks/gcm-analogue.ipynb) and [here](https://m2lines.github.io/L96_demo/notebooks/gcm-parameterization-problem.ipynb). \n",
+    "\n",
+    "As a succinct reminder, the parameterization task requires us to represent the impact of the fast scales on the slow scales as a function of the slow scales. Machine learning provides us a way to learn complex non-linear functional dependecies from a given data set. In this notebook we will show :\n",
+    "- how the data to train a machine learning model is generated. \n",
+    "- how a ML model can be trained to learn functional dependence in this data. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e2c6ba62-1b14-408c-ac05-146c96a130b3",
+   "metadata": {},
+   "source": [
+    "## Generating the data"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "129a18f4-f3d5-4503-8c0b-e81895ee2406",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "import time\n",
+    "from IPython.display import Image\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import torch\n",
+    "import torch.utils.data as Data\n",
+    "from torch import nn, optim\n",
+    "\n",
+    "from L96_model import L96, RK2, RK4, EulerFwd, L96_eq1_xdot, integrate_L96_2t"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "79044f9a-ac01-415a-a88a-0385718c8b02",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Ensuring reproducibility\n",
+    "np.random.seed(14)\n",
+    "torch.manual_seed(14);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d555922c-f58e-4343-8cf0-1ad027046c6a",
+   "metadata": {},
+   "source": [
+    "### Build the *Real World* to Generate the Ground Truth Dataset\n",
+    "\n",
+    "Remeber from the [notebook](https://m2lines.github.io/L96_demo/notebooks/gcm-analogue.ipynb) that described the parameterization problem, the impact of the fast time scales on the slow time scales is captured by the term:\n",
+    "\n",
+    "\\begin{align}\n",
+    "\\underbrace{ \n",
+    "\\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}\n",
+    "}_{=U_k}.\n",
+    "\\end{align}\n",
+    "\n",
+    "The goal of the parameterization is to express $U_k$ purely as a function of the large scale variables $P(X_k)$. In the previous [notebook](https://m2lines.github.io/L96_demo/notebooks/gcm-analogue.ipynb) this was done using simple polynomial functions with specified form, but here we will estimate this function using a more general function - neural network.  \n",
+    "\n",
+    "First we need to run a two time-scale model and generate data for $U_k$ and $X_k$.\n",
+    "We initialise the L96 two time-scale model using $K$ (set to 8) values of $X$ and $J$ (set to 32) values of $Y$ for each $X$. The model is run for 20,000 timesteps to generate the dataset for the neural network."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ae0738e8-fd6f-4f0c-befa-aa5db97f4521",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "time_steps = 20000\n",
+    "forcing, dt, T = 18, 0.01, 0.01 * time_steps\n",
+    "\n",
+    "# Create a \"real world\" with K=8 and J=32\n",
+    "W = L96(8, 32, F=forcing)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a8df4af7-7283-4a64-8a3d-cbb7dada4363",
+   "metadata": {},
+   "source": [
+    "### Run the simulation to generate data\n",
+    "\n",
+    "Using the *real world* model created above we generate the training data (input and output pairs) for the neural network by running the simulation and writing out the large scale state and subgrid tendencies."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8ae3e6b4-269f-4ac5-82e8-366eb2fb6dee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# The effect of Y on X is `xy_true`\n",
+    "X_true, _, _, xy_true = W.run(dt, T, store=True, return_coupling=True)\n",
+    "\n",
+    "# Change the data type to `float32` in order to avoid doing type conversions later on\n",
+    "X_true, xy_true = X_true.astype(np.float32), xy_true.astype(np.float32)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c4586ecb-43a0-4764-9b6f-c968be40eca9",
+   "metadata": {},
+   "source": [
+    "### Split the Data into Training and Testing Set\n",
+    "\n",
+    "We split the data into 80% for training and 20% for testing. In more conventional ML tasks one may split the data into training, validation and testing sets, but for simplicity we do not do this here."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9f99293a-df6d-4d60-b177-03fc2f8ae361",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Number of time steps for validation\n",
+    "val_size = 4000\n",
+    "\n",
+    "# Training Data\n",
+    "X_true_train = X_true[\n",
+    "    :-val_size, :\n",
+    "]  # Flatten because we first use single input as a sample\n",
+    "subgrid_tend_train = xy_true[:-val_size, :]\n",
+    "\n",
+    "# Test Data\n",
+    "X_true_test = X_true[-val_size:, :]\n",
+    "subgrid_tend_test = xy_true[-val_size:, :]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "bded5ed8-7cd8-4c50-940b-93c37d82d984",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_true_train.shape"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7ac52993-712b-4aba-8012-7ca83a4d605e",
+   "metadata": {},
+   "source": [
+    "## The Dataloaders \n",
+    "\n",
+    "In ML work flows it is common practice to create ways to access the data that allow for easily iterating over the whole data set. This is done because usually the datasets are very large, and breaking the data up into smaller chunks (batches) that can be loaded into memory (GPU or CPU) one at a time allow us to train using the whole dataset (see more details [here](https://machinelearningmastery.com/gentle-introduction-mini-batch-gradient-descent-configure-batch-size/) and [here](https://pytorch.org/tutorials/beginner/data_loading_tutorial.html)). \n",
+    "\n",
+    "`Dataset` and `Dataloader` classes in PyTorch provide a very convenient way of iterating over a dataset while training a deep learning model."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1e751231-d506-4bee-982d-c7522a4481e4",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Number of sample in each batch\n",
+    "BATCH_SIZE = 2000\n",
+    "# this batch size would result in 2 test batches and 8 training batches."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4615518a-e6d8-4a9d-ad05-b334fd7f8224",
+   "metadata": {},
+   "source": [
+    "Define the X (state), Y (subgrid tendency) pairs for the linear regression local network."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "060e3982-1791-418e-bf8d-81475da84a50",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "local_dataset = Data.TensorDataset(\n",
+    "    torch.from_numpy(np.reshape(X_true_train, -1)),\n",
+    "    torch.from_numpy(np.reshape(subgrid_tend_train, -1)),\n",
+    ")\n",
+    "\n",
+    "local_loader = Data.DataLoader(\n",
+    "    dataset=local_dataset, batch_size=BATCH_SIZE, shuffle=True\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5d9ef6f0-9441-49c5-8cea-904c992d75c1",
+   "metadata": {},
+   "source": [
+    "Define the dataloader for the test set."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3eec82d6-91e3-4851-993f-fa28836af5d5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "local_dataset_test = Data.TensorDataset(\n",
+    "    torch.from_numpy(np.reshape(X_true_test, -1)),\n",
+    "    torch.from_numpy(np.reshape(subgrid_tend_test, -1)),\n",
+    ")\n",
+    "\n",
+    "local_loader_test = Data.DataLoader(\n",
+    "    dataset=local_dataset_test, batch_size=BATCH_SIZE, shuffle=True\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "33ff24b3-f8c9-4b5e-a36f-82d6b0f177d1",
+   "metadata": {},
+   "source": [
+    "Display a batch of samples from the dataset."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8527ff9a-01c8-419a-9ca4-c28a38a54b88",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Iterating over the data to get one batch\n",
+    "data_iterator = iter(local_loader)\n",
+    "X_iter, subgrid_tend_iter = next(data_iterator)\n",
+    "\n",
+    "print(\"X (State):\\n\", X_iter)\n",
+    "print(\"\\nY (Subgrid Tendency):\\n\", subgrid_tend_iter)\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(X_iter, subgrid_tend_iter, \".\")\n",
+    "plt.xlabel(\"State - X\", fontsize=20)\n",
+    "plt.ylabel(\"Subgrid tendency - U\", fontsize=20)\n",
+    "plt.xlim([-12, 16])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "edfeefb9-cfb5-44d9-b258-a8701ce4e4c7",
+   "metadata": {},
+   "source": [
+    "Clearly there is a relationship between $U_k$ and $X_k$, we plan to learn this with the help of ML models."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5c54b68a-bae5-4bce-b4f0-9bdb7f4235bc",
+   "metadata": {},
+   "source": [
+    "## The *local* ML models\n",
+    "\n",
+    "Here we show how a few different machine learning models can be setup. These models will predict the $U_k$ at a specific $k$ using the $X_k$ as the same $k$, and so may be considered local. Later we will consider non-local models, which predict $U_k$ at specific $k$ using $X_k$ at multiple locations (values of $k$). \n",
+    "\n",
+    "We will consider a linear regression network and a non-linear fully connected neural network (FCNN) here. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ed3e53bd-fb12-4aad-bcff-be495935ef80",
+   "metadata": {},
+   "source": [
+    "### Linear regression network"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "eed1ac73-a213-4d2c-81af-d74b093d2ac2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class LinearRegression(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(1, 1)  # A single input and a single output\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        # This method is automatically executed when\n",
+    "        # we call a object of this class\n",
+    "        x = self.linear1(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3c9a3d2f-7f95-4906-a3b0-dd2ecf81822d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "linear_network = LinearRegression()\n",
+    "linear_network"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1f5abff9-0d16-4bcb-a929-f39c010a4b92",
+   "metadata": {},
+   "source": [
+    "### Fully connected neural network (FCNN)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "10044fa0-de73-4240-be0f-55bb30af2513",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class FCNN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(1, 16)  # 8 inputs\n",
+    "        self.linear2 = nn.Linear(16, 16)\n",
+    "        self.linear3 = nn.Linear(16, 1)  # 8 outputs\n",
+    "\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = self.relu(self.linear1(x))\n",
+    "        x = self.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e4e55f54-c315-4aba-b00f-85e00812f75d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fcnn_network = FCNN()\n",
+    "fcnn_network"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1287a0d4-f7e0-4a4f-bbee-3666ff2c1adc",
+   "metadata": {},
+   "source": [
+    "We do not expect the linear regression to do well in this problem, as the dependence seenin the figure is clearly more complex than a simple line. We include it here as a baseline. \n",
+    "\n",
+    "For the FCNN the non-linearity is introduced by adding the activation function, which is chosen to be ReLU in this case. The number of layers and width of layers are hyperparameters, which the practitioner needs to set based on some trial and error (or systematic hyperparameter sweeps). The goal is to make the network as complex as needed but no more."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b3b259d3-3478-4a9c-b23c-8e9a8042404a",
+   "metadata": {},
+   "source": [
+    "```{admonition} Need for Activation Functions\n",
+    "\n",
+    "**If layers of a neural network contain only fully-connected layers (matrix multiplications), everything would be linear.**\n",
+    "\n",
+    "For example, if we have an input $x$ along with 2 layers of weight matrices $A$ and $B$ then the neural network would compute the output as $A(Bx)$, which is linear (in $x$). Thus, in order to introduce some non-linearity we use activation functions.\n",
+    "\n",
+    "Now the same neural network as above with an activation function $\\phi$ would compute the output as $A(\\phi(Bx))$.\n",
+    "```\n",
+    "\n",
+    "`````{admonition} ReLU Activation Function\n",
+    "\n",
+    "```{figure} figs/relu_activation_function.png\n",
+    ":name: relu-activation\n",
+    "```\n",
+    "\n",
+    "The `ReLU` activation function is just a $max(0,X)$ function (The image is referenced from [this blog](https://analyticsindiamag.com/most-common-activation-functions-in-neural-networks-and-rationale-behind-it/)). Even a function as simple as this enables a typical NN to be a nonlinear function of the inputs!\n",
+    "`````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "419f6fd5-0055-4583-8677-ae31ac9953e9",
+   "metadata": {},
+   "source": [
+    "**Test forward function**  \n",
+    "Below we test to see if the models can return an output given an input. Since at this point the models have not been trained (parameters have not been optimized) the resulting output is just a random number. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2b236b7f-dd79-4f88-8e8a-e60bfbc2a045",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "net_input = torch.randn(1, 1)\n",
+    "out_linear = linear_network(net_input)\n",
+    "out_fcnn = fcnn_network(net_input)\n",
+    "print(\n",
+    "    f\"The output of the random input from the linear network is: {out_linear.item():.4f}\"\n",
+    ")\n",
+    "print(f\"The output of the random input from the fcnn is: {out_fcnn.item():.4f}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6471c571-5ee4-48ef-bfce-e2349bcdfe5b",
+   "metadata": {},
+   "source": [
+    "## The Loss Function\n",
+    "\n",
+    "Apart from the architecutre of the model, the other major component of any machine learning task is the loss function. This is basically some quantitative way to test the skill of the model - whether the model does a good job or not at estimating the output. During the optimization of the weights, the weights are changed such that the model skill is improved.\n",
+    "\n",
+    "For our task, we choose the *Mean Squared Error* metric as our loss function. However, other loss functions may be used or need to be adapted to the problem. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7c42de43-27de-4896-91ee-caf695f212df",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# MSE loss function\n",
+    "loss_fn = torch.nn.MSELoss()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d33c7361-f717-4a74-b296-075f398d7072",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Load the input and output pair from the data loader\n",
+    "X_tmp = next(iter(local_loader))\n",
+    "\n",
+    "# Predict the output\n",
+    "y_tmp_linear = linear_network(torch.unsqueeze(X_tmp[0], 1))\n",
+    "y_tmp_fcnn = fcnn_network(torch.unsqueeze(X_tmp[0], 1))\n",
+    "\n",
+    "# Calculate the MSE loss\n",
+    "loss_linear = loss_fn(y_tmp_linear, torch.unsqueeze(X_tmp[1], 1))\n",
+    "print(f\"Linear network MSE Loss: {loss_linear.item():.4f}\")\n",
+    "\n",
+    "loss_fcnn = loss_fn(y_tmp_fcnn, torch.unsqueeze(X_tmp[1], 1))\n",
+    "print(f\"FCNN MSE Loss: {loss_fcnn.item():.4f}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1eb9cd59-ef25-48e8-9522-355fdda85d07",
+   "metadata": {},
+   "source": [
+    "Since at this point neither of the networks have been trained, the resultant loss is just arbitrary. If one model is doing better than the other, it is just a result of random chance. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "98716e66-ed26-48f4-8509-9f25e0ddeebf",
+   "metadata": {},
+   "source": [
+    "## The Optimizer\n",
+    "\n",
+    "Now in order to make the network learn (optimize the parameters), we need an algorithm that will update its weights depending on the loss function. This is achieved by using an optimizer. The implementation of many optimizers can be found in PyTorch. The choice of which optimizer we choose might be very important as it will determine how fast the network will be able to learn.\n",
+    "\n",
+    "In the example below, we show one of the popular optimizers `Adam`. The main hyperparameter in this optimizer is the **learning rate**. \n",
+    "\n",
+    "\n",
+    "Another popular optimizer that may be useful in some situations is Stochastic Gradient Descent or `SGD`. \n",
+    "The SGD optimizer consists of two major hyperparameters called the **learning rate** and **momentum**. The **learning rate** determines the magnitude with which the weights of the network update thus making it crucial to choose the correct learning rate ($LR$) otherwise the network will either fail to train, or take much longer to converge. **Momentum** determines how much of the loss function gradients are determined based on the past iteration vs the latest iteration. \n",
+    "To read about momentum, check out this [blog post](https://towardsdatascience.com/stochastic-gradient-descent-with-momentum-a84097641a5d).\n",
+    "\n",
+    "The primary task of the optimizer is to update the parameters in little steps such that the loss function is reduced. In equations, the  effective value of the gradient $V$ (direction in which loss function reduces) at step $t$ in SGD with momentum ($\\beta$) is determined by\n",
+    "\n",
+    "\\begin{equation}\n",
+    "V_t = \\beta V_{t-1} + (1-\\beta) \\nabla_w L(W,X,y)\n",
+    "\\end{equation}\n",
+    "\n",
+    "and the updates to the parameters will be\n",
+    "\n",
+    "\\begin{equation}\n",
+    "w^{new} = w^{old} - LR * V_t\n",
+    "\\end{equation}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4b28b5f2-11b1-4c30-b385-471d7a6c9e79",
+   "metadata": {},
+   "source": [
+    "See in the code sample below how taking a step with the optimizer changes the model parameters. In what comes next we will take these steps to reach a parameter state such that the loss function is as small as possible. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d46984b4-14c6-43ff-b143-c936282b6a24",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "learning_rate = 0.003\n",
+    "momentum = 0.1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7ed28fd9-441e-4e01-9e3c-7571ff7a311b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# switch the commenting out below to try a different optimizer.\n",
+    "# optimizer_linear = optim.SGD(linear_network.parameters(), lr=learning_rate, momentum=momentum)\n",
+    "optimizer_linear = optim.Adam(linear_network.parameters(), lr=learning_rate)\n",
+    "print(\"Before backward pass: \\n\", list(linear_network.parameters())[0].data.numpy())\n",
+    "\n",
+    "loss_linear.backward(retain_graph=True)\n",
+    "optimizer_linear.step()\n",
+    "\n",
+    "print(\"\\nAfter backward pass: \\n\", list(linear_network.parameters())[0].data.numpy())"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "72cdeebe-87c1-49fc-9cba-eacb264bcdf6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# optimizer_fcnn = optim.SGD(fcnn_network.parameters(), lr=learning_rate, momentum=momentum)\n",
+    "optimizer_fcnn = optim.Adam(fcnn_network.parameters(), lr=learning_rate)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3f28f566-bb81-4d2d-9acc-b24ae80712c8",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "## Combining it all Together: Training the network"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fa66f8a8-8c11-4897-b71e-ac06d03271ac",
+   "metadata": {},
+   "source": [
+    "### Define the Training and Test Functions\n",
+    "\n",
+    "The `train_model` function update the network weight such that the loss is reduced. A single call to this function iterates over all the batches in the dataset, and this single iteration is called an epoch. To do its job this function needs access to the model that needs to be trained, the criterion or the loss function that determines the skill of the model, the data loader that allows the model to access the data, and the optimizer that appropriately adjusts the weights to reduce the loss. Also notice that the model weights are adjusted multiple times in a single epoch - everytime a batch is processed. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a800f28f-69c5-40eb-9b0c-cb27059f2f32",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def train_model(network, criterion, loader, optimizer):\n",
+    "    \"\"\"Train the network for one epoch\"\"\"\n",
+    "    network.train()\n",
+    "\n",
+    "    train_loss = 0\n",
+    "    for batch_x, batch_y in loader:\n",
+    "        # Get predictions\n",
+    "        if len(batch_x.shape) == 1:\n",
+    "            # This if block is needed to add a dummy dimension if our inputs are 1D\n",
+    "            # (where each number is a different sample)\n",
+    "            prediction = torch.squeeze(network(torch.unsqueeze(batch_x, 1)))\n",
+    "        else:\n",
+    "            prediction = network(batch_x)\n",
+    "\n",
+    "        # Compute the loss\n",
+    "        loss = criterion(prediction, batch_y)\n",
+    "        train_loss += loss.item()\n",
+    "\n",
+    "        # Clear the gradients\n",
+    "        optimizer.zero_grad()\n",
+    "\n",
+    "        # Backpropagation to compute the gradients and update the weights\n",
+    "        loss.backward()\n",
+    "        optimizer.step()\n",
+    "\n",
+    "    return train_loss / len(loader)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "08dd95be-0a9e-4214-aced-ca19460217c0",
+   "metadata": {},
+   "source": [
+    "The `test_model` function is meant to test the skill of the model, but does not update the weights to further improve it. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f963a83c-c7ba-42a6-9642-002e5d254c4b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def test_model(network, criterion, loader):\n",
+    "    \"\"\"Test the network\"\"\"\n",
+    "    network.eval()  # Evaluation mode (important when having dropout layers)\n",
+    "\n",
+    "    test_loss = 0\n",
+    "    with torch.no_grad():\n",
+    "        for batch_x, batch_y in loader:\n",
+    "            # Get predictions\n",
+    "            if len(batch_x.shape) == 1:\n",
+    "                # This if block is needed to add a dummy dimension if our inputs are 1D\n",
+    "                # (where each number is a different sample)\n",
+    "                prediction = torch.squeeze(network(torch.unsqueeze(batch_x, 1)))\n",
+    "            else:\n",
+    "                prediction = network(batch_x)\n",
+    "\n",
+    "            # Compute the loss\n",
+    "            loss = criterion(prediction, batch_y)\n",
+    "            test_loss += loss.item()\n",
+    "\n",
+    "        # Get an average loss for the entire dataset\n",
+    "        test_loss /= len(loader)\n",
+    "\n",
+    "    return test_loss"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3ea21cd1-d42f-44b0-a7b4-4ca1763abeba",
+   "metadata": {},
+   "source": [
+    "The `fit_model` iterates the `train_model` function over multiple epochs, such that the loss is reduced. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "74e8b24a-3107-48b4-8b52-d26776fa95f8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def fit_model(network, criterion, optimizer, train_loader, test_loader, n_epochs):\n",
+    "    \"\"\"Train and validate the network\"\"\"\n",
+    "    train_losses, test_losses = [], []\n",
+    "    start_time = time.time()\n",
+    "    for epoch in range(1, n_epochs + 1):\n",
+    "        train_loss = train_model(network, criterion, train_loader, optimizer)\n",
+    "        test_loss = test_model(network, criterion, test_loader)\n",
+    "        train_losses.append(train_loss)\n",
+    "        test_losses.append(test_loss)\n",
+    "    end_time = time.time()\n",
+    "    print(f\"Training completed in {int(end_time - start_time)} seconds.\")\n",
+    "\n",
+    "    return train_losses, test_losses"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7a3c15e2-c4ea-4de7-9fe6-2f5cbc656f03",
+   "metadata": {},
+   "source": [
+    "### Train the networks\n",
+    "\n",
+    "Finally we have all the components ready and we can train our models. \n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "447887a6-4c7e-4146-b7c9-0962594ad997",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Epochs refer to the number of times we iterate over the entire training data during training.\n",
+    "n_epochs = 15"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7e39cc83-2cb4-4e54-95cf-e11b6240dcfb",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "train_loss_linear, test_loss_linear = fit_model(\n",
+    "    linear_network, loss_fn, optimizer_linear, local_loader, local_loader_test, n_epochs\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e455c22a-5cf7-4d16-b060-49a747612b60",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "train_loss_fcnn, test_loss_fcnn = fit_model(\n",
+    "    fcnn_network, loss_fn, optimizer_fcnn, local_loader, local_loader_test, n_epochs\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "46e10b48-3253-4ef8-8223-945f15726d24",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "\n",
+    "plt.plot(train_loss_linear, label=\"Linear train loss\")\n",
+    "plt.plot(test_loss_linear, linestyle=\"--\", label=\"Linear test loss\")\n",
+    "\n",
+    "plt.plot(train_loss_fcnn, label=\"FCNN train loss\")\n",
+    "plt.plot(test_loss_fcnn, linestyle=\"--\", label=\"FCNN test loss\")\n",
+    "\n",
+    "plt.legend()\n",
+    "plt.xlabel(\"Iteration\")\n",
+    "plt.ylabel(\"Loss\")\n",
+    "plt.yscale(\"log\")\n",
+    "plt.title(\"Loss vs Iteration\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9615e08d-6399-49e4-9c7f-0cbd171d084f",
+   "metadata": {},
+   "source": [
+    "### Compare Predictions with Ground Truth"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "84aab879-0337-4b5e-8dc9-cdb9497c9103",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "predictions_linear = linear_network(\n",
+    "    torch.unsqueeze(torch.from_numpy(np.reshape(X_true_test[:, 1], -1)), 1)\n",
+    ")\n",
+    "predictions_fcnn = fcnn_network(\n",
+    "    torch.unsqueeze(torch.from_numpy(np.reshape(X_true_test[:, 1], -1)), 1)\n",
+    ")\n",
+    "\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(\n",
+    "    predictions_linear.detach().numpy()[0:1000], label=\"Predicted from linear model\"\n",
+    ")\n",
+    "plt.plot(predictions_fcnn.detach().numpy()[0:1000], label=\"Predicted from FCNN model\")\n",
+    "plt.plot(subgrid_tend_test[:1000, 1], label=\"True Values\", color=\"k\", linestyle=\"--\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "20ebb0e1-4e83-4c11-9d90-f3c828c52802",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_points = torch.from_numpy(np.linspace(-10, 15, 31).astype(np.float32))\n",
+    "\n",
+    "linear_pred = linear_network(torch.unsqueeze(X_points, 1)).detach().numpy()\n",
+    "\n",
+    "fcnn_pred = fcnn_network(torch.unsqueeze(X_points, 1)).detach().numpy()\n",
+    "\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.hist2d(np.reshape(X_true, -1), np.reshape(xy_true, -1), bins=91, cmap=\"Reds\")\n",
+    "\n",
+    "plt.plot(X_points, linear_pred, \"-\", label=\"Linear predictions\")\n",
+    "plt.plot(X_points, fcnn_pred, \"-\", label=\"FCNN predictions\", color=\"g\")\n",
+    "\n",
+    "plt.legend()\n",
+    "plt.xlim([-12, 16])\n",
+    "plt.xlabel(\"State - X\", fontsize=20)\n",
+    "plt.ylabel(\"Subgrid tendency - U\", fontsize=20);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7aae0beb-27e2-4db2-95be-875f01cabfb4",
+   "metadata": {},
+   "source": [
+    "As seen above, the FCNN does a better job at learning the relationship between the $X_k$ and $U_k$, when compared to the linear model.\n",
+    "Can we do even better? "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e8732e89-6841-460f-98f4-84b2e7f3d73a",
+   "metadata": {},
+   "source": [
+    "## The *non-local* ML models"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "286cb239-4868-47a1-84e7-35618970d09a",
+   "metadata": {},
+   "source": [
+    "It is possible that sub-grid tendency at $k$ depends on the neighbouring points (as shown in figure below), or points farther away. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f66b3446-9a0f-4ecc-8038-937266a30fda",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(12, 4), dpi=150)\n",
+    "\n",
+    "plt.subplot(131)\n",
+    "plt.hist2d(\n",
+    "    np.reshape(np.roll(X_true, -1, axis=1), -1),\n",
+    "    np.reshape(xy_true, -1),\n",
+    "    bins=91,\n",
+    "    cmap=\"Reds\",\n",
+    ")\n",
+    "plt.xlim([-12, 16])\n",
+    "plt.xlabel(\"State - $X_{k-1}$\", fontsize=20)\n",
+    "plt.ylabel(\"Subgrid tendency - $U_{k}$\", fontsize=20)\n",
+    "\n",
+    "plt.subplot(132)\n",
+    "plt.hist2d(np.reshape(X_true, -1), np.reshape(xy_true, -1), bins=91, cmap=\"Reds\")\n",
+    "plt.xlim([-12, 16])\n",
+    "plt.xlabel(\"State - $X_{k}$\", fontsize=20)\n",
+    "plt.ylabel(\"Subgrid tendency - $U_{k}$\", fontsize=20)\n",
+    "\n",
+    "plt.subplot(133)\n",
+    "plt.hist2d(\n",
+    "    np.reshape(np.roll(X_true, 1, axis=1), -1),\n",
+    "    np.reshape(xy_true, -1),\n",
+    "    bins=91,\n",
+    "    cmap=\"Reds\",\n",
+    ")\n",
+    "plt.xlim([-12, 16])\n",
+    "plt.xlabel(\"State - $X_{k+1}$\", fontsize=20)\n",
+    "plt.ylabel(\"Subgrid tendency - $U_{k}$\", fontsize=20)\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4836e71a-5afe-4d51-bc58-ff6d68c74e7a",
+   "metadata": {},
+   "source": [
+    "So to make more accurate model, one may want to build a model that learns relationship between not only the specific k point but also other k points. \n",
+    "\n",
+    "We will do this with the help of a non-local model, which uses all $X_k$s to predict the $U_k$ at all 8 $k$s.\n",
+    "\n",
+    "To build this model, we need to change two things: \n",
+    "- The data sets need to be slightly altered, such that each sample for the ML model will be composed of all eight k points. \n",
+    "- The architecture of the neural network will have to be altered to take 8 inputs and produce 8 outputs.  \n",
+    "\n",
+    "but apart from these changes, the rest of the training process is the same. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0681dc84-64ca-4c81-9dc5-eae2293cc1d9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Training Dataset\n",
+    "# ----------------\n",
+    "nlocal_data_train = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_train),\n",
+    "    torch.from_numpy(subgrid_tend_train),\n",
+    ")\n",
+    "nlocal_loader_train = Data.DataLoader(\n",
+    "    dataset=nlocal_data_train, batch_size=BATCH_SIZE, shuffle=True\n",
+    ")\n",
+    "\n",
+    "\n",
+    "# Test Dataset\n",
+    "# ------------\n",
+    "nlocal_data_test = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_test), torch.from_numpy(subgrid_tend_test)\n",
+    ")\n",
+    "nlocal_loader_test = Data.DataLoader(\n",
+    "    dataset=nlocal_data_test, batch_size=BATCH_SIZE, shuffle=True\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "89867c3f-914a-4504-bf16-c03f9c26874a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class NonLocal_FCNN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs\n",
+    "        self.linear2 = nn.Linear(16, 16)\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = self.relu(self.linear1(x))\n",
+    "        x = self.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c79eeaa0-2682-4b0a-84bb-89b0b2145cb4",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "nonlocal_fcnn_network = NonLocal_FCNN()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "df7bb279-7eff-4760-b0ba-0e2bad2bc676",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# optimizer_nonlocal_fcnn = optim.SGD(nonlocal_fcnn_network.parameters(),\n",
+    "#                                    lr=learning_rate, momentum=momentum)\n",
+    "optimizer_nonlocal_fcnn = optim.Adam(\n",
+    "    nonlocal_fcnn_network.parameters(), lr=learning_rate\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6222280d-2ecb-4d00-97c6-969fad2360d5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n_epochs = 120\n",
+    "train_loss_nonlocal, test_loss_nonlocal = fit_model(\n",
+    "    nonlocal_fcnn_network,\n",
+    "    loss_fn,\n",
+    "    optimizer_nonlocal_fcnn,\n",
+    "    nlocal_loader_train,\n",
+    "    nlocal_loader_test,\n",
+    "    n_epochs,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "93184419-875c-4b07-b5bd-38f528bea6dd",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "\n",
+    "plt.plot(train_loss_nonlocal, label=\"Non-local model train loss\")\n",
+    "plt.plot(test_loss_nonlocal, linestyle=\"--\", label=\"Non-local model test loss\")\n",
+    "\n",
+    "plt.plot(train_loss_fcnn, label=\"local FCNN train loss\")\n",
+    "plt.plot(test_loss_fcnn, linestyle=\"--\", label=\"local FCNN test loss\")\n",
+    "\n",
+    "plt.legend()\n",
+    "plt.xlabel(\"Epochs\")\n",
+    "plt.ylabel(\"Loss\")\n",
+    "plt.yscale(\"log\")\n",
+    "plt.title(\"Loss vs Epochs\")\n",
+    "plt.grid()\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b4541ce4-fb05-4153-9413-9da1618ca1c9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "k_loc = 1\n",
+    "predictions_linear = linear_network(\n",
+    "    torch.unsqueeze(torch.from_numpy(np.reshape(X_true_test[:, k_loc], -1)), 1)\n",
+    ")\n",
+    "predictions_fcnn = fcnn_network(\n",
+    "    torch.unsqueeze(torch.from_numpy(np.reshape(X_true_test[:, k_loc], -1)), 1)\n",
+    ")\n",
+    "\n",
+    "predictions_nonlocal_fcnn = nonlocal_fcnn_network(torch.from_numpy(X_true_test))\n",
+    "\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(\n",
+    "    predictions_linear.detach().numpy()[0:1000], label=\"Predicted from linear model\"\n",
+    ")\n",
+    "plt.plot(\n",
+    "    predictions_fcnn.detach().numpy()[0:1000], label=\"Predicted from local FCNN model\"\n",
+    ")\n",
+    "plt.plot(\n",
+    "    predictions_nonlocal_fcnn.detach().numpy()[0:1000, k_loc],\n",
+    "    label=\"Predicted from non-local FCNN model\",\n",
+    ")\n",
+    "plt.plot(subgrid_tend_test[:1000, 1], label=\"True Values\", color=\"k\", linestyle=\"--\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dad69d69-960f-4d24-8ba1-676f619da1cf",
+   "metadata": {},
+   "source": [
+    "As seen above, the non-local model does about as good as the local model. It may be possible to improve the skill of the model by adjusting the hyperparameters associated with the training or model architecture, but we did not pursue this much more in this notebook. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d208ea6f-a2ff-49e1-9369-a26a7095483c",
+   "metadata": {},
+   "source": [
+    "## Save the networks\n",
+    "\n",
+    "Now we save the weights that were learnt here, so we can use them directly later without having to retrain the neural network. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "85b2a542-4012-4be3-a897-7219c0ce71ae",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Save network\n",
+    "save_path = \"./networks/non_local_FCNN.pth\"\n",
+    "torch.save(nonlocal_fcnn_network.state_dict(), save_path)\n",
+    "\n",
+    "save_path = \"./networks/local_FCNN.pth\"\n",
+    "torch.save(fcnn_network.state_dict(), save_path)\n",
+    "\n",
+    "save_path = \"./networks/linear.pth\"\n",
+    "torch.save(linear_network.state_dict(), save_path)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4ab018fe-819f-4dd2-a208-161ea0368d82",
+   "metadata": {},
+   "source": [
+    "## Summary\n",
+    "\n",
+    "In this notebook we showed how machine learning based models, particularly fully connected neural networks, can be trained to learn the relationship between the sub-grid fluxes and the slow variables. This pattern learning exercise is sometimes referred to as offline training, as the skill of the ML model in an actual simulation of the slow variables (gcm) has not been evaluated. In the next notebook we show how these models trained offline can be incorporated into the gcm version of the L96 model, and how its skill can be evaluated."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "839b02a1-c8f5-4dbd-91b8-cbe2f8d138aa",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/L96_online_implement_NN.ipynb b/_sources/notebooks/L96_online_implement_NN.ipynb
new file mode 100644
index 00000000..d4caeab8
--- /dev/null
+++ b/_sources/notebooks/L96_online_implement_NN.ipynb
@@ -0,0 +1,526 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# Using Neural Networks for L96 Parameterization: Online Testing"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Outline:\n",
+    "In this notebook, we'll extend upon the concepts learned in the [previous](https://m2lines.github.io/L96_demo/notebooks/L96_offline_nn.html) notebook by implementing the neural networks as a parameterization in the single time scale Lorenz 96, and testing its impact in a simulation. Testing the impacts of parameterizations in a simulation is sometimes also referred to as online testing."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "import time\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import torch\n",
+    "from torch import nn, optim\n",
+    "from torch.utils.data import TensorDataset, DataLoader\n",
+    "\n",
+    "from L96_model import L96, RK4, L96_eq1_xdot"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Ensuring reproducibility\n",
+    "np.random.seed(14)\n",
+    "torch.manual_seed(14);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Load in the pre-trained parameterization networks\n",
+    "\n",
+    "We first load in the networks that were trained in the previous [notebook](https://m2lines.github.io/L96_demo/notebooks/L96_offline_NN.html). "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "linear_weights = torch.load(\"./networks/linear.pth\")\n",
+    "local_FCNN_weights = torch.load(\"./networks/local_FCNN.pth\")\n",
+    "nonlocal_FCNN_weights = torch.load(\"./networks/non_local_FCNN.pth\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We need to define the architecture that each model has before the weights can be assigned. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": [
+     "hide-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "# The model architectures\n",
+    "# ---------------------------\n",
+    "\n",
+    "\n",
+    "class LinearRegression(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(1, 1)  # A single input and a single output\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        # This method is automatically executed when\n",
+    "        # we call a object of this class\n",
+    "        x = self.linear1(x)\n",
+    "        return x\n",
+    "\n",
+    "\n",
+    "class FCNN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(1, 16)  # 8 inputs\n",
+    "        self.linear2 = nn.Linear(16, 16)\n",
+    "        self.linear3 = nn.Linear(16, 1)  # 8 outputs\n",
+    "\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = self.relu(self.linear1(x))\n",
+    "        x = self.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x\n",
+    "\n",
+    "\n",
+    "class NonLocal_FCNN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs\n",
+    "        self.linear2 = nn.Linear(16, 16)\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = self.relu(self.linear1(x))\n",
+    "        x = self.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Initialize network instances and assign pre-trained weights.\n",
+    "linear_network = LinearRegression()\n",
+    "linear_network.load_state_dict(linear_weights)\n",
+    "\n",
+    "local_fcnn_network = FCNN()\n",
+    "local_fcnn_network.load_state_dict(local_FCNN_weights)\n",
+    "\n",
+    "nonlocal_fcnn_network = NonLocal_FCNN()\n",
+    "nonlocal_fcnn_network.load_state_dict(nonlocal_FCNN_weights)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Adding Parameterizations to GCM"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "To recap, {cite}`Lorenz1995` describes a two-time scale dynamical system using two equations which are:\n",
+    "\n",
+    "\\begin{gather*}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}\n",
+    "\\end{gather*}\n",
+    "\n",
+    "\\begin{gather*}\n",
+    "\\frac{d}{dt} Y_{j,k}\n",
+    "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - Y_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
+    "\\end{gather*}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "```{note}\n",
+    "All the GCM networks used in this notebook have been introduced earlier in notebooks [Key aspects of GCMs parameterizations](https://m2lines.github.io/L96_demo/notebooks/gcm-parameterization-problem.html) and [Tuning GCM Parameterizations](https://m2lines.github.io/L96_demo/notebooks/estimating-gcm-parameters.html). To see the definition of those networks, expand the cells in the respective GCM sections below.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "T_test = 10\n",
+    "forcing = 18\n",
+    "dt = 0.01\n",
+    "\n",
+    "k = 8\n",
+    "j = 32\n",
+    "\n",
+    "W = L96(k, j, F=forcing)\n",
+    "\n",
+    "# Full L96 model (two time scale model)\n",
+    "X_full, _, _ = W.run(dt, T_test)\n",
+    "X_full = X_full.astype(np.float32)\n",
+    "\n",
+    "init_conditions = X_full[0, :]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### GCM *Without* Neural Network Parameterization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": [
+     "hide-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "class GCM_without_parameterization:\n",
+    "    \"\"\"GCM without parameterization\n",
+    "\n",
+    "    Args:\n",
+    "        F: Forcing term\n",
+    "        time_stepping: Time stepping method\n",
+    "    \"\"\"\n",
+    "\n",
+    "    def __init__(self, F, time_stepping=RK4):\n",
+    "        self.F = F\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, _):\n",
+    "        \"\"\"Compute right hand side of the the GCM equations\"\"\"\n",
+    "        return L96_eq1_xdot(X, self.F)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        \"\"\"Run GCM\n",
+    "\n",
+    "        Args:\n",
+    "            X0: Initial conditions of X\n",
+    "            dt: Time increment\n",
+    "            nt: Number of forward steps to take\n",
+    "            param: Parameters of closure\n",
+    "\n",
+    "        Returns:\n",
+    "            Model output for all variables of X at each timestep\n",
+    "            along with the corresponding time units\n",
+    "        \"\"\"\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gcm_no_param = GCM_without_parameterization(forcing)\n",
+    "X_no_param, t = gcm_no_param(init_conditions, dt, int(T_test / dt))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### GCM *With* Neural Network Parameterization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": [
+     "hide-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "class GCM_network:\n",
+    "    \"\"\"GCM with neural network parameterization\n",
+    "\n",
+    "    Args:\n",
+    "        F: Forcing term\n",
+    "        network: Neural network\n",
+    "        time_stepping: Time stepping method\n",
+    "    \"\"\"\n",
+    "\n",
+    "    def __init__(self, F, network, time_stepping=RK4):\n",
+    "        self.F = F\n",
+    "        self.network = network\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, _):\n",
+    "        \"\"\"Compute right hand side of the the GCM equations\"\"\"\n",
+    "        if self.network.linear1.in_features == 1:\n",
+    "            X_torch = torch.from_numpy(X)\n",
+    "            X_torch = torch.unsqueeze(X_torch, 1)\n",
+    "        else:\n",
+    "            X_torch = torch.from_numpy(np.expand_dims(X, 0))\n",
+    "\n",
+    "        # Adding NN parameterization\n",
+    "        return L96_eq1_xdot(X, self.F) + np.squeeze(self.network(X_torch).data.numpy())\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        \"\"\"Run GCM\n",
+    "\n",
+    "        Args:\n",
+    "            X0: Initial conditions of X\n",
+    "            dt: Time increment\n",
+    "            nt: Number of forward steps to take\n",
+    "            param: Parameters of closure\n",
+    "\n",
+    "        Returns:\n",
+    "            Model output for all variables of X at each timestep\n",
+    "            along with the corresponding time units\n",
+    "        \"\"\"\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Evaluate with linear network\n",
+    "gcm_linear_net = GCM_network(forcing, linear_network)\n",
+    "Xnn_linear, t = gcm_linear_net(init_conditions, dt, int(T_test / dt), linear_network)\n",
+    "\n",
+    "# Evaluate with local FCNN\n",
+    "gcm_local_net = GCM_network(forcing, local_fcnn_network)\n",
+    "Xnn_local, t = gcm_local_net(init_conditions, dt, int(T_test / dt), local_fcnn_network)\n",
+    "\n",
+    "# Evaluate with nonlocal FCNN\n",
+    "gcm_nonlocal_net = GCM_network(forcing, nonlocal_fcnn_network)\n",
+    "Xnn_nonlocal, t = gcm_nonlocal_net(\n",
+    "    init_conditions, dt, int(T_test / dt), nonlocal_fcnn_network\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Comparing Results\n",
+    "\n",
+    "Comparing the predictions of GCM with different parameterizations."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "time_i = 200\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t[:time_i], X_full[:time_i, 4], label=\"Full L96\")\n",
+    "plt.plot(t[:time_i], X_no_param[:time_i, 4], \"--\", label=\"No parameterization\")\n",
+    "\n",
+    "plt.plot(t[:time_i], Xnn_linear[:time_i, 4], label=\"linear parameterization\")\n",
+    "\n",
+    "plt.plot(t[:time_i], Xnn_local[:time_i, 4], label=\"local NN\")\n",
+    "plt.plot(t[:time_i], Xnn_nonlocal[:time_i, 4], label=\"nonlocal NN\")\n",
+    "plt.legend(loc=\"upper left\", fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "### Checking over many different initial conditions"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "err_linear, err_local, err_nonlocal = [], [], []\n",
+    "T_test = 1\n",
+    "\n",
+    "for i in range(90):\n",
+    "    init_conditions_i = X_full[i * 10, :]\n",
+    "\n",
+    "    # Evaluate with linear network\n",
+    "    gcm_linear_net = GCM_network(forcing, linear_network)\n",
+    "    Xnn_linear, t = gcm_linear_net(\n",
+    "        init_conditions_i, dt, int(T_test / dt), linear_network\n",
+    "    )\n",
+    "\n",
+    "    # Evaluate with local FCNN\n",
+    "    gcm_local_net = GCM_network(forcing, local_fcnn_network)\n",
+    "    Xnn_local, t = gcm_local_net(\n",
+    "        init_conditions_i, dt, int(T_test / dt), local_fcnn_network\n",
+    "    )\n",
+    "\n",
+    "    # Evaluate with nonlocal FCNN\n",
+    "    gcm_nonlocal_net = GCM_network(forcing, nonlocal_fcnn_network)\n",
+    "    Xnn_nonlocal, t = gcm_nonlocal_net(\n",
+    "        init_conditions_i, dt, int(T_test / dt), nonlocal_fcnn_network\n",
+    "    )\n",
+    "\n",
+    "    # GCM parameterized by the global 3-layer network\n",
+    "    # gcm_net_3layers = GCM_network(forcing, nn_3l)\n",
+    "    # Xnn_3layer_i, t = gcm_net_3layers(init_conditions_i, dt, int(T_test / dt), nn_3l)\n",
+    "\n",
+    "    # GCM parameterized by the linear network\n",
+    "    # gcm_net_1layers = GCM_network(forcing, linear_network)\n",
+    "    # Xnn_1layer_i, t = gcm_net_1layers(init_conditions_i, dt, int(T_test / dt), linear_network)\n",
+    "\n",
+    "    err_linear.append(\n",
+    "        np.sum(np.abs(X_full[i * 10 : i * 10 + T_test * 100 + 1] - Xnn_linear))\n",
+    "    )\n",
+    "\n",
+    "    err_local.append(\n",
+    "        np.sum(np.abs(X_full[i * 10 : i * 10 + T_test * 100 + 1] - Xnn_local))\n",
+    "    )\n",
+    "\n",
+    "    err_nonlocal.append(\n",
+    "        np.sum(np.abs(X_full[i * 10 : i * 10 + T_test * 100 + 1] - Xnn_nonlocal))\n",
+    "    )\n",
+    "\n",
+    "\n",
+    "print(f\"Sum of errors for linear: {sum(err_linear):.2f}\")\n",
+    "print(f\"Sum of errors for local neural network: {sum(err_local):.2f}\")\n",
+    "print(f\"Sum of errors for non-local neural network: {sum(err_nonlocal):.2f}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In this evaluation we see that the neural networks perform much better than the linear model. However, incorporating non-locality did not improve model performance over the local model. It may be possible to have improved this by building a better non-local model by a larger parameter sweep. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Summary \n",
+    "\n",
+    "In this notebook we showed how neural networks can be added as a parameterization to the single timescale L96 model (the gcm analogue of the two timescale L96 model). The neural network parameterizations performed better than the linear parameterization, as shown by evaluating the model performance over many initial conditions. \n",
+    "\n",
+    "In these notebook we have followed the strategy of :\n",
+    "- running a realistic (more time scales resolved) simulation.\n",
+    "- evaluating the impact of the fast scales (which could not be resolved if the simulation resolved fewer time scales) on the slow scales in the realistic simulation.\n",
+    "- learning a functional relationship between the impact of the fast scales on the slow scale and the slow scales.\n",
+    "- incoporating this relationship (ML based parameterization) in the model of the slow time scales, and evaluating the success (if any). \n",
+    "\n",
+    "This is a strategy of offline training and online testing. \n",
+    "\n",
+    "A different strategy to have would have been to not necessarily learn the impact of or the patterns in the sub-grid scales. But rather optimize a model such that when it is added to the slow time scale model, it leads to the time evolution of the slow time scale model being closer to the two time scale model (which is our ultimate goal). This strategy is sometimes called online training, and discussed in the next notebook. A short primer to this was presented in [the notebook on gcm tuning](https://m2lines.github.io/L96_demo/notebooks/estimating-gcm-parameters.html), where we tuned parameters to match the time evolution of the more realistic model. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/L96_online_training_NN.ipynb b/_sources/notebooks/L96_online_training_NN.ipynb
new file mode 100644
index 00000000..748fe94a
--- /dev/null
+++ b/_sources/notebooks/L96_online_training_NN.ipynb
@@ -0,0 +1,858 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "c0d2675f-796a-4466-a76a-c9a354518d1b",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# Using Neural Networks for L96 Parameterization: Online Training"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aa918476-598e-4ec0-9d9c-b2fd20d2e699",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Outline:\n",
+    "In the previous two notebooks we showed how a parameterization can be developed for the L96 setup using neural networks using a methodology known as offline training. In this notebook we dive into an alternative training philosophy known as online training.  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c2f632ad-af84-4958-b940-20b96755460d",
+   "metadata": {},
+   "source": [
+    "## Online vs offline training\n",
+    "\n",
+    "The type of ML training done in the previous couple of notebooks, where we estimate the sub-grid terms explicitly as the terms corresponding to the missing scales from a full simulations, is known as offline training. In this training procedure it is assumed that know exactly how to estimate the effect of the missing scales. However, this might not be true in general. For the L96 case there can be differences in the numerics of the one time scale and two time scale models, which will not be accounted for in the sub-grid terms we estimate from the two time-scale more. In more general simulations, e.g. turbulent flows, a clear scale separation into scales is not available, and it is not obvious what the right way to filter the high resolution simulation might be. Additionally, the parameterization learnt using the offline procedure may turn out to be unstable because numerical stability of the parameterized model was never imposed as a constraint during the traing procedure. \n",
+    "\n",
+    "An alternate way to train machine learning models is called online training. In this procedure, rather than training a model that best fits a pre-defined sub-grid tendency, a model is trained which tries to produce a solution of the parameterized low resolution model ($X^{param-LR}$) that is closest to the high resolution model ($X^{HR}$, where this $X$ has been appropriately downscaled to match the resolution of the low resolution model). For the L96 simulation, this would imply that the parameterized one time-scale model match the evolution of the slow variables in the two-time scale model. \n",
+    "\n",
+    "A corresponding loss function may be defined as:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "L^{Online} = \\frac{1}{T}\\int_0^T |X^{param-LR} - X^{HR}|^2 dt\n",
+    "\\end{equation}\n",
+    "\n",
+    "Contrast this to the loss function offline training:\n",
+    "\\begin{equation}\n",
+    "L^{Offline} = |P(X_k^{LR}) - U_k|^2,\n",
+    "\\end{equation}\n",
+    "where $P(X)$ is the parameterization of the sub-grid terms (coupling terms) $U$. *Note that in both of the loss functions above appropriate averages over available samples are taken.*\n",
+    "\n",
+    "So, let's see how this can be done in practice, and what technical changes we need to make. \n",
+    "\n",
+    "Note: Much of this notebook has been inspired by Pavel Perezhogin's [repo](https://github.com/Pperezhogin/L96-predictability). "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "129a18f4-f3d5-4503-8c0b-e81895ee2406",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "import time\n",
+    "from IPython.display import Image\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import torch\n",
+    "import torch.utils.data as Data\n",
+    "from torch import nn, optim\n",
+    "\n",
+    "from L96_model import L96, RK2, RK4, EulerFwd, L96_eq1_xdot, integrate_L96_2t"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "79044f9a-ac01-415a-a88a-0385718c8b02",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Ensuring reproducibility\n",
+    "np.random.seed(14)\n",
+    "torch.manual_seed(14);"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "59c9e8fe-9e64-4a58-8cf9-0223192a0d6e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# device = torch.device('cuda' if torch.cuda.is_available() else 'cpu') # You can try this if you have a GPU. However, speed up on GPU is only guaranteed after overhead cost of moving data between CPU and GPU is accounted for.\n",
+    "device = torch.device(\"cpu\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d555922c-f58e-4343-8cf0-1ad027046c6a",
+   "metadata": {},
+   "source": [
+    "## Generate the Ground Truth Dataset from the *Real World* \n",
+    "\n",
+    "Same as the past notebooks, we first generate some instance from the two-time scale (also called our \"real world\") simulation. \n",
+    "\n",
+    "We initialise the L96 two time-scale model using $K$ (set to 8) values of $X$ and $J$ (set to 32) values of $Y$ for each $X$. The model is run for many timesteps to generate the dataset for training later."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ae0738e8-fd6f-4f0c-befa-aa5db97f4521",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Setup the two time-scale model\n",
+    "time_steps = 32000\n",
+    "forcing, dt, T = 18, 0.01, 0.01 * time_steps\n",
+    "\n",
+    "# Create a \"real world\" with K=8 and J=32\n",
+    "W = L96(8, 32, F=forcing)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8ae3e6b4-269f-4ac5-82e8-366eb2fb6dee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Run the two time-scale model\n",
+    "\n",
+    "# The effect of Y on X is `xy_true`\n",
+    "X_true, _, _, _ = W.run(dt, T, store=True, return_coupling=True)\n",
+    "\n",
+    "# Change the data type to `float32` in order to avoid doing type conversions later on\n",
+    "X_true = X_true.astype(np.float32)\n",
+    "\n",
+    "# Notice that we only output the model trajectory here, as we don't need the coupling terms for training here."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "afd23807-0b39-4a92-9bef-d8edfcfd2871",
+   "metadata": {},
+   "source": [
+    "We now need to set the number of time steps that the training process will use for every sample (how long of a trajectory are we trying to match). "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0367d3f3-037d-4316-a827-bc45f5bb3b04",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "Tstep_train = 128\n",
+    "\n",
+    "# We split the simulation into ensembles members without any overlap,\n",
+    "# and each of these will be used as a separate sample.\n",
+    "N_ens = int(time_steps / Tstep_train)\n",
+    "X_true_ens = np.reshape(X_true[0 : N_ens * Tstep_train, :], (N_ens, Tstep_train, 8))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "97974dc3-09d8-4a41-8b53-e3fa36ba959d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Split the data into training and test\n",
+    "\n",
+    "# Set the number of time series that will be part of test ensemble.\n",
+    "test_ens = 32\n",
+    "\n",
+    "# Training Data\n",
+    "X_true_train = X_true_ens[0:-test_ens, :, :]\n",
+    "\n",
+    "# Test Data\n",
+    "X_true_test = X_true_ens[-test_ens:, :, :]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "69869e4a-2cb3-481c-9070-1385e2c34084",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_true_train.shape, X_true_test.shape"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "26cd66c5-c845-42ad-8d53-dc6a77c9f8f8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Number of sample in each batch\n",
+    "BATCH_SIZE = 32"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "16ceba35-c4ae-452a-820b-5f2bd45105c4",
+   "metadata": {},
+   "source": [
+    "Notice that in the training and testing datasets defined below the input to the model is the initial condition, and the output that the model will be evaluated against is a time series from the simulation. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f988c083-3931-474c-bd73-53e8447d24b1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Training Dataset\n",
+    "# ----------------\n",
+    "nlocal_data_train = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_train[:, 0, :]),  # expected input is an initial condition\n",
+    "    torch.from_numpy(X_true_train),  # expected output as a time series\n",
+    ")\n",
+    "\n",
+    "nlocal_loader_train = Data.DataLoader(\n",
+    "    dataset=nlocal_data_train, batch_size=BATCH_SIZE, shuffle=False\n",
+    ")\n",
+    "\n",
+    "# Test Dataset\n",
+    "# ------------\n",
+    "nlocal_data_test = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_test[:, 0, :]), torch.from_numpy(X_true_test)\n",
+    ")\n",
+    "nlocal_loader_test = Data.DataLoader(\n",
+    "    dataset=nlocal_data_test, batch_size=BATCH_SIZE, shuffle=False\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d68c2036-7b57-4172-a6f0-0e164765510e",
+   "metadata": {},
+   "source": [
+    "## Create a differentiable 1 time-scale L96 model \n",
+    "\n",
+    "One of the key components needed for online training is a differentiable solver. This can be seen by the presence of $X^{param-LR}$ in the loss function, which indicates that derivatives should be able to pass through a function that not only corresponds to the extra terms that are added, but also produces the full solution.\n",
+    "\n",
+    "This relatively to easy to do with modern machine learning frameworks like PyTorch or JAX, as long as the model is simple enough to be rewritten using these frameworks. \n",
+    "\n",
+    "Here we will write a differentiable solver for :\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F.\n",
+    "\\end{align}\n",
+    "\n",
+    "Notice below that in this case this task was as simple as using the word `torch` instead of `np`, basically swapping out numpy calls with PyTorch calls. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "078ad47f-cec7-446a-ad65-8cd900469341",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def L96_eq1_xdot_torch(X, F=20):\n",
+    "    \"\"\"\n",
+    "    Compared to older function works on batches,\n",
+    "    i.e. dimension is Nbatch x K (single time step for many batches is input)\n",
+    "    \"\"\"\n",
+    "    return (\n",
+    "        torch.roll(X, shifts=1, dims=-1)\n",
+    "        * (torch.roll(X, shifts=-1, dims=-1) - torch.roll(X, shifts=2, dims=-1))\n",
+    "        - X\n",
+    "        + F\n",
+    "    )"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2ec5406a-9992-430d-a61f-008ef44b380f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def RK4(fn, dt, X, *kw):\n",
+    "    \"\"\"\n",
+    "    Calculate the new state X(n+1) for d/dt X = fn(X,t,...) using the fourth order Runge-Kutta method.\n",
+    "    Args:\n",
+    "        fn     : The function returning the time rate of change of model variables X\n",
+    "        dt     : The time step\n",
+    "        X      : Values of X variables at the current time, t\n",
+    "        kw     : All other arguments that should be passed to fn, i.e. fn(X, t, *kw)\n",
+    "    Returns:\n",
+    "        X at t+dt\n",
+    "    \"\"\"\n",
+    "\n",
+    "    Xdot1 = fn(X, *kw)\n",
+    "    Xdot2 = fn(X + 0.5 * dt * Xdot1, *kw)\n",
+    "    Xdot3 = fn(X + 0.5 * dt * Xdot2, *kw)\n",
+    "    Xdot4 = fn(X + dt * Xdot3, *kw)\n",
+    "    return X + (dt / 6.0) * ((Xdot1 + Xdot4) + 2.0 * (Xdot2 + Xdot3))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8b7788d6-13cd-42ae-9249-0926be461470",
+   "metadata": {},
+   "source": [
+    "## Define a Neural Network for the parameterization\n",
+    "\n",
+    "Here we use a neural network architecture that is exactly the same as the non-local architecture that was used in the previous notebooks. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6a6a5262-39e2-4d50-a8ea-c8b028e5fe5f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class FCNN_nonlocal(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs\n",
+    "        self.linear2 = nn.Linear(16, 16)\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = self.relu(self.linear1(x))\n",
+    "        x = self.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0d72c643-f0ac-42f8-99ee-615e9e2087fa",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fcnn_online_network = FCNN_nonlocal().to(device)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2f76d74f-26fa-47ba-b19f-819cafeeee03",
+   "metadata": {},
+   "source": [
+    "## Loss function and optimizer for online training\n",
+    "\n",
+    "The target of the loss function in online training is not simply to match the sub-grid fluxes, but instead to also track the solution of the two time-scale model using the single time-scale model. \n",
+    "\n",
+    "The loss function below is where the magic happens. Notice two key elements: (1) the neural network is combined with the numerical solver that produces the tendency at every time step, (2) a time integration is performed over the number some number of time steps. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ea9fffa9-07fa-4b40-84da-470f8202474c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def compute_loss(NN, x_in, y_true):\n",
+    "    \"\"\"\n",
+    "    NN: the neural network that parameterizes the missing terms\n",
+    "    x_in: initial conditions, which are the input. Shape (batch size X K)\n",
+    "    y_true: full solution from the two time scale model. Shape (batch size X Nt X K)\n",
+    "    \"\"\"\n",
+    "    full_xdot = lambda x: NN(x) + L96_eq1_xdot_torch(\n",
+    "        x, F=forcing\n",
+    "    )  # make a function that returns tendency with NN as param.\n",
+    "\n",
+    "    y_pred = 0 * y_true  # Use this to store the model prediction we make\n",
+    "\n",
+    "    y_pred[:, 0, :] = x_in  # intiailize IC (which is the only input to model).\n",
+    "\n",
+    "    for j in range(y_true.shape[1] - 1):  # loop over time steps\n",
+    "        y_pred[:, j + 1, :] = RK4(\n",
+    "            full_xdot, dt, y_pred[:, j, :].clone()\n",
+    "        )  # time step forward.\n",
+    "\n",
+    "    return ((y_pred - y_true) ** 2).mean()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f5c0ad15-5790-4236-8011-fbff56c55910",
+   "metadata": {},
+   "source": [
+    "Since the only free parameters correspond to the weights of the neural network, they are passed to the optimizer. Notice that even though the loss function is much more complex than the offline training case, the parameters that are being optimized are the same."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "dfbc3b3b-46c3-45a5-9d5c-7dab607015fc",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# setup optimizer.\n",
+    "\n",
+    "optimizer_fcnn = optim.Adam(fcnn_online_network.parameters(), lr=0.01)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0ea74669-66f7-4637-acbf-8b8457668c08",
+   "metadata": {},
+   "source": [
+    "## Training\n",
+    "\n",
+    "First we define the helper functions for training, testing and fitting, same as we did in previous notebooks"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "33c342c1-f430-4085-bc10-efcafa1364d8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def train_model(network, criterion, loader, optimizer):\n",
+    "    \"\"\"Train the network for one epoch\"\"\"\n",
+    "    network.train()\n",
+    "\n",
+    "    train_loss = 0\n",
+    "\n",
+    "    for batch_x, batch_y in loader:\n",
+    "        # Compute the loss (now the predictions are done directly in the loss function).\n",
+    "        loss = criterion(network, batch_x.to(device), batch_y.to(device))\n",
+    "        train_loss += loss.item()\n",
+    "\n",
+    "        # Clear the gradients\n",
+    "        optimizer.zero_grad()\n",
+    "\n",
+    "        # Backpropagation to compute the gradients and update the weights\n",
+    "        loss.backward()\n",
+    "        optimizer.step()\n",
+    "\n",
+    "    return train_loss / len(loader)\n",
+    "\n",
+    "\n",
+    "def test_model(network, criterion, loader):\n",
+    "    \"\"\"Test the network\"\"\"\n",
+    "    network.eval()  # Evaluation mode (important when having dropout layers)\n",
+    "\n",
+    "    test_loss = 0\n",
+    "    with torch.no_grad():\n",
+    "        for batch_x, batch_y in loader:\n",
+    "            # Compute the loss\n",
+    "            loss = criterion(network, batch_x.to(device), batch_y.to(device))\n",
+    "            test_loss += loss.item()\n",
+    "\n",
+    "        # Get an average loss for the entire dataset\n",
+    "        test_loss /= len(loader)\n",
+    "\n",
+    "    return test_loss\n",
+    "\n",
+    "\n",
+    "def fit_model(network, criterion, optimizer, train_loader, test_loader, n_epochs):\n",
+    "    \"\"\"Train and validate the network\"\"\"\n",
+    "    train_losses, test_losses = [], []\n",
+    "    start_time = time.time()\n",
+    "    for epoch in range(1, n_epochs + 1):\n",
+    "        train_loss = train_model(network, criterion, train_loader, optimizer)\n",
+    "        test_loss = test_model(network, criterion, test_loader)\n",
+    "        train_losses.append(train_loss)\n",
+    "        test_losses.append(test_loss)\n",
+    "    end_time = time.time()\n",
+    "    print(f\"Training completed in {int(end_time - start_time)} seconds.\")\n",
+    "\n",
+    "    return train_losses, test_losses"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a329dace-bd5f-4625-9cc3-1b9a6604b6e4",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Epochs refer to the number of times we iterate over the entire training data during training.\n",
+    "n_epochs = 100"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cea48b6c-cb48-47fd-af13-99c215cf683f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Train the model.\n",
+    "train_loss_online, test_loss_online = fit_model(\n",
+    "    fcnn_online_network,\n",
+    "    compute_loss,\n",
+    "    optimizer_fcnn,\n",
+    "    nlocal_loader_train,\n",
+    "    nlocal_loader_test,\n",
+    "    n_epochs,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c0e6329c-1b1f-4a85-8b4d-ea5387936a84",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(np.array(train_loss_online), label=\"Training loss\")\n",
+    "plt.plot(np.array(test_loss_online), label=\"Test loss\")\n",
+    "plt.yscale(\"log\")\n",
+    "plt.legend()\n",
+    "plt.ylabel(\"MSE\")\n",
+    "plt.xlabel(\"Epochs\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "30448682-8d16-48ca-b907-92ffaba2ea9b",
+   "metadata": {},
+   "source": [
+    "The loss curve above shows that online model is training well, and parameters have been optimized in some sense. Let us check below how this online trained model compares against the offline trained model from the previous notebook. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f2932cc0-bada-4649-8121-f84b6e0daff3",
+   "metadata": {},
+   "source": [
+    "## Test in a simulation: Online testing"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ec9f9d38-2320-4490-a4a2-803de8df8770",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Load in the previously trained non-local offline network.\n",
+    "nonlocal_FCNN_weights = torch.load(\"./networks/non_local_FCNN.pth\")\n",
+    "\n",
+    "fcnn_offline_network = FCNN_nonlocal()\n",
+    "fcnn_offline_network.load_state_dict(nonlocal_FCNN_weights)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "90aeb502-5f07-4966-8cfe-4082757d8e72",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# The different GCM classes\n",
+    "# ---------------------------\n",
+    "\n",
+    "\n",
+    "class GCM_without_parameterization:\n",
+    "    \"\"\"GCM without parameterization\n",
+    "\n",
+    "    Args:\n",
+    "        F: Forcing term\n",
+    "        time_stepping: Time stepping method\n",
+    "    \"\"\"\n",
+    "\n",
+    "    def __init__(self, F, time_stepping=RK4):\n",
+    "        self.F = F\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, _):\n",
+    "        \"\"\"Compute right hand side of the the GCM equations\"\"\"\n",
+    "        return L96_eq1_xdot(X, self.F)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        \"\"\"Run GCM\n",
+    "\n",
+    "        Args:\n",
+    "            X0: Initial conditions of X\n",
+    "            dt: Time increment\n",
+    "            nt: Number of forward steps to take\n",
+    "            param: Parameters of closure\n",
+    "\n",
+    "        Returns:\n",
+    "            Model output for all variables of X at each timestep\n",
+    "            along with the corresponding time units\n",
+    "        \"\"\"\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time\n",
+    "\n",
+    "\n",
+    "class GCM_network:\n",
+    "    \"\"\"GCM with neural network parameterization\n",
+    "\n",
+    "    Args:\n",
+    "        F: Forcing term\n",
+    "        network: Neural network\n",
+    "        time_stepping: Time stepping method\n",
+    "    \"\"\"\n",
+    "\n",
+    "    def __init__(self, F, network, time_stepping=RK4):\n",
+    "        self.F = F\n",
+    "        self.network = network\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, _):\n",
+    "        \"\"\"Compute right hand side of the the GCM equations\"\"\"\n",
+    "        if self.network.linear1.in_features == 1:\n",
+    "            X_torch = torch.from_numpy(X)\n",
+    "            X_torch = torch.unsqueeze(X_torch, 1)\n",
+    "        else:\n",
+    "            X_torch = torch.from_numpy(np.expand_dims(X, 0))\n",
+    "\n",
+    "        # Adding NN parameterization\n",
+    "        return L96_eq1_xdot(X, self.F) + np.squeeze(self.network(X_torch).data.numpy())\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        \"\"\"Run GCM\n",
+    "\n",
+    "        Args:\n",
+    "            X0: Initial conditions of X\n",
+    "            dt: Time increment\n",
+    "            nt: Number of forward steps to take\n",
+    "            param: Parameters of closure\n",
+    "\n",
+    "        Returns:\n",
+    "            Model output for all variables of X at each timestep\n",
+    "            along with the corresponding time units\n",
+    "        \"\"\"\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5e6a3ac2-d75d-4643-8aa8-33c3b7a353d1",
+   "metadata": {},
+   "source": [
+    "Now let us pick a random point in the simulation as our initial condition, and compare if there is some drastic difference between offline and online parameterization that can be seen visually. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "57d35701-3fb8-4360-af23-01dff0ff28ca",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "start = 2100\n",
+    "init_conditions = X_true[start, :]\n",
+    "T_test = 10\n",
+    "forcing = 18\n",
+    "dt = 0.01"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ef4586d4-55f9-4c28-b7dc-73ea5284d0fc",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gcm_no_param = GCM_without_parameterization(forcing)\n",
+    "X_no_param, t = gcm_no_param(init_conditions, dt, int(T_test / dt))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8f023ad5-b6b0-4e3e-b75b-834d14daa2b2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Evaluate with nonlocal offline FCNN\n",
+    "gcm_nonlocal_offline_net = GCM_network(forcing, fcnn_offline_network)\n",
+    "Xnn_nonlocal_offline, t = gcm_nonlocal_offline_net(\n",
+    "    init_conditions, dt, int(T_test / dt), fcnn_offline_network\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "63fa095e-9d5d-4d99-aa6b-31b8f7e56df1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Evaluate with nonlocal offline FCNN\n",
+    "gcm_nonlocal_online_net = GCM_network(forcing, fcnn_online_network)\n",
+    "Xnn_nonlocal_online, t = gcm_nonlocal_online_net(\n",
+    "    init_conditions, dt, int(T_test / dt), fcnn_online_network\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1b8ff02e-6acc-4c66-b0a5-461e9358c12b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "time_i = 300\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t[:time_i], X_true[start : start + time_i, 4], label=\"Full L96\", color=\"k\")\n",
+    "\n",
+    "plt.plot(t[:time_i], X_no_param[:time_i, 4], \"--\", label=\"No parameterization\")\n",
+    "\n",
+    "plt.plot(t[:time_i], Xnn_nonlocal_offline[:time_i, 4], label=\"Offline parameterization\")\n",
+    "\n",
+    "plt.plot(t[:time_i], Xnn_nonlocal_online[:time_i, 4], label=\"Online parameterization\")\n",
+    "\n",
+    "plt.hlines(\n",
+    "    0,\n",
+    "    t[0],\n",
+    "    t[Tstep_train],\n",
+    "    label=\"time range of online training\",\n",
+    "    linestyle=\"-.\",\n",
+    "    color=\"r\",\n",
+    ")\n",
+    "\n",
+    "plt.legend(loc=\"lower right\", fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ae711913-49b6-43b9-a70b-368fcfcac88e",
+   "metadata": {},
+   "source": [
+    "The above plot shows that both offline and online trained models perform much better than the simulation without any parameterization. However, it is unclear if there is any signficant gain in the online case. To be more precise, we compare the different cases below over many simulation. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "163c3563-51d7-4949-b8ff-14df98afed1a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "err_noparam, err_offline, err_online = [], [], []\n",
+    "T_test = 1\n",
+    "\n",
+    "for i in range(100):\n",
+    "    init_conditions_i = X_true[i * 10, :]\n",
+    "\n",
+    "    # Evaluate with no parameterization\n",
+    "    gcm_no_param = GCM_without_parameterization(forcing)\n",
+    "    X_no_param, t = gcm_no_param(init_conditions, dt, int(T_test / dt))\n",
+    "\n",
+    "    # Evaluate with local FCNN\n",
+    "    gcm_nonlocal_offline_net = GCM_network(forcing, fcnn_offline_network)\n",
+    "    Xnn_nonlocal_offline, t = gcm_nonlocal_offline_net(\n",
+    "        init_conditions, dt, int(T_test / dt), fcnn_offline_network\n",
+    "    )\n",
+    "\n",
+    "    # Evaluate with nonlocal FCNN\n",
+    "    gcm_nonlocal_online_net = GCM_network(forcing, fcnn_online_network)\n",
+    "    Xnn_nonlocal_online, t = gcm_nonlocal_online_net(\n",
+    "        init_conditions, dt, int(T_test / dt), fcnn_online_network\n",
+    "    )\n",
+    "\n",
+    "    err_noparam.append(\n",
+    "        np.sum(np.abs(X_true[i * 10 : i * 10 + T_test * 100 + 1] - X_no_param))\n",
+    "    )\n",
+    "\n",
+    "    err_offline.append(\n",
+    "        np.sum(\n",
+    "            np.abs(X_true[i * 10 : i * 10 + T_test * 100 + 1] - Xnn_nonlocal_offline)\n",
+    "        )\n",
+    "    )\n",
+    "\n",
+    "    err_online.append(\n",
+    "        np.sum(np.abs(X_true[i * 10 : i * 10 + T_test * 100 + 1] - Xnn_nonlocal_online))\n",
+    "    )\n",
+    "\n",
+    "\n",
+    "print(f\"Sum of errors for no parameterization: {sum(err_noparam):.2f}\")\n",
+    "print(f\"Sum of errors for offline parameterization: {sum(err_offline):.2f}\")\n",
+    "print(f\"Sum of errors for online parameterization: {sum(err_online):.2f}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d336de28-783d-4dfe-ae60-8774f39057bf",
+   "metadata": {},
+   "source": [
+    "This assessment shows that the online parameterization performs about the same as offline parameterzation. However, atleast for the L96 model the gains (if any), which come at the cost of signfiant complexity, are not drastic. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4ab018fe-819f-4dd2-a208-161ea0368d82",
+   "metadata": {},
+   "source": [
+    "## Summary\n",
+    "\n",
+    "In this notebook we described how online training differs from offline training, and provided an example of how online training can be done for the L96 model. While for the L96 model the online training procedure did not produce significant improvements, the gains for other models may be much greater. If interested, you may look at this [blog post](https://raspstephan.github.io/blog/lorenz-96-is-too-easy/) arguing that L96 might be too simple a test case, which is why different training procedures do not result in very significant differences.\n",
+    "\n",
+    "In the next few notebooks we show a few tricks to potentially improve performance of neural networks, how to interpret neural networks, and how physics constraints can be added in their architecture."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "839b02a1-c8f5-4dbd-91b8-cbe2f8d138aa",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/Learning-DA-increments.ipynb b/_sources/notebooks/Learning-DA-increments.ipynb
new file mode 100644
index 00000000..75dbf1aa
--- /dev/null
+++ b/_sources/notebooks/Learning-DA-increments.ipynb
@@ -0,0 +1,1433 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "5f25c823",
+   "metadata": {},
+   "source": [
+    "# Learning Data Assimilation Increments\n",
+    "\n",
+    "This notebook is derived from the previous notebooks: {doc}`Neural_network_for_Lorenz96` and {doc}`DA_demo_L96`.\n",
+    "\n",
+    "We've restricted it to only using the 3-layer network (not the linear regression model)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "53133bbf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import time\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import torch\n",
+    "from torch import nn\n",
+    "from torch.optim import Adam\n",
+    "from torch.utils.data import DataLoader, TensorDataset\n",
+    "\n",
+    "import DA_methods\n",
+    "from L96_model import L96, L96_eq1_xdot"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "86ef077a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Ensuring reproducibility\n",
+    "rng = np.random.default_rng()\n",
+    "torch.manual_seed(42);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b97ec0e3-55d5-469c-9178-8e749893e79b",
+   "metadata": {},
+   "source": [
+    "## Defining the Model, its Parameters, and Other Utility Functions"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f05b1f30-b274-4408-a065-6a86e05c06ba",
+   "metadata": {},
+   "source": [
+    "### Defining the General Circulation Model (GCM)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "99c22825-e3e8-4329-8707-0da40b1700a8",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def GCM(X_init, F, dt, nt, param=[0]):\n",
+    "    \"\"\"A toy `General Circulation Model` (GCM) that uses the single\n",
+    "    time-scale Lorenz 1996 model with a parameterised coupling term\n",
+    "    to represent the interaction between the observed coarse scale\n",
+    "    processes `X` and unobserved fine scale processes `Y` of the two\n",
+    "    time-scale model.\n",
+    "\n",
+    "    Args:\n",
+    "        X_init: Initial conditions of X\n",
+    "        F: Forcing term\n",
+    "        dt: Sampling frequency of the model\n",
+    "        nt: Number of timesteps for which to run the model\n",
+    "        param: Weights to give to the coupling term\n",
+    "\n",
+    "    Returns:\n",
+    "        Model output for all variables of X at each timestep along with\n",
+    "        the corresponding time units\n",
+    "    \"\"\"\n",
+    "    time, hist, X = (\n",
+    "        dt * np.arange(nt + 1),\n",
+    "        np.zeros((nt + 1, len(X_init))) * np.nan,\n",
+    "        X_init.copy(),\n",
+    "    )\n",
+    "    hist[0] = X\n",
+    "\n",
+    "    for n in range(nt):\n",
+    "        X = X + dt * (L96_eq1_xdot(X, F) - np.polyval(param, X))\n",
+    "        hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "\n",
+    "    return hist, time"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "99f9d561-6f0b-4cc7-8148-9666c6601e01",
+   "metadata": {},
+   "source": [
+    "### Defining the Utility Functions"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0a810c15-17d4-4b8d-acff-03aeb4177f73",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def s(k, K):\n",
+    "    \"\"\"A non-dimension coordinate from -1..+1 corresponding to k=0..K\"\"\"\n",
+    "    return 2 * (0.5 + k) / K - 1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "bdf41747-d694-4098-930f-85516f2df60a",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def get_dist(i, j, K):\n",
+    "    \"\"\"Compute the absolute distance between two element indices\n",
+    "    within a square matrix of size (K x K)\n",
+    "\n",
+    "    Args:\n",
+    "        i: the ith row index\n",
+    "        j: the jth column index\n",
+    "        K: shape of square array\n",
+    "\n",
+    "    Returns:\n",
+    "        Distance\n",
+    "    \"\"\"\n",
+    "    return abs(i - j) if abs(i - j) <= 0.5 * K / 2 else K - abs(i - j)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "470b52bb-a77a-4635-b104-b60a4279ad3f",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def observation_operator(K, l_obs, t_obs, i_t):\n",
+    "    \"\"\"Observation operator to map between model and observation space,\n",
+    "    assuming linearity and model space observations.\n",
+    "\n",
+    "    Args:\n",
+    "        K: spatial dimension of the model\n",
+    "        l_obs: spatial positions of observations on model grid\n",
+    "        t_obs: time positions of observations\n",
+    "        i_t: the timestep of the current DA cycle\n",
+    "\n",
+    "    Returns:\n",
+    "        Operator matrix (K * observation_density, K)\n",
+    "    \"\"\"\n",
+    "    n = l_obs.shape[-1]\n",
+    "    H = np.zeros((n, K))\n",
+    "    H[range(n), l_obs[t_obs == i_t]] = 1\n",
+    "    return H"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3d1e6168-70a5-4354-a006-f0657f57bf5d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def gaspari_cohn(distance, radius):\n",
+    "    \"\"\"Compute the appropriate distance dependent weighting of a\n",
+    "    covariance matrix, after Gaspari & Cohn, 1999 (https://doi.org/10.1002/qj.49712555417)\n",
+    "\n",
+    "    Args:\n",
+    "        distance: the distance between array elements\n",
+    "        radius: localization radius for DA\n",
+    "\n",
+    "    Returns:\n",
+    "        distance dependent weight of the (i,j) index of a covariance matrix\n",
+    "    \"\"\"\n",
+    "    if distance == 0:\n",
+    "        weight = 1.0\n",
+    "    else:\n",
+    "        if radius == 0:\n",
+    "            weight = 0.0\n",
+    "        else:\n",
+    "            ratio = distance / radius\n",
+    "            weight = 0.0\n",
+    "            if ratio <= 1:\n",
+    "                weight = (\n",
+    "                    -(ratio**5) / 4\n",
+    "                    + ratio**4 / 2\n",
+    "                    + 5 * ratio**3 / 8\n",
+    "                    - 5 * ratio**2 / 3\n",
+    "                    + 1\n",
+    "                )\n",
+    "            elif ratio <= 2:\n",
+    "                weight = (\n",
+    "                    ratio**5 / 12\n",
+    "                    - ratio**4 / 2\n",
+    "                    + 5 * ratio**3 / 8\n",
+    "                    + 5 * ratio**2 / 3\n",
+    "                    - 5 * ratio\n",
+    "                    + 4\n",
+    "                    - 2 / 3 / ratio\n",
+    "                )\n",
+    "    return weight"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f86db1f1-c20a-4906-b473-d2960cfec200",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def localize_covariance(B, loc=0):\n",
+    "    \"\"\"Localize the model climatology covariance matrix, based on\n",
+    "    the Gaspari-Cohn function.\n",
+    "\n",
+    "    Args:\n",
+    "        B: Covariance matrix over a long model run 'M_truth' (K, K)\n",
+    "        loc: spatial localization radius for DA\n",
+    "\n",
+    "    Returns:\n",
+    "        Covariance matrix scaled to zero outside distance 'loc' from diagonal and\n",
+    "        the matrix of weights which are used to scale covariance matrix\n",
+    "    \"\"\"\n",
+    "    M, N = B.shape\n",
+    "    X, Y = np.ix_(np.arange(M), np.arange(N))\n",
+    "    dist = np.vectorize(get_dist)(X, Y, M)\n",
+    "    W = np.vectorize(gaspari_cohn)(dist, loc)\n",
+    "    return B * W, W"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "62cbac5e-a06b-499e-a404-0a00f9d8d1f9",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def running_average(X, N):\n",
+    "    \"\"\"Compute running mean over a user-specified window.\n",
+    "\n",
+    "    Args:\n",
+    "        X: Input vector of arbitrary length 'n'\n",
+    "        N: Size of window over which to compute mean\n",
+    "\n",
+    "    Returns:\n",
+    "        X averaged over window N\n",
+    "    \"\"\"\n",
+    "    if N % 2 == 0:\n",
+    "        N1, N2 = -N / 2, N / 2\n",
+    "    else:\n",
+    "        N1, N2 = -(N - 1) / 2, (N + 1) / 2\n",
+    "    X_sum = np.zeros(X.shape)\n",
+    "    for i in np.arange(N1, N2):\n",
+    "        X_sum = X_sum + np.roll(X, int(i), axis=0)\n",
+    "    return X_sum / N"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d23da076-8d00-466a-9975-4ecccdf0d7d8",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def find_obs(loc, obs, t_obs, l_obs, period):\n",
+    "    \"\"\"NOTE: This function is for plotting purposes only.\"\"\"\n",
+    "    t_period = np.where((t_obs[:, 0] >= period[0]) & (t_obs[:, 0] < period[1]))\n",
+    "    obs_period = np.zeros(t_period[0].shape)\n",
+    "    obs_period[:] = np.nan\n",
+    "    for i in np.arange(len(obs_period)):\n",
+    "        if np.any(l_obs[t_period[0][i]] == loc):\n",
+    "            obs_period[i] = obs[t_period[0][i]][l_obs[t_period[0][i]] == loc]\n",
+    "    return obs_period"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "22367ca5-242e-4d5f-8cf7-9a82699469ac",
+   "metadata": {},
+   "source": [
+    "### Initializing the Lorenz 1996 Model Parameters\n",
+    "\n",
+    "Let's define the parameters of the Lorenz 1996 model that match those of Wilks, 2005; K=8, J=32, F=18."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "54ff8a4a-641a-41ea-ad9a-71f129b760cf",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "class Config:\n",
+    "    # fmt: off\n",
+    "    K = 8                               # Dimension of L96 `X` variables\n",
+    "    J = 32                              # Dimension of L96 `Y` variables\n",
+    "    obs_freq = 10                       # Observation frequency (number of sampling intervals (si) per observation)\n",
+    "    F_truth = 18                        # F for truth signal\n",
+    "    F_fcst = 18                         # F for forecast (DA) model\n",
+    "    GCM_param = np.array([0, 0, 0, 0])  # Polynomial coefficicents for GCM parameterization\n",
+    "    ns_da = 4000                        # Number of time samples for DA\n",
+    "    ns = 4000                           # Number of time samples for truth signal\n",
+    "    ns_spinup = 200                     # Number of time samples for spin up\n",
+    "    dt = 0.005                          # Model timestep\n",
+    "    si = 0.005                          # Truth sampling interval\n",
+    "    B_loc = 0.0                         # Spatial localization radius for DA\n",
+    "    DA = \"EnKF\"                         # DA method\n",
+    "    nens = 50                           # Number of ensemble members for DA\n",
+    "    inflate_opt = \"relaxation\"          # Method for DA model covariance inflation\n",
+    "    inflate_factor = 0.86               # Inflation factor\n",
+    "    obs_density = 1.0                   # Fraction of spatial gridpoints where observations are collected\n",
+    "    DA_freq = 10                        # Assimilation frequency (number of sampling intervals (si) per assimilation step)\n",
+    "    obs_sigma = 0.1                     # Observational error standard deviation\n",
+    "    initial_spread = 0.1                # Initial spread added to initial conditions\n",
+    "    # fmt: on"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "530a7fdf-f1c5-4134-9d12-67c9cd4c8c97",
+   "metadata": {},
+   "source": [
+    "#### Suggestions for Modifying the L96 Model Paramters\n",
+    "\n",
+    "If you want to modify the default model parameters given above, you can use the suggestions below for alternate values depending upon the desired behaviour.\n",
+    "\n",
+    "**Less certain observations**\n",
+    "- `obs_sigma`: 1.0\n",
+    "- `initial_spread`: 1.0\n",
+    "- `inflate_factor`: 0.5\n",
+    "\n",
+    "**Less frequent observations**\n",
+    "- `obs_freq`: 50\n",
+    "- `DA_freq`: 50\n",
+    "- `inflate_factor`: 0.4\n",
+    "\n",
+    "**Very infrequent observations**\n",
+    "- `obs_freq`: 200\n",
+    "- `DA_freq`: 200\n",
+    "- `inflate_factor`: 0.5\n",
+    "\n",
+    "**More frequent observations**\n",
+    "- `obs_freq`: 5\n",
+    "- `DA_freq`: 5\n",
+    "- `inflate_factor`: 0.9\n",
+    "\n",
+    "**Very frequent observations**\n",
+    "- `obs_freq`: 1\n",
+    "- `DA_freq`: 1\n",
+    "- `inflate_factor`: 0.98\n",
+    "\n",
+    "**Very frequent observations but less accurate**\n",
+    "- `obs_freq`: 1\n",
+    "- `DA_freq`: 1\n",
+    "- `obs_sigma`: 1.0\n",
+    "- `initial_spread`: 1.0\n",
+    "- `inflate_factor`: 0.9\n",
+    "\n",
+    "**Different time-scale**\n",
+    "- `F_fcst`: 16"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1e0e4ef9",
+   "metadata": {},
+   "source": [
+    "## Generate `Truth` Run from Two Time-Scale L96 Model\n",
+    "\n",
+    "The L96 two time-scale model acts as the **real world** using which we obtain the unobserved `truth` field from which our observations will be derived.\n",
+    "\n",
+    "We begin by spinning-up a state and then record a series of $X$ and $Y$ at time $t$ in arrays `X_truth`, `Y_truth` and `t_truth` respectively. The initial state $X(t=0$ is recorded in `X_init` (and equal to `X_truth[0]`)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "becd0f6a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Set up the \"truth\" 2-scale L96 model\n",
+    "M_truth = L96(Config.K, Config.J, F=Config.F_truth, dt=Config.dt)\n",
+    "M_truth.set_state(rng.standard_normal((Config.K)), 0 * M_truth.j)\n",
+    "\n",
+    "# The model runs for `ns_spinup` timesteps to spin-up\n",
+    "X_spinup, Y_spinup, t_spinup = M_truth.run(Config.si, Config.si * Config.ns_spinup)\n",
+    "\n",
+    "# Generate the initial conditions of X and Y\n",
+    "X_init = X_spinup[-1, :]\n",
+    "Y_init = Y_spinup[-1, :]\n",
+    "\n",
+    "# Using the initial conditions, generate the truth\n",
+    "M_truth.set_state(X_init, Y_init)\n",
+    "X_truth, Y_truth, t_truth = M_truth.run(Config.si, Config.si * Config.ns)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1df7ce87",
+   "metadata": {},
+   "source": [
+    "## Generate Synthetic Observations\n",
+    "\n",
+    "Now we create some **observations** of the **real world** by sampling at `obs_freq` intervals and adding some noise (observational error)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ff525d9d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Sample the \"truth\" to generate observations at certain times (t_obs) and locations (l_obs)\n",
+    "t_obs = np.tile(\n",
+    "    np.arange(Config.obs_freq, Config.ns_da + Config.obs_freq, Config.obs_freq),\n",
+    "    [int(Config.K * Config.obs_density), 1],\n",
+    ").T\n",
+    "\n",
+    "l_obs = np.zeros(t_obs.shape, dtype=\"int\")\n",
+    "for i in range(l_obs.shape[0]):\n",
+    "    l_obs[i, :] = rng.choice(\n",
+    "        Config.K, int(Config.K * Config.obs_density), replace=False\n",
+    "    )\n",
+    "\n",
+    "X_obs = X_truth[t_obs, l_obs] + Config.obs_sigma * rng.standard_normal(l_obs.shape)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "93879f7c-3c7c-4f83-93cc-8e6c6a43ca1d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Calculated observation covariance matrix, assuming independent observations\n",
+    "R = Config.obs_sigma**2 * np.eye(int(Config.K * Config.obs_density))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1f8ffbbd-2cd8-4db9-82d7-82ac2bbcfcc2",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=[10, 6], dpi=150)\n",
+    "plt.plot(t_truth[:], X_truth[:, 0], label=\"truth\")\n",
+    "plt.scatter(\n",
+    "    t_truth[t_obs[1:, 0]],\n",
+    "    find_obs(0, X_obs, t_obs, l_obs, [t_obs[0, 0], t_obs[-1, 0]]),\n",
+    "    color=\"k\",\n",
+    "    label=\"obs\",\n",
+    ")\n",
+    "plt.legend()\n",
+    "plt.xlabel(\"Time t\")\n",
+    "plt.ylabel(\"X(t)\")\n",
+    "plt.title(\"Observations at k=0\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7d63c749-f646-4455-97cc-aa781a5550d0",
+   "metadata": {},
+   "source": [
+    "## Apply Localization to the Background Model Covariance"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b4f2c954",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "We run the model in forward mode for 5000 steps to calculate the background covariance. The model is the **GCM** function defined above which integrates forward.\n",
+    "\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F\n",
+    "\\end{align}\n",
+    "\n",
+    "The absence of the coupling term to the $Y$ equations makes this a model with **missing physics** that we hope the Ensemble Kalman Filter will correct."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4c18eb58",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Generate climatological background covariance for 1-scale L96 model\n",
+    "X1_clim, _ = GCM(X_init, Config.F_fcst, Config.dt, 5000)\n",
+    "B_clim1 = np.cov(X1_clim.T)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f8e083da-e89d-41e1-89dc-452106797ae6",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Load pre-calculated climatological background covariance matrix from a long simulation\n",
+    "B_loc, W_clim = localize_covariance(B_clim1, loc=Config.B_loc)\n",
+    "\n",
+    "B_corr1 = np.zeros(B_clim1.shape)\n",
+    "for i in range(B_clim1.shape[0]):\n",
+    "    for j in range(B_clim1.shape[1]):\n",
+    "        B_corr1[i, j] = B_clim1[i, j] / np.sqrt(B_clim1[i, i] * B_clim1[j, j])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0919ecf0-375b-4aca-b3ec-38b5d04897e8",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(16, 6), dpi=150)\n",
+    "plt.subplot(131)\n",
+    "plt.contourf(B_corr1, cmap=\"bwr\", extend=\"both\", levels=np.linspace(-0.95, 0.95, 20))\n",
+    "plt.colorbar()\n",
+    "plt.title(\"Background correlation matrix: 1-scale L96\")\n",
+    "plt.subplot(132)\n",
+    "plt.contourf(B_loc)\n",
+    "plt.colorbar()\n",
+    "plt.title(\"B_loc\")\n",
+    "plt.subplot(133)\n",
+    "plt.contourf(W_clim)\n",
+    "plt.colorbar()\n",
+    "plt.title(\"W_clim\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "502fcb30",
+   "metadata": {},
+   "source": [
+    "## Run Data Assimilation\n",
+    "\n",
+    "The algorithms steps through segments of time (**DA cycles**), launching an ensemble of short forecasts from the posterior estimate of the preceding segment, each perturbed by noise in their initial condition (inflation).\n",
+    "\n",
+    "Each ensemble trajectory is stored in `ensX`. The increment added to correct the prior is in `X_inc`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1b30ed0d-10e0-46ef-b0c4-22b44623195a",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Set up array to store DA increments\n",
+    "X_inc = np.zeros((int(Config.ns_da / Config.DA_freq), Config.K, Config.nens))\n",
+    "if Config.DA == \"3DVar\":\n",
+    "    X_inc = np.squeeze(X_inc)\n",
+    "t_DA = np.zeros(int(Config.ns_da / Config.DA_freq))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "80eca511-68f9-401e-9500-567fc61492d1",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Initialize ensemble with perturbations\n",
+    "i_t = 0\n",
+    "ensX = X_init[None, :, None] + rng.standard_normal((1, Config.K, Config.nens))\n",
+    "X_post = ensX[0, ...]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "30694d88-ec62-4c27-86b3-1c0fb97dbeae",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "W = W_clim"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5bdecb3b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "for cycle in np.arange(0, Config.ns_da / Config.DA_freq, dtype=int):\n",
+    "    # Set up array to store model forecast for each DA cycle\n",
+    "    ensX_fcst = np.zeros((Config.DA_freq + 1, Config.K, Config.nens))\n",
+    "\n",
+    "    # Model forecast for next DA cycle\n",
+    "    for n in range(Config.nens):\n",
+    "        ensX_fcst[..., n] = GCM(\n",
+    "            X_post[0 : Config.K, n],\n",
+    "            Config.F_fcst,\n",
+    "            Config.dt,\n",
+    "            Config.DA_freq,\n",
+    "            Config.GCM_param,\n",
+    "        )[0]\n",
+    "\n",
+    "    i_t = i_t + Config.DA_freq\n",
+    "\n",
+    "    # Get prior/background from the forecast\n",
+    "    X_prior = ensX_fcst[-1, ...]\n",
+    "\n",
+    "    # Call DA\n",
+    "    t_DA[cycle] = t_truth[i_t]\n",
+    "    if Config.DA == \"EnKF\":\n",
+    "        H = observation_operator(Config.K, l_obs, t_obs, i_t)\n",
+    "\n",
+    "        # Augment state vector with parameters when doing parameter estimation\n",
+    "        B_ens = np.cov(X_prior)\n",
+    "        B_ens_loc = B_ens * W[0 : Config.K, 0 : Config.K]\n",
+    "        X_post = DA_methods.EnKF(X_prior, X_obs[t_obs == i_t], H, R, B_ens_loc)\n",
+    "        X_post[0 : Config.K, :] = DA_methods.ens_inflate(\n",
+    "            X_post[0 : Config.K, :],\n",
+    "            X_prior[0 : Config.K, :],\n",
+    "            Config.inflate_opt,\n",
+    "            Config.inflate_factor,\n",
+    "        )\n",
+    "    elif Config.DA == \"None\":\n",
+    "        X_post = X_prior\n",
+    "\n",
+    "    if not Config.DA == \"None\":\n",
+    "        # Get current increments\n",
+    "        X_inc[cycle, ...] = (\n",
+    "            np.squeeze(X_post[0 : Config.K, ...]) - X_prior[0 : Config.K, ...]\n",
+    "        )\n",
+    "\n",
+    "    # Reset initial conditions for next DA cycle\n",
+    "    ensX_fcst[-1, :, :] = X_post[0 : Config.K, :]\n",
+    "    ensX = np.concatenate((ensX, ensX_fcst[1:None, ...]))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ecbff140",
+   "metadata": {},
+   "source": [
+    "## Post Processing and Visualization\n",
+    "\n",
+    "`meanX` is the ensemble mean forecast, averaging over all the ensemble members. It has discontinuities due to the increment addition between each DA segment."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5a25d44f-cdc7-417b-b8bb-6513062948cc",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "meanX = np.mean(ensX, axis=-1)\n",
+    "clim = np.max(np.abs(meanX - X_truth[0 : (Config.ns_da + 1), :]))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "53048cb9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, axes = plt.subplots(2, 3, figsize=(15, 10))\n",
+    "ch = axes[0, 0].contourf(\n",
+    "    M_truth.k,\n",
+    "    t_truth[0 : (Config.ns_da + 1)],\n",
+    "    meanX - X_truth[0 : (Config.ns_da + 1), :],\n",
+    "    cmap=\"bwr\",\n",
+    "    levels=np.arange(-6.5, 7, 1),\n",
+    "    extend=\"both\",\n",
+    ")\n",
+    "plt.colorbar(ch, ax=axes[0, 0], orientation=\"horizontal\")\n",
+    "axes[0, 0].set_xlabel(\"s\")\n",
+    "axes[0, 0].set_ylabel(\"t\")\n",
+    "axes[0, 0].set_title(\"X - X_truth\")\n",
+    "axes[0, 1].plot(\n",
+    "    t_truth[0 : (Config.ns_da + 1)],\n",
+    "    np.sqrt(((meanX - X_truth[0 : (Config.ns_da + 1), :]) ** 2).mean(axis=-1)),\n",
+    "    label=\"RMSE\",\n",
+    ")\n",
+    "axes[0, 1].plot(\n",
+    "    t_truth[0 : (Config.ns_da + 1)],\n",
+    "    np.mean(np.std(ensX, axis=-1), axis=-1),\n",
+    "    label=\"Spread\",\n",
+    ")\n",
+    "axes[0, 1].plot(\n",
+    "    t_truth[0 : (Config.ns_da + 1)],\n",
+    "    Config.obs_sigma * np.ones((Config.ns_da + 1)),\n",
+    "    label=\"Obs error\",\n",
+    ")\n",
+    "axes[0, 1].legend()\n",
+    "axes[0, 1].set_xlabel(\"time\")\n",
+    "axes[0, 1].set_title(\"RMSE (X - X_truth)\")\n",
+    "axes[0, 1].grid(which=\"both\", linestyle=\"--\")\n",
+    "\n",
+    "axes[0, 2].plot(\n",
+    "    M_truth.k,\n",
+    "    np.sqrt(((meanX - X_truth[0 : (Config.ns_da + 1), :]) ** 2).mean(axis=0)),\n",
+    "    label=\"RMSE\",\n",
+    ")\n",
+    "X_inc_ave = X_inc / Config.DA_freq / Config.si\n",
+    "axes[0, 2].plot(M_truth.k, X_inc_ave.mean(axis=(0, -1)), label=\"Inc\")\n",
+    "axes[0, 2].plot(\n",
+    "    M_truth.k, running_average(X_inc_ave.mean(axis=(0, -1)), 7), label=\"Inc Ave\"\n",
+    ")\n",
+    "axes[0, 2].plot(\n",
+    "    M_truth.k,\n",
+    "    np.ones(M_truth.k.shape) * (Config.F_fcst - Config.F_truth),\n",
+    "    label=\"F_bias\",\n",
+    ")\n",
+    "axes[0, 2].plot(\n",
+    "    M_truth.k,\n",
+    "    np.ones(M_truth.k.shape) * (X_inc / Config.DA_freq / Config.si).mean(),\n",
+    "    \"k:\",\n",
+    "    label=\"Ave Inc\",\n",
+    ")\n",
+    "axes[0, 2].legend()\n",
+    "axes[0, 2].set_xlabel(\"s\")\n",
+    "axes[0, 2].set_title(\"Increments\")\n",
+    "axes[0, 2].grid(which=\"both\", linestyle=\"--\")\n",
+    "\n",
+    "plot_start, plot_end = 200, 800\n",
+    "plot_start_DA, plot_end_DA = int(plot_start / Config.DA_freq), int(\n",
+    "    plot_end / Config.DA_freq\n",
+    ")\n",
+    "plot_x = 0\n",
+    "axes[1, 0].plot(\n",
+    "    t_truth[plot_start:plot_end], X_truth[plot_start:plot_end, plot_x], label=\"truth\"\n",
+    ")\n",
+    "axes[1, 0].plot(\n",
+    "    t_truth[plot_start:plot_end], meanX[plot_start:plot_end, plot_x], label=\"forecast\"\n",
+    ")\n",
+    "axes[1, 0].scatter(\n",
+    "    t_DA[plot_start_DA - 1 : plot_end_DA - 1],\n",
+    "    find_obs(plot_x, X_obs, t_obs, l_obs, [plot_start, plot_end]),\n",
+    "    label=\"obs\",\n",
+    ")\n",
+    "axes[1, 0].grid(which=\"both\", linestyle=\"--\")\n",
+    "axes[1, 0].set_xlabel(\"time\")\n",
+    "axes[1, 0].set_title(\"k=\" + str(plot_x + 1) + \" truth and forecast\")\n",
+    "axes[1, 0].legend()\n",
+    "\n",
+    "axes[1, 1].axis(\"off\")\n",
+    "\n",
+    "axes[1, 2].text(\n",
+    "    0.1,\n",
+    "    0.1,\n",
+    "    \"RMSE={:3f}\\nSpread={:3f}\\nGCM param={}\\nDA={},{}\\nDA_freq={}\\nB_loc={}\\ninflation={},{}\\nobs_density={}\\nobs_sigma={}\\nobs_freq={}\".format(\n",
+    "        np.sqrt(((meanX - X_truth[0 : (Config.ns_da + 1), :]) ** 2).mean()),\n",
+    "        np.mean(np.std(ensX, axis=-1)),\n",
+    "        Config.DA,\n",
+    "        Config.GCM_param,\n",
+    "        Config.nens,\n",
+    "        Config.DA_freq,\n",
+    "        Config.B_loc,\n",
+    "        Config.inflate_opt,\n",
+    "        Config.inflate_factor,\n",
+    "        Config.obs_density,\n",
+    "        Config.obs_sigma,\n",
+    "        Config.obs_freq,\n",
+    "    ),\n",
+    "    fontsize=15,\n",
+    ")\n",
+    "axes[1, 2].axis(\"off\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e3cdeb59",
+   "metadata": {},
+   "source": [
+    "## Examining the Relationship between the Members and their Increments\n",
+    "\n",
+    "Converting the increment `X_inc` into a tendency, we can examine the relationship between $\\dot{X}$ (due to the missing physics) and the state of the model at the beginning of each DA segment.\n",
+    "\n",
+    "If we are properly correcting the absence of the coupling term then this structure should look like the parameterization of the coupling term, as done in Wilks, 2005."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fe35d752-3a29-4d3d-a073-96d82359d731",
+   "metadata": {},
+   "source": [
+    "### Individual Ensemble Members"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cdc24313-1928-4a43-aa02-ffce97ba5a74",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "jj = np.abs(X_inc_ave[0:, :].flatten()) > -1e-7\n",
+    "\n",
+    "# The offset by DA_freq looks at the previous posterior\n",
+    "x_input = ensX[t_obs[0:, 0] - Config.DA_freq, :].flatten()[jj]\n",
+    "\n",
+    "# Mid-point of trajectory\n",
+    "x_input = 0.5 * (x_input + ensX[t_obs[0:, 0], :].flatten()[jj])\n",
+    "\n",
+    "xinc_output = X_inc_ave[0:, :].flatten()[jj]\n",
+    "\n",
+    "x = np.linspace(-8, 15, 100)\n",
+    "p = np.polyfit(x_input, xinc_output, 4)\n",
+    "p18 = [0.000707, -0.0130, -0.0190, 1.59, 0.275]  # Polynomial from Wilks, 2005"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3d79e858",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(12, 5), dpi=150)\n",
+    "plt.suptitle(\"All time, all individial k, and all ensemble members\")\n",
+    "plt.subplot(121)\n",
+    "plt.plot(x_input, xinc_output, \"k.\")\n",
+    "plt.grid()\n",
+    "plt.plot(x, -np.polyval(p18, x), label=\"$P_4(X_k)$ - Wilks, 2005\")\n",
+    "plt.plot(x, np.polyval(p, x), label=\"$P_4(X_k)$\")\n",
+    "plt.xlabel(\"Ensemble member $X_i(k,t)$\")\n",
+    "plt.ylabel(\"Ensemble member increment $\\dot{X}$\")\n",
+    "plt.subplot(122)\n",
+    "plt.hist2d(\n",
+    "    x_input, xinc_output, bins=(np.linspace(-10, 15, 50), np.linspace(-25, 20, 150))\n",
+    ")\n",
+    "plt.plot(x, -np.polyval(p18, x), label=\"$P_4(X_k)$ - Wilks, 2005\")\n",
+    "plt.plot(x, np.polyval(p, x), label=\"$P_4(X_k)$\")\n",
+    "plt.xlabel(\"Ensemble member $X_i(k,t)$\")\n",
+    "plt.ylabel(\"Ensemble member increment $\\dot{X}$\");"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3ce5f1ce-e50d-4d99-8e99-2e4106bb302b",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "xl = 8, 10\n",
+    "k = 0\n",
+    "e = 19\n",
+    "si = Config.DA_freq\n",
+    "\n",
+    "l = (l_obs == k).max(axis=1)\n",
+    "\n",
+    "plt.figure(figsize=(14, 5), dpi=150)\n",
+    "plt.suptitle(\"Ensemble member %i , k = %i\" % (e, k))\n",
+    "plt.plot(t_truth, X_truth[:, k], \"--\", label=\"Truth\")\n",
+    "plt.plot(t_truth, ensX[:, k, e], label=\"Ensemble member forecast\")\n",
+    "plt.plot(\n",
+    "    t_truth[t_obs[l, 0]],\n",
+    "    ensX[si::si, k, e][l] - X_inc[:, :, e][l, k],\n",
+    "    \".\",\n",
+    "    label=\"Ensemble member prior\",\n",
+    ")\n",
+    "plt.plot(t_truth[t_obs[l, 0]], ensX[si::si, k, e][l], \".\", label=\"Ensemble member post\")\n",
+    "plt.xlim(xl)\n",
+    "plt.xlabel(\"Time, t\")\n",
+    "plt.ylabel(\"$X(t)$\")\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "17ae15ec-d330-4ae2-864a-1a553e5956ee",
+   "metadata": {},
+   "source": [
+    "### Mean over Ensemble Members"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4d970ec4-6d21-41df-9c75-b96efacbf528",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "jj = np.abs(X_inc_ave.mean(axis=-1).flatten()) > -1e-7\n",
+    "\n",
+    "# The offset by DA_freq looks at the previous posterior\n",
+    "x_input = meanX[t_obs[0:, 0] - Config.DA_freq].flatten()[jj]\n",
+    "\n",
+    "# Mid-point of trajectory\n",
+    "x_input = 0.5 * (x_input + meanX[t_obs[0:, 0]].flatten()[jj])\n",
+    "\n",
+    "xinc_output = X_inc_ave.mean(axis=-1).flatten()[jj]\n",
+    "\n",
+    "x = np.linspace(-8, 15, 100)\n",
+    "p = np.polyfit(x_input, xinc_output, 4)\n",
+    "p18 = [0.000707, -0.0130, -0.0190, 1.59, 0.275]  # Polynomial from Wilks, 2005"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9925ad58",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(12, 5), dpi=150)\n",
+    "plt.suptitle(\"All time, all individial k, mean over ensemble members\")\n",
+    "plt.subplot(121)\n",
+    "plt.plot(x_input, xinc_output, \"k.\")\n",
+    "plt.grid()\n",
+    "plt.plot(x, -np.polyval(p18, x), label=\"$P_4(X_k)$ - Wilks, 2005\")\n",
+    "plt.plot(x, np.polyval(p, x), label=\"$P_4(X_k)\")\n",
+    "plt.xlabel(\"Ensemble member $X_i(k,t)$\")\n",
+    "plt.ylabel(\"Ensemble member increment $\\dot{X}$\")\n",
+    "plt.subplot(122)\n",
+    "plt.hist2d(\n",
+    "    x_input, xinc_output, bins=(np.linspace(-10, 15, 50), np.linspace(-25, 20, 150))\n",
+    ")\n",
+    "plt.plot(x, -np.polyval(p18, x), label=\"$P_4(X_k)$ - Wilks, 2005\")\n",
+    "plt.plot(x, np.polyval(p, x), label=\"$P_4(X_k)\")\n",
+    "plt.xlabel(\"Ensemble member $X_i(k,t)$\")\n",
+    "plt.ylabel(\"Ensemble member increment $\\dot{X}$\");"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "12a15ad9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "xl = 8, 10\n",
+    "k = 0\n",
+    "si = Config.DA_freq\n",
+    "\n",
+    "l = (l_obs == k).max(axis=1)\n",
+    "\n",
+    "plt.figure(figsize=(14, 5), dpi=150)\n",
+    "plt.suptitle(\"Ensemble mean, k = %i\" % (k))\n",
+    "plt.plot(t_truth, X_truth[:, k], \"--\", label=\"Truth\")\n",
+    "plt.fill_between(\n",
+    "    t_truth,\n",
+    "    meanX[:, k] - ensX[:, k, :].std(axis=-1),\n",
+    "    meanX[:, k] + ensX[:, k, :].std(axis=-1),\n",
+    "    color=\"grey\",\n",
+    "    alpha=0.2,\n",
+    "    label=\"Ensemble spread\",\n",
+    ")\n",
+    "plt.plot(t_truth, meanX[:, k], label=\"Ensemble mean forecast\")\n",
+    "plt.plot(\n",
+    "    t_truth[t_obs[l, 0]],\n",
+    "    meanX[si::si, k][l] - X_inc.mean(axis=-1)[l, k],\n",
+    "    \".\",\n",
+    "    label=\"Ensemble mean prior\",\n",
+    ")\n",
+    "plt.plot(t_truth[t_obs[l, 0]], meanX[si::si, k][l], \".\", label=\"Ensemble mean post\")\n",
+    "plt.xlim(xl)\n",
+    "plt.xlabel(\"Time, t\")\n",
+    "plt.ylabel(\"$X(t)$\")\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "380408db",
+   "metadata": {},
+   "source": [
+    "With this, we have successfully **created the DA increments**."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "25bedb98-8922-495e-ba7c-34da6cca5f91",
+   "metadata": {},
+   "source": [
+    "## Learning the DA Increments\n",
+    "\n",
+    "With the dataset for DA Increments created above, we now build a neural network which will try to model that dataset so that we can use it for unseen observations."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3f0ee34b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "t_inc = t_truth[\n",
+    "    t_obs[:, 0]\n",
+    "]  # These are the times in the \"real world\" when observations were made\n",
+    "dt_inc = np.diff(t_inc)[0]  # Time interval between increments\n",
+    "\n",
+    "dt = np.diff(t_truth)[0]  # Time-step of \"real world\" model\n",
+    "da_interval = int(dt_inc / dt)\n",
+    "\n",
+    "# Data from DA system (increments for individual ensemble members)\n",
+    "x_input = ensX[:-1:da_interval]\n",
+    "X_tend = X_inc / dt_inc"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7af4eb68",
+   "metadata": {},
+   "source": [
+    "### Observing the Dataset\n",
+    "\n",
+    "As a sanity check, we look at the data for obvious structure. A polyfit to the data will compare well to Wilks 2005, if the data is similar in distribution. We show Wilks 2005 and the 4th order polyfit for reference but neither are used or needed for training the neural network."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1bf0c4eb",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# A simple scatter plot of x_tend against x_input\n",
+    "x = np.linspace(-7, 14, 100)\n",
+    "p = np.polyfit(x_input.flatten(), X_tend.flatten(), 4)\n",
+    "p18 = [0.000707, -0.0130, -0.0190, 1.59, 0.275]  # Polynomial from Wilks, 2005\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(x_input.flatten(), X_tend.flatten(), \"k.\")\n",
+    "plt.plot(x, -np.polyval(p18, x), label=\"$P_4(X_k)$ - Wilks, 2005\")\n",
+    "plt.plot(x, np.polyval(p, x), label=\"$P_4(X_k)$\")\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8ada1933",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# A PDf of x_tend against x_input\n",
+    "plt.figure(dpi=150)\n",
+    "plt.hist2d(\n",
+    "    x_input.flatten(),\n",
+    "    X_tend.flatten(),\n",
+    "    bins=(np.linspace(-10, 15, 50), np.linspace(-25, 20, 150)),\n",
+    "    cmap=plt.cm.Greys,\n",
+    ")\n",
+    "plt.plot(x, -np.polyval(p18, x), \"k--\", label=\"$P_4(X_k)$ - Wilks, 2005\")\n",
+    "plt.plot(x, np.polyval(p, x), label=\"$P_4(X_k)$ lin. regr.\")\n",
+    "plt.legend()\n",
+    "plt.xlabel(\"X$_k$\")\n",
+    "plt.ylabel(\"Missing X$_k$ tendency\")\n",
+    "plt.title(\"Conventional linear regression\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "79538913",
+   "metadata": {},
+   "source": [
+    "### Creating the Dataset Split\n",
+    "\n",
+    "We partition the dataset into **training** (seen by the network during optimization of the weights) and **validation** (used for evaluation) sets."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ea024c20",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "train_size = x_input.size // 2\n",
+    "train_size = int(0.7 * x_input.size)\n",
+    "print(\"Training set size =\", train_size, \"out of\", x_input.size)\n",
+    "\n",
+    "# Convert the data to type float32\n",
+    "x_input, X_tend = x_input.astype(np.float32), X_tend.astype(np.float32)\n",
+    "\n",
+    "X_train = x_input.flatten()[:train_size]\n",
+    "Y_train = X_tend.flatten()[:train_size]\n",
+    "X_val = x_input.flatten()[train_size:]\n",
+    "Y_val = X_tend.flatten()[train_size:]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "576c14cb-1e75-48ee-b74c-21fb7b4afe12",
+   "metadata": {},
+   "source": [
+    "### Building the Dataset"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d4517f34",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "BATCH_SIZE = 1024\n",
+    "\n",
+    "# Training Dataset\n",
+    "train_dataset = TensorDataset(torch.from_numpy(X_train), torch.from_numpy(Y_train))\n",
+    "train_loader = DataLoader(dataset=train_dataset, batch_size=BATCH_SIZE, shuffle=True)\n",
+    "\n",
+    "# Validation Dataset\n",
+    "val_dataset = TensorDataset(torch.from_numpy(X_val), torch.from_numpy(Y_val))\n",
+    "val_loader = DataLoader(dataset=val_dataset, batch_size=BATCH_SIZE, shuffle=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a176df90-1ab9-4bb2-8809-46f52a4cae73",
+   "metadata": {},
+   "source": [
+    "### Define the Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2ec20b34",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class NetANN(nn.Module):\n",
+    "    def __init__(self, W=16):\n",
+    "        super().__init__()\n",
+    "        self.linear1 = nn.Linear(1, W)\n",
+    "        self.linear2 = nn.Linear(W, W)\n",
+    "        self.linear3 = nn.Linear(W, 1)\n",
+    "\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = self.relu(self.linear1(x))\n",
+    "        x = self.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fe90c2a2-a0b8-4f7c-9061-7ac5336cdd64",
+   "metadata": {},
+   "source": [
+    "### Define the Training and Test Functions"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "816fe6b3",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def train_model(network, criterion, loader, optimizer):\n",
+    "    \"\"\"Train the network for one epoch\"\"\"\n",
+    "    network.train()\n",
+    "\n",
+    "    train_loss = 0\n",
+    "    for batch_x, batch_y in loader:\n",
+    "        # Get predictions\n",
+    "        if len(batch_x.shape) == 1:\n",
+    "            # This if block is needed to add a dummy dimension if our inputs are 1D\n",
+    "            # (where each number is a different sample)\n",
+    "            prediction = torch.squeeze(network(torch.unsqueeze(batch_x, 1)))\n",
+    "        else:\n",
+    "            prediction = network(batch_x)\n",
+    "\n",
+    "        # Compute the loss\n",
+    "        loss = criterion(prediction, batch_y)\n",
+    "        train_loss += loss.item()\n",
+    "\n",
+    "        # Clear the gradients\n",
+    "        optimizer.zero_grad()\n",
+    "\n",
+    "        # Backpropagation to compute the gradients and update the weights\n",
+    "        loss.backward()\n",
+    "        optimizer.step()\n",
+    "\n",
+    "    return train_loss / len(loader)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ec59c249-95c9-426d-b409-04f42e7528f8",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def test_model(network, criterion, loader):\n",
+    "    \"\"\"Test the network\"\"\"\n",
+    "    network.eval()  # Evaluation mode (important when having dropout layers)\n",
+    "\n",
+    "    test_loss = 0\n",
+    "    with torch.no_grad():\n",
+    "        for batch_x, batch_y in loader:\n",
+    "            # Get predictions\n",
+    "            if len(batch_x.shape) == 1:\n",
+    "                # This if block is needed to add a dummy dimension if our inputs are 1D\n",
+    "                # (where each number is a different sample)\n",
+    "                prediction = torch.squeeze(network(torch.unsqueeze(batch_x, 1)))\n",
+    "            else:\n",
+    "                prediction = network(batch_x)\n",
+    "\n",
+    "            # Compute the loss\n",
+    "            loss = criterion(prediction, batch_y)\n",
+    "            test_loss += loss.item()\n",
+    "\n",
+    "        # Get an average loss for the entire dataset\n",
+    "        test_loss /= len(loader)\n",
+    "\n",
+    "    return test_loss"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f684cd81-67f9-4ccd-ac08-2ed75e922669",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def fit_model(network, criterion, optimizer, train_loader, val_loader, n_epochs):\n",
+    "    \"\"\"Train and validate the network\"\"\"\n",
+    "    train_losses, val_losses = [], []\n",
+    "    start_time = time.time()\n",
+    "    for epoch in range(1, n_epochs + 1):\n",
+    "        train_loss = train_model(network, criterion, train_loader, optimizer)\n",
+    "        val_loss = test_model(network, criterion, val_loader)\n",
+    "        train_losses.append(train_loss)\n",
+    "        val_losses.append(val_loss)\n",
+    "    end_time = time.time()\n",
+    "    print(f\"Training completed in {int(end_time - start_time)} seconds.\")\n",
+    "\n",
+    "    return train_losses, val_losses"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d0893a8d-86a8-40d4-9cbc-6116200ec943",
+   "metadata": {},
+   "source": [
+    "### Train the Network"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3c5ca08d-3971-4230-b64e-8846ec0b4cda",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Initialize the network\n",
+    "nn_3l = NetANN(W=16)\n",
+    "\n",
+    "# MSE loss function\n",
+    "criterion = torch.nn.MSELoss()\n",
+    "\n",
+    "# Adam optimizer\n",
+    "optimizer = Adam(nn_3l.parameters(), lr=0.003)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "dabaf445-3e69-4e7e-af52-a190b5848d2c",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "n_epochs = 24\n",
+    "train_loss, val_loss = fit_model(\n",
+    "    nn_3l, criterion, optimizer, train_loader, val_loader, n_epochs\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "203109dc-7817-4a68-a468-bcd62515a90c",
+   "metadata": {},
+   "source": [
+    "### Visualizing the Results"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "15f2f411-6f21-4095-972a-76745d5eb02f",
+   "metadata": {},
+   "source": [
+    "Comparing the training and validation loss curves"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cda2716e-26f2-4701-b036-cbf321cb5058",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(train_loss, \"b\", label=\"Training loss\")\n",
+    "plt.plot(val_loss, \"r\", label=\"Validation loss\")\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "26b1e20b",
+   "metadata": {},
+   "source": [
+    "Since the NN has one input and one output, we can plot it as a function $nn(X)$ (orange), and compare it to the polyfit (blue) and Wilks 2005 polynomial (black dashed)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0409ce7a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.hist2d(\n",
+    "    x_input.flatten(),\n",
+    "    X_tend.flatten(),\n",
+    "    bins=(np.linspace(-10, 15, 50), np.linspace(-25, 20, 150)),\n",
+    "    cmap=plt.cm.Greys,\n",
+    ")\n",
+    "plt.plot(x, -np.polyval(p18, x), \"k--\", label=\"$P_4(X_k)$ - Wilks, 2005\")\n",
+    "plt.plot(x, np.polyval(p, x), label=\"$P_4(X_k)$ lin. regr.\")\n",
+    "plt.plot(\n",
+    "    x,\n",
+    "    (nn_3l(torch.unsqueeze(torch.from_numpy(x.astype(np.float32)), 1))).data.numpy(),\n",
+    "    label=\"NN-3L\",\n",
+    ")\n",
+    "plt.legend()\n",
+    "plt.xlabel(\"X$_k$\")\n",
+    "plt.ylabel(\"Missing X$_k$ tendency\")\n",
+    "plt.title(\"NN fit\");"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2b2c4439-b9f0-4139-9568-6d0863f52993",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/Neural-Network-Interpretation.ipynb b/_sources/notebooks/Neural-Network-Interpretation.ipynb
new file mode 100644
index 00000000..611d66ec
--- /dev/null
+++ b/_sources/notebooks/Neural-Network-Interpretation.ipynb
@@ -0,0 +1,668 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Interpreting Neural Networks"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Outline\n",
+    "\n",
+    "Neural networks are great at approximating complex functions, but usually provide little direct insight or interpretation into the functional relationships that are approximated. In this notebook we present some techniques that can be used to probe neural networks and indirectly infer what the network has learnt.  \n",
+    "\n",
+    "We present two class of approaches: feature importance and saliency maps. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "### Set up the model and networks"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import math\n",
+    "\n",
+    "import torch\n",
+    "from torch.autograd import Variable, grad\n",
+    "from torch.autograd.functional import jacobian\n",
+    "import torch.nn.functional as F\n",
+    "import torch.utils.data as Data\n",
+    "from torch import nn, optim\n",
+    "\n",
+    "from sklearn.metrics import r2_score\n",
+    "from sklearn.linear_model import LinearRegression\n",
+    "\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "from L96_model import (\n",
+    "    L96,\n",
+    "    L96_eq1_xdot,\n",
+    "    integrate_L96_2t,\n",
+    "    EulerFwd,\n",
+    "    RK2,\n",
+    "    RK4,\n",
+    ")\n",
+    "\n",
+    "%matplotlib inline"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "np.random.seed(14)  # For reproducibility\n",
+    "torch.manual_seed(14)  # For reproducibility"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# create a confusion matrix like figure\n",
+    "def imshow(x, colorbar_pct=97.5, cmap=\"seismic\", label=None, vlim=None, **kw):\n",
+    "    if vlim is None:\n",
+    "        vlim = np.percentile(np.abs(x), colorbar_pct)\n",
+    "    plt.xlabel(\"Input dimension\", fontsize=14)\n",
+    "    plt.ylabel(\"Output dimension\", fontsize=14)\n",
+    "    plt.xticks(range(8))\n",
+    "    plt.yticks(range(8))\n",
+    "    im = plt.imshow(x, vmin=-vlim, vmax=vlim, cmap=cmap, **kw)\n",
+    "    cb = plt.colorbar()\n",
+    "    if label is not None:\n",
+    "        cb.set_label(label, fontsize=14)\n",
+    "    return im\n",
+    "\n",
+    "\n",
+    "def plot_feature_importance(result, feature_index):\n",
+    "    fig = plt.figure()\n",
+    "    ax = fig.add_axes([0, 0, 1, 1])\n",
+    "    ax.set_xlabel(\"Shift in score\")\n",
+    "    ax.set_ylabel(\"Column\")\n",
+    "    ax.set_title(\"Permutation Feature Importance for k=\" + str(feature_index))\n",
+    "\n",
+    "    predictors = result[:, 0]\n",
+    "    scores = result[:, 1]\n",
+    "    y_pos = range(len(predictors))\n",
+    "\n",
+    "    ax.barh(y_pos, scores)\n",
+    "    plt.yticks(y_pos, predictors)\n",
+    "    plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "### Generate L96 data"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "time_steps = 20000\n",
+    "Forcing, dt, T = 18, 0.01, 0.01 * time_steps\n",
+    "\n",
+    "# Create a \"synthetic world\" with K=8 and J=32\n",
+    "K = 8\n",
+    "J = 32\n",
+    "W = L96(K, J, F=Forcing)\n",
+    "# Get training data for the neural network.\n",
+    "\n",
+    "# - Run the true state and output subgrid tendencies (the effect of Y on X is xytrue):\n",
+    "X_true, _, _, xy_true = W.run(dt, T, store=True, return_coupling=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "### Load a pretrained neural network"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Specify a path\n",
+    "PATH = \"networks/network_3_layers_100_epoches.pth\"\n",
+    "# Load\n",
+    "weights = torch.load(PATH)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class Net_ANN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super(Net_ANN, self).__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs, 16 neurons for first hidden layer\n",
+    "        self.linear2 = nn.Linear(16, 16)  # 16 neurons for second hidden layer\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = x.to(self.linear1.weight.dtype)\n",
+    "        x = F.relu(self.linear1(x))\n",
+    "        x = F.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model = Net_ANN()\n",
+    "model.load_state_dict(weights)\n",
+    "model.eval()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Feature importance\n",
+    "\n",
+    "Feature importance is a simple technique that allows us to identify the contribution that each input feature has towards bringing a neural network's output close to the target (reducing the loss or performing well on some skill metric). \n",
+    "\n",
+    "In this process the importance of each input feature is evaluate by scrambling its value and evaluating how much the network output changed in response. This changed output is compared to the target, to see whether the particular input has small or large effect on the network output. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# First we compute the baeline score (how the model does with original data)\n",
+    "feature_index = 3  # which k do we focus on for output.\n",
+    "baseline_r2 = r2_score(\n",
+    "    model(torch.tensor(X_true)).detach().numpy()[:, feature_index],\n",
+    "    xy_true[:, feature_index],\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "scrambled_r2 = []\n",
+    "for column in range(X_true.shape[1]):\n",
+    "    # Create a copy of X_test\n",
+    "    X_copy = X_true.copy()\n",
+    "\n",
+    "    # Scramble the values of the given predictor\n",
+    "    shuffle_col = X_copy[:, column]\n",
+    "    np.random.shuffle(shuffle_col)\n",
+    "    X_copy[:, column] = shuffle_col\n",
+    "\n",
+    "    # Calculate the new R2\n",
+    "    score = r2_score(\n",
+    "        model(torch.tensor(X_copy)).detach().numpy()[:, feature_index],\n",
+    "        xy_true[:, feature_index],\n",
+    "    )\n",
+    "\n",
+    "    # Append the increase in R2 to the list of results\n",
+    "    scrambled_r2.append([column, abs(score - baseline_r2)])\n",
+    "\n",
+    "# Put the results into a pandas dataframe and rank the predictors by score\n",
+    "scrambled_r2 = np.array(scrambled_r2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plot_feature_importance(scrambled_r2, feature_index)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Notice above that that we are considering the impact that shuffling the inputs has on a particular output feature ( feature index). The plot below shows that for this network the largest deterioration in model skill takes place when the input feature at the same k as the output k is varied, suggesting the local behavior of the dependence between the inputs and outputs.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "## Saliency maps \n",
+    "\n",
+    "Saliency maps are popular visualization techniques for gaining insights on why neural networks made a particular decision on the given input data. They are usually rendered as a heatmap, where hotness corresponds to regions that have a big impact on the model’s final decision. They are helpful, for example, when you are frustrated by your model incorrectly classifying a certain datapoint, because you can look at the input features that led to that decision {cite}`Simonyan2013`, {cite}`Baehrens2010`, {cite}`Adebayo2018`.\n",
+    "\n",
+    "\n",
+    "\n",
+    "In this notebook, we will explore the different ways one could render saliency maps for the L96 data and visualize them."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "## Generate saliency maps using input gradients\n",
+    "\n",
+    "Since neural networks are differentiable, the simplest way of generating saliency maps (i.e. a quantification of the sensitivity of the output to the input) is to just take its first derivative with respect to its inputs (or Jacobian, for networks with multiple inputs and outputs).\n",
+    "\n",
+    "We can do this easily in Pytorch using `torch.autograd.functional.jacobian`."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Full Jacobian method"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%%time\n",
+    "jacobians = np.array(\n",
+    "    [\n",
+    "        jacobian(\n",
+    "            model,\n",
+    "            torch.tensor(np.single(X_true[i, :]), requires_grad=True),\n",
+    "            create_graph=False,\n",
+    "        )\n",
+    "        .detach()\n",
+    "        .numpy()\n",
+    "        for i in range(200)\n",
+    "    ]\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This gives us an array of 8x8 gradients, one for each of 200 input samples. Let's visualize their average value, as well as their standard deviation:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig = plt.figure(figsize=(12, 4))\n",
+    "fig.suptitle(\n",
+    "    \"Mean and standard deviation of NN input gradients across the dataset\",\n",
+    "    fontsize=18,\n",
+    "    y=1.025,\n",
+    ")\n",
+    "\n",
+    "plt.subplot(121)\n",
+    "plt.title(\"Average value\", fontsize=16)\n",
+    "imshow(jacobians.mean(0), label=\"Average input derivative\", vlim=0.85)\n",
+    "\n",
+    "plt.subplot(122)\n",
+    "plt.title(\"Standard deviation\", fontsize=16)\n",
+    "imshow(jacobians.std(0), label=\"Standard deviation\", vlim=0.85)\n",
+    "\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The dominant term in the average gradient is close to -1 along the main diagonal, but there are significant off-diagonal elements, and also significant deviation across samples. It's interesting to compare this to the behavior of a linear regression model:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Create and fit a Linear Regression model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "lr = LinearRegression()\n",
+    "lr.fit(X_true, xy_true)\n",
+    "\n",
+    "plt.title(\"Comparing to linear model\", fontsize=16)\n",
+    "imshow(lr.coef_, label=\"Linear regression weight\", vlim=0.85)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We see that the weights of a linear regression model generally match the average input gradients of the neural network, especially down the main diagonal. This makes some sense given that, at each point, input gradients represent the best _local_ linear model that approximates the nonlinear neural network.\n",
+    "\n",
+    "Although computing full Jacobians works fine for a small example, it can become expensive for large networks and input/output dimensions, so we can also approximate it using finite differences:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Approximate Jacobian method"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%%time\n",
+    "epsilon = 1e-2\n",
+    "approx_jacobians = []\n",
+    "\n",
+    "for case in range(200):\n",
+    "    inputs = X_true[case, :].copy()\n",
+    "    inputs = torch.tensor(np.single(inputs), requires_grad=False)\n",
+    "    pred = model(inputs)\n",
+    "    Js = np.zeros((len(inputs), len(pred)))\n",
+    "    for j in range(len(inputs)):\n",
+    "        perturb = np.zeros_like(inputs)\n",
+    "        perturb[j] = epsilon\n",
+    "        inputs_perturbed = (inputs + perturb).clone().detach().requires_grad_(False)\n",
+    "        Js[j, :] = (\n",
+    "            model(inputs_perturbed).detach().numpy() - pred.detach().numpy()\n",
+    "        ) / epsilon\n",
+    "    approx_jacobians.append(Js.T)\n",
+    "\n",
+    "approx_jacobians = np.array(approx_jacobians)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "(Technically this is slightly slower than the previous example, but it can be more performant for large networks.)\n",
+    "\n",
+    "Let's see how that looks:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.title(\"Interpreting the network with finite differences\", fontsize=16)\n",
+    "imshow(approx_jacobians.mean(0), label=\"Average finite difference\")\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "As expected, it's fairly similar to the full Jacobian method.\n",
+    "\n",
+    "## Generate saliency maps layerwise relevance propagation (LRP)\n",
+    "\n",
+    "Another proposed method for generating saliency maps is {cite}`Bach2015`. With a baseline of 0, this method is akin to multiplying the input gradients by the input itself (per {cite}`Ancona2018`), but it has become popular in the climate community and does support alternative baselines."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "for name in weights.keys():\n",
+    "    print(name)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "epsilon = 0.0  # filtering small values\n",
+    "gamma = 0.0  # give more weights to positive values"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## get the weight and bias of the NN\n",
+    "def get_weight(weightsname):\n",
+    "    Ws = []\n",
+    "    Bs = []\n",
+    "    for i, name in enumerate(weights.keys()):\n",
+    "        if i % 2 == 0:\n",
+    "            Ws.append(np.array(weights[name]))\n",
+    "        else:\n",
+    "            Bs.append(np.array(weights[name]))\n",
+    "    return Ws, Bs  # weights and biases\n",
+    "\n",
+    "\n",
+    "# forward pass to calculate the output of each layer\n",
+    "def forward_pass(data, Ws, Bs):\n",
+    "    L = len(Ws)\n",
+    "    forward = [data] + [None] * L\n",
+    "\n",
+    "    for l in range(L - 1):\n",
+    "        forward[l + 1] = np.maximum(0, Ws[l].dot(forward[l])) + Bs[l]  # ativation ReLu\n",
+    "\n",
+    "    ## for last layer that does not have activation function\n",
+    "    forward[L] = Ws[L - 1].dot(forward[L - 1]) + Bs[L - 1]  # linear last layer\n",
+    "    return forward\n",
+    "\n",
+    "\n",
+    "def rho(w, l):\n",
+    "    w_intermediate = w + [0.0, 0.0, 0.0, 0.0, 0.0][l] * np.maximum(0, w)\n",
+    "    return w_intermediate + gamma * np.maximum(0, w_intermediate)\n",
+    "\n",
+    "\n",
+    "def incr(z, l):\n",
+    "    return z + [0.0, 0.0, 0.0, 0.0, 0.0][l] * (z**2).mean() ** 0.5 + 1e-9"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## backward pass to compute the LRP of each layer. Same rule applied to the first layer (input layer)\n",
+    "def onelayer_LRP(W, B, forward, nz, zz):\n",
+    "    mask = np.zeros((nz))\n",
+    "    mask[zz] = 1\n",
+    "    L = len(W)\n",
+    "    R = [None] * L + [forward[L] * mask]  # start from last layer Relevance\n",
+    "\n",
+    "    for l in range(0, L)[::-1]:\n",
+    "        w = rho(W[l], l)\n",
+    "        b = rho(B[l], l)\n",
+    "        z = incr(w.dot(forward[l]) + b + epsilon, l)  # step 1 - forward pass\n",
+    "        s = np.array(R[l + 1]) / np.array(z)  # step 2 - element-wise division\n",
+    "        c = w.T.dot(s)  # step 3 - backward pass\n",
+    "        R[l] = forward[l] * c  # step 4 - element-wise product\n",
+    "    return R\n",
+    "\n",
+    "\n",
+    "def LRP_alllayer(data, weights):\n",
+    "    \"\"\"inputs:\n",
+    "        data: for single sample, with the right asix, the shape is (nz,naxis)\n",
+    "        weights: dictionary of weights and biases\n",
+    "    output:\n",
+    "        LRP, shape: (nx,L+1) that each of the column consist of L+1 array\n",
+    "        Relevance of fisrt layer's pixels\"\"\"\n",
+    "    nx = data.shape[0]\n",
+    "    ## step 1: get the wieghts\n",
+    "    Ws, Bs = get_weight(weights)\n",
+    "\n",
+    "    ## step 2: call the forward pass to get the intermediate layers output\n",
+    "    inter_layer = forward_pass(data, Ws, Bs)\n",
+    "\n",
+    "    ## loop over all z and get the LRP of each layer\n",
+    "    R_all = [None] * nx\n",
+    "    relevance = np.zeros((nx, nx))\n",
+    "    for xx in range(nx):\n",
+    "        R_all[xx] = onelayer_LRP(Ws, Bs, inter_layer, nx, xx)\n",
+    "        relevance[xx, :] = R_all[xx][0]\n",
+    "\n",
+    "    return np.array(R_all, dtype=object), relevance"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%%time\n",
+    "R_many = []\n",
+    "for case in range(200):\n",
+    "    inputs = np.copy(X_true[case, :])\n",
+    "    _, Rs = LRP_alllayer(inputs, weights)\n",
+    "    R_many.append(Rs)\n",
+    "LRP = np.stack(R_many)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.title(\"Interpreting the network with\\nlayerwise relevance propagation\", fontsize=16)\n",
+    "imshow(LRP.mean(0), label=\"Average LRP\")\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "LRP outputs something qualitatively different from the gradient-based methods, and in fact each element should be interpreted more as an attribution score (i.e. the actual contribution of an input to the output) than as a sensitivity score (i.e. how much an output changes with an input). Below, we'll see that this approximately reduces to multiplying the gradient by the input.\n",
+    "\n",
+    "## Comparing all of the methods\n",
+    "\n",
+    "Let's look at the output of these methods side-by-side:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "comparisons = [\n",
+    "    (jacobians, \"Jacobian (exact)\"),\n",
+    "    (approx_jacobians, \"Jacobian (finite diff)\"),\n",
+    "    (LRP, \"LRP\"),\n",
+    "    (jacobians * X_true[:200][:, np.newaxis, :], \"Jacobian * input\"),\n",
+    "]\n",
+    "\n",
+    "fig = plt.figure(figsize=(10, 8))\n",
+    "fig.suptitle(\n",
+    "    \"Comparing average outputs of saliency methods\", fontsize=20, y=1.0, va=\"bottom\"\n",
+    ")\n",
+    "\n",
+    "for i, (saliency, label) in enumerate(comparisons):\n",
+    "    plt.subplot(2, 2, i + 1)\n",
+    "    plt.title(label, fontsize=16)\n",
+    "    imshow(saliency.mean(0), label=label)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "As expected, LRP and Jacobian * input produce very similar outputs. For more information on the intricacies of saliency maps, see {cite}`Ancona2018` and {cite}`Bach2015`."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Summary\n",
+    "\n",
+    "In this notebook we introduced some approaches that can be used to better understand how neural networks estimate the output from a given input. This is an important area of research, as we would want to test that the neural networks are producing the right answers for the right reasons (physically reasonable models). This becomes particularly important when we want our networks to produce accurate results (extrapolate) for inputs that were not observed in training data. \n",
+    "\n",
+    "In the next notebook we show how physics based constraints can be incorporated into neural networks, in an attempt to further guide the neural network to learn more appropriate relationships between inputs and outputs."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/citing.ipynb b/_sources/notebooks/citing.ipynb
new file mode 100644
index 00000000..7ce0aeb2
--- /dev/null
+++ b/_sources/notebooks/citing.ipynb
@@ -0,0 +1,39 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "bfb1c039-bf51-4923-9ce9-d0e869b3d8c3",
+   "metadata": {},
+   "source": [
+    "# Citing this book\n",
+    "\n",
+    "If you found this work useful and would like to cite it, please use : \n",
+    "Balwada, D., Abernathey, R., Acharya, S., Adcroft, A., Brener, J., Balaji, V., ... & Zanna, L. (2023). Learning Machine Learning with Lorenz-96. Authorea Preprints.\n",
+    "\n",
+    "DOI: 10.22541/essoar.170365239.95851488/v1\n",
+    "PDF available at: https://essopenarchive.org/doi/full/10.22541/essoar.170365239.95851488/v1"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/closing_remarks.ipynb b/_sources/notebooks/closing_remarks.ipynb
new file mode 100644
index 00000000..8fbacbf9
--- /dev/null
+++ b/_sources/notebooks/closing_remarks.ipynb
@@ -0,0 +1,42 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "fe395e90-f86b-4251-9300-b80389a26d69",
+   "metadata": {},
+   "source": [
+    "# Outlook\n",
+    "\n",
+    "With this JupyterBook, we have designed an introduction to the problem of machine learning for improving climate models. The Lorenz 96 model is a perfect benchmark to illustrate several ideas and concepts, but remains too simple to present a real test case for complex problems in climate physics and modeling. \n",
+    "\n",
+    "M2LInES has been pushing the boundaries of climate model development through the innovative use of scientific machine learning models to explore: \n",
+    "- New subgrid physics parameterizations in the ocean {cite}`sane2023parameterizing,Guillaumin2021,perezhogin2023generative`, sea-ice {cite}`Zampieri2023`, and atmosphere {cite}`Janni2023,Shamekh_2023`. \n",
+    "- New corrections from data assimilation for climate model components {cite}`chapman2023benefits,gregory2023deep,gregory2023machine`.\n",
+    "- Development of innovative architecture and algorithms for multiscale machine learning related to  turbulence in oceans and atmosphere {cite}`Frezat2022, pedersen2023reliable, otness2023data, ross2022benchmarking`.  \n",
+    "\n",
+    "These novel techniques and data-driven models are now making their way into global climate models {cite}`zhang2023implementation`, illustrating the importance of bringing climate physics, scientific computing, and modern machine learning techniques together."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/constraints.ipynb b/_sources/notebooks/constraints.ipynb
new file mode 100644
index 00000000..74dd6e92
--- /dev/null
+++ b/_sources/notebooks/constraints.ipynb
@@ -0,0 +1,1421 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "deb0775b",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# Adding constraints to Neural Networks "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "acd1a857-c84c-4fe6-9494-22cc0aeffefd",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Outline\n",
+    "\n",
+    "The goal of this notebook is to show how a physical or mathematical constraint can be built in to the learning process to try and nudge the ML model towards more consistent solutions."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fb0c32a3",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Physical constraints in L96\n",
+    "\n",
+    "The **$1^{st}$ order equation L96 model**, \n",
+    "\\begin{equation}\n",
+    "\\frac{d}{dt} X_k\n",
+    "= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F \n",
+    "\\end{equation}\n",
+    "conserves energy, \n",
+    "$$E_X = \\frac{1}{2}<X_K^2>,$$ \n",
+    "in the unforced and undamped form (last 2 terms on the RHS are zero), which results due to the form of the non-linearity. \n",
+    "\n",
+    "However, the **$2^{nd}$ order equation L96 model**\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}\n",
+    "\\\\\n",
+    "\\frac{d}{dt} Y_{j,k}\n",
+    "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - Y_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
+    "\\end{align}\n",
+    "does not have the same conservation property. Instead, for the unforced and undamped system the total energy, \n",
+    "$$E = E_X + E_Y = \\frac{1}{2}(<X_K^2> + <Y_j^2>),$$ \n",
+    "is conserved, with the \"subgrid tendencies\" being the exchange terms ($<X_kY_{j,k}>$) between the large scale ($\\frac{1}{2}<X_K^2>$) and small scale energy reservoirs $\\frac{1}{2}<Y_j^2>$. \n",
+    "\n",
+    "Using energetic arguments to contrain a parameterization is non-trivial, but can prove to be very beneficial. For example the Gent-McWilliams scheme guarantees that the sub-grid parameterization will reduce APE. The GEOMETRIC parameterization keeps track of sub-grid energy and the exchange between the resolved and sub-grid energy, in an attempt to have total energy conservation. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6acaa8ea",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## L96 model with momentum conserving term\n",
+    "As shown above, the L96 model conserves total energy that is contained over both X and Y variables. However, for demonstration purposes we created a modified L96, where the coupling term is momentum conserving. \n",
+    "This is a bit silly to do, as the L96 model does not conserve momentum (but no one can stop you!). This also changes the energetics of the original L96. The main reason for this choice was because it was the easiest to implement. \n",
+    "\n",
+    "The equations of the new model are: \n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\left( \\frac{hc}{b} \\right) \\left(\\sum_{j=0}^{J-1} Y_{j,k-1} - \\sum_{j=0}^{J-1} Y_{j,k+1}\\right)\n",
+    "\\\\\n",
+    "\\frac{d}{dt} Y_{j,k}\n",
+    "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - Y_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
+    "\\end{align}\n",
+    "\n",
+    "The small scale forcing from the small scale to the large scale is:\n",
+    "\\begin{equation}\n",
+    "G_K = S_{k-1} - S_{k+1} = - \\left( \\frac{hc}{b} \\right) \\left(\\sum_{j=0}^{J-1} Y_{j,k-1} - \\sum_{j=0}^{J-1} Y_{j,k+1}\\right)\n",
+    "\\end{equation}\n",
+    " \n",
+    "It is clear that summing $G_k$ over the full domain is identically zero, since it is in a flux form. This term does not input any net momentum into the system, $<G_k>=0$. We will use this as a contrain in this exercise. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5dcd5a54",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "import xarray as xr\n",
+    "\n",
+    "import torch\n",
+    "from torch.autograd import Variable\n",
+    "import torch.nn.functional as F\n",
+    "import torch.utils.data as Data\n",
+    "from torch import nn, optim\n",
+    "from tqdm import tqdm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1e97474a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Load modules for the model\n",
+    "from L96_conservative_model import (\n",
+    "    L96,\n",
+    "    L96_eq1_xdot,\n",
+    "    integrate_L96_2t,\n",
+    ")  # L96_model_XYtend Adds the option to ouptput the subgrid tendencies (effect of Y on X)\n",
+    "from L96_conservative_model import EulerFwd, RK2, RK4\n",
+    "\n",
+    "# Note: We are using a slightly different L96 model here."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8e603f29",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.style.use(\"seaborn-v0_8\")\n",
+    "plt.rcParams[\"font.size\"] = 16"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "982eda0b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Simulate the world\n",
+    "\n",
+    "time_steps = 20000\n",
+    "Forcing, dt, T = 12, 0.01, 0.01 * time_steps\n",
+    "\n",
+    "# Create a \"real world\" with K=8 and J=32\n",
+    "K = 8\n",
+    "J = 32\n",
+    "W = L96(K, J, F=Forcing)\n",
+    "\n",
+    "# Get training data for the neural network.\n",
+    "# - Run the true state and output subgrid tendencies (the effect of Y on X is xytrue):\n",
+    "X_true, y, T_true, S_true = W.run(dt, T, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "afee3788",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# G_k = S_(k-1) - S_(k+1)\n",
+    "G_true = np.roll(S_true, 1, axis=1) - np.roll(S_true, -1, axis=1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7ca46561",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# convert to xarray for convenience\n",
+    "\n",
+    "X_true_xr = xr.DataArray(\n",
+    "    X_true, dims=[\"time\", \"K\"], coords={\"time\": T_true, \"K\": np.arange(K)}\n",
+    ").rename(\"X\")\n",
+    "Y_xr = xr.DataArray(\n",
+    "    y, dims=[\"time\", \"JK\"], coords={\"time\": T_true, \"JK\": np.arange(K * J)}\n",
+    ").rename(\"Y\")\n",
+    "S_true_xr = xr.DataArray(\n",
+    "    S_true, dims=[\"time\", \"K\"], coords={\"time\": T_true, \"K\": np.arange(K)}\n",
+    ").rename(\"S\")\n",
+    "G_true_xr = xr.DataArray(\n",
+    "    G_true, dims=[\"time\", \"K\"], coords={\"time\": T_true, \"K\": np.arange(K)}\n",
+    ").rename(\"G\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c91c9c96",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(16, 10), dpi=150)\n",
+    "\n",
+    "plt.subplot(141)\n",
+    "X_true_xr.sel(time=slice(0, 30)).plot.contourf(vmin=-16, levels=21)\n",
+    "\n",
+    "plt.subplot(142)\n",
+    "Y_xr.sel(time=slice(0, 30)).plot.contourf(vmin=-2, levels=21)\n",
+    "\n",
+    "plt.subplot(143)\n",
+    "G_true_xr.sel(time=slice(0, 30)).plot.contourf(levels=21)\n",
+    "\n",
+    "plt.subplot(144)\n",
+    "S_true_xr.sel(time=slice(0, 30)).plot.contourf(levels=21)\n",
+    "\n",
+    "plt.tight_layout();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2a3ac2b8",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The modified system looks fairly organized and chaotic. So the essence of the original L96 still remains, which is what we wanted."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "50d2aa0b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(9, 3), dpi=150)\n",
+    "\n",
+    "plt.subplot(131)\n",
+    "plt.plot(X_true_xr.sel(K=0), S_true_xr.sel(K=1), \".\", markersize=1)\n",
+    "plt.xlabel(\"$X_{k-1}$\")\n",
+    "plt.ylabel(\"$S_{k}$\")\n",
+    "\n",
+    "plt.subplot(132)\n",
+    "plt.plot(X_true_xr.sel(K=1), S_true_xr.sel(K=1), \".\", markersize=1)\n",
+    "plt.xlabel(\"$X_{k}$\")\n",
+    "plt.ylabel(\"$S_{k}$\")\n",
+    "\n",
+    "plt.subplot(133)\n",
+    "plt.plot(X_true_xr.sel(K=2), S_true_xr.sel(K=1), \".\", markersize=1)\n",
+    "plt.xlabel(\"$X_{k+1}$\")\n",
+    "plt.ylabel(\"$S_{k}$\")\n",
+    "\n",
+    "plt.tight_layout();"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8f20a1ac",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(9, 3), dpi=150)\n",
+    "\n",
+    "plt.subplot(131)\n",
+    "plt.plot(X_true_xr.sel(K=0), G_true_xr.sel(K=1), \".\", markersize=1)\n",
+    "plt.xlabel(\"$X_{k-1}$\")\n",
+    "plt.ylabel(\"$G_{k}$\")\n",
+    "\n",
+    "plt.subplot(132)\n",
+    "plt.plot(X_true_xr.sel(K=1), G_true_xr.sel(K=1), \".\", markersize=1)\n",
+    "plt.xlabel(\"$X_{k}$\")\n",
+    "plt.ylabel(\"$G_{k}$\")\n",
+    "\n",
+    "plt.subplot(133)\n",
+    "plt.plot(X_true_xr.sel(K=2), G_true_xr.sel(K=1), \".\", markersize=1)\n",
+    "plt.xlabel(\"$X_{k+1}$\")\n",
+    "plt.ylabel(\"$G_{k}$\")\n",
+    "\n",
+    "plt.tight_layout();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fcc3a04e",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Since the dependence of Y on X remains the same, G now depends on the neighbouring points (and not the K itself)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f91a3b74",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(12, 4), dpi=150)\n",
+    "plt.subplot(121)\n",
+    "S_true_xr.mean(\"K\").plot()\n",
+    "plt.ylabel(\"$<S>$\")\n",
+    "# plt.title('Averaged subgrid term')\n",
+    "plt.tight_layout()\n",
+    "\n",
+    "plt.subplot(122)\n",
+    "G_true_xr.mean(\"K\").plot()\n",
+    "plt.ylabel(\"$<G>$\")\n",
+    "# plt.title('Averaged subgrid term')\n",
+    "plt.tight_layout();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "29380dc3",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The momentum input from the subgrid term is essentially zero (to numerical precision), as should be. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a1aa6cd0",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "**What should we learn using ML?**  \n",
+    "We will do three types of learning (all fully non-local):  \n",
+    "a) Learn $G_k$ directly as a function of $X_k$s by minimizing a MSE loss.  \n",
+    "b) Learn $G_k$ directly as a function of $X_k$s by minimizing a MSE loss + minimizing $<G_k>$.   \n",
+    "c) Learn $S_k$ as a function of $X_k$s, and then $<G_k>$ will be zero by design.  \n",
+    "d) Learn $S_k$ but with $G_k$ being the goal. (This will introduce the Gauge problem).\n",
+    "\n",
+    "*We chose MSE as a loss function because as shown earlier, the learning for L96 is not sensitive to the choice of loss function.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8224f915",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "### Split into Training and testing data\n",
+    "\n",
+    "val_size = 4000  # number of time steps for validation\n",
+    "\n",
+    "# train:\n",
+    "X_true_train = X_true[\n",
+    "    :-val_size, :\n",
+    "]  # Flatten because we first use single input as a sample\n",
+    "subgrid_G_tend_train = G_true[:-val_size, :]\n",
+    "subgrid_S_tend_train = S_true[:-val_size, :]\n",
+    "\n",
+    "# test:\n",
+    "X_true_test = X_true[-val_size:, :]\n",
+    "subgrid_G_tend_test = G_true[-val_size:, :]\n",
+    "subgrid_S_tend_test = S_true[-val_size:, :]\n",
+    "\n",
+    "# Create non local training data\n",
+    "# Define a data loader (8 inputs, 8 outputs)\n",
+    "\n",
+    "### For the cases where will learn G_k\n",
+    "\n",
+    "# Define our X,Y pairs (state, subgrid tendency) for the linear regression local network.local_torch_dataset = Data.TensorDataset(\n",
+    "torch_dataset = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_train).double(),\n",
+    "    torch.from_numpy(subgrid_G_tend_train).double(),\n",
+    ")\n",
+    "\n",
+    "BATCH_SIZE = 1024  # Number of sample in each batch\n",
+    "\n",
+    "loader = Data.DataLoader(dataset=torch_dataset, batch_size=BATCH_SIZE, shuffle=True)\n",
+    "\n",
+    "# Define a test dataloader (8 inputs, 8 outputs)\n",
+    "\n",
+    "torch_dataset_test = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_test).double(),\n",
+    "    torch.from_numpy(subgrid_G_tend_test).double(),\n",
+    ")\n",
+    "\n",
+    "loader_test = Data.DataLoader(\n",
+    "    dataset=torch_dataset_test, batch_size=BATCH_SIZE, shuffle=True\n",
+    ")\n",
+    "\n",
+    "# For the cases where will learn S_k\n",
+    "\n",
+    "# Define our X,Y pairs (state, subgrid tendency) for the linear regression local network.local_torch_dataset = Data.TensorDataset(\n",
+    "torch_dataset = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_train).double(),\n",
+    "    torch.from_numpy(subgrid_S_tend_train).double(),\n",
+    ")\n",
+    "\n",
+    "BATCH_SIZE = 1024  # Number of sample in each batch\n",
+    "\n",
+    "loader_S = Data.DataLoader(dataset=torch_dataset, batch_size=BATCH_SIZE, shuffle=True)\n",
+    "\n",
+    "# Define a test dataloader (8 inputs, 8 outputs)\n",
+    "\n",
+    "torch_dataset_test = Data.TensorDataset(\n",
+    "    torch.from_numpy(X_true_test).double(),\n",
+    "    torch.from_numpy(subgrid_S_tend_test).double(),\n",
+    ")\n",
+    "\n",
+    "loader_S_test = Data.DataLoader(\n",
+    "    dataset=torch_dataset_test, batch_size=BATCH_SIZE, shuffle=True\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2c7554df",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Class for the neural network architecture\n",
+    "class Net_ANN(nn.Module):\n",
+    "    def __init__(self, name=\"Default\"):\n",
+    "        super(Net_ANN, self).__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs, 16 neurons for first hidden layer\n",
+    "        self.linear2 = nn.Linear(16, 16)  # 16 neurons for second hidden layer\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "        self.name = name\n",
+    "\n",
+    "    #         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = F.relu(self.linear1(x))\n",
+    "        x = F.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5c136578",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Setup function to take one step in the learning\n",
+    "def train_model(net, criterion, trainloader, optimizer):\n",
+    "    # set model in to training mode\n",
+    "    net.train()\n",
+    "\n",
+    "    # Loop through all the training subsets (notice we batched in size of 1024)\n",
+    "    for step, (batch_x, batch_y) in enumerate(trainloader):  # for each training step\n",
+    "        b_x = Variable(batch_x)  # Inputs\n",
+    "        b_y = Variable(batch_y)  # outputs\n",
+    "\n",
+    "        if (\n",
+    "            len(b_x.shape) == 1\n",
+    "        ):  # If is needed to add a dummy dimension if our inputs are 1D (where each number is a different sample)\n",
+    "            prediction = torch.squeeze(\n",
+    "                net(torch.unsqueeze(b_x, 1))\n",
+    "            )  # input x and predict based on x\n",
+    "        else:\n",
+    "            prediction = net(b_x)\n",
+    "\n",
+    "        loss = criterion(prediction, b_y)  # Calculating loss\n",
+    "        optimizer.zero_grad()  # clear gradients for next train\n",
+    "        loss.backward()  # backpropagation, compute gradients\n",
+    "        optimizer.step()  # apply gradients to update weights"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "42e24fa0",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Setup function that evaluates a number of metrics at each training step\n",
+    "def test_model(\n",
+    "    net, criterion, metric_fun, trainloader, optimizer, text=\"validation\", print_flag=0\n",
+    "):\n",
+    "    net.eval()  # Evaluation mode (important when having dropout layers)\n",
+    "\n",
+    "    test_loss = 0\n",
+    "    test_metric = 0\n",
+    "    test_metric_2 = 0\n",
+    "\n",
+    "    with torch.no_grad():\n",
+    "        for step, (batch_x, batch_y) in enumerate(\n",
+    "            trainloader\n",
+    "        ):  # for each training step\n",
+    "            b_x = Variable(batch_x)  # Inputs\n",
+    "            b_y = Variable(batch_y)  # outputs\n",
+    "            if (\n",
+    "                len(b_x.shape) == 1\n",
+    "            ):  # If is needed to add a dummy dimension if our inputs are 1D (where each number is a different sample)\n",
+    "                prediction = torch.squeeze(\n",
+    "                    net(torch.unsqueeze(b_x, 1))\n",
+    "                )  # input x and predict based on x\n",
+    "            else:\n",
+    "                prediction = net(b_x)\n",
+    "            loss = criterion(prediction, b_y)  # Calculating loss\n",
+    "            metric = metric_fun(prediction, b_y)\n",
+    "\n",
+    "            test_loss = test_loss + loss.data.numpy()  # Keep track of the loss\n",
+    "            test_metric = test_metric + metric.data.numpy()  # Keep track of the loss\n",
+    "            test_metric_2 = (\n",
+    "                test_metric_2\n",
+    "                + torch.mean(torch.abs(torch.mean(prediction, axis=1))).data.numpy()\n",
+    "            )\n",
+    "\n",
+    "        test_loss /= len(trainloader)  # dividing by the number of batches\n",
+    "        test_metric /= len(trainloader)  # dividing by the number of batches\n",
+    "        test_metric_2 /= len(trainloader)\n",
+    "    #         print(len(trainloader))\n",
+    "    # print(test_loss)\n",
+    "\n",
+    "    if print_flag == 1:\n",
+    "        print(\n",
+    "            f\"{net.name} {text} loss: {test_loss}, metric_1: {test_metric}, metric_2: {test_metric_2}\"\n",
+    "        )\n",
+    "\n",
+    "    return test_loss, test_metric, test_metric_2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8c5c2985",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# The ML models\n",
+    "\n",
+    "lr = 0.001  # the learning rate\n",
+    "\n",
+    "# (a) Learn G_k using MSE\n",
+    "criterion_a = torch.nn.MSELoss()\n",
+    "nn_a = Net_ANN(\"Learn Gk\").double()\n",
+    "optimizer_a = optim.Adam(nn_a.parameters(), lr=lr)\n",
+    "\n",
+    "# (b) learn G_k using MSE + condition of reducing momentum forcing\n",
+    "\n",
+    "alpha = 100  # This is the regularization parameter"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c75fc425-813d-417d-b130-8b7231566849",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def custom_loss(Ypred, Ytrue):\n",
+    "    # loss = MSE + net momentum forcing penalty\n",
+    "    loss = torch.mean((Ypred - Ytrue) ** 2) + alpha * torch.mean(\n",
+    "        torch.abs(torch.mean(Ypred, axis=1))\n",
+    "    )\n",
+    "    return loss"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7e6248e2-4606-4cd6-bbe8-b668ecaa2e53",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "criterion_b = custom_loss\n",
+    "nn_b = Net_ANN(\"Learn Gk w/ constrain\").double()\n",
+    "optimizer_b = optim.Adam(nn_b.parameters(), lr=lr)\n",
+    "\n",
+    "# (c) Learn S_k using MSE\n",
+    "criterion_c = torch.nn.MSELoss()\n",
+    "nn_c = Net_ANN(\"Learn Sk\").double()\n",
+    "optimizer_c = optim.Adam(nn_c.parameters(), lr=lr)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "36f29ca4-7b20-4abd-8704-f3a9d52423cf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# (d) Learn S_k but with G_k as the objective\n",
+    "def custom_loss2(Spred, Gtrue):\n",
+    "    # Gtrue = np.roll(Strue, 1, axis=1) - np.roll(Strue, -1, axis=1)\n",
+    "    Gpred = torch.roll(Spred, 1, dims=1) - torch.roll(Spred, -1, dims=1)\n",
+    "    loss = torch.mean((Gpred - Gtrue) ** 2)\n",
+    "    return loss"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fe69ee5e-0019-4a97-9350-7df9ad38e378",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "criterion_d = custom_loss2\n",
+    "nn_d = Net_ANN(\"Learn Sk from Gk\").double()\n",
+    "optimizer_d = optim.Adam(nn_d.parameters(), lr=lr)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a4555c47",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The custom loss function: \n",
+    "$$\n",
+    "loss = (G_{pred} - G_{true})^2 + \\alpha |<G_k>|\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bfa56c74",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "In model (b) we have to pick a value for the degree of regularization $\\alpha$. Here we use a constant value, but this value itself needed to be determined. We determined it using some rough testing with the L-curve criterion (read about it more [here](https://www.sintef.no/globalassets/project/evitameeting/2005/lcurve.pdf)). "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "91e8e9f8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Do the training\n",
+    "\n",
+    "n_epochs = 100\n",
+    "\n",
+    "# Containers\n",
+    "validation_stat_a = {\n",
+    "    \"loss\": np.zeros((n_epochs)),\n",
+    "    \"metric\": np.zeros((n_epochs)),\n",
+    "    \"metric2\": np.zeros((n_epochs)),\n",
+    "}\n",
+    "train_stat_a = {\n",
+    "    \"loss\": np.zeros((n_epochs)),\n",
+    "    \"metric\": np.zeros((n_epochs)),\n",
+    "    \"metric2\": np.zeros((n_epochs)),\n",
+    "}\n",
+    "\n",
+    "validation_stat_b = {\n",
+    "    \"loss\": np.zeros((n_epochs)),\n",
+    "    \"metric\": np.zeros((n_epochs)),\n",
+    "    \"metric2\": np.zeros((n_epochs)),\n",
+    "}\n",
+    "train_stat_b = {\n",
+    "    \"loss\": np.zeros((n_epochs)),\n",
+    "    \"metric\": np.zeros((n_epochs)),\n",
+    "    \"metric2\": np.zeros((n_epochs)),\n",
+    "}\n",
+    "\n",
+    "validation_stat_c = {\n",
+    "    \"loss\": np.zeros((n_epochs)),\n",
+    "    \"metric\": np.zeros((n_epochs)),\n",
+    "    \"metric2\": np.zeros((n_epochs)),\n",
+    "}\n",
+    "train_stat_c = {\n",
+    "    \"loss\": np.zeros((n_epochs)),\n",
+    "    \"metric\": np.zeros((n_epochs)),\n",
+    "    \"metric2\": np.zeros((n_epochs)),\n",
+    "}\n",
+    "\n",
+    "validation_stat_d = {\n",
+    "    \"loss\": np.zeros((n_epochs)),\n",
+    "    \"metric\": np.zeros((n_epochs)),\n",
+    "    \"metric2\": np.zeros((n_epochs)),\n",
+    "}\n",
+    "train_stat_d = {\n",
+    "    \"loss\": np.zeros((n_epochs)),\n",
+    "    \"metric\": np.zeros((n_epochs)),\n",
+    "    \"metric2\": np.zeros((n_epochs)),\n",
+    "}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "dcc11a1b-34b6-422a-889a-1a044d7accd1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def helper_fun(nn_sel, criterion, metric, loader, loader_test, optimizer):\n",
+    "    train_model(nn_sel, criterion, loader, optimizer)\n",
+    "\n",
+    "    return test_model(\n",
+    "        nn_sel, criterion, metric, loader, optimizer, text=\"train\"\n",
+    "    ), test_model(nn_sel, criterion, metric, loader_test, optimizer, print_flag=0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "28337301-42ec-40db-a5be-b0f00543c56c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "for epoch in tqdm(range(n_epochs)):\n",
+    "    # print(\"--------------\")\n",
+    "    # print(\"Epoch num : \" + str(epoch))\n",
+    "\n",
+    "    # (a)\n",
+    "    (\n",
+    "        validation_stat_a[\"loss\"][epoch],\n",
+    "        validation_stat_a[\"metric\"][epoch],\n",
+    "        validation_stat_a[\"metric2\"][epoch],\n",
+    "    ), (\n",
+    "        train_stat_a[\"loss\"][epoch],\n",
+    "        train_stat_a[\"metric\"][epoch],\n",
+    "        train_stat_a[\"metric2\"][epoch],\n",
+    "    ) = helper_fun(\n",
+    "        nn_a, criterion_a, criterion_a, loader, loader_test, optimizer_a\n",
+    "    )\n",
+    "\n",
+    "    # (b)\n",
+    "    (\n",
+    "        validation_stat_b[\"loss\"][epoch],\n",
+    "        validation_stat_b[\"metric\"][epoch],\n",
+    "        validation_stat_b[\"metric2\"][epoch],\n",
+    "    ), (\n",
+    "        train_stat_b[\"loss\"][epoch],\n",
+    "        train_stat_b[\"metric\"][epoch],\n",
+    "        train_stat_b[\"metric2\"][epoch],\n",
+    "    ) = helper_fun(\n",
+    "        nn_b, criterion_b, criterion_a, loader, loader_test, optimizer_b\n",
+    "    )\n",
+    "\n",
+    "    # (c)\n",
+    "    (\n",
+    "        validation_stat_c[\"loss\"][epoch],\n",
+    "        validation_stat_c[\"metric\"][epoch],\n",
+    "        validation_stat_c[\"metric2\"][epoch],\n",
+    "    ), (\n",
+    "        train_stat_c[\"loss\"][epoch],\n",
+    "        train_stat_c[\"metric\"][epoch],\n",
+    "        train_stat_c[\"metric2\"][epoch],\n",
+    "    ) = helper_fun(\n",
+    "        nn_c, criterion_c, criterion_a, loader_S, loader_S_test, optimizer_c\n",
+    "    )\n",
+    "\n",
+    "    # (d)\n",
+    "    # notice that this learns Sk (what is predicted), but the MSE is relative to Gk.\n",
+    "    (\n",
+    "        validation_stat_d[\"loss\"][epoch],\n",
+    "        validation_stat_d[\"metric\"][epoch],\n",
+    "        validation_stat_d[\"metric2\"][epoch],\n",
+    "    ), (\n",
+    "        train_stat_d[\"loss\"][epoch],\n",
+    "        train_stat_d[\"metric\"][epoch],\n",
+    "        train_stat_d[\"metric2\"][epoch],\n",
+    "    ) = helper_fun(\n",
+    "        nn_d, criterion_d, criterion_d, loader, loader_test, optimizer_d\n",
+    "    )"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ba797430",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig = plt.figure(figsize=(12, 5), dpi=150)\n",
+    "\n",
+    "ax = plt.subplot(121)\n",
+    "ax.plot(range(n_epochs), train_stat_a[\"metric\"], label=nn_a.name)\n",
+    "ax.plot(range(n_epochs), train_stat_b[\"metric\"], label=nn_b.name)\n",
+    "ax.plot(range(n_epochs), train_stat_c[\"metric\"], label=nn_c.name)\n",
+    "ax.plot(range(n_epochs), train_stat_d[\"metric\"], label=nn_d.name)\n",
+    "\n",
+    "plt.yscale(\"log\")\n",
+    "# ax.set_xlim([0, 20])\n",
+    "ax.set_ylabel(\"MSE in $G$ or $S$\")\n",
+    "ax.set_xlabel(\"Epoch number\")\n",
+    "ax.set_title(\"Training\")\n",
+    "\n",
+    "\n",
+    "ax = plt.subplot(122)\n",
+    "ax.plot(range(n_epochs), validation_stat_a[\"metric\"], label=nn_a.name)\n",
+    "ax.plot(range(n_epochs), validation_stat_b[\"metric\"], label=nn_b.name)\n",
+    "ax.plot(range(n_epochs), validation_stat_c[\"metric\"], label=nn_c.name)\n",
+    "ax.plot(range(n_epochs), validation_stat_d[\"metric\"], label=nn_d.name)\n",
+    "plt.yscale(\"log\")\n",
+    "ax.legend(fontsize=7)\n",
+    "ax.set_xlabel(\"Epoch number\")\n",
+    "# ax.set_xlim([0, 20])\n",
+    "ax.set_title(\"Validation\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0e3fea6f",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Looking at the plots above we see that the training went well for all the cases, as the loss function is reduced. Note tht we should not compare the MSE of the cases where G_k is learnt with the cases where S_k is learnt, since the learning objective was different. \n",
+    "\n",
+    "We also see that adding a constrain to minimize net momentum input when learning G_k, made the MSE for the model a bit higher, as it is trying to reduce two things at once. \n",
+    "\n",
+    "It is interesting to note that even the model without the constraint is trying to reduce the momentum input over training epochs, which shows that it is something that the learning is trying to learn without help. (there might be situations with limited data where this help is important, but not super helpful for L96)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "987ef53e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# How do the models that directly learn Gk do at mom cons of that term.\n",
+    "fig = plt.figure(dpi=150)\n",
+    "\n",
+    "ax = plt.subplot(111)\n",
+    "ax.plot(range(n_epochs), train_stat_a[\"metric2\"], label=nn_a.name + \" Train\")\n",
+    "ax.plot(range(n_epochs), train_stat_b[\"metric2\"], label=nn_b.name + \" Train\")\n",
+    "\n",
+    "ax.plot(range(n_epochs), validation_stat_a[\"metric2\"], \"--\", label=nn_a.name + \" Test\")\n",
+    "ax.plot(range(n_epochs), validation_stat_b[\"metric2\"], \"--\", label=nn_b.name + \" Test\")\n",
+    "ax.legend(fontsize=7)\n",
+    "plt.yscale(\"log\")\n",
+    "# ax.set_xlim([0, 20])\n",
+    "ax.set_ylabel(\"$<G_k>$\")\n",
+    "ax.set_xlabel(\"Epoch number\")\n",
+    "ax.set_title(\"Training\");"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9cf6224f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Look at predictions\n",
+    "nn_a.eval()\n",
+    "nn_b.eval()\n",
+    "nn_c.eval()\n",
+    "nn_d.eval()\n",
+    "\n",
+    "pred_G_a = nn_a(torch.from_numpy(X_true_test)).detach().numpy()\n",
+    "pred_G_b = nn_b(torch.from_numpy(X_true_test)).detach().numpy()\n",
+    "pred_S_c = nn_c(torch.from_numpy(X_true_test)).detach().numpy()\n",
+    "pred_G_c = np.roll(pred_S_c, 1, axis=1) - np.roll(\n",
+    "    pred_S_c, -1, axis=1\n",
+    ")  # Since the last model predicts the S, we need to calculate G.\n",
+    "pred_S_d = nn_d(torch.from_numpy(X_true_test)).detach().numpy()\n",
+    "pred_G_d = np.roll(pred_S_d, 1, axis=1) - np.roll(\n",
+    "    pred_S_d, -1, axis=1\n",
+    ")  # Since the last model predicts the S, we need to calculate G."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "07c4f1e0",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n = 500\n",
+    "k = 2\n",
+    "\n",
+    "T_test = T_true[-val_size:]\n",
+    "\n",
+    "plt.figure(figsize=(10, 4), dpi=150)\n",
+    "\n",
+    "plt.subplot(121)\n",
+    "plt.plot(T_test[:n], subgrid_G_tend_test[:n, k], label=\"True\")\n",
+    "plt.plot(T_test[:n], pred_G_a[:n, k], label=nn_a.name)\n",
+    "plt.plot(T_test[:n], pred_G_b[:n, k], label=nn_b.name)\n",
+    "plt.plot(T_test[:n], pred_G_c[:n, k], label=nn_b.name)\n",
+    "plt.plot(T_test[:n], pred_G_d[:n, k], label=nn_b.name)\n",
+    "plt.xlabel(\"Time\")\n",
+    "plt.ylabel(\"$G_k$\")\n",
+    "\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "plt.subplot(122)\n",
+    "# plt.plot(Xtrue_test[:,k],  subgrid_tend_test[:,k], '.', markersize=1)\n",
+    "plt.plot(\n",
+    "    subgrid_G_tend_test[:, k], pred_G_a[:, k], \".\", markersize=2.5, label=nn_a.name\n",
+    ")\n",
+    "plt.plot(\n",
+    "    subgrid_G_tend_test[:, k], pred_G_b[:, k], \".\", markersize=2.5, label=nn_b.name\n",
+    ")\n",
+    "plt.plot(\n",
+    "    subgrid_G_tend_test[:, k], pred_G_c[:, k], \".\", markersize=2.5, label=nn_c.name\n",
+    ")\n",
+    "plt.plot(\n",
+    "    subgrid_G_tend_test[:, k], pred_G_d[:, k], \".\", markersize=2.5, label=nn_c.name\n",
+    ")\n",
+    "\n",
+    "plt.xlabel(\"True G\")\n",
+    "plt.ylabel(\"Predicted G\")\n",
+    "plt.legend(fontsize=7)\n",
+    "# plt.plot(Xtrue_test[:,k],  pred_2.detach().numpy()[:,k], '.', markersize=1.5)\n",
+    "# plt.plot(Xtrue_test[:,k],  pred_3.detach().numpy()[:,k], '.', markersize=1.5)\n",
+    "plt.grid()\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "47ee238e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "MSE_a = np.mean((subgrid_G_tend_test - pred_G_a) ** 2)\n",
+    "MSE_b = np.mean((subgrid_G_tend_test - pred_G_b) ** 2)\n",
+    "MSE_c = np.mean((subgrid_G_tend_test - pred_G_c) ** 2)\n",
+    "MSE_d = np.mean((subgrid_G_tend_test - pred_G_d) ** 2)\n",
+    "\n",
+    "MSE_S_c = np.mean((subgrid_S_tend_test - pred_S_c) ** 2)\n",
+    "MSE_S_d = np.mean((subgrid_S_tend_test - pred_S_d) ** 2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "71ce4b88",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "nn_names = [nn_a.name, nn_b.name, nn_c.name, nn_d.name]\n",
+    "MSEs = [MSE_a, MSE_b, MSE_c, MSE_d]\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.bar(nn_names, MSEs)\n",
+    "plt.setp(plt.gca().get_xticklabels(), rotation=45, horizontalalignment=\"right\")\n",
+    "plt.ylabel(\"MSE in $G_k$\");\n",
+    "# plt.xlabel('Model')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7b7c9696",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "nn_names = [nn_c.name, nn_d.name]\n",
+    "MSEs = [MSE_S_c, MSE_S_d]\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.bar(nn_names, MSEs)\n",
+    "plt.setp(plt.gca().get_xticklabels(), rotation=45, horizontalalignment=\"right\")\n",
+    "plt.ylabel(\"MSE in $S_k$\");\n",
+    "# plt.xlabel('Model')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cc58bc76",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(10, 4), dpi=150)\n",
+    "\n",
+    "plt.subplot(121)\n",
+    "plt.plot(T_test, np.mean(subgrid_G_tend_test, axis=1), \"--\", label=\"True\")\n",
+    "plt.plot(T_test, np.mean(pred_G_a, axis=1), label=nn_a.name)\n",
+    "plt.plot(T_test, np.mean(pred_G_b, axis=1), label=nn_b.name)\n",
+    "plt.plot(T_test, np.mean(pred_G_c, axis=1), \"-.\", label=nn_c.name)\n",
+    "plt.plot(T_test, np.mean(pred_G_d, axis=1), \"-.\", label=nn_c.name)\n",
+    "plt.xlim([160, 175])\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.ylabel(\"$<G_k>$\")\n",
+    "plt.xlabel(\"Time step\")\n",
+    "\n",
+    "plt.subplot(122)\n",
+    "plt.hist(0)\n",
+    "plt.hist(np.mean(pred_G_a, axis=1), density=True, label=nn_a.name)\n",
+    "plt.hist(np.mean(pred_G_b, axis=1), density=True, label=nn_b.name)\n",
+    "\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "plt.xlabel(\"$<G_k>$\")\n",
+    "plt.ylabel(\"PDF\")\n",
+    "plt.xlim([-0.2, 0.2])\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6ad595b3",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "- The models do quite a good job (visually) at capturing the patterns in the subgrid forcing. \n",
+    "- The model with the constrain of momentum conservation does a better job of it (pdf width is much smaller), BUT some bias is introduced (median is shifted from 0). \n",
+    "- From our experimenting this bias is not always the same, and can vary if trainined multiple times. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4842bdb0",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(10, 4), dpi=150)\n",
+    "\n",
+    "plt.subplot(121)\n",
+    "plt.plot(T_test, np.mean(subgrid_S_tend_test, axis=1), \"--\", label=\"True\")\n",
+    "plt.plot(T_test, np.mean(pred_S_c, axis=1), \"-.\", label=nn_c.name)\n",
+    "plt.plot(T_test, np.mean(pred_S_d, axis=1), \"-.\", label=nn_d.name)\n",
+    "plt.xlim([160, 175])\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.ylabel(\"$<S_k>$\")\n",
+    "plt.xlabel(\"Time step\")\n",
+    "\n",
+    "plt.subplot(122)\n",
+    "plt.plot(0, 0)\n",
+    "plt.hist(np.mean(pred_S_c, axis=1), density=True, label=nn_c.name)\n",
+    "plt.hist(np.mean(pred_S_d, axis=1), density=True, label=nn_d.name)\n",
+    "\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "plt.xlabel(\"$<S_k>$\")\n",
+    "plt.ylabel(\"PDF\")\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a4c44fc3",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "**When the model learns Sk from Gk, without knowing anything about Gk. It produces the right Gk, but a Sk that looks very different from the real Sk.**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "63aee82d",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Skill of different ML models in prediction"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "596d1e8e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# - a GCM class including a neural network parameterization in rhs of equation for tendency\n",
+    "\n",
+    "time_method = RK4\n",
+    "\n",
+    "# - a GCM class without any parameterization\n",
+    "\n",
+    "\n",
+    "class GCM_no_param:\n",
+    "    def __init__(self, F, time_stepping=time_method):\n",
+    "        self.F = F\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, param):\n",
+    "        return L96_eq1_xdot(X, self.F)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        # X0 - initial conditions, dt - time increment, nt - number of forward steps to take\n",
+    "        # param - parameters of our closure\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "39d140cb-4602-44b3-91e0-06fc15b90866",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class GCM_network:\n",
+    "    def __init__(self, F, network, time_stepping=time_method):\n",
+    "        self.F = F\n",
+    "        self.network = network\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, param):\n",
+    "        if self.network.linear1.in_features == 1:\n",
+    "            X_torch = torch.from_numpy(X).double()\n",
+    "            X_torch = torch.unsqueeze(X_torch, 1)\n",
+    "        else:\n",
+    "            X_torch = torch.from_numpy(np.expand_dims(X, 0)).double()\n",
+    "\n",
+    "        return L96_eq1_xdot(X, self.F) + np.squeeze(\n",
+    "            self.network(X_torch).data.numpy()\n",
+    "        )  # Adding NN parameterization\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        # X0 - initial conditions, dt - time increment, nt - number of forward steps to take\n",
+    "        # param - parameters of our closure\n",
+    "        time, histX, histB, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        histX[0] = X\n",
+    "        histB[0] = 0.0\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            # this next if statement is being called twice\n",
+    "            if self.network.linear1.in_features == 1:\n",
+    "                X_torch = torch.from_numpy(X).double()\n",
+    "                X_torch = torch.unsqueeze(X_torch, 1)\n",
+    "            else:\n",
+    "                X_torch = torch.from_numpy(np.expand_dims(X, 0)).double()\n",
+    "\n",
+    "            histB[n + 1] = np.squeeze(self.network(X_torch).data.numpy())\n",
+    "\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            histX[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return histX, time, histB"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8a7ebac1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class GCM_network_S:\n",
+    "    def __init__(self, F, network, time_stepping=time_method):\n",
+    "        self.F = F\n",
+    "        self.network = network\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, param):\n",
+    "        if self.network.linear1.in_features == 1:\n",
+    "            X_torch = torch.from_numpy(X).double()\n",
+    "            X_torch = torch.unsqueeze(X_torch, 1)\n",
+    "        else:\n",
+    "            X_torch = torch.from_numpy(np.expand_dims(X, 0)).double()\n",
+    "\n",
+    "            predS = self.network(X_torch).data.numpy()\n",
+    "            predG = np.roll(predS, 1, axis=1) - np.roll(predS, -1, axis=1)\n",
+    "        # return L96_eq1_xdot(X, self.F) + np.squeeze(self.network(X_torch).data.numpy()) # Adding NN parameterization\n",
+    "        return L96_eq1_xdot(X, self.F) + np.squeeze(predG)  # Adding NN parameterization\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        # X0 - initial conditions, dt - time increment, nt - number of forward steps to take\n",
+    "        # param - parameters of our closure\n",
+    "        time, histX, histB, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        histX[0] = X\n",
+    "        histB[0] = 0.0\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            # this next if statement is being called twice\n",
+    "            if self.network.linear1.in_features == 1:\n",
+    "                X_torch = torch.from_numpy(X).double()\n",
+    "                X_torch = torch.unsqueeze(X_torch, 1)\n",
+    "            else:\n",
+    "                X_torch = torch.from_numpy(np.expand_dims(X, 0)).double()\n",
+    "\n",
+    "            predS = self.network(X_torch).data.numpy()\n",
+    "            predG = np.roll(predS, 1, axis=1) - np.roll(predS, -1, axis=1)\n",
+    "            histB[n + 1] = predG  # np.squeeze(self.network(X_torch).data.numpy())\n",
+    "\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            histX[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return histX, time, histB"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "59080546",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "T_test = 20\n",
+    "\n",
+    "X_full, _, t, S_full = W.run(dt, T_test)  # Full model\n",
+    "G_full = np.roll(S_full, 1, axis=1) - np.roll(S_full, -1, axis=1)\n",
+    "init_cond = X_true[-1, :]\n",
+    "\n",
+    "gcm_a = GCM_network(Forcing, nn_a)\n",
+    "X_a, t, G_a = gcm_a(init_cond, dt, int(T_test / dt))\n",
+    "\n",
+    "gcm_b = GCM_network(Forcing, nn_b)\n",
+    "X_b, t, G_b = gcm_b(init_cond, dt, int(T_test / dt))\n",
+    "\n",
+    "gcm_c = GCM_network_S(Forcing, nn_c)\n",
+    "X_c, t, G_c = gcm_c(init_cond, dt, int(T_test / dt))\n",
+    "\n",
+    "gcm_d = GCM_network_S(Forcing, nn_d)\n",
+    "X_d, t, G_d = gcm_d(init_cond, dt, int(T_test / dt))\n",
+    "\n",
+    "\n",
+    "gcm_no_param = GCM_no_param(Forcing)\n",
+    "X_no_param, t = gcm_no_param(init_cond, dt, int(T_test / dt))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "868ac884",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# convert to xarray for convenience\n",
+    "X_full_xr = xr.DataArray(\n",
+    "    X_full, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"X_true\")\n",
+    "G_full_xr = xr.DataArray(\n",
+    "    G_full, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"G_true\")\n",
+    "\n",
+    "X_a_xr = xr.DataArray(\n",
+    "    X_a, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"X_a\")\n",
+    "G_a_xr = xr.DataArray(\n",
+    "    G_a, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"G_a\")\n",
+    "\n",
+    "X_b_xr = xr.DataArray(\n",
+    "    X_b, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"X_b\")\n",
+    "G_b_xr = xr.DataArray(\n",
+    "    G_b, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"G_b\")\n",
+    "\n",
+    "X_c_xr = xr.DataArray(\n",
+    "    X_c, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"X_c\")\n",
+    "G_c_xr = xr.DataArray(\n",
+    "    G_c, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"G_c\")\n",
+    "\n",
+    "X_d_xr = xr.DataArray(\n",
+    "    X_d, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"X_d\")\n",
+    "G_d_xr = xr.DataArray(\n",
+    "    G_d, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"G_d\")\n",
+    "\n",
+    "X_noparam_xr = xr.DataArray(\n",
+    "    X_no_param, dims=[\"time\", \"K\"], coords={\"time\": t, \"K\": np.arange(K)}\n",
+    ").rename(\"X_noparam\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "46274421",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(16, 10), dpi=150)\n",
+    "\n",
+    "plt.subplot(161)\n",
+    "X_full_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-15, 15, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Full model\")\n",
+    "\n",
+    "plt.subplot(162)\n",
+    "X_a_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-15, 15, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Learn Gk\")\n",
+    "\n",
+    "plt.subplot(163)\n",
+    "X_b_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-15, 15, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Lrn Gk w/ con\")\n",
+    "\n",
+    "plt.subplot(164)\n",
+    "X_c_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-15, 15, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Lrn Sk\")\n",
+    "\n",
+    "plt.subplot(165)\n",
+    "X_d_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-15, 15, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Lrn Sk f/ Gk\")\n",
+    "\n",
+    "plt.subplot(166)\n",
+    "X_noparam_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-15, 15, 11),\n",
+    "    extend=\"both\",\n",
+    "    cbar_kwargs={\"aspect\": 20, \"shrink\": 0.2},\n",
+    ")\n",
+    "plt.title(\"No param\")\n",
+    "\n",
+    "plt.suptitle(\"X\")\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5e6649bc",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(16, 10), dpi=150)\n",
+    "\n",
+    "plt.subplot(161)\n",
+    "G_full_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-20, 20, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Full model\")\n",
+    "\n",
+    "plt.subplot(162)\n",
+    "G_a_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-20, 20, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Learn Gk\")\n",
+    "\n",
+    "plt.subplot(163)\n",
+    "G_b_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-20, 20, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Lrn Gk w/ con\")\n",
+    "\n",
+    "plt.subplot(164)\n",
+    "G_c_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-20, 20, 11), extend=\"both\", add_colorbar=False\n",
+    ")\n",
+    "plt.title(\"Lrn Sk\")\n",
+    "\n",
+    "plt.subplot(165)\n",
+    "G_d_xr.sel(time=slice(0, 15)).plot.contourf(\n",
+    "    levels=np.linspace(-20, 20, 11),\n",
+    "    extend=\"both\",\n",
+    "    add_colorbar=True,\n",
+    "    cbar_kwargs={\"aspect\": 20, \"shrink\": 0.2},\n",
+    ")\n",
+    "plt.title(\"Lrn Sk\")\n",
+    "\n",
+    "plt.subplot(166)\n",
+    "# X_noparam_xr.sel(time=slice(0, 15)).plot.contourf(levels=np.linspace(-15, 15, 9), extend='both', cbar_kwargs={'aspect':5})\n",
+    "plt.title(\"No param\")\n",
+    "plt.suptitle(\"G\")\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fc7d3fd8-b460-4c9c-a064-2a6c7dccb973",
+   "metadata": {},
+   "source": [
+    "Given the simplicity of the parameterization problem in L96 model, all the different ML parameterizations performed relatively well. We expect to see much larger gains by adding constraints in more complex realistic problems, particularly the ones that are more data limited. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e0ee7ea5",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "## Summary \n",
+    "\n",
+    "In this notebook we learnt how to add constraints to the learning process, and studied the impacts that this has on the trained models. \n",
+    "\n",
+    "With this notebook we wrap up the section on using neural networks to parameterize missing effects in the L96 model. In the next section we will be learning a different approach to improving models - data assimilation."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/estimating-gcm-parameters.ipynb b/_sources/notebooks/estimating-gcm-parameters.ipynb
new file mode 100644
index 00000000..a6933f30
--- /dev/null
+++ b/_sources/notebooks/estimating-gcm-parameters.ipynb
@@ -0,0 +1,577 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "33938dd5-f864-4642-8ce0-c44bb81bd60d",
+   "metadata": {},
+   "source": [
+    "# Tuning GCM Parameterizations"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "686988e8",
+   "metadata": {},
+   "source": [
+    "The objective of this notebook is to show how GCM closures can be tuned in practice. \n",
+    "We will assume a specific formulation of a closure and estimate its parameters through a standard optimisation procedure using a procedure similar to Data Assimilation (DA). "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8cb1f6aa",
+   "metadata": {},
+   "source": [
+    "**Resources** : We have used material from Emmanuel Cosme's nice GitHub [repository](https://github.com/ecosme38/Data-Assimilation-Notebooks). "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "daa836f4",
+   "metadata": {},
+   "source": [
+    "## The GCM Parameterization Problem "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "33a03d0b",
+   "metadata": {},
+   "source": [
+    "Here we quickly reintroduce the problem we are trying to solve, our starting point here is {doc}`gcm-parameterization-problem`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "eb6ac7ba",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import scipy.optimize as opt\n",
+    "\n",
+    "# L96 provides the \"real world\", L96_eq1_xdot is the beginning of rhs of X tendency\n",
+    "from L96_model import L96, RK2, RK4, EulerFwd, L96_eq1_xdot"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "515d089a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Setting the seed gives us reproducible results\n",
+    "np.random.seed(13)\n",
+    "\n",
+    "# Create a \"real world\" with K=8 and J=32\n",
+    "W = L96(8, 32, F=18)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4c697b25",
+   "metadata": {},
+   "source": [
+    "Run \"real world\" for 3 days to forget initial conditons."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d6d7e234",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# (store=True save the final state as an initial condition for the next run)\n",
+    "W.run(0.05, 3.0, store=True);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "777fef0d",
+   "metadata": {},
+   "source": [
+    "From here on we can use `W.X` as perfect initial conditions for a model and sample the real world using `W.run(dt,T)`"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9647aa43-90ce-45f0-9756-02ed10780e2f",
+   "metadata": {},
+   "source": [
+    "The class defined below is just a sophisticated version of the single time-scale L96 gcm analogue that was defined in 'The Lorenz-96 GCM Analog' notebook."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5b97a051",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class GCM:\n",
+    "    def __init__(self, F, parameterization, time_stepping=EulerFwd):\n",
+    "        \"\"\"\n",
+    "        GCM with paramerization\n",
+    "        Args:\n",
+    "            F: forcing\n",
+    "            parameterization: function that takes parameters and returns a tendency\n",
+    "            time_stepping: time stepping method\n",
+    "        \"\"\"\n",
+    "        self.F = F\n",
+    "        self.parameterization = parameterization\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, param):\n",
+    "        \"\"\"\n",
+    "        Args:\n",
+    "            X: state vector\n",
+    "            param: parameters of our closure\n",
+    "        \"\"\"\n",
+    "        return L96_eq1_xdot(X, self.F) - self.parameterization(param, X)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        \"\"\"\n",
+    "        Args:\n",
+    "            X0: initial conditions\n",
+    "            dt: time increment\n",
+    "            nt: number of forward steps to take\n",
+    "            param: parameters of our closure\n",
+    "        \"\"\"\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "66a2385b",
+   "metadata": {},
+   "source": [
+    "We illustrate the concepts here with the help of the simple linear regression parameterization, which was also introduced in the {doc}`gcm-analogue`. Remember that this was not a very good parameterization, but it is used to show how imperfect parameterizations can be tuned in different ways. Here we use parameters that were estimated by fitting the linear regression to the true sub-grid tendencies. Such parameter estimates are sometimes also referred to as \"offline fitting\" estimates."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1ec03cc6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "naive_parameterization = lambda param, X: np.polyval(param, X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "18baf519",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "F, dt, T = 18, 0.005, 5.0\n",
+    "gcm = GCM(F, naive_parameterization)\n",
+    "# We pass the parameters that were obtained by fitting the linear in 'The Lorenz-96 GCM Analog' notebook.\n",
+    "X, t = gcm(W.X, dt, int(T / dt), param=[0.85439536, 0.75218026])\n",
+    "# We also simulate a case with no parameterization, to show that even the imperfect parameterization is helpful.\n",
+    "X_no_param, t = gcm(W.X, dt, int(T / dt), param=[0.0, 0.0])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "235aed7d",
+   "metadata": {},
+   "source": [
+    "Now we compare the model with the parameterization and trajectories from the \"real world\" truth."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1de79307",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# This samples the real world with the same time interval as \"dt\" used by the model\n",
+    "X_true, _, _ = W.run(dt, T)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5d260623",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 4], label=\"Truth\")\n",
+    "plt.plot(t, X[:, 4], label=\"Model with linear param\")\n",
+    "plt.plot(t, X_no_param[:, 4], label=\"Model with no param\")\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_4(t)$\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2c928cf0-72d6-4cc1-9897-fdb141a9f2f9",
+   "metadata": {},
+   "source": [
+    "The figure above show that the model with linear parameterization outperforms the model with no parameterization. Next we discuss if these parameter estimates can be made even better. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3200004c",
+   "metadata": {},
+   "source": [
+    "## Variational estimation of optimal parameters for a predefined closure\n",
+    "\n",
+    "We will try here to estimate the parameters of `naive_parameterization` with a variational approach. This is different from the way the parameters used above were estimated, which had used an offline fitting approach. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9b5141f2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# - assuming the formulation of the parameterization\n",
+    "gcm = GCM(F, naive_parameterization)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "96b14903",
+   "metadata": {},
+   "source": [
+    "### Estimating parameters based on one initial condition and one time step"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a57669f4",
+   "metadata": {},
+   "source": [
+    "#### Cost function "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "91466a50",
+   "metadata": {},
+   "source": [
+    "What we will be doing here is very close to what is done with classical variational data assimilation, where we try to estimate the state of the parameters of a model through the minimization of a cost function $J$. This is also very close to what is done when parameterizations are encoded as neural networks. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "53f9ab28",
+   "metadata": {},
+   "source": [
+    "We introduce a cost function $J(p)$ which depends on the parameters of the closure. \n",
+    "\n",
+    "$$J(p) = ||X_p - X_{true}||_{d}$$\n",
+    "\n",
+    "where $p=[p1,p2]$, $X_p$ is GCM solution computed with with parameters $p$ and $||\\cdot ||_{d}$ is one of the distances above. \n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0f356baf-71c3-4f70-91c6-cd7da6415191",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def norm_initial_tendency(X1, X2):\n",
+    "    T1 = X1[1, :] - X1[0, :]\n",
+    "    T2 = X2[1, :] - X2[0, :]\n",
+    "    return np.sqrt((T1 - T2) ** 2).mean(axis=0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ce939e3c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def cost_function(param):\n",
+    "    F, dt, T = 18, 0.01, 0.01\n",
+    "    X_gcm, t = gcm(W.X, dt, int(T / dt), param=param)\n",
+    "    return norm_initial_tendency(X_true, X_gcm)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c6198df7",
+   "metadata": {},
+   "source": [
+    "#### Minimization "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "35d38b2b",
+   "metadata": {},
+   "source": [
+    "Since the problem dimension is only two ( $p=[p1,p2]$ ), we can use efficient derivative-free optimization methods."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4b23971d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "prior = np.array([0.85439536, 0.75218026])  #  prior\n",
+    "res = opt.minimize(cost_function, prior, method=\"Powell\")\n",
+    "opt_param = res[\"x\"]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4bdffeae",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(f\"Optimized parameters = {opt_param}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e6b43c9e",
+   "metadata": {},
+   "source": [
+    "#### Let's test the closure."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6a8c65cd",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "F, dt, T = 18, 0.01, 100.0\n",
+    "gcm = GCM(F, naive_parameterization)\n",
+    "X_optimized, t = gcm(W.X, dt, int(T / dt), param=opt_param)\n",
+    "X_prior, t = gcm(W.X, dt, int(T / dt), param=prior)\n",
+    "\n",
+    "# - ... the true state\n",
+    "X_true, _, _ = W.run(dt, T)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9172f27e",
+   "metadata": {},
+   "source": [
+    "#### Results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "328e5851",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t[:500], X_true[:500, 4], label=\"Truth\")\n",
+    "plt.plot(t[:500], X_prior[:500, 4], label=\"Initial GCM\")\n",
+    "plt.plot(t[:500], X_optimized[:500, 4], label=\"Optimized GCM\")\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_4(t)$\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fcc951db",
+   "metadata": {},
+   "source": [
+    "The results are better but not great. This problem is related to the question of *a priori* versus *a posteriori* skill in Large Eddy Simulation (LES) closures. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "de43a461",
+   "metadata": {},
+   "source": [
+    "### Estimating parameters which optimize longer trajectories"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f5bb63b3",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "F, dt, T = 18, 0.01, 5.0\n",
+    "gcm = GCM(F, naive_parameterization)\n",
+    "X_true, _, _ = W.run(dt, T)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "dfa2d4aa",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_gcm, t = gcm(W.X, dt, int(T / dt), param=[0, 0])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1722b186-8ba6-4ff0-b369-18320b9381a5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# - pointwise distance :\n",
+    "def pointwise(X1, X2, L=1.0):\n",
+    "    # computed over some window t<L.\n",
+    "    D = (X1 - X2)[np.where(t < L)]\n",
+    "    return np.sqrt(D**2).mean(axis=0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5a48582d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def cost_function(param):\n",
+    "    F, dt, T = 18, 0.01, 5\n",
+    "    X_gcm, t = gcm(W.X, dt, int(T / dt), param=param)\n",
+    "    return pointwise(X_true, X_gcm, L=5.0).sum()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9b152335",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "prior = np.array([0.85439536, 0.75218026])  #  prior\n",
+    "res = opt.minimize(cost_function, prior, method=\"Powell\")\n",
+    "opt_param = res[\"x\"]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c8c7eb69-fcc1-411e-bbb5-480bd7923b00",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(f\"Optimized parameters = {opt_param}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c5613d4a",
+   "metadata": {},
+   "source": [
+    "#### Let's test the closure."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "56a96080",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "F, dt, T = 18, 0.01, 100.0\n",
+    "gcm = GCM(F, naive_parameterization)\n",
+    "X_optimized, t = gcm(W.X, dt, int(T / dt), param=opt_param)\n",
+    "X_prior, t = gcm(W.X, dt, int(T / dt), param=prior)\n",
+    "X_true, _, _ = W.run(dt, T)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a44d0463",
+   "metadata": {},
+   "source": [
+    "#### Results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "72ad455c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t[:500], X_true[:500, 4], label=\"Truth\")\n",
+    "plt.plot(t[:500], X_prior[:500, 4], label=\"Initial GCM\")\n",
+    "plt.plot(t[:500], X_optimized[:500, 4], label=\"Optimized GCM\")\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_4(t)$\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "47764aff",
+   "metadata": {},
+   "source": [
+    "The closure produces better results but it is not clear how this would generalize to unseen initial conditions. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f814ad48",
+   "metadata": {},
+   "source": [
+    "## Discussion and possible next steps:\n",
+    "\n",
+    " - Estimating parameters over one time-step but with an ensemble of initial conditions. \n",
+    " - Use neural networks to encode closures.\n",
+    " - Explore the feasability of differentiable GCMs. "
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/gcm-analogue.ipynb b/_sources/notebooks/gcm-analogue.ipynb
new file mode 100644
index 00000000..8df3312b
--- /dev/null
+++ b/_sources/notebooks/gcm-analogue.ipynb
@@ -0,0 +1,523 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "08d71bbc",
+   "metadata": {},
+   "source": [
+    "# The Lorenz-96 and its GCM Analog\n",
+    "\n",
+    "The physical equations of motion of the system describing the atmosphere dynamics are known. Also in the real atmosphere there is variability over all scales, from the smallest molecular scales to the largest planetary scales.\n",
+    "Thus, it is not feasible to explicitly solve the known physical equations of motion within Global Climate Models (GCMs), as computational resources limit the range of scales that can be resolved. To make the behavior of the GCM better match the real world, we need to effectively parameterize the effects that the unresolved scales have on the resolved (large) scales.\n",
+    "\n",
+    "The two time-scale L96 model, described in the the [previous notebook](https://m2lines.github.io/L96_demo/notebooks/L96-two-scale-description.html#), can be considered as a simplified analog for real world atmosphere, where variability resides at only two discrete scales (a simplification of the continuum in the real world). In this analogy, a GCM would be a model that can only solve the equations for the slow time scale, where the effects of the fast time-scale variables would be missing. We introduce the single time-scale version of the L96 model below. To make the single time scale model match the two time scale model, the effects of the fast time scales need to parameterized into the single time scale model."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "15939db9-43b8-4d20-8a31-84bdbc80fd2d",
+   "metadata": {},
+   "source": [
+    "## The two time-scale model: Analog for the real atmosphere\n",
+    "\n",
+    "We will first describe a simulation with the two time-scale model from the {doc}`L96-two-scale-description`, which is taken as the control simulation that we hope to replicate with the single time-scale model. \n",
+    "\n",
+    "The forcing and resolution parameters, $F$, $J$ and $K$, for the two time-scale model are fixed based on {cite}`Wilks2005`, as $F=18$ or $20$, $K=8$, and $J=32$. Here, the value chosen for the parameter $F$ is set large enough to ensure chaotic behavior. We also use the reference values for the $h$, $b$ and $c$ parameters to be, $h=1$, $b=10$, and $c=10$.\n",
+    "With this particular choice of parameter values, one model time unit (MTU) is approximately equivalent to five atmospheric days. This estimate is obtained by comparing error-doubling times in the Lorenz-96 model and the real atmosphere {cite}`Lorenz1995`.\n",
+    "\n",
+    "In the following code, we solve the L96 system using an accurate time-stepping scheme (RK4 with sufficiently small $\\Delta t$), and plot the time evolution for the $X_k$ and $Y_{j,k}$ variables."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7f0d7a3f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from L96_model import L96"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1d6f4844",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "np.random.seed(23)\n",
+    "W = L96(8, 32)\n",
+    "T = 5.0\n",
+    "%time X_true,Y_true,t = W.run(0.05, T)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fd78d2c3-b1f6-41d9-bc9c-a8da4898b9b0",
+   "metadata": {},
+   "source": [
+    "Here is the what the solution looks like:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1a01ce7b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(5, 4), dpi=150)\n",
+    "\n",
+    "plt.subplot(121)\n",
+    "plt.contourf(W.k, t, X_true)\n",
+    "plt.colorbar()\n",
+    "plt.xlabel(\"k\")\n",
+    "plt.ylabel(\"t\")\n",
+    "plt.title(\"$X_k(t)$\")\n",
+    "\n",
+    "plt.subplot(122)\n",
+    "plt.contourf(W.j / W.J, t, Y_true, levels=np.linspace(-1, 1, 10))\n",
+    "plt.xlabel(\"k+j/J\")\n",
+    "plt.ylabel(\"t\")\n",
+    "plt.title(\"$Y_{j,k}(t)$\")\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f63cb015",
+   "metadata": {},
+   "source": [
+    "## The single time-scale model: Analog for a general circulation model (GCM)\n",
+    "\n",
+    "The two time-scale model discussed above solves a set of equations for the slow and fast variables, where the equations for the slow variables are:\n",
+    "\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\underbrace{ \n",
+    "\\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}\n",
+    "}_{=U_k} \\ .\n",
+    "\\end{align}\n",
+    "\n",
+    "Here the effects of the fast scales on the slow scales are represented by the last term on the RHS, denoted as $U_k$, $k=1,\\ldots,k$. \n",
+    "\n",
+    "In the single time-scale model the explicit equations for the fast scales are not known, and so we do not know what $Y_{j,k}$ or subsequently $U_k$ are. So, in the single time-scale model the effects of the fast scales are either missing, $U_k=0$, or they need to be specified/parameterized purely in terms of the known slow time-scale variables, $U_k \\approx P (X_k)$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2851e9da-c8f9-4411-9db5-241cee04bc79",
+   "metadata": {},
+   "source": [
+    "In the following code, we show how the single time scale model can be solved. We use `L96_eq1_xdot` (code shown below), which returns the tendency (RHS) corresponding to the following equation, where there are no coupling or parameterization terms, \n",
+    "\n",
+    "\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F.\n",
+    "\\end{align}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c9535c3a-4a5f-4268-a285-cc3ae2c02d4e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from L96_model import L96_eq1_xdot\n",
+    "from utils import display_source"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "60f0a64e-1cf3-4ac9-90ac-8b3f4745ab36",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "display_source(L96_eq1_xdot)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2f5deb49-44b3-41ad-97cd-29a099d21169",
+   "metadata": {},
+   "source": [
+    "Now we define `GCM`, which solves for the temporal evolution of $X$s using a simple Euler integration."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "61a2e64f-dc90-43a4-8488-eac7f7c6c995",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#  We define the GCM which solves for X in time and returns its time series\n",
+    "def GCM(X0, F, dt, nt, param=[0]):\n",
+    "    time, hist, X = dt * np.arange(nt), np.zeros((nt, len(X0))) * np.nan, X0.copy()\n",
+    "\n",
+    "    for n in range(nt):\n",
+    "        X = X + dt * (L96_eq1_xdot(X, F) - np.polyval(param, X))\n",
+    "        if np.abs(X).max() > 1e3:\n",
+    "            break\n",
+    "        hist[n], time[n] = X, dt * (n + 1)\n",
+    "    return hist, time"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7304546c-cd9e-4034-948e-4fa17f7e376e",
+   "metadata": {},
+   "source": [
+    "Notice that we have added the possibility of adding a parameterization that may take the form a polynomial function, this will be discussed futher below."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1317602e-a3b1-4250-b837-a68a67a3746e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_init, dt, F_mod = X_true[0], 0.002, W.F\n",
+    "# set the initial condition to be the same\n",
+    "# we also set the dt to be smaller as the Euler scheme is less accurate\n",
+    "\n",
+    "# no parameterization\n",
+    "X_gcm_no_param, T_gcm_no_param = GCM(X_init, F_mod, dt, int(T / dt))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fb291e41-31e7-48bb-9f86-2b7435f8f984",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(2.5, 4), dpi=150)\n",
+    "\n",
+    "plt.subplot(111)\n",
+    "plt.contourf(W.k, T_gcm_no_param, X_gcm_no_param)\n",
+    "plt.colorbar()\n",
+    "plt.xlabel(\"k\")\n",
+    "plt.ylabel(\"t\")\n",
+    "plt.title(\"$X_k(t)$\")\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "18bcf631-1aac-4115-8eda-752bc1244d47",
+   "metadata": {},
+   "source": [
+    "Clearly, the evolution the single time scale system does not even qualitatively match the evolution of the two time scale system. For example, the leftward propagation is completely missing. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fc65effa",
+   "metadata": {},
+   "source": [
+    "## Adding a parametization to the single time-scale model\n",
+    "\n",
+    "The single time-scale model does not behave the same as the two time-scale model, since the effects of the fast time-scales (or \"unresolved scales\" or \"sub-grid effects\" or \"sub-grid tendencies\") are missing. To obtain a closer match to the two time-scale model, we need to introduce these missing effects into the single time-scale model. Adding these missing effects is known as parameterization, where the effect of the missing scales are represented purely in terms of the variables that are known in the gcm ($X$'s only in the single time-scale model).\n",
+    "\n",
+    "The one time-scale equation of the Lorenz-96 model with a parameterization for the effects of the fast time-scale , can be written as:\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - P\\left( X_k \\right) \\ .\n",
+    "\\end{align}\n",
+    "\n",
+    "Here, we consider a particular functional form for modeling the unknown parameterization, using a polynomial approximation. \n",
+    "Example of such a parameterization include the fourth-order polynomial form proposed by\n",
+    "{cite}`Wilks2005`: \n",
+    "\\begin{equation*}\n",
+    "P(X_k)=b_0 + b_1 X_k + b_2 X_k^2 + b_3 X_k^3 + b_4 X_k^4 + e_k.\n",
+    "\\end{equation*}\n",
+    "\n",
+    "Alternatively, we also consider a first-order polynomial as suggested by\n",
+    "{cite}`Arnold2013`: \n",
+    "\\begin{equation*}\n",
+    "P(X_k)=b_0 + b_1 X_k + e_k.\n",
+    "\\end{equation*}\n",
+    "\n",
+    "In both these forms $e_k$ is a stochastic component, which will not be discussed further till the next notebook. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "58a6bada-ad09-4d51-a017-c32cde4d5b55",
+   "metadata": {},
+   "source": [
+    "All parameterizations have some unknown parameters, which need to be determined in some way. These parameters may be guessed based on some intuition about the physics, or determined from data collected in the real system (two time-scale model here), or optimized to make the evolution of the reduced (single time-scale) model match the evolution of the real world or full (two time-scale model) model. In this notebook we will use the second approach, and the last approach will be discussed in {doc}`estimating-gcm-parameters`. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "39830287-61de-4069-b547-e89a9547b2a7",
+   "metadata": {},
+   "source": [
+    "\n",
+    "In summary: with the \"real atmosphere\" or two time-scale system in hand, we can \"observe\" the effect of the sub-grid forcing on the large scale ($U_k$) and test the skill of the polynomial function, $P(X_k)$, models/parameterizations."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "934f3a28",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%time X, Y, t = W.run(0.05, 200.) # We first run the 2 time-scale model again for longer time, to generate more data."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ff240f58",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_copy = X\n",
+    "# Generate U_k samples from 2 time-scale model\n",
+    "# using the longer time series, we now generate samples of U_k that are used to\n",
+    "# estimate the unknown parameters.\n",
+    "U_sample = (W.h * W.c / W.b) * Y.reshape((Y.shape[0], W.K, W.J)).sum(axis=-1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "eae31105-17af-4c41-9d4d-5bef728ff21c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Fit polynomial of order 1.\n",
+    "p1 = np.polyfit(\n",
+    "    X_copy.flatten(), U_sample.flatten(), 1\n",
+    ")  # Fit a linear curve through the data.\n",
+    "print(\"Linear Poly coeffs:\", p1)\n",
+    "\n",
+    "# We use the parameters from from Wilks, 2005\n",
+    "p4 = [\n",
+    "    0.000707,\n",
+    "    -0.0130,\n",
+    "    -0.0190,\n",
+    "    1.59,\n",
+    "    0.275,\n",
+    "]\n",
+    "# We could have just as easily have fit the parameters using the data as well."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ed0a0e4b",
+   "metadata": {},
+   "source": [
+    "We now compare the results obtained with a linear polynomial approximation, {cite:t}`Wilks2005` polynomial parameterization and the \"truth\" values of the coupling terms."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "06794e2b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "# 2D histogram of X vs U\n",
+    "plt.hist2d(X_copy.flatten(), U_sample.flatten(), bins=50, density=True)\n",
+    "plt.xlabel(\"$X_k$\")\n",
+    "plt.ylabel(r\"$U_k = \\frac{hc}{b}\\sum_j Y_{j,k}$\")\n",
+    "plt.colorbar(label=\"PDF\")\n",
+    "\n",
+    "# Fits from polynomials\n",
+    "x = np.linspace(-12, 18, 100)\n",
+    "plt.plot(x, np.polyval(p4, x), label=\"$P_4(X_k)$ - Wilks, 2005\")\n",
+    "plt.plot(x, np.polyval(p1, x), \"--\", label=\"$P_1(X_k)$\")\n",
+    "\n",
+    "\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "02669085-8725-456d-9100-5a31f92f2db0",
+   "metadata": {},
+   "source": [
+    "(sub-grid-hist-label)=\n",
+    "The figure above shows that the the relationship between the slow variables ($X_k$) and the observed coupling term ($U_k$) is non-linear. The higher order polynomlial, since it is more flexible, does a better job at capturing the overall pattern, relative to the linear fit. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cc34b767-cf90-4503-b708-5dd560d903ff",
+   "metadata": {},
+   "source": [
+    "We had already setup the code in `GCM` to accept polynomial parameterizations, which can be turned on by passing the parameters. We will use this in the next section to test the effect that the parameterization has on the single time-scale model."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e42cd286",
+   "metadata": {},
+   "source": [
+    "## Testing the effect of parameterizations in the \"GCM\" model\n",
+    "\n",
+    "Now that we have a couple of different candidate parameterizations that can roughly predict the relationship between the slow variables and sub-grid forcing, we test their impact in a GCM simulation where the parameterization is required. We compare four simulations:\n",
+    "- \"Real world\": corresponding to the \"truth\" model goverened by the full two time-scale Lorenz-96 system.\n",
+    "- GCM without parameterization: corresponding to the one time-scale Lorenz-96 system without any the coupling term. We use a forward-Euler scheme to integrate the model forward.\n",
+    "- GCM with our parameterization: corresponding to the one time-scale Lorenz-96 system with the linear polynomial approximation of the coupling terms as obtained above.\n",
+    "- GCM with {cite}`Wilks2005` parameterization: corresponding to the one time-scale Lorenz-96 system with a third-order polynomial approximation of the coupling terms."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9579b622",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "np.random.seed(13)\n",
+    "T = 5\n",
+    "# Real world\n",
+    "X_true, Y_true, T_true = W.randomize_IC().run(0.05, T)\n",
+    "X_init, dt, F_mod = X_true[0] + 0.0 * np.random.randn(W.K), 0.002, W.F + 0.0\n",
+    "# The reason for adding the zero terms to X and F will become clear below, where the amplitude will be increased.\n",
+    "\n",
+    "# no parameterization\n",
+    "X_gcm1, T_gcm1 = GCM(X_init, F_mod, dt, int(T / dt))\n",
+    "# Linear parameterization\n",
+    "X_gcm2, T_gcm2 = GCM(X_init, F_mod, dt, int(T / dt), param=p1)\n",
+    "# Wilks parameterization - 4th order polynomial.\n",
+    "X_gcm3, T_gcm3 = GCM(X_init, F_mod, dt, int(T / dt), param=p4)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6a3f04e9-78bc-4af1-bb9b-31370693a20a",
+   "metadata": {},
+   "source": [
+    "Now we look at Hovmoeller plots of the different solutions."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1bb832df",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150, figsize=(17, 8))\n",
+    "lims = {\"vmin\": -12, \"vmax\": 12, \"levels\": np.linspace(-12, 12, 12), \"extend\": \"both\"}\n",
+    "plt.subplot(141)\n",
+    "plt.contourf(W.k, T_true + t[-1], X_true, **lims)\n",
+    "plt.xlabel(\"$k$\")\n",
+    "plt.ylabel(\"$t$\")\n",
+    "plt.title(\"Real world\")\n",
+    "plt.subplot(142)\n",
+    "plt.contourf(W.k, T_gcm1 + t[-1], X_gcm1, **lims)\n",
+    "plt.xlabel(\"$k$\")\n",
+    "plt.ylabel(\"$t$\")\n",
+    "plt.title(\"GCM without parameterization\")\n",
+    "plt.subplot(143)\n",
+    "plt.contourf(W.k, T_gcm3 + t[-1], X_gcm2, **lims)\n",
+    "plt.xlabel(\"$k$\")\n",
+    "plt.ylabel(\"$t$\")\n",
+    "plt.title(\"GCM with linear parameterization\")\n",
+    "plt.subplot(144)\n",
+    "plt.contourf(W.k, T_gcm2 + t[-1], X_gcm3, **lims)\n",
+    "plt.xlabel(\"$k$\")\n",
+    "plt.ylabel(\"$t$\")\n",
+    "plt.title(\"GCM with Wilks' parameterization\")\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aed7579a",
+   "metadata": {},
+   "source": [
+    "The next plot shows the temporal evolution of the variable $X_3$ obtained with the four models listed above."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f5815467",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "k = 3\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(T_true + t[-1], X_true[:, k], \"--\", label=\"Real world\")\n",
+    "plt.plot(T_gcm1 + t[-1], X_gcm1[:, k], \":\", label=\"GCM without parameterization\")\n",
+    "plt.plot(T_gcm1 + t[-1], X_gcm2[:, k], label=\"GCM with our parameterization\")\n",
+    "plt.plot(T_gcm1 + t[-1], X_gcm3[:, k], label=\"GCM with Wilks parameterization\")\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_3(t)$\")\n",
+    "plt.legend(fontsize=7, loc=1)\n",
+    "plt.ylim(-10, 25);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "082fcc26-72a5-43bb-a764-159d14155605",
+   "metadata": {},
+   "source": [
+    "As seen above, all the simulation diverge at long times,and the unparameterized simulation diverges very rapidly. On the other hand, the parameterized GCMs track the \"real world\" solution better. The Wilks parameterization does better than the linear fit."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b33704fc",
+   "metadata": {},
+   "source": [
+    "## Summary\n",
+    "\n",
+    "In this chapter:\n",
+    "- We argued that the single time-scale L96 model is a reduced representation of the two time-scale L96 model, which can be considered analogous to the way GCMs are a reduced repseration of the real world. \n",
+    "- We showed that missing effects need to be parameterized, if the reduced representation is required behave like the full model/real world.\n",
+    "- We used the two time-scale L96 model to generate a \"truth\" dataset for the effects of unresolved scales. \n",
+    "- We built a \"GCM\" with a polynomial parameterization of coupling to unresolved processes ($\\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}$)\n",
+    "- Finally we compared the solution from the two time-scale, single time-scale and parameterized single time-scale models, showing that the paramaterized version evolves more similarly to the two time-scale model.\n",
+    "\n",
+    "In the next chapter we will further explore some key properties of parameterizations, and also discuss the other sources of error that can lead to a difference between the reduced and full models (or GCMs and real world)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "06d61ed7-705f-40dc-9d39-0383dcefcd36",
+   "metadata": {},
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/gcm-parameterization-problem.ipynb b/_sources/notebooks/gcm-parameterization-problem.ipynb
new file mode 100644
index 00000000..40ab4baa
--- /dev/null
+++ b/_sources/notebooks/gcm-parameterization-problem.ipynb
@@ -0,0 +1,1155 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b0650213-5f2e-4821-8489-b969f0da4b63",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# GCM parameterizations, skill metrics, and other sources of uncertainity"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2a9645ec",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "In the last notebook we provided a very quick overview of the problem we run into when trying to simulate a real system (e.g. the atmosphere) on a computer using a General Circulation Model (GCM), even when the exact equations to be solved are known in principle. The problem of limited computational resources translates into an inability to resolve all scales of motion in a GCM, and the unresolved scales need to be parameterized. \n",
+    "\n",
+    "The objective of this notebook is to introduce some more of the key aspects of parameterizations in GCMs, illustrating the deterministic vs stochastic approaches, the interplay with numerical errors, and how to measure the skill of a parameterization. We also provide a comprehensive discussion of the different sources of errors that are present in GCMs."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "69c17d19",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## The need for GCM parameterizations"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f4905750",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Let's first quickly review some concepts from the last notebook, using a slightly modified framing that might benefit some readers. \n",
+    "We will assume from now on that the readers are familiar  with the {cite}`Lorenz1995` two-time scale model and its numerical implementation in the `L96_model` module, which was discussed in {doc}`L96-two-scale-description`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f8255144",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "from numba import jit\n",
+    "from L96_model import L96, RK2, RK4, EulerFwd, L96_eq1_xdot"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "338aff9c-2891-4583-9efa-b1a56669fe2f",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Here, `L96` serves as the **\"real world\"** or two time-scale model, whereas `L96_eq1_xdot` represents the **beginning of rhs of X tendency** or the tendency in the single time-scale model."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c5ba242b",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Setting the seed gives us reproducible results\n",
+    "np.random.seed(13)\n",
+    "\n",
+    "# Create a \"real world\" with K = 8 and J = 32\n",
+    "W = L96(8, 32, F=18)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4b4b5606-be7c-4fd3-8f3d-fa7fbdbec74f",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Since we start the model with a random initial condition, there is no reason to expect that these initial conditions are an actual solution to the model. These arbitrary states can result in initial shocks to the system, which will are unrealistic features but get dissipated after some time.\n",
+    "So we *run the \"real world\" for 3 days in order to forget the initial conditons*, and settle into a state that is an actual solution to the model."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c52019dd-0d33-45d2-a10a-08429c6374a2",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# `store=True` saves the final state as an initial condition for the next run.\n",
+    "W.run(0.05, 3.0, store=True);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8e1c62ff",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "From here on we can use `W.X` as perfect initial conditions for a model and sample the \"real world\" using `W.run(dt, T)`.\n",
+    "\n",
+    "**How to think about the real world vs models:**\n",
+    "\n",
+    "Let's call $Z(t)$ the trajectory of the full complexity physical system (say planet earth). Because in practice, for computational or observational reasons, we cannot afford describing and predicting $Z(t)$, we will only focus on a projection of $Z(t)$ in some lower dimension space. Let's call this reduced dimension state $X(t)$.\n",
+    "\n",
+    "In our L96 toy model (analog to the real world),  $Z(t)=(X(t),Y(t))$ is the full complexity physical system, while $X(t)$ is the lower dimension reduction (single time-scale models is analog to the GCM). In real world situations or more complex models (e.g. actual atmosphere or ocean models), the lower dimension representation of the real system may be thought of as a coarse-grained or a subsampled description of the full-scale system.\n",
+    "\n",
+    "Now, a GCM is simply a numerical machine which intends to predict the trajectory $X(t)$ from knowledge of $X(t=0)$ only. A GCM is generally built from first principle physical laws, by discretizing partial differential equations.\n",
+    "\n",
+    "In what follows, we therefore assume that we know a fraction of the terms that govern the evolution of $X$. We also assume that we do not know what governs the evolution of $Y$ nor how $Y$ may affect $X$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fa072bc6-f5de-4917-92a3-2db493e36a35",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "**GCM without parameterization**"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "53afa120",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def GCM_no_parameterization(X0, F, dt, nt):\n",
+    "    \"\"\"GCM without parameterization\n",
+    "\n",
+    "    Args:\n",
+    "        X0: initial conditions\n",
+    "        dt: time increment\n",
+    "        nt: number of forward steps to take\n",
+    "    \"\"\"\n",
+    "    time, hist, X = (\n",
+    "        dt * np.arange(nt + 1),\n",
+    "        np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "        X0.copy(),\n",
+    "    )\n",
+    "    hist[0] = X\n",
+    "\n",
+    "    for n in range(nt):\n",
+    "        X = X + dt * (L96_eq1_xdot(X, F))\n",
+    "        hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "    return hist, time"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0a620522-8a1a-4eda-9b0a-04e304b49255",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "This GCM is unstable due to Euler forward time stepping scheme, so we don't integrate it for too long and compare it to the real world with the same time interval as `dt` used by the model."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6f113734",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "F, dt, T = 18, 0.01, 3.0\n",
+    "X, t = GCM_no_parameterization(W.X, F, dt, int(T / dt))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "66f3c2ea",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Compare with the real world\n",
+    "X_true, _, _ = W.run(dt, T)\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 4], label=\"Truth (L96)\")\n",
+    "plt.plot(t, X[:, 4], label=\"Model (GCM)\")\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_4(t)$\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dad9cbd6",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "There are several reasons why the above model (i.e. single time-scale model) differs from truth (i.e. L96 two time-scale model), which will be discussed below in {ref}`other-sources-affect-gcm`. One of these reasons is missing physics.\n",
+    "\n",
+    "One way, discussed in the previous notebook, to reduce the differences between the Model and the Truth, is to add a *parameterization*: an extra term to the rhs of the Model evolution operator in order to reduce the Model error as compared to the Truth. It may account for missing processes that are present in the truth, but are not included in the reduced model. The missing processed may be a result of unresolved scales (sub-grid processes) or due to physical processes that could not be encoded into the full equations.\n",
+    "*Parameterizations* are also commonly refered to as *closures*, in particular when they encode explicit physical assumptions on how non-represented variables (e.g. $Y$) impact represented variables (e.g. $X$). \n",
+    "\n",
+    "Parameterizations usually involve free parameters that need to be adjusted. The form of the parameterization may be dictated by physical laws, but generally it is unknown as well."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "64e94c31",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "**GCM with parameterization**"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c797ab03",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Here we introduce a class to solve for the one time-scale problem,\n",
+    "# which can take arbitrary parameterizations and time-stepping schemes as input.\n",
+    "class GCM:\n",
+    "    \"\"\"GCM with parameterization in rhs of equation for tendency\"\"\"\n",
+    "\n",
+    "    def __init__(self, F, parameterization, time_stepping=EulerFwd):\n",
+    "        self.F = F\n",
+    "        self.parameterization = parameterization\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, param):\n",
+    "        return L96_eq1_xdot(X, self.F) - self.parameterization(param, X)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        \"\"\"\n",
+    "        Args:\n",
+    "            X0: initial conditions\n",
+    "            dt: time increment\n",
+    "            nt: number of forward steps to take\n",
+    "            param: parameters of our closure\n",
+    "        \"\"\"\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5f6fd74a-3427-4dd3-b8b4-013acf2c49ff",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "As a first step, we illustrate introducing a polynomial parameterization to GCM and then compare the model to the true trajectories obtained from the real world with the same time interval as `dt` used by the model. This is the same as what was done in the previous notebook, but is shown again for completeness."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6d08038a",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "naive_parameterization = lambda param, X: np.polyval(param, X)\n",
+    "F, dt, T = 18, 0.01, 5.0\n",
+    "gcm = GCM(F, naive_parameterization)\n",
+    "X, t = gcm(W.X, dt, int(T / dt), param=[0.85439536, 0.75218026])\n",
+    "# we use the parameters for the linear polynomial parameterization that were learnt in the last notebook."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ca979ba1",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Compare with the real world\n",
+    "X_true, _, _ = W.run(dt, T)\n",
+    "\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 4], label=\"Truth\")\n",
+    "plt.plot(t, X[:, 4], label=\"Model\")\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_4(t)$\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "96a97034",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "While the GCM with parameterization is better than the GCM without parameterization, it is still not very good at reproducing the true evolution of the full system. It also remains to find the most appropriate coefficients of the polynomial parameterization to make the Model as close as possible to the Truth.\n",
+    "\n",
+    "In summary, the parameterization problem boils down to defining the functional form and finding the best parameters in order to minimize the distance between the true trajectory and the model trajectory.\n",
+    "**With M2LINES, we are approaching this problem as a Machine Learning problem. We want to learn parameterizations from objective measures of their skills through an optimization procedure.**\n",
+    "But we are not only interested in learning the parameters of existing functional forms. More generally, we would like to learn the functional forms too.   "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "42ca8ac3",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Should parameterizations  be deterministic or stochastic ? "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "36678885",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The `naive_parameterization` above has no particular physical nor mathematical justification. Most importantly, it relies on a very strong assumption, that the time rate of change of $X$ at time $t$ is a function of $X(t)$. This assumption implies that the future evolution of the reduced dimension system $X(t)$ is *deterministically* related to the initial reduced dimension state $X(0)$.   "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6083799",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "But this is not a good assumption because the two identical reduced dimension states ($X$, macro-state) can be associated with very different fine scale states ($Y$, micro-state). This can also be seen visually by considering the [plot](sub-grid-hist-label) in the previous notebook, which shows that the for each value of $X$ there is a range of possible values for the sub-grid effects.\n",
+    "Given the non-linearity of the evolution equation for $Z$, the two large scale trajectories will diverge at some point due to these small differences in the un-observed states. Let's illustrate that with L96 alone. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "bd825114",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Randomising the initial Ys\n",
+    "np.random.seed(13)\n",
+    "\n",
+    "# Duplicating L96 to create perturbed versions that include random perturbations in Y\n",
+    "Wp1 = W.copy()\n",
+    "Yp1 = W.Y.std() * np.random.rand(Wp1.Y.size)\n",
+    "Wp1.set_state(W.X, Yp1)\n",
+    "\n",
+    "Wp2 = W.copy()\n",
+    "Yp2 = W.Y + 0.0001 * np.random.rand(Wp2.Y.size)\n",
+    "Wp2.set_state(W.X, Yp2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8a856077",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Running L96 and perturbed versions to compare results\n",
+    "X_true, _, _ = W.run(dt, T)\n",
+    "X_pert1, _, _ = Wp1.run(dt, T)\n",
+    "X_pert2, _, _ = Wp2.run(dt, T)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "04d98879-6013-4161-b6b6-12f45ee7227e",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 4], label=\"Truth\")\n",
+    "plt.plot(t, X_pert1[:, 4], label=\"Perturbed 1\")\n",
+    "plt.plot(t, X_pert2[:, 4], label=\"Perturbed 2\")\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_4(t)$\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "81b73260",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "So even very small uncertainties in the micro-state ($Y$) of L96 can lead to large scale changes (i.e. of the variable $X$) over short time.\n",
+    "\n",
+    "In a Model that does not know anything about micro-state $Y$, it is possible to introduce this uncertainty through a stochastic form in the parameterization.\n",
+    "\n",
+    "\n",
+    "In addition, with this illustration, we also highlight that there is a horizon after which pointwise comparisons of the model with the truth are meaningless, hence there is some needed discussion on how to measure the skill of a parameterization."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6cf8c10",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## How to measure parameterization skill ? "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a6f4cd9",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "We would like to build our closures by systematically measuring their skills, so that we can compare different fomulations using these \"skill scores\". \n",
+    "\n",
+    "Since we are interested in matching the evolution of the \"real world\" using a GCM, we define skill scores that measure the distance between the evolution of the true state $X_{true}$ and the simulated state $X_{gcm}$. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "35609fa7",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Defining our GCM\n",
+    "F, dt, T = 18, 0.005, 1000.0\n",
+    "# Use Runge-Kutta time stepping scheme because forward Euler would lead to instabilities\n",
+    "gcm = GCM(F, naive_parameterization, time_stepping=RK4)\n",
+    "\n",
+    "# Evaluate the GCMs\n",
+    "X_gcm, t = gcm(W.X, dt, int(T / dt), param=[0.85439536, 0.75218026])\n",
+    "X_gcm_no_param, _ = gcm(W.X, dt, int(T / dt), param=[0, 0])\n",
+    "\n",
+    "# Evaluate the true state\n",
+    "X_true, _, _ = W.run(dt, T)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "354b8190-af9b-4bc2-a4c4-de6a9266cc56",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Plotting the results\n",
+    "plt.figure(dpi=150)\n",
+    "\n",
+    "plt.subplot(211)\n",
+    "plt.plot(t[:500], X_true[:500, 4], label=\"Truth\", color=\"k\")\n",
+    "plt.plot(t[:500], X_gcm_no_param[:500, 4], label=\"GCM without param\", color=\"tab:green\")\n",
+    "plt.plot(t[:500], X_gcm[:500, 4], label=\"GCM with param\", color=\"tab:orange\")\n",
+    "\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_4(t)$\")\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "plt.subplot(212)\n",
+    "plt.plot(\n",
+    "    t[:500],\n",
+    "    (X_true[:500, 4] - X_gcm_no_param[:500, 4]),\n",
+    "    label=\"GCM without param\",\n",
+    "    color=\"tab:green\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t[:500],\n",
+    "    (X_true[:500, 4] - X_gcm[:500, 4]),\n",
+    "    label=\"GCM with param\",\n",
+    "    color=\"tab:orange\",\n",
+    ")\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_4^{true}(t) - X_4^{gcm}(t)$\")\n",
+    "plt.legend(fontsize=7)\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "baa78beb",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "**Error metric based on model evolution**:\n",
+    "\n",
+    "Clearly, the absolute error between the true and gcm trajectory grows with time, and we would like measure how the error cumulates over time.\n",
+    "A simple error metric is the point-wise root mean square error, which is averaged over time:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "E (t) = \\frac{1}{t} \\int_0^t |X_{true}(t) - X_{gcm}(t)| dt\n",
+    "\\end{equation}\n",
+    "\n",
+    "This can be computed for each $X_k$ separately, or averaged over all $k$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "33368242-30cf-4b30-8023-02ad1b1de22d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def error_model_evolution(X1, X2, L, dt=dt):\n",
+    "    \"\"\"Model evolution error computed over some window t < L.\"\"\"\n",
+    "    D = np.cumsum(\n",
+    "        np.sqrt((X1[: int(L / dt), :] - X2[: int(L / dt), :]) ** 2), axis=0\n",
+    "    ) / (np.expand_dims(np.arange(1, int(L / dt) + 1), axis=1))\n",
+    "    return D"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7ac888c7-9a97-4881-8ffc-ab5a8af49ec1",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "dist = error_model_evolution(X_true, X_gcm, T)\n",
+    "dist_no_param = error_model_evolution(X_true, X_gcm_no_param, T)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b49636d6",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Plotting how this distance grows with the length of the window\n",
+    "# for all the components of X\n",
+    "plt.figure(dpi=150)\n",
+    "plt.plot(\n",
+    "    t[1:],\n",
+    "    np.mean(dist_no_param, 1),\n",
+    "    linewidth=2,\n",
+    "    label=\"GCM without param\",\n",
+    "    color=\"tab:green\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t[1:], np.mean(dist, 1), linewidth=2, label=\"GCM with param\", color=\"tab:orange\"\n",
+    ")\n",
+    "\n",
+    "\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.legend()\n",
+    "plt.ylabel(\"$||X_{true}-X_{gcm}||$\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c6c31304-6c60-4864-a3ad-d40d6e5117b8",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The error grows with time, but saturates to some a constant after the truth and GCM have gotten completely decorrelated. This constant is equal to the sum of the variance of the truth and GCM states. Also, as expected, the error grows much more rapidly without a parameterization, showing that adding the parameterization has resulted in a quantitative improvement in the solution."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4aea7896",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "**Climatology based error metric:**\n",
+    "\n",
+    "Knowing from the above discussion that the system is not predictable after some time, we may as well have decided to measure how well the model captures the mean state. With this we expect to measure the \"climate\" of the system instead of the \"weather\". \n",
+    "\n",
+    "\\begin{equation}\n",
+    "E_{clim}  = | \\overline{X_{true}}  - \\overline{X_{gcm}}|\n",
+    "\\end{equation}\n",
+    "\n",
+    "where the $\\overline{(.)}$ is some suitably defined average that may be variable in time (like a seasonal climatology). \n",
+    "\n",
+    "This metric will likely converge to some non-trivial values which are indicative of how well our model captures the \"climate\" of the system, and the difference indicating the bias in the model."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "572bedd5-4aa2-4441-9555-31261de10aa9",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "\n",
+    "plt.subplot(211)\n",
+    "plt.plot(np.mean(X_true, 0), marker=\"o\", linewidth=2, color=\"k\", label=\"Truth\")\n",
+    "plt.plot(\n",
+    "    np.mean(X_gcm_no_param, 0),\n",
+    "    marker=\"o\",\n",
+    "    linewidth=2,\n",
+    "    color=\"tab:green\",\n",
+    "    label=\"GCM without param\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    np.mean(X_gcm, 0),\n",
+    "    marker=\"o\",\n",
+    "    linewidth=2,\n",
+    "    color=\"tab:orange\",\n",
+    "    label=\"GCM with param\",\n",
+    ")\n",
+    "\n",
+    "plt.xlabel(\"$k$\")\n",
+    "plt.ylabel(\"$\\overline{X_{k}}$\")\n",
+    "plt.ylim([2, 4])\n",
+    "plt.legend()\n",
+    "\n",
+    "plt.subplot(212)\n",
+    "plt.plot(\n",
+    "    np.sqrt((np.mean(X_gcm_no_param, 0) - np.mean(X_true, 0)) ** 2),\n",
+    "    marker=\"o\",\n",
+    "    linewidth=2,\n",
+    "    color=\"tab:green\",\n",
+    "    label=\"GCM without param\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    np.sqrt((np.mean(X_gcm, 0) - np.mean(X_true, 0)) ** 2),\n",
+    "    marker=\"o\",\n",
+    "    linewidth=2,\n",
+    "    color=\"tab:orange\",\n",
+    "    label=\"GCM with param\",\n",
+    ")\n",
+    "\n",
+    "plt.xlabel(\"$k$\")\n",
+    "plt.ylabel(\"$E_{clim}$\")\n",
+    "plt.ylim([0, 0.8])\n",
+    "plt.legend()\n",
+    "\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b9692de-d650-4054-84e2-7e4b2925a7aa",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "As seen in the first figure, the long term mean state of the true solution is different from the two GCM solutions. Surprisingly, the GCM without parameterization has a climatology that is slightly closer to the true climatology than the climatology of the GCM with parameterization. In other words, the error $E_{clim}$ is larger for the GCM with parameterization. This is shown in the second figure.\n",
+    "\n",
+    "The above result is counter-intuitive, and suggests that $E_{clim}$ is not a very powerful metric. This metric takes into account only the means of the underlying probability distributions. Let's look at the full probability distributions next!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "242f83d4-9112-4552-8a54-0cb69a1939cb",
+   "metadata": {},
+   "source": [
+    "**Distribution based error metric**:\n",
+    "\n",
+    "We want to measure differences between the empirical distributions of the true solution vs. GCM solution. There are several options how to define distributional differences, but here we consider the earth mover's distance or Wasserstein distance:\n",
+    "\\begin{align}\n",
+    "E_{distr}  & = \\int_{-\\infty}^\\infty |P(X^{true}\\leq x) - P(X^{GCM}\\leq x) \\, | dx, \\\\\n",
+    "\\end{align}\n",
+    "where $P(X\\leq x)$ is the cumulative distribution function of $X$.\n",
+    "\n",
+    "Before we compute the differences between the distribution $E_{distr}$, let's first look at the empirical distributions themselves. For simplicity, we show the probability density functions (rather than the cumulative distribution functions, where the latter are used in the computation of $E_{distr}$)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8b8ac189-f5e2-4bb6-a807-082c20f44c36",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import seaborn as sns"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "aeb3d098-1d52-4253-a283-10c7a59158ab",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "\n",
+    "sns.kdeplot(X_true.ravel(), color=\"k\", linewidth=2, label=\"Truth\")\n",
+    "plt.axvline(X_true.mean(), color=\"k\", linewidth=1, label=\"_none\")\n",
+    "\n",
+    "sns.kdeplot(\n",
+    "    X_gcm_no_param.ravel(), color=\"tab:green\", linewidth=2, label=\"GCM without param\"\n",
+    ")\n",
+    "plt.axvline(X_gcm_no_param.mean(), color=\"tab:green\", linewidth=1, label=\"_none\")\n",
+    "\n",
+    "sns.kdeplot(X_gcm.ravel(), color=\"tab:orange\", linewidth=2, label=\"GCM with param\")\n",
+    "plt.axvline(X_gcm.mean(), color=\"tab:orange\", linewidth=1, label=\"_none\")\n",
+    "\n",
+    "plt.xlabel(\"$X$\")\n",
+    "plt.legend()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e750512f-1aa0-481f-91c6-a7913464ce59",
+   "metadata": {},
+   "source": [
+    "The vertical lines in the figure above indicate the means of the respective empirical distributions. As we saw earlier, the mean of the GCM without parameterization is slightly closer to the real mean than the mean of the GCM with parameterization. However, if we take into account the entire distribution, the GCM with parameterization seems to be a closer match to the true distribution. Our eyeball estimate is quantitatively confirmed by the next figure, which computes the distributional error, $E_{distr}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e41ec23b-a0d0-44ab-9757-d958b2171e06",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "from scipy.stats import wasserstein_distance\n",
+    "\n",
+    "\n",
+    "def error_distribution(X1, X2):\n",
+    "    \"\"\"Distribution error measured by the Wasserstein distance.\"\"\"\n",
+    "\n",
+    "    diff = wasserstein_distance(X1, X2)\n",
+    "\n",
+    "    return diff"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "10af9a85-fdac-422d-bedf-7dee947be990",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "E_distr_no_param = [error_distribution(X_true[:, k], X_gcm_no_param[:, k]) for k in W.k]\n",
+    "E_distr_param = [error_distribution(X_true[:, k], X_gcm[:, k]) for k in W.k]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2f8c907c-98a7-4499-8a81-323dcd7dadbe",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(\n",
+    "    W.k,\n",
+    "    E_distr_no_param,\n",
+    "    marker=\"o\",\n",
+    "    color=\"tab:green\",\n",
+    "    label=\"GCM without param\",\n",
+    "    linewidth=2,\n",
+    ")\n",
+    "plt.plot(\n",
+    "    W.k,\n",
+    "    E_distr_param,\n",
+    "    marker=\"o\",\n",
+    "    color=\"tab:orange\",\n",
+    "    label=\"GCM with param\",\n",
+    "    linewidth=2,\n",
+    ")\n",
+    "\n",
+    "plt.xlabel(\"$k$\")\n",
+    "plt.ylabel(\"$E_{distr}$\")\n",
+    "plt.ylim([0, 2.1])\n",
+    "plt.legend()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c769ec8e",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "**Tendency based error metric**:\n",
+    "\n",
+    "It is also very common to formulate closures based on databases of initial tendencies.  In the Large Eddy Simulation  community, this is sometimes refered to as *a priori* skill, because you don't need to run the full model to compute it.\n",
+    "\n",
+    "This is the sort of game that several of us have been playing, trying for instance to estimate subgrid fluxes from knowledge of the large scale quantities  $$ \\nabla\\cdot \\mathbf{s} =   \\nabla\\cdot\\big(\\overline{\\mathbf{u}\\,\\Phi} - \\overline{\\mathbf{u}}\\,\\overline{\\Phi}\\big) \\simeq f(\\overline{\\mathbf{u}},\\overline{\\Phi})$$\n",
+    "\n",
+    "*Note that this is not exactly the same problem as the `a priori` LES problem, because of the interplay with time-discretization. Let's neglect that for the moment.*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c75e73f6",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def norm_initial_tendency(X1, X2):\n",
+    "    T1 = X1[1, :] - X1[0, :]\n",
+    "    T2 = X2[1, :] - X2[0, :]\n",
+    "    return np.sqrt((T1 - T2) ** 2).mean(axis=0)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "13a52dd0",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Because this metric is cheap to evaluate, as we do not need to integrate the GCM more than 1 time-step, we can start a sensitivity analysis in order to identify good optimal parameters for the specific formulation `naive_parameterization`"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2d331c18",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "F, dt = 18, 0.01\n",
+    "\n",
+    "# Let's define again the true state\n",
+    "# But only run for 1 time step\n",
+    "X_true, _, _ = W.run(dt, dt)\n",
+    "\n",
+    "# and an ensemble of trajectories\n",
+    "gcm = GCM(F, naive_parameterization)\n",
+    "\n",
+    "n = 100\n",
+    "\n",
+    "_p1 = np.linspace(-20, 20, n + 1)\n",
+    "_p2 = np.linspace(-20, 20, n + 1)\n",
+    "xp1, yp2 = np.meshgrid(_p1, _p2)\n",
+    "\n",
+    "score = np.zeros((n + 1, n + 1))\n",
+    "\n",
+    "for i, p1 in enumerate(_p1):\n",
+    "    for j, p2 in enumerate(_p2):\n",
+    "        X_gcm, t = gcm(W.X, dt, 1, param=[p1, p2])  # run gcm for 1 time step\n",
+    "        score[i, j] = norm_initial_tendency(X_true, X_gcm)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "44ac3775",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.contourf(xp1, yp2, score)\n",
+    "plt.colorbar()\n",
+    "plt.xlabel(\"$p_2$\")\n",
+    "plt.ylabel(\"$p_1$\")\n",
+    "plt.title(\"Score\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b65357a",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "From this analysis, we see that the optimisation problem is probably well posed as the cost function appears pretty smooth. One can also see that the parameter $p_1$ is more important than $p_2$.  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "91d9ad08",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "(other-sources-affect-gcm)=\n",
+    "## Sources of model error\n",
+    "\n",
+    "Missing physics is not the only source of error in GCMs, and many other factors can contribute to the GCM state diverging from the real world. \n",
+    "Here, we consider the following five sources of error:\n",
+    "\n",
+    "1. Missing physics: which is modeled with a GCM without parameterization that corresponds to the one time-scale Lorenz-96 system without any the coupling term.\n",
+    "2. Poorly parameterized unresolved physics: which is studied by considering a first-order and third-order polynomial approximations of the coupling terms:\n",
+    "    \\begin{equation*}\n",
+    "    P_4 \\rightarrow P_1\n",
+    "    \\end{equation*}\n",
+    "3. Unknown forcing: which is modeled by adding an error to the forcing term:\n",
+    "    \\begin{equation*}\n",
+    "    F \\rightarrow F + error\n",
+    "    \\end{equation*}\n",
+    "4. Numerical approximation: which is studied by increasing the time-step:\n",
+    "    \\begin{equation*}\n",
+    "    \\Delta t \\rightarrow 10 \\Delta t\n",
+    "    \\end{equation*}\n",
+    "5. Initialization error: which is modeled by adding an error to the initial condition:\n",
+    "    \\begin{equation*}\n",
+    "    X(t=0) \\rightarrow X(t=0) + error\n",
+    "    \\end{equation*}\n",
+    "  \n",
+    "The first two sources have already been discussed in some detail previously, and included here to contrast against other error sources.\n",
+    "\n",
+    "The next code estimates these sources of error and the figure shows their relative contributions. For reference, we also plot the error of the GCM using {cite}`Wilks2005` polynomial coupling term and without any of the sources of error listed above. All errors are evaluated by comparing the GCMs to the \"truth\" model goverened by the full two time-scale Lorenz-96 system."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8644d36e",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Generic GCM parameters\n",
+    "F, dt, T = 18, 0.002, 40.0\n",
+    "X0 = W.X\n",
+    "\n",
+    "# Remember the optimized parameters from previous notebook\n",
+    "p1 = [0.85439536, 0.75218026]\n",
+    "p4 = [\n",
+    "    0.000707,\n",
+    "    -0.0130,\n",
+    "    -0.0190,\n",
+    "    1.59,\n",
+    "    0.275,\n",
+    "]\n",
+    "\n",
+    "# Sampling real world over a longer period of time\n",
+    "X_true, _, t = W.run(dt, T)\n",
+    "\n",
+    "# GCM with different parameterizations\n",
+    "gcm = GCM(F, naive_parameterization)\n",
+    "X_no_param, t = gcm(W.X, dt, int(T / dt), param=[0])  # Missing physics\n",
+    "X_p1, _ = gcm(W.X, dt, int(T / dt), param=p1)  # Simpler but poorer parameterization\n",
+    "X_p4, _ = gcm(W.X, dt, int(T / dt), param=p4)  # More complex parameterization\n",
+    "\n",
+    "# GCM with perturbed forcing\n",
+    "gcm_pert_F = GCM(F + 1.0, naive_parameterization)\n",
+    "X_frc, _ = gcm_pert_F(W.X, dt, int(T / dt), param=p4)  # Perturbed forcing\n",
+    "\n",
+    "# GCM with perturbed IC\n",
+    "X_IC, _ = gcm(W.X + 0.5, dt, int(T / dt), param=p4)  # Perturbed IC\n",
+    "\n",
+    "# GCM with changed dt\n",
+    "X_dt, t_dt = gcm(W.X, 5 * dt, int(T / (5 * dt)), param=p4)  # Larged dt"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ca94d484-d1e7-464b-98b2-d34bda5cfa40",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "\n",
+    "plt.plot(t, X_true[:, 4], label=\"Truth\")\n",
+    "plt.plot(t, X_no_param[:, 4], label=\"GCM without param\")\n",
+    "plt.plot(t, X_p1[:, 4], label=\"GCM with linear param\")\n",
+    "plt.plot(t, X_p4[:, 4], label=\"GCM with 4th order param\")\n",
+    "plt.plot(t, X_frc[:, 4], label=\"GCM with perturbed forcing\")\n",
+    "plt.plot(t, X_IC[:, 4], label=\"GCM with pertured IC\")\n",
+    "plt.plot(t_dt, X_dt[:, 4], label=\"GCM with increased dt\")\n",
+    "\n",
+    "plt.xlim([0, 4])\n",
+    "plt.legend()\n",
+    "\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$X_{4}(t)$\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "24427fd3-5ea3-4d78-909c-43f9d60396f2",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Clearly perturbing the gcm in different ways results in different solutions. Below we quantify these effects using the `error_model_evolution metric` that was defined earlier. We present the results as an average over all k."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "79aeab71-cf42-4d8d-9e4b-3dbb616a5971",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def dist(X1, X2, T, dt=dt):\n",
+    "    return np.mean(error_model_evolution(X1, X2, T, dt), 1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7326aa7b-e804-41b0-9d52-8bffb19fad6e",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "dist_no_param = dist(X_true, X_no_param, T, dt)\n",
+    "dist_p1 = dist(X_true, X_p1, T, dt)\n",
+    "dist_p4 = dist(X_true, X_p4, T, dt)\n",
+    "dist_frc = dist(X_true, X_frc, T, dt)\n",
+    "dist_IC = dist(X_true, X_IC, T, dt)\n",
+    "dist_dt = dist(X_true[::5], X_dt, T, 5 * dt)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a4ed0a93-9560-4466-a91f-2a048bfea1d1",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t[1:], dist_no_param, label=\"GCM without param\")\n",
+    "plt.plot(t[1:], dist_p1, label=\"GCM with linear param\")\n",
+    "plt.plot(t[1:], dist_p4, label=\"GCM with 4th order param\")\n",
+    "plt.plot(t[1:], dist_frc, label=\"GCM with perturbed forcing\")\n",
+    "plt.plot(t[1:], dist_IC, label=\"GCM with pertured IC\")\n",
+    "plt.plot(t_dt[1:], dist_dt, label=\"GCM with increased dt\")\n",
+    "\n",
+    "plt.legend()\n",
+    "plt.xlabel(\"$t$\")\n",
+    "plt.ylabel(\"$E(t)$\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a885f6f-0055-478d-9093-3d7080ee9ff4",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Under the perturbations considered above, the lack of missing physics contributes the most error to the GCM.  \n",
+    "This can be fixed by adding parameterizations, which have different contributions to error at different times. Specifically, the error grows more rapidly for the higher order parameterization relative to the linear parameterization, but saturates to smaller error at longer time. This might partly be because we optimized the parameters for the linear parameterization, but used parameter estimates for the higher order parameterization from the literature.  \n",
+    "The second largest error source is the numerical approximation (or changed dt), suggesting that we need to be careful about the design and choice of the numerical schemes.\n",
+    "The errors due to the forcing and initial condition reult in smaller errors, but will likely grow as the perturbations to these change. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dcdba996-0dd4-4ef8-adea-b8b221b56d53",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "## Summary\n",
+    "\n",
+    "In this chapter:\n",
+    "- We first reintroduced the GCM parameterization problem, discussing it in terms of limited observability of GCMs. \n",
+    "- Then we discussed the need for stochastic parameterizations, which arises because of the possible differences in the unresolved or hidden states.\n",
+    "- Then we discussed how skill can be measured, and defined four different metrics. \n",
+    "- Finally we discussed the different sources of error that may be presend in numerical GCMs. \n",
+    "\n",
+    "This chapter has made it clear that even models with some degree of tuning will diverge from the true solution, as there are multiple sources of errors. In the next chapter we discuss if and how GCMs and their parameterizations can be further tuned to enhance their range of predictability (which is known to not be in finite)."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/intro_ML_and_NNs.ipynb b/_sources/notebooks/intro_ML_and_NNs.ipynb
new file mode 100644
index 00000000..de43c1d5
--- /dev/null
+++ b/_sources/notebooks/intro_ML_and_NNs.ipynb
@@ -0,0 +1,839 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Introduction to Machine Learning and Neural Networks"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Outline:\n",
+    "\n",
+    "In this notebook we introduce the basics of machine learning, show how to optimize parameters in simple models (linear regression), and finally demonstrate a basic neural network training case. There are many excellent tutorials on machine learning available on the internet, and our goal here is not to be comprehensive but rather provide a gentle introduction that seems necessary to what comes later in the book. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Introduction to machine learning \n",
+    "\n",
+    "Machine learning can be defined as : \"Machine learning is a way for computers to learn and improve from experience without being explicitly programmed.\" This definition was generated with the help of an AI agent called [ChatGPT](https://chat.openai.com/share/4cb73dc4-e372-445c-8aca-94c6199540eb).  \n",
+    "\n",
+    "It has proven to be a useful tool to the scientific community by being able to make advancements in a range of problems, which had traditionally been very challenging or impossible. In our project, [M2LInES](https://m2lines.github.io/), we are improving simulations of the climate system by targeting the parameterization or sub-grid closure problem with the help of ML. \n",
+    "\n",
+    "In simpler terms, ML for our context can be thought of as a sophisticated way to approximate or estimate function. This can be done in a way where the structure of the functions is prescribed and constrained to very limited space of functions (e.g. linear regression) or in a way where the the structure of the function is left relatively free (e.g. neural networks). Regardless of this choice, all ML tasks can be broken down into a few general pieces:\n",
+    "- Data : All ML tasks need data to learn patterns.\n",
+    "- Model : Some choice needs to be made about the model architecture (e.g. neural network, gaussian process, etc). \n",
+    "- Loss functions : The way a ML model gets closer to the true function is by minimizing some appropriately chosen loss function, a measure of error between truth and ML prediction.\n",
+    "- Optimizers : To minize the error or loss function some method is needed to estimate optimal parameters. \n",
+    "\n",
+    "Since all this is being done on a computer, some software architecture is needed. Possible, one of the major reasons for large scale adoption of machine learning has been the availability of easy to use, well documented, open source and general purpose machine learning libraries. Many such library exist, e.g. [Tensorflow](https://www.tensorflow.org/), [PyTorch](https://pytorch.org/), [JAX](https://jax.readthedocs.io/en/latest/index.html), etc. In this book we will primarily use PyTorch. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### [PyTorch](https://github.com/pytorch/pytorch)\n",
+    "\n",
+    "In short, PyTorch is a Python-based machine learning framework that can be used for scientific machine learning tasks. There are many good reasons to use PyTorch, including that it can be used as a replacement for NumPy if we want to do computations on GPUs, it is extremely flexible and fast, but primarily that it has a large community and excellent tutorials to learn from.\n",
+    "\n",
+    "To understand and install PyTorch, we recommend going through the [tutorials](https://pytorch.org/tutorials/), [blogs](https://pytorch.org/blog/), or one of the many online tutorial (e.g. [Deep Learning With Pytorch: A 60 Minute Blitz](https://pytorch.org/tutorials/beginner/deep_learning_60min_blitz.html), [Neural Networks](https://pytorch.org/tutorials/beginner/blitz/neural_networks_tutorial.html) etc).\n",
+    "\n",
+    "\n",
+    "To follow along a single blogpost that goes through most concepts, we recommend [this](https://blog.paperspace.com/pytorch-101-understanding-graphs-and-automatic-differentiation/). \n",
+    "\n",
+    "---"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Having understood how the tools work, we can now use them to model our data. To train any machine learning model, there is a general recipe followed. This recipe can be summarized in three steps:\n",
+    "1. Define a model representation.\n",
+    "2. Choose a suitable loss function that tells us how far apart our model predictions are from the real data\n",
+    "3. Update the parameters in model representation using an optimization algorithm to minimize the loss function.\n",
+    "\n",
+    "In this notebook, we will be working a linear regression model and a neural network. We will measure loss using the Mean Squared Error (MSE) function and will try to optimize or reduce the loss value using a popular optimization algorithm, gradient descent.\n",
+    "\n",
+    "To read further on how to train machine learning models, refer to Prof. Fund's [lecture notes](https://github.com/ffund/intro-ml-tss21) or Prof. Hegde's [blogposts](https://chinmayhegde.github.io/dl-notes/notes/lecture03/)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Linear regression"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Probably, the simplest \"machine learning\" model is [linear regression](https://en.wikipedia.org/wiki/Linear_regression). This model models the relationship between the input and output data as a linear function, which for a single input and single output dimension is equivalent to fitting a line through the data:\n",
+    "$$\n",
+    "y = w_1x + w_0,\n",
+    "$$\n",
+    "where $w_1$ (slope or kernel) and $w_0$ (bias) are parameters that need to best estimated. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "For multiple inputs and single output the model can be visually represented as follows:\n",
+    "```{image} https://miro.medium.com/v2/resize:fit:1032/1*WswH2fPx0bf_JFRMm8V-HA.gif\n",
+    ":alt: nn-output-computation\n",
+    ":width: 500px\n",
+    "```\n",
+    "\n",
+    "*Image is taken from [here](https://blog.insightdatascience.com/a-quick-introduction-to-vanilla-neural-networks-b0998c6216a1).* Similarly, this linear regression could be modified to have multiple inputs and multiple outputs. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "There are many ways to estimate optimal parameters for linear regression, and here we will show how this can be done using PyTorch and gradient descent. \n",
+    "\n",
+    "The best prediction is defined by minimizing some cost or loss function. For linear regression, generally the mean sequare error between the expected output $y$ and the prediction ($\\tilde{y}$), defined as mean of $(y-\\tilde{y})^2$ over all output dimensions and samples, is minimized."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "*Most of this lesson follows a tutorial from Jeremy Howard's fast.ai [lesson zero](https://www.youtube.com/watch?v=ACU-T9L4_lI)*."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We will learn the parameters $w_0$ and $w_1$ of a line using PyTorch."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Import machine-learning packages\n",
+    "import torch\n",
+    "from torch import nn\n",
+    "\n",
+    "# Import plotting packages\n",
+    "from IPython.display import Image, HTML\n",
+    "from matplotlib.animation import FuncAnimation\n",
+    "import matplotlib.pyplot as plt\n",
+    "import time\n",
+    "import base64\n",
+    "import numpy as np\n",
+    "\n",
+    "%matplotlib inline"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Set random seed for reproducibility\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "np.random.seed(42)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Data generation for linear regression\n",
+    "\n",
+    "Since this is a toy problem, we first choose the true parameters so that we can later easily verify the success of our approach. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "w = torch.as_tensor([3.0, 2])\n",
+    "w\n",
+    "# where 3 is the interesection point (sometimes also called bias) and 2 is the slope."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Notice that we are using tensors here, which is a data structure used by PyTorch to easily deploy computations to CPUs or GPUs."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Then we create some data points x and y which are related to each other through the linear relationship and parameters chosen above. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# We create 100 data points.\n",
+    "n = 100\n",
+    "x = torch.ones(n, 2)\n",
+    "\n",
+    "# uniformly sample x points between -1 and 1.\n",
+    "# Underscore functions in pytorch means replace the value (update)\n",
+    "x[:, 1].uniform_(-1.0, 1)\n",
+    "\n",
+    "x[:6]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note above that for each sample we have created two numbers. \n",
+    "The first is 1, to be used with $w_0$, and the second is x, to be used with $w_1$. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Next we generate output data using the equation for linear regression, but also add some random noise to be in the more real world setting where there is always some noise or if linear regression is not the perfect model. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Generate output data.\n",
+    "y = x @ w + torch.rand(n)  # @ is a matrix product (similar to matmul)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.scatter(x[:, 1], y)\n",
+    "plt.xlabel(\"x\")\n",
+    "plt.ylabel(\"y\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Loss function\n",
+    "We define our loss function $L$ as Mean Square Error loss as:\n",
+    "\\begin{equation*}\n",
+    "L_{MSE} = \\frac{1}{n} \\cdot \\sum_{i=1}^{n} (y_i - \\tilde y_i)^2\n",
+    "\\end{equation*}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def mse(y_true, y_pred):\n",
+    "    return ((y_true - y_pred) ** 2).mean()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Written in terms of $w_0$ and $w_1$, our **loss function** $L$ is:\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "L_{MSE} = \\frac{1}{n} \\cdot \\sum_{i=1}^{n} (y_i - (w_0 + \\mathbf{w_1} \\cdot x_i))^2\n",
+    "\\end{equation*}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Initialize parameters with some randomly guessed values.\n",
+    "w_real = torch.as_tensor([-3.0, -5])\n",
+    "y_hat = x @ w_real\n",
+    "# Initial mean-squared error\n",
+    "mse(y_hat, y)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here we chose randomly guessed initial parameter choices. While this choice is not consequential here, in more complex problems these choice prior parameter choice can be extremely important."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.scatter(x[:, 1], y, label=\"y\")\n",
+    "plt.scatter(x[:, 1], y_hat, label=\"$\\\\tilde{y}$\")\n",
+    "plt.xlabel(\"$x$\")\n",
+    "plt.title(\"Prediction of model using non-optimized model\")\n",
+    "plt.legend(fontsize=7);"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Initialize parameter obect that will be optimized.\n",
+    "w = nn.Parameter(w_real)\n",
+    "w"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Gradient descent and optimization\n",
+    "\n",
+    "So far, we have specified the *model* (linear regression) and the *evaluation criteria* (or *loss function*). Now we need to handle *optimization*; that is, how do we find the best values for weights ($w_0$, $w_1$) such that they best fit the linear regression parameters?\n",
+    "\n",
+    "This optimization is done using gradient descent. In this process the gradient of the loss function is estimated relative to the unknown parameters, and then the parameters are updated in a direction that reduces the loss function. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Load the image file\n",
+    "with open(\"figs/Gradient_descent2.png\", \"rb\") as f:\n",
+    "    image_data = f.read()\n",
+    "\n",
+    "# Create the HTML code to display the image\n",
+    "html = f'<div style=\"text-align:center\"><img src=\"data:image/png;base64,{base64.b64encode(image_data).decode()}\" style=\"max-width:700px;\"/></div>'\n",
+    "\n",
+    "# Display the HTML code in the notebook\n",
+    "display(HTML(html))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "To know how to change $w_0$ and $w_1$ to reduce the loss, we compute the derivatives (or gradients):\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\frac{\\partial L_{MSE}}{\\partial w_0} = \\frac{1}{n}\\sum_{i=1}^{n} -2\\cdot [y_i - (w_0 + \\mathbf{w_1}\\cdot x_i)]\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\frac{\\partial L_{MSE}}{\\partial \\mathbf{w_1}} = \\frac{1}{n}\\sum_{i=1}^{n} -2\\cdot [y_i - (w_0 + \\mathbf{w_1}\\cdot x_i)] \\cdot x_i\n",
+    "\\end{equation*}\n",
+    "\n",
+    "Since we know that we can iteratively take little steps down along the gradient to reduce the loss, aka, *gradient descent*, the size of the step is determined by the learning rate ($\\eta$):\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "w_0^{new} = w_0^{current} -   \\eta \\cdot \\frac{\\partial L_{MSE}}{\\partial w_0}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\mathbf{w_1^{new}} = \\mathbf{w_1^{current}} -  \\eta \\cdot \\frac{\\partial L_{MSE}}{\\partial \\mathbf{w_1}}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "The gradients needed above are computed using the ML library's [autograd feature](https://pytorch.org/tutorials/beginner/blitz/autograd_tutorial.html), and described more [here](https://blog.paperspace.com/pytorch-101-understanding-graphs-and-automatic-differentiation/)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def step(lr):\n",
+    "    y_hat = x @ w\n",
+    "    loss = mse(y, y_hat)\n",
+    "    # calculate the gradient of a tensor! It is now stored at w.grad\n",
+    "    loss.backward()\n",
+    "\n",
+    "    # To prevent tracking history and using memory\n",
+    "    # (code block where we don't need to track the gradients but only modify the values of tensors)\n",
+    "    with torch.no_grad():\n",
+    "        # lr is the learning rate. Good learning rate is a key part of Neural Networks.\n",
+    "        w.sub_(lr * w.grad)\n",
+    "        # We want to zero the gradient before we are re-evaluate it.\n",
+    "        w.grad.zero_()\n",
+    "\n",
+    "    return loss.detach().item(), y_hat.detach().numpy()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "> Note: In PyTorch, we need to set the gradients to zero before starting to do back propragation because PyTorch accumulates the gradients on subsequent backward passes. This is convenient while training Recurrent Neural Networks (RNNs). So, the default action is to accumulate or sum the gradients on every `loss.backward()` call.\n",
+    "Because of this, when you start your training loop, ideally you should zero out the gradients so that you do the parameter update correctly. Else the gradient would point in some other direction than the intended direction towards the minimum (or maximum, in case of maximization objectives).\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "w = torch.as_tensor([-2.0, -3])\n",
+    "w = nn.Parameter(w)\n",
+    "lr = 0.1\n",
+    "losses = [float(\"inf\")]\n",
+    "y_hats = []\n",
+    "epoch = 100"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Train model and perform gradient descent\n",
+    "for _ in range(epoch):\n",
+    "    loss, y_hat = step(lr)\n",
+    "    losses.append(loss)\n",
+    "    y_hats.append(y_hat)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(np.array(losses))\n",
+    "plt.xlabel(\"Iteration\")\n",
+    "plt.ylabel(\"Loss\")\n",
+    "plt.yscale(\"log\")\n",
+    "plt.title(\"Loss vs Iteration\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "As seen above the loss function is reduced as more optimization steps are taken. The animation below shows how the prediction gets closer to the data as more steps are taken."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, axs = plt.subplots(1, 2, figsize=(10, 4), dpi=80)\n",
+    "axs[0].scatter(x[:, 1], y, label=\"y\")\n",
+    "scatter_yhat = axs[0].scatter(x[:, 1], y_hat, label=\"$\\\\tilde{y}$\")\n",
+    "axs[0].set_xlabel(\"$x$\")\n",
+    "axs[0].legend(fontsize=7)\n",
+    "\n",
+    "(line,) = axs[1].plot(range(len(losses)), np.array(losses))\n",
+    "axs[1].set_xlabel(\"Iteration\")\n",
+    "axs[1].set_ylabel(\"Loss\")\n",
+    "axs[1].set_title(\"Loss vs Iteration\")\n",
+    "plt.close()\n",
+    "\n",
+    "\n",
+    "def animate(i):\n",
+    "    axs[0].set_title(\"Loss = %.2f\" % losses[i])\n",
+    "    scatter_yhat.set_offsets(np.c_[[], []])\n",
+    "    scatter_yhat.set_offsets(np.c_[x[:, 1], y_hats[i]])\n",
+    "    line.set_data(np.array(range(i + 1)), np.array(losses[: (i + 1)]))\n",
+    "    return scatter_yhat, line\n",
+    "\n",
+    "\n",
+    "# plt.show()\n",
+    "\n",
+    "animation = FuncAnimation(fig, animate, frames=epoch, interval=100, blit=True)\n",
+    "# let animation load\n",
+    "time.sleep(1)\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In case you have some difficulties running the cell below without importing certain packages. Run the following code in a terminal in `L96M2lines` environment.\n",
+    "```shell\n",
+    "conda install -c conda-forge ffmpeg\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "display(HTML(f'<div style=\"text-align:center;\">{animation.to_html5_video()}</div>'))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Neural Networks \n",
+    "\n",
+    "The above discussion focussed on fitting a simple function, a line, through our data. However, data in many problems correspond to underlying functions that are much more complex and their exact form is not known. For these problem we need functions that are much more flexible; neural networks are these more flexible functions. \n",
+    "\n",
+    "This is shown by the **Universal Approximation Theorm**, which states that neural networks can approximate any continuous function. A visual demonstration that neural nets can compute any function can be seen in [this page](http://neuralnetworksanddeeplearning.com/chap4.html).\n",
+    "\n",
+    "Now we give a brief overview of neural networks and how to build them using PyTorch. If you want to go through it in depth, check out these resources: [Deep Learning With Pytorch: A 60 Minute Blitz](https://pytorch.org/tutorials/beginner/deep_learning_60min_blitz.html) or [Neural Networks](https://pytorch.org/tutorials/beginner/blitz/neural_networks_tutorial.html)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Generate some data"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# We create many data points, as neural networks need lot of data to train from scratch.\n",
+    "n = 10000\n",
+    "x = torch.ones(n, 1)\n",
+    "\n",
+    "# uniformly sample x points between -1 and 1.\n",
+    "# Underscore functions in pytorch means replace the value (update)\n",
+    "x = x.uniform_(-1.0, 1)\n",
+    "\n",
+    "y = torch.sin(x * 2 * torch.pi) + 0.1 * torch.rand(n, 1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In this problem we create a more wigly function (sin), which can not be approximated as a line. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.plot(x[:, 0], y, \".\", markersize=0.5)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Define the neural network\n",
+    "\n",
+    "There are many many flavors of neural networks, as can be seen at the [neural network zoo](https://www.asimovinstitute.org/neural-network-zoo/). Here we use the simplest - a fully connected neural net. \n",
+    "\n",
+    "Visually the fully connected neural networks may be represented as:\n",
+    "```{figure} https://miro.medium.com/max/720/1*VHOUViL8dHGfvxCsswPv-Q.png\n",
+    ":name: neural-network\n",
+    ":width: 600\n",
+    "\n",
+    "A neural network with 4 hidden layers and an output layer.\n",
+    "```\n",
+    "\n",
+    "Our particular design consists of 3 hidden dense layers, two of which have a non-linear Relu activation function. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class Simple_Neural_Net(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super().__init__()\n",
+    "        self.Dense1 = nn.Linear(1, 30)\n",
+    "        self.Dense2 = nn.Linear(30, 30)\n",
+    "        self.Dense3 = nn.Linear(30, 1)\n",
+    "\n",
+    "        self.relu = nn.ReLU()\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        # This method is automatically executed when\n",
+    "        # we call a object of this class\n",
+    "        x = self.Dense1(x)\n",
+    "        x = self.relu(x)\n",
+    "        x = self.Dense2(x)\n",
+    "        x = self.relu(x)\n",
+    "        x = self.Dense3(x)\n",
+    "\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "neural_net = Simple_Neural_Net()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "net_input = torch.randn(1, 1)\n",
+    "out = neural_net(net_input)\n",
+    "print(\n",
+    "    f\"The output of the random input {net_input.item():.4f} from untrained network is: {out.item():.4f}\"\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Updating the Weights using an Optimizer\n",
+    "\n",
+    "In the linear regression problem above we used a very simple optimization procedure, just adjusting the weights manually. However, in more complex problems it is sometimes more convenient to use the optimization module available in PyTorch as it can allow the use of many advanced options. The implementation of almost every optimizer that we'll ever need can be found in PyTorch itself. The choice of which optimizer we choose might be very important as it will determine how fast the network will be able to learn.\n",
+    "\n",
+    "In the example below, we show one of the popular optimizers `Adam`.\n",
+    "\n",
+    "#### Adam Optimizer\n",
+    "\n",
+    "A popular optimizer that is used in many neural networks is the Adam optimizer. It is an adaptive learning rate method that computes individual learning rates for different parameters. For further reading, check out this [post](https://towardsdatascience.com/adam-latest-trends-in-deep-learning-optimization-6be9a291375c) about Adam, and this [post](https://www.ruder.io/optimizing-gradient-descent/) about other optimizers."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Here we use the Adam optimizer.\n",
+    "learning_rate = 0.005\n",
+    "optimizer = torch.optim.Adam(neural_net.parameters())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Loss function\n",
+    "We use the mean square error as loss function again. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# MSE loss function\n",
+    "loss_fn = torch.nn.MSELoss()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Training the network\n",
+    "The training step is defined below, where `optimizer.step()` essentially does the job of updating the parameters. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def train_step(model, loss_fn, optimizer):\n",
+    "    # Set the model to training mode - important for batch normalization and dropout layers\n",
+    "    # Unnecessary in this situation but added for best practices\n",
+    "\n",
+    "    pred = model(x)\n",
+    "    loss = loss_fn(pred, y)\n",
+    "\n",
+    "    # Backpropagation\n",
+    "    loss.backward()\n",
+    "    optimizer.step()\n",
+    "    optimizer.zero_grad()\n",
+    "\n",
+    "    loss = loss.item()\n",
+    "    return loss"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "When it comes to real training problems, often the data sets are very large and can not be fit into memory. In these cases the data might be broken up into smaller chunks called batches, and the training is done on one batch at a time. A pass over all the batches in the data set is referred to as an epoch. Since our data set is very small we do not break it up into batches here, but still refer to each training step as an epoch. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "epochs = 1000\n",
+    "Loss = np.zeros(epochs)\n",
+    "for t in range(epochs):\n",
+    "    Loss[t] = train_step(neural_net, loss_fn, optimizer)\n",
+    "    if np.mod(t, 200) == 0:\n",
+    "        print(f\"Loss at Epoch {t+1} is \", Loss[t])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(Loss)\n",
+    "plt.xlabel(\"Iteration\")\n",
+    "plt.ylabel(\"Loss\")\n",
+    "plt.yscale(\"log\")\n",
+    "plt.title(\"Loss vs Iteration\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Compare Predictions with Ground Truth"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Generate some points where the predictions of the model will be tested.\n",
+    "# Here we pick the testing domain to be larger than the training domain to check if the model\n",
+    "# has any skill at extrapolation.\n",
+    "x_test = torch.linspace(-1.5, 1.5, 501).reshape(501, 1)\n",
+    "\n",
+    "# Generate the predictions from the trained model.\n",
+    "pred = neural_net(x_test).detach().numpy()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "\n",
+    "plt.plot(x, y, \".\", markersize=0.5, label=\"Data\")\n",
+    "plt.plot(x_test, pred, markersize=0.5, label=\"Predicted fit\")\n",
+    "\n",
+    "plt.legend()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "As seen above, the neural network is able to approximate the sine function relatively well over the range over which the training data is available. \n",
+    "However, it should also be noted that the neural network possesses no skill to extrapolate beyond the range of the data that it was trained on. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Summary\n",
+    "\n",
+    "In this notebook we introduced the basics of machine learning - fitting flexible functions to data. In the next few notebooks we will show how neural networks can be used to model sub-grid forcing in the Lorenz 96 model, and also some more useful topics related to neural network training and interpretation.  "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/local_map_L96_and_Burgers.ipynb b/_sources/notebooks/local_map_L96_and_Burgers.ipynb
new file mode 100644
index 00000000..01897b9d
--- /dev/null
+++ b/_sources/notebooks/local_map_L96_and_Burgers.ipynb
@@ -0,0 +1,1461 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# CNNs and Polynomial Maps"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import matplotlib\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "import matplotlib\n",
+    "import os\n",
+    "from IPython import display\n",
+    "\n",
+    "import torch\n",
+    "from torch.autograd import Variable\n",
+    "import torch.utils.data as Data\n",
+    "from torch import nn, optim\n",
+    "\n",
+    "from L96_model import L96, EulerFwd, RK2, RK4\n",
+    "from utils import GCM_no_param  # The GCM object based on prior tutorials\n",
+    "from utils import GCM_discrete  # New GCM with tendency by global network\n",
+    "from utils import GCM_local_discrete  # New GCM with tendency by local network\n",
+    "from utils import Net_ANN  # Import a simple neural network\n",
+    "\n",
+    "from utils import (\n",
+    "    train_model,\n",
+    ")  # Train model for each epoch (Adapted from Yani's tutorial)\n",
+    "from utils import test_model  # Calculate loss (Adapted from Yani's tutorial)\n",
+    "from utils import train_network  # Wrapper function for training a neural network\n",
+    "\n",
+    "from utils import (\n",
+    "    gen_local_training_data,\n",
+    ")  # Generate individual patches as training data\n",
+    "from utils import (\n",
+    "    get_poly_features,\n",
+    ")  # Generate 2nd-order polynomial features from patches\n",
+    "\n",
+    "# Import Burger's equation class\n",
+    "from utils import Burgers\n",
+    "\n",
+    "%matplotlib notebook"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# I. Introduction\n",
+    "The petagogical model we use to illustrate different methods at M2LInES is the [Lorenz-96 model](file:///Users/ziweili/Downloads/10829-predictability-problem-partly-solved.pdf) by Edward Lorenz published in 1996. However, Ed Lorenz did not provide much physical reasoning behind the equation, and the dynamics of the slow, large-scale variables and fast, small-scale variables are seemingly ad-hoc. To see how, consider the original L96 equation:\n",
+    "\n",
+    "\\begin{align}\n",
+    "\\dot{X_k} &= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k - (hc/b)\\sum_jY_{j, k}\\\\\n",
+    "\\dot{Y_{k,j}} &= - cb Y_{j+1, k} \\left( Y_{j+2, k} - Y_{j-1, k} \\right) - cY_{j, k} + (hc/b)X_k. \n",
+    "\\end{align}\n",
+    "\n",
+    "\n",
+    "The advection terms of $X$ and $Y$ are similar to the usual 1D advection form:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\dot{X_k} \\sim - X_{k-1} \\left( X_{k-2} - X_{k+1}\\right) \\sim X\\nabla X,\n",
+    "\\end{equation}\n",
+    "\n",
+    "but the advection term in $X$ is negative compared with the usual momentum conservation in fluid dynamics: $Du = 0\\Rightarrow \\partial_t u = - u\\nabla u$. \n",
+    "\n",
+    "To simplify the problem a little bit, and understand the skill and stability of machine-learning algorithms when they are applied to physical systems, we consider the 1-variable version of the L96 system. This approach is similar to Brandon Reichl's approach (see notebook at this [link](https://github.com/m2lines/L96_demo/blob/main/08-Implementation/Neural%20Network%20Advection.ipynb)). Here we use this view to consider the performance of the following networks: \n",
+    "1. Local ANN\n",
+    "2. Local polynomial stencils\n",
+    "3. Global CNN\n",
+    "4. Global ANN "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## One-variable L96 equation\n",
+    "\n",
+    "The one-variable L96 equation writes\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\dot{X_k} = - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F\n",
+    "\\end{equation}\n",
+    "\n",
+    "\n",
+    "To see how machine learning helps us with simulating Earth's dynamics, we use three approaches to simulate the 1-variable L96 system. Our approach is based on the following principles: \n",
+    " 1. The networks' perception fields must be local, as required by physical laws and the practical constraints of implementation. \n",
+    " 2. The network structure must obey symmetries. They should be translational invariant, and is compatible with periodic boundary conditions. \n",
+    " 3. The complexity of the networks must be small for interpretability. \n",
+    "\n",
+    "## Local networks\n",
+    "\n",
+    "In one-dimensional setting, we define local maps as the following approximation problem: given a receptive field of the network of width $(2m+1)$: $\\mathbf{X^{(m)}_k} = (X_{k-m}, ..., X_{k}, ..., X_{k+m})$, we approximate function $f(\\mathbf{X^{(m)}_k})$ such that\n",
+    "\n",
+    "\\begin{equation}\n",
+    "f: \\mathbf{X^{(m)}_k} \\rightarrow dX_k/dt\n",
+    "\\end{equation}\n",
+    "\n",
+    "In this tutorial, we set $m = 2$, and the local map is illustrated in the following figure: \n",
+    "```{figure} figs/local_map.png\n",
+    ":width: 400px\n",
+    "\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "we choose the functional form, $f(\\cdot)$, to be one of the following two choices, simple feedforward neural network, and polynomial stencils. \n",
+    "\n",
+    "### 1. Simple feedforward neural network (ANN)\n",
+    "\n",
+    "The feedforward neural network is embedded in the GCM, replacing all terms on the right-hand-side. For this tutorial, the network is fully connected, has two hidden layers with 10 neurons each, and has $tanh()$ as its activation function. The structure can be seen in the following figure:  \n",
+    "\n",
+    "```{figure} figs/local_map_ANN.png\n",
+    ":width: 400px\n",
+    "\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 2. Polynomial stencil\n",
+    "\n",
+    "Physical dynamical systems are usually written as ODEs and PDEs, and when they are solved numerically, they are discretized in space and time into regular ODE sets (whether it's in the spectral space or in the real space). Take the real space discretization for example, the terms on the right-hand side usually take the form of polynomials of the information from adjacent points. The one-variable L96 system is a nice example: \n",
+    "\n",
+    "```{figure} figs/local_map_polynet.png\n",
+    ":width: 400px\n",
+    "\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Global networks\n",
+    "\n",
+    "### 1. Global convolutional neural network (global CNN)\n",
+    "\n",
+    "We apply a simple global convolutional neural network to predict future tendencies. The network has a filter width of 5 (consistent with $m = 2$ as above), 2 covolutional layers and 2 fully connected-layers with 100 neurons each. The first convolution layer has 1 channel as input, 2 channels as output, and the second convolution layer has 2 channels as input, 4 channels as output.  \n",
+    "\n",
+    "\n",
+    "```{figure} figs/diagram_CNN.png\n",
+    ":width: 400px\n",
+    "\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 2. Global feedforward neural network (global ANN)\n",
+    "\n",
+    "As a point of comparison with the global CNN, we also train a global ANN with takes in global field and predicts global tendencies. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Define GCM objects\n",
+    "Here, we define GCM objects for\n",
+    "1. the 1-variable L96 equation\n",
+    "2. the L96 system with tendency represented by global networks (global ANN, global CNN)\n",
+    "3. the L96 system with tendency represented by local networks (local ANN, polynet)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Define neural network objects"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define a convolutional neural network object\n",
+    "class Net_CNN(nn.Module):\n",
+    "    def __init__(\n",
+    "        self,\n",
+    "        inout_size=[100, 100],\n",
+    "        kernel_size=5,\n",
+    "        hidden_neurons=[20, 20],\n",
+    "        strides=[1, 1],\n",
+    "        channels=[2, 4],\n",
+    "    ):\n",
+    "        # Class of a very simple 1-dimensional convolutional neural network. This network contains\n",
+    "        # 2 convolutional layers with different channels channel each, and 2 fully-connected layers\n",
+    "        # after the convolutional layers.\n",
+    "\n",
+    "        # Input parameters:\n",
+    "        #\n",
+    "        # inout_size:     A 2-element list, the first element contains the input size;\n",
+    "        #                 the second element contains the output size.\n",
+    "        #\n",
+    "        # kernel_size:    The size of the convolutional kernel. Same across all conv layers\n",
+    "        #\n",
+    "        # hidden_neurons: List of hidden neurons of the fully-connected layers after the conv layers\n",
+    "\n",
+    "        # Inherit from default class\n",
+    "        super(Net_CNN, self).__init__()\n",
+    "\n",
+    "        # Set padding to half the size of the kernel, and padding mode to periodic (circular)\n",
+    "        padding = kernel_size // 2\n",
+    "        padding_mode = \"circular\"\n",
+    "\n",
+    "        # Set dilation to 1, i.e., no dilation\n",
+    "        dilation = 1\n",
+    "\n",
+    "        # Define two convolutional layers\n",
+    "        self.conv1 = torch.nn.Conv1d(\n",
+    "            1,\n",
+    "            channels[0],\n",
+    "            kernel_size,\n",
+    "            stride=strides[0],\n",
+    "            padding=padding,\n",
+    "            padding_mode=padding_mode,\n",
+    "        )\n",
+    "        self.conv2 = torch.nn.Conv1d(\n",
+    "            channels[0],\n",
+    "            channels[1],\n",
+    "            kernel_size,\n",
+    "            stride=strides[1],\n",
+    "            padding=padding,\n",
+    "            padding_mode=padding_mode,\n",
+    "        )\n",
+    "\n",
+    "        # Define a function to calculate the dimensionality of the input to the fully-connected layer\n",
+    "        # (see definition of L_out in https://pytorch.org/docs/stable/generated/torch.nn.Conv1d.html)\n",
+    "        output_size = lambda L_in, padding, dilation, kernel_size, stride: int(\n",
+    "            (L_in + 2 * padding - dilation * (kernel_size - 1) - 1) / stride + 1\n",
+    "        )\n",
+    "        # Use the function to calculate the flattened output size of the convolutional layers\n",
+    "        output_size1 = output_size(\n",
+    "            inout_size[0], padding, dilation, kernel_size, strides[0]\n",
+    "        )\n",
+    "        output_size2 = output_size(\n",
+    "            output_size1, padding, dilation, kernel_size, strides[1]\n",
+    "        )\n",
+    "\n",
+    "        # Assign information to be used in the fully connected layer and self.forward()\n",
+    "        self.output_size2 = output_size2\n",
+    "        self.channels = channels\n",
+    "\n",
+    "        # Define fully-connected layers after the convolutional layers\n",
+    "        self.linear1 = nn.Linear(output_size2 * channels[1], hidden_neurons[0])\n",
+    "        self.linear2 = nn.Linear(hidden_neurons[0], hidden_neurons[1])\n",
+    "        self.linear3 = nn.Linear(hidden_neurons[1], inout_size[1])\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = torch.relu(self.conv1(x))  # Convolution layer 1\n",
+    "        x = torch.relu(self.conv2(x))  # Convolution layer 2\n",
+    "        x = x.view(-1, 1, self.channels[1] * self.output_size2)  # Flatten\n",
+    "        x = torch.tanh(self.linear1(x))  # Fully-connected layer 1\n",
+    "        x = torch.tanh(self.linear2(x))  # Fully-connected layer 2\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Simulate 1-variable L96 equation as ground truth\n",
+    "The 1-variable simulation has 100 grid points of $X$, and is run 20000 timesteps with $dt = 0.001$. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "np.random.seed(14)  # For reproducibility\n",
+    "torch.manual_seed(14)  # For reproducibility\n",
+    "plot_path = \"./figs/\"\n",
+    "if not os.path.isdir(plot_path):\n",
+    "    os.mkdir(plot_path)\n",
+    "\n",
+    "# Default case, 1-variable L96\n",
+    "K = 100  # The number of slow variables, and\n",
+    "J = 0  # The number of fast variables per slow variable, set to 0\n",
+    "F = 18  # Constant external forcing\n",
+    "h = 0  # No interaction with subgrid scale\n",
+    "dt = 0.001  # Timestep\n",
+    "\n",
+    "# Define an 1-variable L96 object\n",
+    "W = L96(K, J, F=F, dt=dt, h=h)\n",
+    "\n",
+    "# Define total timesteps, time, and sampling interval\n",
+    "Nt = 20000  # Total timesteps\n",
+    "T = dt * Nt  # Total time of simulation\n",
+    "si = dt  # Sampling interval"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Run the L96 model as the ground truth\n",
+    "X_true_temp = W.run(si, T, store=True)[0]\n",
+    "X_true = X_true_temp[:-1, :]\n",
+    "dX_true = X_true_temp[1:, :] - X_true_temp[:-1, :]\n",
+    "del X_true_temp"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Train neural networks with local and global setup, and evaluate their performance\n",
+    "First, we define some training parameters. These training parameters are the same across all training procedures in this notebook. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "train_size = 2000  # Number of training data\n",
+    "test_size = 400  # Number of test data\n",
+    "batch_size = 100  # Number of training data per batch\n",
+    "n_epochs = 1000  # Total number of epochs"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 1. Train a global CNN\n",
+    "The global CNN has 2 convolution layers with ReLU activation, and 2 fully connected layers with tanh activation. The number of input/output channels of the two convolutional layers are all 1. The fully-connected layers has 100 hidden neurons each. \n",
+    "\n",
+    "First, we define training data for the global network. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define a random index to select training and test data\n",
+    "ind = np.random.choice(X_true.shape[0], train_size + test_size)\n",
+    "\n",
+    "# Add one extra dimension to represent channels\n",
+    "X_feature_train_global = X_true[ind[:train_size], :][:, np.newaxis, :]\n",
+    "X_target_train_global = dX_true[ind[:train_size], :][:, np.newaxis, :]\n",
+    "X_feature_test_global = X_true[ind[train_size:], :][:, np.newaxis, :]\n",
+    "X_target_test_global = dX_true[ind[train_size:], :][:, np.newaxis, :]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define CNN parameters\n",
+    "inout_size = [K, K]  # Input and output dimensions, global field\n",
+    "kernel_size = 5  # Kernel width\n",
+    "channels = [2, 4]  # Number of channels of the first and second layer\n",
+    "hidden_neurons = [100, 100]  # Number of hidden neurons of the fully-connected layers\n",
+    "\n",
+    "# Define a global CNN\n",
+    "net_CNN_global = Net_CNN(\n",
+    "    inout_size=inout_size,\n",
+    "    kernel_size=kernel_size,\n",
+    "    hidden_neurons=hidden_neurons,\n",
+    "    channels=channels,\n",
+    ").double()\n",
+    "\n",
+    "# Train the global CNN using global data\n",
+    "net_CNN_global, train_loss_CNN, test_loss_CNN = train_network(\n",
+    "    net_CNN_global,\n",
+    "    X_feature_train_global,\n",
+    "    X_target_train_global,\n",
+    "    X_feature_test_global,\n",
+    "    X_target_test_global,\n",
+    "    batch_size=batch_size,\n",
+    "    n_epochs=n_epochs,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 2. Train a global ANN\n",
+    "The global ANN has 2 hidden layers with 100 hidden neurons each. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define a global ANN\n",
+    "net_ANN_global = Net_ANN([K, 100, 100, K], filter_loc=[0]).double()\n",
+    "\n",
+    "# Train the global ANN using global data\n",
+    "net_ANN_global, train_loss_ANN_global, test_loss_ANN_global = train_network(\n",
+    "    net_ANN_global,\n",
+    "    X_feature_train_global,\n",
+    "    X_target_train_global,\n",
+    "    X_feature_test_global,\n",
+    "    X_target_test_global,\n",
+    "    batch_size=batch_size,\n",
+    "    n_epochs=n_epochs,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 3. Train a local polynomial map\n",
+    "To train local maps, we first generate training data as local information around a grid point. Here, we define a local stencil that extends 2 grid points to the left and right. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define the local map location\n",
+    "filter_start = -2\n",
+    "filter_end = 2  # 5 points: k+2, k+1, k, k-1, k-2\n",
+    "filter_loc = np.arange(filter_start, filter_end + 1)  # Filter location list\n",
+    "\n",
+    "# Generate local maps consistent with the geometry of the stencil\n",
+    "X_local_train, dX_train, X_local_test, dX_test = gen_local_training_data(\n",
+    "    X_true, dX_true, filter_loc, train_size, test_size\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Generate polynomial features from data\n",
+    "X_feature_train_poly = get_poly_features(X_local_train)  # Training data\n",
+    "X_feature_test_poly = get_poly_features(X_local_test)  # Test data\n",
+    "\n",
+    "# Define a polynet as a regular ANN, with no hidden layers and a single bias\n",
+    "net_poly = Net_ANN([X_feature_train_poly.shape[-1], 1], filter_loc=filter_loc).double()\n",
+    "\n",
+    "# Train the polynet\n",
+    "net_poly, train_loss_poly, test_loss_poly = train_network(\n",
+    "    net_poly,\n",
+    "    X_feature_train_poly,\n",
+    "    dX_train,\n",
+    "    X_feature_test_poly,\n",
+    "    dX_test,\n",
+    "    batch_size=batch_size,\n",
+    "    n_epochs=n_epochs,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### 4. Train a local ANN map\n",
+    "Using the local information from above, we train a local ANN as another point of comparison. This local ANN has two hidden layers with 10 neurons each, and has $tanh()$ as activation function. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define a regular ANN consistent with the local map\n",
+    "net_ANN_local = Net_ANN([len(filter_loc), 10, 10, 1], filter_loc=filter_loc).double()\n",
+    "\n",
+    "# Train the local ANN\n",
+    "net_ANN_local, train_loss_ANN, test_loss_ANN = train_network(\n",
+    "    net_ANN_local,\n",
+    "    X_local_train,\n",
+    "    dX_train,\n",
+    "    X_local_test,\n",
+    "    dX_test,\n",
+    "    batch_size=batch_size,\n",
+    "    n_epochs=n_epochs,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Display training error"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "dX_var = dX_train.var()\n",
+    "fig = plt.figure(figsize=(6, 5), dpi=150)\n",
+    "plt.plot(train_loss_ANN_global / dX_var, color=\"tab:blue\", label=\"global ANN\")\n",
+    "plt.plot(test_loss_ANN_global / dX_var, \":\", color=\"tab:blue\")\n",
+    "plt.plot(train_loss_CNN / dX_var, color=\"tab:red\", label=\"global CNN\")\n",
+    "plt.plot(test_loss_CNN / dX_var, \":\", color=\"tab:red\")\n",
+    "plt.plot(train_loss_ANN / dX_var, color=\"tab:grey\", label=\"local ANN\")\n",
+    "plt.plot(test_loss_ANN / dX_var, \":\", color=\"tab:grey\")\n",
+    "plt.plot(train_loss_poly / dX_var, color=\"tab:orange\", label=\"local polynet\")\n",
+    "plt.plot(test_loss_poly / dX_var, \":\", color=\"tab:orange\")\n",
+    "plt.yscale(\"log\")\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.title(\"Normalized MSE loss of different models\")\n",
+    "plt.xlabel(\"Epochs\")\n",
+    "plt.ylabel(\"Normalized MSE\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# II. Test predictions of the networks\n",
+    "global ANN, global CNN, local ANN, and polynomial stencils"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Prediction parameters\n",
+    "temp_dt = si  # Timestep same as the sampling interval (same as training data)\n",
+    "t_plot = int(T / temp_dt)  # Total testing timesteps\n",
+    "t_series_plot = np.arange(0, t_plot + 1) * temp_dt\n",
+    "\n",
+    "# Define GCM objects with tendencies represented by networks\n",
+    "gcm_ANN = GCM_discrete(net_ANN_global)\n",
+    "gcm_CNN = GCM_discrete(net_CNN_global)\n",
+    "gcm_local_ANN = GCM_local_discrete(net_ANN_local, None)\n",
+    "gcm_local_poly = GCM_local_discrete(net_poly, get_poly_features)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Instantaneous predictions"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Instataneous predictions by polynomial network and tanh network\n",
+    "X_predict_poly = np.zeros(X_true.shape)\n",
+    "X_predict_local_ANN = np.zeros(X_true.shape)\n",
+    "X_predict_global_ANN = np.zeros(X_true.shape)\n",
+    "X_predict_global_CNN = np.zeros(X_true.shape)\n",
+    "for t in range(X_true.shape[0] - 1):\n",
+    "    X_predict_poly[t + 1, :] = gcm_local_poly(X_true[t, :], 1)[1, :]\n",
+    "    X_predict_local_ANN[t + 1, :] = gcm_local_ANN(X_true[t, :], 1)[1, :]\n",
+    "    X_predict_global_ANN[t + 1, :] = gcm_ANN(X_true[t, :], 1)[1, :]\n",
+    "    X_predict_global_CNN[t + 1, :] = gcm_CNN(X_true[t, :], 1)[1, :]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_plot = 9  # Example variable, the 9th grid point\n",
+    "fig = plt.figure(figsize=[5, 5])\n",
+    "t_ind = np.arange(1000, 11000)\n",
+    "plt.figure(dpi=150)\n",
+    "plt.title(\"Instantaneous predictions (shifted)\")\n",
+    "plt.plot(\n",
+    "    dX_true[t_ind, X_plot],\n",
+    "    0.03\n",
+    "    + X_predict_global_ANN[t_ind + 1, X_plot]\n",
+    "    - X_predict_global_ANN[t_ind, X_plot],\n",
+    "    label=\"global ANN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    dX_true[t_ind, X_plot],\n",
+    "    0.02\n",
+    "    + X_predict_global_CNN[t_ind + 1, X_plot]\n",
+    "    - X_predict_global_CNN[t_ind, X_plot],\n",
+    "    label=\"global CNN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    dX_true[t_ind, X_plot],\n",
+    "    0.01 + X_predict_poly[t_ind + 1, X_plot] - X_predict_poly[t_ind, X_plot],\n",
+    "    label=\"local polynet\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    dX_true[t_ind, X_plot],\n",
+    "    X_predict_local_ANN[t_ind + 1, X_plot] - X_predict_local_ANN[t_ind, X_plot],\n",
+    "    label=\"local ANN\",\n",
+    ")\n",
+    "plt.legend(frameon=False, fontsize=7)\n",
+    "plt.ylim([-0.3, 0.2])\n",
+    "plt.xlim([-0.3, 0.2])\n",
+    "plt.xlabel(r\"Ground truth $dX_{0}$\".format(X_plot))\n",
+    "plt.ylabel(r\"Predicted $dX_{0}$\".format(X_plot))\n",
+    "plt.tight_layout();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Time-dependent initial-value simulations"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Perform simulations with the GCM objects\n",
+    "init_cond = X_true[0, :]\n",
+    "X_simulation_global_ANN = gcm_ANN(init_cond, t_plot)\n",
+    "X_simulation_global_CNN = gcm_CNN(init_cond, t_plot)\n",
+    "X_simulation_local_ANN = gcm_local_ANN(init_cond, t_plot)\n",
+    "X_simulation_poly = gcm_local_poly(init_cond, t_plot)\n",
+    "\n",
+    "# Perform the groud truth simulation\n",
+    "gcm_no_param = GCM_no_param(F, time_stepping=RK4)\n",
+    "X_no_param, _ = gcm_no_param(init_cond, temp_dt, t_plot)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Show trajectories of a single variable\n",
+    "temp_t = min([X_no_param.shape[0], 1200])\n",
+    "t_ind = np.arange(0, 1000)\n",
+    "fig = plt.figure(figsize=[9, 4], dpi=150)\n",
+    "plt.subplot(1, 2, 1)\n",
+    "plt.title(\"Initial-value simulations, $X_9$\")\n",
+    "plt.plot(t_series_plot[t_ind], X_no_param[t_ind, X_plot], \"k\", label=\"Truth\")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    X_simulation_global_ANN[t_ind, X_plot],\n",
+    "    \"-.\",\n",
+    "    label=\"Global ANN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    X_simulation_global_CNN[t_ind, X_plot],\n",
+    "    \"-.\",\n",
+    "    label=\"Global CNN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind], X_simulation_poly[t_ind, X_plot], \"--\", label=\"Local polynet\"\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind], X_simulation_local_ANN[t_ind, X_plot], \"--\", label=\"Local ANN\"\n",
+    ")\n",
+    "plt.legend(frameon=False, loc=\"best\", fontsize=7)\n",
+    "plt.xlabel(\"Time\")\n",
+    "plt.ylabel(r\"$X_{0:d}$\".format(X_plot))\n",
+    "fig.tight_layout()\n",
+    "\n",
+    "# Energy conservation\n",
+    "plt.subplot(1, 2, 2)\n",
+    "plt.title(\"Initial-value simulations, energy\")\n",
+    "plt.plot(t_series_plot[t_ind], np.sum(X_no_param[t_ind, :] ** 2, 1), \"k\", label=\"Truth\")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_simulation_global_ANN[t_ind, :] ** 2, 1),\n",
+    "    \"-.\",\n",
+    "    label=\"Global ANN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_simulation_global_CNN[t_ind, :] ** 2, 1),\n",
+    "    \"-.\",\n",
+    "    label=\"Global CNN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_simulation_poly[t_ind, :] ** 2, 1),\n",
+    "    \"--\",\n",
+    "    label=\"Local polynet\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_simulation_local_ANN[t_ind, :] ** 2, 1),\n",
+    "    \"--\",\n",
+    "    label=\"Local ANN\",\n",
+    ")\n",
+    "plt.legend(frameon=False, loc=\"best\", fontsize=7)\n",
+    "plt.ylim(np.array([0, 2]) * np.sum(init_cond**2))\n",
+    "plt.xlabel(\"Time\")\n",
+    "plt.ylabel(r\"Energy ($|\\mathbf{X}|^2$)\")\n",
+    "fig.tight_layout();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Show parameters learned by the polynet"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define a function to show parameters in the polynet\n",
+    "def plot_parameters(net_poly, si, filter_loc, var_str, weights_true):\n",
+    "    # Store the weights of the polynomial networks and normalize by sampling interval\n",
+    "    weights_poly = (list(net_poly.named_parameters())[0][1].data.numpy() / si).squeeze()\n",
+    "\n",
+    "    n = len(filter_loc)\n",
+    "    n_feature = int(n + n + n * (n - 1) / 2)\n",
+    "    from itertools import combinations\n",
+    "\n",
+    "    index_list = list(combinations(np.arange(n) + filter_loc[0], 2))\n",
+    "    x_plot = np.arange(n_feature) * 0.8\n",
+    "\n",
+    "    def label_part(f):\n",
+    "        if f > 0:\n",
+    "            return r\"+{0:d}\".format(f)\n",
+    "        elif f < 0:\n",
+    "            return r\"-{0:d}\".format(-f)\n",
+    "        elif f == 0:\n",
+    "            return r\"\"\n",
+    "\n",
+    "    xlabels = (\n",
+    "        [r\"$\" + var_str + \"_{k\" + label_part(-f) + \"}$\" for f in filter_loc]\n",
+    "        + [r\"$\" + var_str + \"^2_{k\" + label_part(-f) + \"}$\" for f in filter_loc]\n",
+    "        + [\n",
+    "            r\"$\"\n",
+    "            + var_str\n",
+    "            + \"_{k\"\n",
+    "            + label_part(-index[0])\n",
+    "            + \"}\"\n",
+    "            + var_str\n",
+    "            + \"_{k\"\n",
+    "            + label_part(-index[1])\n",
+    "            + \"}$\"\n",
+    "            for index in index_list\n",
+    "        ]\n",
+    "    )\n",
+    "    # Scatter plot\n",
+    "    fig = plt.figure(figsize=[9, 3])\n",
+    "    cmap = matplotlib.colormaps[\"bone_r\"]\n",
+    "    plt.bar(\n",
+    "        x_plot - 0.10,\n",
+    "        weights_poly,\n",
+    "        label=\"Polynet\",\n",
+    "        color=\"tab:blue\",\n",
+    "        edgecolor=\"k\",\n",
+    "        width=0.15,\n",
+    "    )\n",
+    "    plt.bar(\n",
+    "        x_plot + 0.10,\n",
+    "        weights_true,\n",
+    "        label=\"Truth\",\n",
+    "        color=\"tab:brown\",\n",
+    "        edgecolor=\"k\",\n",
+    "        width=0.15,\n",
+    "    )\n",
+    "    plt.plot([x_plot[0] - 5, x_plot[-1] + 5], np.zeros(2), \"--k\", linewidth=1.0)\n",
+    "    plt.legend(frameon=False, loc=\"best\")\n",
+    "    plt.xticks(x_plot, rotation=-30)\n",
+    "    plt.gca().set_xticklabels(xlabels)\n",
+    "    plt.xlim([x_plot[0] - 1, x_plot[-1] + 1])\n",
+    "    fig.tight_layout()\n",
+    "\n",
+    "\n",
+    "# Show parameters learned by the polynet\n",
+    "weights_true = np.array([0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1])\n",
+    "plot_parameters(net_poly, si, filter_loc, \"X\", weights_true)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Compare with the original 1-variable L96 equation: \n",
+    "\\begin{equation}\n",
+    "\\dot{X_k} = - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F\n",
+    "\\end{equation}\n",
+    "we see that the polynomial stencil can reconstruct the advection term and the exponential decay term. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## III. Energy-conserved case\n",
+    "It would be interesting to see the performance of ANN, CNN, and polynomial stencils when the underlying system conserves energy:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\dot{X_k} = - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right)\n",
+    "\\end{equation}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Train networks"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "F = 0  # no forcing\n",
+    "a = 0  # no exponential decay\n",
+    "h = 0  # no interaction with subgrid scale\n",
+    "\n",
+    "Nt = 20000\n",
+    "dt = 0.001\n",
+    "si = dt  # Sampling interval\n",
+    "T = dt * Nt  # Total time of simulation\n",
+    "\n",
+    "K = 100  # The number of slow variables, and\n",
+    "J = 0  # The number of fast variables, set to 0\n",
+    "\n",
+    "# Define the ground truth model to generate training data\n",
+    "W = L96(K, J, F=F, dt=dt, h=h)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from utils import run_analysis\n",
+    "\n",
+    "(\n",
+    "    net_ANN_global_2,\n",
+    "    net_CNN_global_2,\n",
+    "    net_ANN_local_2,\n",
+    "    net_poly_2,\n",
+    "    X_true_2,\n",
+    "    dX_true_2,\n",
+    ") = run_analysis(W, si, T, n_epochs=n_epochs)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Generate simulation data with networks trained on the energy-conserved case"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#### Test predictions\n",
+    "temp_dt = si\n",
+    "t_plot = X_true_2.shape[0] - 1\n",
+    "t_series_plot = np.arange(0, t_plot + 1) * temp_dt\n",
+    "\n",
+    "# Define GCM objects using the trained networks\n",
+    "gcm_ANN = GCM_discrete(net_ANN_global_2)\n",
+    "gcm_CNN = GCM_discrete(net_CNN_global_2)\n",
+    "gcm_local_ANN = GCM_local_discrete(net_ANN_local_2, None)\n",
+    "gcm_local_poly = GCM_local_discrete(net_poly_2, get_poly_features)\n",
+    "\n",
+    "# Instataneous predictions by polynomial network and tanh network\n",
+    "X_predict_poly = np.zeros(X_true_2.shape)\n",
+    "X_predict_local_ANN = np.zeros(X_true_2.shape)\n",
+    "X_predict_global_ANN = np.zeros(X_true_2.shape)\n",
+    "X_predict_global_CNN = np.zeros(X_true_2.shape)\n",
+    "for t in range(X_true_2.shape[0] - 1):\n",
+    "    X_predict_poly[t + 1, :] = gcm_local_poly(X_true_2[t, :], 1)[1, :]\n",
+    "    X_predict_local_ANN[t + 1, :] = gcm_local_ANN(X_true_2[t, :], 1)[1, :]\n",
+    "    X_predict_global_ANN[t + 1, :] = gcm_ANN(X_true_2[t, :], 1)[1, :]\n",
+    "    X_predict_global_CNN[t + 1, :] = gcm_CNN(X_true_2[t, :], 1)[1, :]\n",
+    "\n",
+    "# Perform simulations with the GCM objects\n",
+    "init_cond = X_true_2[0, :]\n",
+    "X_simulation_global_ANN = gcm_ANN(init_cond, t_plot)\n",
+    "X_simulation_global_CNN = gcm_CNN(init_cond, t_plot)\n",
+    "X_simulation_local_ANN = gcm_local_ANN(init_cond, t_plot)\n",
+    "X_simulation_poly = gcm_local_poly(init_cond, t_plot)\n",
+    "\n",
+    "# Perform the groud truth simulation with different forward-stepping schemes\n",
+    "# 1. RK4\n",
+    "gcm_no_param = GCM_no_param(F, time_stepping=RK4)\n",
+    "X_no_param, _ = gcm_no_param(init_cond, temp_dt, t_plot)\n",
+    "# 2. RK2\n",
+    "gcm_no_param_RK2 = GCM_no_param(F, time_stepping=RK2)\n",
+    "X_no_param_RK2, _ = gcm_no_param_RK2(init_cond, temp_dt, t_plot)\n",
+    "# 3. Euler\n",
+    "gcm_no_param_Euler = GCM_no_param(F, time_stepping=EulerFwd)\n",
+    "X_no_param_Euler, _ = gcm_no_param_Euler(init_cond, temp_dt, t_plot)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Plot instantaneous prediction"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_plot = 9  # Example variable, the 9th grid point\n",
+    "fig = plt.figure(figsize=[5, 5])\n",
+    "t_ind = np.arange(1000, 11000)\n",
+    "plt.figure(dpi=150)\n",
+    "plt.title(\"Instantaneous predictions (shifted)\")\n",
+    "plt.plot(\n",
+    "    dX_true_2[t_ind, X_plot],\n",
+    "    0.18\n",
+    "    + X_predict_global_ANN[t_ind + 1, X_plot]\n",
+    "    - X_predict_global_ANN[t_ind, X_plot],\n",
+    "    label=\"global ANN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    dX_true_2[t_ind, X_plot],\n",
+    "    0.12\n",
+    "    + X_predict_global_CNN[t_ind + 1, X_plot]\n",
+    "    - X_predict_global_CNN[t_ind, X_plot],\n",
+    "    label=\"global CNN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    dX_true_2[t_ind, X_plot],\n",
+    "    0.06 + X_predict_poly[t_ind + 1, X_plot] - X_predict_poly[t_ind, X_plot],\n",
+    "    label=\"local polynet\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    dX_true_2[t_ind, X_plot],\n",
+    "    X_predict_local_ANN[t_ind + 1, X_plot] - X_predict_local_ANN[t_ind, X_plot],\n",
+    "    label=\"local ANN\",\n",
+    ")\n",
+    "plt.legend(frameon=False, fontsize=7)\n",
+    "# plt.ylim([-0.3, 0.2])\n",
+    "# plt.xlim([-0.3, 0.2])\n",
+    "plt.xlabel(r\"Ground truth $dX_{0}$\".format(X_plot))\n",
+    "plt.ylabel(r\"Predicted $dX_{0}$\".format(X_plot))\n",
+    "plt.tight_layout();"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Evolution of one variable\n",
+    "temp_t = min([X_no_param.shape[0], 1200])\n",
+    "X_plot = 9\n",
+    "t_ind = np.arange(0, 2000)\n",
+    "fig = plt.figure(figsize=[8, 6])\n",
+    "plt.subplot(2, 1, 1)\n",
+    "plt.title(\"Initial-value simulations, $X_9$\")\n",
+    "plt.plot(t_series_plot[t_ind], X_no_param[t_ind, X_plot], \"k\", label=\"Truth (RK4)\")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind], X_no_param_Euler[t_ind, X_plot], \"grey\", label=\"Truth (Euler)\"\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    X_simulation_global_ANN[t_ind, X_plot],\n",
+    "    \"-.\",\n",
+    "    label=\"global ANN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    X_simulation_global_CNN[t_ind, X_plot],\n",
+    "    \"-.\",\n",
+    "    label=\"global CNN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind], X_simulation_poly[t_ind, X_plot], \"--\", label=\"local polynet\"\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind], X_simulation_local_ANN[t_ind, X_plot], \"--\", label=\"local ANN\"\n",
+    ")\n",
+    "plt.legend(frameon=False, loc=\"best\")\n",
+    "plt.xlabel(\"Time\")\n",
+    "plt.ylabel(r\"$X_{0:d}$\".format(X_plot))\n",
+    "plt.xlim([-0.1, 3])\n",
+    "plt.ylim([-80, 80])\n",
+    "fig.tight_layout()\n",
+    "\n",
+    "# Energy evolution\n",
+    "plt.subplot(2, 1, 2)\n",
+    "plt.title(\"Initial-value simulations, energy\")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind], np.sum(X_no_param[t_ind, :] ** 2, 1), \"k\", label=\"Truth (RK4)\"\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_no_param_Euler[t_ind, :] ** 2, 1),\n",
+    "    \"grey\",\n",
+    "    label=\"Truth (Euler)\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_simulation_global_ANN[t_ind, :] ** 2, 1),\n",
+    "    \"-.\",\n",
+    "    label=\"global ANN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_simulation_global_CNN[t_ind, :] ** 2, 1),\n",
+    "    \"-.\",\n",
+    "    label=\"global CNN\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_simulation_poly[t_ind, :] ** 2, 1),\n",
+    "    \"--\",\n",
+    "    label=\"local polynet\",\n",
+    ")\n",
+    "plt.plot(\n",
+    "    t_series_plot[t_ind],\n",
+    "    np.sum(X_simulation_local_ANN[t_ind, :] ** 2, 1),\n",
+    "    \"--\",\n",
+    "    label=\"local ANN\",\n",
+    ")\n",
+    "plt.legend(frameon=False, loc=\"best\")\n",
+    "plt.ylim(np.array([0, 2]) * np.sum(init_cond**2))\n",
+    "plt.xlabel(\"Time\")\n",
+    "plt.ylabel(r\"Energy ($|\\mathbf{X}|^2$)\")\n",
+    "plt.xlim([-0.1, 3])\n",
+    "fig.tight_layout();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# III. A more realistic model: the Burger's equation\n",
+    "\n",
+    "## Problem with the 1-variable L96 equation\n",
+    "\n",
+    "One problem with the L96 model is that it sometimes lacks the physical basis. If we look at the temporal evolution of the 1-variable L96 equation, we see that its dynamics look quite unstable, and most of its variabilities can be numerical artifacts. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Start from a bump initial condition"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Energy conserving case, 1-variable L96\n",
+    "K = 100  # The number of slow variables, and\n",
+    "J = 0  # The number of fast variables per slow variable, set to 0\n",
+    "F = 0  # no external forcing\n",
+    "a = 0  # no exponential decay\n",
+    "h = 0  # no interaction with subgrid scale\n",
+    "\n",
+    "# Set a bump initial condition\n",
+    "x = np.arange(0, K)\n",
+    "X_init = np.exp(-(((x - x.mean()) / K * 10) ** 2)) * 10\n",
+    "\n",
+    "W_conserve = L96(K, J, F=F, dt=dt, h=h)\n",
+    "(\n",
+    "    X_true_3,\n",
+    "    _,\n",
+    "    _,\n",
+    ") = W_conserve.run(si, T, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Animate conservative 1-variable L96 equation\n",
+    "plt.ioff()  # close notebook mode\n",
+    "from matplotlib.animation import FuncAnimation\n",
+    "\n",
+    "interval = 10\n",
+    "frames = 200\n",
+    "\n",
+    "fig = plt.figure(dpi=150)\n",
+    "ax = plt.axes(xlim=(0, K), ylim=(-25, 25))\n",
+    "plt.title(\"Energy-conserved initial value problem\")\n",
+    "(line,) = ax.plot([], [], lw=3)\n",
+    "plt.xlabel(r\"$k$\")\n",
+    "plt.ylabel(r\"$X$\")\n",
+    "\n",
+    "\n",
+    "def init():\n",
+    "    line.set_data([], [])\n",
+    "    return (line,)\n",
+    "\n",
+    "\n",
+    "def animate(i):\n",
+    "    x = np.arange(0, K)\n",
+    "    y = X_true_3[i * 15, :]\n",
+    "    line.set_data(x, y)\n",
+    "    return (line,)\n",
+    "\n",
+    "\n",
+    "anim = FuncAnimation(fig, animate, init_func=init, frames=frames, interval=1, blit=True)\n",
+    "gif_name = os.path.join(plot_path, \"L96_one_variable.gif\")\n",
+    "anim.save(gif_name, writer=\"pillow\")\n",
+    "# reopen inline mode\n",
+    "%matplotlib notebook"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "![SegmentLocal](figs/L96_one_variable.gif \"segment\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The 1-variable energy-conserved case with the L96 system is just an advection equation with the sign of the sign advection term flipped:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\dot{X_k} = - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right)\n",
+    "\\end{equation}\n",
+    "\n",
+    "We can find much similarity with the Burger's equation. \n",
+    "\n",
+    "## The Burger's equation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "To see the performance of the neural networks on a more realistic model, we use the Burger's equation as an example. The Burger's equation is an advection-diffusion equation. It has a diffusive term which reduces the tendency to develop shock waves and numerical instabilities. \n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\frac{\\partial u}{\\partial t} = -u\\cdot\\nabla u + \\nu\\nabla^2 u. \n",
+    "\\end{equation}\n",
+    "\n",
+    "When discretized with simple finite difference, the tendency of $u$ at a particular location be written as functions of adjacent variables\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\frac{d u_k}{d t} = -\\frac{1}{2\\Delta x}u_k (u_{k+1} - u_{k-1}) + \\frac{\\nu}{\\Delta x^2} (u_{k+1} - 2u_k + u_{k-1}), \n",
+    "\\end{equation}\n",
+    "\n",
+    "which makes it an ideal target for learning local maps with machine-learning algorithms. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Generate training data from Burger's equation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define parameters of the Burger's equation\n",
+    "nu = 0.3\n",
+    "dx = 1\n",
+    "K = 100\n",
+    "dt = 0.01\n",
+    "T = 100\n",
+    "si = dt\n",
+    "x = np.arange(0, K)\n",
+    "\n",
+    "# Use two sets of initial-value simulations for training\n",
+    "W1 = Burgers(K, nu, dx, dt=dt)\n",
+    "init_cond1 = np.exp(-(((x - x.mean()) / K * 10) ** 2))\n",
+    "W1.set_init(init_cond1)\n",
+    "\n",
+    "init_cond2 = -np.exp(-np.abs((x - x.mean()) / K * 15))\n",
+    "W2 = Burgers(K, nu, dx, dt=dt)\n",
+    "W2.set_init(init_cond2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(init_cond1, lw=2, label=\"initial condition 1\")\n",
+    "plt.plot(init_cond2, lw=2, label=\"initial condition 2\")\n",
+    "plt.legend(frameon=False, fontsize=7)\n",
+    "plt.xlabel(\"k\")\n",
+    "plt.ylabel(\"u\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Train the same set of networks with the Burger's equation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Import a wrapper function to perform all of the training above\n",
+    "from utils import run_analysis\n",
+    "\n",
+    "(\n",
+    "    net_ANN_global_burger,\n",
+    "    net_CNN_global_burger,\n",
+    "    net_ANN_local_burger,\n",
+    "    net_poly_burger,\n",
+    "    X_true_burger,\n",
+    "    dX_true_burger,\n",
+    ") = run_analysis([W1, W2], si, T, n_epochs=n_epochs, Poly_init=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#### Test predictions with a different initial condition\n",
+    "gcm_ANN = GCM_discrete(net_ANN_global_burger)\n",
+    "gcm_CNN = GCM_discrete(net_CNN_global_burger)\n",
+    "gcm_local_ANN = GCM_local_discrete(net_ANN_local_burger, None)\n",
+    "gcm_local_poly = GCM_local_discrete(net_poly_burger, get_poly_features)\n",
+    "\n",
+    "# Define a different initial condition\n",
+    "x = np.arange(0, K)\n",
+    "t_plot = int(T / si)\n",
+    "init_cond = np.exp(-np.abs((x - 0.1 * x.mean()) / K * 20))\n",
+    "\n",
+    "# Calculate ground truth using Burger's equation\n",
+    "W1.set_init(init_cond)\n",
+    "X_simulation_truth = W1.run(si, T, store=True)[0]\n",
+    "\n",
+    "# Calculate initial-value simulations with the trained networks\n",
+    "X_simulation_global_ANN = gcm_ANN(init_cond, t_plot)\n",
+    "X_simulation_global_CNN = gcm_CNN(init_cond, t_plot)\n",
+    "X_simulation_local_ANN = gcm_local_ANN(init_cond, t_plot)\n",
+    "X_simulation_poly = gcm_local_poly(init_cond, t_plot)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.ioff()\n",
+    "from utils import animate_Burgers\n",
+    "\n",
+    "animate_Burgers(\n",
+    "    X_simulation_truth,\n",
+    "    X_simulation_global_ANN,\n",
+    "    X_simulation_global_CNN,\n",
+    "    X_simulation_local_ANN,\n",
+    "    X_simulation_poly,\n",
+    "    plot_path,\n",
+    ")\n",
+    "%matplotlib notebook"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "![SegmentLocal](figs/Burgers_simulation.gif \"segment\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "weights_true = np.array(\n",
+    "    [\n",
+    "        0,\n",
+    "        nu / dx**2,\n",
+    "        -2 * nu / dx**2,\n",
+    "        nu / dx**2,\n",
+    "        0,\n",
+    "        0,\n",
+    "        0,\n",
+    "        0,\n",
+    "        0,\n",
+    "        0,\n",
+    "        0,\n",
+    "        0,\n",
+    "        0,\n",
+    "        0,\n",
+    "        -1 / 2 / dx,\n",
+    "        0,\n",
+    "        0,\n",
+    "        1 / 2 / dx,\n",
+    "        0,\n",
+    "        0,\n",
+    "    ]\n",
+    ")\n",
+    "filter_start = -2\n",
+    "filter_end = 2\n",
+    "# this includes k+2, k+1, k, k-1, k-2, note that roll by +2 means looking at location k-2\n",
+    "filter_loc = np.arange(filter_start, filter_end + 1)\n",
+    "plot_parameters(net_poly_burger, si, filter_loc, \"u\", weights_true)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The polynet appropriately recovers the terms in the Burger's equation: \n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\frac{d u_k}{d t} = 0.5 u_k (u_{k+1} - u_{k-1}) + 0.3 (u_{k+1} - 2u_k + u_{k-1}), \n",
+    "\\end{equation}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Conclusions\n",
+    "\n",
+    "- Convolutional neural networks has better error performance during training, partially due to its added complexity\n",
+    "- Local maps is potentially more useful than global maps in the following aspects:\n",
+    "    - It achieves better performance in initial-value simulations with smaller number of parameters, energy-conservation properties, and less training data. \n",
+    "    - It is not constrained by training data geometry -- can be easily generalized to larger simulation domains. \n",
+    "    - It does not need access to global data\n",
+    "    - Local maps are more explanable due to its simplified structure and parameters\n",
+    "- Local polynomial stencils might be useful as an alternative functional class than off-the-shelf ML algorithms."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/random_forest_parameterization.ipynb b/_sources/notebooks/random_forest_parameterization.ipynb
new file mode 100644
index 00000000..5b8cc42e
--- /dev/null
+++ b/_sources/notebooks/random_forest_parameterization.ipynb
@@ -0,0 +1,1170 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Random Forest\n",
+    "\n",
+    "In this notebook we present some basic application of random forest to\n",
+    "the parameterization of the L96 model previously introduced."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "pycharm": {
+     "name": "#%%\n"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "import L96_model\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "from sklearn import tree\n",
+    "from sklearn.ensemble import RandomForestRegressor, GradientBoostingRegressor\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from collections import deque\n",
+    "from sklearn.model_selection import RandomizedSearchCV\n",
+    "from L96_model import RK4, L96_eq1_xdot\n",
+    "\n",
+    "import torch\n",
+    "import torch.nn as nn\n",
+    "import torch.nn.functional as FF"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Main script parameters\n",
+    "K = 8\n",
+    "J = 32\n",
+    "FORCING = 18\n",
+    "dt = 0.01\n",
+    "T_SPINUP = 50\n",
+    "T_SIMULATIONS = 200"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "pycharm": {
+     "name": "#%%\n"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "model = L96_model.L96(K, J, F=FORCING)\n",
+    "# First run for spin up\n",
+    "model.run(dt, T_SPINUP, store=True)\n",
+    "# The data from the run below will be used both for offline training and testing\n",
+    "X_history, Y_history, t, closure = model.run(\n",
+    "    dt, T_SIMULATIONS, store=True, return_coupling=True\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# We will use the quantity below in our online test\n",
+    "var_X = X_history.var()\n",
+    "var_X"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "pycharm": {
+     "name": "#%% md\n"
+    }
+   },
+   "source": [
+    "Scatter-plot of subgrid tendency vs large-scale state X"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "pycharm": {
+     "name": "#%%\n"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "_ = plt.hist2d(X_history.flatten(), closure.flatten(), bins=50, cmap=\"binary\")\n",
+    "plt.xlabel(\"State X\")\n",
+    "plt.ylabel(\"Closure\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "pycharm": {
+     "name": "#%% md\n"
+    }
+   },
+   "source": [
+    "## Starting with a single regression tree"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We start with a very simple approach: we use a single value of the state as our feature (i.e. our feature is a scalar), and the subgrid tendency for the same large-scale variable as the target."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Offline training / fitting and testing"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "closure.shape, X_history.shape"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "First we split the data into training and test. Since the relation between $X_k$ and its associated subgrid tendency is not expected to depend on $k$, we transform the recorded history into a column vector before splitting."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_train, X_test, xy_train, xy_test = train_test_split(\n",
+    "    X_history.flatten().reshape(-1, 1), closure.flatten(), test_size=0.33, shuffle=False\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(f\"X_train has shape: {X_train.shape}\")\n",
+    "print(f\"xy_train has shape: {xy_train.shape}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Then we define our decision tree. For the sake of the example, we limit the number of leaves to 15. Feel free to increase that number and see how the plot below, showing the true and fitted data, changes. In particular, for larger values, we can see that we overfit the data."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "single_tree = tree.DecisionTreeRegressor(max_leaf_nodes=15)\n",
+    "single_tree.fit(X_train, xy_train)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here we plot the true training data (the large scale state values on the x-axis, and the parameterization value on the y-axis) and the fitted data."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(X_train, xy_train, \"b*\")\n",
+    "plt.plot(X_train, single_tree.predict(X_train), \"r*\")\n",
+    "plt.xlabel(\"X\")\n",
+    "plt.ylabel(\"Parameterization\")\n",
+    "plt.legend((\"True\", \"Fitted\"), fontsize=7)\n",
+    "plt.title(\"Fit of a regression tree with 15 leaves\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now, same plot but for the test data. The test data has not been seen during the fitting procedure, so in some sense this is the true test. While we may fit very well the training data, overfitting will result in poor performance on the test dataset."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(X_test, xy_test, \"b*\")\n",
+    "plt.plot(X_test, single_tree.predict(X_test), \"r*\")\n",
+    "plt.xlabel(\"X\")\n",
+    "plt.ylabel(\"Parameterization\")\n",
+    "plt.legend((\"True\", \"Predicted\"), fontsize=7)\n",
+    "plt.title(\"Test of a regression tree with 15 leaves\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Fit and test scores"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can compute the $R^2$ score for both the training data (fit score) and test data.\n",
+    "\\begin{equation*}\n",
+    "R^2 = 1 - \\frac{\\sum_{i}{(\\hat{S}_i - S_i)^2}}{\\sum_{i}{(S_i - S_\\text{mean})^2}}\n",
+    "\\end{equation*}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fit_score = single_tree.score(X_train, xy_train)\n",
+    "test_score = single_tree.score(X_test, xy_test)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(fit_score, test_score)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(xy_test[:8000:K])\n",
+    "plt.plot(single_tree.predict(X_test)[:8000:K])\n",
+    "plt.xlabel(\"time step\")\n",
+    "plt.ylabel(r\"$xy_0$\")\n",
+    "plt.legend((\"True\", \"Predicted\"), fontsize=7)\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Overfitting"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "To illustrate what is meant by overfitting, we will increase the number of leaves, and store the fit loss and test loss for plotting"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "nb_leaves = list(map(lambda x: int(np.exp(x / 2)) + 1, range(1, 20)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fit_scores, test_scores = [], []\n",
+    "for nb in nb_leaves:\n",
+    "    current_tree = tree.DecisionTreeRegressor(max_leaf_nodes=nb)\n",
+    "    current_tree.fit(X_train, xy_train)\n",
+    "    fit_scores.append(current_tree.score(X_train, xy_train))\n",
+    "    test_scores.append(current_tree.score(X_test, xy_test))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.loglog(nb_leaves, list(fit_scores), \"*\")\n",
+    "plt.loglog(nb_leaves, list(test_scores), \"*\")\n",
+    "plt.legend((\"fit_score\", \"test_scores\"), fontsize=7)\n",
+    "plt.xlabel(\"Number of leaf nodes\")\n",
+    "plt.ylabel(\"Score\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Online implementation of the parameterization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now let's implement this as a parameterization for L96."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class Parameterization:\n",
+    "    def __init__(self, predictor):\n",
+    "        self.predictor = predictor\n",
+    "\n",
+    "    def __call__(self, X):\n",
+    "        X = X.reshape(-1, 1)\n",
+    "        return self.predictor.predict(X)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here we use the GCM class defined by Yani in his notebook"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class GCM:\n",
+    "    def __init__(self, F, parameterization, time_stepping=RK4):\n",
+    "        \"\"\"\n",
+    "        GCM with paramerization\n",
+    "        Args:\n",
+    "            F: forcing\n",
+    "            parameterization: function that takes parameters and returns a tendency\n",
+    "            time_stepping: time stepping method\n",
+    "        \"\"\"\n",
+    "        self.F = F\n",
+    "        self.parameterization = parameterization\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X):\n",
+    "        \"\"\"\n",
+    "        Args:\n",
+    "            X: state vector\n",
+    "        \"\"\"\n",
+    "        return L96_eq1_xdot(X, self.F) + self.parameterization(X)\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt):\n",
+    "        \"\"\"\n",
+    "        Args:\n",
+    "            X0: initial conditions\n",
+    "            dt: time increment\n",
+    "            nt: number of forward steps to take\n",
+    "            param: parameters of our closure\n",
+    "        \"\"\"\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gcm = GCM(model.F, Parameterization(single_tree))\n",
+    "gcm_no_param = GCM(model.F, lambda x: 0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n_steps = 2000"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def online_test(param, n_runs, n_steps):\n",
+    "    simulations = []\n",
+    "    gcm = GCM(model.F, param)\n",
+    "    for i in range(n_runs):\n",
+    "        if i % 10 == 0:\n",
+    "            print(\"run \", i)\n",
+    "        X_param, t = gcm(model.X, model.dt, n_steps)\n",
+    "        X_no_param, t = gcm_no_param(model.X, model.dt, n_steps)\n",
+    "        X_true, _, _ = model.run(model.dt, n_steps * model.dt, store=True)\n",
+    "        simulations.append((X_true, X_param))\n",
+    "    squared_errors = np.stack(\n",
+    "        [((true - sim) ** 2).mean(axis=-1) for true, sim in simulations], axis=0\n",
+    "    )\n",
+    "    return np.median(squared_errors, axis=0) / var_X"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gcm = GCM(model.F, Parameterization(single_tree))\n",
+    "gcm_no_param = GCM(model.F, lambda x: 0)\n",
+    "X_param, t = gcm(model.X, model.dt, n_steps)\n",
+    "X_no_param, t = gcm_no_param(model.X, model.dt, n_steps)\n",
+    "X_true, _, _ = model.run(model.dt, n_steps * model.dt, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 0], label=\"true\")\n",
+    "plt.plot(t, X_no_param[:, 0], label=\"GCM without parameterization\")\n",
+    "plt.plot(t, X_param[:, 0], label=\"GCM with paramerization\")\n",
+    "plt.xlabel(\"time\")\n",
+    "plt.ylabel(r\"$X_0$\")\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Multiple features"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In this section, we use the global state."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "single_treeV2 = tree.DecisionTreeRegressor(max_leaf_nodes=300)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_train, X_test, xy_train, xy_test = train_test_split(\n",
+    "    X_history, closure[:, 0], test_size=0.33, shuffle=False\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "single_treeV2.fit(X_train, xy_train)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "single_treeV2.score(X_train, xy_train)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "single_treeV2.score(X_test, xy_test)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(xy_test[:1000])\n",
+    "plt.plot(single_treeV2.predict(X_test)[:1000])\n",
+    "plt.xlabel(\"time steps\")\n",
+    "plt.ylabel(r\"$xy_0$\")\n",
+    "plt.legend((\"True\", \"Predicted\"), fontsize=7)\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We need to adapt our parameterization so that it makes use of the homogeneity of the problem"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class ParameterizationV2:\n",
+    "    def __init__(self, predictor):\n",
+    "        self.predictor = predictor\n",
+    "\n",
+    "    def __call__(self, X):\n",
+    "        X = X.reshape(1, K)\n",
+    "        new_X = []\n",
+    "        for i in range(K):\n",
+    "            new_X.append(np.roll(X, -i, axis=-1))\n",
+    "        new_X = np.concatenate(new_X, axis=0)\n",
+    "        return self.predictor.predict(new_X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gcm = GCM(model.F, ParameterizationV2(single_treeV2))\n",
+    "gcm_no_param = GCM(model.F, lambda x: 0)\n",
+    "X_param, t = gcm(model.X, model.dt, n_steps)\n",
+    "X_no_param, t = gcm_no_param(model.X, model.dt, n_steps)\n",
+    "X_true, _, _ = model.run(model.dt, n_steps * model.dt, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 0], label=\"true\")\n",
+    "plt.plot(t, X_no_param[:, 0], label=\"GCM without parameterization\")\n",
+    "plt.plot(t, X_param[:, 0], label=\"GCM with paramerization\")\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.xlabel(\"time\")\n",
+    "plt.ylabel(r\"$X_0$\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Random Forest: ensemble of decision trees"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here we use a random forest instead of a single decision tree. The idea is simple: to avoid overfitting, one trains several decision trees on the same data.  The prediction of the random forest is then the average prediction of all decision trees within the ensemble.\n",
+    "\n",
+    "There are two main ways in which the fitted trees within the forest might differ:\n",
+    "- the selected features for the splits: at each iteration of the fitting procedure, the algorithm does not consider all features to select the best split, at least not when the number of features is large. Instead, a finite number of features is considered at random. \n",
+    "- the data on which they are trained. Below, by setting bootstrap to True and max_samples to 0.5, we specify that each tree within the forest will only be trained on half of the training data\n",
+    "\n",
+    "Random forest are an efficient way to avoid overfitting compared to using a single tree. However they do come with a computational and memory cost."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# We start with an ensemble of 10 decision trees\n",
+    "rf = RandomForestRegressor(\n",
+    "    n_estimators=50, max_leaf_nodes=300, bootstrap=True, max_samples=0.5\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We fit to the same training data as in the previous section"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rf.fit(X_train, xy_train)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can actually get the prediction of each tree within the forest:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "tree_predictions = []\n",
+    "for est in rf.estimators_:\n",
+    "    tree_predictions.append(est.predict(X_test[:1, :]))\n",
+    "tree_predictions = np.array(tree_predictions)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(f\"tree_predictions = {tree_predictions.flatten()}\")\n",
+    "print(f\"mean of tree_predictions = {tree_predictions.mean()}\")\n",
+    "print(f\"prediction on first row = {rf.predict(X_test[:1, :])[0]}\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rf.score(X_train, xy_train), rf.score(X_test, xy_test)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(xy_test[:1000])\n",
+    "plt.plot(rf.predict(X_test)[:1000])\n",
+    "plt.xlabel(\"time steps\")\n",
+    "plt.ylabel(r\"$xy_0$\")\n",
+    "plt.legend((\"True\", \"Predicted\"), fontsize=7)\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Again, we can test this online."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gcm = GCM(model.F, ParameterizationV2(rf))\n",
+    "gcm_no_param = GCM(model.F, lambda x: 0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_param, t = gcm(model.X, model.dt, n_steps)\n",
+    "X_no_param, t = gcm_no_param(model.X, model.dt, n_steps)\n",
+    "X_true, _, _ = model.run(model.dt, n_steps * model.dt, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 0], label=\"true\")\n",
+    "plt.plot(t, X_no_param[:, 0], label=\"GCM without parameterization\")\n",
+    "plt.plot(t, X_param[:, 0], label=\"GCM with paramerization\")\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.xlabel(\"time\")\n",
+    "plt.ylabel(r\"$X_0$\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Gradient boosting"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Gradient boosting is a type of machine learning [boosting](https://en.wikipedia.org/wiki/Boosting_(machine_learning)). It relies on the intuition that the best possible next model, when combined with previous models, minimizes the overall prediction error. The key idea is to set the target outcomes for this next model in order to minimize the error.\n",
+    "\n",
+    "Further reading [here](https://www.displayr.com/gradient-boosting-the-coolest-kid-on-the-machine-learning-block/)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gbr = GradientBoostingRegressor(n_estimators=200)\n",
+    "gbr.fit(X_train, xy_train)\n",
+    "score_train = gbr.score(X_train, xy_train)\n",
+    "score_test = gbr.score(X_test, xy_test)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(score_train, score_test)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## More features: using past state"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here we further increase the complexity of our approach by incorporating past state into the features used to carry out our prediction."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n_times = 2\n",
+    "new_X = X_history.copy()\n",
+    "for i in range(n_times - 1):\n",
+    "    new_X = np.concatenate((new_X, np.roll(X_history, i + 1, axis=0)), axis=1)\n",
+    "new_X = new_X[n_times:]\n",
+    "new_closure = closure[n_times:]\n",
+    "\n",
+    "X_train, X_test, xy_train, xy_test = train_test_split(\n",
+    "    new_X, new_closure[:, 0], test_size=0.33, shuffle=False\n",
+    ")\n",
+    "X_train.shape"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rf.max_leaf_nodes = 350\n",
+    "rf.max_samples = 0.99\n",
+    "rf.n_estimators = 100"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rf.fit(X_train, xy_train)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rf.score(X_train, xy_train)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rf.score(X_test, xy_test)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Again we need to adapt the form of the parameterization, here so that it records past values of the state for later use."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class Parameterization:\n",
+    "    def __init__(self, predictor, n_times: int = 5):\n",
+    "        self.predictor = predictor\n",
+    "        self.past_states = deque(maxlen=n_times - 1)\n",
+    "        self.n_times = n_times\n",
+    "\n",
+    "    def __call__(self, X):\n",
+    "        X = X.reshape(1, K)\n",
+    "        new_X = []\n",
+    "        for i in range(K):\n",
+    "            new_X.append(np.roll(X, -i, axis=-1))\n",
+    "        new_X = np.concatenate(new_X, 0)\n",
+    "        shape = new_X.shape\n",
+    "        save_X = new_X.copy()\n",
+    "        for i in range(self.n_times - 1):\n",
+    "            try:\n",
+    "                new_X = np.concatenate((new_X, self.past_states[-i - 1]), axis=1)\n",
+    "            except IndexError:\n",
+    "                new_X = np.concatenate((new_X, np.zeros(shape)), axis=-1)\n",
+    "        self.past_states.append(save_X)\n",
+    "        return self.predictor.predict(new_X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n_steps = 2000\n",
+    "gcm = GCM(model.F, Parameterization(rf, n_times))\n",
+    "gcm_no_param = GCM(model.F, lambda x: 0)\n",
+    "X_param, t = gcm(model.X, model.dt, n_steps)\n",
+    "X_no_param, t = gcm_no_param(model.X, model.dt, n_steps)\n",
+    "X_true, _, _ = model.run(model.dt, n_steps * model.dt, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 0], label=\"true\")\n",
+    "plt.plot(t, X_no_param[:, 0], label=\"GCM without parameterization\")\n",
+    "plt.plot(t, X_param[:, 0], label=\"GCM with paramerization\")\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.xlabel(\"time\")\n",
+    "plt.ylabel(r\"$X_0$\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Comparison with NN parameterization (based on Yani's code)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class Net_ANN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super(Net_ANN, self).__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)  # 8 inputs, 16 neurons for first hidden layer\n",
+    "        self.linear2 = nn.Linear(16, 16)  # 16 neurons for second hidden layer\n",
+    "        self.linear3 = nn.Linear(16, 8)  # 8 outputs\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = FF.relu(self.linear1(x))\n",
+    "        x = FF.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "class GCM_network:\n",
+    "    \"\"\"\n",
+    "    GCM class including a neural network parameterization in rhs of equation for tendency\n",
+    "    Args:\n",
+    "        F: forcing\n",
+    "        parameterization: function that takes parameters and returns a tendency\n",
+    "        time_stepping: time stepping method\n",
+    "    \"\"\"\n",
+    "\n",
+    "    def __init__(self, F, network, time_stepping=RK4):\n",
+    "        self.F = F\n",
+    "        self.network = network\n",
+    "        self.time_stepping = time_stepping\n",
+    "\n",
+    "    def rhs(self, X, param=[0]):\n",
+    "        \"\"\"\n",
+    "        Args:\n",
+    "            X: state vector\n",
+    "            param: parameters of our closure, unused here but maintained for consistency of interface\n",
+    "        \"\"\"\n",
+    "        if self.network.linear1.in_features == 1:\n",
+    "            X_torch = torch.from_numpy(X).double()\n",
+    "            X_torch = torch.unsqueeze(X_torch, 1)\n",
+    "        else:\n",
+    "            X_torch = torch.from_numpy(np.expand_dims(X, 0)).double()\n",
+    "        return L96_eq1_xdot(X, self.F) + np.squeeze(\n",
+    "            self.network(X_torch).data.numpy()\n",
+    "        )  # Adding NN parameterization\n",
+    "\n",
+    "    def __call__(self, X0, dt, nt, param=[0]):\n",
+    "        \"\"\"\n",
+    "        Args:\n",
+    "            X0: initial conditions\n",
+    "            dt: time increment\n",
+    "            nt: number of forward steps to take\n",
+    "            param: parameters of our closure\n",
+    "        \"\"\"\n",
+    "        time, hist, X = (\n",
+    "            dt * np.arange(nt + 1),\n",
+    "            np.zeros((nt + 1, len(X0))) * np.nan,\n",
+    "            X0.copy(),\n",
+    "        )\n",
+    "        hist[0] = X\n",
+    "\n",
+    "        for n in range(nt):\n",
+    "            X = self.time_stepping(self.rhs, dt, X, param)\n",
+    "            hist[n + 1], time[n + 1] = X, dt * (n + 1)\n",
+    "        return hist, time"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Load network\n",
+    "path_load = \"networks/network_3_layers_100_epoches.pth\"\n",
+    "nn_3l = Net_ANN().double()\n",
+    "nn_3l.load_state_dict(torch.load(path_load))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n_steps = 2000\n",
+    "gcm = GCM(model.F, ParameterizationV2(single_treeV2))\n",
+    "gcm_network = GCM_network(model.F, nn_3l)\n",
+    "X_param, t = gcm(model.X, model.dt, n_steps)\n",
+    "X_network, t = gcm_network(model.X, model.dt, n_steps)\n",
+    "X_no_param, t = gcm_no_param(model.X, model.dt, n_steps)\n",
+    "X_true, _, _ = model.run(model.dt, n_steps * model.dt, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(t, X_true[:, 0], label=\"true\")\n",
+    "plt.plot(t, X_no_param[:, 0], label=\"GCM without parameterization (tree)\")\n",
+    "plt.plot(t, X_param[:, 0], label=\"GCM with paramerization (network)\")\n",
+    "plt.legend(fontsize=7)\n",
+    "plt.xlabel(\"time\")\n",
+    "plt.ylabel(r\"$X_0$\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Which model / model parameters should we go for?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Standard Machine Learning approach: validation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The Machine Learning approach to this problem is to separate the data into three data sets:\n",
+    "- training dataset\n",
+    "- validation dataset\n",
+    "- test dataset\n",
+    "\n",
+    "Only data from the training dataset is used for fitting. The validation dataset is then used for model selection and hyperparameters tuning: we select the model whose performance on the validation dataset is best.\n",
+    "\n",
+    "In order to choose hyperparameters, one needs to make a decision on what set of values to try. One approach is to try a grid of values. Another approach is to try random values. This is recommanded when the possible number of hyperparameter combinations is large. Finally Bayesian approaches to this problem are also popular. Further reading on grid search, random search and bayesian approach [here](https://towardsdatascience.com/a-practical-introduction-to-grid-search-random-search-and-bayes-search-d5580b1d941d).\n",
+    "\n",
+    "Here we shall implement the random approach: for each model, we will define a range of values for each parameter we wish to tune. We will randomly select a combination of parameter values, fit the model to the training dataset, and assess its performance on the validation dataset. The combination of values leading to the best validation score will be retained. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "single_tree_params_dist = {\n",
+    "    \"max_leaf_nodes\": list(map(int, 1 + np.exp(np.arange(11)))),\n",
+    "}\n",
+    "\n",
+    "rf_params_dist = {\n",
+    "    \"max_leaf_nodes\": list(map(int, 1 + np.exp(np.arange(13)))),\n",
+    "    \"n_estimators\": np.arange(1, 10) * 50,\n",
+    "}\n",
+    "\n",
+    "X_train, X_test, xy_train, xy_test = train_test_split(\n",
+    "    X_history, closure[:, 0], test_size=0.33\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The cells below are very slow to execute. Uncomment them to perform the parameter search."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# best_single_treeV2 = RandomizedSearchCV(\n",
+    "#     single_treeV2, single_tree_params_dist, n_iter=10, verbose=True\n",
+    "# )\n",
+    "# best_rf = RandomizedSearchCV(rf, rf_params_dist, n_iter=20, verbose=True)\n",
+    "\n",
+    "# best_single_treeV2.fit(X_train, xy_train)\n",
+    "# best_rf.fit(X_train, xy_train)\n",
+    "\n",
+    "# print(\"single tree\")\n",
+    "# print(f\"best score = {best_single_treeV2.best_score_}\")\n",
+    "# print(f\"best parameters = {best_single_treeV2.best_params_}\")\n",
+    "# print(\"\\nrandom forest\")\n",
+    "# print(f\"best score = {best_rf.best_params_}\")\n",
+    "# print(f\"best parameters = {best_rf.score(X_test, xy_test)}\")"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/sindy_L96_2scale.ipynb b/_sources/notebooks/sindy_L96_2scale.ipynb
new file mode 100644
index 00000000..e4596235
--- /dev/null
+++ b/_sources/notebooks/sindy_L96_2scale.ipynb
@@ -0,0 +1,800 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "f37e1144-636c-4e98-b40d-1205c665d96f",
+   "metadata": {},
+   "source": [
+    "# Applying SINDy equation identification to L96"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7cd0aa75-b52a-4b0b-b5ec-df2ac8c1bd86",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "## Outline"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "42566028-fe9c-4c50-8582-b5e217d6394d",
+   "metadata": {},
+   "source": [
+    "This notebook provides an example of SINDy equation identification in application to the two-scale L96 model with the use of `pysindy` library. The objective is to try to identify the governing ODEs for the large-scale variable ($X_i$) if we know only their time measurements. In other words, we want to see how well the SINDy model can capture the form of the governing equations for large-scale variables in the presence of the subgrid forcing by small-scale variables ($Y$) when there is no measurements of the small scales."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1f222842-0a8a-4069-addf-55fc44d24dd2",
+   "metadata": {},
+   "source": [
+    "## Import"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "73bbfeec-72e6-4d81-89b0-e5369671dff9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import pysindy as ps\n",
+    "import matplotlib.pyplot as plt\n",
+    "import seaborn as sns\n",
+    "from L96_model import L96, L96_eq1_xdot, integrate_L96_2t, integrate_L96_1t"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "866c8025-f383-4b05-ab30-b75945223edc",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "## Routines for plotting"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "22c47249-5ac0-475e-a28b-54f725ad5ecf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "colors = [\n",
+    "    \"#1f77b4\",\n",
+    "    \"#ff7f0e\",\n",
+    "    \"#2ca02c\",\n",
+    "    \"#d62728\",\n",
+    "    \"#9467bd\",\n",
+    "    \"#8c564b\",\n",
+    "    \"#e377c2\",\n",
+    "    \"#7f7f7f\",\n",
+    "    \"#bcbd22\",\n",
+    "    \"#17becf\",\n",
+    "]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a0461f22-0ffd-4c84-abe0-3e2844a135f6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def plot_coefficients(\n",
+    "    coefficients, input_features=None, feature_names=None, ax=None, **heatmap_kws\n",
+    "):\n",
+    "    if input_features is None:\n",
+    "        input_features = [f\"$\\dot x_{k}$\" for k in range(coefficients.shape[0])]\n",
+    "    else:\n",
+    "        input_features = [f\"$\\dot {fi}$\" for fi in input_features]\n",
+    "\n",
+    "    if feature_names is None:\n",
+    "        feature_names = [f\"f{k}\" for k in range(coefficients.shape[1])]\n",
+    "\n",
+    "    with sns.axes_style(style=\"white\", rc={\"axes.facecolor\": (0, 0, 0, 0)}):\n",
+    "        if ax is None:\n",
+    "            fig, ax = plt.subplots(1, 1)\n",
+    "\n",
+    "        max_mag = np.max(np.abs(coefficients))\n",
+    "        heatmap_args = {\n",
+    "            \"xticklabels\": input_features,\n",
+    "            \"yticklabels\": feature_names,\n",
+    "            \"center\": 0.0,\n",
+    "            \"cmap\": sns.color_palette(\"vlag\", n_colors=20, as_cmap=True),\n",
+    "            \"ax\": ax,\n",
+    "            \"linewidths\": 0.1,\n",
+    "            \"linecolor\": \"whitesmoke\",\n",
+    "        }\n",
+    "        heatmap_args.update(**heatmap_kws)\n",
+    "\n",
+    "        sns.heatmap(coefficients.T, **heatmap_args)\n",
+    "\n",
+    "        ax.tick_params(axis=\"y\", rotation=0)\n",
+    "\n",
+    "    return ax"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d37f0cf2-e597-428a-bb64-4fbe0e424ce4",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def signed_sqrt(x):\n",
+    "    return np.sign(x) * np.sqrt(np.abs(x))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "10ce16bc-e99a-4260-b132-940e6af15bb8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def compare_coefficient_plots(coefficients, input_features=None, feature_names=None):\n",
+    "    n_cols = len(coefficients)\n",
+    "    with sns.axes_style(style=\"white\", rc={\"axes.facecolor\": (0, 0, 0, 0)}):\n",
+    "        max_mag = np.max(np.abs(coefficients))\n",
+    "        fig, axs = plt.subplots(1, 1, figsize=(4, 4), sharey=True, sharex=True)\n",
+    "        plot_coefficients(\n",
+    "            signed_sqrt(coefficients),\n",
+    "            input_features=input_features,\n",
+    "            feature_names=feature_names,\n",
+    "            ax=axs,\n",
+    "            cbar=False,\n",
+    "            vmax=max_mag,\n",
+    "            vmin=-max_mag,\n",
+    "        )\n",
+    "        fig.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "28330e77-e8c5-42ba-b01d-d4088155c397",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "## L96 two-scale model\n",
+    "\n",
+    "```{figure} https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376@1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png\n",
+    ":width: 400px\n",
+    ":name: l96-equation-figure-sindy2scale\n",
+    "\n",
+    "*Visualisation of a two-scale Lorenz '96 system with J = 8 and K = 6. Global-scale variables ($X_k$) are updated based on neighbouring variables and on the local-scale variables ($Y_{j,k}$) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from [Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.](https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436)*.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a5fa3dff-b284-4f82-a641-8c5ef42880f8",
+   "metadata": {},
+   "source": [
+    "Two-time-scale Lorenz 96 model:\n",
+    "$$ \\frac{d}{dt} X_k = - X_{k-2} X_{k-1} + X_{k-1} X_{k+1} - X_k + F - \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k} $$\n",
+    "$$ \\frac{d}{dt} Y_{j,k} = - c b Y_{j+1,k} (Y_{j+2,k} - Y_{j-1,k}) - c Y_{j,k} + \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} X_{k} $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f6611d39-a079-4809-b7ce-a740f0ecf374",
+   "metadata": {},
+   "source": [
+    "We will put K=8 large-scale variables ($X$), for each of them there is J=32 small-scale variables ($Y$). For the first example, we will consider the case of **weak** interaction between the scales, which is defined by the parameter $h$ in the L96 class."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "771cc013-8731-4885-ae69-00cc19161486",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "K = 8  # Number of degrees of freedom for X\n",
+    "J = 32  # Number of degrees of freedom for Y per X\n",
+    "Forcing = 18.0  # Model parameter\n",
+    "h = 0.1  # WEAK coupling"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "32c16a5e-5db9-4a96-88bd-f5e9c39d7179",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def s(k, K):\n",
+    "    \"\"\"A non-dimension coordinate from -1..+1 corresponding to k=0..K\"\"\"\n",
+    "    return 2 * (0.5 + k) / K - 1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "88ba9965-c64f-4194-86b4-d2ddba8081ef",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Initial conditions\n",
+    "k = np.arange(K)  # For coordinate in plots\n",
+    "j = np.arange(J * K)  # For coordinate in plots\n",
+    "# Initial conditions\n",
+    "X_init = s(k, K) * (s(k, K) - 1) * (s(k, K) + 1)\n",
+    "Y_init = 0 * s(j, J * K) * (s(j, J * K) - 1) * (s(j, J * K) + 1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2225caa3-bcd8-4b9f-8049-6fb17a8f7c66",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "steps = 1000\n",
+    "dt = 0.005"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "128356d3-b283-4b07-97f3-bc675ab77c10",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create a \"real world\" with K=8 and J=32\n",
+    "W = L96(K, J, F=Forcing, h=h)\n",
+    "W.set_state(X_init, Y_init)\n",
+    "# Run the true state\n",
+    "X, Y, t = W.run(dt, dt * steps, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8958b6da-690f-451a-8d83-88cbc3403b29",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, ax = plt.subplots(1, 1, figsize=(6, 4), dpi=150)\n",
+    "contours = ax.contourf(k, t, X, levels=20)\n",
+    "fig.colorbar(contours)\n",
+    "ax.set(xlabel=\"k\", ylabel=\"t\")\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "12b86c0a-3618-4a2f-abf8-0c9f74dcc240",
+   "metadata": {},
+   "source": [
+    "The data array $X$ will be used for the system identification by SINDy method. We suppose that we do not dispose measurements for the small-scale variables $Y$, so their contribution to the large-scale dymanics will basically treated as noise."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "48efefd9-6716-43fc-b45e-7428249a366d",
+   "metadata": {},
+   "source": [
+    "## Define SINDy model"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9357c676-12bd-4a64-b609-3fe594201f4e",
+   "metadata": {},
+   "source": [
+    "The goal is to identify the governing equations for the large-scale variables $X$, having the measurement for them."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "71be9da9-d2c4-40d4-b69a-d9eee9c694b7",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "differentiation_method = ps.SINDyDerivative(kind=\"finite_difference\", k=1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a87dddfb-1980-47c2-9d35-622a1e8bcc58",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "feature_library = ps.PolynomialLibrary(degree=2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7e386c41-5682-4b3d-bcf4-399b14b36f2f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# sequential thresholded least squares optimizer\n",
+    "optimizer = ps.STLSQ(threshold=0.3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1ee1517b-85da-4231-a8f6-9c8966aae990",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model = ps.SINDy(\n",
+    "    differentiation_method=differentiation_method,\n",
+    "    feature_library=feature_library,\n",
+    "    optimizer=optimizer,\n",
+    "    feature_names=[(\"x\" + str(i)) for i in k],\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "720353b2-8448-4836-ac52-e462b5afe5ef",
+   "metadata": {},
+   "source": [
+    "# Optimize coefficients"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d0ef6b41-cd32-4cbb-9514-c238b2a7b8ad",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model.fit(X, t=t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "725d681c-b7fb-4d9a-aac0-65ffa9f970fa",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "feature_names = feature_library.get_feature_names()\n",
+    "print(feature_names)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "254779b5-cc18-4280-9a1b-421844bddffc",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model.print()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ed037830-d9ec-4a4a-98d5-227f5395eb72",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model.score(X, t=t[1] - t[0])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d4caf05c-7ae9-44da-b456-63f3db11ea3c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model.coefficients()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9966ea81-0724-4a1b-87d6-79f40ffcc6bf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "with sns.axes_style(style=\"white\", rc={\"axes.facecolor\": (0, 0, 0, 0)}):\n",
+    "    max_mag = np.max(np.abs(model.coefficients()))\n",
+    "    fig, axs = plt.subplots(\n",
+    "        1, 1, figsize=(8, 10), sharey=True, sharex=True, tight_layout=True\n",
+    "    )\n",
+    "    plot_coefficients(\n",
+    "        model.coefficients(),\n",
+    "        input_features=[(\"x\" + str(i)) for i in k],\n",
+    "        feature_names=feature_names,\n",
+    "        ax=axs,\n",
+    "        cbar=True,\n",
+    "        vmax=max_mag,\n",
+    "        vmin=-max_mag,\n",
+    "        cmap=\"seismic\",\n",
+    "    )\n",
+    "    axs.set(title=\"SINDy coefficients\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9da63499-fe14-49c3-a99b-c96416b764b6",
+   "metadata": {},
+   "source": [
+    "We can see that the SINDy model captures correctly the dominant terms and gives quite accurate predictions for the model coefficients and the value of the forcing. The subgrid forcing does not appear in this prediction."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fa412ba4-4c36-494d-a805-638a632ac731",
+   "metadata": {},
+   "source": [
+    "## Compare"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1a6c62ed-60e5-4496-84df-32d10716d874",
+   "metadata": {},
+   "source": [
+    "### Same initial conditions"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2824415c-95bd-4fdb-8b47-2432500108df",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "sindy_simulation = model.simulate(X_init, t=t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a49f38e6-581e-46b0-ab85-1efb33156f08",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, ax = plt.subplots(1, 1, figsize=(6, 4), dpi=150)\n",
+    "for i, component in enumerate(np.linspace(0, K - 1, num=3, dtype=int)):\n",
+    "    ax.plot(t, X[:, component], \"--\", color=colors[i])\n",
+    "    ax.plot(\n",
+    "        t,\n",
+    "        sindy_simulation[:, component],\n",
+    "        color=colors[i],\n",
+    "        label=model.feature_names[component],\n",
+    "    )\n",
+    "ax.legend(fontsize=7)\n",
+    "ax.set(xlabel=\"t\", ylabel=r\"$x_i$\")\n",
+    "ax.grid()\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0757ead7-8fc1-418e-8b31-b72b43ae384f",
+   "metadata": {},
+   "source": [
+    "## Other initial conditions"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9de3b4aa-cf64-4b34-ab4c-1ce89d4763a2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_init_new = 0.5 * s(k, K) * (s(k, K) - 2) * (s(k, K) + 2)\n",
+    "W.set_state(X_init_new, Y_init)\n",
+    "# Run the true state\n",
+    "X_new, Y_new, t = W.run(dt, dt * steps, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "36d64b56-4591-4b85-9402-703a12365f95",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "sindy_simulation_new = model.simulate(X_init_new, t=t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a6790161-afc1-4db0-af1e-aa751f5f72b3",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, ax = plt.subplots(1, 1, figsize=(6, 4), dpi=150)\n",
+    "for i, component in enumerate(np.linspace(0, K - 1, num=3, dtype=int)):\n",
+    "    ax.plot(t, X_new[:, component], \"--\", color=colors[i])\n",
+    "    ax.plot(\n",
+    "        t,\n",
+    "        sindy_simulation_new[:, component],\n",
+    "        color=colors[i],\n",
+    "        label=model.feature_names[component],\n",
+    "    )\n",
+    "ax.legend(fontsize=7)\n",
+    "ax.set(xlabel=\"t\", ylabel=r\"$x_i$\", title=\"New initial conditions\")\n",
+    "ax.grid()\n",
+    "plt.show();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af3d1e9f-ef45-4c2c-9eb7-fee641376e62",
+   "metadata": {},
+   "source": [
+    "Similar to the case with L63 model, the system predicted by SINDy gives an accurate prediction of trajectories for a certain time, then the prediction strts deviating from the original model. It is related to the error in the coefficients obtained in optimization anf the chaotical nature of the Lorenz system."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "74e8f259-a795-47c7-a4b0-cce3faea6c12",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_init_new = s(k, K) * (s(k, K) - 2) * (s(k, K) + 2)  # Initial conditions"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "624b632a-787f-4942-a192-62e516fb4704",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X_new, t = integrate_L96_1t(X_init_new, Forcing, dt, steps)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ed1605e6-809b-46d4-9762-f1e908e15378",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "sindy_simulation_new = model.simulate(X_init_new, t=t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c5cbf9dc-ffdd-4447-ac44-ce098508b5f2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, ax = plt.subplots(1, 1, figsize=(6, 4), dpi=150)\n",
+    "for i, component in enumerate(np.linspace(0, K - 1, num=3, dtype=int)):\n",
+    "    ax.plot(t, X_new[:, component], \"--\", color=colors[i])\n",
+    "    ax.plot(\n",
+    "        t,\n",
+    "        sindy_simulation_new[:, component],\n",
+    "        color=colors[i],\n",
+    "        label=model.feature_names[component],\n",
+    "    )\n",
+    "ax.legend(fontsize=7)\n",
+    "ax.set(xlabel=\"t\", ylabel=r\"$x_i$\")\n",
+    "ax.grid()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b1cad42-fda7-4764-af7a-5217a09ce8c9",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "## Case of strong coupling"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b827bc2c-127d-48ba-a262-7942ff0f5e2b",
+   "metadata": {},
+   "source": [
+    "Now, let us consider the L96 two-scale model with a stronger coupling between the large and small scales. The coupling coefficient here is put to be 10 times higher."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6c79ad20-f6db-43be-9bf5-c69a53ba1956",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create a \"real world\" with K=8 and J=32\n",
+    "W_strong = L96(K, J, F=Forcing, h=1.0)\n",
+    "W_strong.set_state(X_init, Y_init)\n",
+    "# Run the true state\n",
+    "X_s, Y_s, t = W_strong.run(dt, dt * steps, store=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e84bddb0-2a1b-4224-93d9-3d1c5ea48fe8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, ax = plt.subplots(1, 1, figsize=(6, 4), dpi=150)\n",
+    "contours = ax.contourf(k, t, X_s, levels=20)\n",
+    "fig.colorbar(contours)\n",
+    "ax.set(xlabel=\"k\", ylabel=\"t\")\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af054c0e-014b-4dc6-8b4f-b38f918dd943",
+   "metadata": {},
+   "source": [
+    "### Build the SINDy model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3b352eaa-2aa6-4d62-ae70-6494fb83d84a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model_s = ps.SINDy(\n",
+    "    differentiation_method=differentiation_method,\n",
+    "    feature_library=feature_library,\n",
+    "    optimizer=optimizer,\n",
+    "    feature_names=[(\"x\" + str(i)) for i in k],\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "17c9b317-bb0b-4299-9e3f-967836c7d3ef",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model_s.fit(X_s, t=t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "816f9bb0-c168-4bf6-bd98-c939ef929c04",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "model_s.print()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "37168b11-a774-4b61-8300-20b759c84bdf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "with sns.axes_style(style=\"white\", rc={\"axes.facecolor\": (0, 0, 0, 0)}):\n",
+    "    max_mag = np.max(np.abs(model_s.coefficients()))\n",
+    "    fig, axs = plt.subplots(\n",
+    "        1, 1, figsize=(8, 10), sharey=True, sharex=True, tight_layout=True\n",
+    "    )\n",
+    "    plot_coefficients(\n",
+    "        model_s.coefficients(),\n",
+    "        input_features=[(\"x\" + str(i)) for i in k],\n",
+    "        feature_names=feature_library.get_feature_names(),\n",
+    "        ax=axs,\n",
+    "        cbar=True,\n",
+    "        vmax=max_mag,\n",
+    "        vmin=-max_mag,\n",
+    "        cmap=\"seismic\",\n",
+    "    )\n",
+    "    axs.set(title=\"SINDy coefficients\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eceb3597-f4d7-4f43-ba45-5fbaf472b523",
+   "metadata": {},
+   "source": [
+    "We can see that despite the fact that in general SINDy captures the structure of ODEs, the coefficients become different from the original model, due to the influence of the interaction with the small scale. Interestingly, the coefficients in front of linear terms in the ODEs are now all close to 1.7-1.8 (instead of 1.0 in the true model), and the predicted forcing value is systematically smaller than the true value 18. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b8fcce95-8330-4f29-b385-d2943deb5ad2",
+   "metadata": {},
+   "source": [
+    "### Run the predicted model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "89d3d8d3-b82f-433b-b239-c530b6d4b06b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "sindy_simulation_s = model_s.simulate(X_init, t=t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "975e70b0-2bfb-4ab7-8b6c-a4d5d68b5e39",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, ax = plt.subplots(1, 1, figsize=(6, 4), dpi=150)\n",
+    "for i, component in enumerate(np.linspace(0, K - 1, num=3, dtype=int)):\n",
+    "    ax.plot(t, X_s[:, component], \"--\", color=colors[i])\n",
+    "    ax.plot(\n",
+    "        t,\n",
+    "        sindy_simulation_s[:, component],\n",
+    "        color=colors[i],\n",
+    "        label=model.feature_names[component],\n",
+    "    )\n",
+    "ax.legend(fontsize=7)\n",
+    "ax.set(xlabel=\"t\", ylabel=r\"$x_i$\")\n",
+    "ax.grid()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4bb4f383-606b-45e8-99a7-bcbe5a5bb860",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "The resulting trajectories start deviating fast from the true trajectories. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8042a757-85e5-4113-a568-fcb0cfcec4d6",
+   "metadata": {},
+   "source": [
+    "## Conclusions\n",
+    "\n",
+    "Sparse system identification method works well in conditions when we have already have some knowledgement about the equations (about sparsity, the type on non-lineartity) and optimal data representation (good coordinates). "
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/symbolic_methods_comparison.ipynb b/_sources/notebooks/symbolic_methods_comparison.ipynb
new file mode 100644
index 00000000..83e8231c
--- /dev/null
+++ b/_sources/notebooks/symbolic_methods_comparison.ipynb
@@ -0,0 +1,982 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "f65214e4-a677-4ede-871e-5f3fef45f2b1",
+   "metadata": {},
+   "source": [
+    "# Introduction to Equation Discovery - Comparing Symbolic Regression Methods"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "25d66ef6-0c47-4725-8d01-eb8141171d6c",
+   "metadata": {},
+   "source": [
+    "Upto now we have seen that the climate models we developed are using physical equations that are based on our understanding of the physical processes that govern the climate. However, these equations are often complex and difficult to solve, and they can only be used to model the climate at a coarse resolution.\n",
+    "\n",
+    "**Equation discovery is a different approach that uses machine learning to automatically discover equations that can be used to model the climate.** Specifically, equation discovery in climate modeling can be used to:\n",
+    "- **Automate the process of developing subgrid parameterizations.** Subgrid parameterizations are mathematical equations that are used to represent the effects of processes that are too small to be resolved by climate models. These equations are often complex and difficult to develop, but they can be automatically discovered using equation discovery.\n",
+    "- **Identify relationships between variables that are important for understanding climate change.** Machine learning algorithms can learn from large datasets of climate data to identify relationships between variables that are important for understanding climate change.\n",
+    "- **Develop models that are more interpretable and that can be used to make better decisions about how to mitigate and adapt to climate change.** Machine learning algorithms can generate equations that are easier to understand than traditional physical equations. This can help scientists and policymakers to understand how climate models work and to make better decisions about how to mitigate and adapt to climate change."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7a104c71",
+   "metadata": {},
+   "source": [
+    "In this notebook, we'll review some common techniques for **Symbolic Regression (SR)**, a family of methods of discovering (simple) equations that relate inputs to outputs.\n",
+    "\n",
+    "In particular, we'll cover the following methods:\n",
+    "\n",
+    "- Symbolic regression based on {ref}`genetic-programming-section`\n",
+    "    * {ref}`gplearn-sec`\n",
+    "    * {ref}`pysr-sec`\n",
+    "\n",
+    "- Symbolic regression based on {ref}`sparse-regression-sec`\n",
+    "    * {ref}`linear-regression-sec`\n",
+    "    * {ref}`lasso-sec`\n",
+    "    * {ref}`rvm-sec`\n",
+    "    * {ref}`stlsq-sec`\n",
+    "\n",
+    "- Brief overview of other techniques"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9070565d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import pysindy as ps\n",
+    "from sklearn.linear_model import Lasso, LinearRegression\n",
+    "from sympy import Number, latex\n",
+    "\n",
+    "from rvm import RVR"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c74c2213-8d54-477b-b483-0dc38d65a219",
+   "metadata": {},
+   "source": [
+    "## Data\n",
+    "\n",
+    "When explaining the methods, we'll consider the following bivariate system:\n",
+    "\n",
+    "$$\n",
+    "y = x_0^2 - \\frac{1}{2}x_1^2 + \\sin\\left(\\frac{1}{2} x_0 x_1\\right)\n",
+    "$$\n",
+    "\n",
+    "Although the equation is relatively simple, discovering it from $x_0$ and $x_1$ directly is a bit tough, since the $\\sin$ term requires going at least three levels deep in term combinations, and also involves an inner multiplication by a constant (which not all symbolic regression frameworks support)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "58b37b2c",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def toy_function(x):\n",
+    "    x0 = x[:, 0]\n",
+    "    x1 = x[:, 1]\n",
+    "    return x0**2 - 0.5 * x1**2 + np.sin(0.5 * x0 * x1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c04e4072",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def contour(\n",
+    "    f, lines=None, linewidths=6, alpha=0.9, linestyles=\"solid\", label=None, **kw\n",
+    "):\n",
+    "    grid = np.linspace(-4, 4, 150)\n",
+    "    xyg = np.array(np.meshgrid(grid, grid))\n",
+    "    X = xyg.reshape(2, -1).T\n",
+    "    zg = f(X)\n",
+    "    xg, yg = xyg\n",
+    "    vlim = np.abs(zg).max()\n",
+    "    plt.contourf(xg, yg, zg.reshape(xg.shape), 300, vmin=-vlim, vmax=vlim, cmap=\"bwr\")\n",
+    "    plt.colorbar(label=label)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "98e6a5fc-5a49-4db8-b0fd-a4d34806f465",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "x = np.random.normal(size=(1000, 2))\n",
+    "y = toy_function(x)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5930af35",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "contour(toy_function, label=\"Function value\")\n",
+    "plt.title(\"Toy domain for illustrating\\nsymbolic regression\", fontsize=16)\n",
+    "plt.xlabel(\"$x_0$\", fontsize=16)\n",
+    "plt.ylabel(\"$x_1$\", fontsize=16)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "752732b1",
+   "metadata": {},
+   "source": [
+    "(genetic-programming-section)=\n",
+    "## Genetic Programming\n",
+    "\n",
+    "Genetic programming is a classic technique in AI, and interest in the idea dates back at least to Alan Turing (who proposes the idea in the [same paper](https://doi.org/10.1093/mind/LIX.236.433) that introduces the imitation game, or Turing test).\n",
+    "\n",
+    "The general idea is to have a:\n",
+    "- \"Population\" of programs (in our case, possible symbolic expressions, usually represented as trees)\n",
+    "- Some notion of individual \"fitness\" (in our case, how well they model the data, as well as their simplicity)\n",
+    "- Some means of applying random \"mutations\" (e.g. adding a term, deleting a term)\n",
+    "\n",
+    "```{figure} figs/gp_mutation_types.png\n",
+    ":name: genetic-programming-mutation-types\n",
+    "\n",
+    "Some examples for different mutation types\n",
+    "```\n",
+    "\n",
+    "\"Crossover\" and \"subtree\" mutations combine features from multiple candidate equations, imitating the process of genetic recombination from sexual reproduction. \"Hoist\" and \"point\" mutations modify individual equations, imitating the process of genetic mutation from, e.g., random errors in DNA replication.\n",
+    "\n",
+    "Specific mutations aside, programs can have a limited lifespan, and those with high \"fitness\" produce more \"offspring\" (possibly-mutated versions of themselves). However, everything is probabilistic, so programs with lower fitness can still persist. This is critically important, because it allows genetic programming to overcome local minima during optimization.\n",
+    "\n",
+    "Note that although genetic programming can be a robust way to solve extremely difficult non-convex optimization problems (as perhaps evidenced by life itself), it also tends to be slow and resource-intensive."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eb6501f9",
+   "metadata": {},
+   "source": [
+    "(gplearn-sec)=\n",
+    "### gplearn\n",
+    "\n",
+    "One nicely-documented library for \"pure\" genetic programming-based symbolic regression is [gplearn](https://gplearn.readthedocs.io/en/stable/), which provides a scikit-learn style API and a number of configuration options:\n",
+    "\n",
+    "#### Run gplearn"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1100219a-ac1a-4b47-a6f9-c9606b8c852e",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import gplearn\n",
+    "import gplearn.genetic"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1423685b",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "gplearn_sr = gplearn.genetic.SymbolicRegressor(\n",
+    "    population_size=5000,\n",
+    "    generations=50,\n",
+    "    p_crossover=0.7,\n",
+    "    p_subtree_mutation=0.1,\n",
+    "    p_hoist_mutation=0.05,\n",
+    "    p_point_mutation=0.1,\n",
+    "    max_samples=0.9,\n",
+    "    verbose=1,\n",
+    "    parsimony_coefficient=0.01,\n",
+    "    function_set=(\"add\", \"mul\", \"sin\"),\n",
+    ")\n",
+    "gplearn_sr.fit(x, y)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "65cd4ab0",
+   "metadata": {},
+   "source": [
+    "#### Interpret results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fa7a7000",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "print(gplearn_sr._program)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0b9ccf66",
+   "metadata": {},
+   "source": [
+    "This looks very close to the right answer, though the constants are slightly off."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a7fa571b",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "[const for const in gplearn_sr._program.program if isinstance(const, float)]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8e187826-75be-4522-a2eb-46b70e103847",
+   "metadata": {},
+   "source": [
+    "This is likely because gplearn mutations which add or update constants always pick values by drawing random uniform values (within pre-specified ranges, by default -1 to 1). In my limited experience so far, this is one of the major inefficiencies."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7a5c4b50",
+   "metadata": {},
+   "source": [
+    "(pysr-sec)=\n",
+    "### PySR\n",
+    "\n",
+    "[PySR](https://pysr.readthedocs.io/en/latest/) is a different library for symbolic regression which, though still based on genetic programming, includes simulated annealing and gradient-free optimization to set the values of constants as well as performance improvements (although accessible via a Python API, the main library is written in Julia for speed). This seems to make it a bit more reliable and precise.\n",
+    "\n",
+    "#### Run PySR"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8fff8157-5658-4a29-a97e-a5f0e4d3d265",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import pysr"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5f15cf43",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "equations = pysr.PySRRegressor(\n",
+    "    niterations=5,\n",
+    "    binary_operators=[\"+\", \"*\"],  # operators that can combine two terms\n",
+    "    unary_operators=[\"sin\"],  # operators that modify a single term\n",
+    ")\n",
+    "equations.fit(x, y)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2e79683d",
+   "metadata": {},
+   "source": [
+    "#### Interpret results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0037d292",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def round_expr(expr, num_digits=4):\n",
+    "    return expr.xreplace({n: round(n, num_digits) for n in expr.atoms(Number)})"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5adf39e9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "round_expr(equations.sympy())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f1de78e9",
+   "metadata": {},
+   "source": [
+    "It looks like PySR is able to not only discover the correct equation, but also format it for us nicely.\n",
+    "\n",
+    "In addition, it saves a Pareto frontier of possible expressions of varying complexities, which had the lowest error of any other equations of equal or lesser complexity (with a configurable trade-off rule):"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ac318f36",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equations.equations_"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8f3ec9c9-517b-420a-854a-824577068130",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equations.equations_.loss"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a3724a38",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(12, 3), dpi=150)\n",
+    "plt.bar(\n",
+    "    np.arange(len(equations.equations_)),\n",
+    "    equations.equations_.loss,\n",
+    "    width=0.33,\n",
+    "    color=\"blue\",\n",
+    ")\n",
+    "\n",
+    "\n",
+    "plt.yscale(\"log\")\n",
+    "plt.ylabel(\"Mean squared error\", fontsize=14, color=\"blue\")\n",
+    "plt.xticks(\n",
+    "    range(len(equations.equations_)),\n",
+    "    [f\"${latex(round_expr(v,7))}$\" for v in equations.equations_.sympy_format],\n",
+    "    rotation=30,\n",
+    "    ha=\"right\",\n",
+    "    fontsize=12,\n",
+    ")\n",
+    "plt.title(\"PySR Pareto frontier\", fontsize=16)\n",
+    "plt.xlabel(\"Expression (more complex $\\\\to$)\", fontsize=14)\n",
+    "\n",
+    "ax2 = plt.twinx()\n",
+    "ax2.bar(\n",
+    "    np.arange(len(equations.equations_)) + 0.33,\n",
+    "    equations.equations_.score,\n",
+    "    width=0.33,\n",
+    "    color=\"orange\",\n",
+    ")\n",
+    "ax2.set_ylabel(\"PySR score\", color=\"orange\", fontsize=14)\n",
+    "\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "acfbc136",
+   "metadata": {},
+   "source": [
+    "(sparse-regression-sec)=\n",
+    "## Manually Constructed Library + Sparse Regression\n",
+    "\n",
+    "Another approach to performing symbolic regression is to\n",
+    "- Hand-construct a library of basis functions (e.g. by recursively combining all terms with various operations)\n",
+    "- Run sparse linear regression to extract a small set of basis functions\n",
+    "\n",
+    "This approach works well for problems where any constants in the underlying formula can be extracted out to the outermost level (so they can be determined by linear regression). It also works well when we have domain knowledge about which terms could potentially appear in the final sum.\n",
+    "\n",
+    "It does not work well in cases where constants can't be pulled outside certain operations (e.g. in this case, with the sine operator, though powers and derivatives are generally fine). This method can also struggle when basis function values are highly correlated, though it depends on the sparse regression method.\n",
+    "\n",
+    "Let's create an augmentation function which takes an array of features and adds additional basis functions by combining them with binary multiplication and unary sine operations:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c82ac622-e269-4c67-a484-d826b9746c6d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def augment(feats, names):\n",
+    "    existing = set(names)\n",
+    "    new_feats = []\n",
+    "    new_names = []\n",
+    "\n",
+    "    for i in range(feats.shape[1]):\n",
+    "        name = f\"sin({names[i]})\"\n",
+    "        if name not in existing:\n",
+    "            new_feats.append(np.sin(feats[:, i]))\n",
+    "            new_names.append(name)\n",
+    "        for j in range(i, feats.shape[1]):\n",
+    "            name = f\"mul({names[i]}, {names[j]})\"\n",
+    "            if name not in existing:\n",
+    "                new_feats.append(feats[:, i] * feats[:, j])\n",
+    "                new_names.append(name)\n",
+    "\n",
+    "    return np.hstack((feats, np.array(new_feats).T)), names + new_names"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c3df85a0",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def repeatedly_augment(feats, names=None, depth=2):\n",
+    "    if names is None:\n",
+    "        names = [f\"x{i}\" for i in range(len(feats))]\n",
+    "    for _ in range(depth):\n",
+    "        feats, names = augment(feats, names)\n",
+    "    return feats, names"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "59368f96-e85b-4086-bee0-133e31f231eb",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "feats, names = repeatedly_augment(np.array(x), [\"x0\", \"x1\"], depth=2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2b7b11c2",
+   "metadata": {},
+   "source": [
+    "Applying this twice brings us from 2 features up to 37:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1221f8a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "feats.shape"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "598e13c9",
+   "metadata": {},
+   "source": [
+    "We can see that the complexity of this approach starts to blow up exponentially, though, as we increase the maximum depth of an expression:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0c865291",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "augment(feats, names)[0].shape"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "47403518",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "augment(*augment(feats, names))[0].shape"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "094e5f3c",
+   "metadata": {},
+   "source": [
+    "If we need arbitrary polynomials of up to 4th order, we end up with orders of magnitude more features than samples, which will cause problems.\n",
+    "\n",
+    "Note that since this method lacks the ability to apply operations with constants within other operations, $\\sin(0.5 x_0 x_1)$ isn't present in any of these possible sets, though $\\sin(x_0 x_1)$ is.\n",
+    "\n",
+    "Let's now see how different methods for (sparse) linear regression perform on this task:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2e53fc88",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def learned_expr(coef, names, cutoff=1e-2):\n",
+    "    coef_idx = np.argwhere(np.abs(coef) > cutoff)[:, 0]\n",
+    "    sort_order = reversed(np.argsort(np.abs(coef[coef_idx])))\n",
+    "    return \" + \".join(\n",
+    "        [f\"{coef[coef_idx[i]]:.4f}*{names[coef_idx[i]]}\" for i in sort_order]\n",
+    "    )"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1d678e8b",
+   "metadata": {},
+   "source": [
+    "(linear-regression-sec)=\n",
+    "### Linear regression\n",
+    "\n",
+    "We can start with completely unregularized linear regression, which will find the linear combination of basis terms that minimizes mean squared error (to high precision, due to the convexity of the problem and an analytic way of solving it)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a9175e41",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "linreg = LinearRegression()\n",
+    "linreg.fit(feats, y)\n",
+    "learned_expr(linreg.coef_, names)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8cd9b0d9-f8f7-4cc9-985b-daa322894112",
+   "metadata": {},
+   "source": [
+    "Unfortunately, this doesn't seem to give us a very sparse solution in this case."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d6fd81cd",
+   "metadata": {},
+   "source": [
+    "(lasso-sec)=\n",
+    "### L1-regularized linear regression (LASSO)\n",
+    "\n",
+    "A common way to make linear regression return more sparse solutions is to minimize a linear combination of the mean squared error and the sum of the magnitudes of the weights. This is called applying an \"L1 penalty\" since it is the L1-norm of the weight vector, and is also referred to as the \"Least Absolute Shrinkage and Selection Operator\", or LASSO. From a Bayesian perspective, this corresponds to doing Bayesian linear regression with a Laplace or double-exponential prior on the weights.\n",
+    "\n",
+    "Note that although this penalty does tend to encourage sparsity, it's also meant as a regularizer, which can help with the presence of noise. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d485daad",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "lasso = Lasso(alpha=0.01)\n",
+    "lasso.fit(feats, y)\n",
+    "learned_expr(lasso.coef_, names)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "605f5bbc",
+   "metadata": {},
+   "source": [
+    "Here, LASSO ends up doing a pretty good job finding the first two terms of the ground-truth model ($x_0^2 - 0.5x_1^2$), with an approximation of the remaining $\\sin(0.5 x_0 x_1)$ term as $ 0.2(x_0\\sin(x_1)+x_1\\sin(x_0)) + 0.1(\\sin(x_0 x_1) + x_0x_1)$:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "732ab3e2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(4, 4), dpi=150)\n",
+    "x0 = x[:, 0]\n",
+    "x1 = x[:, 1]\n",
+    "sin = np.sin\n",
+    "plt.scatter(\n",
+    "    sin(0.5 * x0 * x1),\n",
+    "    0.2 * (x0 * sin(x1) + x1 * sin(x0)) + 0.1 * (sin(x0 * x1) + x0 * x1),\n",
+    ")\n",
+    "plt.plot([-1, 1], [-1, 1], color=\"red\")\n",
+    "plt.xlabel(r\"$\\sin(0.5 x_0 x_1)$\", fontsize=12)\n",
+    "plt.ylabel(\n",
+    "    r\"$0.2(x_0\\sin(x_1)+x_1\\sin(x_0))$\" + \"\\n\" + r\"$+ 0.1(\\sin(x_0 x_1) + x_0x_1)$\",\n",
+    "    fontsize=12,\n",
+    ")\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "83068241",
+   "metadata": {},
+   "source": [
+    "(rvm-sec)=\n",
+    "### Relevance vector machines (RVM)\n",
+    "\n",
+    "Relevance vector machines, used by [Zanna and Bolton 2020](https://laurezanna.github.io/files/Zanna-Bolton-2020.pdf) for equation discovery, provide another way of learning sparse linear models in a Bayesian manner, though with a different prior that prioritizes sparsity over shrinkage when the data permits it. RVMs can also be used with a kernel trick to learn nonlinear decision boundaries which are sparse in a kernelized feature space, though in this case, we use a linear kernel on top of our manually computed library features."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b96ed603",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rvm = RVR(verbose=False)\n",
+    "rvm.fit(feats, y, names)\n",
+    "learned_expr(rvm.m_, rvm.labels)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "037f08eb-2e3c-4ecd-9f0f-962413e31d8c",
+   "metadata": {},
+   "source": [
+    "RVM does discover the $x_0^2$ and $-0.5x_1^2$ terms almost exactly, but ends up with an even more complex approximate expression for the missing $\\sin(0.5x_0x_1)$ term."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "38dff9b2",
+   "metadata": {},
+   "source": [
+    "(stlsq-sec)=\n",
+    "### Sequentially Thresholded Least Squares (STLSQ) from pysindy\n",
+    "\n",
+    "The PySINDy library provides a number of efficient Python implementations of sparse regression algorithms. Their default regression method is sequentially thresholded least squares (STLSQ), which (as its name might suggest) repeatedly runs L2-regularized linear regression with penalty strength `alpha` and prunes features with weight magnitudes below a given `threshold`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6ae57fce",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "stlsq = ps.optimizers.stlsq.STLSQ(alpha=0.0001, threshold=0.1)\n",
+    "stlsq.fit(feats, y)\n",
+    "learned_expr(stlsq.coef_[0], names)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7c57ec9d",
+   "metadata": {},
+   "source": [
+    "This method ends up finding a sparser solution than Lasso, though I did tweak the parameters a bit to make that happen :)\n",
+    "\n",
+    "It's worth checking out [PySINDy's full suite of sparse regression methods](https://pysindy.readthedocs.io/en/latest/api/pysindy.optimizers.html) for other approaches!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3016804a",
+   "metadata": {},
+   "source": [
+    "Overall, all of these methods were able to find the $x_0^2$ and $-0.5x_1^2$ terms, but all of them also needed to find approximations for the remaining term because it wasn't in the feature library. Meaning the expressions are both less accurate and more complex."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "da109831",
+   "metadata": {},
+   "source": [
+    "### Tweaking the Feature Library\n",
+    "\n",
+    "Now, we can potentially resolve some of these issues by pre-multiplying $x_1$ by 0.5 before the augmentation process (which we'll call $x_{1h}$)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b0e30e71",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "feats2 = np.array(x)\n",
+    "feats2[:, 1] = feats2[:, 1] * 0.5\n",
+    "names2 = [\"x0\", \"x1h\"]\n",
+    "depth = 2\n",
+    "\n",
+    "for _ in range(depth):\n",
+    "    feats2, names2 = augment(feats2, names2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0d31d7c4",
+   "metadata": {},
+   "source": [
+    "In this case, the correct model should be expressed as\n",
+    "\n",
+    "$$\n",
+    "y = x_0^2 - 2x_{1h}^2 + \\sin(x_0 x_{1h})\n",
+    "$$\n",
+    "\n",
+    "which is now a linear combination of our basis terms. Let's see if these regression techniques can find it:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2dc1a7bf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "linreg = LinearRegression()\n",
+    "linreg.fit(feats2, y)\n",
+    "learned_expr(linreg.coef_, names2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9f9c67f7",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "lasso = Lasso(alpha=0.01)\n",
+    "lasso.fit(feats2, y)\n",
+    "learned_expr(lasso.coef_, names2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6ff1bb78",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rvm = RVR(verbose=False)\n",
+    "rvm.fit(feats2, y, names2)\n",
+    "learned_expr(rvm.m_, rvm.labels)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "65cb4ad3",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "stlsq = ps.optimizers.stlsq.STLSQ(alpha=0.0001, threshold=0.1)\n",
+    "stlsq.fit(feats2, y)\n",
+    "learned_expr(stlsq.coef_[0], names2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a1f55c25",
+   "metadata": {},
+   "source": [
+    "In this case, every method except Lasso finds the true model exactly (which held true across many choices of regularization parameter).\n",
+    "\n",
+    "## Handling noise\n",
+    "\n",
+    "What if we add noise to the data?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a9159d34",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "noise = np.random.normal(size=feats2.shape) * 0.01"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a3ab527f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "linreg = LinearRegression()\n",
+    "linreg.fit(feats2 + noise, y)\n",
+    "learned_expr(linreg.coef_, names2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ce01433d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "lasso = Lasso(alpha=0.01)\n",
+    "lasso.fit(feats2 + noise, y)\n",
+    "learned_expr(lasso.coef_, names2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a7f3d131",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "rvm = RVR(verbose=False)\n",
+    "rvm.fit(feats2 + noise, y, names2)\n",
+    "learned_expr(rvm.m_, rvm.labels)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a0e87cc2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "stlsq = ps.optimizers.stlsq.STLSQ(alpha=0.0001, threshold=0.1)\n",
+    "stlsq.fit(feats2 + noise, y)\n",
+    "learned_expr(stlsq.coef_[0], names2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "90e5a4d7",
+   "metadata": {},
+   "source": [
+    "In this case, it actually becomes _Lasso_ which gets closest to the true model (i.e. is the only regression technique whose first three leading terms match the ground-truth model). This reversal illustrates how noise can strongly impact the effectiveness of different methods for learning symbolic models.\n",
+    "\n",
+    "Let's see how genetic programming-based techniques deal with noise."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1de9bcf5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "noisy_equations = pysr.PySRRegressor(\n",
+    "    niterations=5,\n",
+    "    binary_operators=[\"+\", \"*\"],  # operators that can combine two terms\n",
+    "    unary_operators=[\"sin\"],  # operators that modify a single term\n",
+    ")\n",
+    "noisy_equations.fit(x, y + np.random.normal(size=y.shape));"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1834ef0d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "noisy_equations.equations_"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ad12ddb5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "round_expr(noisy_equations.sympy())"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3f4831fe",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "noisy_equations.equations_.sympy_format.values[-4]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6fd761c3",
+   "metadata": {},
+   "source": [
+    "It looks like PySR was still able to discover the true expression as part of its Pareto frontier, though with the noise it's only rated third-best for the default tradeoff between complexity and performance (with a sin-less version taking first)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "68ffe19c",
+   "metadata": {},
+   "source": [
+    "## Other Methods for Symbolic Regression\n",
+    "\n",
+    "To finish the notebook, here are a few other techniques for symbolic regression that don't fit under either previously described methods (and all developed in the last several years):\n",
+    "\n",
+    "- [STreCH](https://arxiv.org/pdf/2102.08351.pdf) (Julia implementation [here](https://github.com/jongeunkim/STreCH)) formulates symbolic regression as a non-convex mixed-integer nonlinear programming (MINLP) technique, which it solves using a heuristic-guided cutting plane formulation.\n",
+    "- [EQL](http://proceedings.mlr.press/v80/sahoo18a/sahoo18a.pdf) (Python implementation [here](https://github.com/martius-lab/EQL)) develops \"equation layers\" for neural networks which can be readily interpreted and also chained together, and are regularized with a thresholded L1 penalty to encourage sparsity.\n",
+    "- [AI Feynman](https://arxiv.org/abs/2006.10782) (hybrid Python-Fortran implementation [here](https://github.com/SJ001/AI-Feynman)) starts by training a neural network, then computes _input gradients_ of the learned network, and then uses attributes of those gradients to decompose the symbolic regression problem into something more tractable using graph theory.\n",
+    "- [Symbolic Metamodeling](https://papers.nips.cc/paper/2019/hash/567b8f5f423af15818a068235807edc0-Abstract.html) (Python implementation [here](https://github.com/ahmedmalaa/Symbolic-Metamodeling)) uses gradient descent to learn compositions of Meijer-G functions, a flexible family that can be programmatically projected back onto a wide set of analytic, closed-form, or even algebriac expressions."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "282ec194-148a-4e70-9570-104e18b007bf",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/notebooks/symbolic_vs_nn_multiscale_L96.ipynb b/_sources/notebooks/symbolic_vs_nn_multiscale_L96.ipynb
new file mode 100644
index 00000000..b89ef174
--- /dev/null
+++ b/_sources/notebooks/symbolic_vs_nn_multiscale_L96.ipynb
@@ -0,0 +1,1184 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "d41242f2",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "# Symbolic Regression vs. Neural Networks on Multiscale L96\n",
+    "\n",
+    "To recap from the previous notebooks, [Lorenz (1996)](https://www.ecmwf.int/en/elibrary/10829-predictability-problem-partly-solved) describes a \"two time-scale\" model in two equations which are:\n",
+    "\\begin{align}\n",
+    "\\frac{d}{dt} X_k\n",
+    "&= - X_{k-1} \\left( X_{k-2} - X_{k+1} \\right) - X_k + F - \\left( \\frac{hc}{b} \\right) \\sum_{j=0}^{J-1} Y_{j,k}\n",
+    "\\\\\n",
+    "\\frac{d}{dt} Y_{j,k}\n",
+    "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - Y_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
+    "\\end{align}\n",
+    "\n",
+    "This model can be visualized as follows:\n",
+    "\n",
+    "```{figure} https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376@1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png\n",
+    ":width: 400px\n",
+    ":name: l96-equation-figure-symbolicvnn\n",
+    "\n",
+    "*Visualisation of a two-scale Lorenz '96 system with J = 8 and K = 6. Global-scale variables ($X_k$) are updated based on neighbouring variables and on the local-scale variables ($Y_{j,k}$) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from [Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.](https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436)*.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9070565d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import os\n",
+    "import time\n",
+    "from IPython.display import display, SVG\n",
+    "\n",
+    "import gplearn\n",
+    "import gplearn.genetic\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import pydot\n",
+    "import pysindy as ps\n",
+    "import torch\n",
+    "import torch.nn.functional as F\n",
+    "from sklearn.linear_model import LinearRegression\n",
+    "from sklearn.metrics import r2_score\n",
+    "from sklearn.preprocessing import PolynomialFeatures\n",
+    "from torch import nn, optim\n",
+    "from torch.utils.data import DataLoader, TensorDataset\n",
+    "\n",
+    "from L96_model import L96"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "bf5ecc5a-b99e-4dbb-8b46-3920ef9e3513",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Ensuring reproducibility\n",
+    "np.random.seed(14)\n",
+    "torch.manual_seed(14);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "379d8bd2-e489-44d2-9791-fe7b453c5636",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Dataset Generation\n",
+    "\n",
+    "Let's generate some data using the same conditions as in the previous notebooks [Introduction to Neural Networks](https://m2lines.github.io/L96_demo/notebooks/Universal_approximation.html) and [Using Neural Networks for L96 Parameterization](https://m2lines.github.io/L96_demo/notebooks/Neural_network_for_Lorenz96.html).\n",
+    "\n",
+    "### Obtaining the Ground Truth"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fbc1437d-188e-427f-8da9-5e955cdc58b0",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "time_steps = 20000\n",
+    "forcing, dt, T = 18, 0.01, 0.01 * time_steps\n",
+    "\n",
+    "W = L96(8, 32, F=forcing)\n",
+    "\n",
+    "X_true, _, _, subgrid_tend = W.run(dt, T, store=True, return_coupling=True)\n",
+    "X_true, subgrid_tend = X_true.astype(np.float32), subgrid_tend.astype(np.float32)\n",
+    "\n",
+    "print(f\"X_true shape: {X_true.shape}\")\n",
+    "print(f\"subgrid_tend shape: {subgrid_tend.shape}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b0350f65",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "In this case, although we have 8 different dimensions in $X$, each is derived in the exact same way from neighboring values of $X$ (and local subdimensions $Y$). Therefore, we can actually reduce this problem from an 8-to-8D regression problem down to an 8-to-1D regression problem:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "54892735",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "Dx = X_true.shape[1]\n",
+    "inputs = np.vstack([X_true[:, range(-i, Dx - i)] for i in range(Dx)])\n",
+    "targets = np.hstack([subgrid_tend[:, -i] for i in range(Dx)])\n",
+    "\n",
+    "print(f\"inputs shape: {inputs.shape}\")\n",
+    "print(f\"targets shape: {targets.shape}\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "351d3e5a",
+   "metadata": {
+    "tags": [
+     "hide-input"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(16, 4), dpi=150)\n",
+    "plt.title(\"Multiscale L96 trajectory\", fontsize=16)\n",
+    "for i in range(Dx):\n",
+    "    plt.plot(inputs[:, i][:500], label=r\"$x\" + str(i) + \"$\")\n",
+    "plt.legend(ncol=2, fontsize=14)\n",
+    "plt.xlabel(\"Time\", fontsize=16)\n",
+    "plt.ylabel(\"Value\", fontsize=16)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b7be290f",
+   "metadata": {
+    "tags": [
+     "hide-input"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "fig = plt.figure(figsize=(12, 6), dpi=150)\n",
+    "fig.suptitle(\"2D histograms of features vs. forcing\", fontsize=16)\n",
+    "for i in range(Dx):\n",
+    "    plt.subplot(2, 4, i + 1)\n",
+    "    plt.hist2d(inputs[:, i], targets, bins=50)\n",
+    "    plt.xlabel(f\"$x_{i}$\", fontsize=16)\n",
+    "    plt.xticks(np.arange(-6, 13, 2))\n",
+    "    plt.xlim(-8, 14)\n",
+    "    if i % 4 == 0:\n",
+    "        plt.ylabel(\"Forcing\", fontsize=16)\n",
+    "plt.tight_layout()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "04e202f0",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Looking at these marginal histograms of each large-scale variable vs. the subgrid forcing, we can see there's a mostly linear relationship between $x_0$ and its forcing, suggesting that the subgrid terms tend to force small values higher and large values lower. This suggests that an extremely simple linear model only containing $x_0$ with negative slope will do reasonably well. However, the relationship isn't _completely_ linear, and also contains a bifurcation at larger $x_0$.\n",
+    "\n",
+    "For the other features, the relationships are less straightforward, but there's clearly high mutual information."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aa935b26-4491-4678-a94a-48675c9dea1b",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Split the Data into Training and Test Sets"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cd24ec2c-f8ad-4481-b81e-397af8239330",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "data_idxs = np.arange(len(inputs))\n",
+    "np.random.shuffle(data_idxs)\n",
+    "\n",
+    "num_train_samples = int(len(inputs) * 0.75)\n",
+    "train_idxs = data_idxs[:num_train_samples]\n",
+    "test_idxs = data_idxs[num_train_samples:]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5b46ea2a-eb12-4be7-822d-bc0f8f93fb58",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "### Build the Dataset and the Dataloaders"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b288a4a4-078b-44d5-8e41-1e058de861ed",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "BATCH_SIZE = 1024\n",
+    "\n",
+    "# Training Data\n",
+    "train_dataset = TensorDataset(\n",
+    "    torch.tensor(inputs[train_idxs]), torch.tensor(targets[train_idxs])\n",
+    ")\n",
+    "loader_train = DataLoader(dataset=train_dataset, batch_size=BATCH_SIZE, shuffle=True)\n",
+    "\n",
+    "# Test Data\n",
+    "test_dataset = TensorDataset(\n",
+    "    torch.tensor(inputs[test_idxs]), torch.tensor(targets[test_idxs])\n",
+    ")\n",
+    "loader_test = DataLoader(dataset=test_dataset, batch_size=BATCH_SIZE, shuffle=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "745bcc69-4d0f-4b3e-a1de-41e341b4eb6d",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Creating and Training the Neural Network\n",
+    "\n",
+    "Now let's create a neural network and train it on the dataset we generated above."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c6bcfcb9-b79e-4777-8e55-a31373e5241b",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Define Functions to Train and Evaluate the Model\n",
+    "\n",
+    "```{note}\n",
+    "The code in this section is taken from the notebook [Introduction to Neural Networks](https://m2lines.github.io/L96_demo/notebooks/Universal_approximation.html).\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a0ed1572-d504-47db-b962-98a1300ec4a1",
+   "metadata": {
+    "tags": [
+     "hide-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "def train_model(network, criterion, loader, optimizer):\n",
+    "    \"\"\"Train the network for one epoch\"\"\"\n",
+    "    network.train()\n",
+    "\n",
+    "    train_loss = 0\n",
+    "    for batch_x, batch_y in loader:\n",
+    "        # Get predictions\n",
+    "        if len(batch_x.shape) == 1:\n",
+    "            # This if block is needed to add a dummy dimension if our inputs are 1D\n",
+    "            # (where each number is a different sample)\n",
+    "            prediction = torch.squeeze(network(torch.unsqueeze(batch_x, 1)))\n",
+    "        else:\n",
+    "            prediction = network(batch_x)\n",
+    "\n",
+    "        # Compute the loss\n",
+    "        loss = criterion(prediction, batch_y)\n",
+    "        train_loss += loss.item()\n",
+    "\n",
+    "        # Clear the gradients\n",
+    "        optimizer.zero_grad()\n",
+    "\n",
+    "        # Backpropagation to compute the gradients and update the weights\n",
+    "        loss.backward()\n",
+    "        optimizer.step()\n",
+    "\n",
+    "    return train_loss / len(loader)\n",
+    "\n",
+    "\n",
+    "def test_model(network, criterion, loader):\n",
+    "    \"\"\"Test the network\"\"\"\n",
+    "    network.eval()  # Evaluation mode (important when having dropout layers)\n",
+    "\n",
+    "    test_loss = 0\n",
+    "    with torch.no_grad():\n",
+    "        for batch_x, batch_y in loader:\n",
+    "            # Get predictions\n",
+    "            if len(batch_x.shape) == 1:\n",
+    "                # This if block is needed to add a dummy dimension if our inputs are 1D\n",
+    "                # (where each number is a different sample)\n",
+    "                prediction = torch.squeeze(network(torch.unsqueeze(batch_x, 1)))\n",
+    "            else:\n",
+    "                prediction = network(batch_x)\n",
+    "\n",
+    "            # Compute the loss\n",
+    "            loss = criterion(prediction, batch_y)\n",
+    "            test_loss += loss.item()\n",
+    "\n",
+    "        # Get an average loss for the entire dataset\n",
+    "        test_loss /= len(loader)\n",
+    "\n",
+    "    return test_loss\n",
+    "\n",
+    "\n",
+    "def fit_model(network, criterion, optimizer, train_loader, val_loader, n_epochs):\n",
+    "    \"\"\"Train and validate the network\"\"\"\n",
+    "    train_losses, val_losses = [], []\n",
+    "    start_time = time.time()\n",
+    "    for epoch in range(1, n_epochs + 1):\n",
+    "        train_loss = train_model(network, criterion, train_loader, optimizer)\n",
+    "        val_loss = test_model(network, criterion, val_loader)\n",
+    "        train_losses.append(train_loss)\n",
+    "        val_losses.append(val_loss)\n",
+    "    end_time = time.time()\n",
+    "    print(f\"Training completed in {int(end_time - start_time)} seconds.\")\n",
+    "\n",
+    "    return train_losses, val_losses"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6526b2d4",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Create the Neural Network"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6445e5d8",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "class Net_ANN(nn.Module):\n",
+    "    def __init__(self):\n",
+    "        super(Net_ANN, self).__init__()\n",
+    "        self.linear1 = nn.Linear(8, 16)\n",
+    "        self.linear2 = nn.Linear(16, 16)\n",
+    "        self.linear3 = nn.Linear(16, 1)\n",
+    "\n",
+    "    def forward(self, x):\n",
+    "        x = F.relu(self.linear1(x))\n",
+    "        x = F.relu(self.linear2(x))\n",
+    "        x = self.linear3(x)[:, 0]\n",
+    "        return x"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "22dadd73-44d8-4aa4-b7a7-72352dc9ccaa",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Train the Network"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ab123e48",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "nn_3l = Net_ANN()\n",
+    "\n",
+    "n_epochs = 20\n",
+    "criterion = torch.nn.MSELoss()\n",
+    "optimizer = optim.Adam(nn_3l.parameters(), lr=0.01)\n",
+    "train_loss, validation_loss = fit_model(\n",
+    "    nn_3l, criterion, optimizer, loader_train, loader_test, n_epochs\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "26fd373a-2f8e-427e-9ac4-67eb365f9675",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Visualize Results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "26a0eec2-007c-49fa-96ca-8b32ac839d4b",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.plot(train_loss, \"b\", label=\"Training loss\")\n",
+    "plt.plot(validation_loss, \"r\", label=\"Validation loss\")\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d5965164",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "preds_nn = nn_3l(torch.from_numpy(inputs[test_idxs])).detach().numpy()\n",
+    "print(\"R2 Score:\", r2_score(preds_nn, targets[test_idxs]))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1ba1dacc",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "The neural network gets an $R^2$ of ~ 0.916, which seems pretty good given the non-determinism of the problem."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4e3aaddc-fcc8-41eb-ab37-cc923a83aeb8",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Comparing with Linear Regression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ebec28d9",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "linear_reg = LinearRegression()\n",
+    "linear_reg.fit(inputs[train_idxs], targets[train_idxs])\n",
+    "print(\"R2 Score:\", linear_reg.score(inputs[test_idxs], targets[test_idxs]))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1677913e",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Linear regression also does fairly well, though about 0.1 lower in terms of $R^2$. Lets look at its coefficients:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4f3f9d6c",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(dpi=150)\n",
+    "plt.bar(range(8), linear_reg.coef_)\n",
+    "plt.xticks(range(8), [f\"x{i}\" for i in range(8)], fontsize=12)\n",
+    "plt.ylabel(\"Coefficient value\", fontsize=12)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "01bb0893",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "This matches the 2D histogram plots from earlier that showed a negative linear relationship between forcing and $x_0$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4f2a4893",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(15, 6), dpi=150)\n",
+    "plt.title(\"Neural Network vs. Linear Regression\", fontsize=16)\n",
+    "\n",
+    "preds_nn = nn_3l(torch.from_numpy(inputs)).detach().numpy()\n",
+    "preds_lr = linear_reg.predict(inputs)\n",
+    "random_idx = np.random.choice(len(targets - 1000))\n",
+    "random_slice = slice(random_idx, random_idx + 1000)\n",
+    "\n",
+    "plt.plot(preds_nn[random_slice], label=\"NN Predicted values\")\n",
+    "plt.plot(preds_lr[random_slice], label=\"LR Predicted values\")\n",
+    "plt.plot(\n",
+    "    targets[random_slice], label=\"True values\", ls=\"--\", color=\"black\", alpha=0.667\n",
+    ")\n",
+    "plt.xlabel(f\"Time (starting at idx {random_idx})\", fontsize=16)\n",
+    "plt.ylabel(\"Subgrid forcing\", fontsize=16)\n",
+    "plt.legend(fontsize=14)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d29faede",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Symbolic regression with genetic programming\n",
+    "\n",
+    "Let's try using one of the genetic programming libraries to discover a parameterization:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3947cb34",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "symbolic_reg = gplearn.genetic.SymbolicRegressor(\n",
+    "    population_size=5000,\n",
+    "    generations=25,\n",
+    "    p_crossover=0.7,\n",
+    "    p_subtree_mutation=0.1,\n",
+    "    p_hoist_mutation=0.05,\n",
+    "    p_point_mutation=0.1,\n",
+    "    max_samples=0.9,\n",
+    "    verbose=1,\n",
+    "    parsimony_coefficient=0.001,\n",
+    "    const_range=(-2, 2),\n",
+    "    function_set=(\"add\", \"mul\", \"sub\", \"div\", \"sin\", \"log\"),\n",
+    ")\n",
+    "\n",
+    "symbolic_reg.fit(inputs[train_idxs][:20000], targets[train_idxs][:20000])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cd991e1b",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "print(symbolic_reg._program)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ccd32a51",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "print(\n",
+    "    \"R2 Score:\", r2_score(symbolic_reg.predict(inputs[test_idxs]), targets[test_idxs])\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6510bdd5",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "While the program does slightly better than linear regression, it's extremely complicated (graph visualization below for slightly easier inspection):"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f88adb60-93a2-44f9-8314-c411e59d43e7",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Build the graph\n",
+    "graph = pydot.graph_from_dot_data(symbolic_reg._program.export_graphviz())[0]\n",
+    "graph.write_svg(\"figs/symbolic_regression.svg\")\n",
+    "\n",
+    "# Display the graph\n",
+    "display(SVG(\"figs/symbolic_regression.svg\"))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "59da9e31",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Explore the Pareto frontier\n",
+    "\n",
+    "We can explore the Pareto frontier of the final population of programs as well, to see if there's a simpler program that still does well"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "bc2f7f44",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "df = pd.DataFrame.from_dict(\n",
+    "    [\n",
+    "        dict(\n",
+    "            idx=i,\n",
+    "            expr=str(p),\n",
+    "            fitness=p.fitness_,\n",
+    "            length=p.length_,\n",
+    "            r2_score=r2_score(p.execute(inputs[test_idxs]), targets[test_idxs]),\n",
+    "        )\n",
+    "        for i, p in enumerate(symbolic_reg._programs[-1])\n",
+    "    ]\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "67789b73",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "pareto_frontier = []\n",
+    "used_programs = set()\n",
+    "\n",
+    "for l in range(1, 40):\n",
+    "    programs = df[df.length <= l]\n",
+    "    best = programs.r2_score.argmax()\n",
+    "    idx = programs.idx.values[best]\n",
+    "    r2 = programs.r2_score.values[best]\n",
+    "    program = symbolic_reg._programs[-1][idx]\n",
+    "    if program not in used_programs:\n",
+    "        pareto_frontier.append(dict(length=l, r2=r2, expr=str(program)))\n",
+    "        used_programs.add(program)\n",
+    "\n",
+    "pd.set_option(\"max_colwidth\", 200)\n",
+    "pareto_frontier = pd.DataFrame.from_dict(pareto_frontier)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "55fca58a-911e-46c3-9a15-4a2ba8590eee",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "pareto_frontier"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "905cfb5c",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Unfortunately, nothing really jumps out here as both simple and performant."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bfecdaba",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Symbolic Regression with Sparse Linear Regression and Polynomial Features\n",
+    "\n",
+    "We can also apply sparse regression methods to this problem with polynomial features"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a30255fd",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "def learned_expr(coef, names, cutoff=1e-3):\n",
+    "    coef_idx = np.argwhere(np.abs(coef) > cutoff)[:, 0]\n",
+    "    sort_order = reversed(np.argsort(np.abs(coef[coef_idx])))\n",
+    "    return \" + \".join(\n",
+    "        [\n",
+    "            f\"{coef[coef_idx[i]]:.4f}*{names[coef_idx[i]].replace(' ','*')}\"\n",
+    "            for i in sort_order\n",
+    "        ]\n",
+    "    )"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b469cac9",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "pf = PolynomialFeatures(2)\n",
+    "feats = pf.fit_transform(inputs)\n",
+    "names = pf.get_feature_names_out([f\"x_{i}\" for i in range(8)])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "30121a3d",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Non-sparse linear regression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "721fd64e-eae8-4b1c-b2f9-053aab36902a",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "linear_reg = LinearRegression()\n",
+    "linear_reg.fit(feats[train_idxs], targets[train_idxs])\n",
+    "learned_expr(linear_reg.coef_, names)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c928dc9b-eb83-45bc-a416-4357f1cb6c30",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "print(\"R2 Score:\", linear_reg.score(feats[test_idxs], targets[test_idxs]))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "222c9020",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Adding polynomial features brings our $R^2$ much closer to that of the neural network, but we have a ton of terms. Let's try making the regression sparser.\n",
+    "\n",
+    "### Sequentially thresholded least squares"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0d60408a",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "stlsq = ps.optimizers.stlsq.STLSQ(alpha=0.001, threshold=0.01)\n",
+    "stlsq.fit(feats[train_idxs], targets[train_idxs])\n",
+    "learned_expr(stlsq.coef_[0], names)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f3918224",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "stlsq.score(feats[test_idxs], targets[test_idxs])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "45196067",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "Looks like we can get a much sparser expression without sacrificing too much performance, though if we increase the threshold slightly, we will see a trade-off:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "210d8e5f",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "stlsq2 = ps.optimizers.stlsq.STLSQ(alpha=0.001, threshold=0.02)\n",
+    "stlsq2.fit(feats[train_idxs], targets[train_idxs])\n",
+    "learned_expr(stlsq2.coef_[0], names)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2b3aaf12",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "stlsq2.score(feats[test_idxs], targets[test_idxs])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d666fd8e",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "This is much simpler, but less performant. It still strongly outperformed the particluar genetic programming method we used, though. Let's visualize some of the predictions:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4165960d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(15, 6), dpi=150)\n",
+    "plt.title(\"Neural network vs. quadratic regressions\", fontsize=16)\n",
+    "\n",
+    "preds_nn = nn_3l(torch.from_numpy(inputs)).detach().numpy()\n",
+    "preds_lr_quad = linear_reg.predict(feats)\n",
+    "preds_st_quad = stlsq.predict(feats)\n",
+    "preds_st_quad2 = stlsq2.predict(feats)\n",
+    "random_idx = np.random.choice(len(targets - 1000))\n",
+    "random_slice = slice(random_idx, random_idx + 1000)\n",
+    "\n",
+    "plt.plot(preds_nn[random_slice], label=\"NN predicted values\")\n",
+    "plt.plot(preds_lr_quad[random_slice], label=\"LR + quadratic\")\n",
+    "plt.plot(preds_st_quad[random_slice], label=r\"STLSQ + quadratic, threshold=$0.01$\")\n",
+    "plt.plot(preds_st_quad2[random_slice], label=r\"STLSQ + quadratic, threshold=$0.02$\")\n",
+    "plt.plot(\n",
+    "    targets[random_slice], label=\"True values\", ls=\"--\", color=\"black\", alpha=0.667\n",
+    ")\n",
+    "plt.xlabel(f\"Time (starting at idx {random_idx})\", fontsize=16)\n",
+    "plt.ylabel(\"Subgrid forcing\", fontsize=16)\n",
+    "plt.legend(fontsize=14)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bdbfaa2c",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "## Add threshold features\n",
+    "\n",
+    "The 2D histogram of $x_0$ looks like it might be potentially modelable by piecewise linear regression:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "24aa3a4b",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(4, 4), dpi=150)\n",
+    "plt.hist2d(inputs[:, 0], targets, bins=50)\n",
+    "plt.xlim(-6, 14)\n",
+    "plt.xlabel(\"$x_0$\", fontsize=16)\n",
+    "plt.ylabel(\"Forcing\", fontsize=16)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0e5bb5ba",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "### Sparse regression\n",
+    "\n",
+    "Let's see if sparse regression will discover that if we provide it with thresholding features and cross-terms:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ddd19c3d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "x0_thresholds = np.array(\n",
+    "    [np.ones_like(inputs[:, 0])] + [inputs[:, 0] > i for i in range(6)]\n",
+    ")\n",
+    "x0_threshold_names = [\"\"] + [f\"*(x0>{t})\" for t in range(6)]\n",
+    "\n",
+    "feats2 = np.array(\n",
+    "    [\n",
+    "        feats[:, i] * x0_thresholds[j]\n",
+    "        for i in range(feats.shape[1])\n",
+    "        for j in range(len(x0_thresholds))\n",
+    "    ]\n",
+    ").T\n",
+    "names2 = [\n",
+    "    f\"{names[i]}{x0_threshold_names[j]}\"\n",
+    "    for i in range(feats.shape[1])\n",
+    "    for j in range(len(x0_thresholds))\n",
+    "]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "da7ab7fb",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "stlsq3 = ps.optimizers.stlsq.STLSQ(alpha=0.001, threshold=0.25)\n",
+    "stlsq3.fit(feats2[train_idxs], targets[train_idxs])\n",
+    "learned_expr(stlsq3.coef_[0], names2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "794ef1ce",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "stlsq3.score(feats2[test_idxs], targets[test_idxs])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4d853e84",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "plt.figure(figsize=(4, 4), dpi=150)\n",
+    "plt.hist2d(inputs[:, 0], targets, bins=50)\n",
+    "\n",
+    "fn = (\n",
+    "    lambda x0: -2.1200 * 1 * (x0 > 2)\n",
+    "    + -1.2695 * 1 * (x0 > 0)\n",
+    "    + -0.9488 * x0\n",
+    "    + -0.6694 * 1 * (x0 > 4)\n",
+    "    + 0.5175 * x0 * (x0 > 1)\n",
+    ")\n",
+    "\n",
+    "xt = np.linspace(-6, 14, 100)\n",
+    "yt = fn(xt)\n",
+    "plt.plot(xt, yt, color=\"red\")\n",
+    "plt.xlim(-6, 14)\n",
+    "plt.xlabel(\"$x_0$\", fontsize=16)\n",
+    "plt.ylabel(\"Forcing\", fontsize=16)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a6f67e0",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "It looks like the model learns a piecewise linear function in a single variable, even though we didn't explicitly require it!\n",
+    "\n",
+    "### Genetic programming\n",
+    "\n",
+    "We can also try using genetic programming here, but with a different function set that removes some of the nonlinear operators we included before and replaces them with min and max, which allow for piecewise behavior:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8310dfba",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "symbolic_reg_pw = gplearn.genetic.SymbolicRegressor(\n",
+    "    population_size=5000,\n",
+    "    generations=50,\n",
+    "    p_crossover=0.7,\n",
+    "    p_subtree_mutation=0.1,\n",
+    "    p_hoist_mutation=0.05,\n",
+    "    p_point_mutation=0.1,\n",
+    "    max_samples=0.9,\n",
+    "    verbose=1,\n",
+    "    parsimony_coefficient=0.001,\n",
+    "    const_range=(-2, 2),\n",
+    "    function_set=(\"add\", \"mul\", \"max\", \"min\"),\n",
+    ")\n",
+    "symbolic_reg_pw.fit(inputs[train_idxs][:20000], targets[train_idxs][:20000])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c9a9027c",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "print(symbolic_reg_pw._program)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f06a133d",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "print(\n",
+    "    \"R2 Score:\",\n",
+    "    r2_score(symbolic_reg_pw.predict(inputs[test_idxs]), targets[test_idxs]),\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e828836f",
+   "metadata": {
+    "user_expressions": []
+   },
+   "source": [
+    "While this leads to slightly higher $R^2$ than before, the model is still pretty complex:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "45225961-2844-4d42-8756-b99eed311443",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# Build the graph\n",
+    "graph = pydot.graph_from_dot_data(symbolic_reg_pw._program.export_graphviz())[0]\n",
+    "graph.write_svg(\"figs/symbolic_regression_pw.svg\")\n",
+    "\n",
+    "# Display the graph\n",
+    "from IPython.display import Image, display\n",
+    "\n",
+    "display(SVG(\"figs/symbolic_regression_pw.svg\"))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "84b55c24",
+   "metadata": {
+    "tags": [],
+    "user_expressions": []
+   },
+   "source": [
+    "Tuning the hyperparameters could help, but requires more compute."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
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diff --git a/_sphinx_design_static/design-tabs.js b/_sphinx_design_static/design-tabs.js
new file mode 100644
index 00000000..36b38cf0
--- /dev/null
+++ b/_sphinx_design_static/design-tabs.js
@@ -0,0 +1,27 @@
+var sd_labels_by_text = {};
+
+function ready() {
+  const li = document.getElementsByClassName("sd-tab-label");
+  for (const label of li) {
+    syncId = label.getAttribute("data-sync-id");
+    if (syncId) {
+      label.onclick = onLabelClick;
+      if (!sd_labels_by_text[syncId]) {
+        sd_labels_by_text[syncId] = [];
+      }
+      sd_labels_by_text[syncId].push(label);
+    }
+  }
+}
+
+function onLabelClick() {
+  // Activate other inputs with the same sync id.
+  syncId = this.getAttribute("data-sync-id");
+  for (label of sd_labels_by_text[syncId]) {
+    if (label === this) continue;
+    label.previousElementSibling.checked = true;
+  }
+  window.localStorage.setItem("sphinx-design-last-tab", syncId);
+}
+
+document.addEventListener("DOMContentLoaded", ready, false);
diff --git a/_static/basic.css b/_static/basic.css
new file mode 100644
index 00000000..d54be806
--- /dev/null
+++ b/_static/basic.css
@@ -0,0 +1,906 @@
+/*
+ * basic.css
+ * ~~~~~~~~~
+ *
+ * Sphinx stylesheet -- basic theme.
+ *
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+/* -- main layout ----------------------------------------------------------- */
+
+div.clearer {
+    clear: both;
+}
+
+div.section::after {
+    display: block;
+    content: '';
+    clear: left;
+}
+
+/* -- relbar ---------------------------------------------------------------- */
+
+div.related {
+    width: 100%;
+    font-size: 90%;
+}
+
+div.related h3 {
+    display: none;
+}
+
+div.related ul {
+    margin: 0;
+    padding: 0 0 0 10px;
+    list-style: none;
+}
+
+div.related li {
+    display: inline;
+}
+
+div.related li.right {
+    float: right;
+    margin-right: 5px;
+}
+
+/* -- sidebar --------------------------------------------------------------- */
+
+div.sphinxsidebarwrapper {
+    padding: 10px 5px 0 10px;
+}
+
+div.sphinxsidebar {
+    float: left;
+    width: 270px;
+    margin-left: -100%;
+    font-size: 90%;
+    word-wrap: break-word;
+    overflow-wrap : break-word;
+}
+
+div.sphinxsidebar ul {
+    list-style: none;
+}
+
+div.sphinxsidebar ul ul,
+div.sphinxsidebar ul.want-points {
+    margin-left: 20px;
+    list-style: square;
+}
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+div.sphinxsidebar ul ul {
+    margin-top: 0;
+    margin-bottom: 0;
+}
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+div.sphinxsidebar form {
+    margin-top: 10px;
+}
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+div.sphinxsidebar input {
+    border: 1px solid #98dbcc;
+    font-family: sans-serif;
+    font-size: 1em;
+}
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+div.sphinxsidebar #searchbox form.search {
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+}
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+div.sphinxsidebar #searchbox input[type="text"] {
+    float: left;
+    width: 80%;
+    padding: 0.25em;
+    box-sizing: border-box;
+}
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+div.sphinxsidebar #searchbox input[type="submit"] {
+    float: left;
+    width: 20%;
+    border-left: none;
+    padding: 0.25em;
+    box-sizing: border-box;
+}
+
+
+img {
+    border: 0;
+    max-width: 100%;
+}
+
+/* -- search page ----------------------------------------------------------- */
+
+ul.search {
+    margin: 10px 0 0 20px;
+    padding: 0;
+}
+
+ul.search li {
+    padding: 5px 0 5px 20px;
+    background-image: url(file.png);
+    background-repeat: no-repeat;
+    background-position: 0 7px;
+}
+
+ul.search li a {
+    font-weight: bold;
+}
+
+ul.search li p.context {
+    color: #888;
+    margin: 2px 0 0 30px;
+    text-align: left;
+}
+
+ul.keywordmatches li.goodmatch a {
+    font-weight: bold;
+}
+
+/* -- index page ------------------------------------------------------------ */
+
+table.contentstable {
+    width: 90%;
+    margin-left: auto;
+    margin-right: auto;
+}
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+table.contentstable p.biglink {
+    line-height: 150%;
+}
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+a.biglink {
+    font-size: 1.3em;
+}
+
+span.linkdescr {
+    font-style: italic;
+    padding-top: 5px;
+    font-size: 90%;
+}
+
+/* -- general index --------------------------------------------------------- */
+
+table.indextable {
+    width: 100%;
+}
+
+table.indextable td {
+    text-align: left;
+    vertical-align: top;
+}
+
+table.indextable ul {
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+    margin-bottom: 0;
+    list-style-type: none;
+}
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+}
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+}
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+table.indextable tr.cap {
+    margin-top: 10px;
+    background-color: #f2f2f2;
+}
+
+img.toggler {
+    margin-right: 3px;
+    margin-top: 3px;
+    cursor: pointer;
+}
+
+div.modindex-jumpbox {
+    border-top: 1px solid #ddd;
+    border-bottom: 1px solid #ddd;
+    margin: 1em 0 1em 0;
+    padding: 0.4em;
+}
+
+div.genindex-jumpbox {
+    border-top: 1px solid #ddd;
+    border-bottom: 1px solid #ddd;
+    margin: 1em 0 1em 0;
+    padding: 0.4em;
+}
+
+/* -- domain module index --------------------------------------------------- */
+
+table.modindextable td {
+    padding: 2px;
+    border-collapse: collapse;
+}
+
+/* -- general body styles --------------------------------------------------- */
+
+div.body {
+    min-width: 450px;
+    max-width: 800px;
+}
+
+div.body p, div.body dd, div.body li, div.body blockquote {
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+    -ms-hyphens: auto;
+    -webkit-hyphens: auto;
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+}
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+}
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+    content: "[";
+}
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+}
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+}
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+div.body p.caption {
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+}
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+div.body td {
+    text-align: left;
+}
+
+.first {
+    margin-top: 0 !important;
+}
+
+p.rubric {
+    margin-top: 30px;
+    font-weight: bold;
+}
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+img.align-left, figure.align-left, .figure.align-left, object.align-left {
+    clear: left;
+    float: left;
+    margin-right: 1em;
+}
+
+img.align-right, figure.align-right, .figure.align-right, object.align-right {
+    clear: right;
+    float: right;
+    margin-left: 1em;
+}
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+img.align-center, figure.align-center, .figure.align-center, object.align-center {
+  display: block;
+  margin-left: auto;
+  margin-right: auto;
+}
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+img.align-default, figure.align-default, .figure.align-default {
+  display: block;
+  margin-left: auto;
+  margin-right: auto;
+}
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+.align-left {
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+}
+
+.align-center {
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+}
+
+.align-default {
+    text-align: center;
+}
+
+.align-right {
+    text-align: right;
+}
+
+/* -- sidebars -------------------------------------------------------------- */
+
+div.sidebar,
+aside.sidebar {
+    margin: 0 0 0.5em 1em;
+    border: 1px solid #ddb;
+    padding: 7px;
+    background-color: #ffe;
+    width: 40%;
+    float: right;
+    clear: right;
+    overflow-x: auto;
+}
+
+p.sidebar-title {
+    font-weight: bold;
+}
+
+div.admonition, div.topic, blockquote {
+    clear: left;
+}
+
+/* -- topics ---------------------------------------------------------------- */
+
+div.topic {
+    border: 1px solid #ccc;
+    padding: 7px;
+    margin: 10px 0 10px 0;
+}
+
+p.topic-title {
+    font-size: 1.1em;
+    font-weight: bold;
+    margin-top: 10px;
+}
+
+/* -- admonitions ----------------------------------------------------------- */
+
+div.admonition {
+    margin-top: 10px;
+    margin-bottom: 10px;
+    padding: 7px;
+}
+
+div.admonition dt {
+    font-weight: bold;
+}
+
+p.admonition-title {
+    margin: 0px 10px 5px 0px;
+    font-weight: bold;
+}
+
+div.body p.centered {
+    text-align: center;
+    margin-top: 25px;
+}
+
+/* -- content of sidebars/topics/admonitions -------------------------------- */
+
+div.sidebar > :last-child,
+aside.sidebar > :last-child,
+div.topic > :last-child,
+div.admonition > :last-child {
+    margin-bottom: 0;
+}
+
+div.sidebar::after,
+aside.sidebar::after,
+div.topic::after,
+div.admonition::after,
+blockquote::after {
+    display: block;
+    content: '';
+    clear: both;
+}
+
+/* -- tables ---------------------------------------------------------------- */
+
+table.docutils {
+    margin-top: 10px;
+    margin-bottom: 10px;
+    border: 0;
+    border-collapse: collapse;
+}
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+}
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+}
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+}
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+}
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+    border-top: 0;
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+    border-bottom: 1px solid #aaa;
+}
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+}
+
+th {
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+    padding-right: 5px;
+}
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+    border-left: solid 1px gray;
+    margin-left: 1px;
+}
+
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+    border-bottom: none;
+}
+
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+    margin-top: 0px;
+}
+
+th > :last-child,
+td > :last-child {
+    margin-bottom: 0px;
+}
+
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+
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+    padding: 0.5em;
+}
+
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+    padding: 0.3em;
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+    font-style: italic;
+}
+
+div.figure p.caption span.caption-text,
+figcaption span.caption-text {
+}
+
+/* -- field list styles ----------------------------------------------------- */
+
+table.field-list td, table.field-list th {
+    border: 0 !important;
+}
+
+.field-list ul {
+    margin: 0;
+    padding-left: 1em;
+}
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+.field-list p {
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+}
+
+.field-name {
+    -moz-hyphens: manual;
+    -ms-hyphens: manual;
+    -webkit-hyphens: manual;
+    hyphens: manual;
+}
+
+/* -- hlist styles ---------------------------------------------------------- */
+
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+    margin: 1em 0;
+}
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+    vertical-align: top;
+}
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+	font-family: 'Consolas', 'Menlo', 'DejaVu Sans Mono', 'Bitstream Vera Sans Mono', monospace;
+}
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+    font-weight: bold;
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+}
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+    font-size: larger;
+}
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+	font-style: italic;
+}
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+
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+	font-family: unset;
+}
+
+.sig.c   .k, .sig.c   .kt,
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+	color: #0033B3;
+}
+
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+	color: #1750EB;
+}
+
+.sig.c   .s, .sig.c   .sc,
+.sig.cpp .s, .sig.cpp .sc {
+	color: #067D17;
+}
+
+
+/* -- other body styles ----------------------------------------------------- */
+
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+}
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+}
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+}
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+}
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+:not(li) > ul > li:first-child > :first-child {
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+}
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+:not(li) > ol > li:last-child > :last-child,
+:not(li) > ul > li:last-child > :last-child {
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+    margin-bottom: 0em;
+}
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+}
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+dd > :first-child {
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+}
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+dd ul, dd table {
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+}
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+    margin-bottom: 10px;
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+}
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+}
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+}
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+}
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+    font-size: 1.1em;
+}
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+.versionmodified {
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+}
+
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+    padding: 5px;
+    border: 3px solid red;
+}
+
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+}
+
+.line-block {
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+    margin-top: 1em;
+    margin-bottom: 1em;
+}
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+}
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+}
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+}
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+}
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+}
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+abbr, acronym {
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+    cursor: help;
+}
+
+/* -- code displays --------------------------------------------------------- */
+
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+    overflow: auto;
+    overflow-y: hidden;  /* fixes display issues on Chrome browsers */
+}
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+    clear: both;
+}
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+    -ms-hyphens: none;
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+    white-space: nowrap;
+}
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+}
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+}
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+}
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+
+div.code-block-caption {
+    margin-top: 1em;
+    padding: 2px 5px;
+    font-size: small;
+}
+
+div.code-block-caption code {
+    background-color: transparent;
+}
+
+table.highlighttable td.linenos,
+span.linenos,
+div.highlight span.gp {  /* gp: Generic.Prompt */
+  user-select: none;
+  -webkit-user-select: text; /* Safari fallback only */
+  -webkit-user-select: none; /* Chrome/Safari */
+  -moz-user-select: none; /* Firefox */
+  -ms-user-select: none; /* IE10+ */
+}
+
+div.code-block-caption span.caption-number {
+    padding: 0.1em 0.3em;
+    font-style: italic;
+}
+
+div.code-block-caption span.caption-text {
+}
+
+div.literal-block-wrapper {
+    margin: 1em 0;
+}
+
+code.xref, a code {
+    background-color: transparent;
+    font-weight: bold;
+}
+
+h1 code, h2 code, h3 code, h4 code, h5 code, h6 code {
+    background-color: transparent;
+}
+
+.viewcode-link {
+    float: right;
+}
+
+.viewcode-back {
+    float: right;
+    font-family: sans-serif;
+}
+
+div.viewcode-block:target {
+    margin: -1px -10px;
+    padding: 0 10px;
+}
+
+/* -- math display ---------------------------------------------------------- */
+
+img.math {
+    vertical-align: middle;
+}
+
+div.body div.math p {
+    text-align: center;
+}
+
+span.eqno {
+    float: right;
+}
+
+span.eqno a.headerlink {
+    position: absolute;
+    z-index: 1;
+}
+
+div.math:hover a.headerlink {
+    visibility: visible;
+}
+
+/* -- printout stylesheet --------------------------------------------------- */
+
+@media print {
+    div.document,
+    div.documentwrapper,
+    div.bodywrapper {
+        margin: 0 !important;
+        width: 100%;
+    }
+
+    div.sphinxsidebar,
+    div.related,
+    div.footer,
+    #top-link {
+        display: none;
+    }
+}
\ No newline at end of file
diff --git a/_static/check-solid.svg b/_static/check-solid.svg
new file mode 100644
index 00000000..92fad4b5
--- /dev/null
+++ b/_static/check-solid.svg
@@ -0,0 +1,4 @@
+<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-check" width="44" height="44" viewBox="0 0 24 24" stroke-width="2" stroke="#22863a" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <path d="M5 12l5 5l10 -10" />
+</svg>
diff --git a/_static/clipboard.min.js b/_static/clipboard.min.js
new file mode 100644
index 00000000..54b3c463
--- /dev/null
+++ b/_static/clipboard.min.js
@@ -0,0 +1,7 @@
+/*!
+ * clipboard.js v2.0.8
+ * https://clipboardjs.com/
+ *
+ * Licensed MIT © Zeno Rocha
+ */
+!function(t,e){"object"==typeof exports&&"object"==typeof module?module.exports=e():"function"==typeof define&&define.amd?define([],e):"object"==typeof exports?exports.ClipboardJS=e():t.ClipboardJS=e()}(this,function(){return n={686:function(t,e,n){"use strict";n.d(e,{default:function(){return o}});var e=n(279),i=n.n(e),e=n(370),u=n.n(e),e=n(817),c=n.n(e);function a(t){try{return document.execCommand(t)}catch(t){return}}var f=function(t){t=c()(t);return a("cut"),t};var l=function(t){var e,n,o,r=1<arguments.length&&void 0!==arguments[1]?arguments[1]:{container:document.body},i="";return"string"==typeof t?(e=t,n="rtl"===document.documentElement.getAttribute("dir"),(o=document.createElement("textarea")).style.fontSize="12pt",o.style.border="0",o.style.padding="0",o.style.margin="0",o.style.position="absolute",o.style[n?"right":"left"]="-9999px",n=window.pageYOffset||document.documentElement.scrollTop,o.style.top="".concat(n,"px"),o.setAttribute("readonly",""),o.value=e,o=o,r.container.appendChild(o),i=c()(o),a("copy"),o.remove()):(i=c()(t),a("copy")),i};function r(t){return(r="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t})(t)}var s=function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:{},e=t.action,n=void 0===e?"copy":e,o=t.container,e=t.target,t=t.text;if("copy"!==n&&"cut"!==n)throw new Error('Invalid "action" value, use either "copy" or "cut"');if(void 0!==e){if(!e||"object"!==r(e)||1!==e.nodeType)throw new Error('Invalid "target" value, use a valid Element');if("copy"===n&&e.hasAttribute("disabled"))throw new Error('Invalid "target" attribute. 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You can\'t cut text from elements with "readonly" or "disabled" attributes')}return t?l(t,{container:o}):e?"cut"===n?f(e):l(e,{container:o}):void 0};function d(t){return(d="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t})(t)}function p(t,e){for(var n=0;n<e.length;n++){var o=e[n];o.enumerable=o.enumerable||!1,o.configurable=!0,"value"in o&&(o.writable=!0),Object.defineProperty(t,o.key,o)}}function y(t,e){return(y=Object.setPrototypeOf||function(t,e){return t.__proto__=e,t})(t,e)}function h(n){var o=function(){if("undefined"==typeof Reflect||!Reflect.construct)return!1;if(Reflect.construct.sham)return!1;if("function"==typeof Proxy)return!0;try{return Date.prototype.toString.call(Reflect.construct(Date,[],function(){})),!0}catch(t){return!1}}();return function(){var t,e=m(n);return t=o?(t=m(this).constructor,Reflect.construct(e,arguments,t)):e.apply(this,arguments),e=this,!(t=t)||"object"!==d(t)&&"function"!=typeof t?function(t){if(void 0!==t)return t;throw new ReferenceError("this hasn't been initialised - super() hasn't been called")}(e):t}}function m(t){return(m=Object.setPrototypeOf?Object.getPrototypeOf:function(t){return t.__proto__||Object.getPrototypeOf(t)})(t)}function v(t,e){t="data-clipboard-".concat(t);if(e.hasAttribute(t))return e.getAttribute(t)}var o=function(){!function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Super expression must either be null or a function");t.prototype=Object.create(e&&e.prototype,{constructor:{value:t,writable:!0,configurable:!0}}),e&&y(t,e)}(r,i());var t,e,n,o=h(r);function r(t,e){var n;return function(t){if(!(t instanceof r))throw new TypeError("Cannot call a class as a function")}(this),(n=o.call(this)).resolveOptions(e),n.listenClick(t),n}return t=r,n=[{key:"copy",value:function(t){var e=1<arguments.length&&void 0!==arguments[1]?arguments[1]:{container:document.body};return l(t,e)}},{key:"cut",value:function(t){return f(t)}},{key:"isSupported",value:function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:["copy","cut"],t="string"==typeof t?[t]:t,e=!!document.queryCommandSupported;return t.forEach(function(t){e=e&&!!document.queryCommandSupported(t)}),e}}],(e=[{key:"resolveOptions",value:function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:{};this.action="function"==typeof t.action?t.action:this.defaultAction,this.target="function"==typeof t.target?t.target:this.defaultTarget,this.text="function"==typeof t.text?t.text:this.defaultText,this.container="object"===d(t.container)?t.container:document.body}},{key:"listenClick",value:function(t){var e=this;this.listener=u()(t,"click",function(t){return e.onClick(t)})}},{key:"onClick",value:function(t){var e=t.delegateTarget||t.currentTarget,t=s({action:this.action(e),container:this.container,target:this.target(e),text:this.text(e)});this.emit(t?"success":"error",{action:this.action,text:t,trigger:e,clearSelection:function(){e&&e.focus(),document.activeElement.blur(),window.getSelection().removeAllRanges()}})}},{key:"defaultAction",value:function(t){return v("action",t)}},{key:"defaultTarget",value:function(t){t=v("target",t);if(t)return document.querySelector(t)}},{key:"defaultText",value:function(t){return v("text",t)}},{key:"destroy",value:function(){this.listener.destroy()}}])&&p(t.prototype,e),n&&p(t,n),r}()},828:function(t){var e;"undefined"==typeof Element||Element.prototype.matches||((e=Element.prototype).matches=e.matchesSelector||e.mozMatchesSelector||e.msMatchesSelector||e.oMatchesSelector||e.webkitMatchesSelector),t.exports=function(t,e){for(;t&&9!==t.nodeType;){if("function"==typeof t.matches&&t.matches(e))return t;t=t.parentNode}}},438:function(t,e,n){var u=n(828);function i(t,e,n,o,r){var i=function(e,n,t,o){return function(t){t.delegateTarget=u(t.target,n),t.delegateTarget&&o.call(e,t)}}.apply(this,arguments);return t.addEventListener(n,i,r),{destroy:function(){t.removeEventListener(n,i,r)}}}t.exports=function(t,e,n,o,r){return"function"==typeof t.addEventListener?i.apply(null,arguments):"function"==typeof n?i.bind(null,document).apply(null,arguments):("string"==typeof t&&(t=document.querySelectorAll(t)),Array.prototype.map.call(t,function(t){return i(t,e,n,o,r)}))}},879:function(t,n){n.node=function(t){return void 0!==t&&t instanceof HTMLElement&&1===t.nodeType},n.nodeList=function(t){var e=Object.prototype.toString.call(t);return void 0!==t&&("[object NodeList]"===e||"[object HTMLCollection]"===e)&&"length"in t&&(0===t.length||n.node(t[0]))},n.string=function(t){return"string"==typeof t||t instanceof String},n.fn=function(t){return"[object Function]"===Object.prototype.toString.call(t)}},370:function(t,e,n){var f=n(879),l=n(438);t.exports=function(t,e,n){if(!t&&!e&&!n)throw new Error("Missing required arguments");if(!f.string(e))throw new TypeError("Second argument must be a String");if(!f.fn(n))throw new TypeError("Third argument must be a Function");if(f.node(t))return c=e,a=n,(u=t).addEventListener(c,a),{destroy:function(){u.removeEventListener(c,a)}};if(f.nodeList(t))return o=t,r=e,i=n,Array.prototype.forEach.call(o,function(t){t.addEventListener(r,i)}),{destroy:function(){Array.prototype.forEach.call(o,function(t){t.removeEventListener(r,i)})}};if(f.string(t))return t=t,e=e,n=n,l(document.body,t,e,n);throw new TypeError("First argument must be a String, HTMLElement, HTMLCollection, or NodeList");var o,r,i,u,c,a}},817:function(t){t.exports=function(t){var e,n="SELECT"===t.nodeName?(t.focus(),t.value):"INPUT"===t.nodeName||"TEXTAREA"===t.nodeName?((e=t.hasAttribute("readonly"))||t.setAttribute("readonly",""),t.select(),t.setSelectionRange(0,t.value.length),e||t.removeAttribute("readonly"),t.value):(t.hasAttribute("contenteditable")&&t.focus(),n=window.getSelection(),(e=document.createRange()).selectNodeContents(t),n.removeAllRanges(),n.addRange(e),n.toString());return n}},279:function(t){function e(){}e.prototype={on:function(t,e,n){var o=this.e||(this.e={});return(o[t]||(o[t]=[])).push({fn:e,ctx:n}),this},once:function(t,e,n){var o=this;function r(){o.off(t,r),e.apply(n,arguments)}return r._=e,this.on(t,r,n)},emit:function(t){for(var e=[].slice.call(arguments,1),n=((this.e||(this.e={}))[t]||[]).slice(),o=0,r=n.length;o<r;o++)n[o].fn.apply(n[o].ctx,e);return this},off:function(t,e){var n=this.e||(this.e={}),o=n[t],r=[];if(o&&e)for(var i=0,u=o.length;i<u;i++)o[i].fn!==e&&o[i].fn._!==e&&r.push(o[i]);return r.length?n[t]=r:delete n[t],this}},t.exports=e,t.exports.TinyEmitter=e}},r={},o.n=function(t){var e=t&&t.__esModule?function(){return t.default}:function(){return t};return o.d(e,{a:e}),e},o.d=function(t,e){for(var n in e)o.o(e,n)&&!o.o(t,n)&&Object.defineProperty(t,n,{enumerable:!0,get:e[n]})},o.o=function(t,e){return Object.prototype.hasOwnProperty.call(t,e)},o(686).default;function o(t){if(r[t])return r[t].exports;var e=r[t]={exports:{}};return n[t](e,e.exports,o),e.exports}var n,r});
\ No newline at end of file
diff --git a/_static/copy-button.svg b/_static/copy-button.svg
new file mode 100644
index 00000000..9c074dae
--- /dev/null
+++ b/_static/copy-button.svg
@@ -0,0 +1,5 @@
+<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-copy" width="44" height="44" viewBox="0 0 24 24" stroke-width="1.5" stroke="#000000" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <rect x="8" y="8" width="12" height="12" rx="2" />
+  <path d="M16 8v-2a2 2 0 0 0 -2 -2h-8a2 2 0 0 0 -2 2v8a2 2 0 0 0 2 2h2" />
+</svg>
diff --git a/_static/copybutton.css b/_static/copybutton.css
new file mode 100644
index 00000000..f1916ec7
--- /dev/null
+++ b/_static/copybutton.css
@@ -0,0 +1,94 @@
+/* Copy buttons */
+button.copybtn {
+    position: absolute;
+    display: flex;
+    top: .3em;
+    right: .3em;
+    width: 1.7em;
+    height: 1.7em;
+	opacity: 0;
+    transition: opacity 0.3s, border .3s, background-color .3s;
+    user-select: none;
+    padding: 0;
+    border: none;
+    outline: none;
+    border-radius: 0.4em;
+    /* The colors that GitHub uses */
+    border: #1b1f2426 1px solid;
+    background-color: #f6f8fa;
+    color: #57606a;
+}
+
+button.copybtn.success {
+    border-color: #22863a;
+    color: #22863a;
+}
+
+button.copybtn svg {
+    stroke: currentColor;
+    width: 1.5em;
+    height: 1.5em;
+    padding: 0.1em;
+}
+
+div.highlight  {
+    position: relative;
+}
+
+/* Show the copybutton */
+.highlight:hover button.copybtn, button.copybtn.success {
+	opacity: 1;
+}
+
+.highlight button.copybtn:hover {
+    background-color: rgb(235, 235, 235);
+}
+
+.highlight button.copybtn:active {
+    background-color: rgb(187, 187, 187);
+}
+
+/**
+ * A minimal CSS-only tooltip copied from:
+ *   https://codepen.io/mildrenben/pen/rVBrpK
+ *
+ * To use, write HTML like the following:
+ *
+ * <p class="o-tooltip--left" data-tooltip="Hey">Short</p>
+ */
+ .o-tooltip--left {
+  position: relative;
+ }
+
+ .o-tooltip--left:after {
+    opacity: 0;
+    visibility: hidden;
+    position: absolute;
+    content: attr(data-tooltip);
+    padding: .2em;
+    font-size: .8em;
+    left: -.2em;
+    background: grey;
+    color: white;
+    white-space: nowrap;
+    z-index: 2;
+    border-radius: 2px;
+    transform: translateX(-102%) translateY(0);
+    transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1);
+}
+
+.o-tooltip--left:hover:after {
+    display: block;
+    opacity: 1;
+    visibility: visible;
+    transform: translateX(-100%) translateY(0);
+    transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1);
+    transition-delay: .5s;
+}
+
+/* By default the copy button shouldn't show up when printing a page */
+@media print {
+    button.copybtn {
+        display: none;
+    }
+}
diff --git a/_static/copybutton.js b/_static/copybutton.js
new file mode 100644
index 00000000..2ea7ff3e
--- /dev/null
+++ b/_static/copybutton.js
@@ -0,0 +1,248 @@
+// Localization support
+const messages = {
+  'en': {
+    'copy': 'Copy',
+    'copy_to_clipboard': 'Copy to clipboard',
+    'copy_success': 'Copied!',
+    'copy_failure': 'Failed to copy',
+  },
+  'es' : {
+    'copy': 'Copiar',
+    'copy_to_clipboard': 'Copiar al portapapeles',
+    'copy_success': '¡Copiado!',
+    'copy_failure': 'Error al copiar',
+  },
+  'de' : {
+    'copy': 'Kopieren',
+    'copy_to_clipboard': 'In die Zwischenablage kopieren',
+    'copy_success': 'Kopiert!',
+    'copy_failure': 'Fehler beim Kopieren',
+  },
+  'fr' : {
+    'copy': 'Copier',
+    'copy_to_clipboard': 'Copier dans le presse-papier',
+    'copy_success': 'Copié !',
+    'copy_failure': 'Échec de la copie',
+  },
+  'ru': {
+    'copy': 'Скопировать',
+    'copy_to_clipboard': 'Скопировать в буфер',
+    'copy_success': 'Скопировано!',
+    'copy_failure': 'Не удалось скопировать',
+  },
+  'zh-CN': {
+    'copy': '复制',
+    'copy_to_clipboard': '复制到剪贴板',
+    'copy_success': '复制成功!',
+    'copy_failure': '复制失败',
+  },
+  'it' : {
+    'copy': 'Copiare',
+    'copy_to_clipboard': 'Copiato negli appunti',
+    'copy_success': 'Copiato!',
+    'copy_failure': 'Errore durante la copia',
+  }
+}
+
+let locale = 'en'
+if( document.documentElement.lang !== undefined
+    && messages[document.documentElement.lang] !== undefined ) {
+  locale = document.documentElement.lang
+}
+
+let doc_url_root = DOCUMENTATION_OPTIONS.URL_ROOT;
+if (doc_url_root == '#') {
+    doc_url_root = '';
+}
+
+/**
+ * SVG files for our copy buttons
+ */
+let iconCheck = `<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-check" width="44" height="44" viewBox="0 0 24 24" stroke-width="2" stroke="#22863a" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <title>${messages[locale]['copy_success']}</title>
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <path d="M5 12l5 5l10 -10" />
+</svg>`
+
+// If the user specified their own SVG use that, otherwise use the default
+let iconCopy = ``;
+if (!iconCopy) {
+  iconCopy = `<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-copy" width="44" height="44" viewBox="0 0 24 24" stroke-width="1.5" stroke="#000000" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <title>${messages[locale]['copy_to_clipboard']}</title>
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <rect x="8" y="8" width="12" height="12" rx="2" />
+  <path d="M16 8v-2a2 2 0 0 0 -2 -2h-8a2 2 0 0 0 -2 2v8a2 2 0 0 0 2 2h2" />
+</svg>`
+}
+
+/**
+ * Set up copy/paste for code blocks
+ */
+
+const runWhenDOMLoaded = cb => {
+  if (document.readyState != 'loading') {
+    cb()
+  } else if (document.addEventListener) {
+    document.addEventListener('DOMContentLoaded', cb)
+  } else {
+    document.attachEvent('onreadystatechange', function() {
+      if (document.readyState == 'complete') cb()
+    })
+  }
+}
+
+const codeCellId = index => `codecell${index}`
+
+// Clears selected text since ClipboardJS will select the text when copying
+const clearSelection = () => {
+  if (window.getSelection) {
+    window.getSelection().removeAllRanges()
+  } else if (document.selection) {
+    document.selection.empty()
+  }
+}
+
+// Changes tooltip text for a moment, then changes it back
+// We want the timeout of our `success` class to be a bit shorter than the
+// tooltip and icon change, so that we can hide the icon before changing back.
+var timeoutIcon = 2000;
+var timeoutSuccessClass = 1500;
+
+const temporarilyChangeTooltip = (el, oldText, newText) => {
+  el.setAttribute('data-tooltip', newText)
+  el.classList.add('success')
+  // Remove success a little bit sooner than we change the tooltip
+  // So that we can use CSS to hide the copybutton first
+  setTimeout(() => el.classList.remove('success'), timeoutSuccessClass)
+  setTimeout(() => el.setAttribute('data-tooltip', oldText), timeoutIcon)
+}
+
+// Changes the copy button icon for two seconds, then changes it back
+const temporarilyChangeIcon = (el) => {
+  el.innerHTML = iconCheck;
+  setTimeout(() => {el.innerHTML = iconCopy}, timeoutIcon)
+}
+
+const addCopyButtonToCodeCells = () => {
+  // If ClipboardJS hasn't loaded, wait a bit and try again. This
+  // happens because we load ClipboardJS asynchronously.
+  if (window.ClipboardJS === undefined) {
+    setTimeout(addCopyButtonToCodeCells, 250)
+    return
+  }
+
+  // Add copybuttons to all of our code cells
+  const COPYBUTTON_SELECTOR = 'div.highlight pre';
+  const codeCells = document.querySelectorAll(COPYBUTTON_SELECTOR)
+  codeCells.forEach((codeCell, index) => {
+    const id = codeCellId(index)
+    codeCell.setAttribute('id', id)
+
+    const clipboardButton = id =>
+    `<button class="copybtn o-tooltip--left" data-tooltip="${messages[locale]['copy']}" data-clipboard-target="#${id}">
+      ${iconCopy}
+    </button>`
+    codeCell.insertAdjacentHTML('afterend', clipboardButton(id))
+  })
+
+function escapeRegExp(string) {
+    return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
+}
+
+/**
+ * Removes excluded text from a Node.
+ *
+ * @param {Node} target Node to filter.
+ * @param {string} exclude CSS selector of nodes to exclude.
+ * @returns {DOMString} Text from `target` with text removed.
+ */
+function filterText(target, exclude) {
+    const clone = target.cloneNode(true);  // clone as to not modify the live DOM
+    if (exclude) {
+        // remove excluded nodes
+        clone.querySelectorAll(exclude).forEach(node => node.remove());
+    }
+    return clone.innerText;
+}
+
+// Callback when a copy button is clicked. Will be passed the node that was clicked
+// should then grab the text and replace pieces of text that shouldn't be used in output
+function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") {
+    var regexp;
+    var match;
+
+    // Do we check for line continuation characters and "HERE-documents"?
+    var useLineCont = !!lineContinuationChar
+    var useHereDoc = !!hereDocDelim
+
+    // create regexp to capture prompt and remaining line
+    if (isRegexp) {
+        regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)')
+    } else {
+        regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)')
+    }
+
+    const outputLines = [];
+    var promptFound = false;
+    var gotLineCont = false;
+    var gotHereDoc = false;
+    const lineGotPrompt = [];
+    for (const line of textContent.split('\n')) {
+        match = line.match(regexp)
+        if (match || gotLineCont || gotHereDoc) {
+            promptFound = regexp.test(line)
+            lineGotPrompt.push(promptFound)
+            if (removePrompts && promptFound) {
+                outputLines.push(match[2])
+            } else {
+                outputLines.push(line)
+            }
+            gotLineCont = line.endsWith(lineContinuationChar) & useLineCont
+            if (line.includes(hereDocDelim) & useHereDoc)
+                gotHereDoc = !gotHereDoc
+        } else if (!onlyCopyPromptLines) {
+            outputLines.push(line)
+        } else if (copyEmptyLines && line.trim() === '') {
+            outputLines.push(line)
+        }
+    }
+
+    // If no lines with the prompt were found then just use original lines
+    if (lineGotPrompt.some(v => v === true)) {
+        textContent = outputLines.join('\n');
+    }
+
+    // Remove a trailing newline to avoid auto-running when pasting
+    if (textContent.endsWith("\n")) {
+        textContent = textContent.slice(0, -1)
+    }
+    return textContent
+}
+
+
+var copyTargetText = (trigger) => {
+  var target = document.querySelector(trigger.attributes['data-clipboard-target'].value);
+
+  // get filtered text
+  let exclude = '.linenos';
+
+  let text = filterText(target, exclude);
+  return formatCopyText(text, '', false, true, true, true, '', '')
+}
+
+  // Initialize with a callback so we can modify the text before copy
+  const clipboard = new ClipboardJS('.copybtn', {text: copyTargetText})
+
+  // Update UI with error/success messages
+  clipboard.on('success', event => {
+    clearSelection()
+    temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_success'])
+    temporarilyChangeIcon(event.trigger)
+  })
+
+  clipboard.on('error', event => {
+    temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_failure'])
+  })
+}
+
+runWhenDOMLoaded(addCopyButtonToCodeCells)
\ No newline at end of file
diff --git a/_static/copybutton_funcs.js b/_static/copybutton_funcs.js
new file mode 100644
index 00000000..dbe1aaad
--- /dev/null
+++ b/_static/copybutton_funcs.js
@@ -0,0 +1,73 @@
+function escapeRegExp(string) {
+    return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
+}
+
+/**
+ * Removes excluded text from a Node.
+ *
+ * @param {Node} target Node to filter.
+ * @param {string} exclude CSS selector of nodes to exclude.
+ * @returns {DOMString} Text from `target` with text removed.
+ */
+export function filterText(target, exclude) {
+    const clone = target.cloneNode(true);  // clone as to not modify the live DOM
+    if (exclude) {
+        // remove excluded nodes
+        clone.querySelectorAll(exclude).forEach(node => node.remove());
+    }
+    return clone.innerText;
+}
+
+// Callback when a copy button is clicked. Will be passed the node that was clicked
+// should then grab the text and replace pieces of text that shouldn't be used in output
+export function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") {
+    var regexp;
+    var match;
+
+    // Do we check for line continuation characters and "HERE-documents"?
+    var useLineCont = !!lineContinuationChar
+    var useHereDoc = !!hereDocDelim
+
+    // create regexp to capture prompt and remaining line
+    if (isRegexp) {
+        regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)')
+    } else {
+        regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)')
+    }
+
+    const outputLines = [];
+    var promptFound = false;
+    var gotLineCont = false;
+    var gotHereDoc = false;
+    const lineGotPrompt = [];
+    for (const line of textContent.split('\n')) {
+        match = line.match(regexp)
+        if (match || gotLineCont || gotHereDoc) {
+            promptFound = regexp.test(line)
+            lineGotPrompt.push(promptFound)
+            if (removePrompts && promptFound) {
+                outputLines.push(match[2])
+            } else {
+                outputLines.push(line)
+            }
+            gotLineCont = line.endsWith(lineContinuationChar) & useLineCont
+            if (line.includes(hereDocDelim) & useHereDoc)
+                gotHereDoc = !gotHereDoc
+        } else if (!onlyCopyPromptLines) {
+            outputLines.push(line)
+        } else if (copyEmptyLines && line.trim() === '') {
+            outputLines.push(line)
+        }
+    }
+
+    // If no lines with the prompt were found then just use original lines
+    if (lineGotPrompt.some(v => v === true)) {
+        textContent = outputLines.join('\n');
+    }
+
+    // Remove a trailing newline to avoid auto-running when pasting
+    if (textContent.endsWith("\n")) {
+        textContent = textContent.slice(0, -1)
+    }
+    return textContent
+}
diff --git a/_static/design-style.4045f2051d55cab465a707391d5b2007.min.css b/_static/design-style.4045f2051d55cab465a707391d5b2007.min.css
new file mode 100644
index 00000000..3225661c
--- /dev/null
+++ b/_static/design-style.4045f2051d55cab465a707391d5b2007.min.css
@@ -0,0 +1 @@
+.sd-bg-primary{background-color:var(--sd-color-primary) !important}.sd-bg-text-primary{color:var(--sd-color-primary-text) !important}button.sd-bg-primary:focus,button.sd-bg-primary:hover{background-color:var(--sd-color-primary-highlight) !important}a.sd-bg-primary:focus,a.sd-bg-primary:hover{background-color:var(--sd-color-primary-highlight) !important}.sd-bg-secondary{background-color:var(--sd-color-secondary) !important}.sd-bg-text-secondary{color:var(--sd-color-secondary-text) !important}button.sd-bg-secondary:focus,button.sd-bg-secondary:hover{background-color:var(--sd-color-secondary-highlight) !important}a.sd-bg-secondary:focus,a.sd-bg-secondary:hover{background-color:var(--sd-color-secondary-highlight) !important}.sd-bg-success{background-color:var(--sd-color-success) !important}.sd-bg-text-success{color:var(--sd-color-success-text) !important}button.sd-bg-success:focus,button.sd-bg-success:hover{background-color:var(--sd-color-success-highlight) !important}a.sd-bg-success:focus,a.sd-bg-success:hover{background-color:var(--sd-color-success-highlight) !important}.sd-bg-info{background-color:var(--sd-color-info) !important}.sd-bg-text-info{color:var(--sd-color-info-text) !important}button.sd-bg-info:focus,button.sd-bg-info:hover{background-color:var(--sd-color-info-highlight) !important}a.sd-bg-info:focus,a.sd-bg-info:hover{background-color:var(--sd-color-info-highlight) !important}.sd-bg-warning{background-color:var(--sd-color-warning) !important}.sd-bg-text-warning{color:var(--sd-color-warning-text) !important}button.sd-bg-warning:focus,button.sd-bg-warning:hover{background-color:var(--sd-color-warning-highlight) !important}a.sd-bg-warning:focus,a.sd-bg-warning:hover{background-color:var(--sd-color-warning-highlight) !important}.sd-bg-danger{background-color:var(--sd-color-danger) !important}.sd-bg-text-danger{color:var(--sd-color-danger-text) 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!important;border-style:solid !important;border-width:1px !important}a.sd-outline-primary:focus,a.sd-outline-primary:hover{border-color:var(--sd-color-primary-highlight) !important}.sd-outline-secondary{border-color:var(--sd-color-secondary) !important;border-style:solid !important;border-width:1px !important}a.sd-outline-secondary:focus,a.sd-outline-secondary:hover{border-color:var(--sd-color-secondary-highlight) !important}.sd-outline-success{border-color:var(--sd-color-success) !important;border-style:solid !important;border-width:1px !important}a.sd-outline-success:focus,a.sd-outline-success:hover{border-color:var(--sd-color-success-highlight) !important}.sd-outline-info{border-color:var(--sd-color-info) !important;border-style:solid !important;border-width:1px !important}a.sd-outline-info:focus,a.sd-outline-info:hover{border-color:var(--sd-color-info-highlight) !important}.sd-outline-warning{border-color:var(--sd-color-warning) !important;border-style:solid 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diff --git a/_static/design-tabs.js b/_static/design-tabs.js
new file mode 100644
index 00000000..36b38cf0
--- /dev/null
+++ b/_static/design-tabs.js
@@ -0,0 +1,27 @@
+var sd_labels_by_text = {};
+
+function ready() {
+  const li = document.getElementsByClassName("sd-tab-label");
+  for (const label of li) {
+    syncId = label.getAttribute("data-sync-id");
+    if (syncId) {
+      label.onclick = onLabelClick;
+      if (!sd_labels_by_text[syncId]) {
+        sd_labels_by_text[syncId] = [];
+      }
+      sd_labels_by_text[syncId].push(label);
+    }
+  }
+}
+
+function onLabelClick() {
+  // Activate other inputs with the same sync id.
+  syncId = this.getAttribute("data-sync-id");
+  for (label of sd_labels_by_text[syncId]) {
+    if (label === this) continue;
+    label.previousElementSibling.checked = true;
+  }
+  window.localStorage.setItem("sphinx-design-last-tab", syncId);
+}
+
+document.addEventListener("DOMContentLoaded", ready, false);
diff --git a/_static/doctools.js b/_static/doctools.js
new file mode 100644
index 00000000..e1bfd708
--- /dev/null
+++ b/_static/doctools.js
@@ -0,0 +1,358 @@
+/*
+ * doctools.js
+ * ~~~~~~~~~~~
+ *
+ * Sphinx JavaScript utilities for all documentation.
+ *
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+/**
+ * select a different prefix for underscore
+ */
+$u = _.noConflict();
+
+/**
+ * make the code below compatible with browsers without
+ * an installed firebug like debugger
+if (!window.console || !console.firebug) {
+  var names = ["log", "debug", "info", "warn", "error", "assert", "dir",
+    "dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace",
+    "profile", "profileEnd"];
+  window.console = {};
+  for (var i = 0; i < names.length; ++i)
+    window.console[names[i]] = function() {};
+}
+ */
+
+/**
+ * small helper function to urldecode strings
+ *
+ * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL
+ */
+jQuery.urldecode = function(x) {
+  if (!x) {
+    return x
+  }
+  return decodeURIComponent(x.replace(/\+/g, ' '));
+};
+
+/**
+ * small helper function to urlencode strings
+ */
+jQuery.urlencode = encodeURIComponent;
+
+/**
+ * This function returns the parsed url parameters of the
+ * current request. Multiple values per key are supported,
+ * it will always return arrays of strings for the value parts.
+ */
+jQuery.getQueryParameters = function(s) {
+  if (typeof s === 'undefined')
+    s = document.location.search;
+  var parts = s.substr(s.indexOf('?') + 1).split('&');
+  var result = {};
+  for (var i = 0; i < parts.length; i++) {
+    var tmp = parts[i].split('=', 2);
+    var key = jQuery.urldecode(tmp[0]);
+    var value = jQuery.urldecode(tmp[1]);
+    if (key in result)
+      result[key].push(value);
+    else
+      result[key] = [value];
+  }
+  return result;
+};
+
+/**
+ * highlight a given string on a jquery object by wrapping it in
+ * span elements with the given class name.
+ */
+jQuery.fn.highlightText = function(text, className) {
+  function highlight(node, addItems) {
+    if (node.nodeType === 3) {
+      var val = node.nodeValue;
+      var pos = val.toLowerCase().indexOf(text);
+      if (pos >= 0 &&
+          !jQuery(node.parentNode).hasClass(className) &&
+          !jQuery(node.parentNode).hasClass("nohighlight")) {
+        var span;
+        var isInSVG = jQuery(node).closest("body, svg, foreignObject").is("svg");
+        if (isInSVG) {
+          span = document.createElementNS("http://www.w3.org/2000/svg", "tspan");
+        } else {
+          span = document.createElement("span");
+          span.className = className;
+        }
+        span.appendChild(document.createTextNode(val.substr(pos, text.length)));
+        node.parentNode.insertBefore(span, node.parentNode.insertBefore(
+          document.createTextNode(val.substr(pos + text.length)),
+          node.nextSibling));
+        node.nodeValue = val.substr(0, pos);
+        if (isInSVG) {
+          var rect = document.createElementNS("http://www.w3.org/2000/svg", "rect");
+          var bbox = node.parentElement.getBBox();
+          rect.x.baseVal.value = bbox.x;
+          rect.y.baseVal.value = bbox.y;
+          rect.width.baseVal.value = bbox.width;
+          rect.height.baseVal.value = bbox.height;
+          rect.setAttribute('class', className);
+          addItems.push({
+              "parent": node.parentNode,
+              "target": rect});
+        }
+      }
+    }
+    else if (!jQuery(node).is("button, select, textarea")) {
+      jQuery.each(node.childNodes, function() {
+        highlight(this, addItems);
+      });
+    }
+  }
+  var addItems = [];
+  var result = this.each(function() {
+    highlight(this, addItems);
+  });
+  for (var i = 0; i < addItems.length; ++i) {
+    jQuery(addItems[i].parent).before(addItems[i].target);
+  }
+  return result;
+};
+
+/*
+ * backward compatibility for jQuery.browser
+ * This will be supported until firefox bug is fixed.
+ */
+if (!jQuery.browser) {
+  jQuery.uaMatch = function(ua) {
+    ua = ua.toLowerCase();
+
+    var match = /(chrome)[ \/]([\w.]+)/.exec(ua) ||
+      /(webkit)[ \/]([\w.]+)/.exec(ua) ||
+      /(opera)(?:.*version|)[ \/]([\w.]+)/.exec(ua) ||
+      /(msie) ([\w.]+)/.exec(ua) ||
+      ua.indexOf("compatible") < 0 && /(mozilla)(?:.*? rv:([\w.]+)|)/.exec(ua) ||
+      [];
+
+    return {
+      browser: match[ 1 ] || "",
+      version: match[ 2 ] || "0"
+    };
+  };
+  jQuery.browser = {};
+  jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true;
+}
+
+/**
+ * Small JavaScript module for the documentation.
+ */
+var Documentation = {
+
+  init : function() {
+    this.fixFirefoxAnchorBug();
+    this.highlightSearchWords();
+    this.initIndexTable();
+    this.initOnKeyListeners();
+  },
+
+  /**
+   * i18n support
+   */
+  TRANSLATIONS : {},
+  PLURAL_EXPR : function(n) { return n === 1 ? 0 : 1; },
+  LOCALE : 'unknown',
+
+  // gettext and ngettext don't access this so that the functions
+  // can safely bound to a different name (_ = Documentation.gettext)
+  gettext : function(string) {
+    var translated = Documentation.TRANSLATIONS[string];
+    if (typeof translated === 'undefined')
+      return string;
+    return (typeof translated === 'string') ? translated : translated[0];
+  },
+
+  ngettext : function(singular, plural, n) {
+    var translated = Documentation.TRANSLATIONS[singular];
+    if (typeof translated === 'undefined')
+      return (n == 1) ? singular : plural;
+    return translated[Documentation.PLURALEXPR(n)];
+  },
+
+  addTranslations : function(catalog) {
+    for (var key in catalog.messages)
+      this.TRANSLATIONS[key] = catalog.messages[key];
+    this.PLURAL_EXPR = new Function('n', 'return +(' + catalog.plural_expr + ')');
+    this.LOCALE = catalog.locale;
+  },
+
+  /**
+   * add context elements like header anchor links
+   */
+  addContextElements : function() {
+    $('div[id] > :header:first').each(function() {
+      $('<a class="headerlink">\u00B6</a>').
+      attr('href', '#' + this.id).
+      attr('title', _('Permalink to this headline')).
+      appendTo(this);
+    });
+    $('dt[id]').each(function() {
+      $('<a class="headerlink">\u00B6</a>').
+      attr('href', '#' + this.id).
+      attr('title', _('Permalink to this definition')).
+      appendTo(this);
+    });
+  },
+
+  /**
+   * workaround a firefox stupidity
+   * see: https://bugzilla.mozilla.org/show_bug.cgi?id=645075
+   */
+  fixFirefoxAnchorBug : function() {
+    if (document.location.hash && $.browser.mozilla)
+      window.setTimeout(function() {
+        document.location.href += '';
+      }, 10);
+  },
+
+  /**
+   * highlight the search words provided in the url in the text
+   */
+  highlightSearchWords : function() {
+    var params = $.getQueryParameters();
+    var terms = (params.highlight) ? params.highlight[0].split(/\s+/) : [];
+    if (terms.length) {
+      var body = $('div.body');
+      if (!body.length) {
+        body = $('body');
+      }
+      window.setTimeout(function() {
+        $.each(terms, function() {
+          body.highlightText(this.toLowerCase(), 'highlighted');
+        });
+      }, 10);
+      $('<p class="highlight-link"><a href="javascript:Documentation.' +
+        'hideSearchWords()">' + _('Hide Search Matches') + '</a></p>')
+          .appendTo($('#searchbox'));
+    }
+  },
+
+  /**
+   * init the domain index toggle buttons
+   */
+  initIndexTable : function() {
+    var togglers = $('img.toggler').click(function() {
+      var src = $(this).attr('src');
+      var idnum = $(this).attr('id').substr(7);
+      $('tr.cg-' + idnum).toggle();
+      if (src.substr(-9) === 'minus.png')
+        $(this).attr('src', src.substr(0, src.length-9) + 'plus.png');
+      else
+        $(this).attr('src', src.substr(0, src.length-8) + 'minus.png');
+    }).css('display', '');
+    if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) {
+        togglers.click();
+    }
+  },
+
+  /**
+   * helper function to hide the search marks again
+   */
+  hideSearchWords : function() {
+    $('#searchbox .highlight-link').fadeOut(300);
+    $('span.highlighted').removeClass('highlighted');
+    var url = new URL(window.location);
+    url.searchParams.delete('highlight');
+    window.history.replaceState({}, '', url);
+  },
+
+   /**
+   * helper function to focus on search bar
+   */
+  focusSearchBar : function() {
+    $('input[name=q]').first().focus();
+  },
+
+  /**
+   * make the url absolute
+   */
+  makeURL : function(relativeURL) {
+    return DOCUMENTATION_OPTIONS.URL_ROOT + '/' + relativeURL;
+  },
+
+  /**
+   * get the current relative url
+   */
+  getCurrentURL : function() {
+    var path = document.location.pathname;
+    var parts = path.split(/\//);
+    $.each(DOCUMENTATION_OPTIONS.URL_ROOT.split(/\//), function() {
+      if (this === '..')
+        parts.pop();
+    });
+    var url = parts.join('/');
+    return path.substring(url.lastIndexOf('/') + 1, path.length - 1);
+  },
+
+  initOnKeyListeners: function() {
+    // only install a listener if it is really needed
+    if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS &&
+        !DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS)
+        return;
+
+    $(document).keydown(function(event) {
+      var activeElementType = document.activeElement.tagName;
+      // don't navigate when in search box, textarea, dropdown or button
+      if (activeElementType !== 'TEXTAREA' && activeElementType !== 'INPUT' && activeElementType !== 'SELECT'
+          && activeElementType !== 'BUTTON') {
+        if (event.altKey || event.ctrlKey || event.metaKey)
+          return;
+
+          if (!event.shiftKey) {
+            switch (event.key) {
+              case 'ArrowLeft':
+                if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS)
+                  break;
+                var prevHref = $('link[rel="prev"]').prop('href');
+                if (prevHref) {
+                  window.location.href = prevHref;
+                  return false;
+                }
+                break;
+              case 'ArrowRight':
+                if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS)
+                  break;
+                var nextHref = $('link[rel="next"]').prop('href');
+                if (nextHref) {
+                  window.location.href = nextHref;
+                  return false;
+                }
+                break;
+              case 'Escape':
+                if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS)
+                  break;
+                Documentation.hideSearchWords();
+                return false;
+          }
+        }
+
+        // some keyboard layouts may need Shift to get /
+        switch (event.key) {
+          case '/':
+            if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS)
+              break;
+            Documentation.focusSearchBar();
+            return false;
+        }
+      }
+    });
+  }
+};
+
+// quick alias for translations
+_ = Documentation.gettext;
+
+$(document).ready(function() {
+  Documentation.init();
+});
diff --git a/_static/documentation_options.js b/_static/documentation_options.js
new file mode 100644
index 00000000..877e3c31
--- /dev/null
+++ b/_static/documentation_options.js
@@ -0,0 +1,14 @@
+var DOCUMENTATION_OPTIONS = {
+    URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'),
+    VERSION: '',
+    LANGUAGE: 'None',
+    COLLAPSE_INDEX: false,
+    BUILDER: 'html',
+    FILE_SUFFIX: '.html',
+    LINK_SUFFIX: '.html',
+    HAS_SOURCE: true,
+    SOURCELINK_SUFFIX: '',
+    NAVIGATION_WITH_KEYS: true,
+    SHOW_SEARCH_SUMMARY: true,
+    ENABLE_SEARCH_SHORTCUTS: true,
+};
\ No newline at end of file
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diff --git a/_static/images/logo_binder.svg b/_static/images/logo_binder.svg
new file mode 100644
index 00000000..45fecf75
--- /dev/null
+++ b/_static/images/logo_binder.svg
@@ -0,0 +1,19 @@
+<?xml version="1.0" encoding="utf-8"?>
+<!-- Generator: Adobe Illustrator 23.0.1, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
+<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
+	 viewBox="0 0 44.4 44.4" style="enable-background:new 0 0 44.4 44.4;" xml:space="preserve">
+<style type="text/css">
+	.st0{fill:none;stroke:#F5A252;stroke-width:5;stroke-miterlimit:10;}
+	.st1{fill:none;stroke:#579ACA;stroke-width:5;stroke-miterlimit:10;}
+	.st2{fill:none;stroke:#E66581;stroke-width:5;stroke-miterlimit:10;}
+</style>
+<title>logo</title>
+<g>
+	<path class="st0" d="M33.9,6.4c3.6,3.9,3.4,9.9-0.5,13.5s-9.9,3.4-13.5-0.5s-3.4-9.9,0.5-13.5l0,0C24.2,2.4,30.2,2.6,33.9,6.4z"/>
+	<path class="st1" d="M35.1,27.3c2.6,4.6,1.1,10.4-3.5,13c-4.6,2.6-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0
+		C26.6,21.2,32.4,22.7,35.1,27.3z"/>
+	<path class="st2" d="M25.9,17.8c2.6,4.6,1.1,10.4-3.5,13s-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0C17.5,11.7,23.3,13.2,25.9,17.8z"/>
+	<path class="st1" d="M19.2,26.4c3.1-4.3,9.1-5.2,13.3-2.1c1.1,0.8,2,1.8,2.7,3"/>
+	<path class="st0" d="M19.9,19.4c-3.6-3.9-3.4-9.9,0.5-13.5s9.9-3.4,13.5,0.5"/>
+</g>
+</svg>
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diff --git a/_static/images/logo_deepnote.svg b/_static/images/logo_deepnote.svg
new file mode 100644
index 00000000..fa77ebfc
--- /dev/null
+++ b/_static/images/logo_deepnote.svg
@@ -0,0 +1 @@
+<svg viewBox="0 0 128 128" fill="none" xmlns="http://www.w3.org/2000/svg"><path fill-rule="evenodd" clip-rule="evenodd" d="M0 128h52.512l29.539-11.077-11.077-43.487-34.051 3.693L0 128Z" fill="#0076D4"/><path fill-rule="evenodd" clip-rule="evenodd" d="M52.513 128s16.6-8.759 19.673-24.277c3.072-15.517-12.091-26.594-35.263-26.594 0-.41 20.343-28.718 20.343-28.718l49.4 1.435L95.71 107.7l-20.452 15.978L52.513 128Z" fill="#002868"/><path fill-rule="evenodd" clip-rule="evenodd" d="M0 60.718 41.025.001s1.006.01 3.282 0c16.082-.068 81.23 3.12 81.23 60.368 0 65.352-73.025 67.631-73.025 67.631s30.495-5.839 30.495-34.816c0-28.978-27.541-32.466-45.264-32.466H0Z" fill="#00A9FF"/></svg>
diff --git a/_static/images/logo_jupyterhub.svg b/_static/images/logo_jupyterhub.svg
new file mode 100644
index 00000000..60cfe9f2
--- /dev/null
+++ b/_static/images/logo_jupyterhub.svg
@@ -0,0 +1 @@
+<svg id="Layer_1" data-name="Layer 1" xmlns="http://www.w3.org/2000/svg" width="38.73" height="50" viewBox="0 0 38.73 50"><defs><style>.cls-1{fill:#767677;}.cls-2{fill:#f37726;}.cls-3{fill:#9e9e9e;}.cls-4{fill:#616262;}.cls-5{font-size:17.07px;fill:#fff;font-family:Roboto-Regular, Roboto;}</style></defs><title>logo_jupyterhub</title><g id="Canvas"><path id="path7_fill" data-name="path7 fill" class="cls-1" d="M39.51,3.53a3,3,0,0,1-1.7,2.9A3,3,0,0,1,34.48,6a3,3,0,0,1-.82-3.26,3,3,0,0,1,1.05-1.41A3,3,0,0,1,37.52.86a2.88,2.88,0,0,1,1,.6,3,3,0,0,1,.7.93,3.18,3.18,0,0,1,.28,1.14Z" transform="translate(-1.87 -0.69)"/><path id="path8_fill" data-name="path8 fill" class="cls-2" d="M21.91,38.39c-8,0-15.06-2.87-18.7-7.12a19.93,19.93,0,0,0,37.39,0C37,35.52,30,38.39,21.91,38.39Z" transform="translate(-1.87 -0.69)"/><path id="path9_fill" data-name="path9 fill" class="cls-2" d="M21.91,10.78c8,0,15.05,2.87,18.69,7.12a19.93,19.93,0,0,0-37.39,0C6.85,13.64,13.86,10.78,21.91,10.78Z" transform="translate(-1.87 -0.69)"/><path id="path10_fill" data-name="path10 fill" class="cls-3" d="M10.88,46.66a3.86,3.86,0,0,1-.52,2.15,3.81,3.81,0,0,1-1.62,1.51,3.93,3.93,0,0,1-2.19.34,3.79,3.79,0,0,1-2-.94,3.73,3.73,0,0,1-1.14-1.9,3.79,3.79,0,0,1,.1-2.21,3.86,3.86,0,0,1,1.33-1.78,3.92,3.92,0,0,1,3.54-.53,3.85,3.85,0,0,1,2.14,1.93,3.74,3.74,0,0,1,.37,1.43Z" transform="translate(-1.87 -0.69)"/><path id="path11_fill" data-name="path11 fill" class="cls-4" d="M4.12,9.81A2.18,2.18,0,0,1,2.9,9.48a2.23,2.23,0,0,1-.84-1A2.26,2.26,0,0,1,1.9,7.26a2.13,2.13,0,0,1,.56-1.13,2.18,2.18,0,0,1,2.36-.56,2.13,2.13,0,0,1,1,.76,2.18,2.18,0,0,1,.42,1.2A2.22,2.22,0,0,1,4.12,9.81Z" transform="translate(-1.87 -0.69)"/></g><text class="cls-5" transform="translate(5.24 30.01)">Hub</text></svg>
diff --git a/_static/jquery-3.5.1.js b/_static/jquery-3.5.1.js
new file mode 100644
index 00000000..50937333
--- /dev/null
+++ b/_static/jquery-3.5.1.js
@@ -0,0 +1,10872 @@
+/*!
+ * jQuery JavaScript Library v3.5.1
+ * https://jquery.com/
+ *
+ * Includes Sizzle.js
+ * https://sizzlejs.com/
+ *
+ * Copyright JS Foundation and other contributors
+ * Released under the MIT license
+ * https://jquery.org/license
+ *
+ * Date: 2020-05-04T22:49Z
+ */
+( function( global, factory ) {
+
+	"use strict";
+
+	if ( typeof module === "object" && typeof module.exports === "object" ) {
+
+		// For CommonJS and CommonJS-like environments where a proper `window`
+		// is present, execute the factory and get jQuery.
+		// For environments that do not have a `window` with a `document`
+		// (such as Node.js), expose a factory as module.exports.
+		// This accentuates the need for the creation of a real `window`.
+		// e.g. var jQuery = require("jquery")(window);
+		// See ticket #14549 for more info.
+		module.exports = global.document ?
+			factory( global, true ) :
+			function( w ) {
+				if ( !w.document ) {
+					throw new Error( "jQuery requires a window with a document" );
+				}
+				return factory( w );
+			};
+	} else {
+		factory( global );
+	}
+
+// Pass this if window is not defined yet
+} )( typeof window !== "undefined" ? window : this, function( window, noGlobal ) {
+
+// Edge <= 12 - 13+, Firefox <=18 - 45+, IE 10 - 11, Safari 5.1 - 9+, iOS 6 - 9.1
+// throw exceptions when non-strict code (e.g., ASP.NET 4.5) accesses strict mode
+// arguments.callee.caller (trac-13335). But as of jQuery 3.0 (2016), strict mode should be common
+// enough that all such attempts are guarded in a try block.
+"use strict";
+
+var arr = [];
+
+var getProto = Object.getPrototypeOf;
+
+var slice = arr.slice;
+
+var flat = arr.flat ? function( array ) {
+	return arr.flat.call( array );
+} : function( array ) {
+	return arr.concat.apply( [], array );
+};
+
+
+var push = arr.push;
+
+var indexOf = arr.indexOf;
+
+var class2type = {};
+
+var toString = class2type.toString;
+
+var hasOwn = class2type.hasOwnProperty;
+
+var fnToString = hasOwn.toString;
+
+var ObjectFunctionString = fnToString.call( Object );
+
+var support = {};
+
+var isFunction = function isFunction( obj ) {
+
+      // Support: Chrome <=57, Firefox <=52
+      // In some browsers, typeof returns "function" for HTML <object> elements
+      // (i.e., `typeof document.createElement( "object" ) === "function"`).
+      // We don't want to classify *any* DOM node as a function.
+      return typeof obj === "function" && typeof obj.nodeType !== "number";
+  };
+
+
+var isWindow = function isWindow( obj ) {
+		return obj != null && obj === obj.window;
+	};
+
+
+var document = window.document;
+
+
+
+	var preservedScriptAttributes = {
+		type: true,
+		src: true,
+		nonce: true,
+		noModule: true
+	};
+
+	function DOMEval( code, node, doc ) {
+		doc = doc || document;
+
+		var i, val,
+			script = doc.createElement( "script" );
+
+		script.text = code;
+		if ( node ) {
+			for ( i in preservedScriptAttributes ) {
+
+				// Support: Firefox 64+, Edge 18+
+				// Some browsers don't support the "nonce" property on scripts.
+				// On the other hand, just using `getAttribute` is not enough as
+				// the `nonce` attribute is reset to an empty string whenever it
+				// becomes browsing-context connected.
+				// See https://github.com/whatwg/html/issues/2369
+				// See https://html.spec.whatwg.org/#nonce-attributes
+				// The `node.getAttribute` check was added for the sake of
+				// `jQuery.globalEval` so that it can fake a nonce-containing node
+				// via an object.
+				val = node[ i ] || node.getAttribute && node.getAttribute( i );
+				if ( val ) {
+					script.setAttribute( i, val );
+				}
+			}
+		}
+		doc.head.appendChild( script ).parentNode.removeChild( script );
+	}
+
+
+function toType( obj ) {
+	if ( obj == null ) {
+		return obj + "";
+	}
+
+	// Support: Android <=2.3 only (functionish RegExp)
+	return typeof obj === "object" || typeof obj === "function" ?
+		class2type[ toString.call( obj ) ] || "object" :
+		typeof obj;
+}
+/* global Symbol */
+// Defining this global in .eslintrc.json would create a danger of using the global
+// unguarded in another place, it seems safer to define global only for this module
+
+
+
+var
+	version = "3.5.1",
+
+	// Define a local copy of jQuery
+	jQuery = function( selector, context ) {
+
+		// The jQuery object is actually just the init constructor 'enhanced'
+		// Need init if jQuery is called (just allow error to be thrown if not included)
+		return new jQuery.fn.init( selector, context );
+	};
+
+jQuery.fn = jQuery.prototype = {
+
+	// The current version of jQuery being used
+	jquery: version,
+
+	constructor: jQuery,
+
+	// The default length of a jQuery object is 0
+	length: 0,
+
+	toArray: function() {
+		return slice.call( this );
+	},
+
+	// Get the Nth element in the matched element set OR
+	// Get the whole matched element set as a clean array
+	get: function( num ) {
+
+		// Return all the elements in a clean array
+		if ( num == null ) {
+			return slice.call( this );
+		}
+
+		// Return just the one element from the set
+		return num < 0 ? this[ num + this.length ] : this[ num ];
+	},
+
+	// Take an array of elements and push it onto the stack
+	// (returning the new matched element set)
+	pushStack: function( elems ) {
+
+		// Build a new jQuery matched element set
+		var ret = jQuery.merge( this.constructor(), elems );
+
+		// Add the old object onto the stack (as a reference)
+		ret.prevObject = this;
+
+		// Return the newly-formed element set
+		return ret;
+	},
+
+	// Execute a callback for every element in the matched set.
+	each: function( callback ) {
+		return jQuery.each( this, callback );
+	},
+
+	map: function( callback ) {
+		return this.pushStack( jQuery.map( this, function( elem, i ) {
+			return callback.call( elem, i, elem );
+		} ) );
+	},
+
+	slice: function() {
+		return this.pushStack( slice.apply( this, arguments ) );
+	},
+
+	first: function() {
+		return this.eq( 0 );
+	},
+
+	last: function() {
+		return this.eq( -1 );
+	},
+
+	even: function() {
+		return this.pushStack( jQuery.grep( this, function( _elem, i ) {
+			return ( i + 1 ) % 2;
+		} ) );
+	},
+
+	odd: function() {
+		return this.pushStack( jQuery.grep( this, function( _elem, i ) {
+			return i % 2;
+		} ) );
+	},
+
+	eq: function( i ) {
+		var len = this.length,
+			j = +i + ( i < 0 ? len : 0 );
+		return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] );
+	},
+
+	end: function() {
+		return this.prevObject || this.constructor();
+	},
+
+	// For internal use only.
+	// Behaves like an Array's method, not like a jQuery method.
+	push: push,
+	sort: arr.sort,
+	splice: arr.splice
+};
+
+jQuery.extend = jQuery.fn.extend = function() {
+	var options, name, src, copy, copyIsArray, clone,
+		target = arguments[ 0 ] || {},
+		i = 1,
+		length = arguments.length,
+		deep = false;
+
+	// Handle a deep copy situation
+	if ( typeof target === "boolean" ) {
+		deep = target;
+
+		// Skip the boolean and the target
+		target = arguments[ i ] || {};
+		i++;
+	}
+
+	// Handle case when target is a string or something (possible in deep copy)
+	if ( typeof target !== "object" && !isFunction( target ) ) {
+		target = {};
+	}
+
+	// Extend jQuery itself if only one argument is passed
+	if ( i === length ) {
+		target = this;
+		i--;
+	}
+
+	for ( ; i < length; i++ ) {
+
+		// Only deal with non-null/undefined values
+		if ( ( options = arguments[ i ] ) != null ) {
+
+			// Extend the base object
+			for ( name in options ) {
+				copy = options[ name ];
+
+				// Prevent Object.prototype pollution
+				// Prevent never-ending loop
+				if ( name === "__proto__" || target === copy ) {
+					continue;
+				}
+
+				// Recurse if we're merging plain objects or arrays
+				if ( deep && copy && ( jQuery.isPlainObject( copy ) ||
+					( copyIsArray = Array.isArray( copy ) ) ) ) {
+					src = target[ name ];
+
+					// Ensure proper type for the source value
+					if ( copyIsArray && !Array.isArray( src ) ) {
+						clone = [];
+					} else if ( !copyIsArray && !jQuery.isPlainObject( src ) ) {
+						clone = {};
+					} else {
+						clone = src;
+					}
+					copyIsArray = false;
+
+					// Never move original objects, clone them
+					target[ name ] = jQuery.extend( deep, clone, copy );
+
+				// Don't bring in undefined values
+				} else if ( copy !== undefined ) {
+					target[ name ] = copy;
+				}
+			}
+		}
+	}
+
+	// Return the modified object
+	return target;
+};
+
+jQuery.extend( {
+
+	// Unique for each copy of jQuery on the page
+	expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ),
+
+	// Assume jQuery is ready without the ready module
+	isReady: true,
+
+	error: function( msg ) {
+		throw new Error( msg );
+	},
+
+	noop: function() {},
+
+	isPlainObject: function( obj ) {
+		var proto, Ctor;
+
+		// Detect obvious negatives
+		// Use toString instead of jQuery.type to catch host objects
+		if ( !obj || toString.call( obj ) !== "[object Object]" ) {
+			return false;
+		}
+
+		proto = getProto( obj );
+
+		// Objects with no prototype (e.g., `Object.create( null )`) are plain
+		if ( !proto ) {
+			return true;
+		}
+
+		// Objects with prototype are plain iff they were constructed by a global Object function
+		Ctor = hasOwn.call( proto, "constructor" ) && proto.constructor;
+		return typeof Ctor === "function" && fnToString.call( Ctor ) === ObjectFunctionString;
+	},
+
+	isEmptyObject: function( obj ) {
+		var name;
+
+		for ( name in obj ) {
+			return false;
+		}
+		return true;
+	},
+
+	// Evaluates a script in a provided context; falls back to the global one
+	// if not specified.
+	globalEval: function( code, options, doc ) {
+		DOMEval( code, { nonce: options && options.nonce }, doc );
+	},
+
+	each: function( obj, callback ) {
+		var length, i = 0;
+
+		if ( isArrayLike( obj ) ) {
+			length = obj.length;
+			for ( ; i < length; i++ ) {
+				if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) {
+					break;
+				}
+			}
+		} else {
+			for ( i in obj ) {
+				if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) {
+					break;
+				}
+			}
+		}
+
+		return obj;
+	},
+
+	// results is for internal usage only
+	makeArray: function( arr, results ) {
+		var ret = results || [];
+
+		if ( arr != null ) {
+			if ( isArrayLike( Object( arr ) ) ) {
+				jQuery.merge( ret,
+					typeof arr === "string" ?
+					[ arr ] : arr
+				);
+			} else {
+				push.call( ret, arr );
+			}
+		}
+
+		return ret;
+	},
+
+	inArray: function( elem, arr, i ) {
+		return arr == null ? -1 : indexOf.call( arr, elem, i );
+	},
+
+	// Support: Android <=4.0 only, PhantomJS 1 only
+	// push.apply(_, arraylike) throws on ancient WebKit
+	merge: function( first, second ) {
+		var len = +second.length,
+			j = 0,
+			i = first.length;
+
+		for ( ; j < len; j++ ) {
+			first[ i++ ] = second[ j ];
+		}
+
+		first.length = i;
+
+		return first;
+	},
+
+	grep: function( elems, callback, invert ) {
+		var callbackInverse,
+			matches = [],
+			i = 0,
+			length = elems.length,
+			callbackExpect = !invert;
+
+		// Go through the array, only saving the items
+		// that pass the validator function
+		for ( ; i < length; i++ ) {
+			callbackInverse = !callback( elems[ i ], i );
+			if ( callbackInverse !== callbackExpect ) {
+				matches.push( elems[ i ] );
+			}
+		}
+
+		return matches;
+	},
+
+	// arg is for internal usage only
+	map: function( elems, callback, arg ) {
+		var length, value,
+			i = 0,
+			ret = [];
+
+		// Go through the array, translating each of the items to their new values
+		if ( isArrayLike( elems ) ) {
+			length = elems.length;
+			for ( ; i < length; i++ ) {
+				value = callback( elems[ i ], i, arg );
+
+				if ( value != null ) {
+					ret.push( value );
+				}
+			}
+
+		// Go through every key on the object,
+		} else {
+			for ( i in elems ) {
+				value = callback( elems[ i ], i, arg );
+
+				if ( value != null ) {
+					ret.push( value );
+				}
+			}
+		}
+
+		// Flatten any nested arrays
+		return flat( ret );
+	},
+
+	// A global GUID counter for objects
+	guid: 1,
+
+	// jQuery.support is not used in Core but other projects attach their
+	// properties to it so it needs to exist.
+	support: support
+} );
+
+if ( typeof Symbol === "function" ) {
+	jQuery.fn[ Symbol.iterator ] = arr[ Symbol.iterator ];
+}
+
+// Populate the class2type map
+jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ),
+function( _i, name ) {
+	class2type[ "[object " + name + "]" ] = name.toLowerCase();
+} );
+
+function isArrayLike( obj ) {
+
+	// Support: real iOS 8.2 only (not reproducible in simulator)
+	// `in` check used to prevent JIT error (gh-2145)
+	// hasOwn isn't used here due to false negatives
+	// regarding Nodelist length in IE
+	var length = !!obj && "length" in obj && obj.length,
+		type = toType( obj );
+
+	if ( isFunction( obj ) || isWindow( obj ) ) {
+		return false;
+	}
+
+	return type === "array" || length === 0 ||
+		typeof length === "number" && length > 0 && ( length - 1 ) in obj;
+}
+var Sizzle =
+/*!
+ * Sizzle CSS Selector Engine v2.3.5
+ * https://sizzlejs.com/
+ *
+ * Copyright JS Foundation and other contributors
+ * Released under the MIT license
+ * https://js.foundation/
+ *
+ * Date: 2020-03-14
+ */
+( function( window ) {
+var i,
+	support,
+	Expr,
+	getText,
+	isXML,
+	tokenize,
+	compile,
+	select,
+	outermostContext,
+	sortInput,
+	hasDuplicate,
+
+	// Local document vars
+	setDocument,
+	document,
+	docElem,
+	documentIsHTML,
+	rbuggyQSA,
+	rbuggyMatches,
+	matches,
+	contains,
+
+	// Instance-specific data
+	expando = "sizzle" + 1 * new Date(),
+	preferredDoc = window.document,
+	dirruns = 0,
+	done = 0,
+	classCache = createCache(),
+	tokenCache = createCache(),
+	compilerCache = createCache(),
+	nonnativeSelectorCache = createCache(),
+	sortOrder = function( a, b ) {
+		if ( a === b ) {
+			hasDuplicate = true;
+		}
+		return 0;
+	},
+
+	// Instance methods
+	hasOwn = ( {} ).hasOwnProperty,
+	arr = [],
+	pop = arr.pop,
+	pushNative = arr.push,
+	push = arr.push,
+	slice = arr.slice,
+
+	// Use a stripped-down indexOf as it's faster than native
+	// https://jsperf.com/thor-indexof-vs-for/5
+	indexOf = function( list, elem ) {
+		var i = 0,
+			len = list.length;
+		for ( ; i < len; i++ ) {
+			if ( list[ i ] === elem ) {
+				return i;
+			}
+		}
+		return -1;
+	},
+
+	booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|" +
+		"ismap|loop|multiple|open|readonly|required|scoped",
+
+	// Regular expressions
+
+	// http://www.w3.org/TR/css3-selectors/#whitespace
+	whitespace = "[\\x20\\t\\r\\n\\f]",
+
+	// https://www.w3.org/TR/css-syntax-3/#ident-token-diagram
+	identifier = "(?:\\\\[\\da-fA-F]{1,6}" + whitespace +
+		"?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+",
+
+	// Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors
+	attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace +
+
+		// Operator (capture 2)
+		"*([*^$|!~]?=)" + whitespace +
+
+		// "Attribute values must be CSS identifiers [capture 5]
+		// or strings [capture 3 or capture 4]"
+		"*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" +
+		whitespace + "*\\]",
+
+	pseudos = ":(" + identifier + ")(?:\\((" +
+
+		// To reduce the number of selectors needing tokenize in the preFilter, prefer arguments:
+		// 1. quoted (capture 3; capture 4 or capture 5)
+		"('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" +
+
+		// 2. simple (capture 6)
+		"((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" +
+
+		// 3. anything else (capture 2)
+		".*" +
+		")\\)|)",
+
+	// Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter
+	rwhitespace = new RegExp( whitespace + "+", "g" ),
+	rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" +
+		whitespace + "+$", "g" ),
+
+	rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ),
+	rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace +
+		"*" ),
+	rdescend = new RegExp( whitespace + "|>" ),
+
+	rpseudo = new RegExp( pseudos ),
+	ridentifier = new RegExp( "^" + identifier + "$" ),
+
+	matchExpr = {
+		"ID": new RegExp( "^#(" + identifier + ")" ),
+		"CLASS": new RegExp( "^\\.(" + identifier + ")" ),
+		"TAG": new RegExp( "^(" + identifier + "|[*])" ),
+		"ATTR": new RegExp( "^" + attributes ),
+		"PSEUDO": new RegExp( "^" + pseudos ),
+		"CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\\(" +
+			whitespace + "*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" +
+			whitespace + "*(\\d+)|))" + whitespace + "*\\)|)", "i" ),
+		"bool": new RegExp( "^(?:" + booleans + ")$", "i" ),
+
+		// For use in libraries implementing .is()
+		// We use this for POS matching in `select`
+		"needsContext": new RegExp( "^" + whitespace +
+			"*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\\(" + whitespace +
+			"*((?:-\\d)?\\d*)" + whitespace + "*\\)|)(?=[^-]|$)", "i" )
+	},
+
+	rhtml = /HTML$/i,
+	rinputs = /^(?:input|select|textarea|button)$/i,
+	rheader = /^h\d$/i,
+
+	rnative = /^[^{]+\{\s*\[native \w/,
+
+	// Easily-parseable/retrievable ID or TAG or CLASS selectors
+	rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/,
+
+	rsibling = /[+~]/,
+
+	// CSS escapes
+	// http://www.w3.org/TR/CSS21/syndata.html#escaped-characters
+	runescape = new RegExp( "\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\([^\\r\\n\\f])", "g" ),
+	funescape = function( escape, nonHex ) {
+		var high = "0x" + escape.slice( 1 ) - 0x10000;
+
+		return nonHex ?
+
+			// Strip the backslash prefix from a non-hex escape sequence
+			nonHex :
+
+			// Replace a hexadecimal escape sequence with the encoded Unicode code point
+			// Support: IE <=11+
+			// For values outside the Basic Multilingual Plane (BMP), manually construct a
+			// surrogate pair
+			high < 0 ?
+				String.fromCharCode( high + 0x10000 ) :
+				String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 );
+	},
+
+	// CSS string/identifier serialization
+	// https://drafts.csswg.org/cssom/#common-serializing-idioms
+	rcssescape = /([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g,
+	fcssescape = function( ch, asCodePoint ) {
+		if ( asCodePoint ) {
+
+			// U+0000 NULL becomes U+FFFD REPLACEMENT CHARACTER
+			if ( ch === "\0" ) {
+				return "\uFFFD";
+			}
+
+			// Control characters and (dependent upon position) numbers get escaped as code points
+			return ch.slice( 0, -1 ) + "\\" +
+				ch.charCodeAt( ch.length - 1 ).toString( 16 ) + " ";
+		}
+
+		// Other potentially-special ASCII characters get backslash-escaped
+		return "\\" + ch;
+	},
+
+	// Used for iframes
+	// See setDocument()
+	// Removing the function wrapper causes a "Permission Denied"
+	// error in IE
+	unloadHandler = function() {
+		setDocument();
+	},
+
+	inDisabledFieldset = addCombinator(
+		function( elem ) {
+			return elem.disabled === true && elem.nodeName.toLowerCase() === "fieldset";
+		},
+		{ dir: "parentNode", next: "legend" }
+	);
+
+// Optimize for push.apply( _, NodeList )
+try {
+	push.apply(
+		( arr = slice.call( preferredDoc.childNodes ) ),
+		preferredDoc.childNodes
+	);
+
+	// Support: Android<4.0
+	// Detect silently failing push.apply
+	// eslint-disable-next-line no-unused-expressions
+	arr[ preferredDoc.childNodes.length ].nodeType;
+} catch ( e ) {
+	push = { apply: arr.length ?
+
+		// Leverage slice if possible
+		function( target, els ) {
+			pushNative.apply( target, slice.call( els ) );
+		} :
+
+		// Support: IE<9
+		// Otherwise append directly
+		function( target, els ) {
+			var j = target.length,
+				i = 0;
+
+			// Can't trust NodeList.length
+			while ( ( target[ j++ ] = els[ i++ ] ) ) {}
+			target.length = j - 1;
+		}
+	};
+}
+
+function Sizzle( selector, context, results, seed ) {
+	var m, i, elem, nid, match, groups, newSelector,
+		newContext = context && context.ownerDocument,
+
+		// nodeType defaults to 9, since context defaults to document
+		nodeType = context ? context.nodeType : 9;
+
+	results = results || [];
+
+	// Return early from calls with invalid selector or context
+	if ( typeof selector !== "string" || !selector ||
+		nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) {
+
+		return results;
+	}
+
+	// Try to shortcut find operations (as opposed to filters) in HTML documents
+	if ( !seed ) {
+		setDocument( context );
+		context = context || document;
+
+		if ( documentIsHTML ) {
+
+			// If the selector is sufficiently simple, try using a "get*By*" DOM method
+			// (excepting DocumentFragment context, where the methods don't exist)
+			if ( nodeType !== 11 && ( match = rquickExpr.exec( selector ) ) ) {
+
+				// ID selector
+				if ( ( m = match[ 1 ] ) ) {
+
+					// Document context
+					if ( nodeType === 9 ) {
+						if ( ( elem = context.getElementById( m ) ) ) {
+
+							// Support: IE, Opera, Webkit
+							// TODO: identify versions
+							// getElementById can match elements by name instead of ID
+							if ( elem.id === m ) {
+								results.push( elem );
+								return results;
+							}
+						} else {
+							return results;
+						}
+
+					// Element context
+					} else {
+
+						// Support: IE, Opera, Webkit
+						// TODO: identify versions
+						// getElementById can match elements by name instead of ID
+						if ( newContext && ( elem = newContext.getElementById( m ) ) &&
+							contains( context, elem ) &&
+							elem.id === m ) {
+
+							results.push( elem );
+							return results;
+						}
+					}
+
+				// Type selector
+				} else if ( match[ 2 ] ) {
+					push.apply( results, context.getElementsByTagName( selector ) );
+					return results;
+
+				// Class selector
+				} else if ( ( m = match[ 3 ] ) && support.getElementsByClassName &&
+					context.getElementsByClassName ) {
+
+					push.apply( results, context.getElementsByClassName( m ) );
+					return results;
+				}
+			}
+
+			// Take advantage of querySelectorAll
+			if ( support.qsa &&
+				!nonnativeSelectorCache[ selector + " " ] &&
+				( !rbuggyQSA || !rbuggyQSA.test( selector ) ) &&
+
+				// Support: IE 8 only
+				// Exclude object elements
+				( nodeType !== 1 || context.nodeName.toLowerCase() !== "object" ) ) {
+
+				newSelector = selector;
+				newContext = context;
+
+				// qSA considers elements outside a scoping root when evaluating child or
+				// descendant combinators, which is not what we want.
+				// In such cases, we work around the behavior by prefixing every selector in the
+				// list with an ID selector referencing the scope context.
+				// The technique has to be used as well when a leading combinator is used
+				// as such selectors are not recognized by querySelectorAll.
+				// Thanks to Andrew Dupont for this technique.
+				if ( nodeType === 1 &&
+					( rdescend.test( selector ) || rcombinators.test( selector ) ) ) {
+
+					// Expand context for sibling selectors
+					newContext = rsibling.test( selector ) && testContext( context.parentNode ) ||
+						context;
+
+					// We can use :scope instead of the ID hack if the browser
+					// supports it & if we're not changing the context.
+					if ( newContext !== context || !support.scope ) {
+
+						// Capture the context ID, setting it first if necessary
+						if ( ( nid = context.getAttribute( "id" ) ) ) {
+							nid = nid.replace( rcssescape, fcssescape );
+						} else {
+							context.setAttribute( "id", ( nid = expando ) );
+						}
+					}
+
+					// Prefix every selector in the list
+					groups = tokenize( selector );
+					i = groups.length;
+					while ( i-- ) {
+						groups[ i ] = ( nid ? "#" + nid : ":scope" ) + " " +
+							toSelector( groups[ i ] );
+					}
+					newSelector = groups.join( "," );
+				}
+
+				try {
+					push.apply( results,
+						newContext.querySelectorAll( newSelector )
+					);
+					return results;
+				} catch ( qsaError ) {
+					nonnativeSelectorCache( selector, true );
+				} finally {
+					if ( nid === expando ) {
+						context.removeAttribute( "id" );
+					}
+				}
+			}
+		}
+	}
+
+	// All others
+	return select( selector.replace( rtrim, "$1" ), context, results, seed );
+}
+
+/**
+ * Create key-value caches of limited size
+ * @returns {function(string, object)} Returns the Object data after storing it on itself with
+ *	property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength)
+ *	deleting the oldest entry
+ */
+function createCache() {
+	var keys = [];
+
+	function cache( key, value ) {
+
+		// Use (key + " ") to avoid collision with native prototype properties (see Issue #157)
+		if ( keys.push( key + " " ) > Expr.cacheLength ) {
+
+			// Only keep the most recent entries
+			delete cache[ keys.shift() ];
+		}
+		return ( cache[ key + " " ] = value );
+	}
+	return cache;
+}
+
+/**
+ * Mark a function for special use by Sizzle
+ * @param {Function} fn The function to mark
+ */
+function markFunction( fn ) {
+	fn[ expando ] = true;
+	return fn;
+}
+
+/**
+ * Support testing using an element
+ * @param {Function} fn Passed the created element and returns a boolean result
+ */
+function assert( fn ) {
+	var el = document.createElement( "fieldset" );
+
+	try {
+		return !!fn( el );
+	} catch ( e ) {
+		return false;
+	} finally {
+
+		// Remove from its parent by default
+		if ( el.parentNode ) {
+			el.parentNode.removeChild( el );
+		}
+
+		// release memory in IE
+		el = null;
+	}
+}
+
+/**
+ * Adds the same handler for all of the specified attrs
+ * @param {String} attrs Pipe-separated list of attributes
+ * @param {Function} handler The method that will be applied
+ */
+function addHandle( attrs, handler ) {
+	var arr = attrs.split( "|" ),
+		i = arr.length;
+
+	while ( i-- ) {
+		Expr.attrHandle[ arr[ i ] ] = handler;
+	}
+}
+
+/**
+ * Checks document order of two siblings
+ * @param {Element} a
+ * @param {Element} b
+ * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b
+ */
+function siblingCheck( a, b ) {
+	var cur = b && a,
+		diff = cur && a.nodeType === 1 && b.nodeType === 1 &&
+			a.sourceIndex - b.sourceIndex;
+
+	// Use IE sourceIndex if available on both nodes
+	if ( diff ) {
+		return diff;
+	}
+
+	// Check if b follows a
+	if ( cur ) {
+		while ( ( cur = cur.nextSibling ) ) {
+			if ( cur === b ) {
+				return -1;
+			}
+		}
+	}
+
+	return a ? 1 : -1;
+}
+
+/**
+ * Returns a function to use in pseudos for input types
+ * @param {String} type
+ */
+function createInputPseudo( type ) {
+	return function( elem ) {
+		var name = elem.nodeName.toLowerCase();
+		return name === "input" && elem.type === type;
+	};
+}
+
+/**
+ * Returns a function to use in pseudos for buttons
+ * @param {String} type
+ */
+function createButtonPseudo( type ) {
+	return function( elem ) {
+		var name = elem.nodeName.toLowerCase();
+		return ( name === "input" || name === "button" ) && elem.type === type;
+	};
+}
+
+/**
+ * Returns a function to use in pseudos for :enabled/:disabled
+ * @param {Boolean} disabled true for :disabled; false for :enabled
+ */
+function createDisabledPseudo( disabled ) {
+
+	// Known :disabled false positives: fieldset[disabled] > legend:nth-of-type(n+2) :can-disable
+	return function( elem ) {
+
+		// Only certain elements can match :enabled or :disabled
+		// https://html.spec.whatwg.org/multipage/scripting.html#selector-enabled
+		// https://html.spec.whatwg.org/multipage/scripting.html#selector-disabled
+		if ( "form" in elem ) {
+
+			// Check for inherited disabledness on relevant non-disabled elements:
+			// * listed form-associated elements in a disabled fieldset
+			//   https://html.spec.whatwg.org/multipage/forms.html#category-listed
+			//   https://html.spec.whatwg.org/multipage/forms.html#concept-fe-disabled
+			// * option elements in a disabled optgroup
+			//   https://html.spec.whatwg.org/multipage/forms.html#concept-option-disabled
+			// All such elements have a "form" property.
+			if ( elem.parentNode && elem.disabled === false ) {
+
+				// Option elements defer to a parent optgroup if present
+				if ( "label" in elem ) {
+					if ( "label" in elem.parentNode ) {
+						return elem.parentNode.disabled === disabled;
+					} else {
+						return elem.disabled === disabled;
+					}
+				}
+
+				// Support: IE 6 - 11
+				// Use the isDisabled shortcut property to check for disabled fieldset ancestors
+				return elem.isDisabled === disabled ||
+
+					// Where there is no isDisabled, check manually
+					/* jshint -W018 */
+					elem.isDisabled !== !disabled &&
+					inDisabledFieldset( elem ) === disabled;
+			}
+
+			return elem.disabled === disabled;
+
+		// Try to winnow out elements that can't be disabled before trusting the disabled property.
+		// Some victims get caught in our net (label, legend, menu, track), but it shouldn't
+		// even exist on them, let alone have a boolean value.
+		} else if ( "label" in elem ) {
+			return elem.disabled === disabled;
+		}
+
+		// Remaining elements are neither :enabled nor :disabled
+		return false;
+	};
+}
+
+/**
+ * Returns a function to use in pseudos for positionals
+ * @param {Function} fn
+ */
+function createPositionalPseudo( fn ) {
+	return markFunction( function( argument ) {
+		argument = +argument;
+		return markFunction( function( seed, matches ) {
+			var j,
+				matchIndexes = fn( [], seed.length, argument ),
+				i = matchIndexes.length;
+
+			// Match elements found at the specified indexes
+			while ( i-- ) {
+				if ( seed[ ( j = matchIndexes[ i ] ) ] ) {
+					seed[ j ] = !( matches[ j ] = seed[ j ] );
+				}
+			}
+		} );
+	} );
+}
+
+/**
+ * Checks a node for validity as a Sizzle context
+ * @param {Element|Object=} context
+ * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value
+ */
+function testContext( context ) {
+	return context && typeof context.getElementsByTagName !== "undefined" && context;
+}
+
+// Expose support vars for convenience
+support = Sizzle.support = {};
+
+/**
+ * Detects XML nodes
+ * @param {Element|Object} elem An element or a document
+ * @returns {Boolean} True iff elem is a non-HTML XML node
+ */
+isXML = Sizzle.isXML = function( elem ) {
+	var namespace = elem.namespaceURI,
+		docElem = ( elem.ownerDocument || elem ).documentElement;
+
+	// Support: IE <=8
+	// Assume HTML when documentElement doesn't yet exist, such as inside loading iframes
+	// https://bugs.jquery.com/ticket/4833
+	return !rhtml.test( namespace || docElem && docElem.nodeName || "HTML" );
+};
+
+/**
+ * Sets document-related variables once based on the current document
+ * @param {Element|Object} [doc] An element or document object to use to set the document
+ * @returns {Object} Returns the current document
+ */
+setDocument = Sizzle.setDocument = function( node ) {
+	var hasCompare, subWindow,
+		doc = node ? node.ownerDocument || node : preferredDoc;
+
+	// Return early if doc is invalid or already selected
+	// Support: IE 11+, Edge 17 - 18+
+	// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+	// two documents; shallow comparisons work.
+	// eslint-disable-next-line eqeqeq
+	if ( doc == document || doc.nodeType !== 9 || !doc.documentElement ) {
+		return document;
+	}
+
+	// Update global variables
+	document = doc;
+	docElem = document.documentElement;
+	documentIsHTML = !isXML( document );
+
+	// Support: IE 9 - 11+, Edge 12 - 18+
+	// Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936)
+	// Support: IE 11+, Edge 17 - 18+
+	// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+	// two documents; shallow comparisons work.
+	// eslint-disable-next-line eqeqeq
+	if ( preferredDoc != document &&
+		( subWindow = document.defaultView ) && subWindow.top !== subWindow ) {
+
+		// Support: IE 11, Edge
+		if ( subWindow.addEventListener ) {
+			subWindow.addEventListener( "unload", unloadHandler, false );
+
+		// Support: IE 9 - 10 only
+		} else if ( subWindow.attachEvent ) {
+			subWindow.attachEvent( "onunload", unloadHandler );
+		}
+	}
+
+	// Support: IE 8 - 11+, Edge 12 - 18+, Chrome <=16 - 25 only, Firefox <=3.6 - 31 only,
+	// Safari 4 - 5 only, Opera <=11.6 - 12.x only
+	// IE/Edge & older browsers don't support the :scope pseudo-class.
+	// Support: Safari 6.0 only
+	// Safari 6.0 supports :scope but it's an alias of :root there.
+	support.scope = assert( function( el ) {
+		docElem.appendChild( el ).appendChild( document.createElement( "div" ) );
+		return typeof el.querySelectorAll !== "undefined" &&
+			!el.querySelectorAll( ":scope fieldset div" ).length;
+	} );
+
+	/* Attributes
+	---------------------------------------------------------------------- */
+
+	// Support: IE<8
+	// Verify that getAttribute really returns attributes and not properties
+	// (excepting IE8 booleans)
+	support.attributes = assert( function( el ) {
+		el.className = "i";
+		return !el.getAttribute( "className" );
+	} );
+
+	/* getElement(s)By*
+	---------------------------------------------------------------------- */
+
+	// Check if getElementsByTagName("*") returns only elements
+	support.getElementsByTagName = assert( function( el ) {
+		el.appendChild( document.createComment( "" ) );
+		return !el.getElementsByTagName( "*" ).length;
+	} );
+
+	// Support: IE<9
+	support.getElementsByClassName = rnative.test( document.getElementsByClassName );
+
+	// Support: IE<10
+	// Check if getElementById returns elements by name
+	// The broken getElementById methods don't pick up programmatically-set names,
+	// so use a roundabout getElementsByName test
+	support.getById = assert( function( el ) {
+		docElem.appendChild( el ).id = expando;
+		return !document.getElementsByName || !document.getElementsByName( expando ).length;
+	} );
+
+	// ID filter and find
+	if ( support.getById ) {
+		Expr.filter[ "ID" ] = function( id ) {
+			var attrId = id.replace( runescape, funescape );
+			return function( elem ) {
+				return elem.getAttribute( "id" ) === attrId;
+			};
+		};
+		Expr.find[ "ID" ] = function( id, context ) {
+			if ( typeof context.getElementById !== "undefined" && documentIsHTML ) {
+				var elem = context.getElementById( id );
+				return elem ? [ elem ] : [];
+			}
+		};
+	} else {
+		Expr.filter[ "ID" ] =  function( id ) {
+			var attrId = id.replace( runescape, funescape );
+			return function( elem ) {
+				var node = typeof elem.getAttributeNode !== "undefined" &&
+					elem.getAttributeNode( "id" );
+				return node && node.value === attrId;
+			};
+		};
+
+		// Support: IE 6 - 7 only
+		// getElementById is not reliable as a find shortcut
+		Expr.find[ "ID" ] = function( id, context ) {
+			if ( typeof context.getElementById !== "undefined" && documentIsHTML ) {
+				var node, i, elems,
+					elem = context.getElementById( id );
+
+				if ( elem ) {
+
+					// Verify the id attribute
+					node = elem.getAttributeNode( "id" );
+					if ( node && node.value === id ) {
+						return [ elem ];
+					}
+
+					// Fall back on getElementsByName
+					elems = context.getElementsByName( id );
+					i = 0;
+					while ( ( elem = elems[ i++ ] ) ) {
+						node = elem.getAttributeNode( "id" );
+						if ( node && node.value === id ) {
+							return [ elem ];
+						}
+					}
+				}
+
+				return [];
+			}
+		};
+	}
+
+	// Tag
+	Expr.find[ "TAG" ] = support.getElementsByTagName ?
+		function( tag, context ) {
+			if ( typeof context.getElementsByTagName !== "undefined" ) {
+				return context.getElementsByTagName( tag );
+
+			// DocumentFragment nodes don't have gEBTN
+			} else if ( support.qsa ) {
+				return context.querySelectorAll( tag );
+			}
+		} :
+
+		function( tag, context ) {
+			var elem,
+				tmp = [],
+				i = 0,
+
+				// By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too
+				results = context.getElementsByTagName( tag );
+
+			// Filter out possible comments
+			if ( tag === "*" ) {
+				while ( ( elem = results[ i++ ] ) ) {
+					if ( elem.nodeType === 1 ) {
+						tmp.push( elem );
+					}
+				}
+
+				return tmp;
+			}
+			return results;
+		};
+
+	// Class
+	Expr.find[ "CLASS" ] = support.getElementsByClassName && function( className, context ) {
+		if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) {
+			return context.getElementsByClassName( className );
+		}
+	};
+
+	/* QSA/matchesSelector
+	---------------------------------------------------------------------- */
+
+	// QSA and matchesSelector support
+
+	// matchesSelector(:active) reports false when true (IE9/Opera 11.5)
+	rbuggyMatches = [];
+
+	// qSa(:focus) reports false when true (Chrome 21)
+	// We allow this because of a bug in IE8/9 that throws an error
+	// whenever `document.activeElement` is accessed on an iframe
+	// So, we allow :focus to pass through QSA all the time to avoid the IE error
+	// See https://bugs.jquery.com/ticket/13378
+	rbuggyQSA = [];
+
+	if ( ( support.qsa = rnative.test( document.querySelectorAll ) ) ) {
+
+		// Build QSA regex
+		// Regex strategy adopted from Diego Perini
+		assert( function( el ) {
+
+			var input;
+
+			// Select is set to empty string on purpose
+			// This is to test IE's treatment of not explicitly
+			// setting a boolean content attribute,
+			// since its presence should be enough
+			// https://bugs.jquery.com/ticket/12359
+			docElem.appendChild( el ).innerHTML = "<a id='" + expando + "'></a>" +
+				"<select id='" + expando + "-\r\\' msallowcapture=''>" +
+				"<option selected=''></option></select>";
+
+			// Support: IE8, Opera 11-12.16
+			// Nothing should be selected when empty strings follow ^= or $= or *=
+			// The test attribute must be unknown in Opera but "safe" for WinRT
+			// https://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section
+			if ( el.querySelectorAll( "[msallowcapture^='']" ).length ) {
+				rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" );
+			}
+
+			// Support: IE8
+			// Boolean attributes and "value" are not treated correctly
+			if ( !el.querySelectorAll( "[selected]" ).length ) {
+				rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" );
+			}
+
+			// Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+
+			if ( !el.querySelectorAll( "[id~=" + expando + "-]" ).length ) {
+				rbuggyQSA.push( "~=" );
+			}
+
+			// Support: IE 11+, Edge 15 - 18+
+			// IE 11/Edge don't find elements on a `[name='']` query in some cases.
+			// Adding a temporary attribute to the document before the selection works
+			// around the issue.
+			// Interestingly, IE 10 & older don't seem to have the issue.
+			input = document.createElement( "input" );
+			input.setAttribute( "name", "" );
+			el.appendChild( input );
+			if ( !el.querySelectorAll( "[name='']" ).length ) {
+				rbuggyQSA.push( "\\[" + whitespace + "*name" + whitespace + "*=" +
+					whitespace + "*(?:''|\"\")" );
+			}
+
+			// Webkit/Opera - :checked should return selected option elements
+			// http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked
+			// IE8 throws error here and will not see later tests
+			if ( !el.querySelectorAll( ":checked" ).length ) {
+				rbuggyQSA.push( ":checked" );
+			}
+
+			// Support: Safari 8+, iOS 8+
+			// https://bugs.webkit.org/show_bug.cgi?id=136851
+			// In-page `selector#id sibling-combinator selector` fails
+			if ( !el.querySelectorAll( "a#" + expando + "+*" ).length ) {
+				rbuggyQSA.push( ".#.+[+~]" );
+			}
+
+			// Support: Firefox <=3.6 - 5 only
+			// Old Firefox doesn't throw on a badly-escaped identifier.
+			el.querySelectorAll( "\\\f" );
+			rbuggyQSA.push( "[\\r\\n\\f]" );
+		} );
+
+		assert( function( el ) {
+			el.innerHTML = "<a href='' disabled='disabled'></a>" +
+				"<select disabled='disabled'><option/></select>";
+
+			// Support: Windows 8 Native Apps
+			// The type and name attributes are restricted during .innerHTML assignment
+			var input = document.createElement( "input" );
+			input.setAttribute( "type", "hidden" );
+			el.appendChild( input ).setAttribute( "name", "D" );
+
+			// Support: IE8
+			// Enforce case-sensitivity of name attribute
+			if ( el.querySelectorAll( "[name=d]" ).length ) {
+				rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" );
+			}
+
+			// FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled)
+			// IE8 throws error here and will not see later tests
+			if ( el.querySelectorAll( ":enabled" ).length !== 2 ) {
+				rbuggyQSA.push( ":enabled", ":disabled" );
+			}
+
+			// Support: IE9-11+
+			// IE's :disabled selector does not pick up the children of disabled fieldsets
+			docElem.appendChild( el ).disabled = true;
+			if ( el.querySelectorAll( ":disabled" ).length !== 2 ) {
+				rbuggyQSA.push( ":enabled", ":disabled" );
+			}
+
+			// Support: Opera 10 - 11 only
+			// Opera 10-11 does not throw on post-comma invalid pseudos
+			el.querySelectorAll( "*,:x" );
+			rbuggyQSA.push( ",.*:" );
+		} );
+	}
+
+	if ( ( support.matchesSelector = rnative.test( ( matches = docElem.matches ||
+		docElem.webkitMatchesSelector ||
+		docElem.mozMatchesSelector ||
+		docElem.oMatchesSelector ||
+		docElem.msMatchesSelector ) ) ) ) {
+
+		assert( function( el ) {
+
+			// Check to see if it's possible to do matchesSelector
+			// on a disconnected node (IE 9)
+			support.disconnectedMatch = matches.call( el, "*" );
+
+			// This should fail with an exception
+			// Gecko does not error, returns false instead
+			matches.call( el, "[s!='']:x" );
+			rbuggyMatches.push( "!=", pseudos );
+		} );
+	}
+
+	rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join( "|" ) );
+	rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join( "|" ) );
+
+	/* Contains
+	---------------------------------------------------------------------- */
+	hasCompare = rnative.test( docElem.compareDocumentPosition );
+
+	// Element contains another
+	// Purposefully self-exclusive
+	// As in, an element does not contain itself
+	contains = hasCompare || rnative.test( docElem.contains ) ?
+		function( a, b ) {
+			var adown = a.nodeType === 9 ? a.documentElement : a,
+				bup = b && b.parentNode;
+			return a === bup || !!( bup && bup.nodeType === 1 && (
+				adown.contains ?
+					adown.contains( bup ) :
+					a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16
+			) );
+		} :
+		function( a, b ) {
+			if ( b ) {
+				while ( ( b = b.parentNode ) ) {
+					if ( b === a ) {
+						return true;
+					}
+				}
+			}
+			return false;
+		};
+
+	/* Sorting
+	---------------------------------------------------------------------- */
+
+	// Document order sorting
+	sortOrder = hasCompare ?
+	function( a, b ) {
+
+		// Flag for duplicate removal
+		if ( a === b ) {
+			hasDuplicate = true;
+			return 0;
+		}
+
+		// Sort on method existence if only one input has compareDocumentPosition
+		var compare = !a.compareDocumentPosition - !b.compareDocumentPosition;
+		if ( compare ) {
+			return compare;
+		}
+
+		// Calculate position if both inputs belong to the same document
+		// Support: IE 11+, Edge 17 - 18+
+		// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+		// two documents; shallow comparisons work.
+		// eslint-disable-next-line eqeqeq
+		compare = ( a.ownerDocument || a ) == ( b.ownerDocument || b ) ?
+			a.compareDocumentPosition( b ) :
+
+			// Otherwise we know they are disconnected
+			1;
+
+		// Disconnected nodes
+		if ( compare & 1 ||
+			( !support.sortDetached && b.compareDocumentPosition( a ) === compare ) ) {
+
+			// Choose the first element that is related to our preferred document
+			// Support: IE 11+, Edge 17 - 18+
+			// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+			// two documents; shallow comparisons work.
+			// eslint-disable-next-line eqeqeq
+			if ( a == document || a.ownerDocument == preferredDoc &&
+				contains( preferredDoc, a ) ) {
+				return -1;
+			}
+
+			// Support: IE 11+, Edge 17 - 18+
+			// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+			// two documents; shallow comparisons work.
+			// eslint-disable-next-line eqeqeq
+			if ( b == document || b.ownerDocument == preferredDoc &&
+				contains( preferredDoc, b ) ) {
+				return 1;
+			}
+
+			// Maintain original order
+			return sortInput ?
+				( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) :
+				0;
+		}
+
+		return compare & 4 ? -1 : 1;
+	} :
+	function( a, b ) {
+
+		// Exit early if the nodes are identical
+		if ( a === b ) {
+			hasDuplicate = true;
+			return 0;
+		}
+
+		var cur,
+			i = 0,
+			aup = a.parentNode,
+			bup = b.parentNode,
+			ap = [ a ],
+			bp = [ b ];
+
+		// Parentless nodes are either documents or disconnected
+		if ( !aup || !bup ) {
+
+			// Support: IE 11+, Edge 17 - 18+
+			// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+			// two documents; shallow comparisons work.
+			/* eslint-disable eqeqeq */
+			return a == document ? -1 :
+				b == document ? 1 :
+				/* eslint-enable eqeqeq */
+				aup ? -1 :
+				bup ? 1 :
+				sortInput ?
+				( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) :
+				0;
+
+		// If the nodes are siblings, we can do a quick check
+		} else if ( aup === bup ) {
+			return siblingCheck( a, b );
+		}
+
+		// Otherwise we need full lists of their ancestors for comparison
+		cur = a;
+		while ( ( cur = cur.parentNode ) ) {
+			ap.unshift( cur );
+		}
+		cur = b;
+		while ( ( cur = cur.parentNode ) ) {
+			bp.unshift( cur );
+		}
+
+		// Walk down the tree looking for a discrepancy
+		while ( ap[ i ] === bp[ i ] ) {
+			i++;
+		}
+
+		return i ?
+
+			// Do a sibling check if the nodes have a common ancestor
+			siblingCheck( ap[ i ], bp[ i ] ) :
+
+			// Otherwise nodes in our document sort first
+			// Support: IE 11+, Edge 17 - 18+
+			// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+			// two documents; shallow comparisons work.
+			/* eslint-disable eqeqeq */
+			ap[ i ] == preferredDoc ? -1 :
+			bp[ i ] == preferredDoc ? 1 :
+			/* eslint-enable eqeqeq */
+			0;
+	};
+
+	return document;
+};
+
+Sizzle.matches = function( expr, elements ) {
+	return Sizzle( expr, null, null, elements );
+};
+
+Sizzle.matchesSelector = function( elem, expr ) {
+	setDocument( elem );
+
+	if ( support.matchesSelector && documentIsHTML &&
+		!nonnativeSelectorCache[ expr + " " ] &&
+		( !rbuggyMatches || !rbuggyMatches.test( expr ) ) &&
+		( !rbuggyQSA     || !rbuggyQSA.test( expr ) ) ) {
+
+		try {
+			var ret = matches.call( elem, expr );
+
+			// IE 9's matchesSelector returns false on disconnected nodes
+			if ( ret || support.disconnectedMatch ||
+
+				// As well, disconnected nodes are said to be in a document
+				// fragment in IE 9
+				elem.document && elem.document.nodeType !== 11 ) {
+				return ret;
+			}
+		} catch ( e ) {
+			nonnativeSelectorCache( expr, true );
+		}
+	}
+
+	return Sizzle( expr, document, null, [ elem ] ).length > 0;
+};
+
+Sizzle.contains = function( context, elem ) {
+
+	// Set document vars if needed
+	// Support: IE 11+, Edge 17 - 18+
+	// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+	// two documents; shallow comparisons work.
+	// eslint-disable-next-line eqeqeq
+	if ( ( context.ownerDocument || context ) != document ) {
+		setDocument( context );
+	}
+	return contains( context, elem );
+};
+
+Sizzle.attr = function( elem, name ) {
+
+	// Set document vars if needed
+	// Support: IE 11+, Edge 17 - 18+
+	// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+	// two documents; shallow comparisons work.
+	// eslint-disable-next-line eqeqeq
+	if ( ( elem.ownerDocument || elem ) != document ) {
+		setDocument( elem );
+	}
+
+	var fn = Expr.attrHandle[ name.toLowerCase() ],
+
+		// Don't get fooled by Object.prototype properties (jQuery #13807)
+		val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ?
+			fn( elem, name, !documentIsHTML ) :
+			undefined;
+
+	return val !== undefined ?
+		val :
+		support.attributes || !documentIsHTML ?
+			elem.getAttribute( name ) :
+			( val = elem.getAttributeNode( name ) ) && val.specified ?
+				val.value :
+				null;
+};
+
+Sizzle.escape = function( sel ) {
+	return ( sel + "" ).replace( rcssescape, fcssescape );
+};
+
+Sizzle.error = function( msg ) {
+	throw new Error( "Syntax error, unrecognized expression: " + msg );
+};
+
+/**
+ * Document sorting and removing duplicates
+ * @param {ArrayLike} results
+ */
+Sizzle.uniqueSort = function( results ) {
+	var elem,
+		duplicates = [],
+		j = 0,
+		i = 0;
+
+	// Unless we *know* we can detect duplicates, assume their presence
+	hasDuplicate = !support.detectDuplicates;
+	sortInput = !support.sortStable && results.slice( 0 );
+	results.sort( sortOrder );
+
+	if ( hasDuplicate ) {
+		while ( ( elem = results[ i++ ] ) ) {
+			if ( elem === results[ i ] ) {
+				j = duplicates.push( i );
+			}
+		}
+		while ( j-- ) {
+			results.splice( duplicates[ j ], 1 );
+		}
+	}
+
+	// Clear input after sorting to release objects
+	// See https://github.com/jquery/sizzle/pull/225
+	sortInput = null;
+
+	return results;
+};
+
+/**
+ * Utility function for retrieving the text value of an array of DOM nodes
+ * @param {Array|Element} elem
+ */
+getText = Sizzle.getText = function( elem ) {
+	var node,
+		ret = "",
+		i = 0,
+		nodeType = elem.nodeType;
+
+	if ( !nodeType ) {
+
+		// If no nodeType, this is expected to be an array
+		while ( ( node = elem[ i++ ] ) ) {
+
+			// Do not traverse comment nodes
+			ret += getText( node );
+		}
+	} else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) {
+
+		// Use textContent for elements
+		// innerText usage removed for consistency of new lines (jQuery #11153)
+		if ( typeof elem.textContent === "string" ) {
+			return elem.textContent;
+		} else {
+
+			// Traverse its children
+			for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) {
+				ret += getText( elem );
+			}
+		}
+	} else if ( nodeType === 3 || nodeType === 4 ) {
+		return elem.nodeValue;
+	}
+
+	// Do not include comment or processing instruction nodes
+
+	return ret;
+};
+
+Expr = Sizzle.selectors = {
+
+	// Can be adjusted by the user
+	cacheLength: 50,
+
+	createPseudo: markFunction,
+
+	match: matchExpr,
+
+	attrHandle: {},
+
+	find: {},
+
+	relative: {
+		">": { dir: "parentNode", first: true },
+		" ": { dir: "parentNode" },
+		"+": { dir: "previousSibling", first: true },
+		"~": { dir: "previousSibling" }
+	},
+
+	preFilter: {
+		"ATTR": function( match ) {
+			match[ 1 ] = match[ 1 ].replace( runescape, funescape );
+
+			// Move the given value to match[3] whether quoted or unquoted
+			match[ 3 ] = ( match[ 3 ] || match[ 4 ] ||
+				match[ 5 ] || "" ).replace( runescape, funescape );
+
+			if ( match[ 2 ] === "~=" ) {
+				match[ 3 ] = " " + match[ 3 ] + " ";
+			}
+
+			return match.slice( 0, 4 );
+		},
+
+		"CHILD": function( match ) {
+
+			/* matches from matchExpr["CHILD"]
+				1 type (only|nth|...)
+				2 what (child|of-type)
+				3 argument (even|odd|\d*|\d*n([+-]\d+)?|...)
+				4 xn-component of xn+y argument ([+-]?\d*n|)
+				5 sign of xn-component
+				6 x of xn-component
+				7 sign of y-component
+				8 y of y-component
+			*/
+			match[ 1 ] = match[ 1 ].toLowerCase();
+
+			if ( match[ 1 ].slice( 0, 3 ) === "nth" ) {
+
+				// nth-* requires argument
+				if ( !match[ 3 ] ) {
+					Sizzle.error( match[ 0 ] );
+				}
+
+				// numeric x and y parameters for Expr.filter.CHILD
+				// remember that false/true cast respectively to 0/1
+				match[ 4 ] = +( match[ 4 ] ?
+					match[ 5 ] + ( match[ 6 ] || 1 ) :
+					2 * ( match[ 3 ] === "even" || match[ 3 ] === "odd" ) );
+				match[ 5 ] = +( ( match[ 7 ] + match[ 8 ] ) || match[ 3 ] === "odd" );
+
+				// other types prohibit arguments
+			} else if ( match[ 3 ] ) {
+				Sizzle.error( match[ 0 ] );
+			}
+
+			return match;
+		},
+
+		"PSEUDO": function( match ) {
+			var excess,
+				unquoted = !match[ 6 ] && match[ 2 ];
+
+			if ( matchExpr[ "CHILD" ].test( match[ 0 ] ) ) {
+				return null;
+			}
+
+			// Accept quoted arguments as-is
+			if ( match[ 3 ] ) {
+				match[ 2 ] = match[ 4 ] || match[ 5 ] || "";
+
+			// Strip excess characters from unquoted arguments
+			} else if ( unquoted && rpseudo.test( unquoted ) &&
+
+				// Get excess from tokenize (recursively)
+				( excess = tokenize( unquoted, true ) ) &&
+
+				// advance to the next closing parenthesis
+				( excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length ) ) {
+
+				// excess is a negative index
+				match[ 0 ] = match[ 0 ].slice( 0, excess );
+				match[ 2 ] = unquoted.slice( 0, excess );
+			}
+
+			// Return only captures needed by the pseudo filter method (type and argument)
+			return match.slice( 0, 3 );
+		}
+	},
+
+	filter: {
+
+		"TAG": function( nodeNameSelector ) {
+			var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase();
+			return nodeNameSelector === "*" ?
+				function() {
+					return true;
+				} :
+				function( elem ) {
+					return elem.nodeName && elem.nodeName.toLowerCase() === nodeName;
+				};
+		},
+
+		"CLASS": function( className ) {
+			var pattern = classCache[ className + " " ];
+
+			return pattern ||
+				( pattern = new RegExp( "(^|" + whitespace +
+					")" + className + "(" + whitespace + "|$)" ) ) && classCache(
+						className, function( elem ) {
+							return pattern.test(
+								typeof elem.className === "string" && elem.className ||
+								typeof elem.getAttribute !== "undefined" &&
+									elem.getAttribute( "class" ) ||
+								""
+							);
+				} );
+		},
+
+		"ATTR": function( name, operator, check ) {
+			return function( elem ) {
+				var result = Sizzle.attr( elem, name );
+
+				if ( result == null ) {
+					return operator === "!=";
+				}
+				if ( !operator ) {
+					return true;
+				}
+
+				result += "";
+
+				/* eslint-disable max-len */
+
+				return operator === "=" ? result === check :
+					operator === "!=" ? result !== check :
+					operator === "^=" ? check && result.indexOf( check ) === 0 :
+					operator === "*=" ? check && result.indexOf( check ) > -1 :
+					operator === "$=" ? check && result.slice( -check.length ) === check :
+					operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 :
+					operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" :
+					false;
+				/* eslint-enable max-len */
+
+			};
+		},
+
+		"CHILD": function( type, what, _argument, first, last ) {
+			var simple = type.slice( 0, 3 ) !== "nth",
+				forward = type.slice( -4 ) !== "last",
+				ofType = what === "of-type";
+
+			return first === 1 && last === 0 ?
+
+				// Shortcut for :nth-*(n)
+				function( elem ) {
+					return !!elem.parentNode;
+				} :
+
+				function( elem, _context, xml ) {
+					var cache, uniqueCache, outerCache, node, nodeIndex, start,
+						dir = simple !== forward ? "nextSibling" : "previousSibling",
+						parent = elem.parentNode,
+						name = ofType && elem.nodeName.toLowerCase(),
+						useCache = !xml && !ofType,
+						diff = false;
+
+					if ( parent ) {
+
+						// :(first|last|only)-(child|of-type)
+						if ( simple ) {
+							while ( dir ) {
+								node = elem;
+								while ( ( node = node[ dir ] ) ) {
+									if ( ofType ?
+										node.nodeName.toLowerCase() === name :
+										node.nodeType === 1 ) {
+
+										return false;
+									}
+								}
+
+								// Reverse direction for :only-* (if we haven't yet done so)
+								start = dir = type === "only" && !start && "nextSibling";
+							}
+							return true;
+						}
+
+						start = [ forward ? parent.firstChild : parent.lastChild ];
+
+						// non-xml :nth-child(...) stores cache data on `parent`
+						if ( forward && useCache ) {
+
+							// Seek `elem` from a previously-cached index
+
+							// ...in a gzip-friendly way
+							node = parent;
+							outerCache = node[ expando ] || ( node[ expando ] = {} );
+
+							// Support: IE <9 only
+							// Defend against cloned attroperties (jQuery gh-1709)
+							uniqueCache = outerCache[ node.uniqueID ] ||
+								( outerCache[ node.uniqueID ] = {} );
+
+							cache = uniqueCache[ type ] || [];
+							nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ];
+							diff = nodeIndex && cache[ 2 ];
+							node = nodeIndex && parent.childNodes[ nodeIndex ];
+
+							while ( ( node = ++nodeIndex && node && node[ dir ] ||
+
+								// Fallback to seeking `elem` from the start
+								( diff = nodeIndex = 0 ) || start.pop() ) ) {
+
+								// When found, cache indexes on `parent` and break
+								if ( node.nodeType === 1 && ++diff && node === elem ) {
+									uniqueCache[ type ] = [ dirruns, nodeIndex, diff ];
+									break;
+								}
+							}
+
+						} else {
+
+							// Use previously-cached element index if available
+							if ( useCache ) {
+
+								// ...in a gzip-friendly way
+								node = elem;
+								outerCache = node[ expando ] || ( node[ expando ] = {} );
+
+								// Support: IE <9 only
+								// Defend against cloned attroperties (jQuery gh-1709)
+								uniqueCache = outerCache[ node.uniqueID ] ||
+									( outerCache[ node.uniqueID ] = {} );
+
+								cache = uniqueCache[ type ] || [];
+								nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ];
+								diff = nodeIndex;
+							}
+
+							// xml :nth-child(...)
+							// or :nth-last-child(...) or :nth(-last)?-of-type(...)
+							if ( diff === false ) {
+
+								// Use the same loop as above to seek `elem` from the start
+								while ( ( node = ++nodeIndex && node && node[ dir ] ||
+									( diff = nodeIndex = 0 ) || start.pop() ) ) {
+
+									if ( ( ofType ?
+										node.nodeName.toLowerCase() === name :
+										node.nodeType === 1 ) &&
+										++diff ) {
+
+										// Cache the index of each encountered element
+										if ( useCache ) {
+											outerCache = node[ expando ] ||
+												( node[ expando ] = {} );
+
+											// Support: IE <9 only
+											// Defend against cloned attroperties (jQuery gh-1709)
+											uniqueCache = outerCache[ node.uniqueID ] ||
+												( outerCache[ node.uniqueID ] = {} );
+
+											uniqueCache[ type ] = [ dirruns, diff ];
+										}
+
+										if ( node === elem ) {
+											break;
+										}
+									}
+								}
+							}
+						}
+
+						// Incorporate the offset, then check against cycle size
+						diff -= last;
+						return diff === first || ( diff % first === 0 && diff / first >= 0 );
+					}
+				};
+		},
+
+		"PSEUDO": function( pseudo, argument ) {
+
+			// pseudo-class names are case-insensitive
+			// http://www.w3.org/TR/selectors/#pseudo-classes
+			// Prioritize by case sensitivity in case custom pseudos are added with uppercase letters
+			// Remember that setFilters inherits from pseudos
+			var args,
+				fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] ||
+					Sizzle.error( "unsupported pseudo: " + pseudo );
+
+			// The user may use createPseudo to indicate that
+			// arguments are needed to create the filter function
+			// just as Sizzle does
+			if ( fn[ expando ] ) {
+				return fn( argument );
+			}
+
+			// But maintain support for old signatures
+			if ( fn.length > 1 ) {
+				args = [ pseudo, pseudo, "", argument ];
+				return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ?
+					markFunction( function( seed, matches ) {
+						var idx,
+							matched = fn( seed, argument ),
+							i = matched.length;
+						while ( i-- ) {
+							idx = indexOf( seed, matched[ i ] );
+							seed[ idx ] = !( matches[ idx ] = matched[ i ] );
+						}
+					} ) :
+					function( elem ) {
+						return fn( elem, 0, args );
+					};
+			}
+
+			return fn;
+		}
+	},
+
+	pseudos: {
+
+		// Potentially complex pseudos
+		"not": markFunction( function( selector ) {
+
+			// Trim the selector passed to compile
+			// to avoid treating leading and trailing
+			// spaces as combinators
+			var input = [],
+				results = [],
+				matcher = compile( selector.replace( rtrim, "$1" ) );
+
+			return matcher[ expando ] ?
+				markFunction( function( seed, matches, _context, xml ) {
+					var elem,
+						unmatched = matcher( seed, null, xml, [] ),
+						i = seed.length;
+
+					// Match elements unmatched by `matcher`
+					while ( i-- ) {
+						if ( ( elem = unmatched[ i ] ) ) {
+							seed[ i ] = !( matches[ i ] = elem );
+						}
+					}
+				} ) :
+				function( elem, _context, xml ) {
+					input[ 0 ] = elem;
+					matcher( input, null, xml, results );
+
+					// Don't keep the element (issue #299)
+					input[ 0 ] = null;
+					return !results.pop();
+				};
+		} ),
+
+		"has": markFunction( function( selector ) {
+			return function( elem ) {
+				return Sizzle( selector, elem ).length > 0;
+			};
+		} ),
+
+		"contains": markFunction( function( text ) {
+			text = text.replace( runescape, funescape );
+			return function( elem ) {
+				return ( elem.textContent || getText( elem ) ).indexOf( text ) > -1;
+			};
+		} ),
+
+		// "Whether an element is represented by a :lang() selector
+		// is based solely on the element's language value
+		// being equal to the identifier C,
+		// or beginning with the identifier C immediately followed by "-".
+		// The matching of C against the element's language value is performed case-insensitively.
+		// The identifier C does not have to be a valid language name."
+		// http://www.w3.org/TR/selectors/#lang-pseudo
+		"lang": markFunction( function( lang ) {
+
+			// lang value must be a valid identifier
+			if ( !ridentifier.test( lang || "" ) ) {
+				Sizzle.error( "unsupported lang: " + lang );
+			}
+			lang = lang.replace( runescape, funescape ).toLowerCase();
+			return function( elem ) {
+				var elemLang;
+				do {
+					if ( ( elemLang = documentIsHTML ?
+						elem.lang :
+						elem.getAttribute( "xml:lang" ) || elem.getAttribute( "lang" ) ) ) {
+
+						elemLang = elemLang.toLowerCase();
+						return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0;
+					}
+				} while ( ( elem = elem.parentNode ) && elem.nodeType === 1 );
+				return false;
+			};
+		} ),
+
+		// Miscellaneous
+		"target": function( elem ) {
+			var hash = window.location && window.location.hash;
+			return hash && hash.slice( 1 ) === elem.id;
+		},
+
+		"root": function( elem ) {
+			return elem === docElem;
+		},
+
+		"focus": function( elem ) {
+			return elem === document.activeElement &&
+				( !document.hasFocus || document.hasFocus() ) &&
+				!!( elem.type || elem.href || ~elem.tabIndex );
+		},
+
+		// Boolean properties
+		"enabled": createDisabledPseudo( false ),
+		"disabled": createDisabledPseudo( true ),
+
+		"checked": function( elem ) {
+
+			// In CSS3, :checked should return both checked and selected elements
+			// http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked
+			var nodeName = elem.nodeName.toLowerCase();
+			return ( nodeName === "input" && !!elem.checked ) ||
+				( nodeName === "option" && !!elem.selected );
+		},
+
+		"selected": function( elem ) {
+
+			// Accessing this property makes selected-by-default
+			// options in Safari work properly
+			if ( elem.parentNode ) {
+				// eslint-disable-next-line no-unused-expressions
+				elem.parentNode.selectedIndex;
+			}
+
+			return elem.selected === true;
+		},
+
+		// Contents
+		"empty": function( elem ) {
+
+			// http://www.w3.org/TR/selectors/#empty-pseudo
+			// :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5),
+			//   but not by others (comment: 8; processing instruction: 7; etc.)
+			// nodeType < 6 works because attributes (2) do not appear as children
+			for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) {
+				if ( elem.nodeType < 6 ) {
+					return false;
+				}
+			}
+			return true;
+		},
+
+		"parent": function( elem ) {
+			return !Expr.pseudos[ "empty" ]( elem );
+		},
+
+		// Element/input types
+		"header": function( elem ) {
+			return rheader.test( elem.nodeName );
+		},
+
+		"input": function( elem ) {
+			return rinputs.test( elem.nodeName );
+		},
+
+		"button": function( elem ) {
+			var name = elem.nodeName.toLowerCase();
+			return name === "input" && elem.type === "button" || name === "button";
+		},
+
+		"text": function( elem ) {
+			var attr;
+			return elem.nodeName.toLowerCase() === "input" &&
+				elem.type === "text" &&
+
+				// Support: IE<8
+				// New HTML5 attribute values (e.g., "search") appear with elem.type === "text"
+				( ( attr = elem.getAttribute( "type" ) ) == null ||
+					attr.toLowerCase() === "text" );
+		},
+
+		// Position-in-collection
+		"first": createPositionalPseudo( function() {
+			return [ 0 ];
+		} ),
+
+		"last": createPositionalPseudo( function( _matchIndexes, length ) {
+			return [ length - 1 ];
+		} ),
+
+		"eq": createPositionalPseudo( function( _matchIndexes, length, argument ) {
+			return [ argument < 0 ? argument + length : argument ];
+		} ),
+
+		"even": createPositionalPseudo( function( matchIndexes, length ) {
+			var i = 0;
+			for ( ; i < length; i += 2 ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		} ),
+
+		"odd": createPositionalPseudo( function( matchIndexes, length ) {
+			var i = 1;
+			for ( ; i < length; i += 2 ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		} ),
+
+		"lt": createPositionalPseudo( function( matchIndexes, length, argument ) {
+			var i = argument < 0 ?
+				argument + length :
+				argument > length ?
+					length :
+					argument;
+			for ( ; --i >= 0; ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		} ),
+
+		"gt": createPositionalPseudo( function( matchIndexes, length, argument ) {
+			var i = argument < 0 ? argument + length : argument;
+			for ( ; ++i < length; ) {
+				matchIndexes.push( i );
+			}
+			return matchIndexes;
+		} )
+	}
+};
+
+Expr.pseudos[ "nth" ] = Expr.pseudos[ "eq" ];
+
+// Add button/input type pseudos
+for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) {
+	Expr.pseudos[ i ] = createInputPseudo( i );
+}
+for ( i in { submit: true, reset: true } ) {
+	Expr.pseudos[ i ] = createButtonPseudo( i );
+}
+
+// Easy API for creating new setFilters
+function setFilters() {}
+setFilters.prototype = Expr.filters = Expr.pseudos;
+Expr.setFilters = new setFilters();
+
+tokenize = Sizzle.tokenize = function( selector, parseOnly ) {
+	var matched, match, tokens, type,
+		soFar, groups, preFilters,
+		cached = tokenCache[ selector + " " ];
+
+	if ( cached ) {
+		return parseOnly ? 0 : cached.slice( 0 );
+	}
+
+	soFar = selector;
+	groups = [];
+	preFilters = Expr.preFilter;
+
+	while ( soFar ) {
+
+		// Comma and first run
+		if ( !matched || ( match = rcomma.exec( soFar ) ) ) {
+			if ( match ) {
+
+				// Don't consume trailing commas as valid
+				soFar = soFar.slice( match[ 0 ].length ) || soFar;
+			}
+			groups.push( ( tokens = [] ) );
+		}
+
+		matched = false;
+
+		// Combinators
+		if ( ( match = rcombinators.exec( soFar ) ) ) {
+			matched = match.shift();
+			tokens.push( {
+				value: matched,
+
+				// Cast descendant combinators to space
+				type: match[ 0 ].replace( rtrim, " " )
+			} );
+			soFar = soFar.slice( matched.length );
+		}
+
+		// Filters
+		for ( type in Expr.filter ) {
+			if ( ( match = matchExpr[ type ].exec( soFar ) ) && ( !preFilters[ type ] ||
+				( match = preFilters[ type ]( match ) ) ) ) {
+				matched = match.shift();
+				tokens.push( {
+					value: matched,
+					type: type,
+					matches: match
+				} );
+				soFar = soFar.slice( matched.length );
+			}
+		}
+
+		if ( !matched ) {
+			break;
+		}
+	}
+
+	// Return the length of the invalid excess
+	// if we're just parsing
+	// Otherwise, throw an error or return tokens
+	return parseOnly ?
+		soFar.length :
+		soFar ?
+			Sizzle.error( selector ) :
+
+			// Cache the tokens
+			tokenCache( selector, groups ).slice( 0 );
+};
+
+function toSelector( tokens ) {
+	var i = 0,
+		len = tokens.length,
+		selector = "";
+	for ( ; i < len; i++ ) {
+		selector += tokens[ i ].value;
+	}
+	return selector;
+}
+
+function addCombinator( matcher, combinator, base ) {
+	var dir = combinator.dir,
+		skip = combinator.next,
+		key = skip || dir,
+		checkNonElements = base && key === "parentNode",
+		doneName = done++;
+
+	return combinator.first ?
+
+		// Check against closest ancestor/preceding element
+		function( elem, context, xml ) {
+			while ( ( elem = elem[ dir ] ) ) {
+				if ( elem.nodeType === 1 || checkNonElements ) {
+					return matcher( elem, context, xml );
+				}
+			}
+			return false;
+		} :
+
+		// Check against all ancestor/preceding elements
+		function( elem, context, xml ) {
+			var oldCache, uniqueCache, outerCache,
+				newCache = [ dirruns, doneName ];
+
+			// We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching
+			if ( xml ) {
+				while ( ( elem = elem[ dir ] ) ) {
+					if ( elem.nodeType === 1 || checkNonElements ) {
+						if ( matcher( elem, context, xml ) ) {
+							return true;
+						}
+					}
+				}
+			} else {
+				while ( ( elem = elem[ dir ] ) ) {
+					if ( elem.nodeType === 1 || checkNonElements ) {
+						outerCache = elem[ expando ] || ( elem[ expando ] = {} );
+
+						// Support: IE <9 only
+						// Defend against cloned attroperties (jQuery gh-1709)
+						uniqueCache = outerCache[ elem.uniqueID ] ||
+							( outerCache[ elem.uniqueID ] = {} );
+
+						if ( skip && skip === elem.nodeName.toLowerCase() ) {
+							elem = elem[ dir ] || elem;
+						} else if ( ( oldCache = uniqueCache[ key ] ) &&
+							oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) {
+
+							// Assign to newCache so results back-propagate to previous elements
+							return ( newCache[ 2 ] = oldCache[ 2 ] );
+						} else {
+
+							// Reuse newcache so results back-propagate to previous elements
+							uniqueCache[ key ] = newCache;
+
+							// A match means we're done; a fail means we have to keep checking
+							if ( ( newCache[ 2 ] = matcher( elem, context, xml ) ) ) {
+								return true;
+							}
+						}
+					}
+				}
+			}
+			return false;
+		};
+}
+
+function elementMatcher( matchers ) {
+	return matchers.length > 1 ?
+		function( elem, context, xml ) {
+			var i = matchers.length;
+			while ( i-- ) {
+				if ( !matchers[ i ]( elem, context, xml ) ) {
+					return false;
+				}
+			}
+			return true;
+		} :
+		matchers[ 0 ];
+}
+
+function multipleContexts( selector, contexts, results ) {
+	var i = 0,
+		len = contexts.length;
+	for ( ; i < len; i++ ) {
+		Sizzle( selector, contexts[ i ], results );
+	}
+	return results;
+}
+
+function condense( unmatched, map, filter, context, xml ) {
+	var elem,
+		newUnmatched = [],
+		i = 0,
+		len = unmatched.length,
+		mapped = map != null;
+
+	for ( ; i < len; i++ ) {
+		if ( ( elem = unmatched[ i ] ) ) {
+			if ( !filter || filter( elem, context, xml ) ) {
+				newUnmatched.push( elem );
+				if ( mapped ) {
+					map.push( i );
+				}
+			}
+		}
+	}
+
+	return newUnmatched;
+}
+
+function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) {
+	if ( postFilter && !postFilter[ expando ] ) {
+		postFilter = setMatcher( postFilter );
+	}
+	if ( postFinder && !postFinder[ expando ] ) {
+		postFinder = setMatcher( postFinder, postSelector );
+	}
+	return markFunction( function( seed, results, context, xml ) {
+		var temp, i, elem,
+			preMap = [],
+			postMap = [],
+			preexisting = results.length,
+
+			// Get initial elements from seed or context
+			elems = seed || multipleContexts(
+				selector || "*",
+				context.nodeType ? [ context ] : context,
+				[]
+			),
+
+			// Prefilter to get matcher input, preserving a map for seed-results synchronization
+			matcherIn = preFilter && ( seed || !selector ) ?
+				condense( elems, preMap, preFilter, context, xml ) :
+				elems,
+
+			matcherOut = matcher ?
+
+				// If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results,
+				postFinder || ( seed ? preFilter : preexisting || postFilter ) ?
+
+					// ...intermediate processing is necessary
+					[] :
+
+					// ...otherwise use results directly
+					results :
+				matcherIn;
+
+		// Find primary matches
+		if ( matcher ) {
+			matcher( matcherIn, matcherOut, context, xml );
+		}
+
+		// Apply postFilter
+		if ( postFilter ) {
+			temp = condense( matcherOut, postMap );
+			postFilter( temp, [], context, xml );
+
+			// Un-match failing elements by moving them back to matcherIn
+			i = temp.length;
+			while ( i-- ) {
+				if ( ( elem = temp[ i ] ) ) {
+					matcherOut[ postMap[ i ] ] = !( matcherIn[ postMap[ i ] ] = elem );
+				}
+			}
+		}
+
+		if ( seed ) {
+			if ( postFinder || preFilter ) {
+				if ( postFinder ) {
+
+					// Get the final matcherOut by condensing this intermediate into postFinder contexts
+					temp = [];
+					i = matcherOut.length;
+					while ( i-- ) {
+						if ( ( elem = matcherOut[ i ] ) ) {
+
+							// Restore matcherIn since elem is not yet a final match
+							temp.push( ( matcherIn[ i ] = elem ) );
+						}
+					}
+					postFinder( null, ( matcherOut = [] ), temp, xml );
+				}
+
+				// Move matched elements from seed to results to keep them synchronized
+				i = matcherOut.length;
+				while ( i-- ) {
+					if ( ( elem = matcherOut[ i ] ) &&
+						( temp = postFinder ? indexOf( seed, elem ) : preMap[ i ] ) > -1 ) {
+
+						seed[ temp ] = !( results[ temp ] = elem );
+					}
+				}
+			}
+
+		// Add elements to results, through postFinder if defined
+		} else {
+			matcherOut = condense(
+				matcherOut === results ?
+					matcherOut.splice( preexisting, matcherOut.length ) :
+					matcherOut
+			);
+			if ( postFinder ) {
+				postFinder( null, results, matcherOut, xml );
+			} else {
+				push.apply( results, matcherOut );
+			}
+		}
+	} );
+}
+
+function matcherFromTokens( tokens ) {
+	var checkContext, matcher, j,
+		len = tokens.length,
+		leadingRelative = Expr.relative[ tokens[ 0 ].type ],
+		implicitRelative = leadingRelative || Expr.relative[ " " ],
+		i = leadingRelative ? 1 : 0,
+
+		// The foundational matcher ensures that elements are reachable from top-level context(s)
+		matchContext = addCombinator( function( elem ) {
+			return elem === checkContext;
+		}, implicitRelative, true ),
+		matchAnyContext = addCombinator( function( elem ) {
+			return indexOf( checkContext, elem ) > -1;
+		}, implicitRelative, true ),
+		matchers = [ function( elem, context, xml ) {
+			var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || (
+				( checkContext = context ).nodeType ?
+					matchContext( elem, context, xml ) :
+					matchAnyContext( elem, context, xml ) );
+
+			// Avoid hanging onto element (issue #299)
+			checkContext = null;
+			return ret;
+		} ];
+
+	for ( ; i < len; i++ ) {
+		if ( ( matcher = Expr.relative[ tokens[ i ].type ] ) ) {
+			matchers = [ addCombinator( elementMatcher( matchers ), matcher ) ];
+		} else {
+			matcher = Expr.filter[ tokens[ i ].type ].apply( null, tokens[ i ].matches );
+
+			// Return special upon seeing a positional matcher
+			if ( matcher[ expando ] ) {
+
+				// Find the next relative operator (if any) for proper handling
+				j = ++i;
+				for ( ; j < len; j++ ) {
+					if ( Expr.relative[ tokens[ j ].type ] ) {
+						break;
+					}
+				}
+				return setMatcher(
+					i > 1 && elementMatcher( matchers ),
+					i > 1 && toSelector(
+
+					// If the preceding token was a descendant combinator, insert an implicit any-element `*`
+					tokens
+						.slice( 0, i - 1 )
+						.concat( { value: tokens[ i - 2 ].type === " " ? "*" : "" } )
+					).replace( rtrim, "$1" ),
+					matcher,
+					i < j && matcherFromTokens( tokens.slice( i, j ) ),
+					j < len && matcherFromTokens( ( tokens = tokens.slice( j ) ) ),
+					j < len && toSelector( tokens )
+				);
+			}
+			matchers.push( matcher );
+		}
+	}
+
+	return elementMatcher( matchers );
+}
+
+function matcherFromGroupMatchers( elementMatchers, setMatchers ) {
+	var bySet = setMatchers.length > 0,
+		byElement = elementMatchers.length > 0,
+		superMatcher = function( seed, context, xml, results, outermost ) {
+			var elem, j, matcher,
+				matchedCount = 0,
+				i = "0",
+				unmatched = seed && [],
+				setMatched = [],
+				contextBackup = outermostContext,
+
+				// We must always have either seed elements or outermost context
+				elems = seed || byElement && Expr.find[ "TAG" ]( "*", outermost ),
+
+				// Use integer dirruns iff this is the outermost matcher
+				dirrunsUnique = ( dirruns += contextBackup == null ? 1 : Math.random() || 0.1 ),
+				len = elems.length;
+
+			if ( outermost ) {
+
+				// Support: IE 11+, Edge 17 - 18+
+				// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+				// two documents; shallow comparisons work.
+				// eslint-disable-next-line eqeqeq
+				outermostContext = context == document || context || outermost;
+			}
+
+			// Add elements passing elementMatchers directly to results
+			// Support: IE<9, Safari
+			// Tolerate NodeList properties (IE: "length"; Safari: <number>) matching elements by id
+			for ( ; i !== len && ( elem = elems[ i ] ) != null; i++ ) {
+				if ( byElement && elem ) {
+					j = 0;
+
+					// Support: IE 11+, Edge 17 - 18+
+					// IE/Edge sometimes throw a "Permission denied" error when strict-comparing
+					// two documents; shallow comparisons work.
+					// eslint-disable-next-line eqeqeq
+					if ( !context && elem.ownerDocument != document ) {
+						setDocument( elem );
+						xml = !documentIsHTML;
+					}
+					while ( ( matcher = elementMatchers[ j++ ] ) ) {
+						if ( matcher( elem, context || document, xml ) ) {
+							results.push( elem );
+							break;
+						}
+					}
+					if ( outermost ) {
+						dirruns = dirrunsUnique;
+					}
+				}
+
+				// Track unmatched elements for set filters
+				if ( bySet ) {
+
+					// They will have gone through all possible matchers
+					if ( ( elem = !matcher && elem ) ) {
+						matchedCount--;
+					}
+
+					// Lengthen the array for every element, matched or not
+					if ( seed ) {
+						unmatched.push( elem );
+					}
+				}
+			}
+
+			// `i` is now the count of elements visited above, and adding it to `matchedCount`
+			// makes the latter nonnegative.
+			matchedCount += i;
+
+			// Apply set filters to unmatched elements
+			// NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount`
+			// equals `i`), unless we didn't visit _any_ elements in the above loop because we have
+			// no element matchers and no seed.
+			// Incrementing an initially-string "0" `i` allows `i` to remain a string only in that
+			// case, which will result in a "00" `matchedCount` that differs from `i` but is also
+			// numerically zero.
+			if ( bySet && i !== matchedCount ) {
+				j = 0;
+				while ( ( matcher = setMatchers[ j++ ] ) ) {
+					matcher( unmatched, setMatched, context, xml );
+				}
+
+				if ( seed ) {
+
+					// Reintegrate element matches to eliminate the need for sorting
+					if ( matchedCount > 0 ) {
+						while ( i-- ) {
+							if ( !( unmatched[ i ] || setMatched[ i ] ) ) {
+								setMatched[ i ] = pop.call( results );
+							}
+						}
+					}
+
+					// Discard index placeholder values to get only actual matches
+					setMatched = condense( setMatched );
+				}
+
+				// Add matches to results
+				push.apply( results, setMatched );
+
+				// Seedless set matches succeeding multiple successful matchers stipulate sorting
+				if ( outermost && !seed && setMatched.length > 0 &&
+					( matchedCount + setMatchers.length ) > 1 ) {
+
+					Sizzle.uniqueSort( results );
+				}
+			}
+
+			// Override manipulation of globals by nested matchers
+			if ( outermost ) {
+				dirruns = dirrunsUnique;
+				outermostContext = contextBackup;
+			}
+
+			return unmatched;
+		};
+
+	return bySet ?
+		markFunction( superMatcher ) :
+		superMatcher;
+}
+
+compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) {
+	var i,
+		setMatchers = [],
+		elementMatchers = [],
+		cached = compilerCache[ selector + " " ];
+
+	if ( !cached ) {
+
+		// Generate a function of recursive functions that can be used to check each element
+		if ( !match ) {
+			match = tokenize( selector );
+		}
+		i = match.length;
+		while ( i-- ) {
+			cached = matcherFromTokens( match[ i ] );
+			if ( cached[ expando ] ) {
+				setMatchers.push( cached );
+			} else {
+				elementMatchers.push( cached );
+			}
+		}
+
+		// Cache the compiled function
+		cached = compilerCache(
+			selector,
+			matcherFromGroupMatchers( elementMatchers, setMatchers )
+		);
+
+		// Save selector and tokenization
+		cached.selector = selector;
+	}
+	return cached;
+};
+
+/**
+ * A low-level selection function that works with Sizzle's compiled
+ *  selector functions
+ * @param {String|Function} selector A selector or a pre-compiled
+ *  selector function built with Sizzle.compile
+ * @param {Element} context
+ * @param {Array} [results]
+ * @param {Array} [seed] A set of elements to match against
+ */
+select = Sizzle.select = function( selector, context, results, seed ) {
+	var i, tokens, token, type, find,
+		compiled = typeof selector === "function" && selector,
+		match = !seed && tokenize( ( selector = compiled.selector || selector ) );
+
+	results = results || [];
+
+	// Try to minimize operations if there is only one selector in the list and no seed
+	// (the latter of which guarantees us context)
+	if ( match.length === 1 ) {
+
+		// Reduce context if the leading compound selector is an ID
+		tokens = match[ 0 ] = match[ 0 ].slice( 0 );
+		if ( tokens.length > 2 && ( token = tokens[ 0 ] ).type === "ID" &&
+			context.nodeType === 9 && documentIsHTML && Expr.relative[ tokens[ 1 ].type ] ) {
+
+			context = ( Expr.find[ "ID" ]( token.matches[ 0 ]
+				.replace( runescape, funescape ), context ) || [] )[ 0 ];
+			if ( !context ) {
+				return results;
+
+			// Precompiled matchers will still verify ancestry, so step up a level
+			} else if ( compiled ) {
+				context = context.parentNode;
+			}
+
+			selector = selector.slice( tokens.shift().value.length );
+		}
+
+		// Fetch a seed set for right-to-left matching
+		i = matchExpr[ "needsContext" ].test( selector ) ? 0 : tokens.length;
+		while ( i-- ) {
+			token = tokens[ i ];
+
+			// Abort if we hit a combinator
+			if ( Expr.relative[ ( type = token.type ) ] ) {
+				break;
+			}
+			if ( ( find = Expr.find[ type ] ) ) {
+
+				// Search, expanding context for leading sibling combinators
+				if ( ( seed = find(
+					token.matches[ 0 ].replace( runescape, funescape ),
+					rsibling.test( tokens[ 0 ].type ) && testContext( context.parentNode ) ||
+						context
+				) ) ) {
+
+					// If seed is empty or no tokens remain, we can return early
+					tokens.splice( i, 1 );
+					selector = seed.length && toSelector( tokens );
+					if ( !selector ) {
+						push.apply( results, seed );
+						return results;
+					}
+
+					break;
+				}
+			}
+		}
+	}
+
+	// Compile and execute a filtering function if one is not provided
+	// Provide `match` to avoid retokenization if we modified the selector above
+	( compiled || compile( selector, match ) )(
+		seed,
+		context,
+		!documentIsHTML,
+		results,
+		!context || rsibling.test( selector ) && testContext( context.parentNode ) || context
+	);
+	return results;
+};
+
+// One-time assignments
+
+// Sort stability
+support.sortStable = expando.split( "" ).sort( sortOrder ).join( "" ) === expando;
+
+// Support: Chrome 14-35+
+// Always assume duplicates if they aren't passed to the comparison function
+support.detectDuplicates = !!hasDuplicate;
+
+// Initialize against the default document
+setDocument();
+
+// Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27)
+// Detached nodes confoundingly follow *each other*
+support.sortDetached = assert( function( el ) {
+
+	// Should return 1, but returns 4 (following)
+	return el.compareDocumentPosition( document.createElement( "fieldset" ) ) & 1;
+} );
+
+// Support: IE<8
+// Prevent attribute/property "interpolation"
+// https://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx
+if ( !assert( function( el ) {
+	el.innerHTML = "<a href='#'></a>";
+	return el.firstChild.getAttribute( "href" ) === "#";
+} ) ) {
+	addHandle( "type|href|height|width", function( elem, name, isXML ) {
+		if ( !isXML ) {
+			return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 );
+		}
+	} );
+}
+
+// Support: IE<9
+// Use defaultValue in place of getAttribute("value")
+if ( !support.attributes || !assert( function( el ) {
+	el.innerHTML = "<input/>";
+	el.firstChild.setAttribute( "value", "" );
+	return el.firstChild.getAttribute( "value" ) === "";
+} ) ) {
+	addHandle( "value", function( elem, _name, isXML ) {
+		if ( !isXML && elem.nodeName.toLowerCase() === "input" ) {
+			return elem.defaultValue;
+		}
+	} );
+}
+
+// Support: IE<9
+// Use getAttributeNode to fetch booleans when getAttribute lies
+if ( !assert( function( el ) {
+	return el.getAttribute( "disabled" ) == null;
+} ) ) {
+	addHandle( booleans, function( elem, name, isXML ) {
+		var val;
+		if ( !isXML ) {
+			return elem[ name ] === true ? name.toLowerCase() :
+				( val = elem.getAttributeNode( name ) ) && val.specified ?
+					val.value :
+					null;
+		}
+	} );
+}
+
+return Sizzle;
+
+} )( window );
+
+
+
+jQuery.find = Sizzle;
+jQuery.expr = Sizzle.selectors;
+
+// Deprecated
+jQuery.expr[ ":" ] = jQuery.expr.pseudos;
+jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort;
+jQuery.text = Sizzle.getText;
+jQuery.isXMLDoc = Sizzle.isXML;
+jQuery.contains = Sizzle.contains;
+jQuery.escapeSelector = Sizzle.escape;
+
+
+
+
+var dir = function( elem, dir, until ) {
+	var matched = [],
+		truncate = until !== undefined;
+
+	while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) {
+		if ( elem.nodeType === 1 ) {
+			if ( truncate && jQuery( elem ).is( until ) ) {
+				break;
+			}
+			matched.push( elem );
+		}
+	}
+	return matched;
+};
+
+
+var siblings = function( n, elem ) {
+	var matched = [];
+
+	for ( ; n; n = n.nextSibling ) {
+		if ( n.nodeType === 1 && n !== elem ) {
+			matched.push( n );
+		}
+	}
+
+	return matched;
+};
+
+
+var rneedsContext = jQuery.expr.match.needsContext;
+
+
+
+function nodeName( elem, name ) {
+
+  return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase();
+
+};
+var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i );
+
+
+
+// Implement the identical functionality for filter and not
+function winnow( elements, qualifier, not ) {
+	if ( isFunction( qualifier ) ) {
+		return jQuery.grep( elements, function( elem, i ) {
+			return !!qualifier.call( elem, i, elem ) !== not;
+		} );
+	}
+
+	// Single element
+	if ( qualifier.nodeType ) {
+		return jQuery.grep( elements, function( elem ) {
+			return ( elem === qualifier ) !== not;
+		} );
+	}
+
+	// Arraylike of elements (jQuery, arguments, Array)
+	if ( typeof qualifier !== "string" ) {
+		return jQuery.grep( elements, function( elem ) {
+			return ( indexOf.call( qualifier, elem ) > -1 ) !== not;
+		} );
+	}
+
+	// Filtered directly for both simple and complex selectors
+	return jQuery.filter( qualifier, elements, not );
+}
+
+jQuery.filter = function( expr, elems, not ) {
+	var elem = elems[ 0 ];
+
+	if ( not ) {
+		expr = ":not(" + expr + ")";
+	}
+
+	if ( elems.length === 1 && elem.nodeType === 1 ) {
+		return jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : [];
+	}
+
+	return jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) {
+		return elem.nodeType === 1;
+	} ) );
+};
+
+jQuery.fn.extend( {
+	find: function( selector ) {
+		var i, ret,
+			len = this.length,
+			self = this;
+
+		if ( typeof selector !== "string" ) {
+			return this.pushStack( jQuery( selector ).filter( function() {
+				for ( i = 0; i < len; i++ ) {
+					if ( jQuery.contains( self[ i ], this ) ) {
+						return true;
+					}
+				}
+			} ) );
+		}
+
+		ret = this.pushStack( [] );
+
+		for ( i = 0; i < len; i++ ) {
+			jQuery.find( selector, self[ i ], ret );
+		}
+
+		return len > 1 ? jQuery.uniqueSort( ret ) : ret;
+	},
+	filter: function( selector ) {
+		return this.pushStack( winnow( this, selector || [], false ) );
+	},
+	not: function( selector ) {
+		return this.pushStack( winnow( this, selector || [], true ) );
+	},
+	is: function( selector ) {
+		return !!winnow(
+			this,
+
+			// If this is a positional/relative selector, check membership in the returned set
+			// so $("p:first").is("p:last") won't return true for a doc with two "p".
+			typeof selector === "string" && rneedsContext.test( selector ) ?
+				jQuery( selector ) :
+				selector || [],
+			false
+		).length;
+	}
+} );
+
+
+// Initialize a jQuery object
+
+
+// A central reference to the root jQuery(document)
+var rootjQuery,
+
+	// A simple way to check for HTML strings
+	// Prioritize #id over <tag> to avoid XSS via location.hash (#9521)
+	// Strict HTML recognition (#11290: must start with <)
+	// Shortcut simple #id case for speed
+	rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/,
+
+	init = jQuery.fn.init = function( selector, context, root ) {
+		var match, elem;
+
+		// HANDLE: $(""), $(null), $(undefined), $(false)
+		if ( !selector ) {
+			return this;
+		}
+
+		// Method init() accepts an alternate rootjQuery
+		// so migrate can support jQuery.sub (gh-2101)
+		root = root || rootjQuery;
+
+		// Handle HTML strings
+		if ( typeof selector === "string" ) {
+			if ( selector[ 0 ] === "<" &&
+				selector[ selector.length - 1 ] === ">" &&
+				selector.length >= 3 ) {
+
+				// Assume that strings that start and end with <> are HTML and skip the regex check
+				match = [ null, selector, null ];
+
+			} else {
+				match = rquickExpr.exec( selector );
+			}
+
+			// Match html or make sure no context is specified for #id
+			if ( match && ( match[ 1 ] || !context ) ) {
+
+				// HANDLE: $(html) -> $(array)
+				if ( match[ 1 ] ) {
+					context = context instanceof jQuery ? context[ 0 ] : context;
+
+					// Option to run scripts is true for back-compat
+					// Intentionally let the error be thrown if parseHTML is not present
+					jQuery.merge( this, jQuery.parseHTML(
+						match[ 1 ],
+						context && context.nodeType ? context.ownerDocument || context : document,
+						true
+					) );
+
+					// HANDLE: $(html, props)
+					if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) {
+						for ( match in context ) {
+
+							// Properties of context are called as methods if possible
+							if ( isFunction( this[ match ] ) ) {
+								this[ match ]( context[ match ] );
+
+							// ...and otherwise set as attributes
+							} else {
+								this.attr( match, context[ match ] );
+							}
+						}
+					}
+
+					return this;
+
+				// HANDLE: $(#id)
+				} else {
+					elem = document.getElementById( match[ 2 ] );
+
+					if ( elem ) {
+
+						// Inject the element directly into the jQuery object
+						this[ 0 ] = elem;
+						this.length = 1;
+					}
+					return this;
+				}
+
+			// HANDLE: $(expr, $(...))
+			} else if ( !context || context.jquery ) {
+				return ( context || root ).find( selector );
+
+			// HANDLE: $(expr, context)
+			// (which is just equivalent to: $(context).find(expr)
+			} else {
+				return this.constructor( context ).find( selector );
+			}
+
+		// HANDLE: $(DOMElement)
+		} else if ( selector.nodeType ) {
+			this[ 0 ] = selector;
+			this.length = 1;
+			return this;
+
+		// HANDLE: $(function)
+		// Shortcut for document ready
+		} else if ( isFunction( selector ) ) {
+			return root.ready !== undefined ?
+				root.ready( selector ) :
+
+				// Execute immediately if ready is not present
+				selector( jQuery );
+		}
+
+		return jQuery.makeArray( selector, this );
+	};
+
+// Give the init function the jQuery prototype for later instantiation
+init.prototype = jQuery.fn;
+
+// Initialize central reference
+rootjQuery = jQuery( document );
+
+
+var rparentsprev = /^(?:parents|prev(?:Until|All))/,
+
+	// Methods guaranteed to produce a unique set when starting from a unique set
+	guaranteedUnique = {
+		children: true,
+		contents: true,
+		next: true,
+		prev: true
+	};
+
+jQuery.fn.extend( {
+	has: function( target ) {
+		var targets = jQuery( target, this ),
+			l = targets.length;
+
+		return this.filter( function() {
+			var i = 0;
+			for ( ; i < l; i++ ) {
+				if ( jQuery.contains( this, targets[ i ] ) ) {
+					return true;
+				}
+			}
+		} );
+	},
+
+	closest: function( selectors, context ) {
+		var cur,
+			i = 0,
+			l = this.length,
+			matched = [],
+			targets = typeof selectors !== "string" && jQuery( selectors );
+
+		// Positional selectors never match, since there's no _selection_ context
+		if ( !rneedsContext.test( selectors ) ) {
+			for ( ; i < l; i++ ) {
+				for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) {
+
+					// Always skip document fragments
+					if ( cur.nodeType < 11 && ( targets ?
+						targets.index( cur ) > -1 :
+
+						// Don't pass non-elements to Sizzle
+						cur.nodeType === 1 &&
+							jQuery.find.matchesSelector( cur, selectors ) ) ) {
+
+						matched.push( cur );
+						break;
+					}
+				}
+			}
+		}
+
+		return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched );
+	},
+
+	// Determine the position of an element within the set
+	index: function( elem ) {
+
+		// No argument, return index in parent
+		if ( !elem ) {
+			return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1;
+		}
+
+		// Index in selector
+		if ( typeof elem === "string" ) {
+			return indexOf.call( jQuery( elem ), this[ 0 ] );
+		}
+
+		// Locate the position of the desired element
+		return indexOf.call( this,
+
+			// If it receives a jQuery object, the first element is used
+			elem.jquery ? elem[ 0 ] : elem
+		);
+	},
+
+	add: function( selector, context ) {
+		return this.pushStack(
+			jQuery.uniqueSort(
+				jQuery.merge( this.get(), jQuery( selector, context ) )
+			)
+		);
+	},
+
+	addBack: function( selector ) {
+		return this.add( selector == null ?
+			this.prevObject : this.prevObject.filter( selector )
+		);
+	}
+} );
+
+function sibling( cur, dir ) {
+	while ( ( cur = cur[ dir ] ) && cur.nodeType !== 1 ) {}
+	return cur;
+}
+
+jQuery.each( {
+	parent: function( elem ) {
+		var parent = elem.parentNode;
+		return parent && parent.nodeType !== 11 ? parent : null;
+	},
+	parents: function( elem ) {
+		return dir( elem, "parentNode" );
+	},
+	parentsUntil: function( elem, _i, until ) {
+		return dir( elem, "parentNode", until );
+	},
+	next: function( elem ) {
+		return sibling( elem, "nextSibling" );
+	},
+	prev: function( elem ) {
+		return sibling( elem, "previousSibling" );
+	},
+	nextAll: function( elem ) {
+		return dir( elem, "nextSibling" );
+	},
+	prevAll: function( elem ) {
+		return dir( elem, "previousSibling" );
+	},
+	nextUntil: function( elem, _i, until ) {
+		return dir( elem, "nextSibling", until );
+	},
+	prevUntil: function( elem, _i, until ) {
+		return dir( elem, "previousSibling", until );
+	},
+	siblings: function( elem ) {
+		return siblings( ( elem.parentNode || {} ).firstChild, elem );
+	},
+	children: function( elem ) {
+		return siblings( elem.firstChild );
+	},
+	contents: function( elem ) {
+		if ( elem.contentDocument != null &&
+
+			// Support: IE 11+
+			// <object> elements with no `data` attribute has an object
+			// `contentDocument` with a `null` prototype.
+			getProto( elem.contentDocument ) ) {
+
+			return elem.contentDocument;
+		}
+
+		// Support: IE 9 - 11 only, iOS 7 only, Android Browser <=4.3 only
+		// Treat the template element as a regular one in browsers that
+		// don't support it.
+		if ( nodeName( elem, "template" ) ) {
+			elem = elem.content || elem;
+		}
+
+		return jQuery.merge( [], elem.childNodes );
+	}
+}, function( name, fn ) {
+	jQuery.fn[ name ] = function( until, selector ) {
+		var matched = jQuery.map( this, fn, until );
+
+		if ( name.slice( -5 ) !== "Until" ) {
+			selector = until;
+		}
+
+		if ( selector && typeof selector === "string" ) {
+			matched = jQuery.filter( selector, matched );
+		}
+
+		if ( this.length > 1 ) {
+
+			// Remove duplicates
+			if ( !guaranteedUnique[ name ] ) {
+				jQuery.uniqueSort( matched );
+			}
+
+			// Reverse order for parents* and prev-derivatives
+			if ( rparentsprev.test( name ) ) {
+				matched.reverse();
+			}
+		}
+
+		return this.pushStack( matched );
+	};
+} );
+var rnothtmlwhite = ( /[^\x20\t\r\n\f]+/g );
+
+
+
+// Convert String-formatted options into Object-formatted ones
+function createOptions( options ) {
+	var object = {};
+	jQuery.each( options.match( rnothtmlwhite ) || [], function( _, flag ) {
+		object[ flag ] = true;
+	} );
+	return object;
+}
+
+/*
+ * Create a callback list using the following parameters:
+ *
+ *	options: an optional list of space-separated options that will change how
+ *			the callback list behaves or a more traditional option object
+ *
+ * By default a callback list will act like an event callback list and can be
+ * "fired" multiple times.
+ *
+ * Possible options:
+ *
+ *	once:			will ensure the callback list can only be fired once (like a Deferred)
+ *
+ *	memory:			will keep track of previous values and will call any callback added
+ *					after the list has been fired right away with the latest "memorized"
+ *					values (like a Deferred)
+ *
+ *	unique:			will ensure a callback can only be added once (no duplicate in the list)
+ *
+ *	stopOnFalse:	interrupt callings when a callback returns false
+ *
+ */
+jQuery.Callbacks = function( options ) {
+
+	// Convert options from String-formatted to Object-formatted if needed
+	// (we check in cache first)
+	options = typeof options === "string" ?
+		createOptions( options ) :
+		jQuery.extend( {}, options );
+
+	var // Flag to know if list is currently firing
+		firing,
+
+		// Last fire value for non-forgettable lists
+		memory,
+
+		// Flag to know if list was already fired
+		fired,
+
+		// Flag to prevent firing
+		locked,
+
+		// Actual callback list
+		list = [],
+
+		// Queue of execution data for repeatable lists
+		queue = [],
+
+		// Index of currently firing callback (modified by add/remove as needed)
+		firingIndex = -1,
+
+		// Fire callbacks
+		fire = function() {
+
+			// Enforce single-firing
+			locked = locked || options.once;
+
+			// Execute callbacks for all pending executions,
+			// respecting firingIndex overrides and runtime changes
+			fired = firing = true;
+			for ( ; queue.length; firingIndex = -1 ) {
+				memory = queue.shift();
+				while ( ++firingIndex < list.length ) {
+
+					// Run callback and check for early termination
+					if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false &&
+						options.stopOnFalse ) {
+
+						// Jump to end and forget the data so .add doesn't re-fire
+						firingIndex = list.length;
+						memory = false;
+					}
+				}
+			}
+
+			// Forget the data if we're done with it
+			if ( !options.memory ) {
+				memory = false;
+			}
+
+			firing = false;
+
+			// Clean up if we're done firing for good
+			if ( locked ) {
+
+				// Keep an empty list if we have data for future add calls
+				if ( memory ) {
+					list = [];
+
+				// Otherwise, this object is spent
+				} else {
+					list = "";
+				}
+			}
+		},
+
+		// Actual Callbacks object
+		self = {
+
+			// Add a callback or a collection of callbacks to the list
+			add: function() {
+				if ( list ) {
+
+					// If we have memory from a past run, we should fire after adding
+					if ( memory && !firing ) {
+						firingIndex = list.length - 1;
+						queue.push( memory );
+					}
+
+					( function add( args ) {
+						jQuery.each( args, function( _, arg ) {
+							if ( isFunction( arg ) ) {
+								if ( !options.unique || !self.has( arg ) ) {
+									list.push( arg );
+								}
+							} else if ( arg && arg.length && toType( arg ) !== "string" ) {
+
+								// Inspect recursively
+								add( arg );
+							}
+						} );
+					} )( arguments );
+
+					if ( memory && !firing ) {
+						fire();
+					}
+				}
+				return this;
+			},
+
+			// Remove a callback from the list
+			remove: function() {
+				jQuery.each( arguments, function( _, arg ) {
+					var index;
+					while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) {
+						list.splice( index, 1 );
+
+						// Handle firing indexes
+						if ( index <= firingIndex ) {
+							firingIndex--;
+						}
+					}
+				} );
+				return this;
+			},
+
+			// Check if a given callback is in the list.
+			// If no argument is given, return whether or not list has callbacks attached.
+			has: function( fn ) {
+				return fn ?
+					jQuery.inArray( fn, list ) > -1 :
+					list.length > 0;
+			},
+
+			// Remove all callbacks from the list
+			empty: function() {
+				if ( list ) {
+					list = [];
+				}
+				return this;
+			},
+
+			// Disable .fire and .add
+			// Abort any current/pending executions
+			// Clear all callbacks and values
+			disable: function() {
+				locked = queue = [];
+				list = memory = "";
+				return this;
+			},
+			disabled: function() {
+				return !list;
+			},
+
+			// Disable .fire
+			// Also disable .add unless we have memory (since it would have no effect)
+			// Abort any pending executions
+			lock: function() {
+				locked = queue = [];
+				if ( !memory && !firing ) {
+					list = memory = "";
+				}
+				return this;
+			},
+			locked: function() {
+				return !!locked;
+			},
+
+			// Call all callbacks with the given context and arguments
+			fireWith: function( context, args ) {
+				if ( !locked ) {
+					args = args || [];
+					args = [ context, args.slice ? args.slice() : args ];
+					queue.push( args );
+					if ( !firing ) {
+						fire();
+					}
+				}
+				return this;
+			},
+
+			// Call all the callbacks with the given arguments
+			fire: function() {
+				self.fireWith( this, arguments );
+				return this;
+			},
+
+			// To know if the callbacks have already been called at least once
+			fired: function() {
+				return !!fired;
+			}
+		};
+
+	return self;
+};
+
+
+function Identity( v ) {
+	return v;
+}
+function Thrower( ex ) {
+	throw ex;
+}
+
+function adoptValue( value, resolve, reject, noValue ) {
+	var method;
+
+	try {
+
+		// Check for promise aspect first to privilege synchronous behavior
+		if ( value && isFunction( ( method = value.promise ) ) ) {
+			method.call( value ).done( resolve ).fail( reject );
+
+		// Other thenables
+		} else if ( value && isFunction( ( method = value.then ) ) ) {
+			method.call( value, resolve, reject );
+
+		// Other non-thenables
+		} else {
+
+			// Control `resolve` arguments by letting Array#slice cast boolean `noValue` to integer:
+			// * false: [ value ].slice( 0 ) => resolve( value )
+			// * true: [ value ].slice( 1 ) => resolve()
+			resolve.apply( undefined, [ value ].slice( noValue ) );
+		}
+
+	// For Promises/A+, convert exceptions into rejections
+	// Since jQuery.when doesn't unwrap thenables, we can skip the extra checks appearing in
+	// Deferred#then to conditionally suppress rejection.
+	} catch ( value ) {
+
+		// Support: Android 4.0 only
+		// Strict mode functions invoked without .call/.apply get global-object context
+		reject.apply( undefined, [ value ] );
+	}
+}
+
+jQuery.extend( {
+
+	Deferred: function( func ) {
+		var tuples = [
+
+				// action, add listener, callbacks,
+				// ... .then handlers, argument index, [final state]
+				[ "notify", "progress", jQuery.Callbacks( "memory" ),
+					jQuery.Callbacks( "memory" ), 2 ],
+				[ "resolve", "done", jQuery.Callbacks( "once memory" ),
+					jQuery.Callbacks( "once memory" ), 0, "resolved" ],
+				[ "reject", "fail", jQuery.Callbacks( "once memory" ),
+					jQuery.Callbacks( "once memory" ), 1, "rejected" ]
+			],
+			state = "pending",
+			promise = {
+				state: function() {
+					return state;
+				},
+				always: function() {
+					deferred.done( arguments ).fail( arguments );
+					return this;
+				},
+				"catch": function( fn ) {
+					return promise.then( null, fn );
+				},
+
+				// Keep pipe for back-compat
+				pipe: function( /* fnDone, fnFail, fnProgress */ ) {
+					var fns = arguments;
+
+					return jQuery.Deferred( function( newDefer ) {
+						jQuery.each( tuples, function( _i, tuple ) {
+
+							// Map tuples (progress, done, fail) to arguments (done, fail, progress)
+							var fn = isFunction( fns[ tuple[ 4 ] ] ) && fns[ tuple[ 4 ] ];
+
+							// deferred.progress(function() { bind to newDefer or newDefer.notify })
+							// deferred.done(function() { bind to newDefer or newDefer.resolve })
+							// deferred.fail(function() { bind to newDefer or newDefer.reject })
+							deferred[ tuple[ 1 ] ]( function() {
+								var returned = fn && fn.apply( this, arguments );
+								if ( returned && isFunction( returned.promise ) ) {
+									returned.promise()
+										.progress( newDefer.notify )
+										.done( newDefer.resolve )
+										.fail( newDefer.reject );
+								} else {
+									newDefer[ tuple[ 0 ] + "With" ](
+										this,
+										fn ? [ returned ] : arguments
+									);
+								}
+							} );
+						} );
+						fns = null;
+					} ).promise();
+				},
+				then: function( onFulfilled, onRejected, onProgress ) {
+					var maxDepth = 0;
+					function resolve( depth, deferred, handler, special ) {
+						return function() {
+							var that = this,
+								args = arguments,
+								mightThrow = function() {
+									var returned, then;
+
+									// Support: Promises/A+ section 2.3.3.3.3
+									// https://promisesaplus.com/#point-59
+									// Ignore double-resolution attempts
+									if ( depth < maxDepth ) {
+										return;
+									}
+
+									returned = handler.apply( that, args );
+
+									// Support: Promises/A+ section 2.3.1
+									// https://promisesaplus.com/#point-48
+									if ( returned === deferred.promise() ) {
+										throw new TypeError( "Thenable self-resolution" );
+									}
+
+									// Support: Promises/A+ sections 2.3.3.1, 3.5
+									// https://promisesaplus.com/#point-54
+									// https://promisesaplus.com/#point-75
+									// Retrieve `then` only once
+									then = returned &&
+
+										// Support: Promises/A+ section 2.3.4
+										// https://promisesaplus.com/#point-64
+										// Only check objects and functions for thenability
+										( typeof returned === "object" ||
+											typeof returned === "function" ) &&
+										returned.then;
+
+									// Handle a returned thenable
+									if ( isFunction( then ) ) {
+
+										// Special processors (notify) just wait for resolution
+										if ( special ) {
+											then.call(
+												returned,
+												resolve( maxDepth, deferred, Identity, special ),
+												resolve( maxDepth, deferred, Thrower, special )
+											);
+
+										// Normal processors (resolve) also hook into progress
+										} else {
+
+											// ...and disregard older resolution values
+											maxDepth++;
+
+											then.call(
+												returned,
+												resolve( maxDepth, deferred, Identity, special ),
+												resolve( maxDepth, deferred, Thrower, special ),
+												resolve( maxDepth, deferred, Identity,
+													deferred.notifyWith )
+											);
+										}
+
+									// Handle all other returned values
+									} else {
+
+										// Only substitute handlers pass on context
+										// and multiple values (non-spec behavior)
+										if ( handler !== Identity ) {
+											that = undefined;
+											args = [ returned ];
+										}
+
+										// Process the value(s)
+										// Default process is resolve
+										( special || deferred.resolveWith )( that, args );
+									}
+								},
+
+								// Only normal processors (resolve) catch and reject exceptions
+								process = special ?
+									mightThrow :
+									function() {
+										try {
+											mightThrow();
+										} catch ( e ) {
+
+											if ( jQuery.Deferred.exceptionHook ) {
+												jQuery.Deferred.exceptionHook( e,
+													process.stackTrace );
+											}
+
+											// Support: Promises/A+ section 2.3.3.3.4.1
+											// https://promisesaplus.com/#point-61
+											// Ignore post-resolution exceptions
+											if ( depth + 1 >= maxDepth ) {
+
+												// Only substitute handlers pass on context
+												// and multiple values (non-spec behavior)
+												if ( handler !== Thrower ) {
+													that = undefined;
+													args = [ e ];
+												}
+
+												deferred.rejectWith( that, args );
+											}
+										}
+									};
+
+							// Support: Promises/A+ section 2.3.3.3.1
+							// https://promisesaplus.com/#point-57
+							// Re-resolve promises immediately to dodge false rejection from
+							// subsequent errors
+							if ( depth ) {
+								process();
+							} else {
+
+								// Call an optional hook to record the stack, in case of exception
+								// since it's otherwise lost when execution goes async
+								if ( jQuery.Deferred.getStackHook ) {
+									process.stackTrace = jQuery.Deferred.getStackHook();
+								}
+								window.setTimeout( process );
+							}
+						};
+					}
+
+					return jQuery.Deferred( function( newDefer ) {
+
+						// progress_handlers.add( ... )
+						tuples[ 0 ][ 3 ].add(
+							resolve(
+								0,
+								newDefer,
+								isFunction( onProgress ) ?
+									onProgress :
+									Identity,
+								newDefer.notifyWith
+							)
+						);
+
+						// fulfilled_handlers.add( ... )
+						tuples[ 1 ][ 3 ].add(
+							resolve(
+								0,
+								newDefer,
+								isFunction( onFulfilled ) ?
+									onFulfilled :
+									Identity
+							)
+						);
+
+						// rejected_handlers.add( ... )
+						tuples[ 2 ][ 3 ].add(
+							resolve(
+								0,
+								newDefer,
+								isFunction( onRejected ) ?
+									onRejected :
+									Thrower
+							)
+						);
+					} ).promise();
+				},
+
+				// Get a promise for this deferred
+				// If obj is provided, the promise aspect is added to the object
+				promise: function( obj ) {
+					return obj != null ? jQuery.extend( obj, promise ) : promise;
+				}
+			},
+			deferred = {};
+
+		// Add list-specific methods
+		jQuery.each( tuples, function( i, tuple ) {
+			var list = tuple[ 2 ],
+				stateString = tuple[ 5 ];
+
+			// promise.progress = list.add
+			// promise.done = list.add
+			// promise.fail = list.add
+			promise[ tuple[ 1 ] ] = list.add;
+
+			// Handle state
+			if ( stateString ) {
+				list.add(
+					function() {
+
+						// state = "resolved" (i.e., fulfilled)
+						// state = "rejected"
+						state = stateString;
+					},
+
+					// rejected_callbacks.disable
+					// fulfilled_callbacks.disable
+					tuples[ 3 - i ][ 2 ].disable,
+
+					// rejected_handlers.disable
+					// fulfilled_handlers.disable
+					tuples[ 3 - i ][ 3 ].disable,
+
+					// progress_callbacks.lock
+					tuples[ 0 ][ 2 ].lock,
+
+					// progress_handlers.lock
+					tuples[ 0 ][ 3 ].lock
+				);
+			}
+
+			// progress_handlers.fire
+			// fulfilled_handlers.fire
+			// rejected_handlers.fire
+			list.add( tuple[ 3 ].fire );
+
+			// deferred.notify = function() { deferred.notifyWith(...) }
+			// deferred.resolve = function() { deferred.resolveWith(...) }
+			// deferred.reject = function() { deferred.rejectWith(...) }
+			deferred[ tuple[ 0 ] ] = function() {
+				deferred[ tuple[ 0 ] + "With" ]( this === deferred ? undefined : this, arguments );
+				return this;
+			};
+
+			// deferred.notifyWith = list.fireWith
+			// deferred.resolveWith = list.fireWith
+			// deferred.rejectWith = list.fireWith
+			deferred[ tuple[ 0 ] + "With" ] = list.fireWith;
+		} );
+
+		// Make the deferred a promise
+		promise.promise( deferred );
+
+		// Call given func if any
+		if ( func ) {
+			func.call( deferred, deferred );
+		}
+
+		// All done!
+		return deferred;
+	},
+
+	// Deferred helper
+	when: function( singleValue ) {
+		var
+
+			// count of uncompleted subordinates
+			remaining = arguments.length,
+
+			// count of unprocessed arguments
+			i = remaining,
+
+			// subordinate fulfillment data
+			resolveContexts = Array( i ),
+			resolveValues = slice.call( arguments ),
+
+			// the master Deferred
+			master = jQuery.Deferred(),
+
+			// subordinate callback factory
+			updateFunc = function( i ) {
+				return function( value ) {
+					resolveContexts[ i ] = this;
+					resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value;
+					if ( !( --remaining ) ) {
+						master.resolveWith( resolveContexts, resolveValues );
+					}
+				};
+			};
+
+		// Single- and empty arguments are adopted like Promise.resolve
+		if ( remaining <= 1 ) {
+			adoptValue( singleValue, master.done( updateFunc( i ) ).resolve, master.reject,
+				!remaining );
+
+			// Use .then() to unwrap secondary thenables (cf. gh-3000)
+			if ( master.state() === "pending" ||
+				isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) {
+
+				return master.then();
+			}
+		}
+
+		// Multiple arguments are aggregated like Promise.all array elements
+		while ( i-- ) {
+			adoptValue( resolveValues[ i ], updateFunc( i ), master.reject );
+		}
+
+		return master.promise();
+	}
+} );
+
+
+// These usually indicate a programmer mistake during development,
+// warn about them ASAP rather than swallowing them by default.
+var rerrorNames = /^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/;
+
+jQuery.Deferred.exceptionHook = function( error, stack ) {
+
+	// Support: IE 8 - 9 only
+	// Console exists when dev tools are open, which can happen at any time
+	if ( window.console && window.console.warn && error && rerrorNames.test( error.name ) ) {
+		window.console.warn( "jQuery.Deferred exception: " + error.message, error.stack, stack );
+	}
+};
+
+
+
+
+jQuery.readyException = function( error ) {
+	window.setTimeout( function() {
+		throw error;
+	} );
+};
+
+
+
+
+// The deferred used on DOM ready
+var readyList = jQuery.Deferred();
+
+jQuery.fn.ready = function( fn ) {
+
+	readyList
+		.then( fn )
+
+		// Wrap jQuery.readyException in a function so that the lookup
+		// happens at the time of error handling instead of callback
+		// registration.
+		.catch( function( error ) {
+			jQuery.readyException( error );
+		} );
+
+	return this;
+};
+
+jQuery.extend( {
+
+	// Is the DOM ready to be used? Set to true once it occurs.
+	isReady: false,
+
+	// A counter to track how many items to wait for before
+	// the ready event fires. See #6781
+	readyWait: 1,
+
+	// Handle when the DOM is ready
+	ready: function( wait ) {
+
+		// Abort if there are pending holds or we're already ready
+		if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) {
+			return;
+		}
+
+		// Remember that the DOM is ready
+		jQuery.isReady = true;
+
+		// If a normal DOM Ready event fired, decrement, and wait if need be
+		if ( wait !== true && --jQuery.readyWait > 0 ) {
+			return;
+		}
+
+		// If there are functions bound, to execute
+		readyList.resolveWith( document, [ jQuery ] );
+	}
+} );
+
+jQuery.ready.then = readyList.then;
+
+// The ready event handler and self cleanup method
+function completed() {
+	document.removeEventListener( "DOMContentLoaded", completed );
+	window.removeEventListener( "load", completed );
+	jQuery.ready();
+}
+
+// Catch cases where $(document).ready() is called
+// after the browser event has already occurred.
+// Support: IE <=9 - 10 only
+// Older IE sometimes signals "interactive" too soon
+if ( document.readyState === "complete" ||
+	( document.readyState !== "loading" && !document.documentElement.doScroll ) ) {
+
+	// Handle it asynchronously to allow scripts the opportunity to delay ready
+	window.setTimeout( jQuery.ready );
+
+} else {
+
+	// Use the handy event callback
+	document.addEventListener( "DOMContentLoaded", completed );
+
+	// A fallback to window.onload, that will always work
+	window.addEventListener( "load", completed );
+}
+
+
+
+
+// Multifunctional method to get and set values of a collection
+// The value/s can optionally be executed if it's a function
+var access = function( elems, fn, key, value, chainable, emptyGet, raw ) {
+	var i = 0,
+		len = elems.length,
+		bulk = key == null;
+
+	// Sets many values
+	if ( toType( key ) === "object" ) {
+		chainable = true;
+		for ( i in key ) {
+			access( elems, fn, i, key[ i ], true, emptyGet, raw );
+		}
+
+	// Sets one value
+	} else if ( value !== undefined ) {
+		chainable = true;
+
+		if ( !isFunction( value ) ) {
+			raw = true;
+		}
+
+		if ( bulk ) {
+
+			// Bulk operations run against the entire set
+			if ( raw ) {
+				fn.call( elems, value );
+				fn = null;
+
+			// ...except when executing function values
+			} else {
+				bulk = fn;
+				fn = function( elem, _key, value ) {
+					return bulk.call( jQuery( elem ), value );
+				};
+			}
+		}
+
+		if ( fn ) {
+			for ( ; i < len; i++ ) {
+				fn(
+					elems[ i ], key, raw ?
+					value :
+					value.call( elems[ i ], i, fn( elems[ i ], key ) )
+				);
+			}
+		}
+	}
+
+	if ( chainable ) {
+		return elems;
+	}
+
+	// Gets
+	if ( bulk ) {
+		return fn.call( elems );
+	}
+
+	return len ? fn( elems[ 0 ], key ) : emptyGet;
+};
+
+
+// Matches dashed string for camelizing
+var rmsPrefix = /^-ms-/,
+	rdashAlpha = /-([a-z])/g;
+
+// Used by camelCase as callback to replace()
+function fcamelCase( _all, letter ) {
+	return letter.toUpperCase();
+}
+
+// Convert dashed to camelCase; used by the css and data modules
+// Support: IE <=9 - 11, Edge 12 - 15
+// Microsoft forgot to hump their vendor prefix (#9572)
+function camelCase( string ) {
+	return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase );
+}
+var acceptData = function( owner ) {
+
+	// Accepts only:
+	//  - Node
+	//    - Node.ELEMENT_NODE
+	//    - Node.DOCUMENT_NODE
+	//  - Object
+	//    - Any
+	return owner.nodeType === 1 || owner.nodeType === 9 || !( +owner.nodeType );
+};
+
+
+
+
+function Data() {
+	this.expando = jQuery.expando + Data.uid++;
+}
+
+Data.uid = 1;
+
+Data.prototype = {
+
+	cache: function( owner ) {
+
+		// Check if the owner object already has a cache
+		var value = owner[ this.expando ];
+
+		// If not, create one
+		if ( !value ) {
+			value = {};
+
+			// We can accept data for non-element nodes in modern browsers,
+			// but we should not, see #8335.
+			// Always return an empty object.
+			if ( acceptData( owner ) ) {
+
+				// If it is a node unlikely to be stringify-ed or looped over
+				// use plain assignment
+				if ( owner.nodeType ) {
+					owner[ this.expando ] = value;
+
+				// Otherwise secure it in a non-enumerable property
+				// configurable must be true to allow the property to be
+				// deleted when data is removed
+				} else {
+					Object.defineProperty( owner, this.expando, {
+						value: value,
+						configurable: true
+					} );
+				}
+			}
+		}
+
+		return value;
+	},
+	set: function( owner, data, value ) {
+		var prop,
+			cache = this.cache( owner );
+
+		// Handle: [ owner, key, value ] args
+		// Always use camelCase key (gh-2257)
+		if ( typeof data === "string" ) {
+			cache[ camelCase( data ) ] = value;
+
+		// Handle: [ owner, { properties } ] args
+		} else {
+
+			// Copy the properties one-by-one to the cache object
+			for ( prop in data ) {
+				cache[ camelCase( prop ) ] = data[ prop ];
+			}
+		}
+		return cache;
+	},
+	get: function( owner, key ) {
+		return key === undefined ?
+			this.cache( owner ) :
+
+			// Always use camelCase key (gh-2257)
+			owner[ this.expando ] && owner[ this.expando ][ camelCase( key ) ];
+	},
+	access: function( owner, key, value ) {
+
+		// In cases where either:
+		//
+		//   1. No key was specified
+		//   2. A string key was specified, but no value provided
+		//
+		// Take the "read" path and allow the get method to determine
+		// which value to return, respectively either:
+		//
+		//   1. The entire cache object
+		//   2. The data stored at the key
+		//
+		if ( key === undefined ||
+				( ( key && typeof key === "string" ) && value === undefined ) ) {
+
+			return this.get( owner, key );
+		}
+
+		// When the key is not a string, or both a key and value
+		// are specified, set or extend (existing objects) with either:
+		//
+		//   1. An object of properties
+		//   2. A key and value
+		//
+		this.set( owner, key, value );
+
+		// Since the "set" path can have two possible entry points
+		// return the expected data based on which path was taken[*]
+		return value !== undefined ? value : key;
+	},
+	remove: function( owner, key ) {
+		var i,
+			cache = owner[ this.expando ];
+
+		if ( cache === undefined ) {
+			return;
+		}
+
+		if ( key !== undefined ) {
+
+			// Support array or space separated string of keys
+			if ( Array.isArray( key ) ) {
+
+				// If key is an array of keys...
+				// We always set camelCase keys, so remove that.
+				key = key.map( camelCase );
+			} else {
+				key = camelCase( key );
+
+				// If a key with the spaces exists, use it.
+				// Otherwise, create an array by matching non-whitespace
+				key = key in cache ?
+					[ key ] :
+					( key.match( rnothtmlwhite ) || [] );
+			}
+
+			i = key.length;
+
+			while ( i-- ) {
+				delete cache[ key[ i ] ];
+			}
+		}
+
+		// Remove the expando if there's no more data
+		if ( key === undefined || jQuery.isEmptyObject( cache ) ) {
+
+			// Support: Chrome <=35 - 45
+			// Webkit & Blink performance suffers when deleting properties
+			// from DOM nodes, so set to undefined instead
+			// https://bugs.chromium.org/p/chromium/issues/detail?id=378607 (bug restricted)
+			if ( owner.nodeType ) {
+				owner[ this.expando ] = undefined;
+			} else {
+				delete owner[ this.expando ];
+			}
+		}
+	},
+	hasData: function( owner ) {
+		var cache = owner[ this.expando ];
+		return cache !== undefined && !jQuery.isEmptyObject( cache );
+	}
+};
+var dataPriv = new Data();
+
+var dataUser = new Data();
+
+
+
+//	Implementation Summary
+//
+//	1. Enforce API surface and semantic compatibility with 1.9.x branch
+//	2. Improve the module's maintainability by reducing the storage
+//		paths to a single mechanism.
+//	3. Use the same single mechanism to support "private" and "user" data.
+//	4. _Never_ expose "private" data to user code (TODO: Drop _data, _removeData)
+//	5. Avoid exposing implementation details on user objects (eg. expando properties)
+//	6. Provide a clear path for implementation upgrade to WeakMap in 2014
+
+var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/,
+	rmultiDash = /[A-Z]/g;
+
+function getData( data ) {
+	if ( data === "true" ) {
+		return true;
+	}
+
+	if ( data === "false" ) {
+		return false;
+	}
+
+	if ( data === "null" ) {
+		return null;
+	}
+
+	// Only convert to a number if it doesn't change the string
+	if ( data === +data + "" ) {
+		return +data;
+	}
+
+	if ( rbrace.test( data ) ) {
+		return JSON.parse( data );
+	}
+
+	return data;
+}
+
+function dataAttr( elem, key, data ) {
+	var name;
+
+	// If nothing was found internally, try to fetch any
+	// data from the HTML5 data-* attribute
+	if ( data === undefined && elem.nodeType === 1 ) {
+		name = "data-" + key.replace( rmultiDash, "-$&" ).toLowerCase();
+		data = elem.getAttribute( name );
+
+		if ( typeof data === "string" ) {
+			try {
+				data = getData( data );
+			} catch ( e ) {}
+
+			// Make sure we set the data so it isn't changed later
+			dataUser.set( elem, key, data );
+		} else {
+			data = undefined;
+		}
+	}
+	return data;
+}
+
+jQuery.extend( {
+	hasData: function( elem ) {
+		return dataUser.hasData( elem ) || dataPriv.hasData( elem );
+	},
+
+	data: function( elem, name, data ) {
+		return dataUser.access( elem, name, data );
+	},
+
+	removeData: function( elem, name ) {
+		dataUser.remove( elem, name );
+	},
+
+	// TODO: Now that all calls to _data and _removeData have been replaced
+	// with direct calls to dataPriv methods, these can be deprecated.
+	_data: function( elem, name, data ) {
+		return dataPriv.access( elem, name, data );
+	},
+
+	_removeData: function( elem, name ) {
+		dataPriv.remove( elem, name );
+	}
+} );
+
+jQuery.fn.extend( {
+	data: function( key, value ) {
+		var i, name, data,
+			elem = this[ 0 ],
+			attrs = elem && elem.attributes;
+
+		// Gets all values
+		if ( key === undefined ) {
+			if ( this.length ) {
+				data = dataUser.get( elem );
+
+				if ( elem.nodeType === 1 && !dataPriv.get( elem, "hasDataAttrs" ) ) {
+					i = attrs.length;
+					while ( i-- ) {
+
+						// Support: IE 11 only
+						// The attrs elements can be null (#14894)
+						if ( attrs[ i ] ) {
+							name = attrs[ i ].name;
+							if ( name.indexOf( "data-" ) === 0 ) {
+								name = camelCase( name.slice( 5 ) );
+								dataAttr( elem, name, data[ name ] );
+							}
+						}
+					}
+					dataPriv.set( elem, "hasDataAttrs", true );
+				}
+			}
+
+			return data;
+		}
+
+		// Sets multiple values
+		if ( typeof key === "object" ) {
+			return this.each( function() {
+				dataUser.set( this, key );
+			} );
+		}
+
+		return access( this, function( value ) {
+			var data;
+
+			// The calling jQuery object (element matches) is not empty
+			// (and therefore has an element appears at this[ 0 ]) and the
+			// `value` parameter was not undefined. An empty jQuery object
+			// will result in `undefined` for elem = this[ 0 ] which will
+			// throw an exception if an attempt to read a data cache is made.
+			if ( elem && value === undefined ) {
+
+				// Attempt to get data from the cache
+				// The key will always be camelCased in Data
+				data = dataUser.get( elem, key );
+				if ( data !== undefined ) {
+					return data;
+				}
+
+				// Attempt to "discover" the data in
+				// HTML5 custom data-* attrs
+				data = dataAttr( elem, key );
+				if ( data !== undefined ) {
+					return data;
+				}
+
+				// We tried really hard, but the data doesn't exist.
+				return;
+			}
+
+			// Set the data...
+			this.each( function() {
+
+				// We always store the camelCased key
+				dataUser.set( this, key, value );
+			} );
+		}, null, value, arguments.length > 1, null, true );
+	},
+
+	removeData: function( key ) {
+		return this.each( function() {
+			dataUser.remove( this, key );
+		} );
+	}
+} );
+
+
+jQuery.extend( {
+	queue: function( elem, type, data ) {
+		var queue;
+
+		if ( elem ) {
+			type = ( type || "fx" ) + "queue";
+			queue = dataPriv.get( elem, type );
+
+			// Speed up dequeue by getting out quickly if this is just a lookup
+			if ( data ) {
+				if ( !queue || Array.isArray( data ) ) {
+					queue = dataPriv.access( elem, type, jQuery.makeArray( data ) );
+				} else {
+					queue.push( data );
+				}
+			}
+			return queue || [];
+		}
+	},
+
+	dequeue: function( elem, type ) {
+		type = type || "fx";
+
+		var queue = jQuery.queue( elem, type ),
+			startLength = queue.length,
+			fn = queue.shift(),
+			hooks = jQuery._queueHooks( elem, type ),
+			next = function() {
+				jQuery.dequeue( elem, type );
+			};
+
+		// If the fx queue is dequeued, always remove the progress sentinel
+		if ( fn === "inprogress" ) {
+			fn = queue.shift();
+			startLength--;
+		}
+
+		if ( fn ) {
+
+			// Add a progress sentinel to prevent the fx queue from being
+			// automatically dequeued
+			if ( type === "fx" ) {
+				queue.unshift( "inprogress" );
+			}
+
+			// Clear up the last queue stop function
+			delete hooks.stop;
+			fn.call( elem, next, hooks );
+		}
+
+		if ( !startLength && hooks ) {
+			hooks.empty.fire();
+		}
+	},
+
+	// Not public - generate a queueHooks object, or return the current one
+	_queueHooks: function( elem, type ) {
+		var key = type + "queueHooks";
+		return dataPriv.get( elem, key ) || dataPriv.access( elem, key, {
+			empty: jQuery.Callbacks( "once memory" ).add( function() {
+				dataPriv.remove( elem, [ type + "queue", key ] );
+			} )
+		} );
+	}
+} );
+
+jQuery.fn.extend( {
+	queue: function( type, data ) {
+		var setter = 2;
+
+		if ( typeof type !== "string" ) {
+			data = type;
+			type = "fx";
+			setter--;
+		}
+
+		if ( arguments.length < setter ) {
+			return jQuery.queue( this[ 0 ], type );
+		}
+
+		return data === undefined ?
+			this :
+			this.each( function() {
+				var queue = jQuery.queue( this, type, data );
+
+				// Ensure a hooks for this queue
+				jQuery._queueHooks( this, type );
+
+				if ( type === "fx" && queue[ 0 ] !== "inprogress" ) {
+					jQuery.dequeue( this, type );
+				}
+			} );
+	},
+	dequeue: function( type ) {
+		return this.each( function() {
+			jQuery.dequeue( this, type );
+		} );
+	},
+	clearQueue: function( type ) {
+		return this.queue( type || "fx", [] );
+	},
+
+	// Get a promise resolved when queues of a certain type
+	// are emptied (fx is the type by default)
+	promise: function( type, obj ) {
+		var tmp,
+			count = 1,
+			defer = jQuery.Deferred(),
+			elements = this,
+			i = this.length,
+			resolve = function() {
+				if ( !( --count ) ) {
+					defer.resolveWith( elements, [ elements ] );
+				}
+			};
+
+		if ( typeof type !== "string" ) {
+			obj = type;
+			type = undefined;
+		}
+		type = type || "fx";
+
+		while ( i-- ) {
+			tmp = dataPriv.get( elements[ i ], type + "queueHooks" );
+			if ( tmp && tmp.empty ) {
+				count++;
+				tmp.empty.add( resolve );
+			}
+		}
+		resolve();
+		return defer.promise( obj );
+	}
+} );
+var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source;
+
+var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" );
+
+
+var cssExpand = [ "Top", "Right", "Bottom", "Left" ];
+
+var documentElement = document.documentElement;
+
+
+
+	var isAttached = function( elem ) {
+			return jQuery.contains( elem.ownerDocument, elem );
+		},
+		composed = { composed: true };
+
+	// Support: IE 9 - 11+, Edge 12 - 18+, iOS 10.0 - 10.2 only
+	// Check attachment across shadow DOM boundaries when possible (gh-3504)
+	// Support: iOS 10.0-10.2 only
+	// Early iOS 10 versions support `attachShadow` but not `getRootNode`,
+	// leading to errors. We need to check for `getRootNode`.
+	if ( documentElement.getRootNode ) {
+		isAttached = function( elem ) {
+			return jQuery.contains( elem.ownerDocument, elem ) ||
+				elem.getRootNode( composed ) === elem.ownerDocument;
+		};
+	}
+var isHiddenWithinTree = function( elem, el ) {
+
+		// isHiddenWithinTree might be called from jQuery#filter function;
+		// in that case, element will be second argument
+		elem = el || elem;
+
+		// Inline style trumps all
+		return elem.style.display === "none" ||
+			elem.style.display === "" &&
+
+			// Otherwise, check computed style
+			// Support: Firefox <=43 - 45
+			// Disconnected elements can have computed display: none, so first confirm that elem is
+			// in the document.
+			isAttached( elem ) &&
+
+			jQuery.css( elem, "display" ) === "none";
+	};
+
+
+
+function adjustCSS( elem, prop, valueParts, tween ) {
+	var adjusted, scale,
+		maxIterations = 20,
+		currentValue = tween ?
+			function() {
+				return tween.cur();
+			} :
+			function() {
+				return jQuery.css( elem, prop, "" );
+			},
+		initial = currentValue(),
+		unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ),
+
+		// Starting value computation is required for potential unit mismatches
+		initialInUnit = elem.nodeType &&
+			( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) &&
+			rcssNum.exec( jQuery.css( elem, prop ) );
+
+	if ( initialInUnit && initialInUnit[ 3 ] !== unit ) {
+
+		// Support: Firefox <=54
+		// Halve the iteration target value to prevent interference from CSS upper bounds (gh-2144)
+		initial = initial / 2;
+
+		// Trust units reported by jQuery.css
+		unit = unit || initialInUnit[ 3 ];
+
+		// Iteratively approximate from a nonzero starting point
+		initialInUnit = +initial || 1;
+
+		while ( maxIterations-- ) {
+
+			// Evaluate and update our best guess (doubling guesses that zero out).
+			// Finish if the scale equals or crosses 1 (making the old*new product non-positive).
+			jQuery.style( elem, prop, initialInUnit + unit );
+			if ( ( 1 - scale ) * ( 1 - ( scale = currentValue() / initial || 0.5 ) ) <= 0 ) {
+				maxIterations = 0;
+			}
+			initialInUnit = initialInUnit / scale;
+
+		}
+
+		initialInUnit = initialInUnit * 2;
+		jQuery.style( elem, prop, initialInUnit + unit );
+
+		// Make sure we update the tween properties later on
+		valueParts = valueParts || [];
+	}
+
+	if ( valueParts ) {
+		initialInUnit = +initialInUnit || +initial || 0;
+
+		// Apply relative offset (+=/-=) if specified
+		adjusted = valueParts[ 1 ] ?
+			initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] :
+			+valueParts[ 2 ];
+		if ( tween ) {
+			tween.unit = unit;
+			tween.start = initialInUnit;
+			tween.end = adjusted;
+		}
+	}
+	return adjusted;
+}
+
+
+var defaultDisplayMap = {};
+
+function getDefaultDisplay( elem ) {
+	var temp,
+		doc = elem.ownerDocument,
+		nodeName = elem.nodeName,
+		display = defaultDisplayMap[ nodeName ];
+
+	if ( display ) {
+		return display;
+	}
+
+	temp = doc.body.appendChild( doc.createElement( nodeName ) );
+	display = jQuery.css( temp, "display" );
+
+	temp.parentNode.removeChild( temp );
+
+	if ( display === "none" ) {
+		display = "block";
+	}
+	defaultDisplayMap[ nodeName ] = display;
+
+	return display;
+}
+
+function showHide( elements, show ) {
+	var display, elem,
+		values = [],
+		index = 0,
+		length = elements.length;
+
+	// Determine new display value for elements that need to change
+	for ( ; index < length; index++ ) {
+		elem = elements[ index ];
+		if ( !elem.style ) {
+			continue;
+		}
+
+		display = elem.style.display;
+		if ( show ) {
+
+			// Since we force visibility upon cascade-hidden elements, an immediate (and slow)
+			// check is required in this first loop unless we have a nonempty display value (either
+			// inline or about-to-be-restored)
+			if ( display === "none" ) {
+				values[ index ] = dataPriv.get( elem, "display" ) || null;
+				if ( !values[ index ] ) {
+					elem.style.display = "";
+				}
+			}
+			if ( elem.style.display === "" && isHiddenWithinTree( elem ) ) {
+				values[ index ] = getDefaultDisplay( elem );
+			}
+		} else {
+			if ( display !== "none" ) {
+				values[ index ] = "none";
+
+				// Remember what we're overwriting
+				dataPriv.set( elem, "display", display );
+			}
+		}
+	}
+
+	// Set the display of the elements in a second loop to avoid constant reflow
+	for ( index = 0; index < length; index++ ) {
+		if ( values[ index ] != null ) {
+			elements[ index ].style.display = values[ index ];
+		}
+	}
+
+	return elements;
+}
+
+jQuery.fn.extend( {
+	show: function() {
+		return showHide( this, true );
+	},
+	hide: function() {
+		return showHide( this );
+	},
+	toggle: function( state ) {
+		if ( typeof state === "boolean" ) {
+			return state ? this.show() : this.hide();
+		}
+
+		return this.each( function() {
+			if ( isHiddenWithinTree( this ) ) {
+				jQuery( this ).show();
+			} else {
+				jQuery( this ).hide();
+			}
+		} );
+	}
+} );
+var rcheckableType = ( /^(?:checkbox|radio)$/i );
+
+var rtagName = ( /<([a-z][^\/\0>\x20\t\r\n\f]*)/i );
+
+var rscriptType = ( /^$|^module$|\/(?:java|ecma)script/i );
+
+
+
+( function() {
+	var fragment = document.createDocumentFragment(),
+		div = fragment.appendChild( document.createElement( "div" ) ),
+		input = document.createElement( "input" );
+
+	// Support: Android 4.0 - 4.3 only
+	// Check state lost if the name is set (#11217)
+	// Support: Windows Web Apps (WWA)
+	// `name` and `type` must use .setAttribute for WWA (#14901)
+	input.setAttribute( "type", "radio" );
+	input.setAttribute( "checked", "checked" );
+	input.setAttribute( "name", "t" );
+
+	div.appendChild( input );
+
+	// Support: Android <=4.1 only
+	// Older WebKit doesn't clone checked state correctly in fragments
+	support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked;
+
+	// Support: IE <=11 only
+	// Make sure textarea (and checkbox) defaultValue is properly cloned
+	div.innerHTML = "<textarea>x</textarea>";
+	support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue;
+
+	// Support: IE <=9 only
+	// IE <=9 replaces <option> tags with their contents when inserted outside of
+	// the select element.
+	div.innerHTML = "<option></option>";
+	support.option = !!div.lastChild;
+} )();
+
+
+// We have to close these tags to support XHTML (#13200)
+var wrapMap = {
+
+	// XHTML parsers do not magically insert elements in the
+	// same way that tag soup parsers do. So we cannot shorten
+	// this by omitting <tbody> or other required elements.
+	thead: [ 1, "<table>", "</table>" ],
+	col: [ 2, "<table><colgroup>", "</colgroup></table>" ],
+	tr: [ 2, "<table><tbody>", "</tbody></table>" ],
+	td: [ 3, "<table><tbody><tr>", "</tr></tbody></table>" ],
+
+	_default: [ 0, "", "" ]
+};
+
+wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead;
+wrapMap.th = wrapMap.td;
+
+// Support: IE <=9 only
+if ( !support.option ) {
+	wrapMap.optgroup = wrapMap.option = [ 1, "<select multiple='multiple'>", "</select>" ];
+}
+
+
+function getAll( context, tag ) {
+
+	// Support: IE <=9 - 11 only
+	// Use typeof to avoid zero-argument method invocation on host objects (#15151)
+	var ret;
+
+	if ( typeof context.getElementsByTagName !== "undefined" ) {
+		ret = context.getElementsByTagName( tag || "*" );
+
+	} else if ( typeof context.querySelectorAll !== "undefined" ) {
+		ret = context.querySelectorAll( tag || "*" );
+
+	} else {
+		ret = [];
+	}
+
+	if ( tag === undefined || tag && nodeName( context, tag ) ) {
+		return jQuery.merge( [ context ], ret );
+	}
+
+	return ret;
+}
+
+
+// Mark scripts as having already been evaluated
+function setGlobalEval( elems, refElements ) {
+	var i = 0,
+		l = elems.length;
+
+	for ( ; i < l; i++ ) {
+		dataPriv.set(
+			elems[ i ],
+			"globalEval",
+			!refElements || dataPriv.get( refElements[ i ], "globalEval" )
+		);
+	}
+}
+
+
+var rhtml = /<|&#?\w+;/;
+
+function buildFragment( elems, context, scripts, selection, ignored ) {
+	var elem, tmp, tag, wrap, attached, j,
+		fragment = context.createDocumentFragment(),
+		nodes = [],
+		i = 0,
+		l = elems.length;
+
+	for ( ; i < l; i++ ) {
+		elem = elems[ i ];
+
+		if ( elem || elem === 0 ) {
+
+			// Add nodes directly
+			if ( toType( elem ) === "object" ) {
+
+				// Support: Android <=4.0 only, PhantomJS 1 only
+				// push.apply(_, arraylike) throws on ancient WebKit
+				jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem );
+
+			// Convert non-html into a text node
+			} else if ( !rhtml.test( elem ) ) {
+				nodes.push( context.createTextNode( elem ) );
+
+			// Convert html into DOM nodes
+			} else {
+				tmp = tmp || fragment.appendChild( context.createElement( "div" ) );
+
+				// Deserialize a standard representation
+				tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase();
+				wrap = wrapMap[ tag ] || wrapMap._default;
+				tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ];
+
+				// Descend through wrappers to the right content
+				j = wrap[ 0 ];
+				while ( j-- ) {
+					tmp = tmp.lastChild;
+				}
+
+				// Support: Android <=4.0 only, PhantomJS 1 only
+				// push.apply(_, arraylike) throws on ancient WebKit
+				jQuery.merge( nodes, tmp.childNodes );
+
+				// Remember the top-level container
+				tmp = fragment.firstChild;
+
+				// Ensure the created nodes are orphaned (#12392)
+				tmp.textContent = "";
+			}
+		}
+	}
+
+	// Remove wrapper from fragment
+	fragment.textContent = "";
+
+	i = 0;
+	while ( ( elem = nodes[ i++ ] ) ) {
+
+		// Skip elements already in the context collection (trac-4087)
+		if ( selection && jQuery.inArray( elem, selection ) > -1 ) {
+			if ( ignored ) {
+				ignored.push( elem );
+			}
+			continue;
+		}
+
+		attached = isAttached( elem );
+
+		// Append to fragment
+		tmp = getAll( fragment.appendChild( elem ), "script" );
+
+		// Preserve script evaluation history
+		if ( attached ) {
+			setGlobalEval( tmp );
+		}
+
+		// Capture executables
+		if ( scripts ) {
+			j = 0;
+			while ( ( elem = tmp[ j++ ] ) ) {
+				if ( rscriptType.test( elem.type || "" ) ) {
+					scripts.push( elem );
+				}
+			}
+		}
+	}
+
+	return fragment;
+}
+
+
+var
+	rkeyEvent = /^key/,
+	rmouseEvent = /^(?:mouse|pointer|contextmenu|drag|drop)|click/,
+	rtypenamespace = /^([^.]*)(?:\.(.+)|)/;
+
+function returnTrue() {
+	return true;
+}
+
+function returnFalse() {
+	return false;
+}
+
+// Support: IE <=9 - 11+
+// focus() and blur() are asynchronous, except when they are no-op.
+// So expect focus to be synchronous when the element is already active,
+// and blur to be synchronous when the element is not already active.
+// (focus and blur are always synchronous in other supported browsers,
+// this just defines when we can count on it).
+function expectSync( elem, type ) {
+	return ( elem === safeActiveElement() ) === ( type === "focus" );
+}
+
+// Support: IE <=9 only
+// Accessing document.activeElement can throw unexpectedly
+// https://bugs.jquery.com/ticket/13393
+function safeActiveElement() {
+	try {
+		return document.activeElement;
+	} catch ( err ) { }
+}
+
+function on( elem, types, selector, data, fn, one ) {
+	var origFn, type;
+
+	// Types can be a map of types/handlers
+	if ( typeof types === "object" ) {
+
+		// ( types-Object, selector, data )
+		if ( typeof selector !== "string" ) {
+
+			// ( types-Object, data )
+			data = data || selector;
+			selector = undefined;
+		}
+		for ( type in types ) {
+			on( elem, type, selector, data, types[ type ], one );
+		}
+		return elem;
+	}
+
+	if ( data == null && fn == null ) {
+
+		// ( types, fn )
+		fn = selector;
+		data = selector = undefined;
+	} else if ( fn == null ) {
+		if ( typeof selector === "string" ) {
+
+			// ( types, selector, fn )
+			fn = data;
+			data = undefined;
+		} else {
+
+			// ( types, data, fn )
+			fn = data;
+			data = selector;
+			selector = undefined;
+		}
+	}
+	if ( fn === false ) {
+		fn = returnFalse;
+	} else if ( !fn ) {
+		return elem;
+	}
+
+	if ( one === 1 ) {
+		origFn = fn;
+		fn = function( event ) {
+
+			// Can use an empty set, since event contains the info
+			jQuery().off( event );
+			return origFn.apply( this, arguments );
+		};
+
+		// Use same guid so caller can remove using origFn
+		fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ );
+	}
+	return elem.each( function() {
+		jQuery.event.add( this, types, fn, data, selector );
+	} );
+}
+
+/*
+ * Helper functions for managing events -- not part of the public interface.
+ * Props to Dean Edwards' addEvent library for many of the ideas.
+ */
+jQuery.event = {
+
+	global: {},
+
+	add: function( elem, types, handler, data, selector ) {
+
+		var handleObjIn, eventHandle, tmp,
+			events, t, handleObj,
+			special, handlers, type, namespaces, origType,
+			elemData = dataPriv.get( elem );
+
+		// Only attach events to objects that accept data
+		if ( !acceptData( elem ) ) {
+			return;
+		}
+
+		// Caller can pass in an object of custom data in lieu of the handler
+		if ( handler.handler ) {
+			handleObjIn = handler;
+			handler = handleObjIn.handler;
+			selector = handleObjIn.selector;
+		}
+
+		// Ensure that invalid selectors throw exceptions at attach time
+		// Evaluate against documentElement in case elem is a non-element node (e.g., document)
+		if ( selector ) {
+			jQuery.find.matchesSelector( documentElement, selector );
+		}
+
+		// Make sure that the handler has a unique ID, used to find/remove it later
+		if ( !handler.guid ) {
+			handler.guid = jQuery.guid++;
+		}
+
+		// Init the element's event structure and main handler, if this is the first
+		if ( !( events = elemData.events ) ) {
+			events = elemData.events = Object.create( null );
+		}
+		if ( !( eventHandle = elemData.handle ) ) {
+			eventHandle = elemData.handle = function( e ) {
+
+				// Discard the second event of a jQuery.event.trigger() and
+				// when an event is called after a page has unloaded
+				return typeof jQuery !== "undefined" && jQuery.event.triggered !== e.type ?
+					jQuery.event.dispatch.apply( elem, arguments ) : undefined;
+			};
+		}
+
+		// Handle multiple events separated by a space
+		types = ( types || "" ).match( rnothtmlwhite ) || [ "" ];
+		t = types.length;
+		while ( t-- ) {
+			tmp = rtypenamespace.exec( types[ t ] ) || [];
+			type = origType = tmp[ 1 ];
+			namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort();
+
+			// There *must* be a type, no attaching namespace-only handlers
+			if ( !type ) {
+				continue;
+			}
+
+			// If event changes its type, use the special event handlers for the changed type
+			special = jQuery.event.special[ type ] || {};
+
+			// If selector defined, determine special event api type, otherwise given type
+			type = ( selector ? special.delegateType : special.bindType ) || type;
+
+			// Update special based on newly reset type
+			special = jQuery.event.special[ type ] || {};
+
+			// handleObj is passed to all event handlers
+			handleObj = jQuery.extend( {
+				type: type,
+				origType: origType,
+				data: data,
+				handler: handler,
+				guid: handler.guid,
+				selector: selector,
+				needsContext: selector && jQuery.expr.match.needsContext.test( selector ),
+				namespace: namespaces.join( "." )
+			}, handleObjIn );
+
+			// Init the event handler queue if we're the first
+			if ( !( handlers = events[ type ] ) ) {
+				handlers = events[ type ] = [];
+				handlers.delegateCount = 0;
+
+				// Only use addEventListener if the special events handler returns false
+				if ( !special.setup ||
+					special.setup.call( elem, data, namespaces, eventHandle ) === false ) {
+
+					if ( elem.addEventListener ) {
+						elem.addEventListener( type, eventHandle );
+					}
+				}
+			}
+
+			if ( special.add ) {
+				special.add.call( elem, handleObj );
+
+				if ( !handleObj.handler.guid ) {
+					handleObj.handler.guid = handler.guid;
+				}
+			}
+
+			// Add to the element's handler list, delegates in front
+			if ( selector ) {
+				handlers.splice( handlers.delegateCount++, 0, handleObj );
+			} else {
+				handlers.push( handleObj );
+			}
+
+			// Keep track of which events have ever been used, for event optimization
+			jQuery.event.global[ type ] = true;
+		}
+
+	},
+
+	// Detach an event or set of events from an element
+	remove: function( elem, types, handler, selector, mappedTypes ) {
+
+		var j, origCount, tmp,
+			events, t, handleObj,
+			special, handlers, type, namespaces, origType,
+			elemData = dataPriv.hasData( elem ) && dataPriv.get( elem );
+
+		if ( !elemData || !( events = elemData.events ) ) {
+			return;
+		}
+
+		// Once for each type.namespace in types; type may be omitted
+		types = ( types || "" ).match( rnothtmlwhite ) || [ "" ];
+		t = types.length;
+		while ( t-- ) {
+			tmp = rtypenamespace.exec( types[ t ] ) || [];
+			type = origType = tmp[ 1 ];
+			namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort();
+
+			// Unbind all events (on this namespace, if provided) for the element
+			if ( !type ) {
+				for ( type in events ) {
+					jQuery.event.remove( elem, type + types[ t ], handler, selector, true );
+				}
+				continue;
+			}
+
+			special = jQuery.event.special[ type ] || {};
+			type = ( selector ? special.delegateType : special.bindType ) || type;
+			handlers = events[ type ] || [];
+			tmp = tmp[ 2 ] &&
+				new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" );
+
+			// Remove matching events
+			origCount = j = handlers.length;
+			while ( j-- ) {
+				handleObj = handlers[ j ];
+
+				if ( ( mappedTypes || origType === handleObj.origType ) &&
+					( !handler || handler.guid === handleObj.guid ) &&
+					( !tmp || tmp.test( handleObj.namespace ) ) &&
+					( !selector || selector === handleObj.selector ||
+						selector === "**" && handleObj.selector ) ) {
+					handlers.splice( j, 1 );
+
+					if ( handleObj.selector ) {
+						handlers.delegateCount--;
+					}
+					if ( special.remove ) {
+						special.remove.call( elem, handleObj );
+					}
+				}
+			}
+
+			// Remove generic event handler if we removed something and no more handlers exist
+			// (avoids potential for endless recursion during removal of special event handlers)
+			if ( origCount && !handlers.length ) {
+				if ( !special.teardown ||
+					special.teardown.call( elem, namespaces, elemData.handle ) === false ) {
+
+					jQuery.removeEvent( elem, type, elemData.handle );
+				}
+
+				delete events[ type ];
+			}
+		}
+
+		// Remove data and the expando if it's no longer used
+		if ( jQuery.isEmptyObject( events ) ) {
+			dataPriv.remove( elem, "handle events" );
+		}
+	},
+
+	dispatch: function( nativeEvent ) {
+
+		var i, j, ret, matched, handleObj, handlerQueue,
+			args = new Array( arguments.length ),
+
+			// Make a writable jQuery.Event from the native event object
+			event = jQuery.event.fix( nativeEvent ),
+
+			handlers = (
+					dataPriv.get( this, "events" ) || Object.create( null )
+				)[ event.type ] || [],
+			special = jQuery.event.special[ event.type ] || {};
+
+		// Use the fix-ed jQuery.Event rather than the (read-only) native event
+		args[ 0 ] = event;
+
+		for ( i = 1; i < arguments.length; i++ ) {
+			args[ i ] = arguments[ i ];
+		}
+
+		event.delegateTarget = this;
+
+		// Call the preDispatch hook for the mapped type, and let it bail if desired
+		if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) {
+			return;
+		}
+
+		// Determine handlers
+		handlerQueue = jQuery.event.handlers.call( this, event, handlers );
+
+		// Run delegates first; they may want to stop propagation beneath us
+		i = 0;
+		while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) {
+			event.currentTarget = matched.elem;
+
+			j = 0;
+			while ( ( handleObj = matched.handlers[ j++ ] ) &&
+				!event.isImmediatePropagationStopped() ) {
+
+				// If the event is namespaced, then each handler is only invoked if it is
+				// specially universal or its namespaces are a superset of the event's.
+				if ( !event.rnamespace || handleObj.namespace === false ||
+					event.rnamespace.test( handleObj.namespace ) ) {
+
+					event.handleObj = handleObj;
+					event.data = handleObj.data;
+
+					ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle ||
+						handleObj.handler ).apply( matched.elem, args );
+
+					if ( ret !== undefined ) {
+						if ( ( event.result = ret ) === false ) {
+							event.preventDefault();
+							event.stopPropagation();
+						}
+					}
+				}
+			}
+		}
+
+		// Call the postDispatch hook for the mapped type
+		if ( special.postDispatch ) {
+			special.postDispatch.call( this, event );
+		}
+
+		return event.result;
+	},
+
+	handlers: function( event, handlers ) {
+		var i, handleObj, sel, matchedHandlers, matchedSelectors,
+			handlerQueue = [],
+			delegateCount = handlers.delegateCount,
+			cur = event.target;
+
+		// Find delegate handlers
+		if ( delegateCount &&
+
+			// Support: IE <=9
+			// Black-hole SVG <use> instance trees (trac-13180)
+			cur.nodeType &&
+
+			// Support: Firefox <=42
+			// Suppress spec-violating clicks indicating a non-primary pointer button (trac-3861)
+			// https://www.w3.org/TR/DOM-Level-3-Events/#event-type-click
+			// Support: IE 11 only
+			// ...but not arrow key "clicks" of radio inputs, which can have `button` -1 (gh-2343)
+			!( event.type === "click" && event.button >= 1 ) ) {
+
+			for ( ; cur !== this; cur = cur.parentNode || this ) {
+
+				// Don't check non-elements (#13208)
+				// Don't process clicks on disabled elements (#6911, #8165, #11382, #11764)
+				if ( cur.nodeType === 1 && !( event.type === "click" && cur.disabled === true ) ) {
+					matchedHandlers = [];
+					matchedSelectors = {};
+					for ( i = 0; i < delegateCount; i++ ) {
+						handleObj = handlers[ i ];
+
+						// Don't conflict with Object.prototype properties (#13203)
+						sel = handleObj.selector + " ";
+
+						if ( matchedSelectors[ sel ] === undefined ) {
+							matchedSelectors[ sel ] = handleObj.needsContext ?
+								jQuery( sel, this ).index( cur ) > -1 :
+								jQuery.find( sel, this, null, [ cur ] ).length;
+						}
+						if ( matchedSelectors[ sel ] ) {
+							matchedHandlers.push( handleObj );
+						}
+					}
+					if ( matchedHandlers.length ) {
+						handlerQueue.push( { elem: cur, handlers: matchedHandlers } );
+					}
+				}
+			}
+		}
+
+		// Add the remaining (directly-bound) handlers
+		cur = this;
+		if ( delegateCount < handlers.length ) {
+			handlerQueue.push( { elem: cur, handlers: handlers.slice( delegateCount ) } );
+		}
+
+		return handlerQueue;
+	},
+
+	addProp: function( name, hook ) {
+		Object.defineProperty( jQuery.Event.prototype, name, {
+			enumerable: true,
+			configurable: true,
+
+			get: isFunction( hook ) ?
+				function() {
+					if ( this.originalEvent ) {
+							return hook( this.originalEvent );
+					}
+				} :
+				function() {
+					if ( this.originalEvent ) {
+							return this.originalEvent[ name ];
+					}
+				},
+
+			set: function( value ) {
+				Object.defineProperty( this, name, {
+					enumerable: true,
+					configurable: true,
+					writable: true,
+					value: value
+				} );
+			}
+		} );
+	},
+
+	fix: function( originalEvent ) {
+		return originalEvent[ jQuery.expando ] ?
+			originalEvent :
+			new jQuery.Event( originalEvent );
+	},
+
+	special: {
+		load: {
+
+			// Prevent triggered image.load events from bubbling to window.load
+			noBubble: true
+		},
+		click: {
+
+			// Utilize native event to ensure correct state for checkable inputs
+			setup: function( data ) {
+
+				// For mutual compressibility with _default, replace `this` access with a local var.
+				// `|| data` is dead code meant only to preserve the variable through minification.
+				var el = this || data;
+
+				// Claim the first handler
+				if ( rcheckableType.test( el.type ) &&
+					el.click && nodeName( el, "input" ) ) {
+
+					// dataPriv.set( el, "click", ... )
+					leverageNative( el, "click", returnTrue );
+				}
+
+				// Return false to allow normal processing in the caller
+				return false;
+			},
+			trigger: function( data ) {
+
+				// For mutual compressibility with _default, replace `this` access with a local var.
+				// `|| data` is dead code meant only to preserve the variable through minification.
+				var el = this || data;
+
+				// Force setup before triggering a click
+				if ( rcheckableType.test( el.type ) &&
+					el.click && nodeName( el, "input" ) ) {
+
+					leverageNative( el, "click" );
+				}
+
+				// Return non-false to allow normal event-path propagation
+				return true;
+			},
+
+			// For cross-browser consistency, suppress native .click() on links
+			// Also prevent it if we're currently inside a leveraged native-event stack
+			_default: function( event ) {
+				var target = event.target;
+				return rcheckableType.test( target.type ) &&
+					target.click && nodeName( target, "input" ) &&
+					dataPriv.get( target, "click" ) ||
+					nodeName( target, "a" );
+			}
+		},
+
+		beforeunload: {
+			postDispatch: function( event ) {
+
+				// Support: Firefox 20+
+				// Firefox doesn't alert if the returnValue field is not set.
+				if ( event.result !== undefined && event.originalEvent ) {
+					event.originalEvent.returnValue = event.result;
+				}
+			}
+		}
+	}
+};
+
+// Ensure the presence of an event listener that handles manually-triggered
+// synthetic events by interrupting progress until reinvoked in response to
+// *native* events that it fires directly, ensuring that state changes have
+// already occurred before other listeners are invoked.
+function leverageNative( el, type, expectSync ) {
+
+	// Missing expectSync indicates a trigger call, which must force setup through jQuery.event.add
+	if ( !expectSync ) {
+		if ( dataPriv.get( el, type ) === undefined ) {
+			jQuery.event.add( el, type, returnTrue );
+		}
+		return;
+	}
+
+	// Register the controller as a special universal handler for all event namespaces
+	dataPriv.set( el, type, false );
+	jQuery.event.add( el, type, {
+		namespace: false,
+		handler: function( event ) {
+			var notAsync, result,
+				saved = dataPriv.get( this, type );
+
+			if ( ( event.isTrigger & 1 ) && this[ type ] ) {
+
+				// Interrupt processing of the outer synthetic .trigger()ed event
+				// Saved data should be false in such cases, but might be a leftover capture object
+				// from an async native handler (gh-4350)
+				if ( !saved.length ) {
+
+					// Store arguments for use when handling the inner native event
+					// There will always be at least one argument (an event object), so this array
+					// will not be confused with a leftover capture object.
+					saved = slice.call( arguments );
+					dataPriv.set( this, type, saved );
+
+					// Trigger the native event and capture its result
+					// Support: IE <=9 - 11+
+					// focus() and blur() are asynchronous
+					notAsync = expectSync( this, type );
+					this[ type ]();
+					result = dataPriv.get( this, type );
+					if ( saved !== result || notAsync ) {
+						dataPriv.set( this, type, false );
+					} else {
+						result = {};
+					}
+					if ( saved !== result ) {
+
+						// Cancel the outer synthetic event
+						event.stopImmediatePropagation();
+						event.preventDefault();
+						return result.value;
+					}
+
+				// If this is an inner synthetic event for an event with a bubbling surrogate
+				// (focus or blur), assume that the surrogate already propagated from triggering the
+				// native event and prevent that from happening again here.
+				// This technically gets the ordering wrong w.r.t. to `.trigger()` (in which the
+				// bubbling surrogate propagates *after* the non-bubbling base), but that seems
+				// less bad than duplication.
+				} else if ( ( jQuery.event.special[ type ] || {} ).delegateType ) {
+					event.stopPropagation();
+				}
+
+			// If this is a native event triggered above, everything is now in order
+			// Fire an inner synthetic event with the original arguments
+			} else if ( saved.length ) {
+
+				// ...and capture the result
+				dataPriv.set( this, type, {
+					value: jQuery.event.trigger(
+
+						// Support: IE <=9 - 11+
+						// Extend with the prototype to reset the above stopImmediatePropagation()
+						jQuery.extend( saved[ 0 ], jQuery.Event.prototype ),
+						saved.slice( 1 ),
+						this
+					)
+				} );
+
+				// Abort handling of the native event
+				event.stopImmediatePropagation();
+			}
+		}
+	} );
+}
+
+jQuery.removeEvent = function( elem, type, handle ) {
+
+	// This "if" is needed for plain objects
+	if ( elem.removeEventListener ) {
+		elem.removeEventListener( type, handle );
+	}
+};
+
+jQuery.Event = function( src, props ) {
+
+	// Allow instantiation without the 'new' keyword
+	if ( !( this instanceof jQuery.Event ) ) {
+		return new jQuery.Event( src, props );
+	}
+
+	// Event object
+	if ( src && src.type ) {
+		this.originalEvent = src;
+		this.type = src.type;
+
+		// Events bubbling up the document may have been marked as prevented
+		// by a handler lower down the tree; reflect the correct value.
+		this.isDefaultPrevented = src.defaultPrevented ||
+				src.defaultPrevented === undefined &&
+
+				// Support: Android <=2.3 only
+				src.returnValue === false ?
+			returnTrue :
+			returnFalse;
+
+		// Create target properties
+		// Support: Safari <=6 - 7 only
+		// Target should not be a text node (#504, #13143)
+		this.target = ( src.target && src.target.nodeType === 3 ) ?
+			src.target.parentNode :
+			src.target;
+
+		this.currentTarget = src.currentTarget;
+		this.relatedTarget = src.relatedTarget;
+
+	// Event type
+	} else {
+		this.type = src;
+	}
+
+	// Put explicitly provided properties onto the event object
+	if ( props ) {
+		jQuery.extend( this, props );
+	}
+
+	// Create a timestamp if incoming event doesn't have one
+	this.timeStamp = src && src.timeStamp || Date.now();
+
+	// Mark it as fixed
+	this[ jQuery.expando ] = true;
+};
+
+// jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding
+// https://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html
+jQuery.Event.prototype = {
+	constructor: jQuery.Event,
+	isDefaultPrevented: returnFalse,
+	isPropagationStopped: returnFalse,
+	isImmediatePropagationStopped: returnFalse,
+	isSimulated: false,
+
+	preventDefault: function() {
+		var e = this.originalEvent;
+
+		this.isDefaultPrevented = returnTrue;
+
+		if ( e && !this.isSimulated ) {
+			e.preventDefault();
+		}
+	},
+	stopPropagation: function() {
+		var e = this.originalEvent;
+
+		this.isPropagationStopped = returnTrue;
+
+		if ( e && !this.isSimulated ) {
+			e.stopPropagation();
+		}
+	},
+	stopImmediatePropagation: function() {
+		var e = this.originalEvent;
+
+		this.isImmediatePropagationStopped = returnTrue;
+
+		if ( e && !this.isSimulated ) {
+			e.stopImmediatePropagation();
+		}
+
+		this.stopPropagation();
+	}
+};
+
+// Includes all common event props including KeyEvent and MouseEvent specific props
+jQuery.each( {
+	altKey: true,
+	bubbles: true,
+	cancelable: true,
+	changedTouches: true,
+	ctrlKey: true,
+	detail: true,
+	eventPhase: true,
+	metaKey: true,
+	pageX: true,
+	pageY: true,
+	shiftKey: true,
+	view: true,
+	"char": true,
+	code: true,
+	charCode: true,
+	key: true,
+	keyCode: true,
+	button: true,
+	buttons: true,
+	clientX: true,
+	clientY: true,
+	offsetX: true,
+	offsetY: true,
+	pointerId: true,
+	pointerType: true,
+	screenX: true,
+	screenY: true,
+	targetTouches: true,
+	toElement: true,
+	touches: true,
+
+	which: function( event ) {
+		var button = event.button;
+
+		// Add which for key events
+		if ( event.which == null && rkeyEvent.test( event.type ) ) {
+			return event.charCode != null ? event.charCode : event.keyCode;
+		}
+
+		// Add which for click: 1 === left; 2 === middle; 3 === right
+		if ( !event.which && button !== undefined && rmouseEvent.test( event.type ) ) {
+			if ( button & 1 ) {
+				return 1;
+			}
+
+			if ( button & 2 ) {
+				return 3;
+			}
+
+			if ( button & 4 ) {
+				return 2;
+			}
+
+			return 0;
+		}
+
+		return event.which;
+	}
+}, jQuery.event.addProp );
+
+jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) {
+	jQuery.event.special[ type ] = {
+
+		// Utilize native event if possible so blur/focus sequence is correct
+		setup: function() {
+
+			// Claim the first handler
+			// dataPriv.set( this, "focus", ... )
+			// dataPriv.set( this, "blur", ... )
+			leverageNative( this, type, expectSync );
+
+			// Return false to allow normal processing in the caller
+			return false;
+		},
+		trigger: function() {
+
+			// Force setup before trigger
+			leverageNative( this, type );
+
+			// Return non-false to allow normal event-path propagation
+			return true;
+		},
+
+		delegateType: delegateType
+	};
+} );
+
+// Create mouseenter/leave events using mouseover/out and event-time checks
+// so that event delegation works in jQuery.
+// Do the same for pointerenter/pointerleave and pointerover/pointerout
+//
+// Support: Safari 7 only
+// Safari sends mouseenter too often; see:
+// https://bugs.chromium.org/p/chromium/issues/detail?id=470258
+// for the description of the bug (it existed in older Chrome versions as well).
+jQuery.each( {
+	mouseenter: "mouseover",
+	mouseleave: "mouseout",
+	pointerenter: "pointerover",
+	pointerleave: "pointerout"
+}, function( orig, fix ) {
+	jQuery.event.special[ orig ] = {
+		delegateType: fix,
+		bindType: fix,
+
+		handle: function( event ) {
+			var ret,
+				target = this,
+				related = event.relatedTarget,
+				handleObj = event.handleObj;
+
+			// For mouseenter/leave call the handler if related is outside the target.
+			// NB: No relatedTarget if the mouse left/entered the browser window
+			if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) {
+				event.type = handleObj.origType;
+				ret = handleObj.handler.apply( this, arguments );
+				event.type = fix;
+			}
+			return ret;
+		}
+	};
+} );
+
+jQuery.fn.extend( {
+
+	on: function( types, selector, data, fn ) {
+		return on( this, types, selector, data, fn );
+	},
+	one: function( types, selector, data, fn ) {
+		return on( this, types, selector, data, fn, 1 );
+	},
+	off: function( types, selector, fn ) {
+		var handleObj, type;
+		if ( types && types.preventDefault && types.handleObj ) {
+
+			// ( event )  dispatched jQuery.Event
+			handleObj = types.handleObj;
+			jQuery( types.delegateTarget ).off(
+				handleObj.namespace ?
+					handleObj.origType + "." + handleObj.namespace :
+					handleObj.origType,
+				handleObj.selector,
+				handleObj.handler
+			);
+			return this;
+		}
+		if ( typeof types === "object" ) {
+
+			// ( types-object [, selector] )
+			for ( type in types ) {
+				this.off( type, selector, types[ type ] );
+			}
+			return this;
+		}
+		if ( selector === false || typeof selector === "function" ) {
+
+			// ( types [, fn] )
+			fn = selector;
+			selector = undefined;
+		}
+		if ( fn === false ) {
+			fn = returnFalse;
+		}
+		return this.each( function() {
+			jQuery.event.remove( this, types, fn, selector );
+		} );
+	}
+} );
+
+
+var
+
+	// Support: IE <=10 - 11, Edge 12 - 13 only
+	// In IE/Edge using regex groups here causes severe slowdowns.
+	// See https://connect.microsoft.com/IE/feedback/details/1736512/
+	rnoInnerhtml = /<script|<style|<link/i,
+
+	// checked="checked" or checked
+	rchecked = /checked\s*(?:[^=]|=\s*.checked.)/i,
+	rcleanScript = /^\s*<!(?:\[CDATA\[|--)|(?:\]\]|--)>\s*$/g;
+
+// Prefer a tbody over its parent table for containing new rows
+function manipulationTarget( elem, content ) {
+	if ( nodeName( elem, "table" ) &&
+		nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ) {
+
+		return jQuery( elem ).children( "tbody" )[ 0 ] || elem;
+	}
+
+	return elem;
+}
+
+// Replace/restore the type attribute of script elements for safe DOM manipulation
+function disableScript( elem ) {
+	elem.type = ( elem.getAttribute( "type" ) !== null ) + "/" + elem.type;
+	return elem;
+}
+function restoreScript( elem ) {
+	if ( ( elem.type || "" ).slice( 0, 5 ) === "true/" ) {
+		elem.type = elem.type.slice( 5 );
+	} else {
+		elem.removeAttribute( "type" );
+	}
+
+	return elem;
+}
+
+function cloneCopyEvent( src, dest ) {
+	var i, l, type, pdataOld, udataOld, udataCur, events;
+
+	if ( dest.nodeType !== 1 ) {
+		return;
+	}
+
+	// 1. Copy private data: events, handlers, etc.
+	if ( dataPriv.hasData( src ) ) {
+		pdataOld = dataPriv.get( src );
+		events = pdataOld.events;
+
+		if ( events ) {
+			dataPriv.remove( dest, "handle events" );
+
+			for ( type in events ) {
+				for ( i = 0, l = events[ type ].length; i < l; i++ ) {
+					jQuery.event.add( dest, type, events[ type ][ i ] );
+				}
+			}
+		}
+	}
+
+	// 2. Copy user data
+	if ( dataUser.hasData( src ) ) {
+		udataOld = dataUser.access( src );
+		udataCur = jQuery.extend( {}, udataOld );
+
+		dataUser.set( dest, udataCur );
+	}
+}
+
+// Fix IE bugs, see support tests
+function fixInput( src, dest ) {
+	var nodeName = dest.nodeName.toLowerCase();
+
+	// Fails to persist the checked state of a cloned checkbox or radio button.
+	if ( nodeName === "input" && rcheckableType.test( src.type ) ) {
+		dest.checked = src.checked;
+
+	// Fails to return the selected option to the default selected state when cloning options
+	} else if ( nodeName === "input" || nodeName === "textarea" ) {
+		dest.defaultValue = src.defaultValue;
+	}
+}
+
+function domManip( collection, args, callback, ignored ) {
+
+	// Flatten any nested arrays
+	args = flat( args );
+
+	var fragment, first, scripts, hasScripts, node, doc,
+		i = 0,
+		l = collection.length,
+		iNoClone = l - 1,
+		value = args[ 0 ],
+		valueIsFunction = isFunction( value );
+
+	// We can't cloneNode fragments that contain checked, in WebKit
+	if ( valueIsFunction ||
+			( l > 1 && typeof value === "string" &&
+				!support.checkClone && rchecked.test( value ) ) ) {
+		return collection.each( function( index ) {
+			var self = collection.eq( index );
+			if ( valueIsFunction ) {
+				args[ 0 ] = value.call( this, index, self.html() );
+			}
+			domManip( self, args, callback, ignored );
+		} );
+	}
+
+	if ( l ) {
+		fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored );
+		first = fragment.firstChild;
+
+		if ( fragment.childNodes.length === 1 ) {
+			fragment = first;
+		}
+
+		// Require either new content or an interest in ignored elements to invoke the callback
+		if ( first || ignored ) {
+			scripts = jQuery.map( getAll( fragment, "script" ), disableScript );
+			hasScripts = scripts.length;
+
+			// Use the original fragment for the last item
+			// instead of the first because it can end up
+			// being emptied incorrectly in certain situations (#8070).
+			for ( ; i < l; i++ ) {
+				node = fragment;
+
+				if ( i !== iNoClone ) {
+					node = jQuery.clone( node, true, true );
+
+					// Keep references to cloned scripts for later restoration
+					if ( hasScripts ) {
+
+						// Support: Android <=4.0 only, PhantomJS 1 only
+						// push.apply(_, arraylike) throws on ancient WebKit
+						jQuery.merge( scripts, getAll( node, "script" ) );
+					}
+				}
+
+				callback.call( collection[ i ], node, i );
+			}
+
+			if ( hasScripts ) {
+				doc = scripts[ scripts.length - 1 ].ownerDocument;
+
+				// Reenable scripts
+				jQuery.map( scripts, restoreScript );
+
+				// Evaluate executable scripts on first document insertion
+				for ( i = 0; i < hasScripts; i++ ) {
+					node = scripts[ i ];
+					if ( rscriptType.test( node.type || "" ) &&
+						!dataPriv.access( node, "globalEval" ) &&
+						jQuery.contains( doc, node ) ) {
+
+						if ( node.src && ( node.type || "" ).toLowerCase()  !== "module" ) {
+
+							// Optional AJAX dependency, but won't run scripts if not present
+							if ( jQuery._evalUrl && !node.noModule ) {
+								jQuery._evalUrl( node.src, {
+									nonce: node.nonce || node.getAttribute( "nonce" )
+								}, doc );
+							}
+						} else {
+							DOMEval( node.textContent.replace( rcleanScript, "" ), node, doc );
+						}
+					}
+				}
+			}
+		}
+	}
+
+	return collection;
+}
+
+function remove( elem, selector, keepData ) {
+	var node,
+		nodes = selector ? jQuery.filter( selector, elem ) : elem,
+		i = 0;
+
+	for ( ; ( node = nodes[ i ] ) != null; i++ ) {
+		if ( !keepData && node.nodeType === 1 ) {
+			jQuery.cleanData( getAll( node ) );
+		}
+
+		if ( node.parentNode ) {
+			if ( keepData && isAttached( node ) ) {
+				setGlobalEval( getAll( node, "script" ) );
+			}
+			node.parentNode.removeChild( node );
+		}
+	}
+
+	return elem;
+}
+
+jQuery.extend( {
+	htmlPrefilter: function( html ) {
+		return html;
+	},
+
+	clone: function( elem, dataAndEvents, deepDataAndEvents ) {
+		var i, l, srcElements, destElements,
+			clone = elem.cloneNode( true ),
+			inPage = isAttached( elem );
+
+		// Fix IE cloning issues
+		if ( !support.noCloneChecked && ( elem.nodeType === 1 || elem.nodeType === 11 ) &&
+				!jQuery.isXMLDoc( elem ) ) {
+
+			// We eschew Sizzle here for performance reasons: https://jsperf.com/getall-vs-sizzle/2
+			destElements = getAll( clone );
+			srcElements = getAll( elem );
+
+			for ( i = 0, l = srcElements.length; i < l; i++ ) {
+				fixInput( srcElements[ i ], destElements[ i ] );
+			}
+		}
+
+		// Copy the events from the original to the clone
+		if ( dataAndEvents ) {
+			if ( deepDataAndEvents ) {
+				srcElements = srcElements || getAll( elem );
+				destElements = destElements || getAll( clone );
+
+				for ( i = 0, l = srcElements.length; i < l; i++ ) {
+					cloneCopyEvent( srcElements[ i ], destElements[ i ] );
+				}
+			} else {
+				cloneCopyEvent( elem, clone );
+			}
+		}
+
+		// Preserve script evaluation history
+		destElements = getAll( clone, "script" );
+		if ( destElements.length > 0 ) {
+			setGlobalEval( destElements, !inPage && getAll( elem, "script" ) );
+		}
+
+		// Return the cloned set
+		return clone;
+	},
+
+	cleanData: function( elems ) {
+		var data, elem, type,
+			special = jQuery.event.special,
+			i = 0;
+
+		for ( ; ( elem = elems[ i ] ) !== undefined; i++ ) {
+			if ( acceptData( elem ) ) {
+				if ( ( data = elem[ dataPriv.expando ] ) ) {
+					if ( data.events ) {
+						for ( type in data.events ) {
+							if ( special[ type ] ) {
+								jQuery.event.remove( elem, type );
+
+							// This is a shortcut to avoid jQuery.event.remove's overhead
+							} else {
+								jQuery.removeEvent( elem, type, data.handle );
+							}
+						}
+					}
+
+					// Support: Chrome <=35 - 45+
+					// Assign undefined instead of using delete, see Data#remove
+					elem[ dataPriv.expando ] = undefined;
+				}
+				if ( elem[ dataUser.expando ] ) {
+
+					// Support: Chrome <=35 - 45+
+					// Assign undefined instead of using delete, see Data#remove
+					elem[ dataUser.expando ] = undefined;
+				}
+			}
+		}
+	}
+} );
+
+jQuery.fn.extend( {
+	detach: function( selector ) {
+		return remove( this, selector, true );
+	},
+
+	remove: function( selector ) {
+		return remove( this, selector );
+	},
+
+	text: function( value ) {
+		return access( this, function( value ) {
+			return value === undefined ?
+				jQuery.text( this ) :
+				this.empty().each( function() {
+					if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) {
+						this.textContent = value;
+					}
+				} );
+		}, null, value, arguments.length );
+	},
+
+	append: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) {
+				var target = manipulationTarget( this, elem );
+				target.appendChild( elem );
+			}
+		} );
+	},
+
+	prepend: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) {
+				var target = manipulationTarget( this, elem );
+				target.insertBefore( elem, target.firstChild );
+			}
+		} );
+	},
+
+	before: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.parentNode ) {
+				this.parentNode.insertBefore( elem, this );
+			}
+		} );
+	},
+
+	after: function() {
+		return domManip( this, arguments, function( elem ) {
+			if ( this.parentNode ) {
+				this.parentNode.insertBefore( elem, this.nextSibling );
+			}
+		} );
+	},
+
+	empty: function() {
+		var elem,
+			i = 0;
+
+		for ( ; ( elem = this[ i ] ) != null; i++ ) {
+			if ( elem.nodeType === 1 ) {
+
+				// Prevent memory leaks
+				jQuery.cleanData( getAll( elem, false ) );
+
+				// Remove any remaining nodes
+				elem.textContent = "";
+			}
+		}
+
+		return this;
+	},
+
+	clone: function( dataAndEvents, deepDataAndEvents ) {
+		dataAndEvents = dataAndEvents == null ? false : dataAndEvents;
+		deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents;
+
+		return this.map( function() {
+			return jQuery.clone( this, dataAndEvents, deepDataAndEvents );
+		} );
+	},
+
+	html: function( value ) {
+		return access( this, function( value ) {
+			var elem = this[ 0 ] || {},
+				i = 0,
+				l = this.length;
+
+			if ( value === undefined && elem.nodeType === 1 ) {
+				return elem.innerHTML;
+			}
+
+			// See if we can take a shortcut and just use innerHTML
+			if ( typeof value === "string" && !rnoInnerhtml.test( value ) &&
+				!wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) {
+
+				value = jQuery.htmlPrefilter( value );
+
+				try {
+					for ( ; i < l; i++ ) {
+						elem = this[ i ] || {};
+
+						// Remove element nodes and prevent memory leaks
+						if ( elem.nodeType === 1 ) {
+							jQuery.cleanData( getAll( elem, false ) );
+							elem.innerHTML = value;
+						}
+					}
+
+					elem = 0;
+
+				// If using innerHTML throws an exception, use the fallback method
+				} catch ( e ) {}
+			}
+
+			if ( elem ) {
+				this.empty().append( value );
+			}
+		}, null, value, arguments.length );
+	},
+
+	replaceWith: function() {
+		var ignored = [];
+
+		// Make the changes, replacing each non-ignored context element with the new content
+		return domManip( this, arguments, function( elem ) {
+			var parent = this.parentNode;
+
+			if ( jQuery.inArray( this, ignored ) < 0 ) {
+				jQuery.cleanData( getAll( this ) );
+				if ( parent ) {
+					parent.replaceChild( elem, this );
+				}
+			}
+
+		// Force callback invocation
+		}, ignored );
+	}
+} );
+
+jQuery.each( {
+	appendTo: "append",
+	prependTo: "prepend",
+	insertBefore: "before",
+	insertAfter: "after",
+	replaceAll: "replaceWith"
+}, function( name, original ) {
+	jQuery.fn[ name ] = function( selector ) {
+		var elems,
+			ret = [],
+			insert = jQuery( selector ),
+			last = insert.length - 1,
+			i = 0;
+
+		for ( ; i <= last; i++ ) {
+			elems = i === last ? this : this.clone( true );
+			jQuery( insert[ i ] )[ original ]( elems );
+
+			// Support: Android <=4.0 only, PhantomJS 1 only
+			// .get() because push.apply(_, arraylike) throws on ancient WebKit
+			push.apply( ret, elems.get() );
+		}
+
+		return this.pushStack( ret );
+	};
+} );
+var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" );
+
+var getStyles = function( elem ) {
+
+		// Support: IE <=11 only, Firefox <=30 (#15098, #14150)
+		// IE throws on elements created in popups
+		// FF meanwhile throws on frame elements through "defaultView.getComputedStyle"
+		var view = elem.ownerDocument.defaultView;
+
+		if ( !view || !view.opener ) {
+			view = window;
+		}
+
+		return view.getComputedStyle( elem );
+	};
+
+var swap = function( elem, options, callback ) {
+	var ret, name,
+		old = {};
+
+	// Remember the old values, and insert the new ones
+	for ( name in options ) {
+		old[ name ] = elem.style[ name ];
+		elem.style[ name ] = options[ name ];
+	}
+
+	ret = callback.call( elem );
+
+	// Revert the old values
+	for ( name in options ) {
+		elem.style[ name ] = old[ name ];
+	}
+
+	return ret;
+};
+
+
+var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" );
+
+
+
+( function() {
+
+	// Executing both pixelPosition & boxSizingReliable tests require only one layout
+	// so they're executed at the same time to save the second computation.
+	function computeStyleTests() {
+
+		// This is a singleton, we need to execute it only once
+		if ( !div ) {
+			return;
+		}
+
+		container.style.cssText = "position:absolute;left:-11111px;width:60px;" +
+			"margin-top:1px;padding:0;border:0";
+		div.style.cssText =
+			"position:relative;display:block;box-sizing:border-box;overflow:scroll;" +
+			"margin:auto;border:1px;padding:1px;" +
+			"width:60%;top:1%";
+		documentElement.appendChild( container ).appendChild( div );
+
+		var divStyle = window.getComputedStyle( div );
+		pixelPositionVal = divStyle.top !== "1%";
+
+		// Support: Android 4.0 - 4.3 only, Firefox <=3 - 44
+		reliableMarginLeftVal = roundPixelMeasures( divStyle.marginLeft ) === 12;
+
+		// Support: Android 4.0 - 4.3 only, Safari <=9.1 - 10.1, iOS <=7.0 - 9.3
+		// Some styles come back with percentage values, even though they shouldn't
+		div.style.right = "60%";
+		pixelBoxStylesVal = roundPixelMeasures( divStyle.right ) === 36;
+
+		// Support: IE 9 - 11 only
+		// Detect misreporting of content dimensions for box-sizing:border-box elements
+		boxSizingReliableVal = roundPixelMeasures( divStyle.width ) === 36;
+
+		// Support: IE 9 only
+		// Detect overflow:scroll screwiness (gh-3699)
+		// Support: Chrome <=64
+		// Don't get tricked when zoom affects offsetWidth (gh-4029)
+		div.style.position = "absolute";
+		scrollboxSizeVal = roundPixelMeasures( div.offsetWidth / 3 ) === 12;
+
+		documentElement.removeChild( container );
+
+		// Nullify the div so it wouldn't be stored in the memory and
+		// it will also be a sign that checks already performed
+		div = null;
+	}
+
+	function roundPixelMeasures( measure ) {
+		return Math.round( parseFloat( measure ) );
+	}
+
+	var pixelPositionVal, boxSizingReliableVal, scrollboxSizeVal, pixelBoxStylesVal,
+		reliableTrDimensionsVal, reliableMarginLeftVal,
+		container = document.createElement( "div" ),
+		div = document.createElement( "div" );
+
+	// Finish early in limited (non-browser) environments
+	if ( !div.style ) {
+		return;
+	}
+
+	// Support: IE <=9 - 11 only
+	// Style of cloned element affects source element cloned (#8908)
+	div.style.backgroundClip = "content-box";
+	div.cloneNode( true ).style.backgroundClip = "";
+	support.clearCloneStyle = div.style.backgroundClip === "content-box";
+
+	jQuery.extend( support, {
+		boxSizingReliable: function() {
+			computeStyleTests();
+			return boxSizingReliableVal;
+		},
+		pixelBoxStyles: function() {
+			computeStyleTests();
+			return pixelBoxStylesVal;
+		},
+		pixelPosition: function() {
+			computeStyleTests();
+			return pixelPositionVal;
+		},
+		reliableMarginLeft: function() {
+			computeStyleTests();
+			return reliableMarginLeftVal;
+		},
+		scrollboxSize: function() {
+			computeStyleTests();
+			return scrollboxSizeVal;
+		},
+
+		// Support: IE 9 - 11+, Edge 15 - 18+
+		// IE/Edge misreport `getComputedStyle` of table rows with width/height
+		// set in CSS while `offset*` properties report correct values.
+		// Behavior in IE 9 is more subtle than in newer versions & it passes
+		// some versions of this test; make sure not to make it pass there!
+		reliableTrDimensions: function() {
+			var table, tr, trChild, trStyle;
+			if ( reliableTrDimensionsVal == null ) {
+				table = document.createElement( "table" );
+				tr = document.createElement( "tr" );
+				trChild = document.createElement( "div" );
+
+				table.style.cssText = "position:absolute;left:-11111px";
+				tr.style.height = "1px";
+				trChild.style.height = "9px";
+
+				documentElement
+					.appendChild( table )
+					.appendChild( tr )
+					.appendChild( trChild );
+
+				trStyle = window.getComputedStyle( tr );
+				reliableTrDimensionsVal = parseInt( trStyle.height ) > 3;
+
+				documentElement.removeChild( table );
+			}
+			return reliableTrDimensionsVal;
+		}
+	} );
+} )();
+
+
+function curCSS( elem, name, computed ) {
+	var width, minWidth, maxWidth, ret,
+
+		// Support: Firefox 51+
+		// Retrieving style before computed somehow
+		// fixes an issue with getting wrong values
+		// on detached elements
+		style = elem.style;
+
+	computed = computed || getStyles( elem );
+
+	// getPropertyValue is needed for:
+	//   .css('filter') (IE 9 only, #12537)
+	//   .css('--customProperty) (#3144)
+	if ( computed ) {
+		ret = computed.getPropertyValue( name ) || computed[ name ];
+
+		if ( ret === "" && !isAttached( elem ) ) {
+			ret = jQuery.style( elem, name );
+		}
+
+		// A tribute to the "awesome hack by Dean Edwards"
+		// Android Browser returns percentage for some values,
+		// but width seems to be reliably pixels.
+		// This is against the CSSOM draft spec:
+		// https://drafts.csswg.org/cssom/#resolved-values
+		if ( !support.pixelBoxStyles() && rnumnonpx.test( ret ) && rboxStyle.test( name ) ) {
+
+			// Remember the original values
+			width = style.width;
+			minWidth = style.minWidth;
+			maxWidth = style.maxWidth;
+
+			// Put in the new values to get a computed value out
+			style.minWidth = style.maxWidth = style.width = ret;
+			ret = computed.width;
+
+			// Revert the changed values
+			style.width = width;
+			style.minWidth = minWidth;
+			style.maxWidth = maxWidth;
+		}
+	}
+
+	return ret !== undefined ?
+
+		// Support: IE <=9 - 11 only
+		// IE returns zIndex value as an integer.
+		ret + "" :
+		ret;
+}
+
+
+function addGetHookIf( conditionFn, hookFn ) {
+
+	// Define the hook, we'll check on the first run if it's really needed.
+	return {
+		get: function() {
+			if ( conditionFn() ) {
+
+				// Hook not needed (or it's not possible to use it due
+				// to missing dependency), remove it.
+				delete this.get;
+				return;
+			}
+
+			// Hook needed; redefine it so that the support test is not executed again.
+			return ( this.get = hookFn ).apply( this, arguments );
+		}
+	};
+}
+
+
+var cssPrefixes = [ "Webkit", "Moz", "ms" ],
+	emptyStyle = document.createElement( "div" ).style,
+	vendorProps = {};
+
+// Return a vendor-prefixed property or undefined
+function vendorPropName( name ) {
+
+	// Check for vendor prefixed names
+	var capName = name[ 0 ].toUpperCase() + name.slice( 1 ),
+		i = cssPrefixes.length;
+
+	while ( i-- ) {
+		name = cssPrefixes[ i ] + capName;
+		if ( name in emptyStyle ) {
+			return name;
+		}
+	}
+}
+
+// Return a potentially-mapped jQuery.cssProps or vendor prefixed property
+function finalPropName( name ) {
+	var final = jQuery.cssProps[ name ] || vendorProps[ name ];
+
+	if ( final ) {
+		return final;
+	}
+	if ( name in emptyStyle ) {
+		return name;
+	}
+	return vendorProps[ name ] = vendorPropName( name ) || name;
+}
+
+
+var
+
+	// Swappable if display is none or starts with table
+	// except "table", "table-cell", or "table-caption"
+	// See here for display values: https://developer.mozilla.org/en-US/docs/CSS/display
+	rdisplayswap = /^(none|table(?!-c[ea]).+)/,
+	rcustomProp = /^--/,
+	cssShow = { position: "absolute", visibility: "hidden", display: "block" },
+	cssNormalTransform = {
+		letterSpacing: "0",
+		fontWeight: "400"
+	};
+
+function setPositiveNumber( _elem, value, subtract ) {
+
+	// Any relative (+/-) values have already been
+	// normalized at this point
+	var matches = rcssNum.exec( value );
+	return matches ?
+
+		// Guard against undefined "subtract", e.g., when used as in cssHooks
+		Math.max( 0, matches[ 2 ] - ( subtract || 0 ) ) + ( matches[ 3 ] || "px" ) :
+		value;
+}
+
+function boxModelAdjustment( elem, dimension, box, isBorderBox, styles, computedVal ) {
+	var i = dimension === "width" ? 1 : 0,
+		extra = 0,
+		delta = 0;
+
+	// Adjustment may not be necessary
+	if ( box === ( isBorderBox ? "border" : "content" ) ) {
+		return 0;
+	}
+
+	for ( ; i < 4; i += 2 ) {
+
+		// Both box models exclude margin
+		if ( box === "margin" ) {
+			delta += jQuery.css( elem, box + cssExpand[ i ], true, styles );
+		}
+
+		// If we get here with a content-box, we're seeking "padding" or "border" or "margin"
+		if ( !isBorderBox ) {
+
+			// Add padding
+			delta += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles );
+
+			// For "border" or "margin", add border
+			if ( box !== "padding" ) {
+				delta += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles );
+
+			// But still keep track of it otherwise
+			} else {
+				extra += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles );
+			}
+
+		// If we get here with a border-box (content + padding + border), we're seeking "content" or
+		// "padding" or "margin"
+		} else {
+
+			// For "content", subtract padding
+			if ( box === "content" ) {
+				delta -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles );
+			}
+
+			// For "content" or "padding", subtract border
+			if ( box !== "margin" ) {
+				delta -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles );
+			}
+		}
+	}
+
+	// Account for positive content-box scroll gutter when requested by providing computedVal
+	if ( !isBorderBox && computedVal >= 0 ) {
+
+		// offsetWidth/offsetHeight is a rounded sum of content, padding, scroll gutter, and border
+		// Assuming integer scroll gutter, subtract the rest and round down
+		delta += Math.max( 0, Math.ceil(
+			elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] -
+			computedVal -
+			delta -
+			extra -
+			0.5
+
+		// If offsetWidth/offsetHeight is unknown, then we can't determine content-box scroll gutter
+		// Use an explicit zero to avoid NaN (gh-3964)
+		) ) || 0;
+	}
+
+	return delta;
+}
+
+function getWidthOrHeight( elem, dimension, extra ) {
+
+	// Start with computed style
+	var styles = getStyles( elem ),
+
+		// To avoid forcing a reflow, only fetch boxSizing if we need it (gh-4322).
+		// Fake content-box until we know it's needed to know the true value.
+		boxSizingNeeded = !support.boxSizingReliable() || extra,
+		isBorderBox = boxSizingNeeded &&
+			jQuery.css( elem, "boxSizing", false, styles ) === "border-box",
+		valueIsBorderBox = isBorderBox,
+
+		val = curCSS( elem, dimension, styles ),
+		offsetProp = "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 );
+
+	// Support: Firefox <=54
+	// Return a confounding non-pixel value or feign ignorance, as appropriate.
+	if ( rnumnonpx.test( val ) ) {
+		if ( !extra ) {
+			return val;
+		}
+		val = "auto";
+	}
+
+
+	// Support: IE 9 - 11 only
+	// Use offsetWidth/offsetHeight for when box sizing is unreliable.
+	// In those cases, the computed value can be trusted to be border-box.
+	if ( ( !support.boxSizingReliable() && isBorderBox ||
+
+		// Support: IE 10 - 11+, Edge 15 - 18+
+		// IE/Edge misreport `getComputedStyle` of table rows with width/height
+		// set in CSS while `offset*` properties report correct values.
+		// Interestingly, in some cases IE 9 doesn't suffer from this issue.
+		!support.reliableTrDimensions() && nodeName( elem, "tr" ) ||
+
+		// Fall back to offsetWidth/offsetHeight when value is "auto"
+		// This happens for inline elements with no explicit setting (gh-3571)
+		val === "auto" ||
+
+		// Support: Android <=4.1 - 4.3 only
+		// Also use offsetWidth/offsetHeight for misreported inline dimensions (gh-3602)
+		!parseFloat( val ) && jQuery.css( elem, "display", false, styles ) === "inline" ) &&
+
+		// Make sure the element is visible & connected
+		elem.getClientRects().length ) {
+
+		isBorderBox = jQuery.css( elem, "boxSizing", false, styles ) === "border-box";
+
+		// Where available, offsetWidth/offsetHeight approximate border box dimensions.
+		// Where not available (e.g., SVG), assume unreliable box-sizing and interpret the
+		// retrieved value as a content box dimension.
+		valueIsBorderBox = offsetProp in elem;
+		if ( valueIsBorderBox ) {
+			val = elem[ offsetProp ];
+		}
+	}
+
+	// Normalize "" and auto
+	val = parseFloat( val ) || 0;
+
+	// Adjust for the element's box model
+	return ( val +
+		boxModelAdjustment(
+			elem,
+			dimension,
+			extra || ( isBorderBox ? "border" : "content" ),
+			valueIsBorderBox,
+			styles,
+
+			// Provide the current computed size to request scroll gutter calculation (gh-3589)
+			val
+		)
+	) + "px";
+}
+
+jQuery.extend( {
+
+	// Add in style property hooks for overriding the default
+	// behavior of getting and setting a style property
+	cssHooks: {
+		opacity: {
+			get: function( elem, computed ) {
+				if ( computed ) {
+
+					// We should always get a number back from opacity
+					var ret = curCSS( elem, "opacity" );
+					return ret === "" ? "1" : ret;
+				}
+			}
+		}
+	},
+
+	// Don't automatically add "px" to these possibly-unitless properties
+	cssNumber: {
+		"animationIterationCount": true,
+		"columnCount": true,
+		"fillOpacity": true,
+		"flexGrow": true,
+		"flexShrink": true,
+		"fontWeight": true,
+		"gridArea": true,
+		"gridColumn": true,
+		"gridColumnEnd": true,
+		"gridColumnStart": true,
+		"gridRow": true,
+		"gridRowEnd": true,
+		"gridRowStart": true,
+		"lineHeight": true,
+		"opacity": true,
+		"order": true,
+		"orphans": true,
+		"widows": true,
+		"zIndex": true,
+		"zoom": true
+	},
+
+	// Add in properties whose names you wish to fix before
+	// setting or getting the value
+	cssProps: {},
+
+	// Get and set the style property on a DOM Node
+	style: function( elem, name, value, extra ) {
+
+		// Don't set styles on text and comment nodes
+		if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) {
+			return;
+		}
+
+		// Make sure that we're working with the right name
+		var ret, type, hooks,
+			origName = camelCase( name ),
+			isCustomProp = rcustomProp.test( name ),
+			style = elem.style;
+
+		// Make sure that we're working with the right name. We don't
+		// want to query the value if it is a CSS custom property
+		// since they are user-defined.
+		if ( !isCustomProp ) {
+			name = finalPropName( origName );
+		}
+
+		// Gets hook for the prefixed version, then unprefixed version
+		hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ];
+
+		// Check if we're setting a value
+		if ( value !== undefined ) {
+			type = typeof value;
+
+			// Convert "+=" or "-=" to relative numbers (#7345)
+			if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) {
+				value = adjustCSS( elem, name, ret );
+
+				// Fixes bug #9237
+				type = "number";
+			}
+
+			// Make sure that null and NaN values aren't set (#7116)
+			if ( value == null || value !== value ) {
+				return;
+			}
+
+			// If a number was passed in, add the unit (except for certain CSS properties)
+			// The isCustomProp check can be removed in jQuery 4.0 when we only auto-append
+			// "px" to a few hardcoded values.
+			if ( type === "number" && !isCustomProp ) {
+				value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" );
+			}
+
+			// background-* props affect original clone's values
+			if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) {
+				style[ name ] = "inherit";
+			}
+
+			// If a hook was provided, use that value, otherwise just set the specified value
+			if ( !hooks || !( "set" in hooks ) ||
+				( value = hooks.set( elem, value, extra ) ) !== undefined ) {
+
+				if ( isCustomProp ) {
+					style.setProperty( name, value );
+				} else {
+					style[ name ] = value;
+				}
+			}
+
+		} else {
+
+			// If a hook was provided get the non-computed value from there
+			if ( hooks && "get" in hooks &&
+				( ret = hooks.get( elem, false, extra ) ) !== undefined ) {
+
+				return ret;
+			}
+
+			// Otherwise just get the value from the style object
+			return style[ name ];
+		}
+	},
+
+	css: function( elem, name, extra, styles ) {
+		var val, num, hooks,
+			origName = camelCase( name ),
+			isCustomProp = rcustomProp.test( name );
+
+		// Make sure that we're working with the right name. We don't
+		// want to modify the value if it is a CSS custom property
+		// since they are user-defined.
+		if ( !isCustomProp ) {
+			name = finalPropName( origName );
+		}
+
+		// Try prefixed name followed by the unprefixed name
+		hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ];
+
+		// If a hook was provided get the computed value from there
+		if ( hooks && "get" in hooks ) {
+			val = hooks.get( elem, true, extra );
+		}
+
+		// Otherwise, if a way to get the computed value exists, use that
+		if ( val === undefined ) {
+			val = curCSS( elem, name, styles );
+		}
+
+		// Convert "normal" to computed value
+		if ( val === "normal" && name in cssNormalTransform ) {
+			val = cssNormalTransform[ name ];
+		}
+
+		// Make numeric if forced or a qualifier was provided and val looks numeric
+		if ( extra === "" || extra ) {
+			num = parseFloat( val );
+			return extra === true || isFinite( num ) ? num || 0 : val;
+		}
+
+		return val;
+	}
+} );
+
+jQuery.each( [ "height", "width" ], function( _i, dimension ) {
+	jQuery.cssHooks[ dimension ] = {
+		get: function( elem, computed, extra ) {
+			if ( computed ) {
+
+				// Certain elements can have dimension info if we invisibly show them
+				// but it must have a current display style that would benefit
+				return rdisplayswap.test( jQuery.css( elem, "display" ) ) &&
+
+					// Support: Safari 8+
+					// Table columns in Safari have non-zero offsetWidth & zero
+					// getBoundingClientRect().width unless display is changed.
+					// Support: IE <=11 only
+					// Running getBoundingClientRect on a disconnected node
+					// in IE throws an error.
+					( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ?
+						swap( elem, cssShow, function() {
+							return getWidthOrHeight( elem, dimension, extra );
+						} ) :
+						getWidthOrHeight( elem, dimension, extra );
+			}
+		},
+
+		set: function( elem, value, extra ) {
+			var matches,
+				styles = getStyles( elem ),
+
+				// Only read styles.position if the test has a chance to fail
+				// to avoid forcing a reflow.
+				scrollboxSizeBuggy = !support.scrollboxSize() &&
+					styles.position === "absolute",
+
+				// To avoid forcing a reflow, only fetch boxSizing if we need it (gh-3991)
+				boxSizingNeeded = scrollboxSizeBuggy || extra,
+				isBorderBox = boxSizingNeeded &&
+					jQuery.css( elem, "boxSizing", false, styles ) === "border-box",
+				subtract = extra ?
+					boxModelAdjustment(
+						elem,
+						dimension,
+						extra,
+						isBorderBox,
+						styles
+					) :
+					0;
+
+			// Account for unreliable border-box dimensions by comparing offset* to computed and
+			// faking a content-box to get border and padding (gh-3699)
+			if ( isBorderBox && scrollboxSizeBuggy ) {
+				subtract -= Math.ceil(
+					elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] -
+					parseFloat( styles[ dimension ] ) -
+					boxModelAdjustment( elem, dimension, "border", false, styles ) -
+					0.5
+				);
+			}
+
+			// Convert to pixels if value adjustment is needed
+			if ( subtract && ( matches = rcssNum.exec( value ) ) &&
+				( matches[ 3 ] || "px" ) !== "px" ) {
+
+				elem.style[ dimension ] = value;
+				value = jQuery.css( elem, dimension );
+			}
+
+			return setPositiveNumber( elem, value, subtract );
+		}
+	};
+} );
+
+jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft,
+	function( elem, computed ) {
+		if ( computed ) {
+			return ( parseFloat( curCSS( elem, "marginLeft" ) ) ||
+				elem.getBoundingClientRect().left -
+					swap( elem, { marginLeft: 0 }, function() {
+						return elem.getBoundingClientRect().left;
+					} )
+				) + "px";
+		}
+	}
+);
+
+// These hooks are used by animate to expand properties
+jQuery.each( {
+	margin: "",
+	padding: "",
+	border: "Width"
+}, function( prefix, suffix ) {
+	jQuery.cssHooks[ prefix + suffix ] = {
+		expand: function( value ) {
+			var i = 0,
+				expanded = {},
+
+				// Assumes a single number if not a string
+				parts = typeof value === "string" ? value.split( " " ) : [ value ];
+
+			for ( ; i < 4; i++ ) {
+				expanded[ prefix + cssExpand[ i ] + suffix ] =
+					parts[ i ] || parts[ i - 2 ] || parts[ 0 ];
+			}
+
+			return expanded;
+		}
+	};
+
+	if ( prefix !== "margin" ) {
+		jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber;
+	}
+} );
+
+jQuery.fn.extend( {
+	css: function( name, value ) {
+		return access( this, function( elem, name, value ) {
+			var styles, len,
+				map = {},
+				i = 0;
+
+			if ( Array.isArray( name ) ) {
+				styles = getStyles( elem );
+				len = name.length;
+
+				for ( ; i < len; i++ ) {
+					map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles );
+				}
+
+				return map;
+			}
+
+			return value !== undefined ?
+				jQuery.style( elem, name, value ) :
+				jQuery.css( elem, name );
+		}, name, value, arguments.length > 1 );
+	}
+} );
+
+
+function Tween( elem, options, prop, end, easing ) {
+	return new Tween.prototype.init( elem, options, prop, end, easing );
+}
+jQuery.Tween = Tween;
+
+Tween.prototype = {
+	constructor: Tween,
+	init: function( elem, options, prop, end, easing, unit ) {
+		this.elem = elem;
+		this.prop = prop;
+		this.easing = easing || jQuery.easing._default;
+		this.options = options;
+		this.start = this.now = this.cur();
+		this.end = end;
+		this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" );
+	},
+	cur: function() {
+		var hooks = Tween.propHooks[ this.prop ];
+
+		return hooks && hooks.get ?
+			hooks.get( this ) :
+			Tween.propHooks._default.get( this );
+	},
+	run: function( percent ) {
+		var eased,
+			hooks = Tween.propHooks[ this.prop ];
+
+		if ( this.options.duration ) {
+			this.pos = eased = jQuery.easing[ this.easing ](
+				percent, this.options.duration * percent, 0, 1, this.options.duration
+			);
+		} else {
+			this.pos = eased = percent;
+		}
+		this.now = ( this.end - this.start ) * eased + this.start;
+
+		if ( this.options.step ) {
+			this.options.step.call( this.elem, this.now, this );
+		}
+
+		if ( hooks && hooks.set ) {
+			hooks.set( this );
+		} else {
+			Tween.propHooks._default.set( this );
+		}
+		return this;
+	}
+};
+
+Tween.prototype.init.prototype = Tween.prototype;
+
+Tween.propHooks = {
+	_default: {
+		get: function( tween ) {
+			var result;
+
+			// Use a property on the element directly when it is not a DOM element,
+			// or when there is no matching style property that exists.
+			if ( tween.elem.nodeType !== 1 ||
+				tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) {
+				return tween.elem[ tween.prop ];
+			}
+
+			// Passing an empty string as a 3rd parameter to .css will automatically
+			// attempt a parseFloat and fallback to a string if the parse fails.
+			// Simple values such as "10px" are parsed to Float;
+			// complex values such as "rotate(1rad)" are returned as-is.
+			result = jQuery.css( tween.elem, tween.prop, "" );
+
+			// Empty strings, null, undefined and "auto" are converted to 0.
+			return !result || result === "auto" ? 0 : result;
+		},
+		set: function( tween ) {
+
+			// Use step hook for back compat.
+			// Use cssHook if its there.
+			// Use .style if available and use plain properties where available.
+			if ( jQuery.fx.step[ tween.prop ] ) {
+				jQuery.fx.step[ tween.prop ]( tween );
+			} else if ( tween.elem.nodeType === 1 && (
+					jQuery.cssHooks[ tween.prop ] ||
+					tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) {
+				jQuery.style( tween.elem, tween.prop, tween.now + tween.unit );
+			} else {
+				tween.elem[ tween.prop ] = tween.now;
+			}
+		}
+	}
+};
+
+// Support: IE <=9 only
+// Panic based approach to setting things on disconnected nodes
+Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = {
+	set: function( tween ) {
+		if ( tween.elem.nodeType && tween.elem.parentNode ) {
+			tween.elem[ tween.prop ] = tween.now;
+		}
+	}
+};
+
+jQuery.easing = {
+	linear: function( p ) {
+		return p;
+	},
+	swing: function( p ) {
+		return 0.5 - Math.cos( p * Math.PI ) / 2;
+	},
+	_default: "swing"
+};
+
+jQuery.fx = Tween.prototype.init;
+
+// Back compat <1.8 extension point
+jQuery.fx.step = {};
+
+
+
+
+var
+	fxNow, inProgress,
+	rfxtypes = /^(?:toggle|show|hide)$/,
+	rrun = /queueHooks$/;
+
+function schedule() {
+	if ( inProgress ) {
+		if ( document.hidden === false && window.requestAnimationFrame ) {
+			window.requestAnimationFrame( schedule );
+		} else {
+			window.setTimeout( schedule, jQuery.fx.interval );
+		}
+
+		jQuery.fx.tick();
+	}
+}
+
+// Animations created synchronously will run synchronously
+function createFxNow() {
+	window.setTimeout( function() {
+		fxNow = undefined;
+	} );
+	return ( fxNow = Date.now() );
+}
+
+// Generate parameters to create a standard animation
+function genFx( type, includeWidth ) {
+	var which,
+		i = 0,
+		attrs = { height: type };
+
+	// If we include width, step value is 1 to do all cssExpand values,
+	// otherwise step value is 2 to skip over Left and Right
+	includeWidth = includeWidth ? 1 : 0;
+	for ( ; i < 4; i += 2 - includeWidth ) {
+		which = cssExpand[ i ];
+		attrs[ "margin" + which ] = attrs[ "padding" + which ] = type;
+	}
+
+	if ( includeWidth ) {
+		attrs.opacity = attrs.width = type;
+	}
+
+	return attrs;
+}
+
+function createTween( value, prop, animation ) {
+	var tween,
+		collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ),
+		index = 0,
+		length = collection.length;
+	for ( ; index < length; index++ ) {
+		if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) {
+
+			// We're done with this property
+			return tween;
+		}
+	}
+}
+
+function defaultPrefilter( elem, props, opts ) {
+	var prop, value, toggle, hooks, oldfire, propTween, restoreDisplay, display,
+		isBox = "width" in props || "height" in props,
+		anim = this,
+		orig = {},
+		style = elem.style,
+		hidden = elem.nodeType && isHiddenWithinTree( elem ),
+		dataShow = dataPriv.get( elem, "fxshow" );
+
+	// Queue-skipping animations hijack the fx hooks
+	if ( !opts.queue ) {
+		hooks = jQuery._queueHooks( elem, "fx" );
+		if ( hooks.unqueued == null ) {
+			hooks.unqueued = 0;
+			oldfire = hooks.empty.fire;
+			hooks.empty.fire = function() {
+				if ( !hooks.unqueued ) {
+					oldfire();
+				}
+			};
+		}
+		hooks.unqueued++;
+
+		anim.always( function() {
+
+			// Ensure the complete handler is called before this completes
+			anim.always( function() {
+				hooks.unqueued--;
+				if ( !jQuery.queue( elem, "fx" ).length ) {
+					hooks.empty.fire();
+				}
+			} );
+		} );
+	}
+
+	// Detect show/hide animations
+	for ( prop in props ) {
+		value = props[ prop ];
+		if ( rfxtypes.test( value ) ) {
+			delete props[ prop ];
+			toggle = toggle || value === "toggle";
+			if ( value === ( hidden ? "hide" : "show" ) ) {
+
+				// Pretend to be hidden if this is a "show" and
+				// there is still data from a stopped show/hide
+				if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) {
+					hidden = true;
+
+				// Ignore all other no-op show/hide data
+				} else {
+					continue;
+				}
+			}
+			orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop );
+		}
+	}
+
+	// Bail out if this is a no-op like .hide().hide()
+	propTween = !jQuery.isEmptyObject( props );
+	if ( !propTween && jQuery.isEmptyObject( orig ) ) {
+		return;
+	}
+
+	// Restrict "overflow" and "display" styles during box animations
+	if ( isBox && elem.nodeType === 1 ) {
+
+		// Support: IE <=9 - 11, Edge 12 - 15
+		// Record all 3 overflow attributes because IE does not infer the shorthand
+		// from identically-valued overflowX and overflowY and Edge just mirrors
+		// the overflowX value there.
+		opts.overflow = [ style.overflow, style.overflowX, style.overflowY ];
+
+		// Identify a display type, preferring old show/hide data over the CSS cascade
+		restoreDisplay = dataShow && dataShow.display;
+		if ( restoreDisplay == null ) {
+			restoreDisplay = dataPriv.get( elem, "display" );
+		}
+		display = jQuery.css( elem, "display" );
+		if ( display === "none" ) {
+			if ( restoreDisplay ) {
+				display = restoreDisplay;
+			} else {
+
+				// Get nonempty value(s) by temporarily forcing visibility
+				showHide( [ elem ], true );
+				restoreDisplay = elem.style.display || restoreDisplay;
+				display = jQuery.css( elem, "display" );
+				showHide( [ elem ] );
+			}
+		}
+
+		// Animate inline elements as inline-block
+		if ( display === "inline" || display === "inline-block" && restoreDisplay != null ) {
+			if ( jQuery.css( elem, "float" ) === "none" ) {
+
+				// Restore the original display value at the end of pure show/hide animations
+				if ( !propTween ) {
+					anim.done( function() {
+						style.display = restoreDisplay;
+					} );
+					if ( restoreDisplay == null ) {
+						display = style.display;
+						restoreDisplay = display === "none" ? "" : display;
+					}
+				}
+				style.display = "inline-block";
+			}
+		}
+	}
+
+	if ( opts.overflow ) {
+		style.overflow = "hidden";
+		anim.always( function() {
+			style.overflow = opts.overflow[ 0 ];
+			style.overflowX = opts.overflow[ 1 ];
+			style.overflowY = opts.overflow[ 2 ];
+		} );
+	}
+
+	// Implement show/hide animations
+	propTween = false;
+	for ( prop in orig ) {
+
+		// General show/hide setup for this element animation
+		if ( !propTween ) {
+			if ( dataShow ) {
+				if ( "hidden" in dataShow ) {
+					hidden = dataShow.hidden;
+				}
+			} else {
+				dataShow = dataPriv.access( elem, "fxshow", { display: restoreDisplay } );
+			}
+
+			// Store hidden/visible for toggle so `.stop().toggle()` "reverses"
+			if ( toggle ) {
+				dataShow.hidden = !hidden;
+			}
+
+			// Show elements before animating them
+			if ( hidden ) {
+				showHide( [ elem ], true );
+			}
+
+			/* eslint-disable no-loop-func */
+
+			anim.done( function() {
+
+			/* eslint-enable no-loop-func */
+
+				// The final step of a "hide" animation is actually hiding the element
+				if ( !hidden ) {
+					showHide( [ elem ] );
+				}
+				dataPriv.remove( elem, "fxshow" );
+				for ( prop in orig ) {
+					jQuery.style( elem, prop, orig[ prop ] );
+				}
+			} );
+		}
+
+		// Per-property setup
+		propTween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim );
+		if ( !( prop in dataShow ) ) {
+			dataShow[ prop ] = propTween.start;
+			if ( hidden ) {
+				propTween.end = propTween.start;
+				propTween.start = 0;
+			}
+		}
+	}
+}
+
+function propFilter( props, specialEasing ) {
+	var index, name, easing, value, hooks;
+
+	// camelCase, specialEasing and expand cssHook pass
+	for ( index in props ) {
+		name = camelCase( index );
+		easing = specialEasing[ name ];
+		value = props[ index ];
+		if ( Array.isArray( value ) ) {
+			easing = value[ 1 ];
+			value = props[ index ] = value[ 0 ];
+		}
+
+		if ( index !== name ) {
+			props[ name ] = value;
+			delete props[ index ];
+		}
+
+		hooks = jQuery.cssHooks[ name ];
+		if ( hooks && "expand" in hooks ) {
+			value = hooks.expand( value );
+			delete props[ name ];
+
+			// Not quite $.extend, this won't overwrite existing keys.
+			// Reusing 'index' because we have the correct "name"
+			for ( index in value ) {
+				if ( !( index in props ) ) {
+					props[ index ] = value[ index ];
+					specialEasing[ index ] = easing;
+				}
+			}
+		} else {
+			specialEasing[ name ] = easing;
+		}
+	}
+}
+
+function Animation( elem, properties, options ) {
+	var result,
+		stopped,
+		index = 0,
+		length = Animation.prefilters.length,
+		deferred = jQuery.Deferred().always( function() {
+
+			// Don't match elem in the :animated selector
+			delete tick.elem;
+		} ),
+		tick = function() {
+			if ( stopped ) {
+				return false;
+			}
+			var currentTime = fxNow || createFxNow(),
+				remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ),
+
+				// Support: Android 2.3 only
+				// Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497)
+				temp = remaining / animation.duration || 0,
+				percent = 1 - temp,
+				index = 0,
+				length = animation.tweens.length;
+
+			for ( ; index < length; index++ ) {
+				animation.tweens[ index ].run( percent );
+			}
+
+			deferred.notifyWith( elem, [ animation, percent, remaining ] );
+
+			// If there's more to do, yield
+			if ( percent < 1 && length ) {
+				return remaining;
+			}
+
+			// If this was an empty animation, synthesize a final progress notification
+			if ( !length ) {
+				deferred.notifyWith( elem, [ animation, 1, 0 ] );
+			}
+
+			// Resolve the animation and report its conclusion
+			deferred.resolveWith( elem, [ animation ] );
+			return false;
+		},
+		animation = deferred.promise( {
+			elem: elem,
+			props: jQuery.extend( {}, properties ),
+			opts: jQuery.extend( true, {
+				specialEasing: {},
+				easing: jQuery.easing._default
+			}, options ),
+			originalProperties: properties,
+			originalOptions: options,
+			startTime: fxNow || createFxNow(),
+			duration: options.duration,
+			tweens: [],
+			createTween: function( prop, end ) {
+				var tween = jQuery.Tween( elem, animation.opts, prop, end,
+						animation.opts.specialEasing[ prop ] || animation.opts.easing );
+				animation.tweens.push( tween );
+				return tween;
+			},
+			stop: function( gotoEnd ) {
+				var index = 0,
+
+					// If we are going to the end, we want to run all the tweens
+					// otherwise we skip this part
+					length = gotoEnd ? animation.tweens.length : 0;
+				if ( stopped ) {
+					return this;
+				}
+				stopped = true;
+				for ( ; index < length; index++ ) {
+					animation.tweens[ index ].run( 1 );
+				}
+
+				// Resolve when we played the last frame; otherwise, reject
+				if ( gotoEnd ) {
+					deferred.notifyWith( elem, [ animation, 1, 0 ] );
+					deferred.resolveWith( elem, [ animation, gotoEnd ] );
+				} else {
+					deferred.rejectWith( elem, [ animation, gotoEnd ] );
+				}
+				return this;
+			}
+		} ),
+		props = animation.props;
+
+	propFilter( props, animation.opts.specialEasing );
+
+	for ( ; index < length; index++ ) {
+		result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts );
+		if ( result ) {
+			if ( isFunction( result.stop ) ) {
+				jQuery._queueHooks( animation.elem, animation.opts.queue ).stop =
+					result.stop.bind( result );
+			}
+			return result;
+		}
+	}
+
+	jQuery.map( props, createTween, animation );
+
+	if ( isFunction( animation.opts.start ) ) {
+		animation.opts.start.call( elem, animation );
+	}
+
+	// Attach callbacks from options
+	animation
+		.progress( animation.opts.progress )
+		.done( animation.opts.done, animation.opts.complete )
+		.fail( animation.opts.fail )
+		.always( animation.opts.always );
+
+	jQuery.fx.timer(
+		jQuery.extend( tick, {
+			elem: elem,
+			anim: animation,
+			queue: animation.opts.queue
+		} )
+	);
+
+	return animation;
+}
+
+jQuery.Animation = jQuery.extend( Animation, {
+
+	tweeners: {
+		"*": [ function( prop, value ) {
+			var tween = this.createTween( prop, value );
+			adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween );
+			return tween;
+		} ]
+	},
+
+	tweener: function( props, callback ) {
+		if ( isFunction( props ) ) {
+			callback = props;
+			props = [ "*" ];
+		} else {
+			props = props.match( rnothtmlwhite );
+		}
+
+		var prop,
+			index = 0,
+			length = props.length;
+
+		for ( ; index < length; index++ ) {
+			prop = props[ index ];
+			Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || [];
+			Animation.tweeners[ prop ].unshift( callback );
+		}
+	},
+
+	prefilters: [ defaultPrefilter ],
+
+	prefilter: function( callback, prepend ) {
+		if ( prepend ) {
+			Animation.prefilters.unshift( callback );
+		} else {
+			Animation.prefilters.push( callback );
+		}
+	}
+} );
+
+jQuery.speed = function( speed, easing, fn ) {
+	var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : {
+		complete: fn || !fn && easing ||
+			isFunction( speed ) && speed,
+		duration: speed,
+		easing: fn && easing || easing && !isFunction( easing ) && easing
+	};
+
+	// Go to the end state if fx are off
+	if ( jQuery.fx.off ) {
+		opt.duration = 0;
+
+	} else {
+		if ( typeof opt.duration !== "number" ) {
+			if ( opt.duration in jQuery.fx.speeds ) {
+				opt.duration = jQuery.fx.speeds[ opt.duration ];
+
+			} else {
+				opt.duration = jQuery.fx.speeds._default;
+			}
+		}
+	}
+
+	// Normalize opt.queue - true/undefined/null -> "fx"
+	if ( opt.queue == null || opt.queue === true ) {
+		opt.queue = "fx";
+	}
+
+	// Queueing
+	opt.old = opt.complete;
+
+	opt.complete = function() {
+		if ( isFunction( opt.old ) ) {
+			opt.old.call( this );
+		}
+
+		if ( opt.queue ) {
+			jQuery.dequeue( this, opt.queue );
+		}
+	};
+
+	return opt;
+};
+
+jQuery.fn.extend( {
+	fadeTo: function( speed, to, easing, callback ) {
+
+		// Show any hidden elements after setting opacity to 0
+		return this.filter( isHiddenWithinTree ).css( "opacity", 0 ).show()
+
+			// Animate to the value specified
+			.end().animate( { opacity: to }, speed, easing, callback );
+	},
+	animate: function( prop, speed, easing, callback ) {
+		var empty = jQuery.isEmptyObject( prop ),
+			optall = jQuery.speed( speed, easing, callback ),
+			doAnimation = function() {
+
+				// Operate on a copy of prop so per-property easing won't be lost
+				var anim = Animation( this, jQuery.extend( {}, prop ), optall );
+
+				// Empty animations, or finishing resolves immediately
+				if ( empty || dataPriv.get( this, "finish" ) ) {
+					anim.stop( true );
+				}
+			};
+			doAnimation.finish = doAnimation;
+
+		return empty || optall.queue === false ?
+			this.each( doAnimation ) :
+			this.queue( optall.queue, doAnimation );
+	},
+	stop: function( type, clearQueue, gotoEnd ) {
+		var stopQueue = function( hooks ) {
+			var stop = hooks.stop;
+			delete hooks.stop;
+			stop( gotoEnd );
+		};
+
+		if ( typeof type !== "string" ) {
+			gotoEnd = clearQueue;
+			clearQueue = type;
+			type = undefined;
+		}
+		if ( clearQueue ) {
+			this.queue( type || "fx", [] );
+		}
+
+		return this.each( function() {
+			var dequeue = true,
+				index = type != null && type + "queueHooks",
+				timers = jQuery.timers,
+				data = dataPriv.get( this );
+
+			if ( index ) {
+				if ( data[ index ] && data[ index ].stop ) {
+					stopQueue( data[ index ] );
+				}
+			} else {
+				for ( index in data ) {
+					if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) {
+						stopQueue( data[ index ] );
+					}
+				}
+			}
+
+			for ( index = timers.length; index--; ) {
+				if ( timers[ index ].elem === this &&
+					( type == null || timers[ index ].queue === type ) ) {
+
+					timers[ index ].anim.stop( gotoEnd );
+					dequeue = false;
+					timers.splice( index, 1 );
+				}
+			}
+
+			// Start the next in the queue if the last step wasn't forced.
+			// Timers currently will call their complete callbacks, which
+			// will dequeue but only if they were gotoEnd.
+			if ( dequeue || !gotoEnd ) {
+				jQuery.dequeue( this, type );
+			}
+		} );
+	},
+	finish: function( type ) {
+		if ( type !== false ) {
+			type = type || "fx";
+		}
+		return this.each( function() {
+			var index,
+				data = dataPriv.get( this ),
+				queue = data[ type + "queue" ],
+				hooks = data[ type + "queueHooks" ],
+				timers = jQuery.timers,
+				length = queue ? queue.length : 0;
+
+			// Enable finishing flag on private data
+			data.finish = true;
+
+			// Empty the queue first
+			jQuery.queue( this, type, [] );
+
+			if ( hooks && hooks.stop ) {
+				hooks.stop.call( this, true );
+			}
+
+			// Look for any active animations, and finish them
+			for ( index = timers.length; index--; ) {
+				if ( timers[ index ].elem === this && timers[ index ].queue === type ) {
+					timers[ index ].anim.stop( true );
+					timers.splice( index, 1 );
+				}
+			}
+
+			// Look for any animations in the old queue and finish them
+			for ( index = 0; index < length; index++ ) {
+				if ( queue[ index ] && queue[ index ].finish ) {
+					queue[ index ].finish.call( this );
+				}
+			}
+
+			// Turn off finishing flag
+			delete data.finish;
+		} );
+	}
+} );
+
+jQuery.each( [ "toggle", "show", "hide" ], function( _i, name ) {
+	var cssFn = jQuery.fn[ name ];
+	jQuery.fn[ name ] = function( speed, easing, callback ) {
+		return speed == null || typeof speed === "boolean" ?
+			cssFn.apply( this, arguments ) :
+			this.animate( genFx( name, true ), speed, easing, callback );
+	};
+} );
+
+// Generate shortcuts for custom animations
+jQuery.each( {
+	slideDown: genFx( "show" ),
+	slideUp: genFx( "hide" ),
+	slideToggle: genFx( "toggle" ),
+	fadeIn: { opacity: "show" },
+	fadeOut: { opacity: "hide" },
+	fadeToggle: { opacity: "toggle" }
+}, function( name, props ) {
+	jQuery.fn[ name ] = function( speed, easing, callback ) {
+		return this.animate( props, speed, easing, callback );
+	};
+} );
+
+jQuery.timers = [];
+jQuery.fx.tick = function() {
+	var timer,
+		i = 0,
+		timers = jQuery.timers;
+
+	fxNow = Date.now();
+
+	for ( ; i < timers.length; i++ ) {
+		timer = timers[ i ];
+
+		// Run the timer and safely remove it when done (allowing for external removal)
+		if ( !timer() && timers[ i ] === timer ) {
+			timers.splice( i--, 1 );
+		}
+	}
+
+	if ( !timers.length ) {
+		jQuery.fx.stop();
+	}
+	fxNow = undefined;
+};
+
+jQuery.fx.timer = function( timer ) {
+	jQuery.timers.push( timer );
+	jQuery.fx.start();
+};
+
+jQuery.fx.interval = 13;
+jQuery.fx.start = function() {
+	if ( inProgress ) {
+		return;
+	}
+
+	inProgress = true;
+	schedule();
+};
+
+jQuery.fx.stop = function() {
+	inProgress = null;
+};
+
+jQuery.fx.speeds = {
+	slow: 600,
+	fast: 200,
+
+	// Default speed
+	_default: 400
+};
+
+
+// Based off of the plugin by Clint Helfers, with permission.
+// https://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/
+jQuery.fn.delay = function( time, type ) {
+	time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time;
+	type = type || "fx";
+
+	return this.queue( type, function( next, hooks ) {
+		var timeout = window.setTimeout( next, time );
+		hooks.stop = function() {
+			window.clearTimeout( timeout );
+		};
+	} );
+};
+
+
+( function() {
+	var input = document.createElement( "input" ),
+		select = document.createElement( "select" ),
+		opt = select.appendChild( document.createElement( "option" ) );
+
+	input.type = "checkbox";
+
+	// Support: Android <=4.3 only
+	// Default value for a checkbox should be "on"
+	support.checkOn = input.value !== "";
+
+	// Support: IE <=11 only
+	// Must access selectedIndex to make default options select
+	support.optSelected = opt.selected;
+
+	// Support: IE <=11 only
+	// An input loses its value after becoming a radio
+	input = document.createElement( "input" );
+	input.value = "t";
+	input.type = "radio";
+	support.radioValue = input.value === "t";
+} )();
+
+
+var boolHook,
+	attrHandle = jQuery.expr.attrHandle;
+
+jQuery.fn.extend( {
+	attr: function( name, value ) {
+		return access( this, jQuery.attr, name, value, arguments.length > 1 );
+	},
+
+	removeAttr: function( name ) {
+		return this.each( function() {
+			jQuery.removeAttr( this, name );
+		} );
+	}
+} );
+
+jQuery.extend( {
+	attr: function( elem, name, value ) {
+		var ret, hooks,
+			nType = elem.nodeType;
+
+		// Don't get/set attributes on text, comment and attribute nodes
+		if ( nType === 3 || nType === 8 || nType === 2 ) {
+			return;
+		}
+
+		// Fallback to prop when attributes are not supported
+		if ( typeof elem.getAttribute === "undefined" ) {
+			return jQuery.prop( elem, name, value );
+		}
+
+		// Attribute hooks are determined by the lowercase version
+		// Grab necessary hook if one is defined
+		if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) {
+			hooks = jQuery.attrHooks[ name.toLowerCase() ] ||
+				( jQuery.expr.match.bool.test( name ) ? boolHook : undefined );
+		}
+
+		if ( value !== undefined ) {
+			if ( value === null ) {
+				jQuery.removeAttr( elem, name );
+				return;
+			}
+
+			if ( hooks && "set" in hooks &&
+				( ret = hooks.set( elem, value, name ) ) !== undefined ) {
+				return ret;
+			}
+
+			elem.setAttribute( name, value + "" );
+			return value;
+		}
+
+		if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) {
+			return ret;
+		}
+
+		ret = jQuery.find.attr( elem, name );
+
+		// Non-existent attributes return null, we normalize to undefined
+		return ret == null ? undefined : ret;
+	},
+
+	attrHooks: {
+		type: {
+			set: function( elem, value ) {
+				if ( !support.radioValue && value === "radio" &&
+					nodeName( elem, "input" ) ) {
+					var val = elem.value;
+					elem.setAttribute( "type", value );
+					if ( val ) {
+						elem.value = val;
+					}
+					return value;
+				}
+			}
+		}
+	},
+
+	removeAttr: function( elem, value ) {
+		var name,
+			i = 0,
+
+			// Attribute names can contain non-HTML whitespace characters
+			// https://html.spec.whatwg.org/multipage/syntax.html#attributes-2
+			attrNames = value && value.match( rnothtmlwhite );
+
+		if ( attrNames && elem.nodeType === 1 ) {
+			while ( ( name = attrNames[ i++ ] ) ) {
+				elem.removeAttribute( name );
+			}
+		}
+	}
+} );
+
+// Hooks for boolean attributes
+boolHook = {
+	set: function( elem, value, name ) {
+		if ( value === false ) {
+
+			// Remove boolean attributes when set to false
+			jQuery.removeAttr( elem, name );
+		} else {
+			elem.setAttribute( name, name );
+		}
+		return name;
+	}
+};
+
+jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( _i, name ) {
+	var getter = attrHandle[ name ] || jQuery.find.attr;
+
+	attrHandle[ name ] = function( elem, name, isXML ) {
+		var ret, handle,
+			lowercaseName = name.toLowerCase();
+
+		if ( !isXML ) {
+
+			// Avoid an infinite loop by temporarily removing this function from the getter
+			handle = attrHandle[ lowercaseName ];
+			attrHandle[ lowercaseName ] = ret;
+			ret = getter( elem, name, isXML ) != null ?
+				lowercaseName :
+				null;
+			attrHandle[ lowercaseName ] = handle;
+		}
+		return ret;
+	};
+} );
+
+
+
+
+var rfocusable = /^(?:input|select|textarea|button)$/i,
+	rclickable = /^(?:a|area)$/i;
+
+jQuery.fn.extend( {
+	prop: function( name, value ) {
+		return access( this, jQuery.prop, name, value, arguments.length > 1 );
+	},
+
+	removeProp: function( name ) {
+		return this.each( function() {
+			delete this[ jQuery.propFix[ name ] || name ];
+		} );
+	}
+} );
+
+jQuery.extend( {
+	prop: function( elem, name, value ) {
+		var ret, hooks,
+			nType = elem.nodeType;
+
+		// Don't get/set properties on text, comment and attribute nodes
+		if ( nType === 3 || nType === 8 || nType === 2 ) {
+			return;
+		}
+
+		if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) {
+
+			// Fix name and attach hooks
+			name = jQuery.propFix[ name ] || name;
+			hooks = jQuery.propHooks[ name ];
+		}
+
+		if ( value !== undefined ) {
+			if ( hooks && "set" in hooks &&
+				( ret = hooks.set( elem, value, name ) ) !== undefined ) {
+				return ret;
+			}
+
+			return ( elem[ name ] = value );
+		}
+
+		if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) {
+			return ret;
+		}
+
+		return elem[ name ];
+	},
+
+	propHooks: {
+		tabIndex: {
+			get: function( elem ) {
+
+				// Support: IE <=9 - 11 only
+				// elem.tabIndex doesn't always return the
+				// correct value when it hasn't been explicitly set
+				// https://web.archive.org/web/20141116233347/http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/
+				// Use proper attribute retrieval(#12072)
+				var tabindex = jQuery.find.attr( elem, "tabindex" );
+
+				if ( tabindex ) {
+					return parseInt( tabindex, 10 );
+				}
+
+				if (
+					rfocusable.test( elem.nodeName ) ||
+					rclickable.test( elem.nodeName ) &&
+					elem.href
+				) {
+					return 0;
+				}
+
+				return -1;
+			}
+		}
+	},
+
+	propFix: {
+		"for": "htmlFor",
+		"class": "className"
+	}
+} );
+
+// Support: IE <=11 only
+// Accessing the selectedIndex property
+// forces the browser to respect setting selected
+// on the option
+// The getter ensures a default option is selected
+// when in an optgroup
+// eslint rule "no-unused-expressions" is disabled for this code
+// since it considers such accessions noop
+if ( !support.optSelected ) {
+	jQuery.propHooks.selected = {
+		get: function( elem ) {
+
+			/* eslint no-unused-expressions: "off" */
+
+			var parent = elem.parentNode;
+			if ( parent && parent.parentNode ) {
+				parent.parentNode.selectedIndex;
+			}
+			return null;
+		},
+		set: function( elem ) {
+
+			/* eslint no-unused-expressions: "off" */
+
+			var parent = elem.parentNode;
+			if ( parent ) {
+				parent.selectedIndex;
+
+				if ( parent.parentNode ) {
+					parent.parentNode.selectedIndex;
+				}
+			}
+		}
+	};
+}
+
+jQuery.each( [
+	"tabIndex",
+	"readOnly",
+	"maxLength",
+	"cellSpacing",
+	"cellPadding",
+	"rowSpan",
+	"colSpan",
+	"useMap",
+	"frameBorder",
+	"contentEditable"
+], function() {
+	jQuery.propFix[ this.toLowerCase() ] = this;
+} );
+
+
+
+
+	// Strip and collapse whitespace according to HTML spec
+	// https://infra.spec.whatwg.org/#strip-and-collapse-ascii-whitespace
+	function stripAndCollapse( value ) {
+		var tokens = value.match( rnothtmlwhite ) || [];
+		return tokens.join( " " );
+	}
+
+
+function getClass( elem ) {
+	return elem.getAttribute && elem.getAttribute( "class" ) || "";
+}
+
+function classesToArray( value ) {
+	if ( Array.isArray( value ) ) {
+		return value;
+	}
+	if ( typeof value === "string" ) {
+		return value.match( rnothtmlwhite ) || [];
+	}
+	return [];
+}
+
+jQuery.fn.extend( {
+	addClass: function( value ) {
+		var classes, elem, cur, curValue, clazz, j, finalValue,
+			i = 0;
+
+		if ( isFunction( value ) ) {
+			return this.each( function( j ) {
+				jQuery( this ).addClass( value.call( this, j, getClass( this ) ) );
+			} );
+		}
+
+		classes = classesToArray( value );
+
+		if ( classes.length ) {
+			while ( ( elem = this[ i++ ] ) ) {
+				curValue = getClass( elem );
+				cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " );
+
+				if ( cur ) {
+					j = 0;
+					while ( ( clazz = classes[ j++ ] ) ) {
+						if ( cur.indexOf( " " + clazz + " " ) < 0 ) {
+							cur += clazz + " ";
+						}
+					}
+
+					// Only assign if different to avoid unneeded rendering.
+					finalValue = stripAndCollapse( cur );
+					if ( curValue !== finalValue ) {
+						elem.setAttribute( "class", finalValue );
+					}
+				}
+			}
+		}
+
+		return this;
+	},
+
+	removeClass: function( value ) {
+		var classes, elem, cur, curValue, clazz, j, finalValue,
+			i = 0;
+
+		if ( isFunction( value ) ) {
+			return this.each( function( j ) {
+				jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) );
+			} );
+		}
+
+		if ( !arguments.length ) {
+			return this.attr( "class", "" );
+		}
+
+		classes = classesToArray( value );
+
+		if ( classes.length ) {
+			while ( ( elem = this[ i++ ] ) ) {
+				curValue = getClass( elem );
+
+				// This expression is here for better compressibility (see addClass)
+				cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " );
+
+				if ( cur ) {
+					j = 0;
+					while ( ( clazz = classes[ j++ ] ) ) {
+
+						// Remove *all* instances
+						while ( cur.indexOf( " " + clazz + " " ) > -1 ) {
+							cur = cur.replace( " " + clazz + " ", " " );
+						}
+					}
+
+					// Only assign if different to avoid unneeded rendering.
+					finalValue = stripAndCollapse( cur );
+					if ( curValue !== finalValue ) {
+						elem.setAttribute( "class", finalValue );
+					}
+				}
+			}
+		}
+
+		return this;
+	},
+
+	toggleClass: function( value, stateVal ) {
+		var type = typeof value,
+			isValidValue = type === "string" || Array.isArray( value );
+
+		if ( typeof stateVal === "boolean" && isValidValue ) {
+			return stateVal ? this.addClass( value ) : this.removeClass( value );
+		}
+
+		if ( isFunction( value ) ) {
+			return this.each( function( i ) {
+				jQuery( this ).toggleClass(
+					value.call( this, i, getClass( this ), stateVal ),
+					stateVal
+				);
+			} );
+		}
+
+		return this.each( function() {
+			var className, i, self, classNames;
+
+			if ( isValidValue ) {
+
+				// Toggle individual class names
+				i = 0;
+				self = jQuery( this );
+				classNames = classesToArray( value );
+
+				while ( ( className = classNames[ i++ ] ) ) {
+
+					// Check each className given, space separated list
+					if ( self.hasClass( className ) ) {
+						self.removeClass( className );
+					} else {
+						self.addClass( className );
+					}
+				}
+
+			// Toggle whole class name
+			} else if ( value === undefined || type === "boolean" ) {
+				className = getClass( this );
+				if ( className ) {
+
+					// Store className if set
+					dataPriv.set( this, "__className__", className );
+				}
+
+				// If the element has a class name or if we're passed `false`,
+				// then remove the whole classname (if there was one, the above saved it).
+				// Otherwise bring back whatever was previously saved (if anything),
+				// falling back to the empty string if nothing was stored.
+				if ( this.setAttribute ) {
+					this.setAttribute( "class",
+						className || value === false ?
+						"" :
+						dataPriv.get( this, "__className__" ) || ""
+					);
+				}
+			}
+		} );
+	},
+
+	hasClass: function( selector ) {
+		var className, elem,
+			i = 0;
+
+		className = " " + selector + " ";
+		while ( ( elem = this[ i++ ] ) ) {
+			if ( elem.nodeType === 1 &&
+				( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) {
+					return true;
+			}
+		}
+
+		return false;
+	}
+} );
+
+
+
+
+var rreturn = /\r/g;
+
+jQuery.fn.extend( {
+	val: function( value ) {
+		var hooks, ret, valueIsFunction,
+			elem = this[ 0 ];
+
+		if ( !arguments.length ) {
+			if ( elem ) {
+				hooks = jQuery.valHooks[ elem.type ] ||
+					jQuery.valHooks[ elem.nodeName.toLowerCase() ];
+
+				if ( hooks &&
+					"get" in hooks &&
+					( ret = hooks.get( elem, "value" ) ) !== undefined
+				) {
+					return ret;
+				}
+
+				ret = elem.value;
+
+				// Handle most common string cases
+				if ( typeof ret === "string" ) {
+					return ret.replace( rreturn, "" );
+				}
+
+				// Handle cases where value is null/undef or number
+				return ret == null ? "" : ret;
+			}
+
+			return;
+		}
+
+		valueIsFunction = isFunction( value );
+
+		return this.each( function( i ) {
+			var val;
+
+			if ( this.nodeType !== 1 ) {
+				return;
+			}
+
+			if ( valueIsFunction ) {
+				val = value.call( this, i, jQuery( this ).val() );
+			} else {
+				val = value;
+			}
+
+			// Treat null/undefined as ""; convert numbers to string
+			if ( val == null ) {
+				val = "";
+
+			} else if ( typeof val === "number" ) {
+				val += "";
+
+			} else if ( Array.isArray( val ) ) {
+				val = jQuery.map( val, function( value ) {
+					return value == null ? "" : value + "";
+				} );
+			}
+
+			hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ];
+
+			// If set returns undefined, fall back to normal setting
+			if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) {
+				this.value = val;
+			}
+		} );
+	}
+} );
+
+jQuery.extend( {
+	valHooks: {
+		option: {
+			get: function( elem ) {
+
+				var val = jQuery.find.attr( elem, "value" );
+				return val != null ?
+					val :
+
+					// Support: IE <=10 - 11 only
+					// option.text throws exceptions (#14686, #14858)
+					// Strip and collapse whitespace
+					// https://html.spec.whatwg.org/#strip-and-collapse-whitespace
+					stripAndCollapse( jQuery.text( elem ) );
+			}
+		},
+		select: {
+			get: function( elem ) {
+				var value, option, i,
+					options = elem.options,
+					index = elem.selectedIndex,
+					one = elem.type === "select-one",
+					values = one ? null : [],
+					max = one ? index + 1 : options.length;
+
+				if ( index < 0 ) {
+					i = max;
+
+				} else {
+					i = one ? index : 0;
+				}
+
+				// Loop through all the selected options
+				for ( ; i < max; i++ ) {
+					option = options[ i ];
+
+					// Support: IE <=9 only
+					// IE8-9 doesn't update selected after form reset (#2551)
+					if ( ( option.selected || i === index ) &&
+
+							// Don't return options that are disabled or in a disabled optgroup
+							!option.disabled &&
+							( !option.parentNode.disabled ||
+								!nodeName( option.parentNode, "optgroup" ) ) ) {
+
+						// Get the specific value for the option
+						value = jQuery( option ).val();
+
+						// We don't need an array for one selects
+						if ( one ) {
+							return value;
+						}
+
+						// Multi-Selects return an array
+						values.push( value );
+					}
+				}
+
+				return values;
+			},
+
+			set: function( elem, value ) {
+				var optionSet, option,
+					options = elem.options,
+					values = jQuery.makeArray( value ),
+					i = options.length;
+
+				while ( i-- ) {
+					option = options[ i ];
+
+					/* eslint-disable no-cond-assign */
+
+					if ( option.selected =
+						jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1
+					) {
+						optionSet = true;
+					}
+
+					/* eslint-enable no-cond-assign */
+				}
+
+				// Force browsers to behave consistently when non-matching value is set
+				if ( !optionSet ) {
+					elem.selectedIndex = -1;
+				}
+				return values;
+			}
+		}
+	}
+} );
+
+// Radios and checkboxes getter/setter
+jQuery.each( [ "radio", "checkbox" ], function() {
+	jQuery.valHooks[ this ] = {
+		set: function( elem, value ) {
+			if ( Array.isArray( value ) ) {
+				return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 );
+			}
+		}
+	};
+	if ( !support.checkOn ) {
+		jQuery.valHooks[ this ].get = function( elem ) {
+			return elem.getAttribute( "value" ) === null ? "on" : elem.value;
+		};
+	}
+} );
+
+
+
+
+// Return jQuery for attributes-only inclusion
+
+
+support.focusin = "onfocusin" in window;
+
+
+var rfocusMorph = /^(?:focusinfocus|focusoutblur)$/,
+	stopPropagationCallback = function( e ) {
+		e.stopPropagation();
+	};
+
+jQuery.extend( jQuery.event, {
+
+	trigger: function( event, data, elem, onlyHandlers ) {
+
+		var i, cur, tmp, bubbleType, ontype, handle, special, lastElement,
+			eventPath = [ elem || document ],
+			type = hasOwn.call( event, "type" ) ? event.type : event,
+			namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : [];
+
+		cur = lastElement = tmp = elem = elem || document;
+
+		// Don't do events on text and comment nodes
+		if ( elem.nodeType === 3 || elem.nodeType === 8 ) {
+			return;
+		}
+
+		// focus/blur morphs to focusin/out; ensure we're not firing them right now
+		if ( rfocusMorph.test( type + jQuery.event.triggered ) ) {
+			return;
+		}
+
+		if ( type.indexOf( "." ) > -1 ) {
+
+			// Namespaced trigger; create a regexp to match event type in handle()
+			namespaces = type.split( "." );
+			type = namespaces.shift();
+			namespaces.sort();
+		}
+		ontype = type.indexOf( ":" ) < 0 && "on" + type;
+
+		// Caller can pass in a jQuery.Event object, Object, or just an event type string
+		event = event[ jQuery.expando ] ?
+			event :
+			new jQuery.Event( type, typeof event === "object" && event );
+
+		// Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true)
+		event.isTrigger = onlyHandlers ? 2 : 3;
+		event.namespace = namespaces.join( "." );
+		event.rnamespace = event.namespace ?
+			new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) :
+			null;
+
+		// Clean up the event in case it is being reused
+		event.result = undefined;
+		if ( !event.target ) {
+			event.target = elem;
+		}
+
+		// Clone any incoming data and prepend the event, creating the handler arg list
+		data = data == null ?
+			[ event ] :
+			jQuery.makeArray( data, [ event ] );
+
+		// Allow special events to draw outside the lines
+		special = jQuery.event.special[ type ] || {};
+		if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) {
+			return;
+		}
+
+		// Determine event propagation path in advance, per W3C events spec (#9951)
+		// Bubble up to document, then to window; watch for a global ownerDocument var (#9724)
+		if ( !onlyHandlers && !special.noBubble && !isWindow( elem ) ) {
+
+			bubbleType = special.delegateType || type;
+			if ( !rfocusMorph.test( bubbleType + type ) ) {
+				cur = cur.parentNode;
+			}
+			for ( ; cur; cur = cur.parentNode ) {
+				eventPath.push( cur );
+				tmp = cur;
+			}
+
+			// Only add window if we got to document (e.g., not plain obj or detached DOM)
+			if ( tmp === ( elem.ownerDocument || document ) ) {
+				eventPath.push( tmp.defaultView || tmp.parentWindow || window );
+			}
+		}
+
+		// Fire handlers on the event path
+		i = 0;
+		while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) {
+			lastElement = cur;
+			event.type = i > 1 ?
+				bubbleType :
+				special.bindType || type;
+
+			// jQuery handler
+			handle = (
+					dataPriv.get( cur, "events" ) || Object.create( null )
+				)[ event.type ] &&
+				dataPriv.get( cur, "handle" );
+			if ( handle ) {
+				handle.apply( cur, data );
+			}
+
+			// Native handler
+			handle = ontype && cur[ ontype ];
+			if ( handle && handle.apply && acceptData( cur ) ) {
+				event.result = handle.apply( cur, data );
+				if ( event.result === false ) {
+					event.preventDefault();
+				}
+			}
+		}
+		event.type = type;
+
+		// If nobody prevented the default action, do it now
+		if ( !onlyHandlers && !event.isDefaultPrevented() ) {
+
+			if ( ( !special._default ||
+				special._default.apply( eventPath.pop(), data ) === false ) &&
+				acceptData( elem ) ) {
+
+				// Call a native DOM method on the target with the same name as the event.
+				// Don't do default actions on window, that's where global variables be (#6170)
+				if ( ontype && isFunction( elem[ type ] ) && !isWindow( elem ) ) {
+
+					// Don't re-trigger an onFOO event when we call its FOO() method
+					tmp = elem[ ontype ];
+
+					if ( tmp ) {
+						elem[ ontype ] = null;
+					}
+
+					// Prevent re-triggering of the same event, since we already bubbled it above
+					jQuery.event.triggered = type;
+
+					if ( event.isPropagationStopped() ) {
+						lastElement.addEventListener( type, stopPropagationCallback );
+					}
+
+					elem[ type ]();
+
+					if ( event.isPropagationStopped() ) {
+						lastElement.removeEventListener( type, stopPropagationCallback );
+					}
+
+					jQuery.event.triggered = undefined;
+
+					if ( tmp ) {
+						elem[ ontype ] = tmp;
+					}
+				}
+			}
+		}
+
+		return event.result;
+	},
+
+	// Piggyback on a donor event to simulate a different one
+	// Used only for `focus(in | out)` events
+	simulate: function( type, elem, event ) {
+		var e = jQuery.extend(
+			new jQuery.Event(),
+			event,
+			{
+				type: type,
+				isSimulated: true
+			}
+		);
+
+		jQuery.event.trigger( e, null, elem );
+	}
+
+} );
+
+jQuery.fn.extend( {
+
+	trigger: function( type, data ) {
+		return this.each( function() {
+			jQuery.event.trigger( type, data, this );
+		} );
+	},
+	triggerHandler: function( type, data ) {
+		var elem = this[ 0 ];
+		if ( elem ) {
+			return jQuery.event.trigger( type, data, elem, true );
+		}
+	}
+} );
+
+
+// Support: Firefox <=44
+// Firefox doesn't have focus(in | out) events
+// Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787
+//
+// Support: Chrome <=48 - 49, Safari <=9.0 - 9.1
+// focus(in | out) events fire after focus & blur events,
+// which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order
+// Related ticket - https://bugs.chromium.org/p/chromium/issues/detail?id=449857
+if ( !support.focusin ) {
+	jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) {
+
+		// Attach a single capturing handler on the document while someone wants focusin/focusout
+		var handler = function( event ) {
+			jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) );
+		};
+
+		jQuery.event.special[ fix ] = {
+			setup: function() {
+
+				// Handle: regular nodes (via `this.ownerDocument`), window
+				// (via `this.document`) & document (via `this`).
+				var doc = this.ownerDocument || this.document || this,
+					attaches = dataPriv.access( doc, fix );
+
+				if ( !attaches ) {
+					doc.addEventListener( orig, handler, true );
+				}
+				dataPriv.access( doc, fix, ( attaches || 0 ) + 1 );
+			},
+			teardown: function() {
+				var doc = this.ownerDocument || this.document || this,
+					attaches = dataPriv.access( doc, fix ) - 1;
+
+				if ( !attaches ) {
+					doc.removeEventListener( orig, handler, true );
+					dataPriv.remove( doc, fix );
+
+				} else {
+					dataPriv.access( doc, fix, attaches );
+				}
+			}
+		};
+	} );
+}
+var location = window.location;
+
+var nonce = { guid: Date.now() };
+
+var rquery = ( /\?/ );
+
+
+
+// Cross-browser xml parsing
+jQuery.parseXML = function( data ) {
+	var xml;
+	if ( !data || typeof data !== "string" ) {
+		return null;
+	}
+
+	// Support: IE 9 - 11 only
+	// IE throws on parseFromString with invalid input.
+	try {
+		xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" );
+	} catch ( e ) {
+		xml = undefined;
+	}
+
+	if ( !xml || xml.getElementsByTagName( "parsererror" ).length ) {
+		jQuery.error( "Invalid XML: " + data );
+	}
+	return xml;
+};
+
+
+var
+	rbracket = /\[\]$/,
+	rCRLF = /\r?\n/g,
+	rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i,
+	rsubmittable = /^(?:input|select|textarea|keygen)/i;
+
+function buildParams( prefix, obj, traditional, add ) {
+	var name;
+
+	if ( Array.isArray( obj ) ) {
+
+		// Serialize array item.
+		jQuery.each( obj, function( i, v ) {
+			if ( traditional || rbracket.test( prefix ) ) {
+
+				// Treat each array item as a scalar.
+				add( prefix, v );
+
+			} else {
+
+				// Item is non-scalar (array or object), encode its numeric index.
+				buildParams(
+					prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]",
+					v,
+					traditional,
+					add
+				);
+			}
+		} );
+
+	} else if ( !traditional && toType( obj ) === "object" ) {
+
+		// Serialize object item.
+		for ( name in obj ) {
+			buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add );
+		}
+
+	} else {
+
+		// Serialize scalar item.
+		add( prefix, obj );
+	}
+}
+
+// Serialize an array of form elements or a set of
+// key/values into a query string
+jQuery.param = function( a, traditional ) {
+	var prefix,
+		s = [],
+		add = function( key, valueOrFunction ) {
+
+			// If value is a function, invoke it and use its return value
+			var value = isFunction( valueOrFunction ) ?
+				valueOrFunction() :
+				valueOrFunction;
+
+			s[ s.length ] = encodeURIComponent( key ) + "=" +
+				encodeURIComponent( value == null ? "" : value );
+		};
+
+	if ( a == null ) {
+		return "";
+	}
+
+	// If an array was passed in, assume that it is an array of form elements.
+	if ( Array.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) {
+
+		// Serialize the form elements
+		jQuery.each( a, function() {
+			add( this.name, this.value );
+		} );
+
+	} else {
+
+		// If traditional, encode the "old" way (the way 1.3.2 or older
+		// did it), otherwise encode params recursively.
+		for ( prefix in a ) {
+			buildParams( prefix, a[ prefix ], traditional, add );
+		}
+	}
+
+	// Return the resulting serialization
+	return s.join( "&" );
+};
+
+jQuery.fn.extend( {
+	serialize: function() {
+		return jQuery.param( this.serializeArray() );
+	},
+	serializeArray: function() {
+		return this.map( function() {
+
+			// Can add propHook for "elements" to filter or add form elements
+			var elements = jQuery.prop( this, "elements" );
+			return elements ? jQuery.makeArray( elements ) : this;
+		} )
+		.filter( function() {
+			var type = this.type;
+
+			// Use .is( ":disabled" ) so that fieldset[disabled] works
+			return this.name && !jQuery( this ).is( ":disabled" ) &&
+				rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) &&
+				( this.checked || !rcheckableType.test( type ) );
+		} )
+		.map( function( _i, elem ) {
+			var val = jQuery( this ).val();
+
+			if ( val == null ) {
+				return null;
+			}
+
+			if ( Array.isArray( val ) ) {
+				return jQuery.map( val, function( val ) {
+					return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) };
+				} );
+			}
+
+			return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) };
+		} ).get();
+	}
+} );
+
+
+var
+	r20 = /%20/g,
+	rhash = /#.*$/,
+	rantiCache = /([?&])_=[^&]*/,
+	rheaders = /^(.*?):[ \t]*([^\r\n]*)$/mg,
+
+	// #7653, #8125, #8152: local protocol detection
+	rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/,
+	rnoContent = /^(?:GET|HEAD)$/,
+	rprotocol = /^\/\//,
+
+	/* Prefilters
+	 * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example)
+	 * 2) These are called:
+	 *    - BEFORE asking for a transport
+	 *    - AFTER param serialization (s.data is a string if s.processData is true)
+	 * 3) key is the dataType
+	 * 4) the catchall symbol "*" can be used
+	 * 5) execution will start with transport dataType and THEN continue down to "*" if needed
+	 */
+	prefilters = {},
+
+	/* Transports bindings
+	 * 1) key is the dataType
+	 * 2) the catchall symbol "*" can be used
+	 * 3) selection will start with transport dataType and THEN go to "*" if needed
+	 */
+	transports = {},
+
+	// Avoid comment-prolog char sequence (#10098); must appease lint and evade compression
+	allTypes = "*/".concat( "*" ),
+
+	// Anchor tag for parsing the document origin
+	originAnchor = document.createElement( "a" );
+	originAnchor.href = location.href;
+
+// Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport
+function addToPrefiltersOrTransports( structure ) {
+
+	// dataTypeExpression is optional and defaults to "*"
+	return function( dataTypeExpression, func ) {
+
+		if ( typeof dataTypeExpression !== "string" ) {
+			func = dataTypeExpression;
+			dataTypeExpression = "*";
+		}
+
+		var dataType,
+			i = 0,
+			dataTypes = dataTypeExpression.toLowerCase().match( rnothtmlwhite ) || [];
+
+		if ( isFunction( func ) ) {
+
+			// For each dataType in the dataTypeExpression
+			while ( ( dataType = dataTypes[ i++ ] ) ) {
+
+				// Prepend if requested
+				if ( dataType[ 0 ] === "+" ) {
+					dataType = dataType.slice( 1 ) || "*";
+					( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func );
+
+				// Otherwise append
+				} else {
+					( structure[ dataType ] = structure[ dataType ] || [] ).push( func );
+				}
+			}
+		}
+	};
+}
+
+// Base inspection function for prefilters and transports
+function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) {
+
+	var inspected = {},
+		seekingTransport = ( structure === transports );
+
+	function inspect( dataType ) {
+		var selected;
+		inspected[ dataType ] = true;
+		jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) {
+			var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR );
+			if ( typeof dataTypeOrTransport === "string" &&
+				!seekingTransport && !inspected[ dataTypeOrTransport ] ) {
+
+				options.dataTypes.unshift( dataTypeOrTransport );
+				inspect( dataTypeOrTransport );
+				return false;
+			} else if ( seekingTransport ) {
+				return !( selected = dataTypeOrTransport );
+			}
+		} );
+		return selected;
+	}
+
+	return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" );
+}
+
+// A special extend for ajax options
+// that takes "flat" options (not to be deep extended)
+// Fixes #9887
+function ajaxExtend( target, src ) {
+	var key, deep,
+		flatOptions = jQuery.ajaxSettings.flatOptions || {};
+
+	for ( key in src ) {
+		if ( src[ key ] !== undefined ) {
+			( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ];
+		}
+	}
+	if ( deep ) {
+		jQuery.extend( true, target, deep );
+	}
+
+	return target;
+}
+
+/* Handles responses to an ajax request:
+ * - finds the right dataType (mediates between content-type and expected dataType)
+ * - returns the corresponding response
+ */
+function ajaxHandleResponses( s, jqXHR, responses ) {
+
+	var ct, type, finalDataType, firstDataType,
+		contents = s.contents,
+		dataTypes = s.dataTypes;
+
+	// Remove auto dataType and get content-type in the process
+	while ( dataTypes[ 0 ] === "*" ) {
+		dataTypes.shift();
+		if ( ct === undefined ) {
+			ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" );
+		}
+	}
+
+	// Check if we're dealing with a known content-type
+	if ( ct ) {
+		for ( type in contents ) {
+			if ( contents[ type ] && contents[ type ].test( ct ) ) {
+				dataTypes.unshift( type );
+				break;
+			}
+		}
+	}
+
+	// Check to see if we have a response for the expected dataType
+	if ( dataTypes[ 0 ] in responses ) {
+		finalDataType = dataTypes[ 0 ];
+	} else {
+
+		// Try convertible dataTypes
+		for ( type in responses ) {
+			if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) {
+				finalDataType = type;
+				break;
+			}
+			if ( !firstDataType ) {
+				firstDataType = type;
+			}
+		}
+
+		// Or just use first one
+		finalDataType = finalDataType || firstDataType;
+	}
+
+	// If we found a dataType
+	// We add the dataType to the list if needed
+	// and return the corresponding response
+	if ( finalDataType ) {
+		if ( finalDataType !== dataTypes[ 0 ] ) {
+			dataTypes.unshift( finalDataType );
+		}
+		return responses[ finalDataType ];
+	}
+}
+
+/* Chain conversions given the request and the original response
+ * Also sets the responseXXX fields on the jqXHR instance
+ */
+function ajaxConvert( s, response, jqXHR, isSuccess ) {
+	var conv2, current, conv, tmp, prev,
+		converters = {},
+
+		// Work with a copy of dataTypes in case we need to modify it for conversion
+		dataTypes = s.dataTypes.slice();
+
+	// Create converters map with lowercased keys
+	if ( dataTypes[ 1 ] ) {
+		for ( conv in s.converters ) {
+			converters[ conv.toLowerCase() ] = s.converters[ conv ];
+		}
+	}
+
+	current = dataTypes.shift();
+
+	// Convert to each sequential dataType
+	while ( current ) {
+
+		if ( s.responseFields[ current ] ) {
+			jqXHR[ s.responseFields[ current ] ] = response;
+		}
+
+		// Apply the dataFilter if provided
+		if ( !prev && isSuccess && s.dataFilter ) {
+			response = s.dataFilter( response, s.dataType );
+		}
+
+		prev = current;
+		current = dataTypes.shift();
+
+		if ( current ) {
+
+			// There's only work to do if current dataType is non-auto
+			if ( current === "*" ) {
+
+				current = prev;
+
+			// Convert response if prev dataType is non-auto and differs from current
+			} else if ( prev !== "*" && prev !== current ) {
+
+				// Seek a direct converter
+				conv = converters[ prev + " " + current ] || converters[ "* " + current ];
+
+				// If none found, seek a pair
+				if ( !conv ) {
+					for ( conv2 in converters ) {
+
+						// If conv2 outputs current
+						tmp = conv2.split( " " );
+						if ( tmp[ 1 ] === current ) {
+
+							// If prev can be converted to accepted input
+							conv = converters[ prev + " " + tmp[ 0 ] ] ||
+								converters[ "* " + tmp[ 0 ] ];
+							if ( conv ) {
+
+								// Condense equivalence converters
+								if ( conv === true ) {
+									conv = converters[ conv2 ];
+
+								// Otherwise, insert the intermediate dataType
+								} else if ( converters[ conv2 ] !== true ) {
+									current = tmp[ 0 ];
+									dataTypes.unshift( tmp[ 1 ] );
+								}
+								break;
+							}
+						}
+					}
+				}
+
+				// Apply converter (if not an equivalence)
+				if ( conv !== true ) {
+
+					// Unless errors are allowed to bubble, catch and return them
+					if ( conv && s.throws ) {
+						response = conv( response );
+					} else {
+						try {
+							response = conv( response );
+						} catch ( e ) {
+							return {
+								state: "parsererror",
+								error: conv ? e : "No conversion from " + prev + " to " + current
+							};
+						}
+					}
+				}
+			}
+		}
+	}
+
+	return { state: "success", data: response };
+}
+
+jQuery.extend( {
+
+	// Counter for holding the number of active queries
+	active: 0,
+
+	// Last-Modified header cache for next request
+	lastModified: {},
+	etag: {},
+
+	ajaxSettings: {
+		url: location.href,
+		type: "GET",
+		isLocal: rlocalProtocol.test( location.protocol ),
+		global: true,
+		processData: true,
+		async: true,
+		contentType: "application/x-www-form-urlencoded; charset=UTF-8",
+
+		/*
+		timeout: 0,
+		data: null,
+		dataType: null,
+		username: null,
+		password: null,
+		cache: null,
+		throws: false,
+		traditional: false,
+		headers: {},
+		*/
+
+		accepts: {
+			"*": allTypes,
+			text: "text/plain",
+			html: "text/html",
+			xml: "application/xml, text/xml",
+			json: "application/json, text/javascript"
+		},
+
+		contents: {
+			xml: /\bxml\b/,
+			html: /\bhtml/,
+			json: /\bjson\b/
+		},
+
+		responseFields: {
+			xml: "responseXML",
+			text: "responseText",
+			json: "responseJSON"
+		},
+
+		// Data converters
+		// Keys separate source (or catchall "*") and destination types with a single space
+		converters: {
+
+			// Convert anything to text
+			"* text": String,
+
+			// Text to html (true = no transformation)
+			"text html": true,
+
+			// Evaluate text as a json expression
+			"text json": JSON.parse,
+
+			// Parse text as xml
+			"text xml": jQuery.parseXML
+		},
+
+		// For options that shouldn't be deep extended:
+		// you can add your own custom options here if
+		// and when you create one that shouldn't be
+		// deep extended (see ajaxExtend)
+		flatOptions: {
+			url: true,
+			context: true
+		}
+	},
+
+	// Creates a full fledged settings object into target
+	// with both ajaxSettings and settings fields.
+	// If target is omitted, writes into ajaxSettings.
+	ajaxSetup: function( target, settings ) {
+		return settings ?
+
+			// Building a settings object
+			ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) :
+
+			// Extending ajaxSettings
+			ajaxExtend( jQuery.ajaxSettings, target );
+	},
+
+	ajaxPrefilter: addToPrefiltersOrTransports( prefilters ),
+	ajaxTransport: addToPrefiltersOrTransports( transports ),
+
+	// Main method
+	ajax: function( url, options ) {
+
+		// If url is an object, simulate pre-1.5 signature
+		if ( typeof url === "object" ) {
+			options = url;
+			url = undefined;
+		}
+
+		// Force options to be an object
+		options = options || {};
+
+		var transport,
+
+			// URL without anti-cache param
+			cacheURL,
+
+			// Response headers
+			responseHeadersString,
+			responseHeaders,
+
+			// timeout handle
+			timeoutTimer,
+
+			// Url cleanup var
+			urlAnchor,
+
+			// Request state (becomes false upon send and true upon completion)
+			completed,
+
+			// To know if global events are to be dispatched
+			fireGlobals,
+
+			// Loop variable
+			i,
+
+			// uncached part of the url
+			uncached,
+
+			// Create the final options object
+			s = jQuery.ajaxSetup( {}, options ),
+
+			// Callbacks context
+			callbackContext = s.context || s,
+
+			// Context for global events is callbackContext if it is a DOM node or jQuery collection
+			globalEventContext = s.context &&
+				( callbackContext.nodeType || callbackContext.jquery ) ?
+					jQuery( callbackContext ) :
+					jQuery.event,
+
+			// Deferreds
+			deferred = jQuery.Deferred(),
+			completeDeferred = jQuery.Callbacks( "once memory" ),
+
+			// Status-dependent callbacks
+			statusCode = s.statusCode || {},
+
+			// Headers (they are sent all at once)
+			requestHeaders = {},
+			requestHeadersNames = {},
+
+			// Default abort message
+			strAbort = "canceled",
+
+			// Fake xhr
+			jqXHR = {
+				readyState: 0,
+
+				// Builds headers hashtable if needed
+				getResponseHeader: function( key ) {
+					var match;
+					if ( completed ) {
+						if ( !responseHeaders ) {
+							responseHeaders = {};
+							while ( ( match = rheaders.exec( responseHeadersString ) ) ) {
+								responseHeaders[ match[ 1 ].toLowerCase() + " " ] =
+									( responseHeaders[ match[ 1 ].toLowerCase() + " " ] || [] )
+										.concat( match[ 2 ] );
+							}
+						}
+						match = responseHeaders[ key.toLowerCase() + " " ];
+					}
+					return match == null ? null : match.join( ", " );
+				},
+
+				// Raw string
+				getAllResponseHeaders: function() {
+					return completed ? responseHeadersString : null;
+				},
+
+				// Caches the header
+				setRequestHeader: function( name, value ) {
+					if ( completed == null ) {
+						name = requestHeadersNames[ name.toLowerCase() ] =
+							requestHeadersNames[ name.toLowerCase() ] || name;
+						requestHeaders[ name ] = value;
+					}
+					return this;
+				},
+
+				// Overrides response content-type header
+				overrideMimeType: function( type ) {
+					if ( completed == null ) {
+						s.mimeType = type;
+					}
+					return this;
+				},
+
+				// Status-dependent callbacks
+				statusCode: function( map ) {
+					var code;
+					if ( map ) {
+						if ( completed ) {
+
+							// Execute the appropriate callbacks
+							jqXHR.always( map[ jqXHR.status ] );
+						} else {
+
+							// Lazy-add the new callbacks in a way that preserves old ones
+							for ( code in map ) {
+								statusCode[ code ] = [ statusCode[ code ], map[ code ] ];
+							}
+						}
+					}
+					return this;
+				},
+
+				// Cancel the request
+				abort: function( statusText ) {
+					var finalText = statusText || strAbort;
+					if ( transport ) {
+						transport.abort( finalText );
+					}
+					done( 0, finalText );
+					return this;
+				}
+			};
+
+		// Attach deferreds
+		deferred.promise( jqXHR );
+
+		// Add protocol if not provided (prefilters might expect it)
+		// Handle falsy url in the settings object (#10093: consistency with old signature)
+		// We also use the url parameter if available
+		s.url = ( ( url || s.url || location.href ) + "" )
+			.replace( rprotocol, location.protocol + "//" );
+
+		// Alias method option to type as per ticket #12004
+		s.type = options.method || options.type || s.method || s.type;
+
+		// Extract dataTypes list
+		s.dataTypes = ( s.dataType || "*" ).toLowerCase().match( rnothtmlwhite ) || [ "" ];
+
+		// A cross-domain request is in order when the origin doesn't match the current origin.
+		if ( s.crossDomain == null ) {
+			urlAnchor = document.createElement( "a" );
+
+			// Support: IE <=8 - 11, Edge 12 - 15
+			// IE throws exception on accessing the href property if url is malformed,
+			// e.g. http://example.com:80x/
+			try {
+				urlAnchor.href = s.url;
+
+				// Support: IE <=8 - 11 only
+				// Anchor's host property isn't correctly set when s.url is relative
+				urlAnchor.href = urlAnchor.href;
+				s.crossDomain = originAnchor.protocol + "//" + originAnchor.host !==
+					urlAnchor.protocol + "//" + urlAnchor.host;
+			} catch ( e ) {
+
+				// If there is an error parsing the URL, assume it is crossDomain,
+				// it can be rejected by the transport if it is invalid
+				s.crossDomain = true;
+			}
+		}
+
+		// Convert data if not already a string
+		if ( s.data && s.processData && typeof s.data !== "string" ) {
+			s.data = jQuery.param( s.data, s.traditional );
+		}
+
+		// Apply prefilters
+		inspectPrefiltersOrTransports( prefilters, s, options, jqXHR );
+
+		// If request was aborted inside a prefilter, stop there
+		if ( completed ) {
+			return jqXHR;
+		}
+
+		// We can fire global events as of now if asked to
+		// Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118)
+		fireGlobals = jQuery.event && s.global;
+
+		// Watch for a new set of requests
+		if ( fireGlobals && jQuery.active++ === 0 ) {
+			jQuery.event.trigger( "ajaxStart" );
+		}
+
+		// Uppercase the type
+		s.type = s.type.toUpperCase();
+
+		// Determine if request has content
+		s.hasContent = !rnoContent.test( s.type );
+
+		// Save the URL in case we're toying with the If-Modified-Since
+		// and/or If-None-Match header later on
+		// Remove hash to simplify url manipulation
+		cacheURL = s.url.replace( rhash, "" );
+
+		// More options handling for requests with no content
+		if ( !s.hasContent ) {
+
+			// Remember the hash so we can put it back
+			uncached = s.url.slice( cacheURL.length );
+
+			// If data is available and should be processed, append data to url
+			if ( s.data && ( s.processData || typeof s.data === "string" ) ) {
+				cacheURL += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data;
+
+				// #9682: remove data so that it's not used in an eventual retry
+				delete s.data;
+			}
+
+			// Add or update anti-cache param if needed
+			if ( s.cache === false ) {
+				cacheURL = cacheURL.replace( rantiCache, "$1" );
+				uncached = ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + ( nonce.guid++ ) +
+					uncached;
+			}
+
+			// Put hash and anti-cache on the URL that will be requested (gh-1732)
+			s.url = cacheURL + uncached;
+
+		// Change '%20' to '+' if this is encoded form body content (gh-2658)
+		} else if ( s.data && s.processData &&
+			( s.contentType || "" ).indexOf( "application/x-www-form-urlencoded" ) === 0 ) {
+			s.data = s.data.replace( r20, "+" );
+		}
+
+		// Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode.
+		if ( s.ifModified ) {
+			if ( jQuery.lastModified[ cacheURL ] ) {
+				jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] );
+			}
+			if ( jQuery.etag[ cacheURL ] ) {
+				jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] );
+			}
+		}
+
+		// Set the correct header, if data is being sent
+		if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) {
+			jqXHR.setRequestHeader( "Content-Type", s.contentType );
+		}
+
+		// Set the Accepts header for the server, depending on the dataType
+		jqXHR.setRequestHeader(
+			"Accept",
+			s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ?
+				s.accepts[ s.dataTypes[ 0 ] ] +
+					( s.dataTypes[ 0 ] !== "*" ? ", " + allTypes + "; q=0.01" : "" ) :
+				s.accepts[ "*" ]
+		);
+
+		// Check for headers option
+		for ( i in s.headers ) {
+			jqXHR.setRequestHeader( i, s.headers[ i ] );
+		}
+
+		// Allow custom headers/mimetypes and early abort
+		if ( s.beforeSend &&
+			( s.beforeSend.call( callbackContext, jqXHR, s ) === false || completed ) ) {
+
+			// Abort if not done already and return
+			return jqXHR.abort();
+		}
+
+		// Aborting is no longer a cancellation
+		strAbort = "abort";
+
+		// Install callbacks on deferreds
+		completeDeferred.add( s.complete );
+		jqXHR.done( s.success );
+		jqXHR.fail( s.error );
+
+		// Get transport
+		transport = inspectPrefiltersOrTransports( transports, s, options, jqXHR );
+
+		// If no transport, we auto-abort
+		if ( !transport ) {
+			done( -1, "No Transport" );
+		} else {
+			jqXHR.readyState = 1;
+
+			// Send global event
+			if ( fireGlobals ) {
+				globalEventContext.trigger( "ajaxSend", [ jqXHR, s ] );
+			}
+
+			// If request was aborted inside ajaxSend, stop there
+			if ( completed ) {
+				return jqXHR;
+			}
+
+			// Timeout
+			if ( s.async && s.timeout > 0 ) {
+				timeoutTimer = window.setTimeout( function() {
+					jqXHR.abort( "timeout" );
+				}, s.timeout );
+			}
+
+			try {
+				completed = false;
+				transport.send( requestHeaders, done );
+			} catch ( e ) {
+
+				// Rethrow post-completion exceptions
+				if ( completed ) {
+					throw e;
+				}
+
+				// Propagate others as results
+				done( -1, e );
+			}
+		}
+
+		// Callback for when everything is done
+		function done( status, nativeStatusText, responses, headers ) {
+			var isSuccess, success, error, response, modified,
+				statusText = nativeStatusText;
+
+			// Ignore repeat invocations
+			if ( completed ) {
+				return;
+			}
+
+			completed = true;
+
+			// Clear timeout if it exists
+			if ( timeoutTimer ) {
+				window.clearTimeout( timeoutTimer );
+			}
+
+			// Dereference transport for early garbage collection
+			// (no matter how long the jqXHR object will be used)
+			transport = undefined;
+
+			// Cache response headers
+			responseHeadersString = headers || "";
+
+			// Set readyState
+			jqXHR.readyState = status > 0 ? 4 : 0;
+
+			// Determine if successful
+			isSuccess = status >= 200 && status < 300 || status === 304;
+
+			// Get response data
+			if ( responses ) {
+				response = ajaxHandleResponses( s, jqXHR, responses );
+			}
+
+			// Use a noop converter for missing script
+			if ( !isSuccess && jQuery.inArray( "script", s.dataTypes ) > -1 ) {
+				s.converters[ "text script" ] = function() {};
+			}
+
+			// Convert no matter what (that way responseXXX fields are always set)
+			response = ajaxConvert( s, response, jqXHR, isSuccess );
+
+			// If successful, handle type chaining
+			if ( isSuccess ) {
+
+				// Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode.
+				if ( s.ifModified ) {
+					modified = jqXHR.getResponseHeader( "Last-Modified" );
+					if ( modified ) {
+						jQuery.lastModified[ cacheURL ] = modified;
+					}
+					modified = jqXHR.getResponseHeader( "etag" );
+					if ( modified ) {
+						jQuery.etag[ cacheURL ] = modified;
+					}
+				}
+
+				// if no content
+				if ( status === 204 || s.type === "HEAD" ) {
+					statusText = "nocontent";
+
+				// if not modified
+				} else if ( status === 304 ) {
+					statusText = "notmodified";
+
+				// If we have data, let's convert it
+				} else {
+					statusText = response.state;
+					success = response.data;
+					error = response.error;
+					isSuccess = !error;
+				}
+			} else {
+
+				// Extract error from statusText and normalize for non-aborts
+				error = statusText;
+				if ( status || !statusText ) {
+					statusText = "error";
+					if ( status < 0 ) {
+						status = 0;
+					}
+				}
+			}
+
+			// Set data for the fake xhr object
+			jqXHR.status = status;
+			jqXHR.statusText = ( nativeStatusText || statusText ) + "";
+
+			// Success/Error
+			if ( isSuccess ) {
+				deferred.resolveWith( callbackContext, [ success, statusText, jqXHR ] );
+			} else {
+				deferred.rejectWith( callbackContext, [ jqXHR, statusText, error ] );
+			}
+
+			// Status-dependent callbacks
+			jqXHR.statusCode( statusCode );
+			statusCode = undefined;
+
+			if ( fireGlobals ) {
+				globalEventContext.trigger( isSuccess ? "ajaxSuccess" : "ajaxError",
+					[ jqXHR, s, isSuccess ? success : error ] );
+			}
+
+			// Complete
+			completeDeferred.fireWith( callbackContext, [ jqXHR, statusText ] );
+
+			if ( fireGlobals ) {
+				globalEventContext.trigger( "ajaxComplete", [ jqXHR, s ] );
+
+				// Handle the global AJAX counter
+				if ( !( --jQuery.active ) ) {
+					jQuery.event.trigger( "ajaxStop" );
+				}
+			}
+		}
+
+		return jqXHR;
+	},
+
+	getJSON: function( url, data, callback ) {
+		return jQuery.get( url, data, callback, "json" );
+	},
+
+	getScript: function( url, callback ) {
+		return jQuery.get( url, undefined, callback, "script" );
+	}
+} );
+
+jQuery.each( [ "get", "post" ], function( _i, method ) {
+	jQuery[ method ] = function( url, data, callback, type ) {
+
+		// Shift arguments if data argument was omitted
+		if ( isFunction( data ) ) {
+			type = type || callback;
+			callback = data;
+			data = undefined;
+		}
+
+		// The url can be an options object (which then must have .url)
+		return jQuery.ajax( jQuery.extend( {
+			url: url,
+			type: method,
+			dataType: type,
+			data: data,
+			success: callback
+		}, jQuery.isPlainObject( url ) && url ) );
+	};
+} );
+
+jQuery.ajaxPrefilter( function( s ) {
+	var i;
+	for ( i in s.headers ) {
+		if ( i.toLowerCase() === "content-type" ) {
+			s.contentType = s.headers[ i ] || "";
+		}
+	}
+} );
+
+
+jQuery._evalUrl = function( url, options, doc ) {
+	return jQuery.ajax( {
+		url: url,
+
+		// Make this explicit, since user can override this through ajaxSetup (#11264)
+		type: "GET",
+		dataType: "script",
+		cache: true,
+		async: false,
+		global: false,
+
+		// Only evaluate the response if it is successful (gh-4126)
+		// dataFilter is not invoked for failure responses, so using it instead
+		// of the default converter is kludgy but it works.
+		converters: {
+			"text script": function() {}
+		},
+		dataFilter: function( response ) {
+			jQuery.globalEval( response, options, doc );
+		}
+	} );
+};
+
+
+jQuery.fn.extend( {
+	wrapAll: function( html ) {
+		var wrap;
+
+		if ( this[ 0 ] ) {
+			if ( isFunction( html ) ) {
+				html = html.call( this[ 0 ] );
+			}
+
+			// The elements to wrap the target around
+			wrap = jQuery( html, this[ 0 ].ownerDocument ).eq( 0 ).clone( true );
+
+			if ( this[ 0 ].parentNode ) {
+				wrap.insertBefore( this[ 0 ] );
+			}
+
+			wrap.map( function() {
+				var elem = this;
+
+				while ( elem.firstElementChild ) {
+					elem = elem.firstElementChild;
+				}
+
+				return elem;
+			} ).append( this );
+		}
+
+		return this;
+	},
+
+	wrapInner: function( html ) {
+		if ( isFunction( html ) ) {
+			return this.each( function( i ) {
+				jQuery( this ).wrapInner( html.call( this, i ) );
+			} );
+		}
+
+		return this.each( function() {
+			var self = jQuery( this ),
+				contents = self.contents();
+
+			if ( contents.length ) {
+				contents.wrapAll( html );
+
+			} else {
+				self.append( html );
+			}
+		} );
+	},
+
+	wrap: function( html ) {
+		var htmlIsFunction = isFunction( html );
+
+		return this.each( function( i ) {
+			jQuery( this ).wrapAll( htmlIsFunction ? html.call( this, i ) : html );
+		} );
+	},
+
+	unwrap: function( selector ) {
+		this.parent( selector ).not( "body" ).each( function() {
+			jQuery( this ).replaceWith( this.childNodes );
+		} );
+		return this;
+	}
+} );
+
+
+jQuery.expr.pseudos.hidden = function( elem ) {
+	return !jQuery.expr.pseudos.visible( elem );
+};
+jQuery.expr.pseudos.visible = function( elem ) {
+	return !!( elem.offsetWidth || elem.offsetHeight || elem.getClientRects().length );
+};
+
+
+
+
+jQuery.ajaxSettings.xhr = function() {
+	try {
+		return new window.XMLHttpRequest();
+	} catch ( e ) {}
+};
+
+var xhrSuccessStatus = {
+
+		// File protocol always yields status code 0, assume 200
+		0: 200,
+
+		// Support: IE <=9 only
+		// #1450: sometimes IE returns 1223 when it should be 204
+		1223: 204
+	},
+	xhrSupported = jQuery.ajaxSettings.xhr();
+
+support.cors = !!xhrSupported && ( "withCredentials" in xhrSupported );
+support.ajax = xhrSupported = !!xhrSupported;
+
+jQuery.ajaxTransport( function( options ) {
+	var callback, errorCallback;
+
+	// Cross domain only allowed if supported through XMLHttpRequest
+	if ( support.cors || xhrSupported && !options.crossDomain ) {
+		return {
+			send: function( headers, complete ) {
+				var i,
+					xhr = options.xhr();
+
+				xhr.open(
+					options.type,
+					options.url,
+					options.async,
+					options.username,
+					options.password
+				);
+
+				// Apply custom fields if provided
+				if ( options.xhrFields ) {
+					for ( i in options.xhrFields ) {
+						xhr[ i ] = options.xhrFields[ i ];
+					}
+				}
+
+				// Override mime type if needed
+				if ( options.mimeType && xhr.overrideMimeType ) {
+					xhr.overrideMimeType( options.mimeType );
+				}
+
+				// X-Requested-With header
+				// For cross-domain requests, seeing as conditions for a preflight are
+				// akin to a jigsaw puzzle, we simply never set it to be sure.
+				// (it can always be set on a per-request basis or even using ajaxSetup)
+				// For same-domain requests, won't change header if already provided.
+				if ( !options.crossDomain && !headers[ "X-Requested-With" ] ) {
+					headers[ "X-Requested-With" ] = "XMLHttpRequest";
+				}
+
+				// Set headers
+				for ( i in headers ) {
+					xhr.setRequestHeader( i, headers[ i ] );
+				}
+
+				// Callback
+				callback = function( type ) {
+					return function() {
+						if ( callback ) {
+							callback = errorCallback = xhr.onload =
+								xhr.onerror = xhr.onabort = xhr.ontimeout =
+									xhr.onreadystatechange = null;
+
+							if ( type === "abort" ) {
+								xhr.abort();
+							} else if ( type === "error" ) {
+
+								// Support: IE <=9 only
+								// On a manual native abort, IE9 throws
+								// errors on any property access that is not readyState
+								if ( typeof xhr.status !== "number" ) {
+									complete( 0, "error" );
+								} else {
+									complete(
+
+										// File: protocol always yields status 0; see #8605, #14207
+										xhr.status,
+										xhr.statusText
+									);
+								}
+							} else {
+								complete(
+									xhrSuccessStatus[ xhr.status ] || xhr.status,
+									xhr.statusText,
+
+									// Support: IE <=9 only
+									// IE9 has no XHR2 but throws on binary (trac-11426)
+									// For XHR2 non-text, let the caller handle it (gh-2498)
+									( xhr.responseType || "text" ) !== "text"  ||
+									typeof xhr.responseText !== "string" ?
+										{ binary: xhr.response } :
+										{ text: xhr.responseText },
+									xhr.getAllResponseHeaders()
+								);
+							}
+						}
+					};
+				};
+
+				// Listen to events
+				xhr.onload = callback();
+				errorCallback = xhr.onerror = xhr.ontimeout = callback( "error" );
+
+				// Support: IE 9 only
+				// Use onreadystatechange to replace onabort
+				// to handle uncaught aborts
+				if ( xhr.onabort !== undefined ) {
+					xhr.onabort = errorCallback;
+				} else {
+					xhr.onreadystatechange = function() {
+
+						// Check readyState before timeout as it changes
+						if ( xhr.readyState === 4 ) {
+
+							// Allow onerror to be called first,
+							// but that will not handle a native abort
+							// Also, save errorCallback to a variable
+							// as xhr.onerror cannot be accessed
+							window.setTimeout( function() {
+								if ( callback ) {
+									errorCallback();
+								}
+							} );
+						}
+					};
+				}
+
+				// Create the abort callback
+				callback = callback( "abort" );
+
+				try {
+
+					// Do send the request (this may raise an exception)
+					xhr.send( options.hasContent && options.data || null );
+				} catch ( e ) {
+
+					// #14683: Only rethrow if this hasn't been notified as an error yet
+					if ( callback ) {
+						throw e;
+					}
+				}
+			},
+
+			abort: function() {
+				if ( callback ) {
+					callback();
+				}
+			}
+		};
+	}
+} );
+
+
+
+
+// Prevent auto-execution of scripts when no explicit dataType was provided (See gh-2432)
+jQuery.ajaxPrefilter( function( s ) {
+	if ( s.crossDomain ) {
+		s.contents.script = false;
+	}
+} );
+
+// Install script dataType
+jQuery.ajaxSetup( {
+	accepts: {
+		script: "text/javascript, application/javascript, " +
+			"application/ecmascript, application/x-ecmascript"
+	},
+	contents: {
+		script: /\b(?:java|ecma)script\b/
+	},
+	converters: {
+		"text script": function( text ) {
+			jQuery.globalEval( text );
+			return text;
+		}
+	}
+} );
+
+// Handle cache's special case and crossDomain
+jQuery.ajaxPrefilter( "script", function( s ) {
+	if ( s.cache === undefined ) {
+		s.cache = false;
+	}
+	if ( s.crossDomain ) {
+		s.type = "GET";
+	}
+} );
+
+// Bind script tag hack transport
+jQuery.ajaxTransport( "script", function( s ) {
+
+	// This transport only deals with cross domain or forced-by-attrs requests
+	if ( s.crossDomain || s.scriptAttrs ) {
+		var script, callback;
+		return {
+			send: function( _, complete ) {
+				script = jQuery( "<script>" )
+					.attr( s.scriptAttrs || {} )
+					.prop( { charset: s.scriptCharset, src: s.url } )
+					.on( "load error", callback = function( evt ) {
+						script.remove();
+						callback = null;
+						if ( evt ) {
+							complete( evt.type === "error" ? 404 : 200, evt.type );
+						}
+					} );
+
+				// Use native DOM manipulation to avoid our domManip AJAX trickery
+				document.head.appendChild( script[ 0 ] );
+			},
+			abort: function() {
+				if ( callback ) {
+					callback();
+				}
+			}
+		};
+	}
+} );
+
+
+
+
+var oldCallbacks = [],
+	rjsonp = /(=)\?(?=&|$)|\?\?/;
+
+// Default jsonp settings
+jQuery.ajaxSetup( {
+	jsonp: "callback",
+	jsonpCallback: function() {
+		var callback = oldCallbacks.pop() || ( jQuery.expando + "_" + ( nonce.guid++ ) );
+		this[ callback ] = true;
+		return callback;
+	}
+} );
+
+// Detect, normalize options and install callbacks for jsonp requests
+jQuery.ajaxPrefilter( "json jsonp", function( s, originalSettings, jqXHR ) {
+
+	var callbackName, overwritten, responseContainer,
+		jsonProp = s.jsonp !== false && ( rjsonp.test( s.url ) ?
+			"url" :
+			typeof s.data === "string" &&
+				( s.contentType || "" )
+					.indexOf( "application/x-www-form-urlencoded" ) === 0 &&
+				rjsonp.test( s.data ) && "data"
+		);
+
+	// Handle iff the expected data type is "jsonp" or we have a parameter to set
+	if ( jsonProp || s.dataTypes[ 0 ] === "jsonp" ) {
+
+		// Get callback name, remembering preexisting value associated with it
+		callbackName = s.jsonpCallback = isFunction( s.jsonpCallback ) ?
+			s.jsonpCallback() :
+			s.jsonpCallback;
+
+		// Insert callback into url or form data
+		if ( jsonProp ) {
+			s[ jsonProp ] = s[ jsonProp ].replace( rjsonp, "$1" + callbackName );
+		} else if ( s.jsonp !== false ) {
+			s.url += ( rquery.test( s.url ) ? "&" : "?" ) + s.jsonp + "=" + callbackName;
+		}
+
+		// Use data converter to retrieve json after script execution
+		s.converters[ "script json" ] = function() {
+			if ( !responseContainer ) {
+				jQuery.error( callbackName + " was not called" );
+			}
+			return responseContainer[ 0 ];
+		};
+
+		// Force json dataType
+		s.dataTypes[ 0 ] = "json";
+
+		// Install callback
+		overwritten = window[ callbackName ];
+		window[ callbackName ] = function() {
+			responseContainer = arguments;
+		};
+
+		// Clean-up function (fires after converters)
+		jqXHR.always( function() {
+
+			// If previous value didn't exist - remove it
+			if ( overwritten === undefined ) {
+				jQuery( window ).removeProp( callbackName );
+
+			// Otherwise restore preexisting value
+			} else {
+				window[ callbackName ] = overwritten;
+			}
+
+			// Save back as free
+			if ( s[ callbackName ] ) {
+
+				// Make sure that re-using the options doesn't screw things around
+				s.jsonpCallback = originalSettings.jsonpCallback;
+
+				// Save the callback name for future use
+				oldCallbacks.push( callbackName );
+			}
+
+			// Call if it was a function and we have a response
+			if ( responseContainer && isFunction( overwritten ) ) {
+				overwritten( responseContainer[ 0 ] );
+			}
+
+			responseContainer = overwritten = undefined;
+		} );
+
+		// Delegate to script
+		return "script";
+	}
+} );
+
+
+
+
+// Support: Safari 8 only
+// In Safari 8 documents created via document.implementation.createHTMLDocument
+// collapse sibling forms: the second one becomes a child of the first one.
+// Because of that, this security measure has to be disabled in Safari 8.
+// https://bugs.webkit.org/show_bug.cgi?id=137337
+support.createHTMLDocument = ( function() {
+	var body = document.implementation.createHTMLDocument( "" ).body;
+	body.innerHTML = "<form></form><form></form>";
+	return body.childNodes.length === 2;
+} )();
+
+
+// Argument "data" should be string of html
+// context (optional): If specified, the fragment will be created in this context,
+// defaults to document
+// keepScripts (optional): If true, will include scripts passed in the html string
+jQuery.parseHTML = function( data, context, keepScripts ) {
+	if ( typeof data !== "string" ) {
+		return [];
+	}
+	if ( typeof context === "boolean" ) {
+		keepScripts = context;
+		context = false;
+	}
+
+	var base, parsed, scripts;
+
+	if ( !context ) {
+
+		// Stop scripts or inline event handlers from being executed immediately
+		// by using document.implementation
+		if ( support.createHTMLDocument ) {
+			context = document.implementation.createHTMLDocument( "" );
+
+			// Set the base href for the created document
+			// so any parsed elements with URLs
+			// are based on the document's URL (gh-2965)
+			base = context.createElement( "base" );
+			base.href = document.location.href;
+			context.head.appendChild( base );
+		} else {
+			context = document;
+		}
+	}
+
+	parsed = rsingleTag.exec( data );
+	scripts = !keepScripts && [];
+
+	// Single tag
+	if ( parsed ) {
+		return [ context.createElement( parsed[ 1 ] ) ];
+	}
+
+	parsed = buildFragment( [ data ], context, scripts );
+
+	if ( scripts && scripts.length ) {
+		jQuery( scripts ).remove();
+	}
+
+	return jQuery.merge( [], parsed.childNodes );
+};
+
+
+/**
+ * Load a url into a page
+ */
+jQuery.fn.load = function( url, params, callback ) {
+	var selector, type, response,
+		self = this,
+		off = url.indexOf( " " );
+
+	if ( off > -1 ) {
+		selector = stripAndCollapse( url.slice( off ) );
+		url = url.slice( 0, off );
+	}
+
+	// If it's a function
+	if ( isFunction( params ) ) {
+
+		// We assume that it's the callback
+		callback = params;
+		params = undefined;
+
+	// Otherwise, build a param string
+	} else if ( params && typeof params === "object" ) {
+		type = "POST";
+	}
+
+	// If we have elements to modify, make the request
+	if ( self.length > 0 ) {
+		jQuery.ajax( {
+			url: url,
+
+			// If "type" variable is undefined, then "GET" method will be used.
+			// Make value of this field explicit since
+			// user can override it through ajaxSetup method
+			type: type || "GET",
+			dataType: "html",
+			data: params
+		} ).done( function( responseText ) {
+
+			// Save response for use in complete callback
+			response = arguments;
+
+			self.html( selector ?
+
+				// If a selector was specified, locate the right elements in a dummy div
+				// Exclude scripts to avoid IE 'Permission Denied' errors
+				jQuery( "<div>" ).append( jQuery.parseHTML( responseText ) ).find( selector ) :
+
+				// Otherwise use the full result
+				responseText );
+
+		// If the request succeeds, this function gets "data", "status", "jqXHR"
+		// but they are ignored because response was set above.
+		// If it fails, this function gets "jqXHR", "status", "error"
+		} ).always( callback && function( jqXHR, status ) {
+			self.each( function() {
+				callback.apply( this, response || [ jqXHR.responseText, status, jqXHR ] );
+			} );
+		} );
+	}
+
+	return this;
+};
+
+
+
+
+jQuery.expr.pseudos.animated = function( elem ) {
+	return jQuery.grep( jQuery.timers, function( fn ) {
+		return elem === fn.elem;
+	} ).length;
+};
+
+
+
+
+jQuery.offset = {
+	setOffset: function( elem, options, i ) {
+		var curPosition, curLeft, curCSSTop, curTop, curOffset, curCSSLeft, calculatePosition,
+			position = jQuery.css( elem, "position" ),
+			curElem = jQuery( elem ),
+			props = {};
+
+		// Set position first, in-case top/left are set even on static elem
+		if ( position === "static" ) {
+			elem.style.position = "relative";
+		}
+
+		curOffset = curElem.offset();
+		curCSSTop = jQuery.css( elem, "top" );
+		curCSSLeft = jQuery.css( elem, "left" );
+		calculatePosition = ( position === "absolute" || position === "fixed" ) &&
+			( curCSSTop + curCSSLeft ).indexOf( "auto" ) > -1;
+
+		// Need to be able to calculate position if either
+		// top or left is auto and position is either absolute or fixed
+		if ( calculatePosition ) {
+			curPosition = curElem.position();
+			curTop = curPosition.top;
+			curLeft = curPosition.left;
+
+		} else {
+			curTop = parseFloat( curCSSTop ) || 0;
+			curLeft = parseFloat( curCSSLeft ) || 0;
+		}
+
+		if ( isFunction( options ) ) {
+
+			// Use jQuery.extend here to allow modification of coordinates argument (gh-1848)
+			options = options.call( elem, i, jQuery.extend( {}, curOffset ) );
+		}
+
+		if ( options.top != null ) {
+			props.top = ( options.top - curOffset.top ) + curTop;
+		}
+		if ( options.left != null ) {
+			props.left = ( options.left - curOffset.left ) + curLeft;
+		}
+
+		if ( "using" in options ) {
+			options.using.call( elem, props );
+
+		} else {
+			if ( typeof props.top === "number" ) {
+				props.top += "px";
+			}
+			if ( typeof props.left === "number" ) {
+				props.left += "px";
+			}
+			curElem.css( props );
+		}
+	}
+};
+
+jQuery.fn.extend( {
+
+	// offset() relates an element's border box to the document origin
+	offset: function( options ) {
+
+		// Preserve chaining for setter
+		if ( arguments.length ) {
+			return options === undefined ?
+				this :
+				this.each( function( i ) {
+					jQuery.offset.setOffset( this, options, i );
+				} );
+		}
+
+		var rect, win,
+			elem = this[ 0 ];
+
+		if ( !elem ) {
+			return;
+		}
+
+		// Return zeros for disconnected and hidden (display: none) elements (gh-2310)
+		// Support: IE <=11 only
+		// Running getBoundingClientRect on a
+		// disconnected node in IE throws an error
+		if ( !elem.getClientRects().length ) {
+			return { top: 0, left: 0 };
+		}
+
+		// Get document-relative position by adding viewport scroll to viewport-relative gBCR
+		rect = elem.getBoundingClientRect();
+		win = elem.ownerDocument.defaultView;
+		return {
+			top: rect.top + win.pageYOffset,
+			left: rect.left + win.pageXOffset
+		};
+	},
+
+	// position() relates an element's margin box to its offset parent's padding box
+	// This corresponds to the behavior of CSS absolute positioning
+	position: function() {
+		if ( !this[ 0 ] ) {
+			return;
+		}
+
+		var offsetParent, offset, doc,
+			elem = this[ 0 ],
+			parentOffset = { top: 0, left: 0 };
+
+		// position:fixed elements are offset from the viewport, which itself always has zero offset
+		if ( jQuery.css( elem, "position" ) === "fixed" ) {
+
+			// Assume position:fixed implies availability of getBoundingClientRect
+			offset = elem.getBoundingClientRect();
+
+		} else {
+			offset = this.offset();
+
+			// Account for the *real* offset parent, which can be the document or its root element
+			// when a statically positioned element is identified
+			doc = elem.ownerDocument;
+			offsetParent = elem.offsetParent || doc.documentElement;
+			while ( offsetParent &&
+				( offsetParent === doc.body || offsetParent === doc.documentElement ) &&
+				jQuery.css( offsetParent, "position" ) === "static" ) {
+
+				offsetParent = offsetParent.parentNode;
+			}
+			if ( offsetParent && offsetParent !== elem && offsetParent.nodeType === 1 ) {
+
+				// Incorporate borders into its offset, since they are outside its content origin
+				parentOffset = jQuery( offsetParent ).offset();
+				parentOffset.top += jQuery.css( offsetParent, "borderTopWidth", true );
+				parentOffset.left += jQuery.css( offsetParent, "borderLeftWidth", true );
+			}
+		}
+
+		// Subtract parent offsets and element margins
+		return {
+			top: offset.top - parentOffset.top - jQuery.css( elem, "marginTop", true ),
+			left: offset.left - parentOffset.left - jQuery.css( elem, "marginLeft", true )
+		};
+	},
+
+	// This method will return documentElement in the following cases:
+	// 1) For the element inside the iframe without offsetParent, this method will return
+	//    documentElement of the parent window
+	// 2) For the hidden or detached element
+	// 3) For body or html element, i.e. in case of the html node - it will return itself
+	//
+	// but those exceptions were never presented as a real life use-cases
+	// and might be considered as more preferable results.
+	//
+	// This logic, however, is not guaranteed and can change at any point in the future
+	offsetParent: function() {
+		return this.map( function() {
+			var offsetParent = this.offsetParent;
+
+			while ( offsetParent && jQuery.css( offsetParent, "position" ) === "static" ) {
+				offsetParent = offsetParent.offsetParent;
+			}
+
+			return offsetParent || documentElement;
+		} );
+	}
+} );
+
+// Create scrollLeft and scrollTop methods
+jQuery.each( { scrollLeft: "pageXOffset", scrollTop: "pageYOffset" }, function( method, prop ) {
+	var top = "pageYOffset" === prop;
+
+	jQuery.fn[ method ] = function( val ) {
+		return access( this, function( elem, method, val ) {
+
+			// Coalesce documents and windows
+			var win;
+			if ( isWindow( elem ) ) {
+				win = elem;
+			} else if ( elem.nodeType === 9 ) {
+				win = elem.defaultView;
+			}
+
+			if ( val === undefined ) {
+				return win ? win[ prop ] : elem[ method ];
+			}
+
+			if ( win ) {
+				win.scrollTo(
+					!top ? val : win.pageXOffset,
+					top ? val : win.pageYOffset
+				);
+
+			} else {
+				elem[ method ] = val;
+			}
+		}, method, val, arguments.length );
+	};
+} );
+
+// Support: Safari <=7 - 9.1, Chrome <=37 - 49
+// Add the top/left cssHooks using jQuery.fn.position
+// Webkit bug: https://bugs.webkit.org/show_bug.cgi?id=29084
+// Blink bug: https://bugs.chromium.org/p/chromium/issues/detail?id=589347
+// getComputedStyle returns percent when specified for top/left/bottom/right;
+// rather than make the css module depend on the offset module, just check for it here
+jQuery.each( [ "top", "left" ], function( _i, prop ) {
+	jQuery.cssHooks[ prop ] = addGetHookIf( support.pixelPosition,
+		function( elem, computed ) {
+			if ( computed ) {
+				computed = curCSS( elem, prop );
+
+				// If curCSS returns percentage, fallback to offset
+				return rnumnonpx.test( computed ) ?
+					jQuery( elem ).position()[ prop ] + "px" :
+					computed;
+			}
+		}
+	);
+} );
+
+
+// Create innerHeight, innerWidth, height, width, outerHeight and outerWidth methods
+jQuery.each( { Height: "height", Width: "width" }, function( name, type ) {
+	jQuery.each( { padding: "inner" + name, content: type, "": "outer" + name },
+		function( defaultExtra, funcName ) {
+
+		// Margin is only for outerHeight, outerWidth
+		jQuery.fn[ funcName ] = function( margin, value ) {
+			var chainable = arguments.length && ( defaultExtra || typeof margin !== "boolean" ),
+				extra = defaultExtra || ( margin === true || value === true ? "margin" : "border" );
+
+			return access( this, function( elem, type, value ) {
+				var doc;
+
+				if ( isWindow( elem ) ) {
+
+					// $( window ).outerWidth/Height return w/h including scrollbars (gh-1729)
+					return funcName.indexOf( "outer" ) === 0 ?
+						elem[ "inner" + name ] :
+						elem.document.documentElement[ "client" + name ];
+				}
+
+				// Get document width or height
+				if ( elem.nodeType === 9 ) {
+					doc = elem.documentElement;
+
+					// Either scroll[Width/Height] or offset[Width/Height] or client[Width/Height],
+					// whichever is greatest
+					return Math.max(
+						elem.body[ "scroll" + name ], doc[ "scroll" + name ],
+						elem.body[ "offset" + name ], doc[ "offset" + name ],
+						doc[ "client" + name ]
+					);
+				}
+
+				return value === undefined ?
+
+					// Get width or height on the element, requesting but not forcing parseFloat
+					jQuery.css( elem, type, extra ) :
+
+					// Set width or height on the element
+					jQuery.style( elem, type, value, extra );
+			}, type, chainable ? margin : undefined, chainable );
+		};
+	} );
+} );
+
+
+jQuery.each( [
+	"ajaxStart",
+	"ajaxStop",
+	"ajaxComplete",
+	"ajaxError",
+	"ajaxSuccess",
+	"ajaxSend"
+], function( _i, type ) {
+	jQuery.fn[ type ] = function( fn ) {
+		return this.on( type, fn );
+	};
+} );
+
+
+
+
+jQuery.fn.extend( {
+
+	bind: function( types, data, fn ) {
+		return this.on( types, null, data, fn );
+	},
+	unbind: function( types, fn ) {
+		return this.off( types, null, fn );
+	},
+
+	delegate: function( selector, types, data, fn ) {
+		return this.on( types, selector, data, fn );
+	},
+	undelegate: function( selector, types, fn ) {
+
+		// ( namespace ) or ( selector, types [, fn] )
+		return arguments.length === 1 ?
+			this.off( selector, "**" ) :
+			this.off( types, selector || "**", fn );
+	},
+
+	hover: function( fnOver, fnOut ) {
+		return this.mouseenter( fnOver ).mouseleave( fnOut || fnOver );
+	}
+} );
+
+jQuery.each( ( "blur focus focusin focusout resize scroll click dblclick " +
+	"mousedown mouseup mousemove mouseover mouseout mouseenter mouseleave " +
+	"change select submit keydown keypress keyup contextmenu" ).split( " " ),
+	function( _i, name ) {
+
+		// Handle event binding
+		jQuery.fn[ name ] = function( data, fn ) {
+			return arguments.length > 0 ?
+				this.on( name, null, data, fn ) :
+				this.trigger( name );
+		};
+	} );
+
+
+
+
+// Support: Android <=4.0 only
+// Make sure we trim BOM and NBSP
+var rtrim = /^[\s\uFEFF\xA0]+|[\s\uFEFF\xA0]+$/g;
+
+// Bind a function to a context, optionally partially applying any
+// arguments.
+// jQuery.proxy is deprecated to promote standards (specifically Function#bind)
+// However, it is not slated for removal any time soon
+jQuery.proxy = function( fn, context ) {
+	var tmp, args, proxy;
+
+	if ( typeof context === "string" ) {
+		tmp = fn[ context ];
+		context = fn;
+		fn = tmp;
+	}
+
+	// Quick check to determine if target is callable, in the spec
+	// this throws a TypeError, but we will just return undefined.
+	if ( !isFunction( fn ) ) {
+		return undefined;
+	}
+
+	// Simulated bind
+	args = slice.call( arguments, 2 );
+	proxy = function() {
+		return fn.apply( context || this, args.concat( slice.call( arguments ) ) );
+	};
+
+	// Set the guid of unique handler to the same of original handler, so it can be removed
+	proxy.guid = fn.guid = fn.guid || jQuery.guid++;
+
+	return proxy;
+};
+
+jQuery.holdReady = function( hold ) {
+	if ( hold ) {
+		jQuery.readyWait++;
+	} else {
+		jQuery.ready( true );
+	}
+};
+jQuery.isArray = Array.isArray;
+jQuery.parseJSON = JSON.parse;
+jQuery.nodeName = nodeName;
+jQuery.isFunction = isFunction;
+jQuery.isWindow = isWindow;
+jQuery.camelCase = camelCase;
+jQuery.type = toType;
+
+jQuery.now = Date.now;
+
+jQuery.isNumeric = function( obj ) {
+
+	// As of jQuery 3.0, isNumeric is limited to
+	// strings and numbers (primitives or objects)
+	// that can be coerced to finite numbers (gh-2662)
+	var type = jQuery.type( obj );
+	return ( type === "number" || type === "string" ) &&
+
+		// parseFloat NaNs numeric-cast false positives ("")
+		// ...but misinterprets leading-number strings, particularly hex literals ("0x...")
+		// subtraction forces infinities to NaN
+		!isNaN( obj - parseFloat( obj ) );
+};
+
+jQuery.trim = function( text ) {
+	return text == null ?
+		"" :
+		( text + "" ).replace( rtrim, "" );
+};
+
+
+
+// Register as a named AMD module, since jQuery can be concatenated with other
+// files that may use define, but not via a proper concatenation script that
+// understands anonymous AMD modules. A named AMD is safest and most robust
+// way to register. Lowercase jquery is used because AMD module names are
+// derived from file names, and jQuery is normally delivered in a lowercase
+// file name. Do this after creating the global so that if an AMD module wants
+// to call noConflict to hide this version of jQuery, it will work.
+
+// Note that for maximum portability, libraries that are not jQuery should
+// declare themselves as anonymous modules, and avoid setting a global if an
+// AMD loader is present. jQuery is a special case. For more information, see
+// https://github.com/jrburke/requirejs/wiki/Updating-existing-libraries#wiki-anon
+
+if ( typeof define === "function" && define.amd ) {
+	define( "jquery", [], function() {
+		return jQuery;
+	} );
+}
+
+
+
+
+var
+
+	// Map over jQuery in case of overwrite
+	_jQuery = window.jQuery,
+
+	// Map over the $ in case of overwrite
+	_$ = window.$;
+
+jQuery.noConflict = function( deep ) {
+	if ( window.$ === jQuery ) {
+		window.$ = _$;
+	}
+
+	if ( deep && window.jQuery === jQuery ) {
+		window.jQuery = _jQuery;
+	}
+
+	return jQuery;
+};
+
+// Expose jQuery and $ identifiers, even in AMD
+// (#7102#comment:10, https://github.com/jquery/jquery/pull/557)
+// and CommonJS for browser emulators (#13566)
+if ( typeof noGlobal === "undefined" ) {
+	window.jQuery = window.$ = jQuery;
+}
+
+
+
+
+return jQuery;
+} );
diff --git a/_static/jquery.js b/_static/jquery.js
new file mode 100644
index 00000000..b0614034
--- /dev/null
+++ b/_static/jquery.js
@@ -0,0 +1,2 @@
+/*! jQuery v3.5.1 | (c) JS Foundation and other contributors | jquery.org/license */
+!function(e,t){"use strict";"object"==typeof module&&"object"==typeof module.exports?module.exports=e.document?t(e,!0):function(e){if(!e.document)throw new Error("jQuery requires a window with a document");return t(e)}:t(e)}("undefined"!=typeof window?window:this,function(C,e){"use strict";var t=[],r=Object.getPrototypeOf,s=t.slice,g=t.flat?function(e){return t.flat.call(e)}:function(e){return t.concat.apply([],e)},u=t.push,i=t.indexOf,n={},o=n.toString,v=n.hasOwnProperty,a=v.toString,l=a.call(Object),y={},m=function(e){return"function"==typeof e&&"number"!=typeof e.nodeType},x=function(e){return null!=e&&e===e.window},E=C.document,c={type:!0,src:!0,nonce:!0,noModule:!0};function b(e,t,n){var r,i,o=(n=n||E).createElement("script");if(o.text=e,t)for(r in c)(i=t[r]||t.getAttribute&&t.getAttribute(r))&&o.setAttribute(r,i);n.head.appendChild(o).parentNode.removeChild(o)}function w(e){return null==e?e+"":"object"==typeof e||"function"==typeof e?n[o.call(e)]||"object":typeof e}var 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diff --git a/_static/language_data.js b/_static/language_data.js
new file mode 100644
index 00000000..ebe2f03b
--- /dev/null
+++ b/_static/language_data.js
@@ -0,0 +1,297 @@
+/*
+ * language_data.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * This script contains the language-specific data used by searchtools.js,
+ * namely the list of stopwords, stemmer, scorer and splitter.
+ *
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+var stopwords = ["a","and","are","as","at","be","but","by","for","if","in","into","is","it","near","no","not","of","on","or","such","that","the","their","then","there","these","they","this","to","was","will","with"];
+
+
+/* Non-minified version is copied as a separate JS file, is available */
+
+/**
+ * Porter Stemmer
+ */
+var Stemmer = function() {
+
+  var step2list = {
+    ational: 'ate',
+    tional: 'tion',
+    enci: 'ence',
+    anci: 'ance',
+    izer: 'ize',
+    bli: 'ble',
+    alli: 'al',
+    entli: 'ent',
+    eli: 'e',
+    ousli: 'ous',
+    ization: 'ize',
+    ation: 'ate',
+    ator: 'ate',
+    alism: 'al',
+    iveness: 'ive',
+    fulness: 'ful',
+    ousness: 'ous',
+    aliti: 'al',
+    iviti: 'ive',
+    biliti: 'ble',
+    logi: 'log'
+  };
+
+  var step3list = {
+    icate: 'ic',
+    ative: '',
+    alize: 'al',
+    iciti: 'ic',
+    ical: 'ic',
+    ful: '',
+    ness: ''
+  };
+
+  var c = "[^aeiou]";          // consonant
+  var v = "[aeiouy]";          // vowel
+  var C = c + "[^aeiouy]*";    // consonant sequence
+  var V = v + "[aeiou]*";      // vowel sequence
+
+  var mgr0 = "^(" + C + ")?" + V + C;                      // [C]VC... is m>0
+  var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$";    // [C]VC[V] is m=1
+  var mgr1 = "^(" + C + ")?" + V + C + V + C;              // [C]VCVC... is m>1
+  var s_v   = "^(" + C + ")?" + v;                         // vowel in stem
+
+  this.stemWord = function (w) {
+    var stem;
+    var suffix;
+    var firstch;
+    var origword = w;
+
+    if (w.length < 3)
+      return w;
+
+    var re;
+    var re2;
+    var re3;
+    var re4;
+
+    firstch = w.substr(0,1);
+    if (firstch == "y")
+      w = firstch.toUpperCase() + w.substr(1);
+
+    // Step 1a
+    re = /^(.+?)(ss|i)es$/;
+    re2 = /^(.+?)([^s])s$/;
+
+    if (re.test(w))
+      w = w.replace(re,"$1$2");
+    else if (re2.test(w))
+      w = w.replace(re2,"$1$2");
+
+    // Step 1b
+    re = /^(.+?)eed$/;
+    re2 = /^(.+?)(ed|ing)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      re = new RegExp(mgr0);
+      if (re.test(fp[1])) {
+        re = /.$/;
+        w = w.replace(re,"");
+      }
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1];
+      re2 = new RegExp(s_v);
+      if (re2.test(stem)) {
+        w = stem;
+        re2 = /(at|bl|iz)$/;
+        re3 = new RegExp("([^aeiouylsz])\\1$");
+        re4 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+        if (re2.test(w))
+          w = w + "e";
+        else if (re3.test(w)) {
+          re = /.$/;
+          w = w.replace(re,"");
+        }
+        else if (re4.test(w))
+          w = w + "e";
+      }
+    }
+
+    // Step 1c
+    re = /^(.+?)y$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(s_v);
+      if (re.test(stem))
+        w = stem + "i";
+    }
+
+    // Step 2
+    re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step2list[suffix];
+    }
+
+    // Step 3
+    re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step3list[suffix];
+    }
+
+    // Step 4
+    re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;
+    re2 = /^(.+?)(s|t)(ion)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      if (re.test(stem))
+        w = stem;
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1] + fp[2];
+      re2 = new RegExp(mgr1);
+      if (re2.test(stem))
+        w = stem;
+    }
+
+    // Step 5
+    re = /^(.+?)e$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      re2 = new RegExp(meq1);
+      re3 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem))))
+        w = stem;
+    }
+    re = /ll$/;
+    re2 = new RegExp(mgr1);
+    if (re.test(w) && re2.test(w)) {
+      re = /.$/;
+      w = w.replace(re,"");
+    }
+
+    // and turn initial Y back to y
+    if (firstch == "y")
+      w = firstch.toLowerCase() + w.substr(1);
+    return w;
+  }
+}
+
+
+
+
+var splitChars = (function() {
+    var result = {};
+    var singles = [96, 180, 187, 191, 215, 247, 749, 885, 903, 907, 909, 930, 1014, 1648,
+         1748, 1809, 2416, 2473, 2481, 2526, 2601, 2609, 2612, 2615, 2653, 2702,
+         2706, 2729, 2737, 2740, 2857, 2865, 2868, 2910, 2928, 2948, 2961, 2971,
+         2973, 3085, 3089, 3113, 3124, 3213, 3217, 3241, 3252, 3295, 3341, 3345,
+         3369, 3506, 3516, 3633, 3715, 3721, 3736, 3744, 3748, 3750, 3756, 3761,
+         3781, 3912, 4239, 4347, 4681, 4695, 4697, 4745, 4785, 4799, 4801, 4823,
+         4881, 5760, 5901, 5997, 6313, 7405, 8024, 8026, 8028, 8030, 8117, 8125,
+         8133, 8181, 8468, 8485, 8487, 8489, 8494, 8527, 11311, 11359, 11687, 11695,
+         11703, 11711, 11719, 11727, 11735, 12448, 12539, 43010, 43014, 43019, 43587,
+         43696, 43713, 64286, 64297, 64311, 64317, 64319, 64322, 64325, 65141];
+    var i, j, start, end;
+    for (i = 0; i < singles.length; i++) {
+        result[singles[i]] = true;
+    }
+    var ranges = [[0, 47], [58, 64], [91, 94], [123, 169], [171, 177], [182, 184], [706, 709],
+         [722, 735], [741, 747], [751, 879], [888, 889], [894, 901], [1154, 1161],
+         [1318, 1328], [1367, 1368], [1370, 1376], [1416, 1487], [1515, 1519], [1523, 1568],
+         [1611, 1631], [1642, 1645], [1750, 1764], [1767, 1773], [1789, 1790], [1792, 1807],
+         [1840, 1868], [1958, 1968], [1970, 1983], [2027, 2035], [2038, 2041], [2043, 2047],
+         [2070, 2073], [2075, 2083], [2085, 2087], [2089, 2307], [2362, 2364], [2366, 2383],
+         [2385, 2391], [2402, 2405], [2419, 2424], [2432, 2436], [2445, 2446], [2449, 2450],
+         [2483, 2485], [2490, 2492], [2494, 2509], [2511, 2523], [2530, 2533], [2546, 2547],
+         [2554, 2564], [2571, 2574], [2577, 2578], [2618, 2648], [2655, 2661], [2672, 2673],
+         [2677, 2692], [2746, 2748], [2750, 2767], [2769, 2783], [2786, 2789], [2800, 2820],
+         [2829, 2830], [2833, 2834], [2874, 2876], [2878, 2907], [2914, 2917], [2930, 2946],
+         [2955, 2957], [2966, 2968], [2976, 2978], [2981, 2983], [2987, 2989], [3002, 3023],
+         [3025, 3045], [3059, 3076], [3130, 3132], [3134, 3159], [3162, 3167], [3170, 3173],
+         [3184, 3191], [3199, 3204], [3258, 3260], [3262, 3293], [3298, 3301], [3312, 3332],
+         [3386, 3388], [3390, 3423], [3426, 3429], [3446, 3449], [3456, 3460], [3479, 3481],
+         [3518, 3519], [3527, 3584], [3636, 3647], [3655, 3663], [3674, 3712], [3717, 3718],
+         [3723, 3724], [3726, 3731], [3752, 3753], [3764, 3772], [3774, 3775], [3783, 3791],
+         [3802, 3803], [3806, 3839], [3841, 3871], [3892, 3903], [3949, 3975], [3980, 4095],
+         [4139, 4158], [4170, 4175], [4182, 4185], [4190, 4192], [4194, 4196], [4199, 4205],
+         [4209, 4212], [4226, 4237], [4250, 4255], [4294, 4303], [4349, 4351], [4686, 4687],
+         [4702, 4703], [4750, 4751], [4790, 4791], [4806, 4807], [4886, 4887], [4955, 4968],
+         [4989, 4991], [5008, 5023], [5109, 5120], [5741, 5742], [5787, 5791], [5867, 5869],
+         [5873, 5887], [5906, 5919], [5938, 5951], [5970, 5983], [6001, 6015], [6068, 6102],
+         [6104, 6107], [6109, 6111], [6122, 6127], [6138, 6159], [6170, 6175], [6264, 6271],
+         [6315, 6319], [6390, 6399], [6429, 6469], [6510, 6511], [6517, 6527], [6572, 6592],
+         [6600, 6607], [6619, 6655], [6679, 6687], [6741, 6783], [6794, 6799], [6810, 6822],
+         [6824, 6916], [6964, 6980], [6988, 6991], [7002, 7042], [7073, 7085], [7098, 7167],
+         [7204, 7231], [7242, 7244], [7294, 7400], [7410, 7423], [7616, 7679], [7958, 7959],
+         [7966, 7967], [8006, 8007], [8014, 8015], [8062, 8063], [8127, 8129], [8141, 8143],
+         [8148, 8149], [8156, 8159], [8173, 8177], [8189, 8303], [8306, 8307], [8314, 8318],
+         [8330, 8335], [8341, 8449], [8451, 8454], [8456, 8457], [8470, 8472], [8478, 8483],
+         [8506, 8507], [8512, 8516], [8522, 8525], [8586, 9311], [9372, 9449], [9472, 10101],
+         [10132, 11263], [11493, 11498], [11503, 11516], [11518, 11519], [11558, 11567],
+         [11622, 11630], [11632, 11647], [11671, 11679], [11743, 11822], [11824, 12292],
+         [12296, 12320], [12330, 12336], [12342, 12343], [12349, 12352], [12439, 12444],
+         [12544, 12548], [12590, 12592], [12687, 12689], [12694, 12703], [12728, 12783],
+         [12800, 12831], [12842, 12880], [12896, 12927], [12938, 12976], [12992, 13311],
+         [19894, 19967], [40908, 40959], [42125, 42191], [42238, 42239], [42509, 42511],
+         [42540, 42559], [42592, 42593], [42607, 42622], [42648, 42655], [42736, 42774],
+         [42784, 42785], [42889, 42890], [42893, 43002], [43043, 43055], [43062, 43071],
+         [43124, 43137], [43188, 43215], [43226, 43249], [43256, 43258], [43260, 43263],
+         [43302, 43311], [43335, 43359], [43389, 43395], [43443, 43470], [43482, 43519],
+         [43561, 43583], [43596, 43599], [43610, 43615], [43639, 43641], [43643, 43647],
+         [43698, 43700], [43703, 43704], [43710, 43711], [43715, 43738], [43742, 43967],
+         [44003, 44015], [44026, 44031], [55204, 55215], [55239, 55242], [55292, 55295],
+         [57344, 63743], [64046, 64047], [64110, 64111], [64218, 64255], [64263, 64274],
+         [64280, 64284], [64434, 64466], [64830, 64847], [64912, 64913], [64968, 65007],
+         [65020, 65135], [65277, 65295], [65306, 65312], [65339, 65344], [65371, 65381],
+         [65471, 65473], [65480, 65481], [65488, 65489], [65496, 65497]];
+    for (i = 0; i < ranges.length; i++) {
+        start = ranges[i][0];
+        end = ranges[i][1];
+        for (j = start; j <= end; j++) {
+            result[j] = true;
+        }
+    }
+    return result;
+})();
+
+function splitQuery(query) {
+    var result = [];
+    var start = -1;
+    for (var i = 0; i < query.length; i++) {
+        if (splitChars[query.charCodeAt(i)]) {
+            if (start !== -1) {
+                result.push(query.slice(start, i));
+                start = -1;
+            }
+        } else if (start === -1) {
+            start = i;
+        }
+    }
+    if (start !== -1) {
+        result.push(query.slice(start));
+    }
+    return result;
+}
+
+
diff --git a/_static/locales/ar/LC_MESSAGES/booktheme.mo b/_static/locales/ar/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..15541a6a375f93efed3f410c2f892174ba91aaa0
GIT binary patch
literal 1541
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literal 0
HcmV?d00001

diff --git a/_static/locales/ar/LC_MESSAGES/booktheme.po b/_static/locales/ar/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..2e8d6820
--- /dev/null
+++ b/_static/locales/ar/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ar\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "أقترح تحرير"
+
+msgid "Last updated on"
+msgstr "آخر تحديث في"
+
+msgid "Edit this page"
+msgstr "قم بتحرير هذه الصفحة"
+
+msgid "Launch"
+msgstr "إطلاق"
+
+msgid "Print to PDF"
+msgstr "طباعة إلى PDF"
+
+msgid "open issue"
+msgstr "قضية مفتوحة"
+
+msgid "Download notebook file"
+msgstr "تنزيل ملف دفتر الملاحظات"
+
+msgid "Toggle navigation"
+msgstr "تبديل التنقل"
+
+msgid "Source repository"
+msgstr "مستودع المصدر"
+
+msgid "By the"
+msgstr "بواسطة"
+
+msgid "next page"
+msgstr "الصفحة التالية"
+
+msgid "repository"
+msgstr "مخزن"
+
+msgid "Sphinx Book Theme"
+msgstr "موضوع كتاب أبو الهول"
+
+msgid "Download source file"
+msgstr "تنزيل ملف المصدر"
+
+msgid "Contents"
+msgstr "محتويات"
+
+msgid "By"
+msgstr "بواسطة"
+
+msgid "Copyright"
+msgstr "حقوق النشر"
+
+msgid "Fullscreen mode"
+msgstr "وضع ملء الشاشة"
+
+msgid "Open an issue"
+msgstr "افتح قضية"
+
+msgid "previous page"
+msgstr "الصفحة السابقة"
+
+msgid "Download this page"
+msgstr "قم بتنزيل هذه الصفحة"
+
+msgid "Theme by the"
+msgstr "موضوع بواسطة"
diff --git a/_static/locales/bg/LC_MESSAGES/booktheme.mo b/_static/locales/bg/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..da95120037f8d7c52ad1f166a7682ea55ca47a01
GIT binary patch
literal 1708
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literal 0
HcmV?d00001

diff --git a/_static/locales/bg/LC_MESSAGES/booktheme.po b/_static/locales/bg/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..56ef0ebd
--- /dev/null
+++ b/_static/locales/bg/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: bg\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "предложи редактиране"
+
+msgid "Last updated on"
+msgstr "Последна актуализация на"
+
+msgid "Edit this page"
+msgstr "Редактирайте тази страница"
+
+msgid "Launch"
+msgstr "Стартиране"
+
+msgid "Print to PDF"
+msgstr "Печат в PDF"
+
+msgid "open issue"
+msgstr "отворен брой"
+
+msgid "Download notebook file"
+msgstr "Изтеглете файла на бележника"
+
+msgid "Toggle navigation"
+msgstr "Превключване на навигацията"
+
+msgid "Source repository"
+msgstr "Хранилище на източника"
+
+msgid "By the"
+msgstr "По"
+
+msgid "next page"
+msgstr "Следваща страница"
+
+msgid "repository"
+msgstr "хранилище"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема на книгата Sphinx"
+
+msgid "Download source file"
+msgstr "Изтеглете изходния файл"
+
+msgid "Contents"
+msgstr "Съдържание"
+
+msgid "By"
+msgstr "От"
+
+msgid "Copyright"
+msgstr "Авторско право"
+
+msgid "Fullscreen mode"
+msgstr "Режим на цял екран"
+
+msgid "Open an issue"
+msgstr "Отворете проблем"
+
+msgid "previous page"
+msgstr "предишна страница"
+
+msgid "Download this page"
+msgstr "Изтеглете тази страница"
+
+msgid "Theme by the"
+msgstr "Тема от"
diff --git a/_static/locales/bn/LC_MESSAGES/booktheme.mo b/_static/locales/bn/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..6b96639b726a2fa959a6419b6f8b7e0dfcce33ae
GIT binary patch
literal 1646
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literal 0
HcmV?d00001

diff --git a/_static/locales/bn/LC_MESSAGES/booktheme.po b/_static/locales/bn/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..243ca31f
--- /dev/null
+++ b/_static/locales/bn/LC_MESSAGES/booktheme.po
@@ -0,0 +1,63 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: bn\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Last updated on"
+msgstr "সর্বশেষ আপডেট"
+
+msgid "Edit this page"
+msgstr "এই পৃষ্ঠাটি সম্পাদনা করুন"
+
+msgid "Launch"
+msgstr "শুরু করা"
+
+msgid "Print to PDF"
+msgstr "পিডিএফ প্রিন্ট করুন"
+
+msgid "open issue"
+msgstr "খোলা সমস্যা"
+
+msgid "Download notebook file"
+msgstr "নোটবুক ফাইল ডাউনলোড করুন"
+
+msgid "Toggle navigation"
+msgstr "নেভিগেশন টগল করুন"
+
+msgid "Source repository"
+msgstr "উত্স সংগ্রহস্থল"
+
+msgid "By the"
+msgstr "দ্বারা"
+
+msgid "next page"
+msgstr "পরবর্তী পৃষ্ঠা"
+
+msgid "Sphinx Book Theme"
+msgstr "স্পিনিক্স বুক থিম"
+
+msgid "Download source file"
+msgstr "উত্স ফাইল ডাউনলোড করুন"
+
+msgid "By"
+msgstr "দ্বারা"
+
+msgid "Copyright"
+msgstr "কপিরাইট"
+
+msgid "Open an issue"
+msgstr "একটি সমস্যা খুলুন"
+
+msgid "previous page"
+msgstr "আগের পৃষ্ঠা"
+
+msgid "Download this page"
+msgstr "এই পৃষ্ঠাটি ডাউনলোড করুন"
+
+msgid "Theme by the"
+msgstr "থিম দ্বারা"
diff --git a/_static/locales/ca/LC_MESSAGES/booktheme.mo b/_static/locales/ca/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..a4dd30e9bd762578b119c4401d082ea4fa430613
GIT binary patch
literal 1246
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literal 0
HcmV?d00001

diff --git a/_static/locales/ca/LC_MESSAGES/booktheme.po b/_static/locales/ca/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..b27a13db
--- /dev/null
+++ b/_static/locales/ca/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ca\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "suggerir edició"
+
+msgid "Last updated on"
+msgstr "Darrera actualització el"
+
+msgid "Edit this page"
+msgstr "Editeu aquesta pàgina"
+
+msgid "Launch"
+msgstr "Llançament"
+
+msgid "Print to PDF"
+msgstr "Imprimeix a PDF"
+
+msgid "open issue"
+msgstr "número obert"
+
+msgid "Download notebook file"
+msgstr "Descarregar fitxer de quadern"
+
+msgid "Toggle navigation"
+msgstr "Commuta la navegació"
+
+msgid "Source repository"
+msgstr "Dipòsit de fonts"
+
+msgid "By the"
+msgstr "Per la"
+
+msgid "next page"
+msgstr "pàgina següent"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del llibre Esfinx"
+
+msgid "Download source file"
+msgstr "Baixeu el fitxer font"
+
+msgid "By"
+msgstr "Per"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Open an issue"
+msgstr "Obriu un número"
+
+msgid "previous page"
+msgstr "Pàgina anterior"
+
+msgid "Download this page"
+msgstr "Descarregueu aquesta pàgina"
+
+msgid "Theme by the"
+msgstr "Tema del"
diff --git a/_static/locales/cs/LC_MESSAGES/booktheme.mo b/_static/locales/cs/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..c39e01a6ae9ae5242f04566ffa2cda96b328ca18
GIT binary patch
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LZS{&p7>oY^Ev$2?

literal 0
HcmV?d00001

diff --git a/_static/locales/cs/LC_MESSAGES/booktheme.po b/_static/locales/cs/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..3818df97
--- /dev/null
+++ b/_static/locales/cs/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: cs\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "navrhnout úpravy"
+
+msgid "Last updated on"
+msgstr "Naposledy aktualizováno"
+
+msgid "Edit this page"
+msgstr "Upravit tuto stránku"
+
+msgid "Launch"
+msgstr "Zahájení"
+
+msgid "Print to PDF"
+msgstr "Tisk do PDF"
+
+msgid "open issue"
+msgstr "otevřené číslo"
+
+msgid "Download notebook file"
+msgstr "Stáhnout soubor poznámkového bloku"
+
+msgid "Toggle navigation"
+msgstr "Přepnout navigaci"
+
+msgid "Source repository"
+msgstr "Zdrojové úložiště"
+
+msgid "By the"
+msgstr "Podle"
+
+msgid "next page"
+msgstr "další strana"
+
+msgid "repository"
+msgstr "úložiště"
+
+msgid "Sphinx Book Theme"
+msgstr "Téma knihy Sfinga"
+
+msgid "Download source file"
+msgstr "Stáhněte si zdrojový soubor"
+
+msgid "Contents"
+msgstr "Obsah"
+
+msgid "By"
+msgstr "Podle"
+
+msgid "Copyright"
+msgstr "autorská práva"
+
+msgid "Fullscreen mode"
+msgstr "Režim celé obrazovky"
+
+msgid "Open an issue"
+msgstr "Otevřete problém"
+
+msgid "previous page"
+msgstr "předchozí stránka"
+
+msgid "Download this page"
+msgstr "Stáhněte si tuto stránku"
+
+msgid "Theme by the"
+msgstr "Téma od"
diff --git a/_static/locales/da/LC_MESSAGES/booktheme.mo b/_static/locales/da/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..f43157d70c63ff21b4385dc36cb73f5b14eb6b01
GIT binary patch
literal 1304
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literal 0
HcmV?d00001

diff --git a/_static/locales/da/LC_MESSAGES/booktheme.po b/_static/locales/da/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..7f20a3bd
--- /dev/null
+++ b/_static/locales/da/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: da\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "foreslå redigering"
+
+msgid "Last updated on"
+msgstr "Sidst opdateret den"
+
+msgid "Edit this page"
+msgstr "Rediger denne side"
+
+msgid "Launch"
+msgstr "Start"
+
+msgid "Print to PDF"
+msgstr "Udskriv til PDF"
+
+msgid "open issue"
+msgstr "åbent nummer"
+
+msgid "Download notebook file"
+msgstr "Download notesbog-fil"
+
+msgid "Toggle navigation"
+msgstr "Skift navigation"
+
+msgid "Source repository"
+msgstr "Kildelager"
+
+msgid "By the"
+msgstr "Ved"
+
+msgid "next page"
+msgstr "Næste side"
+
+msgid "repository"
+msgstr "lager"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx bogtema"
+
+msgid "Download source file"
+msgstr "Download kildefil"
+
+msgid "Contents"
+msgstr "Indhold"
+
+msgid "By"
+msgstr "Ved"
+
+msgid "Copyright"
+msgstr "ophavsret"
+
+msgid "Fullscreen mode"
+msgstr "Fuldskærmstilstand"
+
+msgid "Open an issue"
+msgstr "Åbn et problem"
+
+msgid "previous page"
+msgstr "forrige side"
+
+msgid "Download this page"
+msgstr "Download denne side"
+
+msgid "Theme by the"
+msgstr "Tema af"
diff --git a/_static/locales/de/LC_MESSAGES/booktheme.mo b/_static/locales/de/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..648b565c7829535bc5f26f86909f458d0cd0fe1b
GIT binary patch
literal 1388
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literal 0
HcmV?d00001

diff --git a/_static/locales/de/LC_MESSAGES/booktheme.po b/_static/locales/de/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..c0027d3a
--- /dev/null
+++ b/_static/locales/de/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: de\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "vorschlagen zu bearbeiten"
+
+msgid "Last updated on"
+msgstr "Zuletzt aktualisiert am"
+
+msgid "Edit this page"
+msgstr "Bearbeite diese Seite"
+
+msgid "Launch"
+msgstr "Starten"
+
+msgid "Print to PDF"
+msgstr "In PDF drucken"
+
+msgid "open issue"
+msgstr "offenes Thema"
+
+msgid "Download notebook file"
+msgstr "Notebook-Datei herunterladen"
+
+msgid "Toggle navigation"
+msgstr "Navigation umschalten"
+
+msgid "Source repository"
+msgstr "Quell-Repository"
+
+msgid "By the"
+msgstr "Bis zum"
+
+msgid "next page"
+msgstr "Nächste Seite"
+
+msgid "repository"
+msgstr "Repository"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-Buch-Thema"
+
+msgid "Download source file"
+msgstr "Quelldatei herunterladen"
+
+msgid "Contents"
+msgstr "Inhalt"
+
+msgid "By"
+msgstr "Durch"
+
+msgid "Copyright"
+msgstr "Urheberrechte ©"
+
+msgid "Fullscreen mode"
+msgstr "Vollbildmodus"
+
+msgid "Open an issue"
+msgstr "Öffnen Sie ein Problem"
+
+msgid "previous page"
+msgstr "vorherige Seite"
+
+msgid "Download this page"
+msgstr "Laden Sie diese Seite herunter"
+
+msgid "Theme by the"
+msgstr "Thema von der"
diff --git a/_static/locales/el/LC_MESSAGES/booktheme.mo b/_static/locales/el/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..fca6e9355f314677f2890f5679fd79dd67a7a5a2
GIT binary patch
literal 1722
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literal 0
HcmV?d00001

diff --git a/_static/locales/el/LC_MESSAGES/booktheme.po b/_static/locales/el/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..bdeb3270
--- /dev/null
+++ b/_static/locales/el/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: el\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "προτείνω επεξεργασία"
+
+msgid "Last updated on"
+msgstr "Τελευταία ενημέρωση στις"
+
+msgid "Edit this page"
+msgstr "Επεξεργαστείτε αυτήν τη σελίδα"
+
+msgid "Launch"
+msgstr "Εκτόξευση"
+
+msgid "Print to PDF"
+msgstr "Εκτύπωση σε PDF"
+
+msgid "open issue"
+msgstr "ανοιχτό ζήτημα"
+
+msgid "Download notebook file"
+msgstr "Λήψη αρχείου σημειωματάριου"
+
+msgid "Toggle navigation"
+msgstr "Εναλλαγή πλοήγησης"
+
+msgid "Source repository"
+msgstr "Αποθήκη πηγής"
+
+msgid "By the"
+msgstr "Από το"
+
+msgid "next page"
+msgstr "επόμενη σελίδα"
+
+msgid "repository"
+msgstr "αποθήκη"
+
+msgid "Sphinx Book Theme"
+msgstr "Θέμα βιβλίου Sphinx"
+
+msgid "Download source file"
+msgstr "Λήψη αρχείου προέλευσης"
+
+msgid "Contents"
+msgstr "Περιεχόμενα"
+
+msgid "By"
+msgstr "Με"
+
+msgid "Copyright"
+msgstr "Πνευματική ιδιοκτησία"
+
+msgid "Fullscreen mode"
+msgstr "ΛΕΙΤΟΥΡΓΙΑ ΠΛΗΡΟΥΣ ΟΘΟΝΗΣ"
+
+msgid "Open an issue"
+msgstr "Ανοίξτε ένα ζήτημα"
+
+msgid "previous page"
+msgstr "προηγούμενη σελίδα"
+
+msgid "Download this page"
+msgstr "Λήψη αυτής της σελίδας"
+
+msgid "Theme by the"
+msgstr "Θέμα από το"
diff --git a/_static/locales/eo/LC_MESSAGES/booktheme.mo b/_static/locales/eo/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..d1072bbec62959524a87953d3f41b7c272ecbe37
GIT binary patch
literal 1337
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z=%6?-<TiX@#hgl!_#l@i9~5D<9eFKPTC~5#u|zStZ)nMi`9*E|UddU#;B3?(i1Q+2
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literal 0
HcmV?d00001

diff --git a/_static/locales/eo/LC_MESSAGES/booktheme.po b/_static/locales/eo/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..6749f3a3
--- /dev/null
+++ b/_static/locales/eo/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: eo\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "sugesti redaktadon"
+
+msgid "Last updated on"
+msgstr "Laste ĝisdatigita la"
+
+msgid "Edit this page"
+msgstr "Redaktu ĉi tiun paĝon"
+
+msgid "Launch"
+msgstr "Lanĉo"
+
+msgid "Print to PDF"
+msgstr "Presi al PDF"
+
+msgid "open issue"
+msgstr "malferma numero"
+
+msgid "Download notebook file"
+msgstr "Elŝutu kajeran dosieron"
+
+msgid "Toggle navigation"
+msgstr "Ŝalti navigadon"
+
+msgid "Source repository"
+msgstr "Fonto-deponejo"
+
+msgid "By the"
+msgstr "Per la"
+
+msgid "next page"
+msgstr "sekva paĝo"
+
+msgid "repository"
+msgstr "deponejo"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksa Libro-Temo"
+
+msgid "Download source file"
+msgstr "Elŝutu fontodosieron"
+
+msgid "Contents"
+msgstr "Enhavo"
+
+msgid "By"
+msgstr "De"
+
+msgid "Copyright"
+msgstr "Kopirajto"
+
+msgid "Fullscreen mode"
+msgstr "Plenekrana reĝimo"
+
+msgid "Open an issue"
+msgstr "Malfermu numeron"
+
+msgid "previous page"
+msgstr "antaŭa paĝo"
+
+msgid "Download this page"
+msgstr "Elŝutu ĉi tiun paĝon"
+
+msgid "Theme by the"
+msgstr "Temo de la"
diff --git a/_static/locales/es/LC_MESSAGES/booktheme.mo b/_static/locales/es/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..ba2ee4dc22148ed53f2aeba32c5965654d4a5a2f
GIT binary patch
literal 1396
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literal 0
HcmV?d00001

diff --git a/_static/locales/es/LC_MESSAGES/booktheme.po b/_static/locales/es/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..71dde37f
--- /dev/null
+++ b/_static/locales/es/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: es\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "sugerir editar"
+
+msgid "Last updated on"
+msgstr "Ultima actualización en"
+
+msgid "Edit this page"
+msgstr "Edita esta página"
+
+msgid "Launch"
+msgstr "Lanzamiento"
+
+msgid "Print to PDF"
+msgstr "Imprimir en PDF"
+
+msgid "open issue"
+msgstr "Tema abierto"
+
+msgid "Download notebook file"
+msgstr "Descargar archivo de cuaderno"
+
+msgid "Toggle navigation"
+msgstr "Navegación de palanca"
+
+msgid "Source repository"
+msgstr "Repositorio de origen"
+
+msgid "By the"
+msgstr "Por el"
+
+msgid "next page"
+msgstr "siguiente página"
+
+msgid "repository"
+msgstr "repositorio"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del libro de la esfinge"
+
+msgid "Download source file"
+msgstr "Descargar archivo fuente"
+
+msgid "Contents"
+msgstr "Contenido"
+
+msgid "By"
+msgstr "Por"
+
+msgid "Copyright"
+msgstr "Derechos de autor"
+
+msgid "Fullscreen mode"
+msgstr "Modo de pantalla completa"
+
+msgid "Open an issue"
+msgstr "Abrir un problema"
+
+msgid "previous page"
+msgstr "pagina anterior"
+
+msgid "Download this page"
+msgstr "Descarga esta pagina"
+
+msgid "Theme by the"
+msgstr "Tema por el"
diff --git a/_static/locales/et/LC_MESSAGES/booktheme.mo b/_static/locales/et/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..983b82391f499b67a9046c15d0dd8744650ad925
GIT binary patch
literal 1341
zcmZ9Kzi%8x6vqb=^5gOYNC*@$VrY^ev<X2J(OFkGi7iF3g@unuDaW_ZyDv98vzi}A
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P_MW2m5}a<3ol*Y+f5$~k

literal 0
HcmV?d00001

diff --git a/_static/locales/et/LC_MESSAGES/booktheme.po b/_static/locales/et/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..cdcd07c7
--- /dev/null
+++ b/_static/locales/et/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: et\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "soovita muuta"
+
+msgid "Last updated on"
+msgstr "Viimati uuendatud"
+
+msgid "Edit this page"
+msgstr "Muutke seda lehte"
+
+msgid "Launch"
+msgstr "Käivitage"
+
+msgid "Print to PDF"
+msgstr "Prindi PDF-i"
+
+msgid "open issue"
+msgstr "avatud küsimus"
+
+msgid "Download notebook file"
+msgstr "Laadige sülearvuti fail alla"
+
+msgid "Toggle navigation"
+msgstr "Lülita navigeerimine sisse"
+
+msgid "Source repository"
+msgstr "Allikahoidla"
+
+msgid "By the"
+msgstr "Autor"
+
+msgid "next page"
+msgstr "järgmine leht"
+
+msgid "repository"
+msgstr "hoidla"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksiraamatu teema"
+
+msgid "Download source file"
+msgstr "Laadige alla lähtefail"
+
+msgid "Contents"
+msgstr "Sisu"
+
+msgid "By"
+msgstr "Kõrval"
+
+msgid "Copyright"
+msgstr "Autoriõigus"
+
+msgid "Fullscreen mode"
+msgstr "Täisekraanirežiim"
+
+msgid "Open an issue"
+msgstr "Avage probleem"
+
+msgid "previous page"
+msgstr "eelmine leht"
+
+msgid "Download this page"
+msgstr "Laadige see leht alla"
+
+msgid "Theme by the"
+msgstr "Teema"
diff --git a/_static/locales/fi/LC_MESSAGES/booktheme.mo b/_static/locales/fi/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..d8ac054597c924e3010f629caeac1c748b7211cd
GIT binary patch
literal 1368
zcmZ9KJ#5r46vquMU&lA4Af!r|g^wXX5kqz52B58kprWEjclYv|xbfLh>>S*}#K;Cr
ztS2O9B*cCTVrBw~l?gF1AR)y6B`K}2;{5K}=lB1=7k@o5`BGq<#+<?Ygn1V8*)F^=
z_U{$~k2rwWUhojOADjXYfHUAJ@H}`D?0_f02jCHqz@uPj_!Q(kFTlg#EASBb20RYF
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z2rdbgz>3?XUWo3*(Ha#V+1j$!k61sdlv~SpmKV48XD?rEHI0qB!<^bOz`&K<C~fCu
zFI9!7(0SNhisrWeS19W@sE8JA&&Ap%Z8^8DL+hThNr}F;(eT!aDGOzyC0F!)TiUz|
zedpS|tbcYcT6wM`XD`jSM3*wXIUb4SGE>TmRqe+kV}^cf!#bT&>G1)AQnFuaA9SXR
zO>~T6=Ywl@%y^W>^xt9_kNV@0^m<T2z-5q-Si@7TDRI|jPTKKk7{>3c4B9BMrgfh(
zuVp?S>46Ru!ccTe7eieYS*GF0;G*3eAeplXnZCllffvOWA=>i5w+5yH-zgk~ao1dl
zs44(@M0}Rt9Mp|iQ#yrKbuAq3msY|P`l$x0E<_Em{7-o?CS$aPMpfmO5T&sTb;r2u
H66VBTf3Z^G

literal 0
HcmV?d00001

diff --git a/_static/locales/fi/LC_MESSAGES/booktheme.po b/_static/locales/fi/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..3c3dd089
--- /dev/null
+++ b/_static/locales/fi/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: fi\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "ehdottaa muokkausta"
+
+msgid "Last updated on"
+msgstr "Viimeksi päivitetty"
+
+msgid "Edit this page"
+msgstr "Muokkaa tätä sivua"
+
+msgid "Launch"
+msgstr "Tuoda markkinoille"
+
+msgid "Print to PDF"
+msgstr "Tulosta PDF-tiedostoon"
+
+msgid "open issue"
+msgstr "avoin ongelma"
+
+msgid "Download notebook file"
+msgstr "Lataa muistikirjatiedosto"
+
+msgid "Toggle navigation"
+msgstr "Vaihda navigointia"
+
+msgid "Source repository"
+msgstr "Lähteen arkisto"
+
+msgid "By the"
+msgstr "Mukaan"
+
+msgid "next page"
+msgstr "seuraava sivu"
+
+msgid "repository"
+msgstr "arkisto"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-kirjan teema"
+
+msgid "Download source file"
+msgstr "Lataa lähdetiedosto"
+
+msgid "Contents"
+msgstr "Sisällys"
+
+msgid "By"
+msgstr "Tekijä"
+
+msgid "Copyright"
+msgstr "Tekijänoikeus"
+
+msgid "Fullscreen mode"
+msgstr "Koko näytön tila"
+
+msgid "Open an issue"
+msgstr "Avaa ongelma"
+
+msgid "previous page"
+msgstr "Edellinen sivu"
+
+msgid "Download this page"
+msgstr "Lataa tämä sivu"
+
+msgid "Theme by the"
+msgstr "Teeman tekijä"
diff --git a/_static/locales/fr/LC_MESSAGES/booktheme.mo b/_static/locales/fr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..f663d39f0faa76c5b9bd504c51252eef74cca5de
GIT binary patch
literal 1412
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T<A7>A2+cw{Vw>h7Yg+MtLz81?

literal 0
HcmV?d00001

diff --git a/_static/locales/fr/LC_MESSAGES/booktheme.po b/_static/locales/fr/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..b57d2fe7
--- /dev/null
+++ b/_static/locales/fr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: fr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "suggestion de modification"
+
+msgid "Last updated on"
+msgstr "Dernière mise à jour le"
+
+msgid "Edit this page"
+msgstr "Modifier cette page"
+
+msgid "Launch"
+msgstr "lancement"
+
+msgid "Print to PDF"
+msgstr "Imprimer au format PDF"
+
+msgid "open issue"
+msgstr "signaler un problème"
+
+msgid "Download notebook file"
+msgstr "Télécharger le fichier notebook"
+
+msgid "Toggle navigation"
+msgstr "Basculer la navigation"
+
+msgid "Source repository"
+msgstr "Dépôt source"
+
+msgid "By the"
+msgstr "Par le"
+
+msgid "next page"
+msgstr "page suivante"
+
+msgid "repository"
+msgstr "dépôt"
+
+msgid "Sphinx Book Theme"
+msgstr "Thème du livre Sphinx"
+
+msgid "Download source file"
+msgstr "Télécharger le fichier source"
+
+msgid "Contents"
+msgstr "Contenu"
+
+msgid "By"
+msgstr "Par"
+
+msgid "Copyright"
+msgstr "droits d'auteur"
+
+msgid "Fullscreen mode"
+msgstr "Mode plein écran"
+
+msgid "Open an issue"
+msgstr "Ouvrez un problème"
+
+msgid "previous page"
+msgstr "page précédente"
+
+msgid "Download this page"
+msgstr "Téléchargez cette page"
+
+msgid "Theme by the"
+msgstr "Thème par le"
diff --git a/_static/locales/hr/LC_MESSAGES/booktheme.mo b/_static/locales/hr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..eca4a1a2842830f06bd5f6235bf01d07bdd313d2
GIT binary patch
literal 1402
zcmZ9KJ!~9B6vu}+kmLv<I0Psx@u1+NI80C?MQ2?>h)-Bbu!YZ&sL*)#Wq0oOW>)iY
z#I7UJKoJQlD$3MIh{}~uqNJdrprN2bqC$d-|Jz;1r;K*?xAU?8_kVBZ_tmrCDzsP8
z*U*1Oe;xg+XYfKhe?}=h>H=QRffvE^;1%!!xCXugz6HJv4#8FMQ}7bV;EQ1F_z)DG
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zY`gv@DF5B)#v@RCe+FI!b5KeyIf$NkxPmU8B<^y9bfYDCNE01tQa`D+)KxSg*?1KH
z%T4MpwK#{4ax`^Pw+`@7ELnZzY-Ae)KI#K+iZZI(?h9Kvos)Hu_nn*4eN*x2ba2UM
z{M2MDO`uvAtTuBKA$>9&rK$><XSOtRIjcK5L`rq8Bj@BS#>r--+ODxdTQVUeRy*EU
zh&bB0JyN@EJkPZYCOUthcI(pE2XsrkjZ2=fYTmTp<}`LiQ88IPH-(M{K`cLrtve@i
z+90*(xpAqpy(Aw}Q7~LGGJzF0;VkyI^8P*c0crzk<M$<gzlq$tyLET-bp6_U@AtaG
z_QwY`4=BRGdR=K_Hz+H$4;+Vk<5B<S={~P*xX-@7X)~7_TMX#tzKOj%+7=0Q9Z>0e
zJ5}<!>W`eCg#p>R8HV8p8`ORd-?8+@kgmPE(NjB4ZRZpsbXo1{+%JywM17c0!7$ZS
zdp+04j?>}H5RL<fJx#Q4stGTStjXjQ+C_6XciyHan@$!YlbfeDgvpZH^HQ;=CO^@~
zUmJ^XJav?r30JP5Lmet-5#9xa%k6|oFOHHoFgxJ^=BJ)5wA6M)Y!NkwxvZ(JP~&Q)
z=4Na)#vBEx&&=X`>>u+?)6`B(!QC;nQ(E%YYHwOi5)9EgF2Tundr{ymsm2Z;f2mKh
k(#nYgw(f}olqHrfw@Z7L48TfWxy6y8<oWoKnK!Qg2ay|IDF6Tf

literal 0
HcmV?d00001

diff --git a/_static/locales/hr/LC_MESSAGES/booktheme.po b/_static/locales/hr/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..4c425e89
--- /dev/null
+++ b/_static/locales/hr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: hr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "predloži uređivanje"
+
+msgid "Last updated on"
+msgstr "Posljednje ažuriranje:"
+
+msgid "Edit this page"
+msgstr "Uredite ovu stranicu"
+
+msgid "Launch"
+msgstr "Pokrenite"
+
+msgid "Print to PDF"
+msgstr "Ispis u PDF"
+
+msgid "open issue"
+msgstr "otvoreno izdanje"
+
+msgid "Download notebook file"
+msgstr "Preuzmi datoteku bilježnice"
+
+msgid "Toggle navigation"
+msgstr "Uključi / isključi navigaciju"
+
+msgid "Source repository"
+msgstr "Izvorno spremište"
+
+msgid "By the"
+msgstr "Od strane"
+
+msgid "next page"
+msgstr "sljedeća stranica"
+
+msgid "repository"
+msgstr "spremište"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema knjige Sphinx"
+
+msgid "Download source file"
+msgstr "Preuzmi izvornu datoteku"
+
+msgid "Contents"
+msgstr "Sadržaj"
+
+msgid "By"
+msgstr "Po"
+
+msgid "Copyright"
+msgstr "Autorska prava"
+
+msgid "Fullscreen mode"
+msgstr "Način preko cijelog zaslona"
+
+msgid "Open an issue"
+msgstr "Otvorite izdanje"
+
+msgid "previous page"
+msgstr "Prethodna stranica"
+
+msgid "Download this page"
+msgstr "Preuzmite ovu stranicu"
+
+msgid "Theme by the"
+msgstr "Tema autora"
diff --git a/_static/locales/id/LC_MESSAGES/booktheme.mo b/_static/locales/id/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..d07a06a9d27aaf6d67234c227c22d61d5a23203b
GIT binary patch
literal 1323
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literal 0
HcmV?d00001

diff --git a/_static/locales/id/LC_MESSAGES/booktheme.po b/_static/locales/id/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..5db2ae14
--- /dev/null
+++ b/_static/locales/id/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: id\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "menyarankan edit"
+
+msgid "Last updated on"
+msgstr "Terakhir diperbarui saat"
+
+msgid "Edit this page"
+msgstr "Edit halaman ini"
+
+msgid "Launch"
+msgstr "Meluncurkan"
+
+msgid "Print to PDF"
+msgstr "Cetak ke PDF"
+
+msgid "open issue"
+msgstr "masalah terbuka"
+
+msgid "Download notebook file"
+msgstr "Unduh file notebook"
+
+msgid "Toggle navigation"
+msgstr "Alihkan navigasi"
+
+msgid "Source repository"
+msgstr "Repositori sumber"
+
+msgid "By the"
+msgstr "Oleh"
+
+msgid "next page"
+msgstr "halaman selanjutnya"
+
+msgid "repository"
+msgstr "gudang"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Buku Sphinx"
+
+msgid "Download source file"
+msgstr "Unduh file sumber"
+
+msgid "Contents"
+msgstr "Isi"
+
+msgid "By"
+msgstr "Oleh"
+
+msgid "Copyright"
+msgstr "hak cipta"
+
+msgid "Fullscreen mode"
+msgstr "Mode layar penuh"
+
+msgid "Open an issue"
+msgstr "Buka masalah"
+
+msgid "previous page"
+msgstr "halaman sebelumnya"
+
+msgid "Download this page"
+msgstr "Unduh halaman ini"
+
+msgid "Theme by the"
+msgstr "Tema oleh"
diff --git a/_static/locales/it/LC_MESSAGES/booktheme.mo b/_static/locales/it/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..53ba476edd2df2a802917e9df402257ceca1a130
GIT binary patch
literal 1403
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literal 0
HcmV?d00001

diff --git a/_static/locales/it/LC_MESSAGES/booktheme.po b/_static/locales/it/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..7d54fdef
--- /dev/null
+++ b/_static/locales/it/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: it\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "suggerisci modifica"
+
+msgid "Last updated on"
+msgstr "Ultimo aggiornamento il"
+
+msgid "Edit this page"
+msgstr "Modifica questa pagina"
+
+msgid "Launch"
+msgstr "Lanciare"
+
+msgid "Print to PDF"
+msgstr "Stampa in PDF"
+
+msgid "open issue"
+msgstr "questione aperta"
+
+msgid "Download notebook file"
+msgstr "Scarica il file del taccuino"
+
+msgid "Toggle navigation"
+msgstr "Attiva / disattiva la navigazione"
+
+msgid "Source repository"
+msgstr "Repository di origine"
+
+msgid "By the"
+msgstr "Dal"
+
+msgid "next page"
+msgstr "pagina successiva"
+
+msgid "repository"
+msgstr "repository"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del libro della Sfinge"
+
+msgid "Download source file"
+msgstr "Scarica il file sorgente"
+
+msgid "Contents"
+msgstr "Contenuti"
+
+msgid "By"
+msgstr "Di"
+
+msgid "Copyright"
+msgstr "Diritto d'autore"
+
+msgid "Fullscreen mode"
+msgstr "Modalità schermo intero"
+
+msgid "Open an issue"
+msgstr "Apri un problema"
+
+msgid "previous page"
+msgstr "pagina precedente"
+
+msgid "Download this page"
+msgstr "Scarica questa pagina"
+
+msgid "Theme by the"
+msgstr "Tema di"
diff --git a/_static/locales/iw/LC_MESSAGES/booktheme.mo b/_static/locales/iw/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..a45c6575e469914582f8223ac54793afc2f26575
GIT binary patch
literal 1445
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F;x7>$;pPAU

literal 0
HcmV?d00001

diff --git a/_static/locales/iw/LC_MESSAGES/booktheme.po b/_static/locales/iw/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..32b017cf
--- /dev/null
+++ b/_static/locales/iw/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: iw\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "מציע לערוך"
+
+msgid "Last updated on"
+msgstr "עודכן לאחרונה ב"
+
+msgid "Edit this page"
+msgstr "ערוך דף זה"
+
+msgid "Launch"
+msgstr "לְהַשִׁיק"
+
+msgid "Print to PDF"
+msgstr "הדפס לקובץ PDF"
+
+msgid "open issue"
+msgstr "בעיה פתוחה"
+
+msgid "Download notebook file"
+msgstr "הורד קובץ מחברת"
+
+msgid "Toggle navigation"
+msgstr "החלף ניווט"
+
+msgid "Source repository"
+msgstr "מאגר המקורות"
+
+msgid "By the"
+msgstr "דרך"
+
+msgid "next page"
+msgstr "עמוד הבא"
+
+msgid "repository"
+msgstr "מאגר"
+
+msgid "Sphinx Book Theme"
+msgstr "נושא ספר ספינקס"
+
+msgid "Download source file"
+msgstr "הורד את קובץ המקור"
+
+msgid "Contents"
+msgstr "תוכן"
+
+msgid "By"
+msgstr "על ידי"
+
+msgid "Copyright"
+msgstr "זכויות יוצרים"
+
+msgid "Fullscreen mode"
+msgstr "מצב מסך מלא"
+
+msgid "Open an issue"
+msgstr "פתח גיליון"
+
+msgid "previous page"
+msgstr "עמוד קודם"
+
+msgid "Download this page"
+msgstr "הורד דף זה"
+
+msgid "Theme by the"
+msgstr "נושא מאת"
diff --git a/_static/locales/ja/LC_MESSAGES/booktheme.mo b/_static/locales/ja/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..1cefd29ce3cc08792667a82dc7ff47e9843107be
GIT binary patch
literal 1471
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literal 0
HcmV?d00001

diff --git a/_static/locales/ja/LC_MESSAGES/booktheme.po b/_static/locales/ja/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..16924e19
--- /dev/null
+++ b/_static/locales/ja/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ja\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "編集を提案する"
+
+msgid "Last updated on"
+msgstr "最終更新日"
+
+msgid "Edit this page"
+msgstr "このページを編集"
+
+msgid "Launch"
+msgstr "起動"
+
+msgid "Print to PDF"
+msgstr "PDFに印刷"
+
+msgid "open issue"
+msgstr "未解決の問題"
+
+msgid "Download notebook file"
+msgstr "ノートブックファイルをダウンロード"
+
+msgid "Toggle navigation"
+msgstr "ナビゲーションを切り替え"
+
+msgid "Source repository"
+msgstr "ソースリポジトリ"
+
+msgid "By the"
+msgstr "によって"
+
+msgid "next page"
+msgstr "次のページ"
+
+msgid "repository"
+msgstr "リポジトリ"
+
+msgid "Sphinx Book Theme"
+msgstr "スフィンクスの本のテーマ"
+
+msgid "Download source file"
+msgstr "ソースファイルをダウンロード"
+
+msgid "Contents"
+msgstr "目次"
+
+msgid "By"
+msgstr "著者"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Fullscreen mode"
+msgstr "全画面モード"
+
+msgid "Open an issue"
+msgstr "問題を報告"
+
+msgid "previous page"
+msgstr "前のページ"
+
+msgid "Download this page"
+msgstr "このページをダウンロード"
+
+msgid "Theme by the"
+msgstr "のテーマ"
diff --git a/_static/locales/ko/LC_MESSAGES/booktheme.mo b/_static/locales/ko/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..06c7ec938b4975c721caff60b956e343eef11570
GIT binary patch
literal 1375
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literal 0
HcmV?d00001

diff --git a/_static/locales/ko/LC_MESSAGES/booktheme.po b/_static/locales/ko/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..69dd18f7
--- /dev/null
+++ b/_static/locales/ko/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ko\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "편집 제안"
+
+msgid "Last updated on"
+msgstr "마지막 업데이트"
+
+msgid "Edit this page"
+msgstr "이 페이지 편집"
+
+msgid "Launch"
+msgstr "시작하다"
+
+msgid "Print to PDF"
+msgstr "PDF로 인쇄"
+
+msgid "open issue"
+msgstr "열린 문제"
+
+msgid "Download notebook file"
+msgstr "노트북 파일 다운로드"
+
+msgid "Toggle navigation"
+msgstr "탐색 전환"
+
+msgid "Source repository"
+msgstr "소스 저장소"
+
+msgid "By the"
+msgstr "에 의해"
+
+msgid "next page"
+msgstr "다음 페이지"
+
+msgid "repository"
+msgstr "저장소"
+
+msgid "Sphinx Book Theme"
+msgstr "스핑크스 도서 테마"
+
+msgid "Download source file"
+msgstr "소스 파일 다운로드"
+
+msgid "Contents"
+msgstr "내용"
+
+msgid "By"
+msgstr "으로"
+
+msgid "Copyright"
+msgstr "저작권"
+
+msgid "Fullscreen mode"
+msgstr "전체 화면으로보기"
+
+msgid "Open an issue"
+msgstr "이슈 열기"
+
+msgid "previous page"
+msgstr "이전 페이지"
+
+msgid "Download this page"
+msgstr "이 페이지 다운로드"
+
+msgid "Theme by the"
+msgstr "테마별"
diff --git a/_static/locales/lt/LC_MESSAGES/booktheme.mo b/_static/locales/lt/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..4468ba04bc134a84fea5e3c973461cf02c9c2da3
GIT binary patch
literal 1413
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ClWZvf

literal 0
HcmV?d00001

diff --git a/_static/locales/lt/LC_MESSAGES/booktheme.po b/_static/locales/lt/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..9f037752
--- /dev/null
+++ b/_static/locales/lt/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: lt\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "pasiūlyti redaguoti"
+
+msgid "Last updated on"
+msgstr "Paskutinį kartą atnaujinta"
+
+msgid "Edit this page"
+msgstr "Redaguoti šį puslapį"
+
+msgid "Launch"
+msgstr "Paleiskite"
+
+msgid "Print to PDF"
+msgstr "Spausdinti į PDF"
+
+msgid "open issue"
+msgstr "atviras klausimas"
+
+msgid "Download notebook file"
+msgstr "Atsisiųsti nešiojamojo kompiuterio failą"
+
+msgid "Toggle navigation"
+msgstr "Perjungti naršymą"
+
+msgid "Source repository"
+msgstr "Šaltinio saugykla"
+
+msgid "By the"
+msgstr "Prie"
+
+msgid "next page"
+msgstr "Kitas puslapis"
+
+msgid "repository"
+msgstr "saugykla"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinkso knygos tema"
+
+msgid "Download source file"
+msgstr "Atsisiųsti šaltinio failą"
+
+msgid "Contents"
+msgstr "Turinys"
+
+msgid "By"
+msgstr "Iki"
+
+msgid "Copyright"
+msgstr "Autorių teisės"
+
+msgid "Fullscreen mode"
+msgstr "Pilno ekrano režimas"
+
+msgid "Open an issue"
+msgstr "Atidarykite problemą"
+
+msgid "previous page"
+msgstr "Ankstesnis puslapis"
+
+msgid "Download this page"
+msgstr "Atsisiųskite šį puslapį"
+
+msgid "Theme by the"
+msgstr "Tema"
diff --git a/_static/locales/lv/LC_MESSAGES/booktheme.mo b/_static/locales/lv/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..74aa4d8985d96a07c4c9be96f006f3b58d884342
GIT binary patch
literal 1404
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A2mk;8

literal 0
HcmV?d00001

diff --git a/_static/locales/lv/LC_MESSAGES/booktheme.po b/_static/locales/lv/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..c9633b54
--- /dev/null
+++ b/_static/locales/lv/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: lv\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "ieteikt rediģēt"
+
+msgid "Last updated on"
+msgstr "Pēdējoreiz atjaunināts"
+
+msgid "Edit this page"
+msgstr "Rediģēt šo lapu"
+
+msgid "Launch"
+msgstr "Uzsākt"
+
+msgid "Print to PDF"
+msgstr "Drukāt PDF formātā"
+
+msgid "open issue"
+msgstr "atklāts jautājums"
+
+msgid "Download notebook file"
+msgstr "Lejupielādēt piezīmju grāmatiņu"
+
+msgid "Toggle navigation"
+msgstr "Pārslēgt navigāciju"
+
+msgid "Source repository"
+msgstr "Avota krātuve"
+
+msgid "By the"
+msgstr "Ar"
+
+msgid "next page"
+msgstr "nākamā lapaspuse"
+
+msgid "repository"
+msgstr "krātuve"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksa grāmatas tēma"
+
+msgid "Download source file"
+msgstr "Lejupielādēt avota failu"
+
+msgid "Contents"
+msgstr "Saturs"
+
+msgid "By"
+msgstr "Autors"
+
+msgid "Copyright"
+msgstr "Autortiesības"
+
+msgid "Fullscreen mode"
+msgstr "Pilnekrāna režīms"
+
+msgid "Open an issue"
+msgstr "Atveriet problēmu"
+
+msgid "previous page"
+msgstr "iepriekšējā lapa"
+
+msgid "Download this page"
+msgstr "Lejupielādējiet šo lapu"
+
+msgid "Theme by the"
+msgstr "Autora tēma"
diff --git a/_static/locales/ml/LC_MESSAGES/booktheme.mo b/_static/locales/ml/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..2736e8fcf6f9e923c2403307f0b366086d37b335
GIT binary patch
literal 1883
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literal 0
HcmV?d00001

diff --git a/_static/locales/ml/LC_MESSAGES/booktheme.po b/_static/locales/ml/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..9a6a41e8
--- /dev/null
+++ b/_static/locales/ml/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ml\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "എഡിറ്റുചെയ്യാൻ നിർദ്ദേശിക്കുക"
+
+msgid "Last updated on"
+msgstr "അവസാനം അപ്‌ഡേറ്റുചെയ്‌തത്"
+
+msgid "Edit this page"
+msgstr "ഈ പേജ് എഡിറ്റുചെയ്യുക"
+
+msgid "Launch"
+msgstr "സമാരംഭിക്കുക"
+
+msgid "Print to PDF"
+msgstr "PDF- ലേക്ക് പ്രിന്റുചെയ്യുക"
+
+msgid "open issue"
+msgstr "തുറന്ന പ്രശ്നം"
+
+msgid "Download notebook file"
+msgstr "നോട്ട്ബുക്ക് ഫയൽ ഡൺലോഡ് ചെയ്യുക"
+
+msgid "Toggle navigation"
+msgstr "നാവിഗേഷൻ ടോഗിൾ ചെയ്യുക"
+
+msgid "Source repository"
+msgstr "ഉറവിട ശേഖരം"
+
+msgid "By the"
+msgstr "എഴുതിയത്"
+
+msgid "next page"
+msgstr "അടുത്ത പേജ്"
+
+msgid "Sphinx Book Theme"
+msgstr "സ്ഫിങ്ക്സ് പുസ്തക തീം"
+
+msgid "Download source file"
+msgstr "ഉറവിട ഫയൽ ഡൗൺലോഡുചെയ്യുക"
+
+msgid "By"
+msgstr "എഴുതിയത്"
+
+msgid "Copyright"
+msgstr "പകർപ്പവകാശം"
+
+msgid "Open an issue"
+msgstr "ഒരു പ്രശ്നം തുറക്കുക"
+
+msgid "previous page"
+msgstr "മുൻപത്തെ താൾ"
+
+msgid "Download this page"
+msgstr "ഈ പേജ് ഡൗൺലോഡുചെയ്യുക"
+
+msgid "Theme by the"
+msgstr "പ്രമേയം"
diff --git a/_static/locales/mr/LC_MESSAGES/booktheme.mo b/_static/locales/mr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..fe530100d7715cdc19a6f9db33a971665835f3c4
GIT binary patch
literal 1674
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A6951J

literal 0
HcmV?d00001

diff --git a/_static/locales/mr/LC_MESSAGES/booktheme.po b/_static/locales/mr/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..ef72d8c6
--- /dev/null
+++ b/_static/locales/mr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: mr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "संपादन सुचवा"
+
+msgid "Last updated on"
+msgstr "अखेरचे अद्यतनित"
+
+msgid "Edit this page"
+msgstr "हे पृष्ठ संपादित करा"
+
+msgid "Launch"
+msgstr "लाँच करा"
+
+msgid "Print to PDF"
+msgstr "पीडीएफवर मुद्रित करा"
+
+msgid "open issue"
+msgstr "खुला मुद्दा"
+
+msgid "Download notebook file"
+msgstr "नोटबुक फाईल डाउनलोड करा"
+
+msgid "Toggle navigation"
+msgstr "नेव्हिगेशन टॉगल करा"
+
+msgid "Source repository"
+msgstr "स्त्रोत भांडार"
+
+msgid "By the"
+msgstr "द्वारा"
+
+msgid "next page"
+msgstr "पुढील पृष्ठ"
+
+msgid "Sphinx Book Theme"
+msgstr "स्फिंक्स बुक थीम"
+
+msgid "Download source file"
+msgstr "स्त्रोत फाइल डाउनलोड करा"
+
+msgid "By"
+msgstr "द्वारा"
+
+msgid "Copyright"
+msgstr "कॉपीराइट"
+
+msgid "Open an issue"
+msgstr "एक मुद्दा उघडा"
+
+msgid "previous page"
+msgstr "मागील पान"
+
+msgid "Download this page"
+msgstr "हे पृष्ठ डाउनलोड करा"
+
+msgid "Theme by the"
+msgstr "द्वारा थीम"
diff --git a/_static/locales/ms/LC_MESSAGES/booktheme.mo b/_static/locales/ms/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..f02603fa2522a40060bd3f1b5d65052c77530de6
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/ms/LC_MESSAGES/booktheme.po b/_static/locales/ms/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..e29cbe2e
--- /dev/null
+++ b/_static/locales/ms/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ms\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "cadangkan edit"
+
+msgid "Last updated on"
+msgstr "Terakhir dikemas kini pada"
+
+msgid "Edit this page"
+msgstr "Edit halaman ini"
+
+msgid "Launch"
+msgstr "Lancarkan"
+
+msgid "Print to PDF"
+msgstr "Cetak ke PDF"
+
+msgid "open issue"
+msgstr "isu terbuka"
+
+msgid "Download notebook file"
+msgstr "Muat turun fail buku nota"
+
+msgid "Toggle navigation"
+msgstr "Togol navigasi"
+
+msgid "Source repository"
+msgstr "Repositori sumber"
+
+msgid "By the"
+msgstr "Oleh"
+
+msgid "next page"
+msgstr "muka surat seterusnya"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Buku Sphinx"
+
+msgid "Download source file"
+msgstr "Muat turun fail sumber"
+
+msgid "By"
+msgstr "Oleh"
+
+msgid "Copyright"
+msgstr "hak cipta"
+
+msgid "Open an issue"
+msgstr "Buka masalah"
+
+msgid "previous page"
+msgstr "halaman sebelumnya"
+
+msgid "Download this page"
+msgstr "Muat turun halaman ini"
+
+msgid "Theme by the"
+msgstr "Tema oleh"
diff --git a/_static/locales/nl/LC_MESSAGES/booktheme.mo b/_static/locales/nl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..e59e7ecb308a7648cd23aa6342a9ad98a61d0009
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/nl/LC_MESSAGES/booktheme.po b/_static/locales/nl/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..e4844d7c
--- /dev/null
+++ b/_static/locales/nl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: nl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "suggereren bewerken"
+
+msgid "Last updated on"
+msgstr "Laatst geupdate op"
+
+msgid "Edit this page"
+msgstr "bewerk deze pagina"
+
+msgid "Launch"
+msgstr "Lancering"
+
+msgid "Print to PDF"
+msgstr "Afdrukken naar pdf"
+
+msgid "open issue"
+msgstr "open probleem"
+
+msgid "Download notebook file"
+msgstr "Download notebookbestand"
+
+msgid "Toggle navigation"
+msgstr "Schakel navigatie"
+
+msgid "Source repository"
+msgstr "Bronopslagplaats"
+
+msgid "By the"
+msgstr "Door de"
+
+msgid "next page"
+msgstr "volgende bladzijde"
+
+msgid "repository"
+msgstr "repository"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-boekthema"
+
+msgid "Download source file"
+msgstr "Download het bronbestand"
+
+msgid "Contents"
+msgstr "Inhoud"
+
+msgid "By"
+msgstr "Door"
+
+msgid "Copyright"
+msgstr "auteursrechten"
+
+msgid "Fullscreen mode"
+msgstr "Volledig scherm"
+
+msgid "Open an issue"
+msgstr "Open een probleem"
+
+msgid "previous page"
+msgstr "vorige pagina"
+
+msgid "Download this page"
+msgstr "Download deze pagina"
+
+msgid "Theme by the"
+msgstr "Thema door de"
diff --git a/_static/locales/no/LC_MESSAGES/booktheme.mo b/_static/locales/no/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..6cd15c88de675226c00ca4d0430171075e5559ff
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/no/LC_MESSAGES/booktheme.po b/_static/locales/no/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..d079dd9b
--- /dev/null
+++ b/_static/locales/no/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: no\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "foreslå redigering"
+
+msgid "Last updated on"
+msgstr "Sist oppdatert den"
+
+msgid "Edit this page"
+msgstr "Rediger denne siden"
+
+msgid "Launch"
+msgstr "Start"
+
+msgid "Print to PDF"
+msgstr "Skriv ut til PDF"
+
+msgid "open issue"
+msgstr "åpent nummer"
+
+msgid "Download notebook file"
+msgstr "Last ned notatbokfilen"
+
+msgid "Toggle navigation"
+msgstr "Bytt navigasjon"
+
+msgid "Source repository"
+msgstr "Kildedepot"
+
+msgid "By the"
+msgstr "Ved"
+
+msgid "next page"
+msgstr "neste side"
+
+msgid "repository"
+msgstr "oppbevaringssted"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx boktema"
+
+msgid "Download source file"
+msgstr "Last ned kildefilen"
+
+msgid "Contents"
+msgstr "Innhold"
+
+msgid "By"
+msgstr "Av"
+
+msgid "Copyright"
+msgstr "opphavsrett"
+
+msgid "Fullscreen mode"
+msgstr "Fullskjerm-modus"
+
+msgid "Open an issue"
+msgstr "Åpne et problem"
+
+msgid "previous page"
+msgstr "forrige side"
+
+msgid "Download this page"
+msgstr "Last ned denne siden"
+
+msgid "Theme by the"
+msgstr "Tema av"
diff --git a/_static/locales/pl/LC_MESSAGES/booktheme.mo b/_static/locales/pl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..9ebb584f77091f5666db4194ecd4f707396cbbca
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/pl/LC_MESSAGES/booktheme.po b/_static/locales/pl/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..fcac51d3
--- /dev/null
+++ b/_static/locales/pl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: pl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "zaproponuj edycję"
+
+msgid "Last updated on"
+msgstr "Ostatnia aktualizacja"
+
+msgid "Edit this page"
+msgstr "Edytuj tę strone"
+
+msgid "Launch"
+msgstr "Uruchomić"
+
+msgid "Print to PDF"
+msgstr "Drukuj do PDF"
+
+msgid "open issue"
+msgstr "otwarty problem"
+
+msgid "Download notebook file"
+msgstr "Pobierz plik notatnika"
+
+msgid "Toggle navigation"
+msgstr "Przełącz nawigację"
+
+msgid "Source repository"
+msgstr "Repozytorium źródłowe"
+
+msgid "By the"
+msgstr "Przez"
+
+msgid "next page"
+msgstr "Następna strona"
+
+msgid "repository"
+msgstr "magazyn"
+
+msgid "Sphinx Book Theme"
+msgstr "Motyw książki Sphinx"
+
+msgid "Download source file"
+msgstr "Pobierz plik źródłowy"
+
+msgid "Contents"
+msgstr "Zawartość"
+
+msgid "By"
+msgstr "Przez"
+
+msgid "Copyright"
+msgstr "prawa autorskie"
+
+msgid "Fullscreen mode"
+msgstr "Pełny ekran"
+
+msgid "Open an issue"
+msgstr "Otwórz problem"
+
+msgid "previous page"
+msgstr "Poprzednia strona"
+
+msgid "Download this page"
+msgstr "Pobierz tę stronę"
+
+msgid "Theme by the"
+msgstr "Motyw autorstwa"
diff --git a/_static/locales/pt/LC_MESSAGES/booktheme.mo b/_static/locales/pt/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..d0ddb8728e1d5ef72dddc1537f66aa32251a83c8
GIT binary patch
literal 1364
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literal 0
HcmV?d00001

diff --git a/_static/locales/pt/LC_MESSAGES/booktheme.po b/_static/locales/pt/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..1761db08
--- /dev/null
+++ b/_static/locales/pt/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: pt\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "sugerir edição"
+
+msgid "Last updated on"
+msgstr "Última atualização em"
+
+msgid "Edit this page"
+msgstr "Edite essa página"
+
+msgid "Launch"
+msgstr "Lançamento"
+
+msgid "Print to PDF"
+msgstr "Imprimir em PDF"
+
+msgid "open issue"
+msgstr "questão aberta"
+
+msgid "Download notebook file"
+msgstr "Baixar arquivo de notebook"
+
+msgid "Toggle navigation"
+msgstr "Alternar de navegação"
+
+msgid "Source repository"
+msgstr "Repositório fonte"
+
+msgid "By the"
+msgstr "Pelo"
+
+msgid "next page"
+msgstr "próxima página"
+
+msgid "repository"
+msgstr "repositório"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema do livro Sphinx"
+
+msgid "Download source file"
+msgstr "Baixar arquivo fonte"
+
+msgid "Contents"
+msgstr "Conteúdo"
+
+msgid "By"
+msgstr "De"
+
+msgid "Copyright"
+msgstr "direito autoral"
+
+msgid "Fullscreen mode"
+msgstr "Modo tela cheia"
+
+msgid "Open an issue"
+msgstr "Abra um problema"
+
+msgid "previous page"
+msgstr "página anterior"
+
+msgid "Download this page"
+msgstr "Baixe esta página"
+
+msgid "Theme by the"
+msgstr "Tema por"
diff --git a/_static/locales/ro/LC_MESSAGES/booktheme.mo b/_static/locales/ro/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..3c36ab1df7c589ad82614599b8c91f072a25d4a6
GIT binary patch
literal 1390
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literal 0
HcmV?d00001

diff --git a/_static/locales/ro/LC_MESSAGES/booktheme.po b/_static/locales/ro/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..db865c8f
--- /dev/null
+++ b/_static/locales/ro/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ro\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "sugerează editare"
+
+msgid "Last updated on"
+msgstr "Ultima actualizare la"
+
+msgid "Edit this page"
+msgstr "Editați această pagină"
+
+msgid "Launch"
+msgstr "Lansa"
+
+msgid "Print to PDF"
+msgstr "Imprimați în PDF"
+
+msgid "open issue"
+msgstr "problema deschisă"
+
+msgid "Download notebook file"
+msgstr "Descărcați fișierul notebook"
+
+msgid "Toggle navigation"
+msgstr "Comutare navigare"
+
+msgid "Source repository"
+msgstr "Depozit sursă"
+
+msgid "By the"
+msgstr "Langa"
+
+msgid "next page"
+msgstr "pagina următoare"
+
+msgid "repository"
+msgstr "repertoriu"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Sphinx Book"
+
+msgid "Download source file"
+msgstr "Descărcați fișierul sursă"
+
+msgid "Contents"
+msgstr "Cuprins"
+
+msgid "By"
+msgstr "De"
+
+msgid "Copyright"
+msgstr "Drepturi de autor"
+
+msgid "Fullscreen mode"
+msgstr "Modul ecran întreg"
+
+msgid "Open an issue"
+msgstr "Deschideți o problemă"
+
+msgid "previous page"
+msgstr "pagina anterioară"
+
+msgid "Download this page"
+msgstr "Descarcă această pagină"
+
+msgid "Theme by the"
+msgstr "Tema de"
diff --git a/_static/locales/ru/LC_MESSAGES/booktheme.mo b/_static/locales/ru/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..6b8ca41f36ebf869818399a9584cdb15619bea78
GIT binary patch
literal 1722
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literal 0
HcmV?d00001

diff --git a/_static/locales/ru/LC_MESSAGES/booktheme.po b/_static/locales/ru/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..84ab6eb5
--- /dev/null
+++ b/_static/locales/ru/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ru\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "предложить редактировать"
+
+msgid "Last updated on"
+msgstr "Последнее обновление"
+
+msgid "Edit this page"
+msgstr "Редактировать эту страницу"
+
+msgid "Launch"
+msgstr "Запуск"
+
+msgid "Print to PDF"
+msgstr "Распечатать в PDF"
+
+msgid "open issue"
+msgstr "открытый вопрос"
+
+msgid "Download notebook file"
+msgstr "Скачать файл записной книжки"
+
+msgid "Toggle navigation"
+msgstr "Переключить навигацию"
+
+msgid "Source repository"
+msgstr "Исходный репозиторий"
+
+msgid "By the"
+msgstr "Посредством"
+
+msgid "next page"
+msgstr "Следующая страница"
+
+msgid "repository"
+msgstr "хранилище"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема книги Сфинкс"
+
+msgid "Download source file"
+msgstr "Скачать исходный файл"
+
+msgid "Contents"
+msgstr "Содержание"
+
+msgid "By"
+msgstr "По"
+
+msgid "Copyright"
+msgstr "авторское право"
+
+msgid "Fullscreen mode"
+msgstr "Полноэкранный режим"
+
+msgid "Open an issue"
+msgstr "Открыть вопрос"
+
+msgid "previous page"
+msgstr "Предыдущая страница"
+
+msgid "Download this page"
+msgstr "Загрузите эту страницу"
+
+msgid "Theme by the"
+msgstr "Тема от"
diff --git a/_static/locales/sk/LC_MESSAGES/booktheme.mo b/_static/locales/sk/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..59bd0ddfa341477cf5ccfdc88f4c3e6127046f7a
GIT binary patch
literal 1393
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Di}h+h

literal 0
HcmV?d00001

diff --git a/_static/locales/sk/LC_MESSAGES/booktheme.po b/_static/locales/sk/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..e44878b5
--- /dev/null
+++ b/_static/locales/sk/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sk\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "navrhnúť úpravu"
+
+msgid "Last updated on"
+msgstr "Posledná aktualizácia dňa"
+
+msgid "Edit this page"
+msgstr "Upraviť túto stránku"
+
+msgid "Launch"
+msgstr "Spustiť"
+
+msgid "Print to PDF"
+msgstr "Tlač do PDF"
+
+msgid "open issue"
+msgstr "otvorené vydanie"
+
+msgid "Download notebook file"
+msgstr "Stiahnite si zošit"
+
+msgid "Toggle navigation"
+msgstr "Prepnúť navigáciu"
+
+msgid "Source repository"
+msgstr "Zdrojové úložisko"
+
+msgid "By the"
+msgstr "Podľa"
+
+msgid "next page"
+msgstr "ďalšia strana"
+
+msgid "repository"
+msgstr "Úložisko"
+
+msgid "Sphinx Book Theme"
+msgstr "Téma knihy Sfinga"
+
+msgid "Download source file"
+msgstr "Stiahnite si zdrojový súbor"
+
+msgid "Contents"
+msgstr "Obsah"
+
+msgid "By"
+msgstr "Autor:"
+
+msgid "Copyright"
+msgstr "Autorské práva"
+
+msgid "Fullscreen mode"
+msgstr "Režim celej obrazovky"
+
+msgid "Open an issue"
+msgstr "Otvorte problém"
+
+msgid "previous page"
+msgstr "predchádzajúca strana"
+
+msgid "Download this page"
+msgstr "Stiahnite si túto stránku"
+
+msgid "Theme by the"
+msgstr "Téma od"
diff --git a/_static/locales/sl/LC_MESSAGES/booktheme.mo b/_static/locales/sl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..87bf26de683cb18c73bd23c2b4c57ef6a02545ec
GIT binary patch
literal 1374
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L??Y>WljW5^x#&@4

literal 0
HcmV?d00001

diff --git a/_static/locales/sl/LC_MESSAGES/booktheme.po b/_static/locales/sl/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..228939bc
--- /dev/null
+++ b/_static/locales/sl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "predlagajte urejanje"
+
+msgid "Last updated on"
+msgstr "Nazadnje posodobljeno dne"
+
+msgid "Edit this page"
+msgstr "Uredite to stran"
+
+msgid "Launch"
+msgstr "Kosilo"
+
+msgid "Print to PDF"
+msgstr "Natisni v PDF"
+
+msgid "open issue"
+msgstr "odprto vprašanje"
+
+msgid "Download notebook file"
+msgstr "Prenesite datoteko zvezka"
+
+msgid "Toggle navigation"
+msgstr "Preklopi navigacijo"
+
+msgid "Source repository"
+msgstr "Izvorno skladišče"
+
+msgid "By the"
+msgstr "Avtor"
+
+msgid "next page"
+msgstr "Naslednja stran"
+
+msgid "repository"
+msgstr "odlagališče"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema knjige Sphinx"
+
+msgid "Download source file"
+msgstr "Prenesite izvorno datoteko"
+
+msgid "Contents"
+msgstr "Vsebina"
+
+msgid "By"
+msgstr "Avtor"
+
+msgid "Copyright"
+msgstr "avtorske pravice"
+
+msgid "Fullscreen mode"
+msgstr "Celozaslonski način"
+
+msgid "Open an issue"
+msgstr "Odprite številko"
+
+msgid "previous page"
+msgstr "Prejšnja stran"
+
+msgid "Download this page"
+msgstr "Prenesite to stran"
+
+msgid "Theme by the"
+msgstr "Tema avtorja"
diff --git a/_static/locales/sr/LC_MESSAGES/booktheme.mo b/_static/locales/sr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..ec740f4852f3973fa72ed0c7f7cf59273b8dba41
GIT binary patch
literal 1679
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literal 0
HcmV?d00001

diff --git a/_static/locales/sr/LC_MESSAGES/booktheme.po b/_static/locales/sr/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..1a712a18
--- /dev/null
+++ b/_static/locales/sr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "предложи уређивање"
+
+msgid "Last updated on"
+msgstr "Последње ажурирање"
+
+msgid "Edit this page"
+msgstr "Уредите ову страницу"
+
+msgid "Launch"
+msgstr "Лансирање"
+
+msgid "Print to PDF"
+msgstr "Испис у ПДФ"
+
+msgid "open issue"
+msgstr "отворено издање"
+
+msgid "Download notebook file"
+msgstr "Преузмите датотеку бележнице"
+
+msgid "Toggle navigation"
+msgstr "Укључи / искључи навигацију"
+
+msgid "Source repository"
+msgstr "Изворно спремиште"
+
+msgid "By the"
+msgstr "Од"
+
+msgid "next page"
+msgstr "Следећа страна"
+
+msgid "repository"
+msgstr "спремиште"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема књиге Спхинк"
+
+msgid "Download source file"
+msgstr "Преузми изворну датотеку"
+
+msgid "Contents"
+msgstr "Садржај"
+
+msgid "By"
+msgstr "Од стране"
+
+msgid "Copyright"
+msgstr "Ауторско право"
+
+msgid "Fullscreen mode"
+msgstr "Режим целог екрана"
+
+msgid "Open an issue"
+msgstr "Отворите издање"
+
+msgid "previous page"
+msgstr "Претходна страница"
+
+msgid "Download this page"
+msgstr "Преузмите ову страницу"
+
+msgid "Theme by the"
+msgstr "Тхеме би"
diff --git a/_static/locales/sv/LC_MESSAGES/booktheme.mo b/_static/locales/sv/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..be951bec201ccd86d534499749af1df8938a14e3
GIT binary patch
literal 1349
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literal 0
HcmV?d00001

diff --git a/_static/locales/sv/LC_MESSAGES/booktheme.po b/_static/locales/sv/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..7d2b56d9
--- /dev/null
+++ b/_static/locales/sv/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sv\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "föreslå redigering"
+
+msgid "Last updated on"
+msgstr "Senast uppdaterad den"
+
+msgid "Edit this page"
+msgstr "Redigera den här sidan"
+
+msgid "Launch"
+msgstr "Lansera"
+
+msgid "Print to PDF"
+msgstr "Skriv ut till PDF"
+
+msgid "open issue"
+msgstr "öppet problem"
+
+msgid "Download notebook file"
+msgstr "Ladda ner anteckningsbokfilen"
+
+msgid "Toggle navigation"
+msgstr "Växla navigering"
+
+msgid "Source repository"
+msgstr "Källförvar"
+
+msgid "By the"
+msgstr "Vid"
+
+msgid "next page"
+msgstr "nästa sida"
+
+msgid "repository"
+msgstr "förvar"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx boktema"
+
+msgid "Download source file"
+msgstr "Ladda ner källfil"
+
+msgid "Contents"
+msgstr "Innehåll"
+
+msgid "By"
+msgstr "Förbi"
+
+msgid "Copyright"
+msgstr "upphovsrätt"
+
+msgid "Fullscreen mode"
+msgstr "Fullskärmsläge"
+
+msgid "Open an issue"
+msgstr "Öppna ett problem"
+
+msgid "previous page"
+msgstr "föregående sida"
+
+msgid "Download this page"
+msgstr "Ladda ner den här sidan"
+
+msgid "Theme by the"
+msgstr "Tema av"
diff --git a/_static/locales/ta/LC_MESSAGES/booktheme.mo b/_static/locales/ta/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..29f52e1f6f1ec96aceb0f883d8ccc82c23eb35c2
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/ta/LC_MESSAGES/booktheme.po b/_static/locales/ta/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..c75ffe19
--- /dev/null
+++ b/_static/locales/ta/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ta\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "திருத்த பரிந்துரைக்கவும்"
+
+msgid "Last updated on"
+msgstr "கடைசியாக புதுப்பிக்கப்பட்டது"
+
+msgid "Edit this page"
+msgstr "இந்தப் பக்கத்தைத் திருத்தவும்"
+
+msgid "Launch"
+msgstr "தொடங்க"
+
+msgid "Print to PDF"
+msgstr "PDF இல் அச்சிடுக"
+
+msgid "open issue"
+msgstr "திறந்த பிரச்சினை"
+
+msgid "Download notebook file"
+msgstr "நோட்புக் கோப்பைப் பதிவிறக்கவும்"
+
+msgid "Toggle navigation"
+msgstr "வழிசெலுத்தலை நிலைமாற்று"
+
+msgid "Source repository"
+msgstr "மூல களஞ்சியம்"
+
+msgid "By the"
+msgstr "மூலம்"
+
+msgid "next page"
+msgstr "அடுத்த பக்கம்"
+
+msgid "Sphinx Book Theme"
+msgstr "ஸ்பிங்க்ஸ் புத்தக தீம்"
+
+msgid "Download source file"
+msgstr "மூல கோப்பைப் பதிவிறக்குக"
+
+msgid "By"
+msgstr "வழங்கியவர்"
+
+msgid "Copyright"
+msgstr "பதிப்புரிமை"
+
+msgid "Open an issue"
+msgstr "சிக்கலைத் திறக்கவும்"
+
+msgid "previous page"
+msgstr "முந்தைய பக்கம்"
+
+msgid "Download this page"
+msgstr "இந்தப் பக்கத்தைப் பதிவிறக்கவும்"
+
+msgid "Theme by the"
+msgstr "வழங்கிய தீம்"
diff --git a/_static/locales/te/LC_MESSAGES/booktheme.mo b/_static/locales/te/LC_MESSAGES/booktheme.mo
new file mode 100644
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GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/te/LC_MESSAGES/booktheme.po b/_static/locales/te/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..2595c035
--- /dev/null
+++ b/_static/locales/te/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: te\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "సవరించమని సూచించండి"
+
+msgid "Last updated on"
+msgstr "చివరిగా నవీకరించబడింది"
+
+msgid "Edit this page"
+msgstr "ఈ పేజీని సవరించండి"
+
+msgid "Launch"
+msgstr "ప్రారంభించండి"
+
+msgid "Print to PDF"
+msgstr "PDF కి ముద్రించండి"
+
+msgid "open issue"
+msgstr "ఓపెన్ ఇష్యూ"
+
+msgid "Download notebook file"
+msgstr "నోట్బుక్ ఫైల్ను డౌన్లోడ్ చేయండి"
+
+msgid "Toggle navigation"
+msgstr "నావిగేషన్‌ను టోగుల్ చేయండి"
+
+msgid "Source repository"
+msgstr "మూల రిపోజిటరీ"
+
+msgid "By the"
+msgstr "ద్వారా"
+
+msgid "next page"
+msgstr "తరువాతి పేజీ"
+
+msgid "Sphinx Book Theme"
+msgstr "సింహిక పుస్తక థీమ్"
+
+msgid "Download source file"
+msgstr "మూల ఫైల్‌ను డౌన్‌లోడ్ చేయండి"
+
+msgid "By"
+msgstr "ద్వారా"
+
+msgid "Copyright"
+msgstr "కాపీరైట్"
+
+msgid "Open an issue"
+msgstr "సమస్యను తెరవండి"
+
+msgid "previous page"
+msgstr "ముందు పేజి"
+
+msgid "Download this page"
+msgstr "ఈ పేజీని డౌన్‌లోడ్ చేయండి"
+
+msgid "Theme by the"
+msgstr "ద్వారా థీమ్"
diff --git a/_static/locales/tg/LC_MESSAGES/booktheme.mo b/_static/locales/tg/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..b21c6c6340194fdf35f7e5660deb42479c8dbff9
GIT binary patch
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zGdJL5Bs-5o<jz(Csb>xB+T`zIGRrAt`^lPLZT?M?g!0angPB{%GR5vxHp0oR8Arqi
zbZiRp&%w__8!lf*NbDpe;WA2%r?Xi$atSF8VXr;q_FB@_W_A(5kPfr&0p+;^8?eY=
pg1vVOUqyoB&tBYSn9xh&7BjPW`v%MukJU629MMs5Ow+8v{s(GpYQg{j

literal 0
HcmV?d00001

diff --git a/_static/locales/tg/LC_MESSAGES/booktheme.po b/_static/locales/tg/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..73cd30ea
--- /dev/null
+++ b/_static/locales/tg/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tg\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "пешниҳод вироиш"
+
+msgid "Last updated on"
+msgstr "Last навсозӣ дар"
+
+msgid "Edit this page"
+msgstr "Ин саҳифаро таҳрир кунед"
+
+msgid "Launch"
+msgstr "Оғоз"
+
+msgid "Print to PDF"
+msgstr "Чоп ба PDF"
+
+msgid "open issue"
+msgstr "барориши кушод"
+
+msgid "Download notebook file"
+msgstr "Файли дафтарро зеркашӣ кунед"
+
+msgid "Toggle navigation"
+msgstr "Гузаришро иваз кунед"
+
+msgid "Source repository"
+msgstr "Анбори манбаъ"
+
+msgid "By the"
+msgstr "Бо"
+
+msgid "next page"
+msgstr "саҳифаи оянда"
+
+msgid "repository"
+msgstr "анбор"
+
+msgid "Sphinx Book Theme"
+msgstr "Сфинкс Мавзӯи китоб"
+
+msgid "Download source file"
+msgstr "Файли манбаъро зеркашӣ кунед"
+
+msgid "Contents"
+msgstr "Мундариҷа"
+
+msgid "By"
+msgstr "Бо"
+
+msgid "Copyright"
+msgstr "Ҳуқуқи муаллиф"
+
+msgid "Fullscreen mode"
+msgstr "Ҳолати экрани пурра"
+
+msgid "Open an issue"
+msgstr "Масъаларо кушоед"
+
+msgid "previous page"
+msgstr "саҳифаи қаблӣ"
+
+msgid "Download this page"
+msgstr "Ин саҳифаро зеркашӣ кунед"
+
+msgid "Theme by the"
+msgstr "Мавзӯъи аз"
diff --git a/_static/locales/th/LC_MESSAGES/booktheme.mo b/_static/locales/th/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..abede98aa11b163e580a26b545980ace31e61ccf
GIT binary patch
literal 1766
zcma)*-)|K~6vqcarB^@&)CU_K663GrYB9uM>wQ7mgal$EMW46#4trN`XEwWgD~&H;
zNI=2(Qi7!k28|(VOaz2Z{{sC3d@_d21Bp*Q8{hn#*{fZQQ8%6a%*>u2-*e9G?VVdL
zD4xey<E$T9PqNN$;>EM|9;NtGW4!Kz+u&BX3y#5Y_$YiD?uH$>6TSg=Km#9y)rhB|
z#94;h;VRq)zl0CNTO<B3y8mf({sl_D-(d@G-dy|dfN~y(66YD{;XWw#zmE7D6kVIP
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z?B5!(gnKxD3?G5l;1lqBC@q&9;xE12#gg7i9nA-6>LHf&S01Tf9_gR-TKX!nk)j@8
z$!wbssp1UphqNWmR~=a5Pgi}TUURlGwkp}=OGTFUD|OJlYX>grY3nL8@7z23cs4K_
z*QFa4UGulgs-KlQPg0|%ds#*3KaVrR!JzCG##p`JdPW^i%1RIOUQ(H!c9!$dcKhn}
z9D~H_tSpB{%@&y@h|{wNXVlR;Utw}rW|b?J)X}`3*%SJJXq)Ss1*76y&(}50xilRZ
zZIi_;O{$C})|{wnbx!isN%F!hW^Oo=-3VU})6}5SPzh4p+ooH!5B1t_nxZ7yq^|R~
zC4W0+ZXG#vWP0QK-WMiXqsF%9mU1(xD+Iil50cDI>25zMN>g=?&CRs;Z`c)yEsvX`
zJ#D+Lm)Ueu@1M`A*5SmaL;6aLert9xERsQc#uW?Yq_%lHmYtWTbiM67YxSOv-uv8C
zONI6KgI^B*v*yl!75w$!KM(#=@YjO>NCp2z@ZSZGG2hr+4E__@te@u_{+M44euZ<4
z_;qwx4(q3b|5Wzei|3W#eYCFwuf#yNf<N{Dqg)8<A4n$8yN+_{U!6T#(1kS`al6T$
zVo#A&Zbczg{j*X10nt=)GI&y-i8VB>&j$Yu0#;+cXcn64J%wUT=~$4O>SO5=h=^<P
z9#1*rKu?$NI>cLPeiV0|6qlnFzL#U0qn`*`iT?nxl@xb2Mx-{rpyeF$L|PmGL73Xj
zjT&US*r9DL+Oc6=XyNi4J`IP)4~3RpiSp}N5WkMWZ=`O(hjnrBo?i;ppCMASYj{fI
qV*6MBqwc03(drtRAFXAoSBd=B?a4S*sk|OB^ZqtY{YJQZ;_45RC*i9A

literal 0
HcmV?d00001

diff --git a/_static/locales/th/LC_MESSAGES/booktheme.po b/_static/locales/th/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..0392b4ad
--- /dev/null
+++ b/_static/locales/th/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: th\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "แนะนำแก้ไข"
+
+msgid "Last updated on"
+msgstr "ปรับปรุงล่าสุดเมื่อ"
+
+msgid "Edit this page"
+msgstr "แก้ไขหน้านี้"
+
+msgid "Launch"
+msgstr "เปิด"
+
+msgid "Print to PDF"
+msgstr "พิมพ์เป็น PDF"
+
+msgid "open issue"
+msgstr "เปิดปัญหา"
+
+msgid "Download notebook file"
+msgstr "ดาวน์โหลดไฟล์สมุดบันทึก"
+
+msgid "Toggle navigation"
+msgstr "ไม่ต้องสลับช่องทาง"
+
+msgid "Source repository"
+msgstr "ที่เก็บซอร์ส"
+
+msgid "By the"
+msgstr "โดย"
+
+msgid "next page"
+msgstr "หน้าต่อไป"
+
+msgid "repository"
+msgstr "ที่เก็บ"
+
+msgid "Sphinx Book Theme"
+msgstr "ธีมหนังสือสฟิงซ์"
+
+msgid "Download source file"
+msgstr "ดาวน์โหลดไฟล์ต้นฉบับ"
+
+msgid "Contents"
+msgstr "สารบัญ"
+
+msgid "By"
+msgstr "โดย"
+
+msgid "Copyright"
+msgstr "ลิขสิทธิ์"
+
+msgid "Fullscreen mode"
+msgstr "โหมดเต็มหน้าจอ"
+
+msgid "Open an issue"
+msgstr "เปิดปัญหา"
+
+msgid "previous page"
+msgstr "หน้าที่แล้ว"
+
+msgid "Download this page"
+msgstr "ดาวน์โหลดหน้านี้"
+
+msgid "Theme by the"
+msgstr "ธีมโดย"
diff --git a/_static/locales/tl/LC_MESSAGES/booktheme.mo b/_static/locales/tl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..8df1b73310e0c606251d1abff7e6033d1f6b273f
GIT binary patch
literal 1273
zcmZ{iJ#P~+7{?8ixA4}MHz7bWh@lJz6qYJggw!_(YD=U9)PcpxxsJ)jKAr8h8TbH<
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z_Gk|th%ew?a0@&DeghAKKfp8KZ;;mh1;@cLOg;`?0Z)LpKw38o9s}=zN5K_v43vGY
z_s5_1`!7J!c@37pw;-+m2-5n`;Ct{ph(FuhSLAyJ(z;h52j75~!7cDS_y;@-j*b-m
zC%}W~-v&p)J0SUA0Z)SpJPAGlY2GGy4txoc|92qy`UKMYuOOYrFAzsE2s#`3PH_Yc
zHrRgZAP%9?9w{GbQ&0{H+8gOp(7q^W|Kx>&wCSAbj3LR`Ob>tV+mg-Nw3lhs_H53s
zo5YHk8|!7$+74gUi5w0)8?u(%HR#*g@l+_u=40)Val5-HoaZ5pg_kk6hAoO<T5Wbe
zm4*w$wR1tTTBZ#aSzeo4VD%z@CR6LQw^@(X)3!Dne1^O<+OjKI{^ZTVdc!J}NN&WM
zR>EtD7`fpKH<ok?ku;NQ+J?SO7nG8)CSe!)_OWdFs2WEPW#*8g%nQ+oM5A1;v{YT1
zAAX;_cB3?FjhDtpjb19t+(Y$xnh0&Cd8;ilCw=8XV<DOv&dY>xt1^q`P0PmGs4|~w
zYF}CuMg<%mawMf%5;BoQ3pVSzGB;`7b(Ndbym+ll7=Ee3C$CPI*s^AIft_~oQ`+M|
zjoMMlC845tP)?QC3^N9mW&fLyY9mxPB*L)jFaumd3$>sZlm_4E_)y1P<99=X&7mSu
zQF#bj7^^11I1!U-GjxO_E86t}@}&6<a3a3}wSjs*quUf67-`6^p#F~AuswzZzPJR1
s6&3X_gJkvKg0Z;FbWCKO^`%82y^P$M818`St~Q|?jOt)I*ZB_FKRsDD>i_@%

literal 0
HcmV?d00001

diff --git a/_static/locales/tl/LC_MESSAGES/booktheme.po b/_static/locales/tl/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..c8375b54
--- /dev/null
+++ b/_static/locales/tl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "iminumungkahi i-edit"
+
+msgid "Last updated on"
+msgstr "Huling na-update noong"
+
+msgid "Edit this page"
+msgstr "I-edit ang pahinang ito"
+
+msgid "Launch"
+msgstr "Ilunsad"
+
+msgid "Print to PDF"
+msgstr "I-print sa PDF"
+
+msgid "open issue"
+msgstr "bukas na isyu"
+
+msgid "Download notebook file"
+msgstr "Mag-download ng file ng notebook"
+
+msgid "Toggle navigation"
+msgstr "I-toggle ang pag-navigate"
+
+msgid "Source repository"
+msgstr "Pinagmulan ng imbakan"
+
+msgid "By the"
+msgstr "Sa pamamagitan ng"
+
+msgid "next page"
+msgstr "Susunod na pahina"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema ng Sphinx Book"
+
+msgid "Download source file"
+msgstr "Mag-download ng file ng pinagmulan"
+
+msgid "By"
+msgstr "Ni"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Open an issue"
+msgstr "Magbukas ng isyu"
+
+msgid "previous page"
+msgstr "Nakaraang pahina"
+
+msgid "Download this page"
+msgstr "I-download ang pahinang ito"
+
+msgid "Theme by the"
+msgstr "Tema ng"
diff --git a/_static/locales/tr/LC_MESSAGES/booktheme.mo b/_static/locales/tr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..029ae18afb5360a8c238a052e8d3971761759d3d
GIT binary patch
literal 1373
zcmZ{izi%8x6vqdWKynFz!~~?sQXUOIiX{o62)3@sVLOWA2n!$d<J)(;liizH&CG_Z
zMVT%slPZNJnur1kHxWf7in~bEh>C&=i7J19yL{j6T0W2%?d)gY?96-L@67yl{`g~s
z_CESD`XA_*&>y~r7uw0!mBOP=;q?Z120RHafv3P_@ICND@LjMEo(I1M&w>oT4aSc5
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zP4-{T@J)1V%lv%Rpus~iXZ5ACk!=k4s2XpwJgPPKoh_YC$-2lp&J{E^B_9n3SNjqF
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zo6h$Ap{&BCl*;`;;xXGI&@B^nMMIv5kwyPFiiuV?c-PnoHwMdQd*;PnG+qT~k=xL=
f7*f}zXhDUuwVFPIQ2d!&RZ>YE8iRKH#ZJBgJpfrQ

literal 0
HcmV?d00001

diff --git a/_static/locales/tr/LC_MESSAGES/booktheme.po b/_static/locales/tr/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..47d7bdf7
--- /dev/null
+++ b/_static/locales/tr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "düzenleme öner"
+
+msgid "Last updated on"
+msgstr "Son güncelleme tarihi"
+
+msgid "Edit this page"
+msgstr "Bu sayfayı düzenle"
+
+msgid "Launch"
+msgstr "Başlatmak"
+
+msgid "Print to PDF"
+msgstr "PDF olarak yazdır"
+
+msgid "open issue"
+msgstr "Açık konu"
+
+msgid "Download notebook file"
+msgstr "Defter dosyasını indirin"
+
+msgid "Toggle navigation"
+msgstr "Gezinmeyi değiştir"
+
+msgid "Source repository"
+msgstr "Kaynak kod deposu"
+
+msgid "By the"
+msgstr "Tarafından"
+
+msgid "next page"
+msgstr "sonraki Sayfa"
+
+msgid "repository"
+msgstr "depo"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfenks Kitap Teması"
+
+msgid "Download source file"
+msgstr "Kaynak dosyayı indirin"
+
+msgid "Contents"
+msgstr "İçindekiler"
+
+msgid "By"
+msgstr "Tarafından"
+
+msgid "Copyright"
+msgstr "Telif hakkı"
+
+msgid "Fullscreen mode"
+msgstr "Tam ekran modu"
+
+msgid "Open an issue"
+msgstr "Bir sorunu açın"
+
+msgid "previous page"
+msgstr "önceki sayfa"
+
+msgid "Download this page"
+msgstr "Bu sayfayı indirin"
+
+msgid "Theme by the"
+msgstr "Tarafından tema"
diff --git a/_static/locales/uk/LC_MESSAGES/booktheme.mo b/_static/locales/uk/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..16ab78909cfbaba7fa199f47fdfc2934426ac699
GIT binary patch
literal 1681
zcmZvaO>7%Q6vv0QKx#^WmXAw-X;52=<n%)vDo4JQCQ_wVL}|)>V^7yhotb5Kwlue-
zPJ@uD1fdWPh|e20*TKPc;zZ)a@r}d@A+B8DUJjM|f4dvvq>McK+j;xuef*#QJbLI0
zf%O{p5cY4_Z(`qj3J<J-147^@4&(6*cmx~(kAa85A@Ei3B=|B|29JW1;2@~L7eJeH
z0pvI<;Pc>D;1Tdg@Fj3P=dZc__x$`9$oc*S3*f<n+5JI~&qE-`IR$pWcXRtc;31qJ
z=I2M?>o^a<I0Bvo&w|&%cfiG*_rVi5{{aqz2cFL2p8#2(^WYocB*^hT0@-gF<bGS=
zY48{DIQTcn_Xg4YTi^)D`b>gnKo4Zexd+ydwLFH++H&9h3nhscvH3nP?w=Ry$GKV8
zSFn*%JcrG*?O#Zh&hS|fa-{PW7iaO4wx+~;Mq8zAjF)gW3jCTCm(0hyZoI10#;R#!
zZps@$UG17<lSCD@&uD8wEJM#%Vyqfigx)!hC-r(<iImcE##EKK;>A`bVb!y$DvicD
z(UqEbKZJv)We~@S5))CN5yZ%eOXK2NmM>DFi34k*S#d3_1^TwU$hA$?)Ql48Q%+|+
zO&Q;>E2+I(f$!M>Nwm6cv+4}z$&!STx)qotFS{>%ocO*%l?t7p6!W2~*y81C@dFjb
zh*pwW{vziurkM*@FJB$o{eEubY#|r6I5iuplC&t`Oj!2<Jt`|TFN&2dU!NK;UfA`E
zJRRRqQE^OHOf}GcNnV%^Y~hNh{RFy}WOB1GQBNYTUK}^kOk9#WOpkH-{HV-c%WrFW
zye!Y19xVv^j9TsvZMiu?-_n+#FKN@Qx_Nh}?-P#qs7q~0+is4U)XE(BrAa+pZ_+xR
zZRr;9`hYeh-KP!O!kr#M<2%RQ<4zrzI}#B(?yg&<pWxY{2meKQ;zAaG(Je_Xjtj=E
z9(7$q(09~EfF`ZE2A7mZcK`2+9^rzjaHX3z)7!~B2h%!=*c9|lnghL|yA=s%zR*HT
z?bNhf%F5ez4L46+L7%4`a6I&oR?3Y*t^E56s%=nLx&}9h+G&%^bgSuKmsXIs4P%={
z<NofZ={oLHR_+Jb;Vx4(JG2JbhUt5^koP}7+~?BbLU@u{xGrtDCHKjW*gc*BJl2qW
z5mR0g^fg|gC#u^<Yt4P)a{4r6s#Q8N%xG_Ktmt;04^wo@PsZ5OQCrZ@sDuAWd=c9A
LPmM*`H%jps*u#;t

literal 0
HcmV?d00001

diff --git a/_static/locales/uk/LC_MESSAGES/booktheme.po b/_static/locales/uk/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..e85f6f16
--- /dev/null
+++ b/_static/locales/uk/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: uk\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "запропонувати редагувати"
+
+msgid "Last updated on"
+msgstr "Останнє оновлення:"
+
+msgid "Edit this page"
+msgstr "Редагувати цю сторінку"
+
+msgid "Launch"
+msgstr "Запуск"
+
+msgid "Print to PDF"
+msgstr "Друк у форматі PDF"
+
+msgid "open issue"
+msgstr "відкритий випуск"
+
+msgid "Download notebook file"
+msgstr "Завантажте файл блокнота"
+
+msgid "Toggle navigation"
+msgstr "Переключити навігацію"
+
+msgid "Source repository"
+msgstr "Джерело сховища"
+
+msgid "By the"
+msgstr "По"
+
+msgid "next page"
+msgstr "Наступна сторінка"
+
+msgid "repository"
+msgstr "сховище"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема книги \"Сфінкс\""
+
+msgid "Download source file"
+msgstr "Завантажити вихідний файл"
+
+msgid "Contents"
+msgstr "Зміст"
+
+msgid "By"
+msgstr "Автор"
+
+msgid "Copyright"
+msgstr "Авторське право"
+
+msgid "Fullscreen mode"
+msgstr "Повноекранний режим"
+
+msgid "Open an issue"
+msgstr "Відкрийте випуск"
+
+msgid "previous page"
+msgstr "Попередня сторінка"
+
+msgid "Download this page"
+msgstr "Завантажте цю сторінку"
+
+msgid "Theme by the"
+msgstr "Тема від"
diff --git a/_static/locales/ur/LC_MESSAGES/booktheme.mo b/_static/locales/ur/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..de8c84b935ed5c3db23e304015d45b638a3d2db3
GIT binary patch
literal 1462
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literal 0
HcmV?d00001

diff --git a/_static/locales/ur/LC_MESSAGES/booktheme.po b/_static/locales/ur/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..0f90726c
--- /dev/null
+++ b/_static/locales/ur/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ur\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "ترمیم کی تجویز کریں"
+
+msgid "Last updated on"
+msgstr "آخری بار تازہ کاری ہوئی"
+
+msgid "Edit this page"
+msgstr "اس صفحے میں ترمیم کریں"
+
+msgid "Launch"
+msgstr "لانچ کریں"
+
+msgid "Print to PDF"
+msgstr "پی ڈی ایف پرنٹ کریں"
+
+msgid "open issue"
+msgstr "کھلا مسئلہ"
+
+msgid "Download notebook file"
+msgstr "نوٹ بک فائل ڈاؤن لوڈ کریں"
+
+msgid "Toggle navigation"
+msgstr "نیویگیشن ٹوگل کریں"
+
+msgid "Source repository"
+msgstr "ماخذ ذخیرہ"
+
+msgid "By the"
+msgstr "کی طرف"
+
+msgid "next page"
+msgstr "اگلا صفحہ"
+
+msgid "Sphinx Book Theme"
+msgstr "سپنکس بک تھیم"
+
+msgid "Download source file"
+msgstr "سورس فائل ڈاؤن لوڈ کریں"
+
+msgid "By"
+msgstr "بذریعہ"
+
+msgid "Copyright"
+msgstr "کاپی رائٹ"
+
+msgid "Open an issue"
+msgstr "ایک مسئلہ کھولیں"
+
+msgid "previous page"
+msgstr "سابقہ ​​صفحہ"
+
+msgid "Download this page"
+msgstr "اس صفحے کو ڈاؤن لوڈ کریں"
+
+msgid "Theme by the"
+msgstr "کے ذریعہ تھیم"
diff --git a/_static/locales/vi/LC_MESSAGES/booktheme.mo b/_static/locales/vi/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..2bb32555c3ea0dab08ed37b71cb863fbe75698c6
GIT binary patch
literal 1431
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ze1pUH?uya95XS<na80h3K{qt+X_P1<b?<JGZ*p}M{3_8NrsOrUN%%ix#fTSncUzQM
Z`*e6S6ql71ocB7M!O~590wXmNPXTtKmQ4Tv

literal 0
HcmV?d00001

diff --git a/_static/locales/vi/LC_MESSAGES/booktheme.po b/_static/locales/vi/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..2cb5cf3b
--- /dev/null
+++ b/_static/locales/vi/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: vi\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "đề nghị chỉnh sửa"
+
+msgid "Last updated on"
+msgstr "Cập nhật lần cuối vào"
+
+msgid "Edit this page"
+msgstr "chỉnh sửa trang này"
+
+msgid "Launch"
+msgstr "Phóng"
+
+msgid "Print to PDF"
+msgstr "In sang PDF"
+
+msgid "open issue"
+msgstr "vấn đề mở"
+
+msgid "Download notebook file"
+msgstr "Tải xuống tệp sổ tay"
+
+msgid "Toggle navigation"
+msgstr "Chuyển đổi điều hướng thành"
+
+msgid "Source repository"
+msgstr "Kho nguồn"
+
+msgid "By the"
+msgstr "Bằng"
+
+msgid "next page"
+msgstr "Trang tiếp theo"
+
+msgid "repository"
+msgstr "kho"
+
+msgid "Sphinx Book Theme"
+msgstr "Chủ đề sách nhân sư"
+
+msgid "Download source file"
+msgstr "Tải xuống tệp nguồn"
+
+msgid "Contents"
+msgstr "Nội dung"
+
+msgid "By"
+msgstr "Bởi"
+
+msgid "Copyright"
+msgstr "Bản quyền"
+
+msgid "Fullscreen mode"
+msgstr "Chế độ toàn màn hình"
+
+msgid "Open an issue"
+msgstr "Mở một vấn đề"
+
+msgid "previous page"
+msgstr "trang trước"
+
+msgid "Download this page"
+msgstr "Tải xuống trang này"
+
+msgid "Theme by the"
+msgstr "Chủ đề của"
diff --git a/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo b/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..0e3235d090c2a05c0e278d38520f0437847dc79f
GIT binary patch
literal 1310
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zu3Y;tdw<bab+qWu4YKNuTrl=0tIXW0&COLO2WrKs;K5`tbiGoZj%I^WVQK#N<NElk
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literal 0
HcmV?d00001

diff --git a/_static/locales/zh_CN/LC_MESSAGES/booktheme.po b/_static/locales/zh_CN/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..f91f3ba0
--- /dev/null
+++ b/_static/locales/zh_CN/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: zh_CN\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "提出修改建议"
+
+msgid "Last updated on"
+msgstr "上次更新时间："
+
+msgid "Edit this page"
+msgstr "编辑此页面"
+
+msgid "Launch"
+msgstr "启动"
+
+msgid "Print to PDF"
+msgstr "列印成 PDF"
+
+msgid "open issue"
+msgstr "创建议题"
+
+msgid "Download notebook file"
+msgstr "下载笔记本文件"
+
+msgid "Toggle navigation"
+msgstr "显示或隐藏导航栏"
+
+msgid "Source repository"
+msgstr "源码库"
+
+msgid "By the"
+msgstr "作者："
+
+msgid "next page"
+msgstr "下一页"
+
+msgid "repository"
+msgstr "仓库"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Book 主题"
+
+msgid "Download source file"
+msgstr "下载源文件"
+
+msgid "Contents"
+msgstr "目录"
+
+msgid "By"
+msgstr "作者："
+
+msgid "Copyright"
+msgstr "版权"
+
+msgid "Fullscreen mode"
+msgstr "全屏模式"
+
+msgid "Open an issue"
+msgstr "创建议题"
+
+msgid "previous page"
+msgstr "上一页"
+
+msgid "Download this page"
+msgstr "下载此页面"
+
+msgid "Theme by the"
+msgstr "主题作者："
diff --git a/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo b/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..9116fa95d0a752665e0f1d59857a32abfc0c6bce
GIT binary patch
literal 1341
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literal 0
HcmV?d00001

diff --git a/_static/locales/zh_TW/LC_MESSAGES/booktheme.po b/_static/locales/zh_TW/LC_MESSAGES/booktheme.po
new file mode 100644
index 00000000..7833d904
--- /dev/null
+++ b/_static/locales/zh_TW/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: zh_TW\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "suggest edit"
+msgstr "提出修改建議"
+
+msgid "Last updated on"
+msgstr "最後更新時間："
+
+msgid "Edit this page"
+msgstr "編輯此頁面"
+
+msgid "Launch"
+msgstr "啟動"
+
+msgid "Print to PDF"
+msgstr "列印成 PDF"
+
+msgid "open issue"
+msgstr "公開的問題"
+
+msgid "Download notebook file"
+msgstr "下載 Notebook 檔案"
+
+msgid "Toggle navigation"
+msgstr "顯示或隱藏導覽列"
+
+msgid "Source repository"
+msgstr "來源儲存庫"
+
+msgid "By the"
+msgstr "作者："
+
+msgid "next page"
+msgstr "下一頁"
+
+msgid "repository"
+msgstr "儲存庫"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Book 佈景主題"
+
+msgid "Download source file"
+msgstr "下載原始檔"
+
+msgid "Contents"
+msgstr "目錄"
+
+msgid "By"
+msgstr "作者："
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Fullscreen mode"
+msgstr "全螢幕模式"
+
+msgid "Open an issue"
+msgstr "開啟議題"
+
+msgid "previous page"
+msgstr "上一頁"
+
+msgid "Download this page"
+msgstr "下載此頁面"
+
+msgid "Theme by the"
+msgstr "佈景主題作者："
diff --git a/_static/minus.png b/_static/minus.png
new file mode 100644
index 0000000000000000000000000000000000000000..d96755fdaf8bb2214971e0db9c1fd3077d7c419d
GIT binary patch
literal 90
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literal 0
HcmV?d00001

diff --git a/_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css b/_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css
new file mode 100644
index 00000000..33566310
--- /dev/null
+++ b/_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css
@@ -0,0 +1,2342 @@
+/* Variables */
+:root {
+  --mystnb-source-bg-color: #f7f7f7;
+  --mystnb-stdout-bg-color: #fcfcfc;
+  --mystnb-stderr-bg-color: #fdd;
+  --mystnb-traceback-bg-color: #fcfcfc;
+  --mystnb-source-border-color: #ccc;
+  --mystnb-source-margin-color: green;
+  --mystnb-stdout-border-color: #f7f7f7;
+  --mystnb-stderr-border-color: #f7f7f7;
+  --mystnb-traceback-border-color: #ffd6d6;
+  --mystnb-hide-prompt-opacity: 70%;
+  --mystnb-source-border-radius: .4em;
+  --mystnb-source-border-width: 1px;
+}
+
+/* Whole cell */
+div.container.cell {
+  padding-left: 0;
+  margin-bottom: 1em;
+}
+
+/* Removing all background formatting so we can control at the div level */
+.cell_input div.highlight,
+.cell_output pre,
+.cell_input pre,
+.cell_output .output {
+  border: none;
+  box-shadow: none;
+}
+
+.cell_output .output pre,
+.cell_input pre {
+  margin: 0px;
+}
+
+/* Input cells */
+div.cell div.cell_input,
+div.cell details.above-input>summary {
+  padding-left: 0em;
+  padding-right: 0em;
+  border: var(--mystnb-source-border-width) var(--mystnb-source-border-color) solid;
+  background-color: var(--mystnb-source-bg-color);
+  border-left-color: var(--mystnb-source-margin-color);
+  border-left-width: medium;
+  border-radius: var(--mystnb-source-border-radius);
+}
+
+div.cell_input>div,
+div.cell_output div.output>div.highlight {
+  margin: 0em !important;
+  border: none !important;
+}
+
+/* All cell outputs */
+.cell_output {
+  padding-left: 1em;
+  padding-right: 0em;
+  margin-top: 1em;
+}
+
+/* Text outputs from cells */
+.cell_output .output.text_plain,
+.cell_output .output.traceback,
+.cell_output .output.stream,
+.cell_output .output.stderr {
+  margin-top: 1em;
+  margin-bottom: 0em;
+  box-shadow: none;
+}
+
+.cell_output .output.text_plain,
+.cell_output .output.stream {
+  background: var(--mystnb-stdout-bg-color);
+  border: 1px solid var(--mystnb-stdout-border-color);
+}
+
+.cell_output .output.stderr {
+  background: var(--mystnb-stderr-bg-color);
+  border: 1px solid var(--mystnb-stderr-border-color);
+}
+
+.cell_output .output.traceback {
+  background: var(--mystnb-traceback-bg-color);
+  border: 1px solid var(--mystnb-traceback-border-color);
+}
+
+/* Collapsible cell content */
+div.cell details.above-input div.cell_input {
+  border-top-left-radius: 0;
+  border-top-right-radius: 0;
+  border-top: var(--mystnb-source-border-width) var(--mystnb-source-border-color) dashed;
+}
+
+div.cell div.cell_input.above-output-prompt {
+  border-bottom-left-radius: 0;
+  border-bottom-right-radius: 0;
+}
+
+div.cell details.above-input>summary {
+  border-bottom-left-radius: 0;
+  border-bottom-right-radius: 0;
+  border-bottom: var(--mystnb-source-border-width) var(--mystnb-source-border-color) dashed;
+  padding-left: 1em;
+  margin-bottom: 0;
+}
+
+div.cell details.above-output>summary {
+  background-color: var(--mystnb-source-bg-color);
+  padding-left: 1em;
+  padding-right: 0em;
+  border: var(--mystnb-source-border-width) var(--mystnb-source-border-color) solid;
+  border-radius: var(--mystnb-source-border-radius);
+  border-left-color: var(--mystnb-source-margin-color);
+  border-left-width: medium;
+}
+
+div.cell details.below-input>summary {
+  background-color: var(--mystnb-source-bg-color);
+  padding-left: 1em;
+  padding-right: 0em;
+  border: var(--mystnb-source-border-width) var(--mystnb-source-border-color) solid;
+  border-top: none;
+  border-bottom-left-radius: var(--mystnb-source-border-radius);
+  border-bottom-right-radius: var(--mystnb-source-border-radius);
+  border-left-color: var(--mystnb-source-margin-color);
+  border-left-width: medium;
+}
+
+div.cell details.hide>summary>span {
+  opacity: var(--mystnb-hide-prompt-opacity);
+}
+
+div.cell details.hide[open]>summary>span.collapsed {
+  display: none;
+}
+
+div.cell details.hide:not([open])>summary>span.expanded {
+  display: none;
+}
+
+@keyframes collapsed-fade-in {
+  0% {
+    opacity: 0;
+  }
+
+  100% {
+    opacity: 1;
+  }
+}
+div.cell details.hide[open]>summary~* {
+  -moz-animation: collapsed-fade-in 0.3s ease-in-out;
+  -webkit-animation: collapsed-fade-in 0.3s ease-in-out;
+  animation: collapsed-fade-in 0.3s ease-in-out;
+}
+
+/* Math align to the left */
+.cell_output .MathJax_Display {
+  text-align: left !important;
+}
+
+/* Pandas tables. Pulled from the Jupyter / nbsphinx CSS */
+div.cell_output table {
+  border: none;
+  border-collapse: collapse;
+  border-spacing: 0;
+  color: black;
+  font-size: 1em;
+  table-layout: fixed;
+}
+
+div.cell_output thead {
+  border-bottom: 1px solid black;
+  vertical-align: bottom;
+}
+
+div.cell_output tr,
+div.cell_output th,
+div.cell_output td {
+  text-align: right;
+  vertical-align: middle;
+  padding: 0.5em 0.5em;
+  line-height: normal;
+  white-space: normal;
+  max-width: none;
+  border: none;
+}
+
+div.cell_output th {
+  font-weight: bold;
+}
+
+div.cell_output tbody tr:nth-child(odd) {
+  background: #f5f5f5;
+}
+
+div.cell_output tbody tr:hover {
+  background: rgba(66, 165, 245, 0.2);
+}
+
+/** source code line numbers **/
+span.linenos {
+  opacity: 0.5;
+}
+
+/* Inline text from `paste` operation */
+
+span.pasted-text {
+  font-weight: bold;
+}
+
+span.pasted-inline img {
+  max-height: 2em;
+}
+
+tbody span.pasted-inline img {
+  max-height: none;
+}
+
+/* Font colors for translated ANSI escape sequences
+Color values are copied from Jupyter Notebook
+https://github.com/jupyter/notebook/blob/52581f8eda9b319eb0390ac77fe5903c38f81e3e/notebook/static/notebook/less/ansicolors.less#L14-L21
+Background colors from
+https://nbsphinx.readthedocs.io/en/latest/code-cells.html#ANSI-Colors
+*/
+div.highlight .-Color-Bold {
+  font-weight: bold;
+}
+
+div.highlight .-Color[class*=-Black] {
+  color: #3E424D
+}
+
+div.highlight .-Color[class*=-Red] {
+  color: #E75C58
+}
+
+div.highlight .-Color[class*=-Green] {
+  color: #00A250
+}
+
+div.highlight .-Color[class*=-Yellow] {
+  color: #DDB62B
+}
+
+div.highlight .-Color[class*=-Blue] {
+  color: #208FFB
+}
+
+div.highlight .-Color[class*=-Magenta] {
+  color: #D160C4
+}
+
+div.highlight .-Color[class*=-Cyan] {
+  color: #60C6C8
+}
+
+div.highlight .-Color[class*=-White] {
+  color: #C5C1B4
+}
+
+div.highlight .-Color[class*=-BGBlack] {
+  background-color: #3E424D
+}
+
+div.highlight .-Color[class*=-BGRed] {
+  background-color: #E75C58
+}
+
+div.highlight .-Color[class*=-BGGreen] {
+  background-color: #00A250
+}
+
+div.highlight .-Color[class*=-BGYellow] {
+  background-color: #DDB62B
+}
+
+div.highlight .-Color[class*=-BGBlue] {
+  background-color: #208FFB
+}
+
+div.highlight .-Color[class*=-BGMagenta] {
+  background-color: #D160C4
+}
+
+div.highlight .-Color[class*=-BGCyan] {
+  background-color: #60C6C8
+}
+
+div.highlight .-Color[class*=-BGWhite] {
+  background-color: #C5C1B4
+}
+
+/* Font colors for 8-bit ANSI */
+
+div.highlight .-Color[class*=-C0] {
+  color: #000000
+}
+
+div.highlight .-Color[class*=-BGC0] {
+  background-color: #000000
+}
+
+div.highlight .-Color[class*=-C1] {
+  color: #800000
+}
+
+div.highlight .-Color[class*=-BGC1] {
+  background-color: #800000
+}
+
+div.highlight .-Color[class*=-C2] {
+  color: #008000
+}
+
+div.highlight .-Color[class*=-BGC2] {
+  background-color: #008000
+}
+
+div.highlight .-Color[class*=-C3] {
+  color: #808000
+}
+
+div.highlight .-Color[class*=-BGC3] {
+  background-color: #808000
+}
+
+div.highlight .-Color[class*=-C4] {
+  color: #000080
+}
+
+div.highlight .-Color[class*=-BGC4] {
+  background-color: #000080
+}
+
+div.highlight .-Color[class*=-C5] {
+  color: #800080
+}
+
+div.highlight .-Color[class*=-BGC5] {
+  background-color: #800080
+}
+
+div.highlight .-Color[class*=-C6] {
+  color: #008080
+}
+
+div.highlight .-Color[class*=-BGC6] {
+  background-color: #008080
+}
+
+div.highlight .-Color[class*=-C7] {
+  color: #C0C0C0
+}
+
+div.highlight .-Color[class*=-BGC7] {
+  background-color: #C0C0C0
+}
+
+div.highlight .-Color[class*=-C8] {
+  color: #808080
+}
+
+div.highlight .-Color[class*=-BGC8] {
+  background-color: #808080
+}
+
+div.highlight .-Color[class*=-C9] {
+  color: #FF0000
+}
+
+div.highlight .-Color[class*=-BGC9] {
+  background-color: #FF0000
+}
+
+div.highlight .-Color[class*=-C10] {
+  color: #00FF00
+}
+
+div.highlight .-Color[class*=-BGC10] {
+  background-color: #00FF00
+}
+
+div.highlight .-Color[class*=-C11] {
+  color: #FFFF00
+}
+
+div.highlight .-Color[class*=-BGC11] {
+  background-color: #FFFF00
+}
+
+div.highlight .-Color[class*=-C12] {
+  color: #0000FF
+}
+
+div.highlight .-Color[class*=-BGC12] {
+  background-color: #0000FF
+}
+
+div.highlight .-Color[class*=-C13] {
+  color: #FF00FF
+}
+
+div.highlight .-Color[class*=-BGC13] {
+  background-color: #FF00FF
+}
+
+div.highlight .-Color[class*=-C14] {
+  color: #00FFFF
+}
+
+div.highlight .-Color[class*=-BGC14] {
+  background-color: #00FFFF
+}
+
+div.highlight .-Color[class*=-C15] {
+  color: #FFFFFF
+}
+
+div.highlight .-Color[class*=-BGC15] {
+  background-color: #FFFFFF
+}
+
+div.highlight .-Color[class*=-C16] {
+  color: #000000
+}
+
+div.highlight .-Color[class*=-BGC16] {
+  background-color: #000000
+}
+
+div.highlight .-Color[class*=-C17] {
+  color: #00005F
+}
+
+div.highlight .-Color[class*=-BGC17] {
+  background-color: #00005F
+}
+
+div.highlight .-Color[class*=-C18] {
+  color: #000087
+}
+
+div.highlight .-Color[class*=-BGC18] {
+  background-color: #000087
+}
+
+div.highlight .-Color[class*=-C19] {
+  color: #0000AF
+}
+
+div.highlight .-Color[class*=-BGC19] {
+  background-color: #0000AF
+}
+
+div.highlight .-Color[class*=-C20] {
+  color: #0000D7
+}
+
+div.highlight .-Color[class*=-BGC20] {
+  background-color: #0000D7
+}
+
+div.highlight .-Color[class*=-C21] {
+  color: #0000FF
+}
+
+div.highlight .-Color[class*=-BGC21] {
+  background-color: #0000FF
+}
+
+div.highlight .-Color[class*=-C22] {
+  color: #005F00
+}
+
+div.highlight .-Color[class*=-BGC22] {
+  background-color: #005F00
+}
+
+div.highlight .-Color[class*=-C23] {
+  color: #005F5F
+}
+
+div.highlight .-Color[class*=-BGC23] {
+  background-color: #005F5F
+}
+
+div.highlight .-Color[class*=-C24] {
+  color: #005F87
+}
+
+div.highlight .-Color[class*=-BGC24] {
+  background-color: #005F87
+}
+
+div.highlight .-Color[class*=-C25] {
+  color: #005FAF
+}
+
+div.highlight .-Color[class*=-BGC25] {
+  background-color: #005FAF
+}
+
+div.highlight .-Color[class*=-C26] {
+  color: #005FD7
+}
+
+div.highlight .-Color[class*=-BGC26] {
+  background-color: #005FD7
+}
+
+div.highlight .-Color[class*=-C27] {
+  color: #005FFF
+}
+
+div.highlight .-Color[class*=-BGC27] {
+  background-color: #005FFF
+}
+
+div.highlight .-Color[class*=-C28] {
+  color: #008700
+}
+
+div.highlight .-Color[class*=-BGC28] {
+  background-color: #008700
+}
+
+div.highlight .-Color[class*=-C29] {
+  color: #00875F
+}
+
+div.highlight .-Color[class*=-BGC29] {
+  background-color: #00875F
+}
+
+div.highlight .-Color[class*=-C30] {
+  color: #008787
+}
+
+div.highlight .-Color[class*=-BGC30] {
+  background-color: #008787
+}
+
+div.highlight .-Color[class*=-C31] {
+  color: #0087AF
+}
+
+div.highlight .-Color[class*=-BGC31] {
+  background-color: #0087AF
+}
+
+div.highlight .-Color[class*=-C32] {
+  color: #0087D7
+}
+
+div.highlight .-Color[class*=-BGC32] {
+  background-color: #0087D7
+}
+
+div.highlight .-Color[class*=-C33] {
+  color: #0087FF
+}
+
+div.highlight .-Color[class*=-BGC33] {
+  background-color: #0087FF
+}
+
+div.highlight .-Color[class*=-C34] {
+  color: #00AF00
+}
+
+div.highlight .-Color[class*=-BGC34] {
+  background-color: #00AF00
+}
+
+div.highlight .-Color[class*=-C35] {
+  color: #00AF5F
+}
+
+div.highlight .-Color[class*=-BGC35] {
+  background-color: #00AF5F
+}
+
+div.highlight .-Color[class*=-C36] {
+  color: #00AF87
+}
+
+div.highlight .-Color[class*=-BGC36] {
+  background-color: #00AF87
+}
+
+div.highlight .-Color[class*=-C37] {
+  color: #00AFAF
+}
+
+div.highlight .-Color[class*=-BGC37] {
+  background-color: #00AFAF
+}
+
+div.highlight .-Color[class*=-C38] {
+  color: #00AFD7
+}
+
+div.highlight .-Color[class*=-BGC38] {
+  background-color: #00AFD7
+}
+
+div.highlight .-Color[class*=-C39] {
+  color: #00AFFF
+}
+
+div.highlight .-Color[class*=-BGC39] {
+  background-color: #00AFFF
+}
+
+div.highlight .-Color[class*=-C40] {
+  color: #00D700
+}
+
+div.highlight .-Color[class*=-BGC40] {
+  background-color: #00D700
+}
+
+div.highlight .-Color[class*=-C41] {
+  color: #00D75F
+}
+
+div.highlight .-Color[class*=-BGC41] {
+  background-color: #00D75F
+}
+
+div.highlight .-Color[class*=-C42] {
+  color: #00D787
+}
+
+div.highlight .-Color[class*=-BGC42] {
+  background-color: #00D787
+}
+
+div.highlight .-Color[class*=-C43] {
+  color: #00D7AF
+}
+
+div.highlight .-Color[class*=-BGC43] {
+  background-color: #00D7AF
+}
+
+div.highlight .-Color[class*=-C44] {
+  color: #00D7D7
+}
+
+div.highlight .-Color[class*=-BGC44] {
+  background-color: #00D7D7
+}
+
+div.highlight .-Color[class*=-C45] {
+  color: #00D7FF
+}
+
+div.highlight .-Color[class*=-BGC45] {
+  background-color: #00D7FF
+}
+
+div.highlight .-Color[class*=-C46] {
+  color: #00FF00
+}
+
+div.highlight .-Color[class*=-BGC46] {
+  background-color: #00FF00
+}
+
+div.highlight .-Color[class*=-C47] {
+  color: #00FF5F
+}
+
+div.highlight .-Color[class*=-BGC47] {
+  background-color: #00FF5F
+}
+
+div.highlight .-Color[class*=-C48] {
+  color: #00FF87
+}
+
+div.highlight .-Color[class*=-BGC48] {
+  background-color: #00FF87
+}
+
+div.highlight .-Color[class*=-C49] {
+  color: #00FFAF
+}
+
+div.highlight .-Color[class*=-BGC49] {
+  background-color: #00FFAF
+}
+
+div.highlight .-Color[class*=-C50] {
+  color: #00FFD7
+}
+
+div.highlight .-Color[class*=-BGC50] {
+  background-color: #00FFD7
+}
+
+div.highlight .-Color[class*=-C51] {
+  color: #00FFFF
+}
+
+div.highlight .-Color[class*=-BGC51] {
+  background-color: #00FFFF
+}
+
+div.highlight .-Color[class*=-C52] {
+  color: #5F0000
+}
+
+div.highlight .-Color[class*=-BGC52] {
+  background-color: #5F0000
+}
+
+div.highlight .-Color[class*=-C53] {
+  color: #5F005F
+}
+
+div.highlight .-Color[class*=-BGC53] {
+  background-color: #5F005F
+}
+
+div.highlight .-Color[class*=-C54] {
+  color: #5F0087
+}
+
+div.highlight .-Color[class*=-BGC54] {
+  background-color: #5F0087
+}
+
+div.highlight .-Color[class*=-C55] {
+  color: #5F00AF
+}
+
+div.highlight .-Color[class*=-BGC55] {
+  background-color: #5F00AF
+}
+
+div.highlight .-Color[class*=-C56] {
+  color: #5F00D7
+}
+
+div.highlight .-Color[class*=-BGC56] {
+  background-color: #5F00D7
+}
+
+div.highlight .-Color[class*=-C57] {
+  color: #5F00FF
+}
+
+div.highlight .-Color[class*=-BGC57] {
+  background-color: #5F00FF
+}
+
+div.highlight .-Color[class*=-C58] {
+  color: #5F5F00
+}
+
+div.highlight .-Color[class*=-BGC58] {
+  background-color: #5F5F00
+}
+
+div.highlight .-Color[class*=-C59] {
+  color: #5F5F5F
+}
+
+div.highlight .-Color[class*=-BGC59] {
+  background-color: #5F5F5F
+}
+
+div.highlight .-Color[class*=-C60] {
+  color: #5F5F87
+}
+
+div.highlight .-Color[class*=-BGC60] {
+  background-color: #5F5F87
+}
+
+div.highlight .-Color[class*=-C61] {
+  color: #5F5FAF
+}
+
+div.highlight .-Color[class*=-BGC61] {
+  background-color: #5F5FAF
+}
+
+div.highlight .-Color[class*=-C62] {
+  color: #5F5FD7
+}
+
+div.highlight .-Color[class*=-BGC62] {
+  background-color: #5F5FD7
+}
+
+div.highlight .-Color[class*=-C63] {
+  color: #5F5FFF
+}
+
+div.highlight .-Color[class*=-BGC63] {
+  background-color: #5F5FFF
+}
+
+div.highlight .-Color[class*=-C64] {
+  color: #5F8700
+}
+
+div.highlight .-Color[class*=-BGC64] {
+  background-color: #5F8700
+}
+
+div.highlight .-Color[class*=-C65] {
+  color: #5F875F
+}
+
+div.highlight .-Color[class*=-BGC65] {
+  background-color: #5F875F
+}
+
+div.highlight .-Color[class*=-C66] {
+  color: #5F8787
+}
+
+div.highlight .-Color[class*=-BGC66] {
+  background-color: #5F8787
+}
+
+div.highlight .-Color[class*=-C67] {
+  color: #5F87AF
+}
+
+div.highlight .-Color[class*=-BGC67] {
+  background-color: #5F87AF
+}
+
+div.highlight .-Color[class*=-C68] {
+  color: #5F87D7
+}
+
+div.highlight .-Color[class*=-BGC68] {
+  background-color: #5F87D7
+}
+
+div.highlight .-Color[class*=-C69] {
+  color: #5F87FF
+}
+
+div.highlight .-Color[class*=-BGC69] {
+  background-color: #5F87FF
+}
+
+div.highlight .-Color[class*=-C70] {
+  color: #5FAF00
+}
+
+div.highlight .-Color[class*=-BGC70] {
+  background-color: #5FAF00
+}
+
+div.highlight .-Color[class*=-C71] {
+  color: #5FAF5F
+}
+
+div.highlight .-Color[class*=-BGC71] {
+  background-color: #5FAF5F
+}
+
+div.highlight .-Color[class*=-C72] {
+  color: #5FAF87
+}
+
+div.highlight .-Color[class*=-BGC72] {
+  background-color: #5FAF87
+}
+
+div.highlight .-Color[class*=-C73] {
+  color: #5FAFAF
+}
+
+div.highlight .-Color[class*=-BGC73] {
+  background-color: #5FAFAF
+}
+
+div.highlight .-Color[class*=-C74] {
+  color: #5FAFD7
+}
+
+div.highlight .-Color[class*=-BGC74] {
+  background-color: #5FAFD7
+}
+
+div.highlight .-Color[class*=-C75] {
+  color: #5FAFFF
+}
+
+div.highlight .-Color[class*=-BGC75] {
+  background-color: #5FAFFF
+}
+
+div.highlight .-Color[class*=-C76] {
+  color: #5FD700
+}
+
+div.highlight .-Color[class*=-BGC76] {
+  background-color: #5FD700
+}
+
+div.highlight .-Color[class*=-C77] {
+  color: #5FD75F
+}
+
+div.highlight .-Color[class*=-BGC77] {
+  background-color: #5FD75F
+}
+
+div.highlight .-Color[class*=-C78] {
+  color: #5FD787
+}
+
+div.highlight .-Color[class*=-BGC78] {
+  background-color: #5FD787
+}
+
+div.highlight .-Color[class*=-C79] {
+  color: #5FD7AF
+}
+
+div.highlight .-Color[class*=-BGC79] {
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+html[data-theme="dark"] .highlight .mf { color: #ffd900 } /* Literal.Number.Float */
+html[data-theme="dark"] .highlight .mh { color: #ffd900 } /* Literal.Number.Hex */
+html[data-theme="dark"] .highlight .mi { color: #ffd900 } /* Literal.Number.Integer */
+html[data-theme="dark"] .highlight .mo { color: #ffd900 } /* Literal.Number.Oct */
+html[data-theme="dark"] .highlight .sa { color: #abe338 } /* Literal.String.Affix */
+html[data-theme="dark"] .highlight .sb { color: #abe338 } /* Literal.String.Backtick */
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diff --git a/_static/sbt-webpack-macros.html b/_static/sbt-webpack-macros.html
new file mode 100644
index 00000000..6cbf559f
--- /dev/null
+++ b/_static/sbt-webpack-macros.html
@@ -0,0 +1,11 @@
+<!--
+  All these macros are auto-generated and must **NOT** be edited by hand.
+  See the webpack.config.js file, to learn more about how this is generated.
+-->
+{% macro head_pre_bootstrap() %}
+  <link href="{{ pathto('_static/styles/sphinx-book-theme.css', 1) }}" rel="stylesheet">
+{% endmacro %}
+
+{% macro body_post() %}
+  <script src="{{ pathto('_static/scripts/sphinx-book-theme.js', 1) }}"></script>
+{% endmacro %}
diff --git a/_static/scripts/bootstrap.js b/_static/scripts/bootstrap.js
new file mode 100644
index 00000000..766173ab
--- /dev/null
+++ b/_static/scripts/bootstrap.js
@@ -0,0 +1,3 @@
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NAME(){return"button"}toggle(){this._element.setAttribute("aria-pressed",this._element.classList.toggle("active"))}static jQueryInterface(t){return this.each((function(){const e=xe.getOrCreateInstance(this);"toggle"===t&&e[t]()}))}}de.on(document,"click.bs.button.data-api",Oe,(t=>{t.preventDefault();const e=t.target.closest(Oe);xe.getOrCreateInstance(e).toggle()})),Qt(xe);const ke={find:(t,e=document.documentElement)=>[].concat(...Element.prototype.querySelectorAll.call(e,t)),findOne:(t,e=document.documentElement)=>Element.prototype.querySelector.call(e,t),children:(t,e)=>[].concat(...t.children).filter((t=>t.matches(e))),parents(t,e){const i=[];let n=t.parentNode.closest(e);for(;n;)i.push(n),n=n.parentNode.closest(e);return i},prev(t,e){let i=t.previousElementSibling;for(;i;){if(i.matches(e))return[i];i=i.previousElementSibling}return[]},next(t,e){let i=t.nextElementSibling;for(;i;){if(i.matches(e))return[i];i=i.nextElementSibling}return[]},focusableChildren(t){const e=["a","button","input","textarea","select","details","[tabindex]",'[contenteditable="true"]'].map((t=>`${t}:not([tabindex^="-"])`)).join(",");return this.find(e,t).filter((t=>!Ft(t)&&Wt(t)))}},Le=".bs.swipe",De=`touchstart${Le}`,$e=`touchmove${Le}`,Se=`touchend${Le}`,Ie=`pointerdown${Le}`,Ne=`pointerup${Le}`,Pe={endCallback:null,leftCallback:null,rightCallback:null},je={endCallback:"(function|null)",leftCallback:"(function|null)",rightCallback:"(function|null)"};class Me extends ve{constructor(t,e){super(),this._element=t,t&&Me.isSupported()&&(this._config=this._getConfig(e),this._deltaX=0,this._supportPointerEvents=Boolean(window.PointerEvent),this._initEvents())}static get Default(){return Pe}static get DefaultType(){return je}static get NAME(){return"swipe"}dispose(){de.off(this._element,Le)}_start(t){this._supportPointerEvents?this._eventIsPointerPenTouch(t)&&(this._deltaX=t.clientX):this._deltaX=t.touches[0].clientX}_end(t){this._eventIsPointerPenTouch(t)&&(this._deltaX=t.clientX-this._deltaX),this._handleSwipe(),Xt(this._config.endCallback)}_move(t){this._deltaX=t.touches&&t.touches.length>1?0:t.touches[0].clientX-this._deltaX}_handleSwipe(){const t=Math.abs(this._deltaX);if(t<=40)return;const e=t/this._deltaX;this._deltaX=0,e&&Xt(e>0?this._config.rightCallback:this._config.leftCallback)}_initEvents(){this._supportPointerEvents?(de.on(this._element,Ie,(t=>this._start(t))),de.on(this._element,Ne,(t=>this._end(t))),this._element.classList.add("pointer-event")):(de.on(this._element,De,(t=>this._start(t))),de.on(this._element,$e,(t=>this._move(t))),de.on(this._element,Se,(t=>this._end(t))))}_eventIsPointerPenTouch(t){return this._supportPointerEvents&&("pen"===t.pointerType||"touch"===t.pointerType)}static isSupported(){return"ontouchstart"in document.documentElement||navigator.maxTouchPoints>0}}const He=".bs.carousel",We=".data-api",Fe="next",Be="prev",ze="left",qe="right",Re=`slide${He}`,Ve=`slid${He}`,Ke=`keydown${He}`,Qe=`mouseenter${He}`,Xe=`mouseleave${He}`,Ye=`dragstart${He}`,Ue=`load${He}${We}`,Ge=`click${He}${We}`,Je="carousel",Ze="active",ti=".active",ei=".carousel-item",ii=ti+ei,ni={ArrowLeft:qe,ArrowRight:ze},si={interval:5e3,keyboard:!0,pause:"hover",ride:!1,touch:!0,wrap:!0},oi={interval:"(number|boolean)",keyboard:"boolean",pause:"(string|boolean)",ride:"(boolean|string)",touch:"boolean",wrap:"boolean"};class ri extends ye{constructor(t,e){super(t,e),this._interval=null,this._activeElement=null,this._isSliding=!1,this.touchTimeout=null,this._swipeHelper=null,this._indicatorsElement=ke.findOne(".carousel-indicators",this._element),this._addEventListeners(),this._config.ride===Je&&this.cycle()}static get Default(){return si}static get DefaultType(){return oi}static get NAME(){return"carousel"}next(){this._slide(Fe)}nextWhenVisible(){!document.hidden&&Wt(this._element)&&this.next()}prev(){this._slide(Be)}pause(){this._isSliding&&jt(this._element),this._clearInterval()}cycle(){this._clearInterval(),this._updateInterval(),this._interval=setInterval((()=>this.nextWhenVisible()),this._config.interval)}_maybeEnableCycle(){this._config.ride&&(this._isSliding?de.one(this._element,Ve,(()=>this.cycle())):this.cycle())}to(t){const e=this._getItems();if(t>e.length-1||t<0)return;if(this._isSliding)return void de.one(this._element,Ve,(()=>this.to(t)));const i=this._getItemIndex(this._getActive());if(i===t)return;const n=t>i?Fe:Be;this._slide(n,e[t])}dispose(){this._swipeHelper&&this._swipeHelper.dispose(),super.dispose()}_configAfterMerge(t){return t.defaultInterval=t.interval,t}_addEventListeners(){this._config.keyboard&&de.on(this._element,Ke,(t=>this._keydown(t))),"hover"===this._config.pause&&(de.on(this._element,Qe,(()=>this.pause())),de.on(this._element,Xe,(()=>this._maybeEnableCycle()))),this._config.touch&&Me.isSupported()&&this._addTouchEventListeners()}_addTouchEventListeners(){for(const t of ke.find(".carousel-item img",this._element))de.on(t,Ye,(t=>t.preventDefault()));const t={leftCallback:()=>this._slide(this._directionToOrder(ze)),rightCallback:()=>this._slide(this._directionToOrder(qe)),endCallback:()=>{"hover"===this._config.pause&&(this.pause(),this.touchTimeout&&clearTimeout(this.touchTimeout),this.touchTimeout=setTimeout((()=>this._maybeEnableCycle()),500+this._config.interval))}};this._swipeHelper=new Me(this._element,t)}_keydown(t){if(/input|textarea/i.test(t.target.tagName))return;const e=ni[t.key];e&&(t.preventDefault(),this._slide(this._directionToOrder(e)))}_getItemIndex(t){return this._getItems().indexOf(t)}_setActiveIndicatorElement(t){if(!this._indicatorsElement)return;const e=ke.findOne(ti,this._indicatorsElement);e.classList.remove(Ze),e.removeAttribute("aria-current");const i=ke.findOne(`[data-bs-slide-to="${t}"]`,this._indicatorsElement);i&&(i.classList.add(Ze),i.setAttribute("aria-current","true"))}_updateInterval(){const t=this._activeElement||this._getActive();if(!t)return;const e=Number.parseInt(t.getAttribute("data-bs-interval"),10);this._config.interval=e||this._config.defaultInterval}_slide(t,e=null){if(this._isSliding)return;const i=this._getActive(),n=t===Fe,s=e||Ut(this._getItems(),i,n,this._config.wrap);if(s===i)return;const o=this._getItemIndex(s),r=e=>de.trigger(this._element,e,{relatedTarget:s,direction:this._orderToDirection(t),from:this._getItemIndex(i),to:o});if(r(Re).defaultPrevented)return;if(!i||!s)return;const a=Boolean(this._interval);this.pause(),this._isSliding=!0,this._setActiveIndicatorElement(o),this._activeElement=s;const l=n?"carousel-item-start":"carousel-item-end",c=n?"carousel-item-next":"carousel-item-prev";s.classList.add(c),qt(s),i.classList.add(l),s.classList.add(l),this._queueCallback((()=>{s.classList.remove(l,c),s.classList.add(Ze),i.classList.remove(Ze,c,l),this._isSliding=!1,r(Ve)}),i,this._isAnimated()),a&&this.cycle()}_isAnimated(){return this._element.classList.contains("slide")}_getActive(){return ke.findOne(ii,this._element)}_getItems(){return ke.find(ei,this._element)}_clearInterval(){this._interval&&(clearInterval(this._interval),this._interval=null)}_directionToOrder(t){return Kt()?t===ze?Be:Fe:t===ze?Fe:Be}_orderToDirection(t){return Kt()?t===Be?ze:qe:t===Be?qe:ze}static jQueryInterface(t){return this.each((function(){const e=ri.getOrCreateInstance(this,t);if("number"!=typeof t){if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}else e.to(t)}))}}de.on(document,Ge,"[data-bs-slide], [data-bs-slide-to]",(function(t){const e=Pt(this);if(!e||!e.classList.contains(Je))return;t.preventDefault();const i=ri.getOrCreateInstance(e),n=this.getAttribute("data-bs-slide-to");return n?(i.to(n),void i._maybeEnableCycle()):"next"===be.getDataAttribute(this,"slide")?(i.next(),void i._maybeEnableCycle()):(i.prev(),void i._maybeEnableCycle())})),de.on(window,Ue,(()=>{const t=ke.find('[data-bs-ride="carousel"]');for(const e of t)ri.getOrCreateInstance(e)})),Qt(ri);const ai=".bs.collapse",li=`show${ai}`,ci=`shown${ai}`,hi=`hide${ai}`,ui=`hidden${ai}`,di=`click${ai}.data-api`,fi="show",pi="collapse",gi="collapsing",mi=`:scope .${pi} .${pi}`,_i='[data-bs-toggle="collapse"]',bi={parent:null,toggle:!0},vi={parent:"(null|element)",toggle:"boolean"};class yi extends ye{constructor(t,e){super(t,e),this._isTransitioning=!1,this._triggerArray=[];const i=ke.find(_i);for(const t of i){const e=Nt(t),i=ke.find(e).filter((t=>t===this._element));null!==e&&i.length&&this._triggerArray.push(t)}this._initializeChildren(),this._config.parent||this._addAriaAndCollapsedClass(this._triggerArray,this._isShown()),this._config.toggle&&this.toggle()}static get Default(){return bi}static get DefaultType(){return vi}static get NAME(){return"collapse"}toggle(){this._isShown()?this.hide():this.show()}show(){if(this._isTransitioning||this._isShown())return;let t=[];if(this._config.parent&&(t=this._getFirstLevelChildren(".collapse.show, .collapse.collapsing").filter((t=>t!==this._element)).map((t=>yi.getOrCreateInstance(t,{toggle:!1})))),t.length&&t[0]._isTransitioning)return;if(de.trigger(this._element,li).defaultPrevented)return;for(const e of t)e.hide();const e=this._getDimension();this._element.classList.remove(pi),this._element.classList.add(gi),this._element.style[e]=0,this._addAriaAndCollapsedClass(this._triggerArray,!0),this._isTransitioning=!0;const i=`scroll${e[0].toUpperCase()+e.slice(1)}`;this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(gi),this._element.classList.add(pi,fi),this._element.style[e]="",de.trigger(this._element,ci)}),this._element,!0),this._element.style[e]=`${this._element[i]}px`}hide(){if(this._isTransitioning||!this._isShown())return;if(de.trigger(this._element,hi).defaultPrevented)return;const t=this._getDimension();this._element.style[t]=`${this._element.getBoundingClientRect()[t]}px`,qt(this._element),this._element.classList.add(gi),this._element.classList.remove(pi,fi);for(const t of this._triggerArray){const e=Pt(t);e&&!this._isShown(e)&&this._addAriaAndCollapsedClass([t],!1)}this._isTransitioning=!0,this._element.style[t]="",this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(gi),this._element.classList.add(pi),de.trigger(this._element,ui)}),this._element,!0)}_isShown(t=this._element){return t.classList.contains(fi)}_configAfterMerge(t){return t.toggle=Boolean(t.toggle),t.parent=Ht(t.parent),t}_getDimension(){return this._element.classList.contains("collapse-horizontal")?"width":"height"}_initializeChildren(){if(!this._config.parent)return;const t=this._getFirstLevelChildren(_i);for(const e of t){const t=Pt(e);t&&this._addAriaAndCollapsedClass([e],this._isShown(t))}}_getFirstLevelChildren(t){const e=ke.find(mi,this._config.parent);return ke.find(t,this._config.parent).filter((t=>!e.includes(t)))}_addAriaAndCollapsedClass(t,e){if(t.length)for(const i of t)i.classList.toggle("collapsed",!e),i.setAttribute("aria-expanded",e)}static jQueryInterface(t){const e={};return"string"==typeof t&&/show|hide/.test(t)&&(e.toggle=!1),this.each((function(){const i=yi.getOrCreateInstance(this,e);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t]()}}))}}de.on(document,di,_i,(function(t){("A"===t.target.tagName||t.delegateTarget&&"A"===t.delegateTarget.tagName)&&t.preventDefault();const e=Nt(this),i=ke.find(e);for(const t of i)yi.getOrCreateInstance(t,{toggle:!1}).toggle()})),Qt(yi);const wi="dropdown",Ai=".bs.dropdown",Ei=".data-api",Ci="ArrowUp",Ti="ArrowDown",Oi=`hide${Ai}`,xi=`hidden${Ai}`,ki=`show${Ai}`,Li=`shown${Ai}`,Di=`click${Ai}${Ei}`,$i=`keydown${Ai}${Ei}`,Si=`keyup${Ai}${Ei}`,Ii="show",Ni='[data-bs-toggle="dropdown"]:not(.disabled):not(:disabled)',Pi=`${Ni}.${Ii}`,ji=".dropdown-menu",Mi=Kt()?"top-end":"top-start",Hi=Kt()?"top-start":"top-end",Wi=Kt()?"bottom-end":"bottom-start",Fi=Kt()?"bottom-start":"bottom-end",Bi=Kt()?"left-start":"right-start",zi=Kt()?"right-start":"left-start",qi={autoClose:!0,boundary:"clippingParents",display:"dynamic",offset:[0,2],popperConfig:null,reference:"toggle"},Ri={autoClose:"(boolean|string)",boundary:"(string|element)",display:"string",offset:"(array|string|function)",popperConfig:"(null|object|function)",reference:"(string|element|object)"};class Vi extends ye{constructor(t,e){super(t,e),this._popper=null,this._parent=this._element.parentNode,this._menu=ke.next(this._element,ji)[0]||ke.prev(this._element,ji)[0]||ke.findOne(ji,this._parent),this._inNavbar=this._detectNavbar()}static get Default(){return qi}static get DefaultType(){return Ri}static get NAME(){return wi}toggle(){return this._isShown()?this.hide():this.show()}show(){if(Ft(this._element)||this._isShown())return;const t={relatedTarget:this._element};if(!de.trigger(this._element,ki,t).defaultPrevented){if(this._createPopper(),"ontouchstart"in document.documentElement&&!this._parent.closest(".navbar-nav"))for(const t of[].concat(...document.body.children))de.on(t,"mouseover",zt);this._element.focus(),this._element.setAttribute("aria-expanded",!0),this._menu.classList.add(Ii),this._element.classList.add(Ii),de.trigger(this._element,Li,t)}}hide(){if(Ft(this._element)||!this._isShown())return;const t={relatedTarget:this._element};this._completeHide(t)}dispose(){this._popper&&this._popper.destroy(),super.dispose()}update(){this._inNavbar=this._detectNavbar(),this._popper&&this._popper.update()}_completeHide(t){if(!de.trigger(this._element,Oi,t).defaultPrevented){if("ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))de.off(t,"mouseover",zt);this._popper&&this._popper.destroy(),this._menu.classList.remove(Ii),this._element.classList.remove(Ii),this._element.setAttribute("aria-expanded","false"),be.removeDataAttribute(this._menu,"popper"),de.trigger(this._element,xi,t)}}_getConfig(t){if("object"==typeof(t=super._getConfig(t)).reference&&!Mt(t.reference)&&"function"!=typeof t.reference.getBoundingClientRect)throw new TypeError(`${wi.toUpperCase()}: Option "reference" provided type "object" without a required "getBoundingClientRect" method.`);return t}_createPopper(){if(void 0===e)throw new TypeError("Bootstrap's dropdowns require Popper (https://popper.js.org)");let t=this._element;"parent"===this._config.reference?t=this._parent:Mt(this._config.reference)?t=Ht(this._config.reference):"object"==typeof this._config.reference&&(t=this._config.reference);const i=this._getPopperConfig();this._popper=Dt(t,this._menu,i)}_isShown(){return this._menu.classList.contains(Ii)}_getPlacement(){const t=this._parent;if(t.classList.contains("dropend"))return Bi;if(t.classList.contains("dropstart"))return zi;if(t.classList.contains("dropup-center"))return"top";if(t.classList.contains("dropdown-center"))return"bottom";const e="end"===getComputedStyle(this._menu).getPropertyValue("--bs-position").trim();return t.classList.contains("dropup")?e?Hi:Mi:e?Fi:Wi}_detectNavbar(){return null!==this._element.closest(".navbar")}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_getPopperConfig(){const t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return(this._inNavbar||"static"===this._config.display)&&(be.setDataAttribute(this._menu,"popper","static"),t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,..."function"==typeof this._config.popperConfig?this._config.popperConfig(t):this._config.popperConfig}}_selectMenuItem({key:t,target:e}){const i=ke.find(".dropdown-menu .dropdown-item:not(.disabled):not(:disabled)",this._menu).filter((t=>Wt(t)));i.length&&Ut(i,e,t===Ti,!i.includes(e)).focus()}static jQueryInterface(t){return this.each((function(){const e=Vi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}static clearMenus(t){if(2===t.button||"keyup"===t.type&&"Tab"!==t.key)return;const e=ke.find(Pi);for(const i of e){const e=Vi.getInstance(i);if(!e||!1===e._config.autoClose)continue;const n=t.composedPath(),s=n.includes(e._menu);if(n.includes(e._element)||"inside"===e._config.autoClose&&!s||"outside"===e._config.autoClose&&s)continue;if(e._menu.contains(t.target)&&("keyup"===t.type&&"Tab"===t.key||/input|select|option|textarea|form/i.test(t.target.tagName)))continue;const o={relatedTarget:e._element};"click"===t.type&&(o.clickEvent=t),e._completeHide(o)}}static dataApiKeydownHandler(t){const e=/input|textarea/i.test(t.target.tagName),i="Escape"===t.key,n=[Ci,Ti].includes(t.key);if(!n&&!i)return;if(e&&!i)return;t.preventDefault();const s=this.matches(Ni)?this:ke.prev(this,Ni)[0]||ke.next(this,Ni)[0]||ke.findOne(Ni,t.delegateTarget.parentNode),o=Vi.getOrCreateInstance(s);if(n)return t.stopPropagation(),o.show(),void o._selectMenuItem(t);o._isShown()&&(t.stopPropagation(),o.hide(),s.focus())}}de.on(document,$i,Ni,Vi.dataApiKeydownHandler),de.on(document,$i,ji,Vi.dataApiKeydownHandler),de.on(document,Di,Vi.clearMenus),de.on(document,Si,Vi.clearMenus),de.on(document,Di,Ni,(function(t){t.preventDefault(),Vi.getOrCreateInstance(this).toggle()})),Qt(Vi);const Ki=".fixed-top, .fixed-bottom, .is-fixed, .sticky-top",Qi=".sticky-top",Xi="padding-right",Yi="margin-right";class Ui{constructor(){this._element=document.body}getWidth(){const t=document.documentElement.clientWidth;return Math.abs(window.innerWidth-t)}hide(){const t=this.getWidth();this._disableOverFlow(),this._setElementAttributes(this._element,Xi,(e=>e+t)),this._setElementAttributes(Ki,Xi,(e=>e+t)),this._setElementAttributes(Qi,Yi,(e=>e-t))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,Xi),this._resetElementAttributes(Ki,Xi),this._resetElementAttributes(Qi,Yi)}isOverflowing(){return this.getWidth()>0}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t).getPropertyValue(e);t.style.setProperty(e,`${i(Number.parseFloat(s))}px`)}))}_saveInitialAttribute(t,e){const i=t.style.getPropertyValue(e);i&&be.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=be.getDataAttribute(t,e);null!==i?(be.removeDataAttribute(t,e),t.style.setProperty(e,i)):t.style.removeProperty(e)}))}_applyManipulationCallback(t,e){if(Mt(t))e(t);else for(const i of ke.find(t,this._element))e(i)}}const Gi="backdrop",Ji="show",Zi=`mousedown.bs.${Gi}`,tn={className:"modal-backdrop",clickCallback:null,isAnimated:!1,isVisible:!0,rootElement:"body"},en={className:"string",clickCallback:"(function|null)",isAnimated:"boolean",isVisible:"boolean",rootElement:"(element|string)"};class nn extends ve{constructor(t){super(),this._config=this._getConfig(t),this._isAppended=!1,this._element=null}static get Default(){return tn}static get DefaultType(){return en}static get NAME(){return Gi}show(t){if(!this._config.isVisible)return void Xt(t);this._append();const e=this._getElement();this._config.isAnimated&&qt(e),e.classList.add(Ji),this._emulateAnimation((()=>{Xt(t)}))}hide(t){this._config.isVisible?(this._getElement().classList.remove(Ji),this._emulateAnimation((()=>{this.dispose(),Xt(t)}))):Xt(t)}dispose(){this._isAppended&&(de.off(this._element,Zi),this._element.remove(),this._isAppended=!1)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_configAfterMerge(t){return t.rootElement=Ht(t.rootElement),t}_append(){if(this._isAppended)return;const t=this._getElement();this._config.rootElement.append(t),de.on(t,Zi,(()=>{Xt(this._config.clickCallback)})),this._isAppended=!0}_emulateAnimation(t){Yt(t,this._getElement(),this._config.isAnimated)}}const sn=".bs.focustrap",on=`focusin${sn}`,rn=`keydown.tab${sn}`,an="backward",ln={autofocus:!0,trapElement:null},cn={autofocus:"boolean",trapElement:"element"};class hn extends ve{constructor(t){super(),this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}static get Default(){return ln}static get DefaultType(){return cn}static get NAME(){return"focustrap"}activate(){this._isActive||(this._config.autofocus&&this._config.trapElement.focus(),de.off(document,sn),de.on(document,on,(t=>this._handleFocusin(t))),de.on(document,rn,(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,de.off(document,sn))}_handleFocusin(t){const{trapElement:e}=this._config;if(t.target===document||t.target===e||e.contains(t.target))return;const i=ke.focusableChildren(e);0===i.length?e.focus():this._lastTabNavDirection===an?i[i.length-1].focus():i[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?an:"forward")}}const un=".bs.modal",dn=`hide${un}`,fn=`hidePrevented${un}`,pn=`hidden${un}`,gn=`show${un}`,mn=`shown${un}`,_n=`resize${un}`,bn=`click.dismiss${un}`,vn=`mousedown.dismiss${un}`,yn=`keydown.dismiss${un}`,wn=`click${un}.data-api`,An="modal-open",En="show",Cn="modal-static",Tn={backdrop:!0,focus:!0,keyboard:!0},On={backdrop:"(boolean|string)",focus:"boolean",keyboard:"boolean"};class xn extends ye{constructor(t,e){super(t,e),this._dialog=ke.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._isTransitioning=!1,this._scrollBar=new Ui,this._addEventListeners()}static get Default(){return Tn}static get DefaultType(){return On}static get NAME(){return"modal"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||de.trigger(this._element,gn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isTransitioning=!0,this._scrollBar.hide(),document.body.classList.add(An),this._adjustDialog(),this._backdrop.show((()=>this._showElement(t))))}hide(){this._isShown&&!this._isTransitioning&&(de.trigger(this._element,dn).defaultPrevented||(this._isShown=!1,this._isTransitioning=!0,this._focustrap.deactivate(),this._element.classList.remove(En),this._queueCallback((()=>this._hideModal()),this._element,this._isAnimated())))}dispose(){for(const t of[window,this._dialog])de.off(t,un);this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new nn({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new hn({trapElement:this._element})}_showElement(t){document.body.contains(this._element)||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0;const e=ke.findOne(".modal-body",this._dialog);e&&(e.scrollTop=0),qt(this._element),this._element.classList.add(En),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,de.trigger(this._element,mn,{relatedTarget:t})}),this._dialog,this._isAnimated())}_addEventListeners(){de.on(this._element,yn,(t=>{if("Escape"===t.key)return this._config.keyboard?(t.preventDefault(),void this.hide()):void this._triggerBackdropTransition()})),de.on(window,_n,(()=>{this._isShown&&!this._isTransitioning&&this._adjustDialog()})),de.on(this._element,vn,(t=>{de.one(this._element,bn,(e=>{this._element===t.target&&this._element===e.target&&("static"!==this._config.backdrop?this._config.backdrop&&this.hide():this._triggerBackdropTransition())}))}))}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(An),this._resetAdjustments(),this._scrollBar.reset(),de.trigger(this._element,pn)}))}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(de.trigger(this._element,fn).defaultPrevented)return;const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._element.style.overflowY;"hidden"===e||this._element.classList.contains(Cn)||(t||(this._element.style.overflowY="hidden"),this._element.classList.add(Cn),this._queueCallback((()=>{this._element.classList.remove(Cn),this._queueCallback((()=>{this._element.style.overflowY=e}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;if(i&&!t){const t=Kt()?"paddingLeft":"paddingRight";this._element.style[t]=`${e}px`}if(!i&&t){const t=Kt()?"paddingRight":"paddingLeft";this._element.style[t]=`${e}px`}}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=xn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}de.on(document,wn,'[data-bs-toggle="modal"]',(function(t){const e=Pt(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),de.one(e,gn,(t=>{t.defaultPrevented||de.one(e,pn,(()=>{Wt(this)&&this.focus()}))}));const i=ke.findOne(".modal.show");i&&xn.getInstance(i).hide(),xn.getOrCreateInstance(e).toggle(this)})),we(xn),Qt(xn);const kn=".bs.offcanvas",Ln=".data-api",Dn=`load${kn}${Ln}`,$n="show",Sn="showing",In="hiding",Nn=".offcanvas.show",Pn=`show${kn}`,jn=`shown${kn}`,Mn=`hide${kn}`,Hn=`hidePrevented${kn}`,Wn=`hidden${kn}`,Fn=`resize${kn}`,Bn=`click${kn}${Ln}`,zn=`keydown.dismiss${kn}`,qn={backdrop:!0,keyboard:!0,scroll:!1},Rn={backdrop:"(boolean|string)",keyboard:"boolean",scroll:"boolean"};class Vn extends ye{constructor(t,e){super(t,e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get Default(){return qn}static get DefaultType(){return Rn}static get NAME(){return"offcanvas"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||de.trigger(this._element,Pn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._backdrop.show(),this._config.scroll||(new Ui).hide(),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add(Sn),this._queueCallback((()=>{this._config.scroll&&!this._config.backdrop||this._focustrap.activate(),this._element.classList.add($n),this._element.classList.remove(Sn),de.trigger(this._element,jn,{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(de.trigger(this._element,Mn).defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.add(In),this._backdrop.hide(),this._queueCallback((()=>{this._element.classList.remove($n,In),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._config.scroll||(new Ui).reset(),de.trigger(this._element,Wn)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_initializeBackDrop(){const t=Boolean(this._config.backdrop);return new nn({className:"offcanvas-backdrop",isVisible:t,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:t?()=>{"static"!==this._config.backdrop?this.hide():de.trigger(this._element,Hn)}:null})}_initializeFocusTrap(){return new hn({trapElement:this._element})}_addEventListeners(){de.on(this._element,zn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():de.trigger(this._element,Hn))}))}static jQueryInterface(t){return this.each((function(){const e=Vn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}de.on(document,Bn,'[data-bs-toggle="offcanvas"]',(function(t){const e=Pt(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),Ft(this))return;de.one(e,Wn,(()=>{Wt(this)&&this.focus()}));const i=ke.findOne(Nn);i&&i!==e&&Vn.getInstance(i).hide(),Vn.getOrCreateInstance(e).toggle(this)})),de.on(window,Dn,(()=>{for(const t of ke.find(Nn))Vn.getOrCreateInstance(t).show()})),de.on(window,Fn,(()=>{for(const t of ke.find("[aria-modal][class*=show][class*=offcanvas-]"))"fixed"!==getComputedStyle(t).position&&Vn.getOrCreateInstance(t).hide()})),we(Vn),Qt(Vn);const Kn=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Qn=/^(?:(?:https?|mailto|ftp|tel|file|sms):|[^#&/:?]*(?:[#/?]|$))/i,Xn=/^data:(?:image\/(?:bmp|gif|jpeg|jpg|png|tiff|webp)|video\/(?:mpeg|mp4|ogg|webm)|audio\/(?:mp3|oga|ogg|opus));base64,[\d+/a-z]+=*$/i,Yn=(t,e)=>{const i=t.nodeName.toLowerCase();return e.includes(i)?!Kn.has(i)||Boolean(Qn.test(t.nodeValue)||Xn.test(t.nodeValue)):e.filter((t=>t instanceof RegExp)).some((t=>t.test(i)))},Un={"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],div:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},Gn={allowList:Un,content:{},extraClass:"",html:!1,sanitize:!0,sanitizeFn:null,template:"<div></div>"},Jn={allowList:"object",content:"object",extraClass:"(string|function)",html:"boolean",sanitize:"boolean",sanitizeFn:"(null|function)",template:"string"},Zn={entry:"(string|element|function|null)",selector:"(string|element)"};class ts extends ve{constructor(t){super(),this._config=this._getConfig(t)}static get Default(){return Gn}static get DefaultType(){return Jn}static get NAME(){return"TemplateFactory"}getContent(){return Object.values(this._config.content).map((t=>this._resolvePossibleFunction(t))).filter(Boolean)}hasContent(){return this.getContent().length>0}changeContent(t){return this._checkContent(t),this._config.content={...this._config.content,...t},this}toHtml(){const t=document.createElement("div");t.innerHTML=this._maybeSanitize(this._config.template);for(const[e,i]of Object.entries(this._config.content))this._setContent(t,i,e);const e=t.children[0],i=this._resolvePossibleFunction(this._config.extraClass);return i&&e.classList.add(...i.split(" ")),e}_typeCheckConfig(t){super._typeCheckConfig(t),this._checkContent(t.content)}_checkContent(t){for(const[e,i]of Object.entries(t))super._typeCheckConfig({selector:e,entry:i},Zn)}_setContent(t,e,i){const n=ke.findOne(i,t);n&&((e=this._resolvePossibleFunction(e))?Mt(e)?this._putElementInTemplate(Ht(e),n):this._config.html?n.innerHTML=this._maybeSanitize(e):n.textContent=e:n.remove())}_maybeSanitize(t){return this._config.sanitize?function(t,e,i){if(!t.length)return t;if(i&&"function"==typeof i)return i(t);const n=(new window.DOMParser).parseFromString(t,"text/html"),s=[].concat(...n.body.querySelectorAll("*"));for(const t of s){const i=t.nodeName.toLowerCase();if(!Object.keys(e).includes(i)){t.remove();continue}const n=[].concat(...t.attributes),s=[].concat(e["*"]||[],e[i]||[]);for(const e of n)Yn(e,s)||t.removeAttribute(e.nodeName)}return n.body.innerHTML}(t,this._config.allowList,this._config.sanitizeFn):t}_resolvePossibleFunction(t){return"function"==typeof t?t(this):t}_putElementInTemplate(t,e){if(this._config.html)return e.innerHTML="",void e.append(t);e.textContent=t.textContent}}const es=new Set(["sanitize","allowList","sanitizeFn"]),is="fade",ns="show",ss=".modal",os="hide.bs.modal",rs="hover",as="focus",ls={AUTO:"auto",TOP:"top",RIGHT:Kt()?"left":"right",BOTTOM:"bottom",LEFT:Kt()?"right":"left"},cs={allowList:Un,animation:!0,boundary:"clippingParents",container:!1,customClass:"",delay:0,fallbackPlacements:["top","right","bottom","left"],html:!1,offset:[0,0],placement:"top",popperConfig:null,sanitize:!0,sanitizeFn:null,selector:!1,template:'<div class="tooltip" role="tooltip"><div class="tooltip-arrow"></div><div class="tooltip-inner"></div></div>',title:"",trigger:"hover focus"},hs={allowList:"object",animation:"boolean",boundary:"(string|element)",container:"(string|element|boolean)",customClass:"(string|function)",delay:"(number|object)",fallbackPlacements:"array",html:"boolean",offset:"(array|string|function)",placement:"(string|function)",popperConfig:"(null|object|function)",sanitize:"boolean",sanitizeFn:"(null|function)",selector:"(string|boolean)",template:"string",title:"(string|element|function)",trigger:"string"};class us extends ye{constructor(t,i){if(void 0===e)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t,i),this._isEnabled=!0,this._timeout=0,this._isHovered=null,this._activeTrigger={},this._popper=null,this._templateFactory=null,this._newContent=null,this.tip=null,this._setListeners(),this._config.selector||this._fixTitle()}static get Default(){return cs}static get DefaultType(){return hs}static get NAME(){return"tooltip"}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(){this._isEnabled&&(this._activeTrigger.click=!this._activeTrigger.click,this._isShown()?this._leave():this._enter())}dispose(){clearTimeout(this._timeout),de.off(this._element.closest(ss),os,this._hideModalHandler),this._element.getAttribute("data-bs-original-title")&&this._element.setAttribute("title",this._element.getAttribute("data-bs-original-title")),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this._isWithContent()||!this._isEnabled)return;const t=de.trigger(this._element,this.constructor.eventName("show")),e=(Bt(this._element)||this._element.ownerDocument.documentElement).contains(this._element);if(t.defaultPrevented||!e)return;this._disposePopper();const i=this._getTipElement();this._element.setAttribute("aria-describedby",i.getAttribute("id"));const{container:n}=this._config;if(this._element.ownerDocument.documentElement.contains(this.tip)||(n.append(i),de.trigger(this._element,this.constructor.eventName("inserted"))),this._popper=this._createPopper(i),i.classList.add(ns),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))de.on(t,"mouseover",zt);this._queueCallback((()=>{de.trigger(this._element,this.constructor.eventName("shown")),!1===this._isHovered&&this._leave(),this._isHovered=!1}),this.tip,this._isAnimated())}hide(){if(this._isShown()&&!de.trigger(this._element,this.constructor.eventName("hide")).defaultPrevented){if(this._getTipElement().classList.remove(ns),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))de.off(t,"mouseover",zt);this._activeTrigger.click=!1,this._activeTrigger[as]=!1,this._activeTrigger[rs]=!1,this._isHovered=null,this._queueCallback((()=>{this._isWithActiveTrigger()||(this._isHovered||this._disposePopper(),this._element.removeAttribute("aria-describedby"),de.trigger(this._element,this.constructor.eventName("hidden")))}),this.tip,this._isAnimated())}}update(){this._popper&&this._popper.update()}_isWithContent(){return Boolean(this._getTitle())}_getTipElement(){return this.tip||(this.tip=this._createTipElement(this._newContent||this._getContentForTemplate())),this.tip}_createTipElement(t){const e=this._getTemplateFactory(t).toHtml();if(!e)return null;e.classList.remove(is,ns),e.classList.add(`bs-${this.constructor.NAME}-auto`);const i=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME).toString();return e.setAttribute("id",i),this._isAnimated()&&e.classList.add(is),e}setContent(t){this._newContent=t,this._isShown()&&(this._disposePopper(),this.show())}_getTemplateFactory(t){return this._templateFactory?this._templateFactory.changeContent(t):this._templateFactory=new ts({...this._config,content:t,extraClass:this._resolvePossibleFunction(this._config.customClass)}),this._templateFactory}_getContentForTemplate(){return{".tooltip-inner":this._getTitle()}}_getTitle(){return this._resolvePossibleFunction(this._config.title)||this._element.getAttribute("data-bs-original-title")}_initializeOnDelegatedTarget(t){return this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_isAnimated(){return this._config.animation||this.tip&&this.tip.classList.contains(is)}_isShown(){return this.tip&&this.tip.classList.contains(ns)}_createPopper(t){const e="function"==typeof this._config.placement?this._config.placement.call(this,t,this._element):this._config.placement,i=ls[e.toUpperCase()];return Dt(this._element,t,this._getPopperConfig(i))}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_resolvePossibleFunction(t){return"function"==typeof t?t.call(this._element):t}_getPopperConfig(t){const e={placement:t,modifiers:[{name:"flip",options:{fallbackPlacements:this._config.fallbackPlacements}},{name:"offset",options:{offset:this._getOffset()}},{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"arrow",options:{element:`.${this.constructor.NAME}-arrow`}},{name:"preSetPlacement",enabled:!0,phase:"beforeMain",fn:t=>{this._getTipElement().setAttribute("data-popper-placement",t.state.placement)}}]};return{...e,..."function"==typeof this._config.popperConfig?this._config.popperConfig(e):this._config.popperConfig}}_setListeners(){const t=this._config.trigger.split(" ");for(const e of t)if("click"===e)de.on(this._element,this.constructor.eventName("click"),this._config.selector,(t=>{this._initializeOnDelegatedTarget(t).toggle()}));else if("manual"!==e){const t=e===rs?this.constructor.eventName("mouseenter"):this.constructor.eventName("focusin"),i=e===rs?this.constructor.eventName("mouseleave"):this.constructor.eventName("focusout");de.on(this._element,t,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusin"===t.type?as:rs]=!0,e._enter()})),de.on(this._element,i,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusout"===t.type?as:rs]=e._element.contains(t.relatedTarget),e._leave()}))}this._hideModalHandler=()=>{this._element&&this.hide()},de.on(this._element.closest(ss),os,this._hideModalHandler)}_fixTitle(){const t=this._element.getAttribute("title");t&&(this._element.getAttribute("aria-label")||this._element.textContent.trim()||this._element.setAttribute("aria-label",t),this._element.setAttribute("data-bs-original-title",t),this._element.removeAttribute("title"))}_enter(){this._isShown()||this._isHovered?this._isHovered=!0:(this._isHovered=!0,this._setTimeout((()=>{this._isHovered&&this.show()}),this._config.delay.show))}_leave(){this._isWithActiveTrigger()||(this._isHovered=!1,this._setTimeout((()=>{this._isHovered||this.hide()}),this._config.delay.hide))}_setTimeout(t,e){clearTimeout(this._timeout),this._timeout=setTimeout(t,e)}_isWithActiveTrigger(){return Object.values(this._activeTrigger).includes(!0)}_getConfig(t){const e=be.getDataAttributes(this._element);for(const t of Object.keys(e))es.has(t)&&delete e[t];return t={...e,..."object"==typeof t&&t?t:{}},t=this._mergeConfigObj(t),t=this._configAfterMerge(t),this._typeCheckConfig(t),t}_configAfterMerge(t){return t.container=!1===t.container?document.body:Ht(t.container),"number"==typeof t.delay&&(t.delay={show:t.delay,hide:t.delay}),"number"==typeof t.title&&(t.title=t.title.toString()),"number"==typeof t.content&&(t.content=t.content.toString()),t}_getDelegateConfig(){const t={};for(const e in this._config)this.constructor.Default[e]!==this._config[e]&&(t[e]=this._config[e]);return t.selector=!1,t.trigger="manual",t}_disposePopper(){this._popper&&(this._popper.destroy(),this._popper=null),this.tip&&(this.tip.remove(),this.tip=null)}static jQueryInterface(t){return this.each((function(){const e=us.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}Qt(us);const ds={...us.Default,content:"",offset:[0,8],placement:"right",template:'<div class="popover" role="tooltip"><div class="popover-arrow"></div><h3 class="popover-header"></h3><div class="popover-body"></div></div>',trigger:"click"},fs={...us.DefaultType,content:"(null|string|element|function)"};class ps extends us{static get Default(){return ds}static get DefaultType(){return fs}static get NAME(){return"popover"}_isWithContent(){return this._getTitle()||this._getContent()}_getContentForTemplate(){return{".popover-header":this._getTitle(),".popover-body":this._getContent()}}_getContent(){return this._resolvePossibleFunction(this._config.content)}static jQueryInterface(t){return this.each((function(){const e=ps.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}Qt(ps);const gs=".bs.scrollspy",ms=`activate${gs}`,_s=`click${gs}`,bs=`load${gs}.data-api`,vs="active",ys="[href]",ws=".nav-link",As=`${ws}, .nav-item > ${ws}, .list-group-item`,Es={offset:null,rootMargin:"0px 0px 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Ks=".bs.toast",Qs=`mouseover${Ks}`,Xs=`mouseout${Ks}`,Ys=`focusin${Ks}`,Us=`focusout${Ks}`,Gs=`hide${Ks}`,Js=`hidden${Ks}`,Zs=`show${Ks}`,to=`shown${Ks}`,eo="hide",io="show",no="showing",so={animation:"boolean",autohide:"boolean",delay:"number"},oo={animation:!0,autohide:!0,delay:5e3};class ro extends ye{constructor(t,e){super(t,e),this._timeout=null,this._hasMouseInteraction=!1,this._hasKeyboardInteraction=!1,this._setListeners()}static get Default(){return oo}static get DefaultType(){return so}static get 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ao;we(ro),Qt(ro),ao=function(){[].slice.call(document.querySelectorAll('[data-bs-toggle="tooltip"]')).map((function(t){return new us(t,{delay:{show:500,hide:100}})}))},"loading"!=document.readyState?ao():document.addEventListener("DOMContentLoaded",ao)})();
+//# sourceMappingURL=bootstrap.js.map
\ No newline at end of file
diff --git a/_static/scripts/bootstrap.js.LICENSE.txt b/_static/scripts/bootstrap.js.LICENSE.txt
new file mode 100644
index 00000000..91ad10aa
--- /dev/null
+++ b/_static/scripts/bootstrap.js.LICENSE.txt
@@ -0,0 +1,5 @@
+/*!
+  * Bootstrap v5.2.3 (https://getbootstrap.com/)
+  * Copyright 2011-2022 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)
+  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+  */
diff --git a/_static/scripts/bootstrap.js.map b/_static/scripts/bootstrap.js.map
new file mode 100644
index 00000000..d83e2f7c
--- /dev/null
+++ b/_static/scripts/bootstrap.js.map
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= 'end';\nexport var clippingParents = 'clippingParents';\nexport var viewport = 'viewport';\nexport var popper = 'popper';\nexport var reference = 'reference';\nexport var variationPlacements = /*#__PURE__*/basePlacements.reduce(function (acc, placement) {\n  return acc.concat([placement + \"-\" + start, placement + \"-\" + end]);\n}, []);\nexport var placements = /*#__PURE__*/[].concat(basePlacements, [auto]).reduce(function (acc, placement) {\n  return acc.concat([placement, placement + \"-\" + start, placement + \"-\" + end]);\n}, []); // modifiers that need to read the DOM\n\nexport var beforeRead = 'beforeRead';\nexport var read = 'read';\nexport var afterRead = 'afterRead'; // pure-logic modifiers\n\nexport var beforeMain = 'beforeMain';\nexport var main = 'main';\nexport var afterMain = 'afterMain'; // modifier with the purpose to write to the DOM (or write into a framework state)\n\nexport var beforeWrite = 'beforeWrite';\nexport var write = 'write';\nexport var afterWrite = 'afterWrite';\nexport var modifierPhases = [beforeRead, read, afterRead, beforeMain, main, afterMain, beforeWrite, write, afterWrite];","export default function getNodeName(element) {\n  return element ? (element.nodeName || '').toLowerCase() : null;\n}","export default function getWindow(node) {\n  if (node == null) {\n    return window;\n  }\n\n  if (node.toString() !== '[object Window]') {\n    var ownerDocument = node.ownerDocument;\n    return ownerDocument ? ownerDocument.defaultView || window : window;\n  }\n\n  return node;\n}","import getWindow from \"./getWindow.js\";\n\nfunction isElement(node) {\n  var OwnElement = getWindow(node).Element;\n  return node instanceof OwnElement || node instanceof Element;\n}\n\nfunction isHTMLElement(node) {\n  var OwnElement = getWindow(node).HTMLElement;\n  return node instanceof OwnElement || node instanceof HTMLElement;\n}\n\nfunction isShadowRoot(node) {\n  // IE 11 has no ShadowRoot\n  if (typeof ShadowRoot === 'undefined') {\n    return false;\n  }\n\n  var OwnElement = getWindow(node).ShadowRoot;\n  return node instanceof OwnElement || node instanceof ShadowRoot;\n}\n\nexport { isElement, isHTMLElement, isShadowRoot };","import getNodeName from \"../dom-utils/getNodeName.js\";\nimport { isHTMLElement } from \"../dom-utils/instanceOf.js\"; // This modifier takes the styles prepared by the `computeStyles` modifier\n// and applies them to the HTMLElements such as popper and arrow\n\nfunction applyStyles(_ref) {\n  var state = _ref.state;\n  Object.keys(state.elements).forEach(function (name) {\n    var style = state.styles[name] || {};\n    var attributes = state.attributes[name] || {};\n    var element = state.elements[name]; // arrow is optional + virtual elements\n\n    if (!isHTMLElement(element) || !getNodeName(element)) {\n      return;\n    } // Flow doesn't support to extend this property, but it's the most\n    // effective way to apply styles to an HTMLElement\n    // $FlowFixMe[cannot-write]\n\n\n    Object.assign(element.style, style);\n    Object.keys(attributes).forEach(function (name) {\n      var value = attributes[name];\n\n      if (value === false) {\n        element.removeAttribute(name);\n      } else {\n        element.setAttribute(name, value === true ? '' : value);\n      }\n    });\n  });\n}\n\nfunction effect(_ref2) {\n  var state = _ref2.state;\n  var initialStyles = {\n    popper: {\n      position: state.options.strategy,\n      left: '0',\n      top: '0',\n      margin: '0'\n    },\n    arrow: {\n      position: 'absolute'\n    },\n    reference: {}\n  };\n  Object.assign(state.elements.popper.style, initialStyles.popper);\n  state.styles = initialStyles;\n\n  if (state.elements.arrow) {\n    Object.assign(state.elements.arrow.style, initialStyles.arrow);\n  }\n\n  return function () {\n    Object.keys(state.elements).forEach(function (name) {\n      var element = state.elements[name];\n      var attributes = state.attributes[name] || {};\n      var styleProperties = Object.keys(state.styles.hasOwnProperty(name) ? state.styles[name] : initialStyles[name]); // Set all values to an empty string to unset them\n\n      var style = styleProperties.reduce(function (style, property) {\n        style[property] = '';\n        return style;\n      }, {}); // arrow is optional + virtual elements\n\n      if (!isHTMLElement(element) || !getNodeName(element)) {\n        return;\n      }\n\n      Object.assign(element.style, style);\n      Object.keys(attributes).forEach(function (attribute) {\n        element.removeAttribute(attribute);\n      });\n    });\n  };\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'applyStyles',\n  enabled: true,\n  phase: 'write',\n  fn: applyStyles,\n  effect: effect,\n  requires: ['computeStyles']\n};","import { auto } from \"../enums.js\";\nexport default function getBasePlacement(placement) {\n  return placement.split('-')[0];\n}","export var max = Math.max;\nexport var min = Math.min;\nexport var round = Math.round;","export default function getUAString() {\n  var uaData = navigator.userAgentData;\n\n  if (uaData != null && uaData.brands && Array.isArray(uaData.brands)) {\n    return uaData.brands.map(function (item) {\n      return item.brand + \"/\" + item.version;\n    }).join(' ');\n  }\n\n  return navigator.userAgent;\n}","import getUAString from \"../utils/userAgent.js\";\nexport default function isLayoutViewport() {\n  return !/^((?!chrome|android).)*safari/i.test(getUAString());\n}","import { isElement, isHTMLElement } from \"./instanceOf.js\";\nimport { round } from \"../utils/math.js\";\nimport getWindow from \"./getWindow.js\";\nimport isLayoutViewport from \"./isLayoutViewport.js\";\nexport default function getBoundingClientRect(element, includeScale, isFixedStrategy) {\n  if (includeScale === void 0) {\n    includeScale = false;\n  }\n\n  if (isFixedStrategy === void 0) {\n    isFixedStrategy = false;\n  }\n\n  var clientRect = element.getBoundingClientRect();\n  var scaleX = 1;\n  var scaleY = 1;\n\n  if (includeScale && isHTMLElement(element)) {\n    scaleX = element.offsetWidth > 0 ? round(clientRect.width) / element.offsetWidth || 1 : 1;\n    scaleY = element.offsetHeight > 0 ? round(clientRect.height) / element.offsetHeight || 1 : 1;\n  }\n\n  var _ref = isElement(element) ? getWindow(element) : window,\n      visualViewport = _ref.visualViewport;\n\n  var addVisualOffsets = !isLayoutViewport() && isFixedStrategy;\n  var x = (clientRect.left + (addVisualOffsets && visualViewport ? visualViewport.offsetLeft : 0)) / scaleX;\n  var y = (clientRect.top + (addVisualOffsets && visualViewport ? visualViewport.offsetTop : 0)) / scaleY;\n  var width = clientRect.width / scaleX;\n  var height = clientRect.height / scaleY;\n  return {\n    width: width,\n    height: height,\n    top: y,\n    right: x + width,\n    bottom: y + height,\n    left: x,\n    x: x,\n    y: y\n  };\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\"; // Returns the layout rect of an element relative to its offsetParent. Layout\n// means it doesn't take into account transforms.\n\nexport default function getLayoutRect(element) {\n  var clientRect = getBoundingClientRect(element); // Use the clientRect sizes if it's not been transformed.\n  // Fixes https://github.com/popperjs/popper-core/issues/1223\n\n  var width = element.offsetWidth;\n  var height = element.offsetHeight;\n\n  if (Math.abs(clientRect.width - width) <= 1) {\n    width = clientRect.width;\n  }\n\n  if (Math.abs(clientRect.height - height) <= 1) {\n    height = clientRect.height;\n  }\n\n  return {\n    x: element.offsetLeft,\n    y: element.offsetTop,\n    width: width,\n    height: height\n  };\n}","import { isShadowRoot } from \"./instanceOf.js\";\nexport default function contains(parent, child) {\n  var rootNode = child.getRootNode && child.getRootNode(); // First, attempt with faster native method\n\n  if (parent.contains(child)) {\n    return true;\n  } // then fallback to custom implementation with Shadow DOM support\n  else if (rootNode && isShadowRoot(rootNode)) {\n      var next = child;\n\n      do {\n        if (next && parent.isSameNode(next)) {\n          return true;\n        } // $FlowFixMe[prop-missing]: need a better way to handle this...\n\n\n        next = next.parentNode || next.host;\n      } while (next);\n    } // Give up, the result is false\n\n\n  return false;\n}","import getWindow from \"./getWindow.js\";\nexport default function getComputedStyle(element) {\n  return getWindow(element).getComputedStyle(element);\n}","import getNodeName from \"./getNodeName.js\";\nexport default function isTableElement(element) {\n  return ['table', 'td', 'th'].indexOf(getNodeName(element)) >= 0;\n}","import { isElement } from \"./instanceOf.js\";\nexport default function getDocumentElement(element) {\n  // $FlowFixMe[incompatible-return]: assume body is always available\n  return ((isElement(element) ? element.ownerDocument : // $FlowFixMe[prop-missing]\n  element.document) || window.document).documentElement;\n}","import getNodeName from \"./getNodeName.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport { isShadowRoot } from \"./instanceOf.js\";\nexport default function getParentNode(element) {\n  if (getNodeName(element) === 'html') {\n    return element;\n  }\n\n  return (// this is a quicker (but less type safe) way to save quite some bytes from the bundle\n    // $FlowFixMe[incompatible-return]\n    // $FlowFixMe[prop-missing]\n    element.assignedSlot || // step into the shadow DOM of the parent of a slotted node\n    element.parentNode || ( // DOM Element detected\n    isShadowRoot(element) ? element.host : null) || // ShadowRoot detected\n    // $FlowFixMe[incompatible-call]: HTMLElement is a Node\n    getDocumentElement(element) // fallback\n\n  );\n}","import getWindow from \"./getWindow.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport { isHTMLElement, isShadowRoot } from \"./instanceOf.js\";\nimport isTableElement from \"./isTableElement.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport getUAString from \"../utils/userAgent.js\";\n\nfunction getTrueOffsetParent(element) {\n  if (!isHTMLElement(element) || // https://github.com/popperjs/popper-core/issues/837\n  getComputedStyle(element).position === 'fixed') {\n    return null;\n  }\n\n  return element.offsetParent;\n} // `.offsetParent` reports `null` for fixed elements, while absolute elements\n// return the containing block\n\n\nfunction getContainingBlock(element) {\n  var isFirefox = /firefox/i.test(getUAString());\n  var isIE = /Trident/i.test(getUAString());\n\n  if (isIE && isHTMLElement(element)) {\n    // In IE 9, 10 and 11 fixed elements containing block is always established by the viewport\n    var elementCss = getComputedStyle(element);\n\n    if (elementCss.position === 'fixed') {\n      return null;\n    }\n  }\n\n  var currentNode = getParentNode(element);\n\n  if (isShadowRoot(currentNode)) {\n    currentNode = currentNode.host;\n  }\n\n  while (isHTMLElement(currentNode) && ['html', 'body'].indexOf(getNodeName(currentNode)) < 0) {\n    var css = getComputedStyle(currentNode); // This is non-exhaustive but covers the most common CSS properties that\n    // create a containing block.\n    // https://developer.mozilla.org/en-US/docs/Web/CSS/Containing_block#identifying_the_containing_block\n\n    if (css.transform !== 'none' || css.perspective !== 'none' || css.contain === 'paint' || ['transform', 'perspective'].indexOf(css.willChange) !== -1 || isFirefox && css.willChange === 'filter' || isFirefox && css.filter && css.filter !== 'none') {\n      return currentNode;\n    } else {\n      currentNode = currentNode.parentNode;\n    }\n  }\n\n  return null;\n} // Gets the closest ancestor positioned element. Handles some edge cases,\n// such as table ancestors and cross browser bugs.\n\n\nexport default function getOffsetParent(element) {\n  var window = getWindow(element);\n  var offsetParent = getTrueOffsetParent(element);\n\n  while (offsetParent && isTableElement(offsetParent) && getComputedStyle(offsetParent).position === 'static') {\n    offsetParent = getTrueOffsetParent(offsetParent);\n  }\n\n  if (offsetParent && (getNodeName(offsetParent) === 'html' || getNodeName(offsetParent) === 'body' && getComputedStyle(offsetParent).position === 'static')) {\n    return window;\n  }\n\n  return offsetParent || getContainingBlock(element) || window;\n}","export default function getMainAxisFromPlacement(placement) {\n  return ['top', 'bottom'].indexOf(placement) >= 0 ? 'x' : 'y';\n}","import { max as mathMax, min as mathMin } from \"./math.js\";\nexport function within(min, value, max) {\n  return mathMax(min, mathMin(value, max));\n}\nexport function withinMaxClamp(min, value, max) {\n  var v = within(min, value, max);\n  return v > max ? max : v;\n}","import getFreshSideObject from \"./getFreshSideObject.js\";\nexport default function mergePaddingObject(paddingObject) {\n  return Object.assign({}, getFreshSideObject(), paddingObject);\n}","export default function getFreshSideObject() {\n  return {\n    top: 0,\n    right: 0,\n    bottom: 0,\n    left: 0\n  };\n}","export default function expandToHashMap(value, keys) {\n  return keys.reduce(function (hashMap, key) {\n    hashMap[key] = value;\n    return hashMap;\n  }, {});\n}","import getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getLayoutRect from \"../dom-utils/getLayoutRect.js\";\nimport contains from \"../dom-utils/contains.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport getMainAxisFromPlacement from \"../utils/getMainAxisFromPlacement.js\";\nimport { within } from \"../utils/within.js\";\nimport mergePaddingObject from \"../utils/mergePaddingObject.js\";\nimport expandToHashMap from \"../utils/expandToHashMap.js\";\nimport { left, right, basePlacements, top, bottom } from \"../enums.js\";\nimport { isHTMLElement } from \"../dom-utils/instanceOf.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar toPaddingObject = function toPaddingObject(padding, state) {\n  padding = typeof padding === 'function' ? padding(Object.assign({}, state.rects, {\n    placement: state.placement\n  })) : padding;\n  return mergePaddingObject(typeof padding !== 'number' ? padding : expandToHashMap(padding, basePlacements));\n};\n\nfunction arrow(_ref) {\n  var _state$modifiersData$;\n\n  var state = _ref.state,\n      name = _ref.name,\n      options = _ref.options;\n  var arrowElement = state.elements.arrow;\n  var popperOffsets = state.modifiersData.popperOffsets;\n  var basePlacement = getBasePlacement(state.placement);\n  var axis = getMainAxisFromPlacement(basePlacement);\n  var isVertical = [left, right].indexOf(basePlacement) >= 0;\n  var len = isVertical ? 'height' : 'width';\n\n  if (!arrowElement || !popperOffsets) {\n    return;\n  }\n\n  var paddingObject = toPaddingObject(options.padding, state);\n  var arrowRect = getLayoutRect(arrowElement);\n  var minProp = axis === 'y' ? top : left;\n  var maxProp = axis === 'y' ? bottom : right;\n  var endDiff = state.rects.reference[len] + state.rects.reference[axis] - popperOffsets[axis] - state.rects.popper[len];\n  var startDiff = popperOffsets[axis] - state.rects.reference[axis];\n  var arrowOffsetParent = getOffsetParent(arrowElement);\n  var clientSize = arrowOffsetParent ? axis === 'y' ? arrowOffsetParent.clientHeight || 0 : arrowOffsetParent.clientWidth || 0 : 0;\n  var centerToReference = endDiff / 2 - startDiff / 2; // Make sure the arrow doesn't overflow the popper if the center point is\n  // outside of the popper bounds\n\n  var min = paddingObject[minProp];\n  var max = clientSize - arrowRect[len] - paddingObject[maxProp];\n  var center = clientSize / 2 - arrowRect[len] / 2 + centerToReference;\n  var offset = within(min, center, max); // Prevents breaking syntax highlighting...\n\n  var axisProp = axis;\n  state.modifiersData[name] = (_state$modifiersData$ = {}, _state$modifiersData$[axisProp] = offset, _state$modifiersData$.centerOffset = offset - center, _state$modifiersData$);\n}\n\nfunction effect(_ref2) {\n  var state = _ref2.state,\n      options = _ref2.options;\n  var _options$element = options.element,\n      arrowElement = _options$element === void 0 ? '[data-popper-arrow]' : _options$element;\n\n  if (arrowElement == null) {\n    return;\n  } // CSS selector\n\n\n  if (typeof arrowElement === 'string') {\n    arrowElement = state.elements.popper.querySelector(arrowElement);\n\n    if (!arrowElement) {\n      return;\n    }\n  }\n\n  if (process.env.NODE_ENV !== \"production\") {\n    if (!isHTMLElement(arrowElement)) {\n      console.error(['Popper: \"arrow\" element must be an HTMLElement (not an SVGElement).', 'To use an SVG arrow, wrap it in an HTMLElement that will be used as', 'the arrow.'].join(' '));\n    }\n  }\n\n  if (!contains(state.elements.popper, arrowElement)) {\n    if (process.env.NODE_ENV !== \"production\") {\n      console.error(['Popper: \"arrow\" modifier\\'s `element` must be a child of the popper', 'element.'].join(' '));\n    }\n\n    return;\n  }\n\n  state.elements.arrow = arrowElement;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'arrow',\n  enabled: true,\n  phase: 'main',\n  fn: arrow,\n  effect: effect,\n  requires: ['popperOffsets'],\n  requiresIfExists: ['preventOverflow']\n};","export default function getVariation(placement) {\n  return placement.split('-')[1];\n}","import { top, left, right, bottom, end } from \"../enums.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport getWindow from \"../dom-utils/getWindow.js\";\nimport getDocumentElement from \"../dom-utils/getDocumentElement.js\";\nimport getComputedStyle from \"../dom-utils/getComputedStyle.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getVariation from \"../utils/getVariation.js\";\nimport { round } from \"../utils/math.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar unsetSides = {\n  top: 'auto',\n  right: 'auto',\n  bottom: 'auto',\n  left: 'auto'\n}; // Round the offsets to the nearest suitable subpixel based on the DPR.\n// Zooming can change the DPR, but it seems to report a value that will\n// cleanly divide the values into the appropriate subpixels.\n\nfunction roundOffsetsByDPR(_ref, win) {\n  var x = _ref.x,\n      y = _ref.y;\n  var dpr = win.devicePixelRatio || 1;\n  return {\n    x: round(x * dpr) / dpr || 0,\n    y: round(y * dpr) / dpr || 0\n  };\n}\n\nexport function mapToStyles(_ref2) {\n  var _Object$assign2;\n\n  var popper = _ref2.popper,\n      popperRect = _ref2.popperRect,\n      placement = _ref2.placement,\n      variation = _ref2.variation,\n      offsets = _ref2.offsets,\n      position = _ref2.position,\n      gpuAcceleration = _ref2.gpuAcceleration,\n      adaptive = _ref2.adaptive,\n      roundOffsets = _ref2.roundOffsets,\n      isFixed = _ref2.isFixed;\n  var _offsets$x = offsets.x,\n      x = _offsets$x === void 0 ? 0 : _offsets$x,\n      _offsets$y = offsets.y,\n      y = _offsets$y === void 0 ? 0 : _offsets$y;\n\n  var _ref3 = typeof roundOffsets === 'function' ? roundOffsets({\n    x: x,\n    y: y\n  }) : {\n    x: x,\n    y: y\n  };\n\n  x = _ref3.x;\n  y = _ref3.y;\n  var hasX = offsets.hasOwnProperty('x');\n  var hasY = offsets.hasOwnProperty('y');\n  var sideX = left;\n  var sideY = top;\n  var win = window;\n\n  if (adaptive) {\n    var offsetParent = getOffsetParent(popper);\n    var heightProp = 'clientHeight';\n    var widthProp = 'clientWidth';\n\n    if (offsetParent === getWindow(popper)) {\n      offsetParent = getDocumentElement(popper);\n\n      if (getComputedStyle(offsetParent).position !== 'static' && position === 'absolute') {\n        heightProp = 'scrollHeight';\n        widthProp = 'scrollWidth';\n      }\n    } // $FlowFixMe[incompatible-cast]: force type refinement, we compare offsetParent with window above, but Flow doesn't detect it\n\n\n    offsetParent = offsetParent;\n\n    if (placement === top || (placement === left || placement === right) && variation === end) {\n      sideY = bottom;\n      var offsetY = isFixed && offsetParent === win && win.visualViewport ? win.visualViewport.height : // $FlowFixMe[prop-missing]\n      offsetParent[heightProp];\n      y -= offsetY - popperRect.height;\n      y *= gpuAcceleration ? 1 : -1;\n    }\n\n    if (placement === left || (placement === top || placement === bottom) && variation === end) {\n      sideX = right;\n      var offsetX = isFixed && offsetParent === win && win.visualViewport ? win.visualViewport.width : // $FlowFixMe[prop-missing]\n      offsetParent[widthProp];\n      x -= offsetX - popperRect.width;\n      x *= gpuAcceleration ? 1 : -1;\n    }\n  }\n\n  var commonStyles = Object.assign({\n    position: position\n  }, adaptive && unsetSides);\n\n  var _ref4 = roundOffsets === true ? roundOffsetsByDPR({\n    x: x,\n    y: y\n  }, getWindow(popper)) : {\n    x: x,\n    y: y\n  };\n\n  x = _ref4.x;\n  y = _ref4.y;\n\n  if (gpuAcceleration) {\n    var _Object$assign;\n\n    return Object.assign({}, commonStyles, (_Object$assign = {}, _Object$assign[sideY] = hasY ? '0' : '', _Object$assign[sideX] = hasX ? '0' : '', _Object$assign.transform = (win.devicePixelRatio || 1) <= 1 ? \"translate(\" + x + \"px, \" + y + \"px)\" : \"translate3d(\" + x + \"px, \" + y + \"px, 0)\", _Object$assign));\n  }\n\n  return Object.assign({}, commonStyles, (_Object$assign2 = {}, _Object$assign2[sideY] = hasY ? y + \"px\" : '', _Object$assign2[sideX] = hasX ? x + \"px\" : '', _Object$assign2.transform = '', _Object$assign2));\n}\n\nfunction computeStyles(_ref5) {\n  var state = _ref5.state,\n      options = _ref5.options;\n  var _options$gpuAccelerat = options.gpuAcceleration,\n      gpuAcceleration = _options$gpuAccelerat === void 0 ? true : _options$gpuAccelerat,\n      _options$adaptive = options.adaptive,\n      adaptive = _options$adaptive === void 0 ? true : _options$adaptive,\n      _options$roundOffsets = options.roundOffsets,\n      roundOffsets = _options$roundOffsets === void 0 ? true : _options$roundOffsets;\n\n  if (process.env.NODE_ENV !== \"production\") {\n    var transitionProperty = getComputedStyle(state.elements.popper).transitionProperty || '';\n\n    if (adaptive && ['transform', 'top', 'right', 'bottom', 'left'].some(function (property) {\n      return transitionProperty.indexOf(property) >= 0;\n    })) {\n      console.warn(['Popper: Detected CSS transitions on at least one of the following', 'CSS properties: \"transform\", \"top\", \"right\", \"bottom\", \"left\".', '\\n\\n', 'Disable the \"computeStyles\" modifier\\'s `adaptive` option to allow', 'for smooth transitions, or remove these properties from the CSS', 'transition declaration on the popper element if only transitioning', 'opacity or background-color for example.', '\\n\\n', 'We recommend using the popper element as a wrapper around an inner', 'element that can have any CSS property transitioned for animations.'].join(' '));\n    }\n  }\n\n  var commonStyles = {\n    placement: getBasePlacement(state.placement),\n    variation: getVariation(state.placement),\n    popper: state.elements.popper,\n    popperRect: state.rects.popper,\n    gpuAcceleration: gpuAcceleration,\n    isFixed: state.options.strategy === 'fixed'\n  };\n\n  if (state.modifiersData.popperOffsets != null) {\n    state.styles.popper = Object.assign({}, state.styles.popper, mapToStyles(Object.assign({}, commonStyles, {\n      offsets: state.modifiersData.popperOffsets,\n      position: state.options.strategy,\n      adaptive: adaptive,\n      roundOffsets: roundOffsets\n    })));\n  }\n\n  if (state.modifiersData.arrow != null) {\n    state.styles.arrow = Object.assign({}, state.styles.arrow, mapToStyles(Object.assign({}, commonStyles, {\n      offsets: state.modifiersData.arrow,\n      position: 'absolute',\n      adaptive: false,\n      roundOffsets: roundOffsets\n    })));\n  }\n\n  state.attributes.popper = Object.assign({}, state.attributes.popper, {\n    'data-popper-placement': state.placement\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'computeStyles',\n  enabled: true,\n  phase: 'beforeWrite',\n  fn: computeStyles,\n  data: {}\n};","import getWindow from \"../dom-utils/getWindow.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar passive = {\n  passive: true\n};\n\nfunction effect(_ref) {\n  var state = _ref.state,\n      instance = _ref.instance,\n      options = _ref.options;\n  var _options$scroll = options.scroll,\n      scroll = _options$scroll === void 0 ? true : _options$scroll,\n      _options$resize = options.resize,\n      resize = _options$resize === void 0 ? true : _options$resize;\n  var window = getWindow(state.elements.popper);\n  var scrollParents = [].concat(state.scrollParents.reference, state.scrollParents.popper);\n\n  if (scroll) {\n    scrollParents.forEach(function (scrollParent) {\n      scrollParent.addEventListener('scroll', instance.update, passive);\n    });\n  }\n\n  if (resize) {\n    window.addEventListener('resize', instance.update, passive);\n  }\n\n  return function () {\n    if (scroll) {\n      scrollParents.forEach(function (scrollParent) {\n        scrollParent.removeEventListener('scroll', instance.update, passive);\n      });\n    }\n\n    if (resize) {\n      window.removeEventListener('resize', instance.update, passive);\n    }\n  };\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'eventListeners',\n  enabled: true,\n  phase: 'write',\n  fn: function fn() {},\n  effect: effect,\n  data: {}\n};","var hash = {\n  left: 'right',\n  right: 'left',\n  bottom: 'top',\n  top: 'bottom'\n};\nexport default function getOppositePlacement(placement) {\n  return placement.replace(/left|right|bottom|top/g, function (matched) {\n    return hash[matched];\n  });\n}","var hash = {\n  start: 'end',\n  end: 'start'\n};\nexport default function getOppositeVariationPlacement(placement) {\n  return placement.replace(/start|end/g, function (matched) {\n    return hash[matched];\n  });\n}","import getWindow from \"./getWindow.js\";\nexport default function getWindowScroll(node) {\n  var win = getWindow(node);\n  var scrollLeft = win.pageXOffset;\n  var scrollTop = win.pageYOffset;\n  return {\n    scrollLeft: scrollLeft,\n    scrollTop: scrollTop\n  };\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getWindowScroll from \"./getWindowScroll.js\";\nexport default function getWindowScrollBarX(element) {\n  // If <html> has a CSS width greater than the viewport, then this will be\n  // incorrect for RTL.\n  // Popper 1 is broken in this case and never had a bug report so let's assume\n  // it's not an issue. I don't think anyone ever specifies width on <html>\n  // anyway.\n  // Browsers where the left scrollbar doesn't cause an issue report `0` for\n  // this (e.g. Edge 2019, IE11, Safari)\n  return getBoundingClientRect(getDocumentElement(element)).left + getWindowScroll(element).scrollLeft;\n}","import getComputedStyle from \"./getComputedStyle.js\";\nexport default function isScrollParent(element) {\n  // Firefox wants us to check `-x` and `-y` variations as well\n  var _getComputedStyle = getComputedStyle(element),\n      overflow = _getComputedStyle.overflow,\n      overflowX = _getComputedStyle.overflowX,\n      overflowY = _getComputedStyle.overflowY;\n\n  return /auto|scroll|overlay|hidden/.test(overflow + overflowY + overflowX);\n}","import getParentNode from \"./getParentNode.js\";\nimport isScrollParent from \"./isScrollParent.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nexport default function getScrollParent(node) {\n  if (['html', 'body', '#document'].indexOf(getNodeName(node)) >= 0) {\n    // $FlowFixMe[incompatible-return]: assume body is always available\n    return node.ownerDocument.body;\n  }\n\n  if (isHTMLElement(node) && isScrollParent(node)) {\n    return node;\n  }\n\n  return getScrollParent(getParentNode(node));\n}","import getScrollParent from \"./getScrollParent.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport getWindow from \"./getWindow.js\";\nimport isScrollParent from \"./isScrollParent.js\";\n/*\ngiven a DOM element, return the list of all scroll parents, up the list of ancesors\nuntil we get to the top window object. This list is what we attach scroll listeners\nto, because if any of these parent elements scroll, we'll need to re-calculate the\nreference element's position.\n*/\n\nexport default function listScrollParents(element, list) {\n  var _element$ownerDocumen;\n\n  if (list === void 0) {\n    list = [];\n  }\n\n  var scrollParent = getScrollParent(element);\n  var isBody = scrollParent === ((_element$ownerDocumen = element.ownerDocument) == null ? void 0 : _element$ownerDocumen.body);\n  var win = getWindow(scrollParent);\n  var target = isBody ? [win].concat(win.visualViewport || [], isScrollParent(scrollParent) ? scrollParent : []) : scrollParent;\n  var updatedList = list.concat(target);\n  return isBody ? updatedList : // $FlowFixMe[incompatible-call]: isBody tells us target will be an HTMLElement here\n  updatedList.concat(listScrollParents(getParentNode(target)));\n}","export default function rectToClientRect(rect) {\n  return Object.assign({}, rect, {\n    left: rect.x,\n    top: rect.y,\n    right: rect.x + rect.width,\n    bottom: rect.y + rect.height\n  });\n}","import { viewport } from \"../enums.js\";\nimport getViewportRect from \"./getViewportRect.js\";\nimport getDocumentRect from \"./getDocumentRect.js\";\nimport listScrollParents from \"./listScrollParents.js\";\nimport getOffsetParent from \"./getOffsetParent.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport { isElement, isHTMLElement } from \"./instanceOf.js\";\nimport getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport contains from \"./contains.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport rectToClientRect from \"../utils/rectToClientRect.js\";\nimport { max, min } from \"../utils/math.js\";\n\nfunction getInnerBoundingClientRect(element, strategy) {\n  var rect = getBoundingClientRect(element, false, strategy === 'fixed');\n  rect.top = rect.top + element.clientTop;\n  rect.left = rect.left + element.clientLeft;\n  rect.bottom = rect.top + element.clientHeight;\n  rect.right = rect.left + element.clientWidth;\n  rect.width = element.clientWidth;\n  rect.height = element.clientHeight;\n  rect.x = rect.left;\n  rect.y = rect.top;\n  return rect;\n}\n\nfunction getClientRectFromMixedType(element, clippingParent, strategy) {\n  return clippingParent === viewport ? rectToClientRect(getViewportRect(element, strategy)) : isElement(clippingParent) ? getInnerBoundingClientRect(clippingParent, strategy) : rectToClientRect(getDocumentRect(getDocumentElement(element)));\n} // A \"clipping parent\" is an overflowable container with the characteristic of\n// clipping (or hiding) overflowing elements with a position different from\n// `initial`\n\n\nfunction getClippingParents(element) {\n  var clippingParents = listScrollParents(getParentNode(element));\n  var canEscapeClipping = ['absolute', 'fixed'].indexOf(getComputedStyle(element).position) >= 0;\n  var clipperElement = canEscapeClipping && isHTMLElement(element) ? getOffsetParent(element) : element;\n\n  if (!isElement(clipperElement)) {\n    return [];\n  } // $FlowFixMe[incompatible-return]: https://github.com/facebook/flow/issues/1414\n\n\n  return clippingParents.filter(function (clippingParent) {\n    return isElement(clippingParent) && contains(clippingParent, clipperElement) && getNodeName(clippingParent) !== 'body';\n  });\n} // Gets the maximum area that the element is visible in due to any number of\n// clipping parents\n\n\nexport default function getClippingRect(element, boundary, rootBoundary, strategy) {\n  var mainClippingParents = boundary === 'clippingParents' ? getClippingParents(element) : [].concat(boundary);\n  var clippingParents = [].concat(mainClippingParents, [rootBoundary]);\n  var firstClippingParent = clippingParents[0];\n  var clippingRect = clippingParents.reduce(function (accRect, clippingParent) {\n    var rect = getClientRectFromMixedType(element, clippingParent, strategy);\n    accRect.top = max(rect.top, accRect.top);\n    accRect.right = min(rect.right, accRect.right);\n    accRect.bottom = min(rect.bottom, accRect.bottom);\n    accRect.left = max(rect.left, accRect.left);\n    return accRect;\n  }, getClientRectFromMixedType(element, firstClippingParent, strategy));\n  clippingRect.width = clippingRect.right - clippingRect.left;\n  clippingRect.height = clippingRect.bottom - clippingRect.top;\n  clippingRect.x = clippingRect.left;\n  clippingRect.y = clippingRect.top;\n  return clippingRect;\n}","import getWindow from \"./getWindow.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport isLayoutViewport from \"./isLayoutViewport.js\";\nexport default function getViewportRect(element, strategy) {\n  var win = getWindow(element);\n  var html = getDocumentElement(element);\n  var visualViewport = win.visualViewport;\n  var width = html.clientWidth;\n  var height = html.clientHeight;\n  var x = 0;\n  var y = 0;\n\n  if (visualViewport) {\n    width = visualViewport.width;\n    height = visualViewport.height;\n    var layoutViewport = isLayoutViewport();\n\n    if (layoutViewport || !layoutViewport && strategy === 'fixed') {\n      x = visualViewport.offsetLeft;\n      y = visualViewport.offsetTop;\n    }\n  }\n\n  return {\n    width: width,\n    height: height,\n    x: x + getWindowScrollBarX(element),\n    y: y\n  };\n}","import getDocumentElement from \"./getDocumentElement.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport getWindowScroll from \"./getWindowScroll.js\";\nimport { max } from \"../utils/math.js\"; // Gets the entire size of the scrollable document area, even extending outside\n// of the `<html>` and `<body>` rect bounds if horizontally scrollable\n\nexport default function getDocumentRect(element) {\n  var _element$ownerDocumen;\n\n  var html = getDocumentElement(element);\n  var winScroll = getWindowScroll(element);\n  var body = (_element$ownerDocumen = element.ownerDocument) == null ? void 0 : _element$ownerDocumen.body;\n  var width = max(html.scrollWidth, html.clientWidth, body ? body.scrollWidth : 0, body ? body.clientWidth : 0);\n  var height = max(html.scrollHeight, html.clientHeight, body ? body.scrollHeight : 0, body ? body.clientHeight : 0);\n  var x = -winScroll.scrollLeft + getWindowScrollBarX(element);\n  var y = -winScroll.scrollTop;\n\n  if (getComputedStyle(body || html).direction === 'rtl') {\n    x += max(html.clientWidth, body ? body.clientWidth : 0) - width;\n  }\n\n  return {\n    width: width,\n    height: height,\n    x: x,\n    y: y\n  };\n}","import getBasePlacement from \"./getBasePlacement.js\";\nimport getVariation from \"./getVariation.js\";\nimport getMainAxisFromPlacement from \"./getMainAxisFromPlacement.js\";\nimport { top, right, bottom, left, start, end } from \"../enums.js\";\nexport default function computeOffsets(_ref) {\n  var reference = _ref.reference,\n      element = _ref.element,\n      placement = _ref.placement;\n  var basePlacement = placement ? getBasePlacement(placement) : null;\n  var variation = placement ? getVariation(placement) : null;\n  var commonX = reference.x + reference.width / 2 - element.width / 2;\n  var commonY = reference.y + reference.height / 2 - element.height / 2;\n  var offsets;\n\n  switch (basePlacement) {\n    case top:\n      offsets = {\n        x: commonX,\n        y: reference.y - element.height\n      };\n      break;\n\n    case bottom:\n      offsets = {\n        x: commonX,\n        y: reference.y + reference.height\n      };\n      break;\n\n    case right:\n      offsets = {\n        x: reference.x + reference.width,\n        y: commonY\n      };\n      break;\n\n    case left:\n      offsets = {\n        x: reference.x - element.width,\n        y: commonY\n      };\n      break;\n\n    default:\n      offsets = {\n        x: reference.x,\n        y: reference.y\n      };\n  }\n\n  var mainAxis = basePlacement ? getMainAxisFromPlacement(basePlacement) : null;\n\n  if (mainAxis != null) {\n    var len = mainAxis === 'y' ? 'height' : 'width';\n\n    switch (variation) {\n      case start:\n        offsets[mainAxis] = offsets[mainAxis] - (reference[len] / 2 - element[len] / 2);\n        break;\n\n      case end:\n        offsets[mainAxis] = offsets[mainAxis] + (reference[len] / 2 - element[len] / 2);\n        break;\n\n      default:\n    }\n  }\n\n  return offsets;\n}","import getClippingRect from \"../dom-utils/getClippingRect.js\";\nimport getDocumentElement from \"../dom-utils/getDocumentElement.js\";\nimport getBoundingClientRect from \"../dom-utils/getBoundingClientRect.js\";\nimport computeOffsets from \"./computeOffsets.js\";\nimport rectToClientRect from \"./rectToClientRect.js\";\nimport { clippingParents, reference, popper, bottom, top, right, basePlacements, viewport } from \"../enums.js\";\nimport { isElement } from \"../dom-utils/instanceOf.js\";\nimport mergePaddingObject from \"./mergePaddingObject.js\";\nimport expandToHashMap from \"./expandToHashMap.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport default function detectOverflow(state, options) {\n  if (options === void 0) {\n    options = {};\n  }\n\n  var _options = options,\n      _options$placement = _options.placement,\n      placement = _options$placement === void 0 ? state.placement : _options$placement,\n      _options$strategy = _options.strategy,\n      strategy = _options$strategy === void 0 ? state.strategy : _options$strategy,\n      _options$boundary = _options.boundary,\n      boundary = _options$boundary === void 0 ? clippingParents : _options$boundary,\n      _options$rootBoundary = _options.rootBoundary,\n      rootBoundary = _options$rootBoundary === void 0 ? viewport : _options$rootBoundary,\n      _options$elementConte = _options.elementContext,\n      elementContext = _options$elementConte === void 0 ? popper : _options$elementConte,\n      _options$altBoundary = _options.altBoundary,\n      altBoundary = _options$altBoundary === void 0 ? false : _options$altBoundary,\n      _options$padding = _options.padding,\n      padding = _options$padding === void 0 ? 0 : _options$padding;\n  var paddingObject = mergePaddingObject(typeof padding !== 'number' ? padding : expandToHashMap(padding, basePlacements));\n  var altContext = elementContext === popper ? reference : popper;\n  var popperRect = state.rects.popper;\n  var element = state.elements[altBoundary ? altContext : elementContext];\n  var clippingClientRect = getClippingRect(isElement(element) ? element : element.contextElement || getDocumentElement(state.elements.popper), boundary, rootBoundary, strategy);\n  var referenceClientRect = getBoundingClientRect(state.elements.reference);\n  var popperOffsets = computeOffsets({\n    reference: referenceClientRect,\n    element: popperRect,\n    strategy: 'absolute',\n    placement: placement\n  });\n  var popperClientRect = rectToClientRect(Object.assign({}, popperRect, popperOffsets));\n  var elementClientRect = elementContext === popper ? popperClientRect : referenceClientRect; // positive = overflowing the clipping rect\n  // 0 or negative = within the clipping rect\n\n  var overflowOffsets = {\n    top: clippingClientRect.top - elementClientRect.top + paddingObject.top,\n    bottom: elementClientRect.bottom - clippingClientRect.bottom + paddingObject.bottom,\n    left: clippingClientRect.left - elementClientRect.left + paddingObject.left,\n    right: elementClientRect.right - clippingClientRect.right + paddingObject.right\n  };\n  var offsetData = state.modifiersData.offset; // Offsets can be applied only to the popper element\n\n  if (elementContext === popper && offsetData) {\n    var offset = offsetData[placement];\n    Object.keys(overflowOffsets).forEach(function (key) {\n      var multiply = [right, bottom].indexOf(key) >= 0 ? 1 : -1;\n      var axis = [top, bottom].indexOf(key) >= 0 ? 'y' : 'x';\n      overflowOffsets[key] += offset[axis] * multiply;\n    });\n  }\n\n  return overflowOffsets;\n}","import getOppositePlacement from \"../utils/getOppositePlacement.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getOppositeVariationPlacement from \"../utils/getOppositeVariationPlacement.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\nimport computeAutoPlacement from \"../utils/computeAutoPlacement.js\";\nimport { bottom, top, start, right, left, auto } from \"../enums.js\";\nimport getVariation from \"../utils/getVariation.js\"; // eslint-disable-next-line import/no-unused-modules\n\nfunction getExpandedFallbackPlacements(placement) {\n  if (getBasePlacement(placement) === auto) {\n    return [];\n  }\n\n  var oppositePlacement = getOppositePlacement(placement);\n  return [getOppositeVariationPlacement(placement), oppositePlacement, getOppositeVariationPlacement(oppositePlacement)];\n}\n\nfunction flip(_ref) {\n  var state = _ref.state,\n      options = _ref.options,\n      name = _ref.name;\n\n  if (state.modifiersData[name]._skip) {\n    return;\n  }\n\n  var _options$mainAxis = options.mainAxis,\n      checkMainAxis = _options$mainAxis === void 0 ? true : _options$mainAxis,\n      _options$altAxis = options.altAxis,\n      checkAltAxis = _options$altAxis === void 0 ? true : _options$altAxis,\n      specifiedFallbackPlacements = options.fallbackPlacements,\n      padding = options.padding,\n      boundary = options.boundary,\n      rootBoundary = options.rootBoundary,\n      altBoundary = options.altBoundary,\n      _options$flipVariatio = options.flipVariations,\n      flipVariations = _options$flipVariatio === void 0 ? true : _options$flipVariatio,\n      allowedAutoPlacements = options.allowedAutoPlacements;\n  var preferredPlacement = state.options.placement;\n  var basePlacement = getBasePlacement(preferredPlacement);\n  var isBasePlacement = basePlacement === preferredPlacement;\n  var fallbackPlacements = specifiedFallbackPlacements || (isBasePlacement || !flipVariations ? [getOppositePlacement(preferredPlacement)] : getExpandedFallbackPlacements(preferredPlacement));\n  var placements = [preferredPlacement].concat(fallbackPlacements).reduce(function (acc, placement) {\n    return acc.concat(getBasePlacement(placement) === auto ? computeAutoPlacement(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      padding: padding,\n      flipVariations: flipVariations,\n      allowedAutoPlacements: allowedAutoPlacements\n    }) : placement);\n  }, []);\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var checksMap = new Map();\n  var makeFallbackChecks = true;\n  var firstFittingPlacement = placements[0];\n\n  for (var i = 0; i < placements.length; i++) {\n    var placement = placements[i];\n\n    var _basePlacement = getBasePlacement(placement);\n\n    var isStartVariation = getVariation(placement) === start;\n    var isVertical = [top, bottom].indexOf(_basePlacement) >= 0;\n    var len = isVertical ? 'width' : 'height';\n    var overflow = detectOverflow(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      altBoundary: altBoundary,\n      padding: padding\n    });\n    var mainVariationSide = isVertical ? isStartVariation ? right : left : isStartVariation ? bottom : top;\n\n    if (referenceRect[len] > popperRect[len]) {\n      mainVariationSide = getOppositePlacement(mainVariationSide);\n    }\n\n    var altVariationSide = getOppositePlacement(mainVariationSide);\n    var checks = [];\n\n    if (checkMainAxis) {\n      checks.push(overflow[_basePlacement] <= 0);\n    }\n\n    if (checkAltAxis) {\n      checks.push(overflow[mainVariationSide] <= 0, overflow[altVariationSide] <= 0);\n    }\n\n    if (checks.every(function (check) {\n      return check;\n    })) {\n      firstFittingPlacement = placement;\n      makeFallbackChecks = false;\n      break;\n    }\n\n    checksMap.set(placement, checks);\n  }\n\n  if (makeFallbackChecks) {\n    // `2` may be desired in some cases – research later\n    var numberOfChecks = flipVariations ? 3 : 1;\n\n    var _loop = function _loop(_i) {\n      var fittingPlacement = placements.find(function (placement) {\n        var checks = checksMap.get(placement);\n\n        if (checks) {\n          return checks.slice(0, _i).every(function (check) {\n            return check;\n          });\n        }\n      });\n\n      if (fittingPlacement) {\n        firstFittingPlacement = fittingPlacement;\n        return \"break\";\n      }\n    };\n\n    for (var _i = numberOfChecks; _i > 0; _i--) {\n      var _ret = _loop(_i);\n\n      if (_ret === \"break\") break;\n    }\n  }\n\n  if (state.placement !== firstFittingPlacement) {\n    state.modifiersData[name]._skip = true;\n    state.placement = firstFittingPlacement;\n    state.reset = true;\n  }\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'flip',\n  enabled: true,\n  phase: 'main',\n  fn: flip,\n  requiresIfExists: ['offset'],\n  data: {\n    _skip: false\n  }\n};","import getVariation from \"./getVariation.js\";\nimport { variationPlacements, basePlacements, placements as allPlacements } from \"../enums.js\";\nimport detectOverflow from \"./detectOverflow.js\";\nimport getBasePlacement from \"./getBasePlacement.js\";\nexport default function computeAutoPlacement(state, options) {\n  if (options === void 0) {\n    options = {};\n  }\n\n  var _options = options,\n      placement = _options.placement,\n      boundary = _options.boundary,\n      rootBoundary = _options.rootBoundary,\n      padding = _options.padding,\n      flipVariations = _options.flipVariations,\n      _options$allowedAutoP = _options.allowedAutoPlacements,\n      allowedAutoPlacements = _options$allowedAutoP === void 0 ? allPlacements : _options$allowedAutoP;\n  var variation = getVariation(placement);\n  var placements = variation ? flipVariations ? variationPlacements : variationPlacements.filter(function (placement) {\n    return getVariation(placement) === variation;\n  }) : basePlacements;\n  var allowedPlacements = placements.filter(function (placement) {\n    return allowedAutoPlacements.indexOf(placement) >= 0;\n  });\n\n  if (allowedPlacements.length === 0) {\n    allowedPlacements = placements;\n\n    if (process.env.NODE_ENV !== \"production\") {\n      console.error(['Popper: The `allowedAutoPlacements` option did not allow any', 'placements. Ensure the `placement` option matches the variation', 'of the allowed placements.', 'For example, \"auto\" cannot be used to allow \"bottom-start\".', 'Use \"auto-start\" instead.'].join(' '));\n    }\n  } // $FlowFixMe[incompatible-type]: Flow seems to have problems with two array unions...\n\n\n  var overflows = allowedPlacements.reduce(function (acc, placement) {\n    acc[placement] = detectOverflow(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      padding: padding\n    })[getBasePlacement(placement)];\n    return acc;\n  }, {});\n  return Object.keys(overflows).sort(function (a, b) {\n    return overflows[a] - overflows[b];\n  });\n}","import { top, bottom, left, right } from \"../enums.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\n\nfunction getSideOffsets(overflow, rect, preventedOffsets) {\n  if (preventedOffsets === void 0) {\n    preventedOffsets = {\n      x: 0,\n      y: 0\n    };\n  }\n\n  return {\n    top: overflow.top - rect.height - preventedOffsets.y,\n    right: overflow.right - rect.width + preventedOffsets.x,\n    bottom: overflow.bottom - rect.height + preventedOffsets.y,\n    left: overflow.left - rect.width - preventedOffsets.x\n  };\n}\n\nfunction isAnySideFullyClipped(overflow) {\n  return [top, right, bottom, left].some(function (side) {\n    return overflow[side] >= 0;\n  });\n}\n\nfunction hide(_ref) {\n  var state = _ref.state,\n      name = _ref.name;\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var preventedOffsets = state.modifiersData.preventOverflow;\n  var referenceOverflow = detectOverflow(state, {\n    elementContext: 'reference'\n  });\n  var popperAltOverflow = detectOverflow(state, {\n    altBoundary: true\n  });\n  var referenceClippingOffsets = getSideOffsets(referenceOverflow, referenceRect);\n  var popperEscapeOffsets = getSideOffsets(popperAltOverflow, popperRect, preventedOffsets);\n  var isReferenceHidden = isAnySideFullyClipped(referenceClippingOffsets);\n  var hasPopperEscaped = isAnySideFullyClipped(popperEscapeOffsets);\n  state.modifiersData[name] = {\n    referenceClippingOffsets: referenceClippingOffsets,\n    popperEscapeOffsets: popperEscapeOffsets,\n    isReferenceHidden: isReferenceHidden,\n    hasPopperEscaped: hasPopperEscaped\n  };\n  state.attributes.popper = Object.assign({}, state.attributes.popper, {\n    'data-popper-reference-hidden': isReferenceHidden,\n    'data-popper-escaped': hasPopperEscaped\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'hide',\n  enabled: true,\n  phase: 'main',\n  requiresIfExists: ['preventOverflow'],\n  fn: hide\n};","import getBasePlacement from \"../utils/getBasePlacement.js\";\nimport { top, left, right, placements } from \"../enums.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport function distanceAndSkiddingToXY(placement, rects, offset) {\n  var basePlacement = getBasePlacement(placement);\n  var invertDistance = [left, top].indexOf(basePlacement) >= 0 ? -1 : 1;\n\n  var _ref = typeof offset === 'function' ? offset(Object.assign({}, rects, {\n    placement: placement\n  })) : offset,\n      skidding = _ref[0],\n      distance = _ref[1];\n\n  skidding = skidding || 0;\n  distance = (distance || 0) * invertDistance;\n  return [left, right].indexOf(basePlacement) >= 0 ? {\n    x: distance,\n    y: skidding\n  } : {\n    x: skidding,\n    y: distance\n  };\n}\n\nfunction offset(_ref2) {\n  var state = _ref2.state,\n      options = _ref2.options,\n      name = _ref2.name;\n  var _options$offset = options.offset,\n      offset = _options$offset === void 0 ? [0, 0] : _options$offset;\n  var data = placements.reduce(function (acc, placement) {\n    acc[placement] = distanceAndSkiddingToXY(placement, state.rects, offset);\n    return acc;\n  }, {});\n  var _data$state$placement = data[state.placement],\n      x = _data$state$placement.x,\n      y = _data$state$placement.y;\n\n  if (state.modifiersData.popperOffsets != null) {\n    state.modifiersData.popperOffsets.x += x;\n    state.modifiersData.popperOffsets.y += y;\n  }\n\n  state.modifiersData[name] = data;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'offset',\n  enabled: true,\n  phase: 'main',\n  requires: ['popperOffsets'],\n  fn: offset\n};","import computeOffsets from \"../utils/computeOffsets.js\";\n\nfunction popperOffsets(_ref) {\n  var state = _ref.state,\n      name = _ref.name;\n  // Offsets are the actual position the popper needs to have to be\n  // properly positioned near its reference element\n  // This is the most basic placement, and will be adjusted by\n  // the modifiers in the next step\n  state.modifiersData[name] = computeOffsets({\n    reference: state.rects.reference,\n    element: state.rects.popper,\n    strategy: 'absolute',\n    placement: state.placement\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'popperOffsets',\n  enabled: true,\n  phase: 'read',\n  fn: popperOffsets,\n  data: {}\n};","import { top, left, right, bottom, start } from \"../enums.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getMainAxisFromPlacement from \"../utils/getMainAxisFromPlacement.js\";\nimport getAltAxis from \"../utils/getAltAxis.js\";\nimport { within, withinMaxClamp } from \"../utils/within.js\";\nimport getLayoutRect from \"../dom-utils/getLayoutRect.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\nimport getVariation from \"../utils/getVariation.js\";\nimport getFreshSideObject from \"../utils/getFreshSideObject.js\";\nimport { min as mathMin, max as mathMax } from \"../utils/math.js\";\n\nfunction preventOverflow(_ref) {\n  var state = _ref.state,\n      options = _ref.options,\n      name = _ref.name;\n  var _options$mainAxis = options.mainAxis,\n      checkMainAxis = _options$mainAxis === void 0 ? true : _options$mainAxis,\n      _options$altAxis = options.altAxis,\n      checkAltAxis = _options$altAxis === void 0 ? false : _options$altAxis,\n      boundary = options.boundary,\n      rootBoundary = options.rootBoundary,\n      altBoundary = options.altBoundary,\n      padding = options.padding,\n      _options$tether = options.tether,\n      tether = _options$tether === void 0 ? true : _options$tether,\n      _options$tetherOffset = options.tetherOffset,\n      tetherOffset = _options$tetherOffset === void 0 ? 0 : _options$tetherOffset;\n  var overflow = detectOverflow(state, {\n    boundary: boundary,\n    rootBoundary: rootBoundary,\n    padding: padding,\n    altBoundary: altBoundary\n  });\n  var basePlacement = getBasePlacement(state.placement);\n  var variation = getVariation(state.placement);\n  var isBasePlacement = !variation;\n  var mainAxis = getMainAxisFromPlacement(basePlacement);\n  var altAxis = getAltAxis(mainAxis);\n  var popperOffsets = state.modifiersData.popperOffsets;\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var tetherOffsetValue = typeof tetherOffset === 'function' ? tetherOffset(Object.assign({}, state.rects, {\n    placement: state.placement\n  })) : tetherOffset;\n  var normalizedTetherOffsetValue = typeof tetherOffsetValue === 'number' ? {\n    mainAxis: tetherOffsetValue,\n    altAxis: tetherOffsetValue\n  } : Object.assign({\n    mainAxis: 0,\n    altAxis: 0\n  }, tetherOffsetValue);\n  var offsetModifierState = state.modifiersData.offset ? state.modifiersData.offset[state.placement] : null;\n  var data = {\n    x: 0,\n    y: 0\n  };\n\n  if (!popperOffsets) {\n    return;\n  }\n\n  if (checkMainAxis) {\n    var _offsetModifierState$;\n\n    var mainSide = mainAxis === 'y' ? top : left;\n    var altSide = mainAxis === 'y' ? bottom : right;\n    var len = mainAxis === 'y' ? 'height' : 'width';\n    var offset = popperOffsets[mainAxis];\n    var min = offset + overflow[mainSide];\n    var max = offset - overflow[altSide];\n    var additive = tether ? -popperRect[len] / 2 : 0;\n    var minLen = variation === start ? referenceRect[len] : popperRect[len];\n    var maxLen = variation === start ? -popperRect[len] : -referenceRect[len]; // We need to include the arrow in the calculation so the arrow doesn't go\n    // outside the reference bounds\n\n    var arrowElement = state.elements.arrow;\n    var arrowRect = tether && arrowElement ? getLayoutRect(arrowElement) : {\n      width: 0,\n      height: 0\n    };\n    var arrowPaddingObject = state.modifiersData['arrow#persistent'] ? state.modifiersData['arrow#persistent'].padding : getFreshSideObject();\n    var arrowPaddingMin = arrowPaddingObject[mainSide];\n    var arrowPaddingMax = arrowPaddingObject[altSide]; // If the reference length is smaller than the arrow length, we don't want\n    // to include its full size in the calculation. If the reference is small\n    // and near the edge of a boundary, the popper can overflow even if the\n    // reference is not overflowing as well (e.g. virtual elements with no\n    // width or height)\n\n    var arrowLen = within(0, referenceRect[len], arrowRect[len]);\n    var minOffset = isBasePlacement ? referenceRect[len] / 2 - additive - arrowLen - arrowPaddingMin - normalizedTetherOffsetValue.mainAxis : minLen - arrowLen - arrowPaddingMin - normalizedTetherOffsetValue.mainAxis;\n    var maxOffset = isBasePlacement ? -referenceRect[len] / 2 + additive + arrowLen + arrowPaddingMax + normalizedTetherOffsetValue.mainAxis : maxLen + arrowLen + arrowPaddingMax + normalizedTetherOffsetValue.mainAxis;\n    var arrowOffsetParent = state.elements.arrow && getOffsetParent(state.elements.arrow);\n    var clientOffset = arrowOffsetParent ? mainAxis === 'y' ? arrowOffsetParent.clientTop || 0 : arrowOffsetParent.clientLeft || 0 : 0;\n    var offsetModifierValue = (_offsetModifierState$ = offsetModifierState == null ? void 0 : offsetModifierState[mainAxis]) != null ? _offsetModifierState$ : 0;\n    var tetherMin = offset + minOffset - offsetModifierValue - clientOffset;\n    var tetherMax = offset + maxOffset - offsetModifierValue;\n    var preventedOffset = within(tether ? mathMin(min, tetherMin) : min, offset, tether ? mathMax(max, tetherMax) : max);\n    popperOffsets[mainAxis] = preventedOffset;\n    data[mainAxis] = preventedOffset - offset;\n  }\n\n  if (checkAltAxis) {\n    var _offsetModifierState$2;\n\n    var _mainSide = mainAxis === 'x' ? top : left;\n\n    var _altSide = mainAxis === 'x' ? bottom : right;\n\n    var _offset = popperOffsets[altAxis];\n\n    var _len = altAxis === 'y' ? 'height' : 'width';\n\n    var _min = _offset + overflow[_mainSide];\n\n    var _max = _offset - overflow[_altSide];\n\n    var isOriginSide = [top, left].indexOf(basePlacement) !== -1;\n\n    var _offsetModifierValue = (_offsetModifierState$2 = offsetModifierState == null ? void 0 : offsetModifierState[altAxis]) != null ? _offsetModifierState$2 : 0;\n\n    var _tetherMin = isOriginSide ? _min : _offset - referenceRect[_len] - popperRect[_len] - _offsetModifierValue + normalizedTetherOffsetValue.altAxis;\n\n    var _tetherMax = isOriginSide ? _offset + referenceRect[_len] + popperRect[_len] - _offsetModifierValue - normalizedTetherOffsetValue.altAxis : _max;\n\n    var _preventedOffset = tether && isOriginSide ? withinMaxClamp(_tetherMin, _offset, _tetherMax) : within(tether ? _tetherMin : _min, _offset, tether ? _tetherMax : _max);\n\n    popperOffsets[altAxis] = _preventedOffset;\n    data[altAxis] = _preventedOffset - _offset;\n  }\n\n  state.modifiersData[name] = data;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'preventOverflow',\n  enabled: true,\n  phase: 'main',\n  fn: preventOverflow,\n  requiresIfExists: ['offset']\n};","export default function getAltAxis(axis) {\n  return axis === 'x' ? 'y' : 'x';\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getNodeScroll from \"./getNodeScroll.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport isScrollParent from \"./isScrollParent.js\";\nimport { round } from \"../utils/math.js\";\n\nfunction isElementScaled(element) {\n  var rect = element.getBoundingClientRect();\n  var scaleX = round(rect.width) / element.offsetWidth || 1;\n  var scaleY = round(rect.height) / element.offsetHeight || 1;\n  return scaleX !== 1 || scaleY !== 1;\n} // Returns the composite rect of an element relative to its offsetParent.\n// Composite means it takes into account transforms as well as layout.\n\n\nexport default function getCompositeRect(elementOrVirtualElement, offsetParent, isFixed) {\n  if (isFixed === void 0) {\n    isFixed = false;\n  }\n\n  var isOffsetParentAnElement = isHTMLElement(offsetParent);\n  var offsetParentIsScaled = isHTMLElement(offsetParent) && isElementScaled(offsetParent);\n  var documentElement = getDocumentElement(offsetParent);\n  var rect = getBoundingClientRect(elementOrVirtualElement, offsetParentIsScaled, isFixed);\n  var scroll = {\n    scrollLeft: 0,\n    scrollTop: 0\n  };\n  var offsets = {\n    x: 0,\n    y: 0\n  };\n\n  if (isOffsetParentAnElement || !isOffsetParentAnElement && !isFixed) {\n    if (getNodeName(offsetParent) !== 'body' || // https://github.com/popperjs/popper-core/issues/1078\n    isScrollParent(documentElement)) {\n      scroll = getNodeScroll(offsetParent);\n    }\n\n    if (isHTMLElement(offsetParent)) {\n      offsets = getBoundingClientRect(offsetParent, true);\n      offsets.x += offsetParent.clientLeft;\n      offsets.y += offsetParent.clientTop;\n    } else if (documentElement) {\n      offsets.x = getWindowScrollBarX(documentElement);\n    }\n  }\n\n  return {\n    x: rect.left + scroll.scrollLeft - offsets.x,\n    y: rect.top + scroll.scrollTop - offsets.y,\n    width: rect.width,\n    height: rect.height\n  };\n}","import getWindowScroll from \"./getWindowScroll.js\";\nimport getWindow from \"./getWindow.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nimport getHTMLElementScroll from \"./getHTMLElementScroll.js\";\nexport default function getNodeScroll(node) {\n  if (node === getWindow(node) || !isHTMLElement(node)) {\n    return getWindowScroll(node);\n  } else {\n    return getHTMLElementScroll(node);\n  }\n}","export default function getHTMLElementScroll(element) {\n  return {\n    scrollLeft: element.scrollLeft,\n    scrollTop: element.scrollTop\n  };\n}","import { modifierPhases } from \"../enums.js\"; // source: https://stackoverflow.com/questions/49875255\n\nfunction order(modifiers) {\n  var map = new Map();\n  var visited = new Set();\n  var result = [];\n  modifiers.forEach(function (modifier) {\n    map.set(modifier.name, modifier);\n  }); // On visiting object, check for its dependencies and visit them recursively\n\n  function sort(modifier) {\n    visited.add(modifier.name);\n    var requires = [].concat(modifier.requires || [], modifier.requiresIfExists || []);\n    requires.forEach(function (dep) {\n      if (!visited.has(dep)) {\n        var depModifier = map.get(dep);\n\n        if (depModifier) {\n          sort(depModifier);\n        }\n      }\n    });\n    result.push(modifier);\n  }\n\n  modifiers.forEach(function (modifier) {\n    if (!visited.has(modifier.name)) {\n      // check for visited object\n      sort(modifier);\n    }\n  });\n  return result;\n}\n\nexport default function orderModifiers(modifiers) {\n  // order based on dependencies\n  var orderedModifiers = order(modifiers); // order based on phase\n\n  return modifierPhases.reduce(function (acc, phase) {\n    return acc.concat(orderedModifiers.filter(function (modifier) {\n      return modifier.phase === phase;\n    }));\n  }, []);\n}","import getCompositeRect from \"./dom-utils/getCompositeRect.js\";\nimport getLayoutRect from \"./dom-utils/getLayoutRect.js\";\nimport listScrollParents from \"./dom-utils/listScrollParents.js\";\nimport getOffsetParent from \"./dom-utils/getOffsetParent.js\";\nimport getComputedStyle from \"./dom-utils/getComputedStyle.js\";\nimport orderModifiers from \"./utils/orderModifiers.js\";\nimport debounce from \"./utils/debounce.js\";\nimport validateModifiers from \"./utils/validateModifiers.js\";\nimport uniqueBy from \"./utils/uniqueBy.js\";\nimport getBasePlacement from \"./utils/getBasePlacement.js\";\nimport mergeByName from \"./utils/mergeByName.js\";\nimport detectOverflow from \"./utils/detectOverflow.js\";\nimport { isElement } from \"./dom-utils/instanceOf.js\";\nimport { auto } from \"./enums.js\";\nvar INVALID_ELEMENT_ERROR = 'Popper: Invalid reference or popper argument provided. They must be either a DOM element or virtual element.';\nvar INFINITE_LOOP_ERROR = 'Popper: An infinite loop in the modifiers cycle has been detected! The cycle has been interrupted to prevent a browser crash.';\nvar DEFAULT_OPTIONS = {\n  placement: 'bottom',\n  modifiers: [],\n  strategy: 'absolute'\n};\n\nfunction areValidElements() {\n  for (var _len = arguments.length, args = new Array(_len), _key = 0; _key < _len; _key++) {\n    args[_key] = arguments[_key];\n  }\n\n  return !args.some(function (element) {\n    return !(element && typeof element.getBoundingClientRect === 'function');\n  });\n}\n\nexport function popperGenerator(generatorOptions) {\n  if (generatorOptions === void 0) {\n    generatorOptions = {};\n  }\n\n  var _generatorOptions = generatorOptions,\n      _generatorOptions$def = _generatorOptions.defaultModifiers,\n      defaultModifiers = _generatorOptions$def === void 0 ? [] : _generatorOptions$def,\n      _generatorOptions$def2 = _generatorOptions.defaultOptions,\n      defaultOptions = _generatorOptions$def2 === void 0 ? DEFAULT_OPTIONS : _generatorOptions$def2;\n  return function createPopper(reference, popper, options) {\n    if (options === void 0) {\n      options = defaultOptions;\n    }\n\n    var state = {\n      placement: 'bottom',\n      orderedModifiers: [],\n      options: Object.assign({}, DEFAULT_OPTIONS, defaultOptions),\n      modifiersData: {},\n      elements: {\n        reference: reference,\n        popper: popper\n      },\n      attributes: {},\n      styles: {}\n    };\n    var effectCleanupFns = [];\n    var isDestroyed = false;\n    var instance = {\n      state: state,\n      setOptions: function setOptions(setOptionsAction) {\n        var options = typeof setOptionsAction === 'function' ? setOptionsAction(state.options) : setOptionsAction;\n        cleanupModifierEffects();\n        state.options = Object.assign({}, defaultOptions, state.options, options);\n        state.scrollParents = {\n          reference: isElement(reference) ? listScrollParents(reference) : reference.contextElement ? listScrollParents(reference.contextElement) : [],\n          popper: listScrollParents(popper)\n        }; // Orders the modifiers based on their dependencies and `phase`\n        // properties\n\n        var orderedModifiers = orderModifiers(mergeByName([].concat(defaultModifiers, state.options.modifiers))); // Strip out disabled modifiers\n\n        state.orderedModifiers = orderedModifiers.filter(function (m) {\n          return m.enabled;\n        }); // Validate the provided modifiers so that the consumer will get warned\n        // if one of the modifiers is invalid for any reason\n\n        if (process.env.NODE_ENV !== \"production\") {\n          var modifiers = uniqueBy([].concat(orderedModifiers, state.options.modifiers), function (_ref) {\n            var name = _ref.name;\n            return name;\n          });\n          validateModifiers(modifiers);\n\n          if (getBasePlacement(state.options.placement) === auto) {\n            var flipModifier = state.orderedModifiers.find(function (_ref2) {\n              var name = _ref2.name;\n              return name === 'flip';\n            });\n\n            if (!flipModifier) {\n              console.error(['Popper: \"auto\" placements require the \"flip\" modifier be', 'present and enabled to work.'].join(' '));\n            }\n          }\n\n          var _getComputedStyle = getComputedStyle(popper),\n              marginTop = _getComputedStyle.marginTop,\n              marginRight = _getComputedStyle.marginRight,\n              marginBottom = _getComputedStyle.marginBottom,\n              marginLeft = _getComputedStyle.marginLeft; // We no longer take into account `margins` on the popper, and it can\n          // cause bugs with positioning, so we'll warn the consumer\n\n\n          if ([marginTop, marginRight, marginBottom, marginLeft].some(function (margin) {\n            return parseFloat(margin);\n          })) {\n            console.warn(['Popper: CSS \"margin\" styles cannot be used to apply padding', 'between the popper and its reference element or boundary.', 'To replicate margin, use the `offset` modifier, as well as', 'the `padding` option in the `preventOverflow` and `flip`', 'modifiers.'].join(' '));\n          }\n        }\n\n        runModifierEffects();\n        return instance.update();\n      },\n      // Sync update – it will always be executed, even if not necessary. This\n      // is useful for low frequency updates where sync behavior simplifies the\n      // logic.\n      // For high frequency updates (e.g. `resize` and `scroll` events), always\n      // prefer the async Popper#update method\n      forceUpdate: function forceUpdate() {\n        if (isDestroyed) {\n          return;\n        }\n\n        var _state$elements = state.elements,\n            reference = _state$elements.reference,\n            popper = _state$elements.popper; // Don't proceed if `reference` or `popper` are not valid elements\n        // anymore\n\n        if (!areValidElements(reference, popper)) {\n          if (process.env.NODE_ENV !== \"production\") {\n            console.error(INVALID_ELEMENT_ERROR);\n          }\n\n          return;\n        } // Store the reference and popper rects to be read by modifiers\n\n\n        state.rects = {\n          reference: getCompositeRect(reference, getOffsetParent(popper), state.options.strategy === 'fixed'),\n          popper: getLayoutRect(popper)\n        }; // Modifiers have the ability to reset the current update cycle. The\n        // most common use case for this is the `flip` modifier changing the\n        // placement, which then needs to re-run all the modifiers, because the\n        // logic was previously ran for the previous placement and is therefore\n        // stale/incorrect\n\n        state.reset = false;\n        state.placement = state.options.placement; // On each update cycle, the `modifiersData` property for each modifier\n        // is filled with the initial data specified by the modifier. This means\n        // it doesn't persist and is fresh on each update.\n        // To ensure persistent data, use `${name}#persistent`\n\n        state.orderedModifiers.forEach(function (modifier) {\n          return state.modifiersData[modifier.name] = Object.assign({}, modifier.data);\n        });\n        var __debug_loops__ = 0;\n\n        for (var index = 0; index < state.orderedModifiers.length; index++) {\n          if (process.env.NODE_ENV !== \"production\") {\n            __debug_loops__ += 1;\n\n            if (__debug_loops__ > 100) {\n              console.error(INFINITE_LOOP_ERROR);\n              break;\n            }\n          }\n\n          if (state.reset === true) {\n            state.reset = false;\n            index = -1;\n            continue;\n          }\n\n          var _state$orderedModifie = state.orderedModifiers[index],\n              fn = _state$orderedModifie.fn,\n              _state$orderedModifie2 = _state$orderedModifie.options,\n              _options = _state$orderedModifie2 === void 0 ? {} : _state$orderedModifie2,\n              name = _state$orderedModifie.name;\n\n          if (typeof fn === 'function') {\n            state = fn({\n              state: state,\n              options: _options,\n              name: name,\n              instance: instance\n            }) || state;\n          }\n        }\n      },\n      // Async and optimistically optimized update – it will not be executed if\n      // not necessary (debounced to run at most once-per-tick)\n      update: debounce(function () {\n        return new Promise(function (resolve) {\n          instance.forceUpdate();\n          resolve(state);\n        });\n      }),\n      destroy: function destroy() {\n        cleanupModifierEffects();\n        isDestroyed = true;\n      }\n    };\n\n    if (!areValidElements(reference, popper)) {\n      if (process.env.NODE_ENV !== \"production\") {\n        console.error(INVALID_ELEMENT_ERROR);\n      }\n\n      return instance;\n    }\n\n    instance.setOptions(options).then(function (state) {\n      if (!isDestroyed && options.onFirstUpdate) {\n        options.onFirstUpdate(state);\n      }\n    }); // Modifiers have the ability to execute arbitrary code before the first\n    // update cycle runs. They will be executed in the same order as the update\n    // cycle. This is useful when a modifier adds some persistent data that\n    // other modifiers need to use, but the modifier is run after the dependent\n    // one.\n\n    function runModifierEffects() {\n      state.orderedModifiers.forEach(function (_ref3) {\n        var name = _ref3.name,\n            _ref3$options = _ref3.options,\n            options = _ref3$options === void 0 ? {} : _ref3$options,\n            effect = _ref3.effect;\n\n        if (typeof effect === 'function') {\n          var cleanupFn = effect({\n            state: state,\n            name: name,\n            instance: instance,\n            options: options\n          });\n\n          var noopFn = function noopFn() {};\n\n          effectCleanupFns.push(cleanupFn || noopFn);\n        }\n      });\n    }\n\n    function cleanupModifierEffects() {\n      effectCleanupFns.forEach(function (fn) {\n        return fn();\n      });\n      effectCleanupFns = [];\n    }\n\n    return instance;\n  };\n}\nexport var createPopper = /*#__PURE__*/popperGenerator(); // eslint-disable-next-line import/no-unused-modules\n\nexport { detectOverflow };","export default function debounce(fn) {\n  var pending;\n  return function () {\n    if (!pending) {\n      pending = new Promise(function (resolve) {\n        Promise.resolve().then(function () {\n          pending = undefined;\n          resolve(fn());\n        });\n      });\n    }\n\n    return pending;\n  };\n}","export default function mergeByName(modifiers) {\n  var merged = modifiers.reduce(function (merged, current) {\n    var existing = merged[current.name];\n    merged[current.name] = existing ? Object.assign({}, existing, current, {\n      options: Object.assign({}, existing.options, current.options),\n      data: Object.assign({}, existing.data, current.data)\n    }) : current;\n    return merged;\n  }, {}); // IE11 does not support Object.values\n\n  return Object.keys(merged).map(function (key) {\n    return merged[key];\n  });\n}","import { popperGenerator, detectOverflow } from \"./createPopper.js\";\nimport eventListeners from \"./modifiers/eventListeners.js\";\nimport popperOffsets from \"./modifiers/popperOffsets.js\";\nimport computeStyles from \"./modifiers/computeStyles.js\";\nimport applyStyles from \"./modifiers/applyStyles.js\";\nimport offset from \"./modifiers/offset.js\";\nimport flip from \"./modifiers/flip.js\";\nimport preventOverflow from \"./modifiers/preventOverflow.js\";\nimport arrow from \"./modifiers/arrow.js\";\nimport hide from \"./modifiers/hide.js\";\nvar defaultModifiers = [eventListeners, popperOffsets, computeStyles, applyStyles, offset, flip, preventOverflow, arrow, hide];\nvar createPopper = /*#__PURE__*/popperGenerator({\n  defaultModifiers: defaultModifiers\n}); // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper, popperGenerator, defaultModifiers, detectOverflow }; // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper as createPopperLite } from \"./popper-lite.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport * from \"./modifiers/index.js\";","import { popperGenerator, detectOverflow } from \"./createPopper.js\";\nimport eventListeners from \"./modifiers/eventListeners.js\";\nimport popperOffsets from \"./modifiers/popperOffsets.js\";\nimport computeStyles from \"./modifiers/computeStyles.js\";\nimport applyStyles from \"./modifiers/applyStyles.js\";\nvar defaultModifiers = [eventListeners, popperOffsets, computeStyles, applyStyles];\nvar createPopper = /*#__PURE__*/popperGenerator({\n  defaultModifiers: defaultModifiers\n}); // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper, popperGenerator, defaultModifiers, detectOverflow };","/*!\n  * Bootstrap v5.2.3 (https://getbootstrap.com/)\n  * Copyright 2011-2022 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)\n  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n  */\nimport * as Popper from '@popperjs/core';\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/index.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\nconst MAX_UID = 1000000;\nconst MILLISECONDS_MULTIPLIER = 1000;\nconst TRANSITION_END = 'transitionend'; // Shout-out Angus Croll (https://goo.gl/pxwQGp)\n\nconst toType = object => {\n  if (object === null || object === undefined) {\n    return `${object}`;\n  }\n\n  return Object.prototype.toString.call(object).match(/\\s([a-z]+)/i)[1].toLowerCase();\n};\n/**\n * Public Util API\n */\n\n\nconst getUID = prefix => {\n  do {\n    prefix += Math.floor(Math.random() * MAX_UID);\n  } while (document.getElementById(prefix));\n\n  return prefix;\n};\n\nconst getSelector = element => {\n  let selector = element.getAttribute('data-bs-target');\n\n  if (!selector || selector === '#') {\n    let hrefAttribute = element.getAttribute('href'); // The only valid content that could double as a selector are IDs or classes,\n    // so everything starting with `#` or `.`. If a \"real\" URL is used as the selector,\n    // `document.querySelector` will rightfully complain it is invalid.\n    // See https://github.com/twbs/bootstrap/issues/32273\n\n    if (!hrefAttribute || !hrefAttribute.includes('#') && !hrefAttribute.startsWith('.')) {\n      return null;\n    } // Just in case some CMS puts out a full URL with the anchor appended\n\n\n    if (hrefAttribute.includes('#') && !hrefAttribute.startsWith('#')) {\n      hrefAttribute = `#${hrefAttribute.split('#')[1]}`;\n    }\n\n    selector = hrefAttribute && hrefAttribute !== '#' ? hrefAttribute.trim() : null;\n  }\n\n  return selector;\n};\n\nconst getSelectorFromElement = element => {\n  const selector = getSelector(element);\n\n  if (selector) {\n    return document.querySelector(selector) ? selector : null;\n  }\n\n  return null;\n};\n\nconst getElementFromSelector = element => {\n  const selector = getSelector(element);\n  return selector ? document.querySelector(selector) : null;\n};\n\nconst getTransitionDurationFromElement = element => {\n  if (!element) {\n    return 0;\n  } // Get transition-duration of the element\n\n\n  let {\n    transitionDuration,\n    transitionDelay\n  } = window.getComputedStyle(element);\n  const floatTransitionDuration = Number.parseFloat(transitionDuration);\n  const floatTransitionDelay = Number.parseFloat(transitionDelay); // Return 0 if element or transition duration is not found\n\n  if (!floatTransitionDuration && !floatTransitionDelay) {\n    return 0;\n  } // If multiple durations are defined, take the first\n\n\n  transitionDuration = transitionDuration.split(',')[0];\n  transitionDelay = transitionDelay.split(',')[0];\n  return (Number.parseFloat(transitionDuration) + Number.parseFloat(transitionDelay)) * MILLISECONDS_MULTIPLIER;\n};\n\nconst triggerTransitionEnd = element => {\n  element.dispatchEvent(new Event(TRANSITION_END));\n};\n\nconst isElement = object => {\n  if (!object || typeof object !== 'object') {\n    return false;\n  }\n\n  if (typeof object.jquery !== 'undefined') {\n    object = object[0];\n  }\n\n  return typeof object.nodeType !== 'undefined';\n};\n\nconst getElement = object => {\n  // it's a jQuery object or a node element\n  if (isElement(object)) {\n    return object.jquery ? object[0] : object;\n  }\n\n  if (typeof object === 'string' && object.length > 0) {\n    return document.querySelector(object);\n  }\n\n  return null;\n};\n\nconst isVisible = element => {\n  if (!isElement(element) || element.getClientRects().length === 0) {\n    return false;\n  }\n\n  const elementIsVisible = getComputedStyle(element).getPropertyValue('visibility') === 'visible'; // Handle `details` element as its content may falsie appear visible when it is closed\n\n  const closedDetails = element.closest('details:not([open])');\n\n  if (!closedDetails) {\n    return elementIsVisible;\n  }\n\n  if (closedDetails !== element) {\n    const summary = element.closest('summary');\n\n    if (summary && summary.parentNode !== closedDetails) {\n      return false;\n    }\n\n    if (summary === null) {\n      return false;\n    }\n  }\n\n  return elementIsVisible;\n};\n\nconst isDisabled = element => {\n  if (!element || element.nodeType !== Node.ELEMENT_NODE) {\n    return true;\n  }\n\n  if (element.classList.contains('disabled')) {\n    return true;\n  }\n\n  if (typeof element.disabled !== 'undefined') {\n    return element.disabled;\n  }\n\n  return element.hasAttribute('disabled') && element.getAttribute('disabled') !== 'false';\n};\n\nconst findShadowRoot = element => {\n  if (!document.documentElement.attachShadow) {\n    return null;\n  } // Can find the shadow root otherwise it'll return the document\n\n\n  if (typeof element.getRootNode === 'function') {\n    const root = element.getRootNode();\n    return root instanceof ShadowRoot ? root : null;\n  }\n\n  if (element instanceof ShadowRoot) {\n    return element;\n  } // when we don't find a shadow root\n\n\n  if (!element.parentNode) {\n    return null;\n  }\n\n  return findShadowRoot(element.parentNode);\n};\n\nconst noop = () => {};\n/**\n * Trick to restart an element's animation\n *\n * @param {HTMLElement} element\n * @return void\n *\n * @see https://www.charistheo.io/blog/2021/02/restart-a-css-animation-with-javascript/#restarting-a-css-animation\n */\n\n\nconst reflow = element => {\n  element.offsetHeight; // eslint-disable-line no-unused-expressions\n};\n\nconst getjQuery = () => {\n  if (window.jQuery && !document.body.hasAttribute('data-bs-no-jquery')) {\n    return window.jQuery;\n  }\n\n  return null;\n};\n\nconst DOMContentLoadedCallbacks = [];\n\nconst onDOMContentLoaded = callback => {\n  if (document.readyState === 'loading') {\n    // add listener on the first call when the document is in loading state\n    if (!DOMContentLoadedCallbacks.length) {\n      document.addEventListener('DOMContentLoaded', () => {\n        for (const callback of DOMContentLoadedCallbacks) {\n          callback();\n        }\n      });\n    }\n\n    DOMContentLoadedCallbacks.push(callback);\n  } else {\n    callback();\n  }\n};\n\nconst isRTL = () => document.documentElement.dir === 'rtl';\n\nconst defineJQueryPlugin = plugin => {\n  onDOMContentLoaded(() => {\n    const $ = getjQuery();\n    /* istanbul ignore if */\n\n    if ($) {\n      const name = plugin.NAME;\n      const JQUERY_NO_CONFLICT = $.fn[name];\n      $.fn[name] = plugin.jQueryInterface;\n      $.fn[name].Constructor = plugin;\n\n      $.fn[name].noConflict = () => {\n        $.fn[name] = JQUERY_NO_CONFLICT;\n        return plugin.jQueryInterface;\n      };\n    }\n  });\n};\n\nconst execute = callback => {\n  if (typeof callback === 'function') {\n    callback();\n  }\n};\n\nconst executeAfterTransition = (callback, transitionElement, waitForTransition = true) => {\n  if (!waitForTransition) {\n    execute(callback);\n    return;\n  }\n\n  const durationPadding = 5;\n  const emulatedDuration = getTransitionDurationFromElement(transitionElement) + durationPadding;\n  let called = false;\n\n  const handler = ({\n    target\n  }) => {\n    if (target !== transitionElement) {\n      return;\n    }\n\n    called = true;\n    transitionElement.removeEventListener(TRANSITION_END, handler);\n    execute(callback);\n  };\n\n  transitionElement.addEventListener(TRANSITION_END, handler);\n  setTimeout(() => {\n    if (!called) {\n      triggerTransitionEnd(transitionElement);\n    }\n  }, emulatedDuration);\n};\n/**\n * Return the previous/next element of a list.\n *\n * @param {array} list    The list of elements\n * @param activeElement   The active element\n * @param shouldGetNext   Choose to get next or previous element\n * @param isCycleAllowed\n * @return {Element|elem} The proper element\n */\n\n\nconst getNextActiveElement = (list, activeElement, shouldGetNext, isCycleAllowed) => {\n  const listLength = list.length;\n  let index = list.indexOf(activeElement); // if the element does not exist in the list return an element\n  // depending on the direction and if cycle is allowed\n\n  if (index === -1) {\n    return !shouldGetNext && isCycleAllowed ? list[listLength - 1] : list[0];\n  }\n\n  index += shouldGetNext ? 1 : -1;\n\n  if (isCycleAllowed) {\n    index = (index + listLength) % listLength;\n  }\n\n  return list[Math.max(0, Math.min(index, listLength - 1))];\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): dom/event-handler.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst namespaceRegex = /[^.]*(?=\\..*)\\.|.*/;\nconst stripNameRegex = /\\..*/;\nconst stripUidRegex = /::\\d+$/;\nconst eventRegistry = {}; // Events storage\n\nlet uidEvent = 1;\nconst customEvents = {\n  mouseenter: 'mouseover',\n  mouseleave: 'mouseout'\n};\nconst nativeEvents = new Set(['click', 'dblclick', 'mouseup', 'mousedown', 'contextmenu', 'mousewheel', 'DOMMouseScroll', 'mouseover', 'mouseout', 'mousemove', 'selectstart', 'selectend', 'keydown', 'keypress', 'keyup', 'orientationchange', 'touchstart', 'touchmove', 'touchend', 'touchcancel', 'pointerdown', 'pointermove', 'pointerup', 'pointerleave', 'pointercancel', 'gesturestart', 'gesturechange', 'gestureend', 'focus', 'blur', 'change', 'reset', 'select', 'submit', 'focusin', 'focusout', 'load', 'unload', 'beforeunload', 'resize', 'move', 'DOMContentLoaded', 'readystatechange', 'error', 'abort', 'scroll']);\n/**\n * Private methods\n */\n\nfunction makeEventUid(element, uid) {\n  return uid && `${uid}::${uidEvent++}` || element.uidEvent || uidEvent++;\n}\n\nfunction getElementEvents(element) {\n  const uid = makeEventUid(element);\n  element.uidEvent = uid;\n  eventRegistry[uid] = eventRegistry[uid] || {};\n  return eventRegistry[uid];\n}\n\nfunction bootstrapHandler(element, fn) {\n  return function handler(event) {\n    hydrateObj(event, {\n      delegateTarget: element\n    });\n\n    if (handler.oneOff) {\n      EventHandler.off(element, event.type, fn);\n    }\n\n    return fn.apply(element, [event]);\n  };\n}\n\nfunction bootstrapDelegationHandler(element, selector, fn) {\n  return function handler(event) {\n    const domElements = element.querySelectorAll(selector);\n\n    for (let {\n      target\n    } = event; target && target !== this; target = target.parentNode) {\n      for (const domElement of domElements) {\n        if (domElement !== target) {\n          continue;\n        }\n\n        hydrateObj(event, {\n          delegateTarget: target\n        });\n\n        if (handler.oneOff) {\n          EventHandler.off(element, event.type, selector, fn);\n        }\n\n        return fn.apply(target, [event]);\n      }\n    }\n  };\n}\n\nfunction findHandler(events, callable, delegationSelector = null) {\n  return Object.values(events).find(event => event.callable === callable && event.delegationSelector === delegationSelector);\n}\n\nfunction normalizeParameters(originalTypeEvent, handler, delegationFunction) {\n  const isDelegated = typeof handler === 'string'; // todo: tooltip passes `false` instead of selector, so we need to check\n\n  const callable = isDelegated ? delegationFunction : handler || delegationFunction;\n  let typeEvent = getTypeEvent(originalTypeEvent);\n\n  if (!nativeEvents.has(typeEvent)) {\n    typeEvent = originalTypeEvent;\n  }\n\n  return [isDelegated, callable, typeEvent];\n}\n\nfunction addHandler(element, originalTypeEvent, handler, delegationFunction, oneOff) {\n  if (typeof originalTypeEvent !== 'string' || !element) {\n    return;\n  }\n\n  let [isDelegated, callable, typeEvent] = normalizeParameters(originalTypeEvent, handler, delegationFunction); // in case of mouseenter or mouseleave wrap the handler within a function that checks for its DOM position\n  // this prevents the handler from being dispatched the same way as mouseover or mouseout does\n\n  if (originalTypeEvent in customEvents) {\n    const wrapFunction = fn => {\n      return function (event) {\n        if (!event.relatedTarget || event.relatedTarget !== event.delegateTarget && !event.delegateTarget.contains(event.relatedTarget)) {\n          return fn.call(this, event);\n        }\n      };\n    };\n\n    callable = wrapFunction(callable);\n  }\n\n  const events = getElementEvents(element);\n  const handlers = events[typeEvent] || (events[typeEvent] = {});\n  const previousFunction = findHandler(handlers, callable, isDelegated ? handler : null);\n\n  if (previousFunction) {\n    previousFunction.oneOff = previousFunction.oneOff && oneOff;\n    return;\n  }\n\n  const uid = makeEventUid(callable, originalTypeEvent.replace(namespaceRegex, ''));\n  const fn = isDelegated ? bootstrapDelegationHandler(element, handler, callable) : bootstrapHandler(element, callable);\n  fn.delegationSelector = isDelegated ? handler : null;\n  fn.callable = callable;\n  fn.oneOff = oneOff;\n  fn.uidEvent = uid;\n  handlers[uid] = fn;\n  element.addEventListener(typeEvent, fn, isDelegated);\n}\n\nfunction removeHandler(element, events, typeEvent, handler, delegationSelector) {\n  const fn = findHandler(events[typeEvent], handler, delegationSelector);\n\n  if (!fn) {\n    return;\n  }\n\n  element.removeEventListener(typeEvent, fn, Boolean(delegationSelector));\n  delete events[typeEvent][fn.uidEvent];\n}\n\nfunction removeNamespacedHandlers(element, events, typeEvent, namespace) {\n  const storeElementEvent = events[typeEvent] || {};\n\n  for (const handlerKey of Object.keys(storeElementEvent)) {\n    if (handlerKey.includes(namespace)) {\n      const event = storeElementEvent[handlerKey];\n      removeHandler(element, events, typeEvent, event.callable, event.delegationSelector);\n    }\n  }\n}\n\nfunction getTypeEvent(event) {\n  // allow to get the native events from namespaced events ('click.bs.button' --> 'click')\n  event = event.replace(stripNameRegex, '');\n  return customEvents[event] || event;\n}\n\nconst EventHandler = {\n  on(element, event, handler, delegationFunction) {\n    addHandler(element, event, handler, delegationFunction, false);\n  },\n\n  one(element, event, handler, delegationFunction) {\n    addHandler(element, event, handler, delegationFunction, true);\n  },\n\n  off(element, originalTypeEvent, handler, delegationFunction) {\n    if (typeof originalTypeEvent !== 'string' || !element) {\n      return;\n    }\n\n    const [isDelegated, callable, typeEvent] = normalizeParameters(originalTypeEvent, handler, delegationFunction);\n    const inNamespace = typeEvent !== originalTypeEvent;\n    const events = getElementEvents(element);\n    const storeElementEvent = events[typeEvent] || {};\n    const isNamespace = originalTypeEvent.startsWith('.');\n\n    if (typeof callable !== 'undefined') {\n      // Simplest case: handler is passed, remove that listener ONLY.\n      if (!Object.keys(storeElementEvent).length) {\n        return;\n      }\n\n      removeHandler(element, events, typeEvent, callable, isDelegated ? handler : null);\n      return;\n    }\n\n    if (isNamespace) {\n      for (const elementEvent of Object.keys(events)) {\n        removeNamespacedHandlers(element, events, elementEvent, originalTypeEvent.slice(1));\n      }\n    }\n\n    for (const keyHandlers of Object.keys(storeElementEvent)) {\n      const handlerKey = keyHandlers.replace(stripUidRegex, '');\n\n      if (!inNamespace || originalTypeEvent.includes(handlerKey)) {\n        const event = storeElementEvent[keyHandlers];\n        removeHandler(element, events, typeEvent, event.callable, event.delegationSelector);\n      }\n    }\n  },\n\n  trigger(element, event, args) {\n    if (typeof event !== 'string' || !element) {\n      return null;\n    }\n\n    const $ = getjQuery();\n    const typeEvent = getTypeEvent(event);\n    const inNamespace = event !== typeEvent;\n    let jQueryEvent = null;\n    let bubbles = true;\n    let nativeDispatch = true;\n    let defaultPrevented = false;\n\n    if (inNamespace && $) {\n      jQueryEvent = $.Event(event, args);\n      $(element).trigger(jQueryEvent);\n      bubbles = !jQueryEvent.isPropagationStopped();\n      nativeDispatch = !jQueryEvent.isImmediatePropagationStopped();\n      defaultPrevented = jQueryEvent.isDefaultPrevented();\n    }\n\n    let evt = new Event(event, {\n      bubbles,\n      cancelable: true\n    });\n    evt = hydrateObj(evt, args);\n\n    if (defaultPrevented) {\n      evt.preventDefault();\n    }\n\n    if (nativeDispatch) {\n      element.dispatchEvent(evt);\n    }\n\n    if (evt.defaultPrevented && jQueryEvent) {\n      jQueryEvent.preventDefault();\n    }\n\n    return evt;\n  }\n\n};\n\nfunction hydrateObj(obj, meta) {\n  for (const [key, value] of Object.entries(meta || {})) {\n    try {\n      obj[key] = value;\n    } catch (_unused) {\n      Object.defineProperty(obj, key, {\n        configurable: true,\n\n        get() {\n          return value;\n        }\n\n      });\n    }\n  }\n\n  return obj;\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): dom/data.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n/**\n * Constants\n */\nconst elementMap = new Map();\nconst Data = {\n  set(element, key, instance) {\n    if (!elementMap.has(element)) {\n      elementMap.set(element, new Map());\n    }\n\n    const instanceMap = elementMap.get(element); // make it clear we only want one instance per element\n    // can be removed later when multiple key/instances are fine to be used\n\n    if (!instanceMap.has(key) && instanceMap.size !== 0) {\n      // eslint-disable-next-line no-console\n      console.error(`Bootstrap doesn't allow more than one instance per element. Bound instance: ${Array.from(instanceMap.keys())[0]}.`);\n      return;\n    }\n\n    instanceMap.set(key, instance);\n  },\n\n  get(element, key) {\n    if (elementMap.has(element)) {\n      return elementMap.get(element).get(key) || null;\n    }\n\n    return null;\n  },\n\n  remove(element, key) {\n    if (!elementMap.has(element)) {\n      return;\n    }\n\n    const instanceMap = elementMap.get(element);\n    instanceMap.delete(key); // free up element references if there are no instances left for an element\n\n    if (instanceMap.size === 0) {\n      elementMap.delete(element);\n    }\n  }\n\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): dom/manipulator.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\nfunction normalizeData(value) {\n  if (value === 'true') {\n    return true;\n  }\n\n  if (value === 'false') {\n    return false;\n  }\n\n  if (value === Number(value).toString()) {\n    return Number(value);\n  }\n\n  if (value === '' || value === 'null') {\n    return null;\n  }\n\n  if (typeof value !== 'string') {\n    return value;\n  }\n\n  try {\n    return JSON.parse(decodeURIComponent(value));\n  } catch (_unused) {\n    return value;\n  }\n}\n\nfunction normalizeDataKey(key) {\n  return key.replace(/[A-Z]/g, chr => `-${chr.toLowerCase()}`);\n}\n\nconst Manipulator = {\n  setDataAttribute(element, key, value) {\n    element.setAttribute(`data-bs-${normalizeDataKey(key)}`, value);\n  },\n\n  removeDataAttribute(element, key) {\n    element.removeAttribute(`data-bs-${normalizeDataKey(key)}`);\n  },\n\n  getDataAttributes(element) {\n    if (!element) {\n      return {};\n    }\n\n    const attributes = {};\n    const bsKeys = Object.keys(element.dataset).filter(key => key.startsWith('bs') && !key.startsWith('bsConfig'));\n\n    for (const key of bsKeys) {\n      let pureKey = key.replace(/^bs/, '');\n      pureKey = pureKey.charAt(0).toLowerCase() + pureKey.slice(1, pureKey.length);\n      attributes[pureKey] = normalizeData(element.dataset[key]);\n    }\n\n    return attributes;\n  },\n\n  getDataAttribute(element, key) {\n    return normalizeData(element.getAttribute(`data-bs-${normalizeDataKey(key)}`));\n  }\n\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/config.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Class definition\n */\n\nclass Config {\n  // Getters\n  static get Default() {\n    return {};\n  }\n\n  static get DefaultType() {\n    return {};\n  }\n\n  static get NAME() {\n    throw new Error('You have to implement the static method \"NAME\", for each component!');\n  }\n\n  _getConfig(config) {\n    config = this._mergeConfigObj(config);\n    config = this._configAfterMerge(config);\n\n    this._typeCheckConfig(config);\n\n    return config;\n  }\n\n  _configAfterMerge(config) {\n    return config;\n  }\n\n  _mergeConfigObj(config, element) {\n    const jsonConfig = isElement(element) ? Manipulator.getDataAttribute(element, 'config') : {}; // try to parse\n\n    return { ...this.constructor.Default,\n      ...(typeof jsonConfig === 'object' ? jsonConfig : {}),\n      ...(isElement(element) ? Manipulator.getDataAttributes(element) : {}),\n      ...(typeof config === 'object' ? config : {})\n    };\n  }\n\n  _typeCheckConfig(config, configTypes = this.constructor.DefaultType) {\n    for (const property of Object.keys(configTypes)) {\n      const expectedTypes = configTypes[property];\n      const value = config[property];\n      const valueType = isElement(value) ? 'element' : toType(value);\n\n      if (!new RegExp(expectedTypes).test(valueType)) {\n        throw new TypeError(`${this.constructor.NAME.toUpperCase()}: Option \"${property}\" provided type \"${valueType}\" but expected type \"${expectedTypes}\".`);\n      }\n    }\n  }\n\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): base-component.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst VERSION = '5.2.3';\n/**\n * Class definition\n */\n\nclass BaseComponent extends Config {\n  constructor(element, config) {\n    super();\n    element = getElement(element);\n\n    if (!element) {\n      return;\n    }\n\n    this._element = element;\n    this._config = this._getConfig(config);\n    Data.set(this._element, this.constructor.DATA_KEY, this);\n  } // Public\n\n\n  dispose() {\n    Data.remove(this._element, this.constructor.DATA_KEY);\n    EventHandler.off(this._element, this.constructor.EVENT_KEY);\n\n    for (const propertyName of Object.getOwnPropertyNames(this)) {\n      this[propertyName] = null;\n    }\n  }\n\n  _queueCallback(callback, element, isAnimated = true) {\n    executeAfterTransition(callback, element, isAnimated);\n  }\n\n  _getConfig(config) {\n    config = this._mergeConfigObj(config, this._element);\n    config = this._configAfterMerge(config);\n\n    this._typeCheckConfig(config);\n\n    return config;\n  } // Static\n\n\n  static getInstance(element) {\n    return Data.get(getElement(element), this.DATA_KEY);\n  }\n\n  static getOrCreateInstance(element, config = {}) {\n    return this.getInstance(element) || new this(element, typeof config === 'object' ? config : null);\n  }\n\n  static get VERSION() {\n    return VERSION;\n  }\n\n  static get DATA_KEY() {\n    return `bs.${this.NAME}`;\n  }\n\n  static get EVENT_KEY() {\n    return `.${this.DATA_KEY}`;\n  }\n\n  static eventName(name) {\n    return `${name}${this.EVENT_KEY}`;\n  }\n\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/component-functions.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst enableDismissTrigger = (component, method = 'hide') => {\n  const clickEvent = `click.dismiss${component.EVENT_KEY}`;\n  const name = component.NAME;\n  EventHandler.on(document, clickEvent, `[data-bs-dismiss=\"${name}\"]`, function (event) {\n    if (['A', 'AREA'].includes(this.tagName)) {\n      event.preventDefault();\n    }\n\n    if (isDisabled(this)) {\n      return;\n    }\n\n    const target = getElementFromSelector(this) || this.closest(`.${name}`);\n    const instance = component.getOrCreateInstance(target); // Method argument is left, for Alert and only, as it doesn't implement the 'hide' method\n\n    instance[method]();\n  });\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): alert.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$f = 'alert';\nconst DATA_KEY$a = 'bs.alert';\nconst EVENT_KEY$b = `.${DATA_KEY$a}`;\nconst EVENT_CLOSE = `close${EVENT_KEY$b}`;\nconst EVENT_CLOSED = `closed${EVENT_KEY$b}`;\nconst CLASS_NAME_FADE$5 = 'fade';\nconst CLASS_NAME_SHOW$8 = 'show';\n/**\n * Class definition\n */\n\nclass Alert extends BaseComponent {\n  // Getters\n  static get NAME() {\n    return NAME$f;\n  } // Public\n\n\n  close() {\n    const closeEvent = EventHandler.trigger(this._element, EVENT_CLOSE);\n\n    if (closeEvent.defaultPrevented) {\n      return;\n    }\n\n    this._element.classList.remove(CLASS_NAME_SHOW$8);\n\n    const isAnimated = this._element.classList.contains(CLASS_NAME_FADE$5);\n\n    this._queueCallback(() => this._destroyElement(), this._element, isAnimated);\n  } // Private\n\n\n  _destroyElement() {\n    this._element.remove();\n\n    EventHandler.trigger(this._element, EVENT_CLOSED);\n    this.dispose();\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Alert.getOrCreateInstance(this);\n\n      if (typeof config !== 'string') {\n        return;\n      }\n\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n\n      data[config](this);\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nenableDismissTrigger(Alert, 'close');\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Alert);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): button.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$e = 'button';\nconst DATA_KEY$9 = 'bs.button';\nconst EVENT_KEY$a = `.${DATA_KEY$9}`;\nconst DATA_API_KEY$6 = '.data-api';\nconst CLASS_NAME_ACTIVE$3 = 'active';\nconst SELECTOR_DATA_TOGGLE$5 = '[data-bs-toggle=\"button\"]';\nconst EVENT_CLICK_DATA_API$6 = `click${EVENT_KEY$a}${DATA_API_KEY$6}`;\n/**\n * Class definition\n */\n\nclass Button extends BaseComponent {\n  // Getters\n  static get NAME() {\n    return NAME$e;\n  } // Public\n\n\n  toggle() {\n    // Toggle class and sync the `aria-pressed` attribute with the return value of the `.toggle()` method\n    this._element.setAttribute('aria-pressed', this._element.classList.toggle(CLASS_NAME_ACTIVE$3));\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Button.getOrCreateInstance(this);\n\n      if (config === 'toggle') {\n        data[config]();\n      }\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$6, SELECTOR_DATA_TOGGLE$5, event => {\n  event.preventDefault();\n  const button = event.target.closest(SELECTOR_DATA_TOGGLE$5);\n  const data = Button.getOrCreateInstance(button);\n  data.toggle();\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Button);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): dom/selector-engine.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst SelectorEngine = {\n  find(selector, element = document.documentElement) {\n    return [].concat(...Element.prototype.querySelectorAll.call(element, selector));\n  },\n\n  findOne(selector, element = document.documentElement) {\n    return Element.prototype.querySelector.call(element, selector);\n  },\n\n  children(element, selector) {\n    return [].concat(...element.children).filter(child => child.matches(selector));\n  },\n\n  parents(element, selector) {\n    const parents = [];\n    let ancestor = element.parentNode.closest(selector);\n\n    while (ancestor) {\n      parents.push(ancestor);\n      ancestor = ancestor.parentNode.closest(selector);\n    }\n\n    return parents;\n  },\n\n  prev(element, selector) {\n    let previous = element.previousElementSibling;\n\n    while (previous) {\n      if (previous.matches(selector)) {\n        return [previous];\n      }\n\n      previous = previous.previousElementSibling;\n    }\n\n    return [];\n  },\n\n  // TODO: this is now unused; remove later along with prev()\n  next(element, selector) {\n    let next = element.nextElementSibling;\n\n    while (next) {\n      if (next.matches(selector)) {\n        return [next];\n      }\n\n      next = next.nextElementSibling;\n    }\n\n    return [];\n  },\n\n  focusableChildren(element) {\n    const focusables = ['a', 'button', 'input', 'textarea', 'select', 'details', '[tabindex]', '[contenteditable=\"true\"]'].map(selector => `${selector}:not([tabindex^=\"-\"])`).join(',');\n    return this.find(focusables, element).filter(el => !isDisabled(el) && isVisible(el));\n  }\n\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/swipe.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$d = 'swipe';\nconst EVENT_KEY$9 = '.bs.swipe';\nconst EVENT_TOUCHSTART = `touchstart${EVENT_KEY$9}`;\nconst EVENT_TOUCHMOVE = `touchmove${EVENT_KEY$9}`;\nconst EVENT_TOUCHEND = `touchend${EVENT_KEY$9}`;\nconst EVENT_POINTERDOWN = `pointerdown${EVENT_KEY$9}`;\nconst EVENT_POINTERUP = `pointerup${EVENT_KEY$9}`;\nconst POINTER_TYPE_TOUCH = 'touch';\nconst POINTER_TYPE_PEN = 'pen';\nconst CLASS_NAME_POINTER_EVENT = 'pointer-event';\nconst SWIPE_THRESHOLD = 40;\nconst Default$c = {\n  endCallback: null,\n  leftCallback: null,\n  rightCallback: null\n};\nconst DefaultType$c = {\n  endCallback: '(function|null)',\n  leftCallback: '(function|null)',\n  rightCallback: '(function|null)'\n};\n/**\n * Class definition\n */\n\nclass Swipe extends Config {\n  constructor(element, config) {\n    super();\n    this._element = element;\n\n    if (!element || !Swipe.isSupported()) {\n      return;\n    }\n\n    this._config = this._getConfig(config);\n    this._deltaX = 0;\n    this._supportPointerEvents = Boolean(window.PointerEvent);\n\n    this._initEvents();\n  } // Getters\n\n\n  static get Default() {\n    return Default$c;\n  }\n\n  static get DefaultType() {\n    return DefaultType$c;\n  }\n\n  static get NAME() {\n    return NAME$d;\n  } // Public\n\n\n  dispose() {\n    EventHandler.off(this._element, EVENT_KEY$9);\n  } // Private\n\n\n  _start(event) {\n    if (!this._supportPointerEvents) {\n      this._deltaX = event.touches[0].clientX;\n      return;\n    }\n\n    if (this._eventIsPointerPenTouch(event)) {\n      this._deltaX = event.clientX;\n    }\n  }\n\n  _end(event) {\n    if (this._eventIsPointerPenTouch(event)) {\n      this._deltaX = event.clientX - this._deltaX;\n    }\n\n    this._handleSwipe();\n\n    execute(this._config.endCallback);\n  }\n\n  _move(event) {\n    this._deltaX = event.touches && event.touches.length > 1 ? 0 : event.touches[0].clientX - this._deltaX;\n  }\n\n  _handleSwipe() {\n    const absDeltaX = Math.abs(this._deltaX);\n\n    if (absDeltaX <= SWIPE_THRESHOLD) {\n      return;\n    }\n\n    const direction = absDeltaX / this._deltaX;\n    this._deltaX = 0;\n\n    if (!direction) {\n      return;\n    }\n\n    execute(direction > 0 ? this._config.rightCallback : this._config.leftCallback);\n  }\n\n  _initEvents() {\n    if (this._supportPointerEvents) {\n      EventHandler.on(this._element, EVENT_POINTERDOWN, event => this._start(event));\n      EventHandler.on(this._element, EVENT_POINTERUP, event => this._end(event));\n\n      this._element.classList.add(CLASS_NAME_POINTER_EVENT);\n    } else {\n      EventHandler.on(this._element, EVENT_TOUCHSTART, event => this._start(event));\n      EventHandler.on(this._element, EVENT_TOUCHMOVE, event => this._move(event));\n      EventHandler.on(this._element, EVENT_TOUCHEND, event => this._end(event));\n    }\n  }\n\n  _eventIsPointerPenTouch(event) {\n    return this._supportPointerEvents && (event.pointerType === POINTER_TYPE_PEN || event.pointerType === POINTER_TYPE_TOUCH);\n  } // Static\n\n\n  static isSupported() {\n    return 'ontouchstart' in document.documentElement || navigator.maxTouchPoints > 0;\n  }\n\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): carousel.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$c = 'carousel';\nconst DATA_KEY$8 = 'bs.carousel';\nconst EVENT_KEY$8 = `.${DATA_KEY$8}`;\nconst DATA_API_KEY$5 = '.data-api';\nconst ARROW_LEFT_KEY$1 = 'ArrowLeft';\nconst ARROW_RIGHT_KEY$1 = 'ArrowRight';\nconst TOUCHEVENT_COMPAT_WAIT = 500; // Time for mouse compat events to fire after touch\n\nconst ORDER_NEXT = 'next';\nconst ORDER_PREV = 'prev';\nconst DIRECTION_LEFT = 'left';\nconst DIRECTION_RIGHT = 'right';\nconst EVENT_SLIDE = `slide${EVENT_KEY$8}`;\nconst EVENT_SLID = `slid${EVENT_KEY$8}`;\nconst EVENT_KEYDOWN$1 = `keydown${EVENT_KEY$8}`;\nconst EVENT_MOUSEENTER$1 = `mouseenter${EVENT_KEY$8}`;\nconst EVENT_MOUSELEAVE$1 = `mouseleave${EVENT_KEY$8}`;\nconst EVENT_DRAG_START = `dragstart${EVENT_KEY$8}`;\nconst EVENT_LOAD_DATA_API$3 = `load${EVENT_KEY$8}${DATA_API_KEY$5}`;\nconst EVENT_CLICK_DATA_API$5 = `click${EVENT_KEY$8}${DATA_API_KEY$5}`;\nconst CLASS_NAME_CAROUSEL = 'carousel';\nconst CLASS_NAME_ACTIVE$2 = 'active';\nconst CLASS_NAME_SLIDE = 'slide';\nconst CLASS_NAME_END = 'carousel-item-end';\nconst CLASS_NAME_START = 'carousel-item-start';\nconst CLASS_NAME_NEXT = 'carousel-item-next';\nconst CLASS_NAME_PREV = 'carousel-item-prev';\nconst SELECTOR_ACTIVE = '.active';\nconst SELECTOR_ITEM = '.carousel-item';\nconst SELECTOR_ACTIVE_ITEM = SELECTOR_ACTIVE + SELECTOR_ITEM;\nconst SELECTOR_ITEM_IMG = '.carousel-item img';\nconst SELECTOR_INDICATORS = '.carousel-indicators';\nconst SELECTOR_DATA_SLIDE = '[data-bs-slide], [data-bs-slide-to]';\nconst SELECTOR_DATA_RIDE = '[data-bs-ride=\"carousel\"]';\nconst KEY_TO_DIRECTION = {\n  [ARROW_LEFT_KEY$1]: DIRECTION_RIGHT,\n  [ARROW_RIGHT_KEY$1]: DIRECTION_LEFT\n};\nconst Default$b = {\n  interval: 5000,\n  keyboard: true,\n  pause: 'hover',\n  ride: false,\n  touch: true,\n  wrap: true\n};\nconst DefaultType$b = {\n  interval: '(number|boolean)',\n  // TODO:v6 remove boolean support\n  keyboard: 'boolean',\n  pause: '(string|boolean)',\n  ride: '(boolean|string)',\n  touch: 'boolean',\n  wrap: 'boolean'\n};\n/**\n * Class definition\n */\n\nclass Carousel extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._interval = null;\n    this._activeElement = null;\n    this._isSliding = false;\n    this.touchTimeout = null;\n    this._swipeHelper = null;\n    this._indicatorsElement = SelectorEngine.findOne(SELECTOR_INDICATORS, this._element);\n\n    this._addEventListeners();\n\n    if (this._config.ride === CLASS_NAME_CAROUSEL) {\n      this.cycle();\n    }\n  } // Getters\n\n\n  static get Default() {\n    return Default$b;\n  }\n\n  static get DefaultType() {\n    return DefaultType$b;\n  }\n\n  static get NAME() {\n    return NAME$c;\n  } // Public\n\n\n  next() {\n    this._slide(ORDER_NEXT);\n  }\n\n  nextWhenVisible() {\n    // FIXME TODO use `document.visibilityState`\n    // Don't call next when the page isn't visible\n    // or the carousel or its parent isn't visible\n    if (!document.hidden && isVisible(this._element)) {\n      this.next();\n    }\n  }\n\n  prev() {\n    this._slide(ORDER_PREV);\n  }\n\n  pause() {\n    if (this._isSliding) {\n      triggerTransitionEnd(this._element);\n    }\n\n    this._clearInterval();\n  }\n\n  cycle() {\n    this._clearInterval();\n\n    this._updateInterval();\n\n    this._interval = setInterval(() => this.nextWhenVisible(), this._config.interval);\n  }\n\n  _maybeEnableCycle() {\n    if (!this._config.ride) {\n      return;\n    }\n\n    if (this._isSliding) {\n      EventHandler.one(this._element, EVENT_SLID, () => this.cycle());\n      return;\n    }\n\n    this.cycle();\n  }\n\n  to(index) {\n    const items = this._getItems();\n\n    if (index > items.length - 1 || index < 0) {\n      return;\n    }\n\n    if (this._isSliding) {\n      EventHandler.one(this._element, EVENT_SLID, () => this.to(index));\n      return;\n    }\n\n    const activeIndex = this._getItemIndex(this._getActive());\n\n    if (activeIndex === index) {\n      return;\n    }\n\n    const order = index > activeIndex ? ORDER_NEXT : ORDER_PREV;\n\n    this._slide(order, items[index]);\n  }\n\n  dispose() {\n    if (this._swipeHelper) {\n      this._swipeHelper.dispose();\n    }\n\n    super.dispose();\n  } // Private\n\n\n  _configAfterMerge(config) {\n    config.defaultInterval = config.interval;\n    return config;\n  }\n\n  _addEventListeners() {\n    if (this._config.keyboard) {\n      EventHandler.on(this._element, EVENT_KEYDOWN$1, event => this._keydown(event));\n    }\n\n    if (this._config.pause === 'hover') {\n      EventHandler.on(this._element, EVENT_MOUSEENTER$1, () => this.pause());\n      EventHandler.on(this._element, EVENT_MOUSELEAVE$1, () => this._maybeEnableCycle());\n    }\n\n    if (this._config.touch && Swipe.isSupported()) {\n      this._addTouchEventListeners();\n    }\n  }\n\n  _addTouchEventListeners() {\n    for (const img of SelectorEngine.find(SELECTOR_ITEM_IMG, this._element)) {\n      EventHandler.on(img, EVENT_DRAG_START, event => event.preventDefault());\n    }\n\n    const endCallBack = () => {\n      if (this._config.pause !== 'hover') {\n        return;\n      } // If it's a touch-enabled device, mouseenter/leave are fired as\n      // part of the mouse compatibility events on first tap - the carousel\n      // would stop cycling until user tapped out of it;\n      // here, we listen for touchend, explicitly pause the carousel\n      // (as if it's the second time we tap on it, mouseenter compat event\n      // is NOT fired) and after a timeout (to allow for mouse compatibility\n      // events to fire) we explicitly restart cycling\n\n\n      this.pause();\n\n      if (this.touchTimeout) {\n        clearTimeout(this.touchTimeout);\n      }\n\n      this.touchTimeout = setTimeout(() => this._maybeEnableCycle(), TOUCHEVENT_COMPAT_WAIT + this._config.interval);\n    };\n\n    const swipeConfig = {\n      leftCallback: () => this._slide(this._directionToOrder(DIRECTION_LEFT)),\n      rightCallback: () => this._slide(this._directionToOrder(DIRECTION_RIGHT)),\n      endCallback: endCallBack\n    };\n    this._swipeHelper = new Swipe(this._element, swipeConfig);\n  }\n\n  _keydown(event) {\n    if (/input|textarea/i.test(event.target.tagName)) {\n      return;\n    }\n\n    const direction = KEY_TO_DIRECTION[event.key];\n\n    if (direction) {\n      event.preventDefault();\n\n      this._slide(this._directionToOrder(direction));\n    }\n  }\n\n  _getItemIndex(element) {\n    return this._getItems().indexOf(element);\n  }\n\n  _setActiveIndicatorElement(index) {\n    if (!this._indicatorsElement) {\n      return;\n    }\n\n    const activeIndicator = SelectorEngine.findOne(SELECTOR_ACTIVE, this._indicatorsElement);\n    activeIndicator.classList.remove(CLASS_NAME_ACTIVE$2);\n    activeIndicator.removeAttribute('aria-current');\n    const newActiveIndicator = SelectorEngine.findOne(`[data-bs-slide-to=\"${index}\"]`, this._indicatorsElement);\n\n    if (newActiveIndicator) {\n      newActiveIndicator.classList.add(CLASS_NAME_ACTIVE$2);\n      newActiveIndicator.setAttribute('aria-current', 'true');\n    }\n  }\n\n  _updateInterval() {\n    const element = this._activeElement || this._getActive();\n\n    if (!element) {\n      return;\n    }\n\n    const elementInterval = Number.parseInt(element.getAttribute('data-bs-interval'), 10);\n    this._config.interval = elementInterval || this._config.defaultInterval;\n  }\n\n  _slide(order, element = null) {\n    if (this._isSliding) {\n      return;\n    }\n\n    const activeElement = this._getActive();\n\n    const isNext = order === ORDER_NEXT;\n    const nextElement = element || getNextActiveElement(this._getItems(), activeElement, isNext, this._config.wrap);\n\n    if (nextElement === activeElement) {\n      return;\n    }\n\n    const nextElementIndex = this._getItemIndex(nextElement);\n\n    const triggerEvent = eventName => {\n      return EventHandler.trigger(this._element, eventName, {\n        relatedTarget: nextElement,\n        direction: this._orderToDirection(order),\n        from: this._getItemIndex(activeElement),\n        to: nextElementIndex\n      });\n    };\n\n    const slideEvent = triggerEvent(EVENT_SLIDE);\n\n    if (slideEvent.defaultPrevented) {\n      return;\n    }\n\n    if (!activeElement || !nextElement) {\n      // Some weirdness is happening, so we bail\n      // todo: change tests that use empty divs to avoid this check\n      return;\n    }\n\n    const isCycling = Boolean(this._interval);\n    this.pause();\n    this._isSliding = true;\n\n    this._setActiveIndicatorElement(nextElementIndex);\n\n    this._activeElement = nextElement;\n    const directionalClassName = isNext ? CLASS_NAME_START : CLASS_NAME_END;\n    const orderClassName = isNext ? CLASS_NAME_NEXT : CLASS_NAME_PREV;\n    nextElement.classList.add(orderClassName);\n    reflow(nextElement);\n    activeElement.classList.add(directionalClassName);\n    nextElement.classList.add(directionalClassName);\n\n    const completeCallBack = () => {\n      nextElement.classList.remove(directionalClassName, orderClassName);\n      nextElement.classList.add(CLASS_NAME_ACTIVE$2);\n      activeElement.classList.remove(CLASS_NAME_ACTIVE$2, orderClassName, directionalClassName);\n      this._isSliding = false;\n      triggerEvent(EVENT_SLID);\n    };\n\n    this._queueCallback(completeCallBack, activeElement, this._isAnimated());\n\n    if (isCycling) {\n      this.cycle();\n    }\n  }\n\n  _isAnimated() {\n    return this._element.classList.contains(CLASS_NAME_SLIDE);\n  }\n\n  _getActive() {\n    return SelectorEngine.findOne(SELECTOR_ACTIVE_ITEM, this._element);\n  }\n\n  _getItems() {\n    return SelectorEngine.find(SELECTOR_ITEM, this._element);\n  }\n\n  _clearInterval() {\n    if (this._interval) {\n      clearInterval(this._interval);\n      this._interval = null;\n    }\n  }\n\n  _directionToOrder(direction) {\n    if (isRTL()) {\n      return direction === DIRECTION_LEFT ? ORDER_PREV : ORDER_NEXT;\n    }\n\n    return direction === DIRECTION_LEFT ? ORDER_NEXT : ORDER_PREV;\n  }\n\n  _orderToDirection(order) {\n    if (isRTL()) {\n      return order === ORDER_PREV ? DIRECTION_LEFT : DIRECTION_RIGHT;\n    }\n\n    return order === ORDER_PREV ? DIRECTION_RIGHT : DIRECTION_LEFT;\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Carousel.getOrCreateInstance(this, config);\n\n      if (typeof config === 'number') {\n        data.to(config);\n        return;\n      }\n\n      if (typeof config === 'string') {\n        if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n\n        data[config]();\n      }\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$5, SELECTOR_DATA_SLIDE, function (event) {\n  const target = getElementFromSelector(this);\n\n  if (!target || !target.classList.contains(CLASS_NAME_CAROUSEL)) {\n    return;\n  }\n\n  event.preventDefault();\n  const carousel = Carousel.getOrCreateInstance(target);\n  const slideIndex = this.getAttribute('data-bs-slide-to');\n\n  if (slideIndex) {\n    carousel.to(slideIndex);\n\n    carousel._maybeEnableCycle();\n\n    return;\n  }\n\n  if (Manipulator.getDataAttribute(this, 'slide') === 'next') {\n    carousel.next();\n\n    carousel._maybeEnableCycle();\n\n    return;\n  }\n\n  carousel.prev();\n\n  carousel._maybeEnableCycle();\n});\nEventHandler.on(window, EVENT_LOAD_DATA_API$3, () => {\n  const carousels = SelectorEngine.find(SELECTOR_DATA_RIDE);\n\n  for (const carousel of carousels) {\n    Carousel.getOrCreateInstance(carousel);\n  }\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Carousel);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): collapse.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$b = 'collapse';\nconst DATA_KEY$7 = 'bs.collapse';\nconst EVENT_KEY$7 = `.${DATA_KEY$7}`;\nconst DATA_API_KEY$4 = '.data-api';\nconst EVENT_SHOW$6 = `show${EVENT_KEY$7}`;\nconst EVENT_SHOWN$6 = `shown${EVENT_KEY$7}`;\nconst EVENT_HIDE$6 = `hide${EVENT_KEY$7}`;\nconst EVENT_HIDDEN$6 = `hidden${EVENT_KEY$7}`;\nconst EVENT_CLICK_DATA_API$4 = `click${EVENT_KEY$7}${DATA_API_KEY$4}`;\nconst CLASS_NAME_SHOW$7 = 'show';\nconst CLASS_NAME_COLLAPSE = 'collapse';\nconst CLASS_NAME_COLLAPSING = 'collapsing';\nconst CLASS_NAME_COLLAPSED = 'collapsed';\nconst CLASS_NAME_DEEPER_CHILDREN = `:scope .${CLASS_NAME_COLLAPSE} .${CLASS_NAME_COLLAPSE}`;\nconst CLASS_NAME_HORIZONTAL = 'collapse-horizontal';\nconst WIDTH = 'width';\nconst HEIGHT = 'height';\nconst SELECTOR_ACTIVES = '.collapse.show, .collapse.collapsing';\nconst SELECTOR_DATA_TOGGLE$4 = '[data-bs-toggle=\"collapse\"]';\nconst Default$a = {\n  parent: null,\n  toggle: true\n};\nconst DefaultType$a = {\n  parent: '(null|element)',\n  toggle: 'boolean'\n};\n/**\n * Class definition\n */\n\nclass Collapse extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._isTransitioning = false;\n    this._triggerArray = [];\n    const toggleList = SelectorEngine.find(SELECTOR_DATA_TOGGLE$4);\n\n    for (const elem of toggleList) {\n      const selector = getSelectorFromElement(elem);\n      const filterElement = SelectorEngine.find(selector).filter(foundElement => foundElement === this._element);\n\n      if (selector !== null && filterElement.length) {\n        this._triggerArray.push(elem);\n      }\n    }\n\n    this._initializeChildren();\n\n    if (!this._config.parent) {\n      this._addAriaAndCollapsedClass(this._triggerArray, this._isShown());\n    }\n\n    if (this._config.toggle) {\n      this.toggle();\n    }\n  } // Getters\n\n\n  static get Default() {\n    return Default$a;\n  }\n\n  static get DefaultType() {\n    return DefaultType$a;\n  }\n\n  static get NAME() {\n    return NAME$b;\n  } // Public\n\n\n  toggle() {\n    if (this._isShown()) {\n      this.hide();\n    } else {\n      this.show();\n    }\n  }\n\n  show() {\n    if (this._isTransitioning || this._isShown()) {\n      return;\n    }\n\n    let activeChildren = []; // find active children\n\n    if (this._config.parent) {\n      activeChildren = this._getFirstLevelChildren(SELECTOR_ACTIVES).filter(element => element !== this._element).map(element => Collapse.getOrCreateInstance(element, {\n        toggle: false\n      }));\n    }\n\n    if (activeChildren.length && activeChildren[0]._isTransitioning) {\n      return;\n    }\n\n    const startEvent = EventHandler.trigger(this._element, EVENT_SHOW$6);\n\n    if (startEvent.defaultPrevented) {\n      return;\n    }\n\n    for (const activeInstance of activeChildren) {\n      activeInstance.hide();\n    }\n\n    const dimension = this._getDimension();\n\n    this._element.classList.remove(CLASS_NAME_COLLAPSE);\n\n    this._element.classList.add(CLASS_NAME_COLLAPSING);\n\n    this._element.style[dimension] = 0;\n\n    this._addAriaAndCollapsedClass(this._triggerArray, true);\n\n    this._isTransitioning = true;\n\n    const complete = () => {\n      this._isTransitioning = false;\n\n      this._element.classList.remove(CLASS_NAME_COLLAPSING);\n\n      this._element.classList.add(CLASS_NAME_COLLAPSE, CLASS_NAME_SHOW$7);\n\n      this._element.style[dimension] = '';\n      EventHandler.trigger(this._element, EVENT_SHOWN$6);\n    };\n\n    const capitalizedDimension = dimension[0].toUpperCase() + dimension.slice(1);\n    const scrollSize = `scroll${capitalizedDimension}`;\n\n    this._queueCallback(complete, this._element, true);\n\n    this._element.style[dimension] = `${this._element[scrollSize]}px`;\n  }\n\n  hide() {\n    if (this._isTransitioning || !this._isShown()) {\n      return;\n    }\n\n    const startEvent = EventHandler.trigger(this._element, EVENT_HIDE$6);\n\n    if (startEvent.defaultPrevented) {\n      return;\n    }\n\n    const dimension = this._getDimension();\n\n    this._element.style[dimension] = `${this._element.getBoundingClientRect()[dimension]}px`;\n    reflow(this._element);\n\n    this._element.classList.add(CLASS_NAME_COLLAPSING);\n\n    this._element.classList.remove(CLASS_NAME_COLLAPSE, CLASS_NAME_SHOW$7);\n\n    for (const trigger of this._triggerArray) {\n      const element = getElementFromSelector(trigger);\n\n      if (element && !this._isShown(element)) {\n        this._addAriaAndCollapsedClass([trigger], false);\n      }\n    }\n\n    this._isTransitioning = true;\n\n    const complete = () => {\n      this._isTransitioning = false;\n\n      this._element.classList.remove(CLASS_NAME_COLLAPSING);\n\n      this._element.classList.add(CLASS_NAME_COLLAPSE);\n\n      EventHandler.trigger(this._element, EVENT_HIDDEN$6);\n    };\n\n    this._element.style[dimension] = '';\n\n    this._queueCallback(complete, this._element, true);\n  }\n\n  _isShown(element = this._element) {\n    return element.classList.contains(CLASS_NAME_SHOW$7);\n  } // Private\n\n\n  _configAfterMerge(config) {\n    config.toggle = Boolean(config.toggle); // Coerce string values\n\n    config.parent = getElement(config.parent);\n    return config;\n  }\n\n  _getDimension() {\n    return this._element.classList.contains(CLASS_NAME_HORIZONTAL) ? WIDTH : HEIGHT;\n  }\n\n  _initializeChildren() {\n    if (!this._config.parent) {\n      return;\n    }\n\n    const children = this._getFirstLevelChildren(SELECTOR_DATA_TOGGLE$4);\n\n    for (const element of children) {\n      const selected = getElementFromSelector(element);\n\n      if (selected) {\n        this._addAriaAndCollapsedClass([element], this._isShown(selected));\n      }\n    }\n  }\n\n  _getFirstLevelChildren(selector) {\n    const children = SelectorEngine.find(CLASS_NAME_DEEPER_CHILDREN, this._config.parent); // remove children if greater depth\n\n    return SelectorEngine.find(selector, this._config.parent).filter(element => !children.includes(element));\n  }\n\n  _addAriaAndCollapsedClass(triggerArray, isOpen) {\n    if (!triggerArray.length) {\n      return;\n    }\n\n    for (const element of triggerArray) {\n      element.classList.toggle(CLASS_NAME_COLLAPSED, !isOpen);\n      element.setAttribute('aria-expanded', isOpen);\n    }\n  } // Static\n\n\n  static jQueryInterface(config) {\n    const _config = {};\n\n    if (typeof config === 'string' && /show|hide/.test(config)) {\n      _config.toggle = false;\n    }\n\n    return this.each(function () {\n      const data = Collapse.getOrCreateInstance(this, _config);\n\n      if (typeof config === 'string') {\n        if (typeof data[config] === 'undefined') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n\n        data[config]();\n      }\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$4, SELECTOR_DATA_TOGGLE$4, function (event) {\n  // preventDefault only for <a> elements (which change the URL) not inside the collapsible element\n  if (event.target.tagName === 'A' || event.delegateTarget && event.delegateTarget.tagName === 'A') {\n    event.preventDefault();\n  }\n\n  const selector = getSelectorFromElement(this);\n  const selectorElements = SelectorEngine.find(selector);\n\n  for (const element of selectorElements) {\n    Collapse.getOrCreateInstance(element, {\n      toggle: false\n    }).toggle();\n  }\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Collapse);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): dropdown.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$a = 'dropdown';\nconst DATA_KEY$6 = 'bs.dropdown';\nconst EVENT_KEY$6 = `.${DATA_KEY$6}`;\nconst DATA_API_KEY$3 = '.data-api';\nconst ESCAPE_KEY$2 = 'Escape';\nconst TAB_KEY$1 = 'Tab';\nconst ARROW_UP_KEY$1 = 'ArrowUp';\nconst ARROW_DOWN_KEY$1 = 'ArrowDown';\nconst RIGHT_MOUSE_BUTTON = 2; // MouseEvent.button value for the secondary button, usually the right button\n\nconst EVENT_HIDE$5 = `hide${EVENT_KEY$6}`;\nconst EVENT_HIDDEN$5 = `hidden${EVENT_KEY$6}`;\nconst EVENT_SHOW$5 = `show${EVENT_KEY$6}`;\nconst EVENT_SHOWN$5 = `shown${EVENT_KEY$6}`;\nconst EVENT_CLICK_DATA_API$3 = `click${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst EVENT_KEYDOWN_DATA_API = `keydown${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst EVENT_KEYUP_DATA_API = `keyup${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst CLASS_NAME_SHOW$6 = 'show';\nconst CLASS_NAME_DROPUP = 'dropup';\nconst CLASS_NAME_DROPEND = 'dropend';\nconst CLASS_NAME_DROPSTART = 'dropstart';\nconst CLASS_NAME_DROPUP_CENTER = 'dropup-center';\nconst CLASS_NAME_DROPDOWN_CENTER = 'dropdown-center';\nconst SELECTOR_DATA_TOGGLE$3 = '[data-bs-toggle=\"dropdown\"]:not(.disabled):not(:disabled)';\nconst SELECTOR_DATA_TOGGLE_SHOWN = `${SELECTOR_DATA_TOGGLE$3}.${CLASS_NAME_SHOW$6}`;\nconst SELECTOR_MENU = '.dropdown-menu';\nconst SELECTOR_NAVBAR = '.navbar';\nconst SELECTOR_NAVBAR_NAV = '.navbar-nav';\nconst SELECTOR_VISIBLE_ITEMS = '.dropdown-menu .dropdown-item:not(.disabled):not(:disabled)';\nconst PLACEMENT_TOP = isRTL() ? 'top-end' : 'top-start';\nconst PLACEMENT_TOPEND = isRTL() ? 'top-start' : 'top-end';\nconst PLACEMENT_BOTTOM = isRTL() ? 'bottom-end' : 'bottom-start';\nconst PLACEMENT_BOTTOMEND = isRTL() ? 'bottom-start' : 'bottom-end';\nconst PLACEMENT_RIGHT = isRTL() ? 'left-start' : 'right-start';\nconst PLACEMENT_LEFT = isRTL() ? 'right-start' : 'left-start';\nconst PLACEMENT_TOPCENTER = 'top';\nconst PLACEMENT_BOTTOMCENTER = 'bottom';\nconst Default$9 = {\n  autoClose: true,\n  boundary: 'clippingParents',\n  display: 'dynamic',\n  offset: [0, 2],\n  popperConfig: null,\n  reference: 'toggle'\n};\nconst DefaultType$9 = {\n  autoClose: '(boolean|string)',\n  boundary: '(string|element)',\n  display: 'string',\n  offset: '(array|string|function)',\n  popperConfig: '(null|object|function)',\n  reference: '(string|element|object)'\n};\n/**\n * Class definition\n */\n\nclass Dropdown extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._popper = null;\n    this._parent = this._element.parentNode; // dropdown wrapper\n    // todo: v6 revert #37011 & change markup https://getbootstrap.com/docs/5.2/forms/input-group/\n\n    this._menu = SelectorEngine.next(this._element, SELECTOR_MENU)[0] || SelectorEngine.prev(this._element, SELECTOR_MENU)[0] || SelectorEngine.findOne(SELECTOR_MENU, this._parent);\n    this._inNavbar = this._detectNavbar();\n  } // Getters\n\n\n  static get Default() {\n    return Default$9;\n  }\n\n  static get DefaultType() {\n    return DefaultType$9;\n  }\n\n  static get NAME() {\n    return NAME$a;\n  } // Public\n\n\n  toggle() {\n    return this._isShown() ? this.hide() : this.show();\n  }\n\n  show() {\n    if (isDisabled(this._element) || this._isShown()) {\n      return;\n    }\n\n    const relatedTarget = {\n      relatedTarget: this._element\n    };\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$5, relatedTarget);\n\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n\n    this._createPopper(); // If this is a touch-enabled device we add extra\n    // empty mouseover listeners to the body's immediate children;\n    // only needed because of broken event delegation on iOS\n    // https://www.quirksmode.org/blog/archives/2014/02/mouse_event_bub.html\n\n\n    if ('ontouchstart' in document.documentElement && !this._parent.closest(SELECTOR_NAVBAR_NAV)) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.on(element, 'mouseover', noop);\n      }\n    }\n\n    this._element.focus();\n\n    this._element.setAttribute('aria-expanded', true);\n\n    this._menu.classList.add(CLASS_NAME_SHOW$6);\n\n    this._element.classList.add(CLASS_NAME_SHOW$6);\n\n    EventHandler.trigger(this._element, EVENT_SHOWN$5, relatedTarget);\n  }\n\n  hide() {\n    if (isDisabled(this._element) || !this._isShown()) {\n      return;\n    }\n\n    const relatedTarget = {\n      relatedTarget: this._element\n    };\n\n    this._completeHide(relatedTarget);\n  }\n\n  dispose() {\n    if (this._popper) {\n      this._popper.destroy();\n    }\n\n    super.dispose();\n  }\n\n  update() {\n    this._inNavbar = this._detectNavbar();\n\n    if (this._popper) {\n      this._popper.update();\n    }\n  } // Private\n\n\n  _completeHide(relatedTarget) {\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$5, relatedTarget);\n\n    if (hideEvent.defaultPrevented) {\n      return;\n    } // If this is a touch-enabled device we remove the extra\n    // empty mouseover listeners we added for iOS support\n\n\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.off(element, 'mouseover', noop);\n      }\n    }\n\n    if (this._popper) {\n      this._popper.destroy();\n    }\n\n    this._menu.classList.remove(CLASS_NAME_SHOW$6);\n\n    this._element.classList.remove(CLASS_NAME_SHOW$6);\n\n    this._element.setAttribute('aria-expanded', 'false');\n\n    Manipulator.removeDataAttribute(this._menu, 'popper');\n    EventHandler.trigger(this._element, EVENT_HIDDEN$5, relatedTarget);\n  }\n\n  _getConfig(config) {\n    config = super._getConfig(config);\n\n    if (typeof config.reference === 'object' && !isElement(config.reference) && typeof config.reference.getBoundingClientRect !== 'function') {\n      // Popper virtual elements require a getBoundingClientRect method\n      throw new TypeError(`${NAME$a.toUpperCase()}: Option \"reference\" provided type \"object\" without a required \"getBoundingClientRect\" method.`);\n    }\n\n    return config;\n  }\n\n  _createPopper() {\n    if (typeof Popper === 'undefined') {\n      throw new TypeError('Bootstrap\\'s dropdowns require Popper (https://popper.js.org)');\n    }\n\n    let referenceElement = this._element;\n\n    if (this._config.reference === 'parent') {\n      referenceElement = this._parent;\n    } else if (isElement(this._config.reference)) {\n      referenceElement = getElement(this._config.reference);\n    } else if (typeof this._config.reference === 'object') {\n      referenceElement = this._config.reference;\n    }\n\n    const popperConfig = this._getPopperConfig();\n\n    this._popper = Popper.createPopper(referenceElement, this._menu, popperConfig);\n  }\n\n  _isShown() {\n    return this._menu.classList.contains(CLASS_NAME_SHOW$6);\n  }\n\n  _getPlacement() {\n    const parentDropdown = this._parent;\n\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPEND)) {\n      return PLACEMENT_RIGHT;\n    }\n\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPSTART)) {\n      return PLACEMENT_LEFT;\n    }\n\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPUP_CENTER)) {\n      return PLACEMENT_TOPCENTER;\n    }\n\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPDOWN_CENTER)) {\n      return PLACEMENT_BOTTOMCENTER;\n    } // We need to trim the value because custom properties can also include spaces\n\n\n    const isEnd = getComputedStyle(this._menu).getPropertyValue('--bs-position').trim() === 'end';\n\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPUP)) {\n      return isEnd ? PLACEMENT_TOPEND : PLACEMENT_TOP;\n    }\n\n    return isEnd ? PLACEMENT_BOTTOMEND : PLACEMENT_BOTTOM;\n  }\n\n  _detectNavbar() {\n    return this._element.closest(SELECTOR_NAVBAR) !== null;\n  }\n\n  _getOffset() {\n    const {\n      offset\n    } = this._config;\n\n    if (typeof offset === 'string') {\n      return offset.split(',').map(value => Number.parseInt(value, 10));\n    }\n\n    if (typeof offset === 'function') {\n      return popperData => offset(popperData, this._element);\n    }\n\n    return offset;\n  }\n\n  _getPopperConfig() {\n    const defaultBsPopperConfig = {\n      placement: this._getPlacement(),\n      modifiers: [{\n        name: 'preventOverflow',\n        options: {\n          boundary: this._config.boundary\n        }\n      }, {\n        name: 'offset',\n        options: {\n          offset: this._getOffset()\n        }\n      }]\n    }; // Disable Popper if we have a static display or Dropdown is in Navbar\n\n    if (this._inNavbar || this._config.display === 'static') {\n      Manipulator.setDataAttribute(this._menu, 'popper', 'static'); // todo:v6 remove\n\n      defaultBsPopperConfig.modifiers = [{\n        name: 'applyStyles',\n        enabled: false\n      }];\n    }\n\n    return { ...defaultBsPopperConfig,\n      ...(typeof this._config.popperConfig === 'function' ? this._config.popperConfig(defaultBsPopperConfig) : this._config.popperConfig)\n    };\n  }\n\n  _selectMenuItem({\n    key,\n    target\n  }) {\n    const items = SelectorEngine.find(SELECTOR_VISIBLE_ITEMS, this._menu).filter(element => isVisible(element));\n\n    if (!items.length) {\n      return;\n    } // if target isn't included in items (e.g. when expanding the dropdown)\n    // allow cycling to get the last item in case key equals ARROW_UP_KEY\n\n\n    getNextActiveElement(items, target, key === ARROW_DOWN_KEY$1, !items.includes(target)).focus();\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Dropdown.getOrCreateInstance(this, config);\n\n      if (typeof config !== 'string') {\n        return;\n      }\n\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n\n      data[config]();\n    });\n  }\n\n  static clearMenus(event) {\n    if (event.button === RIGHT_MOUSE_BUTTON || event.type === 'keyup' && event.key !== TAB_KEY$1) {\n      return;\n    }\n\n    const openToggles = SelectorEngine.find(SELECTOR_DATA_TOGGLE_SHOWN);\n\n    for (const toggle of openToggles) {\n      const context = Dropdown.getInstance(toggle);\n\n      if (!context || context._config.autoClose === false) {\n        continue;\n      }\n\n      const composedPath = event.composedPath();\n      const isMenuTarget = composedPath.includes(context._menu);\n\n      if (composedPath.includes(context._element) || context._config.autoClose === 'inside' && !isMenuTarget || context._config.autoClose === 'outside' && isMenuTarget) {\n        continue;\n      } // Tab navigation through the dropdown menu or events from contained inputs shouldn't close the menu\n\n\n      if (context._menu.contains(event.target) && (event.type === 'keyup' && event.key === TAB_KEY$1 || /input|select|option|textarea|form/i.test(event.target.tagName))) {\n        continue;\n      }\n\n      const relatedTarget = {\n        relatedTarget: context._element\n      };\n\n      if (event.type === 'click') {\n        relatedTarget.clickEvent = event;\n      }\n\n      context._completeHide(relatedTarget);\n    }\n  }\n\n  static dataApiKeydownHandler(event) {\n    // If not an UP | DOWN | ESCAPE key => not a dropdown command\n    // If input/textarea && if key is other than ESCAPE => not a dropdown command\n    const isInput = /input|textarea/i.test(event.target.tagName);\n    const isEscapeEvent = event.key === ESCAPE_KEY$2;\n    const isUpOrDownEvent = [ARROW_UP_KEY$1, ARROW_DOWN_KEY$1].includes(event.key);\n\n    if (!isUpOrDownEvent && !isEscapeEvent) {\n      return;\n    }\n\n    if (isInput && !isEscapeEvent) {\n      return;\n    }\n\n    event.preventDefault(); // todo: v6 revert #37011 & change markup https://getbootstrap.com/docs/5.2/forms/input-group/\n\n    const getToggleButton = this.matches(SELECTOR_DATA_TOGGLE$3) ? this : SelectorEngine.prev(this, SELECTOR_DATA_TOGGLE$3)[0] || SelectorEngine.next(this, SELECTOR_DATA_TOGGLE$3)[0] || SelectorEngine.findOne(SELECTOR_DATA_TOGGLE$3, event.delegateTarget.parentNode);\n    const instance = Dropdown.getOrCreateInstance(getToggleButton);\n\n    if (isUpOrDownEvent) {\n      event.stopPropagation();\n      instance.show();\n\n      instance._selectMenuItem(event);\n\n      return;\n    }\n\n    if (instance._isShown()) {\n      // else is escape and we check if it is shown\n      event.stopPropagation();\n      instance.hide();\n      getToggleButton.focus();\n    }\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nEventHandler.on(document, EVENT_KEYDOWN_DATA_API, SELECTOR_DATA_TOGGLE$3, Dropdown.dataApiKeydownHandler);\nEventHandler.on(document, EVENT_KEYDOWN_DATA_API, SELECTOR_MENU, Dropdown.dataApiKeydownHandler);\nEventHandler.on(document, EVENT_CLICK_DATA_API$3, Dropdown.clearMenus);\nEventHandler.on(document, EVENT_KEYUP_DATA_API, Dropdown.clearMenus);\nEventHandler.on(document, EVENT_CLICK_DATA_API$3, SELECTOR_DATA_TOGGLE$3, function (event) {\n  event.preventDefault();\n  Dropdown.getOrCreateInstance(this).toggle();\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Dropdown);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/scrollBar.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst SELECTOR_FIXED_CONTENT = '.fixed-top, .fixed-bottom, .is-fixed, .sticky-top';\nconst SELECTOR_STICKY_CONTENT = '.sticky-top';\nconst PROPERTY_PADDING = 'padding-right';\nconst PROPERTY_MARGIN = 'margin-right';\n/**\n * Class definition\n */\n\nclass ScrollBarHelper {\n  constructor() {\n    this._element = document.body;\n  } // Public\n\n\n  getWidth() {\n    // https://developer.mozilla.org/en-US/docs/Web/API/Window/innerWidth#usage_notes\n    const documentWidth = document.documentElement.clientWidth;\n    return Math.abs(window.innerWidth - documentWidth);\n  }\n\n  hide() {\n    const width = this.getWidth();\n\n    this._disableOverFlow(); // give padding to element to balance the hidden scrollbar width\n\n\n    this._setElementAttributes(this._element, PROPERTY_PADDING, calculatedValue => calculatedValue + width); // trick: We adjust positive paddingRight and negative marginRight to sticky-top elements to keep showing fullwidth\n\n\n    this._setElementAttributes(SELECTOR_FIXED_CONTENT, PROPERTY_PADDING, calculatedValue => calculatedValue + width);\n\n    this._setElementAttributes(SELECTOR_STICKY_CONTENT, PROPERTY_MARGIN, calculatedValue => calculatedValue - width);\n  }\n\n  reset() {\n    this._resetElementAttributes(this._element, 'overflow');\n\n    this._resetElementAttributes(this._element, PROPERTY_PADDING);\n\n    this._resetElementAttributes(SELECTOR_FIXED_CONTENT, PROPERTY_PADDING);\n\n    this._resetElementAttributes(SELECTOR_STICKY_CONTENT, PROPERTY_MARGIN);\n  }\n\n  isOverflowing() {\n    return this.getWidth() > 0;\n  } // Private\n\n\n  _disableOverFlow() {\n    this._saveInitialAttribute(this._element, 'overflow');\n\n    this._element.style.overflow = 'hidden';\n  }\n\n  _setElementAttributes(selector, styleProperty, callback) {\n    const scrollbarWidth = this.getWidth();\n\n    const manipulationCallBack = element => {\n      if (element !== this._element && window.innerWidth > element.clientWidth + scrollbarWidth) {\n        return;\n      }\n\n      this._saveInitialAttribute(element, styleProperty);\n\n      const calculatedValue = window.getComputedStyle(element).getPropertyValue(styleProperty);\n      element.style.setProperty(styleProperty, `${callback(Number.parseFloat(calculatedValue))}px`);\n    };\n\n    this._applyManipulationCallback(selector, manipulationCallBack);\n  }\n\n  _saveInitialAttribute(element, styleProperty) {\n    const actualValue = element.style.getPropertyValue(styleProperty);\n\n    if (actualValue) {\n      Manipulator.setDataAttribute(element, styleProperty, actualValue);\n    }\n  }\n\n  _resetElementAttributes(selector, styleProperty) {\n    const manipulationCallBack = element => {\n      const value = Manipulator.getDataAttribute(element, styleProperty); // We only want to remove the property if the value is `null`; the value can also be zero\n\n      if (value === null) {\n        element.style.removeProperty(styleProperty);\n        return;\n      }\n\n      Manipulator.removeDataAttribute(element, styleProperty);\n      element.style.setProperty(styleProperty, value);\n    };\n\n    this._applyManipulationCallback(selector, manipulationCallBack);\n  }\n\n  _applyManipulationCallback(selector, callBack) {\n    if (isElement(selector)) {\n      callBack(selector);\n      return;\n    }\n\n    for (const sel of SelectorEngine.find(selector, this._element)) {\n      callBack(sel);\n    }\n  }\n\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/backdrop.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$9 = 'backdrop';\nconst CLASS_NAME_FADE$4 = 'fade';\nconst CLASS_NAME_SHOW$5 = 'show';\nconst EVENT_MOUSEDOWN = `mousedown.bs.${NAME$9}`;\nconst Default$8 = {\n  className: 'modal-backdrop',\n  clickCallback: null,\n  isAnimated: false,\n  isVisible: true,\n  // if false, we use the backdrop helper without adding any element to the dom\n  rootElement: 'body' // give the choice to place backdrop under different elements\n\n};\nconst DefaultType$8 = {\n  className: 'string',\n  clickCallback: '(function|null)',\n  isAnimated: 'boolean',\n  isVisible: 'boolean',\n  rootElement: '(element|string)'\n};\n/**\n * Class definition\n */\n\nclass Backdrop extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n    this._isAppended = false;\n    this._element = null;\n  } // Getters\n\n\n  static get Default() {\n    return Default$8;\n  }\n\n  static get DefaultType() {\n    return DefaultType$8;\n  }\n\n  static get NAME() {\n    return NAME$9;\n  } // Public\n\n\n  show(callback) {\n    if (!this._config.isVisible) {\n      execute(callback);\n      return;\n    }\n\n    this._append();\n\n    const element = this._getElement();\n\n    if (this._config.isAnimated) {\n      reflow(element);\n    }\n\n    element.classList.add(CLASS_NAME_SHOW$5);\n\n    this._emulateAnimation(() => {\n      execute(callback);\n    });\n  }\n\n  hide(callback) {\n    if (!this._config.isVisible) {\n      execute(callback);\n      return;\n    }\n\n    this._getElement().classList.remove(CLASS_NAME_SHOW$5);\n\n    this._emulateAnimation(() => {\n      this.dispose();\n      execute(callback);\n    });\n  }\n\n  dispose() {\n    if (!this._isAppended) {\n      return;\n    }\n\n    EventHandler.off(this._element, EVENT_MOUSEDOWN);\n\n    this._element.remove();\n\n    this._isAppended = false;\n  } // Private\n\n\n  _getElement() {\n    if (!this._element) {\n      const backdrop = document.createElement('div');\n      backdrop.className = this._config.className;\n\n      if (this._config.isAnimated) {\n        backdrop.classList.add(CLASS_NAME_FADE$4);\n      }\n\n      this._element = backdrop;\n    }\n\n    return this._element;\n  }\n\n  _configAfterMerge(config) {\n    // use getElement() with the default \"body\" to get a fresh Element on each instantiation\n    config.rootElement = getElement(config.rootElement);\n    return config;\n  }\n\n  _append() {\n    if (this._isAppended) {\n      return;\n    }\n\n    const element = this._getElement();\n\n    this._config.rootElement.append(element);\n\n    EventHandler.on(element, EVENT_MOUSEDOWN, () => {\n      execute(this._config.clickCallback);\n    });\n    this._isAppended = true;\n  }\n\n  _emulateAnimation(callback) {\n    executeAfterTransition(callback, this._getElement(), this._config.isAnimated);\n  }\n\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/focustrap.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$8 = 'focustrap';\nconst DATA_KEY$5 = 'bs.focustrap';\nconst EVENT_KEY$5 = `.${DATA_KEY$5}`;\nconst EVENT_FOCUSIN$2 = `focusin${EVENT_KEY$5}`;\nconst EVENT_KEYDOWN_TAB = `keydown.tab${EVENT_KEY$5}`;\nconst TAB_KEY = 'Tab';\nconst TAB_NAV_FORWARD = 'forward';\nconst TAB_NAV_BACKWARD = 'backward';\nconst Default$7 = {\n  autofocus: true,\n  trapElement: null // The element to trap focus inside of\n\n};\nconst DefaultType$7 = {\n  autofocus: 'boolean',\n  trapElement: 'element'\n};\n/**\n * Class definition\n */\n\nclass FocusTrap extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n    this._isActive = false;\n    this._lastTabNavDirection = null;\n  } // Getters\n\n\n  static get Default() {\n    return Default$7;\n  }\n\n  static get DefaultType() {\n    return DefaultType$7;\n  }\n\n  static get NAME() {\n    return NAME$8;\n  } // Public\n\n\n  activate() {\n    if (this._isActive) {\n      return;\n    }\n\n    if (this._config.autofocus) {\n      this._config.trapElement.focus();\n    }\n\n    EventHandler.off(document, EVENT_KEY$5); // guard against infinite focus loop\n\n    EventHandler.on(document, EVENT_FOCUSIN$2, event => this._handleFocusin(event));\n    EventHandler.on(document, EVENT_KEYDOWN_TAB, event => this._handleKeydown(event));\n    this._isActive = true;\n  }\n\n  deactivate() {\n    if (!this._isActive) {\n      return;\n    }\n\n    this._isActive = false;\n    EventHandler.off(document, EVENT_KEY$5);\n  } // Private\n\n\n  _handleFocusin(event) {\n    const {\n      trapElement\n    } = this._config;\n\n    if (event.target === document || event.target === trapElement || trapElement.contains(event.target)) {\n      return;\n    }\n\n    const elements = SelectorEngine.focusableChildren(trapElement);\n\n    if (elements.length === 0) {\n      trapElement.focus();\n    } else if (this._lastTabNavDirection === TAB_NAV_BACKWARD) {\n      elements[elements.length - 1].focus();\n    } else {\n      elements[0].focus();\n    }\n  }\n\n  _handleKeydown(event) {\n    if (event.key !== TAB_KEY) {\n      return;\n    }\n\n    this._lastTabNavDirection = event.shiftKey ? TAB_NAV_BACKWARD : TAB_NAV_FORWARD;\n  }\n\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): modal.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$7 = 'modal';\nconst DATA_KEY$4 = 'bs.modal';\nconst EVENT_KEY$4 = `.${DATA_KEY$4}`;\nconst DATA_API_KEY$2 = '.data-api';\nconst ESCAPE_KEY$1 = 'Escape';\nconst EVENT_HIDE$4 = `hide${EVENT_KEY$4}`;\nconst EVENT_HIDE_PREVENTED$1 = `hidePrevented${EVENT_KEY$4}`;\nconst EVENT_HIDDEN$4 = `hidden${EVENT_KEY$4}`;\nconst EVENT_SHOW$4 = `show${EVENT_KEY$4}`;\nconst EVENT_SHOWN$4 = `shown${EVENT_KEY$4}`;\nconst EVENT_RESIZE$1 = `resize${EVENT_KEY$4}`;\nconst EVENT_CLICK_DISMISS = `click.dismiss${EVENT_KEY$4}`;\nconst EVENT_MOUSEDOWN_DISMISS = `mousedown.dismiss${EVENT_KEY$4}`;\nconst EVENT_KEYDOWN_DISMISS$1 = `keydown.dismiss${EVENT_KEY$4}`;\nconst EVENT_CLICK_DATA_API$2 = `click${EVENT_KEY$4}${DATA_API_KEY$2}`;\nconst CLASS_NAME_OPEN = 'modal-open';\nconst CLASS_NAME_FADE$3 = 'fade';\nconst CLASS_NAME_SHOW$4 = 'show';\nconst CLASS_NAME_STATIC = 'modal-static';\nconst OPEN_SELECTOR$1 = '.modal.show';\nconst SELECTOR_DIALOG = '.modal-dialog';\nconst SELECTOR_MODAL_BODY = '.modal-body';\nconst SELECTOR_DATA_TOGGLE$2 = '[data-bs-toggle=\"modal\"]';\nconst Default$6 = {\n  backdrop: true,\n  focus: true,\n  keyboard: true\n};\nconst DefaultType$6 = {\n  backdrop: '(boolean|string)',\n  focus: 'boolean',\n  keyboard: 'boolean'\n};\n/**\n * Class definition\n */\n\nclass Modal extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._dialog = SelectorEngine.findOne(SELECTOR_DIALOG, this._element);\n    this._backdrop = this._initializeBackDrop();\n    this._focustrap = this._initializeFocusTrap();\n    this._isShown = false;\n    this._isTransitioning = false;\n    this._scrollBar = new ScrollBarHelper();\n\n    this._addEventListeners();\n  } // Getters\n\n\n  static get Default() {\n    return Default$6;\n  }\n\n  static get DefaultType() {\n    return DefaultType$6;\n  }\n\n  static get NAME() {\n    return NAME$7;\n  } // Public\n\n\n  toggle(relatedTarget) {\n    return this._isShown ? this.hide() : this.show(relatedTarget);\n  }\n\n  show(relatedTarget) {\n    if (this._isShown || this._isTransitioning) {\n      return;\n    }\n\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$4, {\n      relatedTarget\n    });\n\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n\n    this._isShown = true;\n    this._isTransitioning = true;\n\n    this._scrollBar.hide();\n\n    document.body.classList.add(CLASS_NAME_OPEN);\n\n    this._adjustDialog();\n\n    this._backdrop.show(() => this._showElement(relatedTarget));\n  }\n\n  hide() {\n    if (!this._isShown || this._isTransitioning) {\n      return;\n    }\n\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$4);\n\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n\n    this._isShown = false;\n    this._isTransitioning = true;\n\n    this._focustrap.deactivate();\n\n    this._element.classList.remove(CLASS_NAME_SHOW$4);\n\n    this._queueCallback(() => this._hideModal(), this._element, this._isAnimated());\n  }\n\n  dispose() {\n    for (const htmlElement of [window, this._dialog]) {\n      EventHandler.off(htmlElement, EVENT_KEY$4);\n    }\n\n    this._backdrop.dispose();\n\n    this._focustrap.deactivate();\n\n    super.dispose();\n  }\n\n  handleUpdate() {\n    this._adjustDialog();\n  } // Private\n\n\n  _initializeBackDrop() {\n    return new Backdrop({\n      isVisible: Boolean(this._config.backdrop),\n      // 'static' option will be translated to true, and booleans will keep their value,\n      isAnimated: this._isAnimated()\n    });\n  }\n\n  _initializeFocusTrap() {\n    return new FocusTrap({\n      trapElement: this._element\n    });\n  }\n\n  _showElement(relatedTarget) {\n    // try to append dynamic modal\n    if (!document.body.contains(this._element)) {\n      document.body.append(this._element);\n    }\n\n    this._element.style.display = 'block';\n\n    this._element.removeAttribute('aria-hidden');\n\n    this._element.setAttribute('aria-modal', true);\n\n    this._element.setAttribute('role', 'dialog');\n\n    this._element.scrollTop = 0;\n    const modalBody = SelectorEngine.findOne(SELECTOR_MODAL_BODY, this._dialog);\n\n    if (modalBody) {\n      modalBody.scrollTop = 0;\n    }\n\n    reflow(this._element);\n\n    this._element.classList.add(CLASS_NAME_SHOW$4);\n\n    const transitionComplete = () => {\n      if (this._config.focus) {\n        this._focustrap.activate();\n      }\n\n      this._isTransitioning = false;\n      EventHandler.trigger(this._element, EVENT_SHOWN$4, {\n        relatedTarget\n      });\n    };\n\n    this._queueCallback(transitionComplete, this._dialog, this._isAnimated());\n  }\n\n  _addEventListeners() {\n    EventHandler.on(this._element, EVENT_KEYDOWN_DISMISS$1, event => {\n      if (event.key !== ESCAPE_KEY$1) {\n        return;\n      }\n\n      if (this._config.keyboard) {\n        event.preventDefault();\n        this.hide();\n        return;\n      }\n\n      this._triggerBackdropTransition();\n    });\n    EventHandler.on(window, EVENT_RESIZE$1, () => {\n      if (this._isShown && !this._isTransitioning) {\n        this._adjustDialog();\n      }\n    });\n    EventHandler.on(this._element, EVENT_MOUSEDOWN_DISMISS, event => {\n      // a bad trick to segregate clicks that may start inside dialog but end outside, and avoid listen to scrollbar clicks\n      EventHandler.one(this._element, EVENT_CLICK_DISMISS, event2 => {\n        if (this._element !== event.target || this._element !== event2.target) {\n          return;\n        }\n\n        if (this._config.backdrop === 'static') {\n          this._triggerBackdropTransition();\n\n          return;\n        }\n\n        if (this._config.backdrop) {\n          this.hide();\n        }\n      });\n    });\n  }\n\n  _hideModal() {\n    this._element.style.display = 'none';\n\n    this._element.setAttribute('aria-hidden', true);\n\n    this._element.removeAttribute('aria-modal');\n\n    this._element.removeAttribute('role');\n\n    this._isTransitioning = false;\n\n    this._backdrop.hide(() => {\n      document.body.classList.remove(CLASS_NAME_OPEN);\n\n      this._resetAdjustments();\n\n      this._scrollBar.reset();\n\n      EventHandler.trigger(this._element, EVENT_HIDDEN$4);\n    });\n  }\n\n  _isAnimated() {\n    return this._element.classList.contains(CLASS_NAME_FADE$3);\n  }\n\n  _triggerBackdropTransition() {\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED$1);\n\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n\n    const isModalOverflowing = this._element.scrollHeight > document.documentElement.clientHeight;\n    const initialOverflowY = this._element.style.overflowY; // return if the following background transition hasn't yet completed\n\n    if (initialOverflowY === 'hidden' || this._element.classList.contains(CLASS_NAME_STATIC)) {\n      return;\n    }\n\n    if (!isModalOverflowing) {\n      this._element.style.overflowY = 'hidden';\n    }\n\n    this._element.classList.add(CLASS_NAME_STATIC);\n\n    this._queueCallback(() => {\n      this._element.classList.remove(CLASS_NAME_STATIC);\n\n      this._queueCallback(() => {\n        this._element.style.overflowY = initialOverflowY;\n      }, this._dialog);\n    }, this._dialog);\n\n    this._element.focus();\n  }\n  /**\n   * The following methods are used to handle overflowing modals\n   */\n\n\n  _adjustDialog() {\n    const isModalOverflowing = this._element.scrollHeight > document.documentElement.clientHeight;\n\n    const scrollbarWidth = this._scrollBar.getWidth();\n\n    const isBodyOverflowing = scrollbarWidth > 0;\n\n    if (isBodyOverflowing && !isModalOverflowing) {\n      const property = isRTL() ? 'paddingLeft' : 'paddingRight';\n      this._element.style[property] = `${scrollbarWidth}px`;\n    }\n\n    if (!isBodyOverflowing && isModalOverflowing) {\n      const property = isRTL() ? 'paddingRight' : 'paddingLeft';\n      this._element.style[property] = `${scrollbarWidth}px`;\n    }\n  }\n\n  _resetAdjustments() {\n    this._element.style.paddingLeft = '';\n    this._element.style.paddingRight = '';\n  } // Static\n\n\n  static jQueryInterface(config, relatedTarget) {\n    return this.each(function () {\n      const data = Modal.getOrCreateInstance(this, config);\n\n      if (typeof config !== 'string') {\n        return;\n      }\n\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n\n      data[config](relatedTarget);\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$2, SELECTOR_DATA_TOGGLE$2, function (event) {\n  const target = getElementFromSelector(this);\n\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n\n  EventHandler.one(target, EVENT_SHOW$4, showEvent => {\n    if (showEvent.defaultPrevented) {\n      // only register focus restorer if modal will actually get shown\n      return;\n    }\n\n    EventHandler.one(target, EVENT_HIDDEN$4, () => {\n      if (isVisible(this)) {\n        this.focus();\n      }\n    });\n  }); // avoid conflict when clicking modal toggler while another one is open\n\n  const alreadyOpen = SelectorEngine.findOne(OPEN_SELECTOR$1);\n\n  if (alreadyOpen) {\n    Modal.getInstance(alreadyOpen).hide();\n  }\n\n  const data = Modal.getOrCreateInstance(target);\n  data.toggle(this);\n});\nenableDismissTrigger(Modal);\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Modal);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): offcanvas.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$6 = 'offcanvas';\nconst DATA_KEY$3 = 'bs.offcanvas';\nconst EVENT_KEY$3 = `.${DATA_KEY$3}`;\nconst DATA_API_KEY$1 = '.data-api';\nconst EVENT_LOAD_DATA_API$2 = `load${EVENT_KEY$3}${DATA_API_KEY$1}`;\nconst ESCAPE_KEY = 'Escape';\nconst CLASS_NAME_SHOW$3 = 'show';\nconst CLASS_NAME_SHOWING$1 = 'showing';\nconst CLASS_NAME_HIDING = 'hiding';\nconst CLASS_NAME_BACKDROP = 'offcanvas-backdrop';\nconst OPEN_SELECTOR = '.offcanvas.show';\nconst EVENT_SHOW$3 = `show${EVENT_KEY$3}`;\nconst EVENT_SHOWN$3 = `shown${EVENT_KEY$3}`;\nconst EVENT_HIDE$3 = `hide${EVENT_KEY$3}`;\nconst EVENT_HIDE_PREVENTED = `hidePrevented${EVENT_KEY$3}`;\nconst EVENT_HIDDEN$3 = `hidden${EVENT_KEY$3}`;\nconst EVENT_RESIZE = `resize${EVENT_KEY$3}`;\nconst EVENT_CLICK_DATA_API$1 = `click${EVENT_KEY$3}${DATA_API_KEY$1}`;\nconst EVENT_KEYDOWN_DISMISS = `keydown.dismiss${EVENT_KEY$3}`;\nconst SELECTOR_DATA_TOGGLE$1 = '[data-bs-toggle=\"offcanvas\"]';\nconst Default$5 = {\n  backdrop: true,\n  keyboard: true,\n  scroll: false\n};\nconst DefaultType$5 = {\n  backdrop: '(boolean|string)',\n  keyboard: 'boolean',\n  scroll: 'boolean'\n};\n/**\n * Class definition\n */\n\nclass Offcanvas extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._isShown = false;\n    this._backdrop = this._initializeBackDrop();\n    this._focustrap = this._initializeFocusTrap();\n\n    this._addEventListeners();\n  } // Getters\n\n\n  static get Default() {\n    return Default$5;\n  }\n\n  static get DefaultType() {\n    return DefaultType$5;\n  }\n\n  static get NAME() {\n    return NAME$6;\n  } // Public\n\n\n  toggle(relatedTarget) {\n    return this._isShown ? this.hide() : this.show(relatedTarget);\n  }\n\n  show(relatedTarget) {\n    if (this._isShown) {\n      return;\n    }\n\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$3, {\n      relatedTarget\n    });\n\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n\n    this._isShown = true;\n\n    this._backdrop.show();\n\n    if (!this._config.scroll) {\n      new ScrollBarHelper().hide();\n    }\n\n    this._element.setAttribute('aria-modal', true);\n\n    this._element.setAttribute('role', 'dialog');\n\n    this._element.classList.add(CLASS_NAME_SHOWING$1);\n\n    const completeCallBack = () => {\n      if (!this._config.scroll || this._config.backdrop) {\n        this._focustrap.activate();\n      }\n\n      this._element.classList.add(CLASS_NAME_SHOW$3);\n\n      this._element.classList.remove(CLASS_NAME_SHOWING$1);\n\n      EventHandler.trigger(this._element, EVENT_SHOWN$3, {\n        relatedTarget\n      });\n    };\n\n    this._queueCallback(completeCallBack, this._element, true);\n  }\n\n  hide() {\n    if (!this._isShown) {\n      return;\n    }\n\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$3);\n\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n\n    this._focustrap.deactivate();\n\n    this._element.blur();\n\n    this._isShown = false;\n\n    this._element.classList.add(CLASS_NAME_HIDING);\n\n    this._backdrop.hide();\n\n    const completeCallback = () => {\n      this._element.classList.remove(CLASS_NAME_SHOW$3, CLASS_NAME_HIDING);\n\n      this._element.removeAttribute('aria-modal');\n\n      this._element.removeAttribute('role');\n\n      if (!this._config.scroll) {\n        new ScrollBarHelper().reset();\n      }\n\n      EventHandler.trigger(this._element, EVENT_HIDDEN$3);\n    };\n\n    this._queueCallback(completeCallback, this._element, true);\n  }\n\n  dispose() {\n    this._backdrop.dispose();\n\n    this._focustrap.deactivate();\n\n    super.dispose();\n  } // Private\n\n\n  _initializeBackDrop() {\n    const clickCallback = () => {\n      if (this._config.backdrop === 'static') {\n        EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED);\n        return;\n      }\n\n      this.hide();\n    }; // 'static' option will be translated to true, and booleans will keep their value\n\n\n    const isVisible = Boolean(this._config.backdrop);\n    return new Backdrop({\n      className: CLASS_NAME_BACKDROP,\n      isVisible,\n      isAnimated: true,\n      rootElement: this._element.parentNode,\n      clickCallback: isVisible ? clickCallback : null\n    });\n  }\n\n  _initializeFocusTrap() {\n    return new FocusTrap({\n      trapElement: this._element\n    });\n  }\n\n  _addEventListeners() {\n    EventHandler.on(this._element, EVENT_KEYDOWN_DISMISS, event => {\n      if (event.key !== ESCAPE_KEY) {\n        return;\n      }\n\n      if (!this._config.keyboard) {\n        EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED);\n        return;\n      }\n\n      this.hide();\n    });\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Offcanvas.getOrCreateInstance(this, config);\n\n      if (typeof config !== 'string') {\n        return;\n      }\n\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n\n      data[config](this);\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$1, SELECTOR_DATA_TOGGLE$1, function (event) {\n  const target = getElementFromSelector(this);\n\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n\n  if (isDisabled(this)) {\n    return;\n  }\n\n  EventHandler.one(target, EVENT_HIDDEN$3, () => {\n    // focus on trigger when it is closed\n    if (isVisible(this)) {\n      this.focus();\n    }\n  }); // avoid conflict when clicking a toggler of an offcanvas, while another is open\n\n  const alreadyOpen = SelectorEngine.findOne(OPEN_SELECTOR);\n\n  if (alreadyOpen && alreadyOpen !== target) {\n    Offcanvas.getInstance(alreadyOpen).hide();\n  }\n\n  const data = Offcanvas.getOrCreateInstance(target);\n  data.toggle(this);\n});\nEventHandler.on(window, EVENT_LOAD_DATA_API$2, () => {\n  for (const selector of SelectorEngine.find(OPEN_SELECTOR)) {\n    Offcanvas.getOrCreateInstance(selector).show();\n  }\n});\nEventHandler.on(window, EVENT_RESIZE, () => {\n  for (const element of SelectorEngine.find('[aria-modal][class*=show][class*=offcanvas-]')) {\n    if (getComputedStyle(element).position !== 'fixed') {\n      Offcanvas.getOrCreateInstance(element).hide();\n    }\n  }\n});\nenableDismissTrigger(Offcanvas);\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Offcanvas);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/sanitizer.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\nconst uriAttributes = new Set(['background', 'cite', 'href', 'itemtype', 'longdesc', 'poster', 'src', 'xlink:href']);\nconst ARIA_ATTRIBUTE_PATTERN = /^aria-[\\w-]*$/i;\n/**\n * A pattern that recognizes a commonly useful subset of URLs that are safe.\n *\n * Shout-out to Angular https://github.com/angular/angular/blob/12.2.x/packages/core/src/sanitization/url_sanitizer.ts\n */\n\nconst SAFE_URL_PATTERN = /^(?:(?:https?|mailto|ftp|tel|file|sms):|[^#&/:?]*(?:[#/?]|$))/i;\n/**\n * A pattern that matches safe data URLs. Only matches image, video and audio types.\n *\n * Shout-out to Angular https://github.com/angular/angular/blob/12.2.x/packages/core/src/sanitization/url_sanitizer.ts\n */\n\nconst DATA_URL_PATTERN = /^data:(?:image\\/(?:bmp|gif|jpeg|jpg|png|tiff|webp)|video\\/(?:mpeg|mp4|ogg|webm)|audio\\/(?:mp3|oga|ogg|opus));base64,[\\d+/a-z]+=*$/i;\n\nconst allowedAttribute = (attribute, allowedAttributeList) => {\n  const attributeName = attribute.nodeName.toLowerCase();\n\n  if (allowedAttributeList.includes(attributeName)) {\n    if (uriAttributes.has(attributeName)) {\n      return Boolean(SAFE_URL_PATTERN.test(attribute.nodeValue) || DATA_URL_PATTERN.test(attribute.nodeValue));\n    }\n\n    return true;\n  } // Check if a regular expression validates the attribute.\n\n\n  return allowedAttributeList.filter(attributeRegex => attributeRegex instanceof RegExp).some(regex => regex.test(attributeName));\n};\n\nconst DefaultAllowlist = {\n  // Global attributes allowed on any supplied element below.\n  '*': ['class', 'dir', 'id', 'lang', 'role', ARIA_ATTRIBUTE_PATTERN],\n  a: ['target', 'href', 'title', 'rel'],\n  area: [],\n  b: [],\n  br: [],\n  col: [],\n  code: [],\n  div: [],\n  em: [],\n  hr: [],\n  h1: [],\n  h2: [],\n  h3: [],\n  h4: [],\n  h5: [],\n  h6: [],\n  i: [],\n  img: ['src', 'srcset', 'alt', 'title', 'width', 'height'],\n  li: [],\n  ol: [],\n  p: [],\n  pre: [],\n  s: [],\n  small: [],\n  span: [],\n  sub: [],\n  sup: [],\n  strong: [],\n  u: [],\n  ul: []\n};\nfunction sanitizeHtml(unsafeHtml, allowList, sanitizeFunction) {\n  if (!unsafeHtml.length) {\n    return unsafeHtml;\n  }\n\n  if (sanitizeFunction && typeof sanitizeFunction === 'function') {\n    return sanitizeFunction(unsafeHtml);\n  }\n\n  const domParser = new window.DOMParser();\n  const createdDocument = domParser.parseFromString(unsafeHtml, 'text/html');\n  const elements = [].concat(...createdDocument.body.querySelectorAll('*'));\n\n  for (const element of elements) {\n    const elementName = element.nodeName.toLowerCase();\n\n    if (!Object.keys(allowList).includes(elementName)) {\n      element.remove();\n      continue;\n    }\n\n    const attributeList = [].concat(...element.attributes);\n    const allowedAttributes = [].concat(allowList['*'] || [], allowList[elementName] || []);\n\n    for (const attribute of attributeList) {\n      if (!allowedAttribute(attribute, allowedAttributes)) {\n        element.removeAttribute(attribute.nodeName);\n      }\n    }\n  }\n\n  return createdDocument.body.innerHTML;\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): util/template-factory.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$5 = 'TemplateFactory';\nconst Default$4 = {\n  allowList: DefaultAllowlist,\n  content: {},\n  // { selector : text ,  selector2 : text2 , }\n  extraClass: '',\n  html: false,\n  sanitize: true,\n  sanitizeFn: null,\n  template: '<div></div>'\n};\nconst DefaultType$4 = {\n  allowList: 'object',\n  content: 'object',\n  extraClass: '(string|function)',\n  html: 'boolean',\n  sanitize: 'boolean',\n  sanitizeFn: '(null|function)',\n  template: 'string'\n};\nconst DefaultContentType = {\n  entry: '(string|element|function|null)',\n  selector: '(string|element)'\n};\n/**\n * Class definition\n */\n\nclass TemplateFactory extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n  } // Getters\n\n\n  static get Default() {\n    return Default$4;\n  }\n\n  static get DefaultType() {\n    return DefaultType$4;\n  }\n\n  static get NAME() {\n    return NAME$5;\n  } // Public\n\n\n  getContent() {\n    return Object.values(this._config.content).map(config => this._resolvePossibleFunction(config)).filter(Boolean);\n  }\n\n  hasContent() {\n    return this.getContent().length > 0;\n  }\n\n  changeContent(content) {\n    this._checkContent(content);\n\n    this._config.content = { ...this._config.content,\n      ...content\n    };\n    return this;\n  }\n\n  toHtml() {\n    const templateWrapper = document.createElement('div');\n    templateWrapper.innerHTML = this._maybeSanitize(this._config.template);\n\n    for (const [selector, text] of Object.entries(this._config.content)) {\n      this._setContent(templateWrapper, text, selector);\n    }\n\n    const template = templateWrapper.children[0];\n\n    const extraClass = this._resolvePossibleFunction(this._config.extraClass);\n\n    if (extraClass) {\n      template.classList.add(...extraClass.split(' '));\n    }\n\n    return template;\n  } // Private\n\n\n  _typeCheckConfig(config) {\n    super._typeCheckConfig(config);\n\n    this._checkContent(config.content);\n  }\n\n  _checkContent(arg) {\n    for (const [selector, content] of Object.entries(arg)) {\n      super._typeCheckConfig({\n        selector,\n        entry: content\n      }, DefaultContentType);\n    }\n  }\n\n  _setContent(template, content, selector) {\n    const templateElement = SelectorEngine.findOne(selector, template);\n\n    if (!templateElement) {\n      return;\n    }\n\n    content = this._resolvePossibleFunction(content);\n\n    if (!content) {\n      templateElement.remove();\n      return;\n    }\n\n    if (isElement(content)) {\n      this._putElementInTemplate(getElement(content), templateElement);\n\n      return;\n    }\n\n    if (this._config.html) {\n      templateElement.innerHTML = this._maybeSanitize(content);\n      return;\n    }\n\n    templateElement.textContent = content;\n  }\n\n  _maybeSanitize(arg) {\n    return this._config.sanitize ? sanitizeHtml(arg, this._config.allowList, this._config.sanitizeFn) : arg;\n  }\n\n  _resolvePossibleFunction(arg) {\n    return typeof arg === 'function' ? arg(this) : arg;\n  }\n\n  _putElementInTemplate(element, templateElement) {\n    if (this._config.html) {\n      templateElement.innerHTML = '';\n      templateElement.append(element);\n      return;\n    }\n\n    templateElement.textContent = element.textContent;\n  }\n\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): tooltip.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$4 = 'tooltip';\nconst DISALLOWED_ATTRIBUTES = new Set(['sanitize', 'allowList', 'sanitizeFn']);\nconst CLASS_NAME_FADE$2 = 'fade';\nconst CLASS_NAME_MODAL = 'modal';\nconst CLASS_NAME_SHOW$2 = 'show';\nconst SELECTOR_TOOLTIP_INNER = '.tooltip-inner';\nconst SELECTOR_MODAL = `.${CLASS_NAME_MODAL}`;\nconst EVENT_MODAL_HIDE = 'hide.bs.modal';\nconst TRIGGER_HOVER = 'hover';\nconst TRIGGER_FOCUS = 'focus';\nconst TRIGGER_CLICK = 'click';\nconst TRIGGER_MANUAL = 'manual';\nconst EVENT_HIDE$2 = 'hide';\nconst EVENT_HIDDEN$2 = 'hidden';\nconst EVENT_SHOW$2 = 'show';\nconst EVENT_SHOWN$2 = 'shown';\nconst EVENT_INSERTED = 'inserted';\nconst EVENT_CLICK$1 = 'click';\nconst EVENT_FOCUSIN$1 = 'focusin';\nconst EVENT_FOCUSOUT$1 = 'focusout';\nconst EVENT_MOUSEENTER = 'mouseenter';\nconst EVENT_MOUSELEAVE = 'mouseleave';\nconst AttachmentMap = {\n  AUTO: 'auto',\n  TOP: 'top',\n  RIGHT: isRTL() ? 'left' : 'right',\n  BOTTOM: 'bottom',\n  LEFT: isRTL() ? 'right' : 'left'\n};\nconst Default$3 = {\n  allowList: DefaultAllowlist,\n  animation: true,\n  boundary: 'clippingParents',\n  container: false,\n  customClass: '',\n  delay: 0,\n  fallbackPlacements: ['top', 'right', 'bottom', 'left'],\n  html: false,\n  offset: [0, 0],\n  placement: 'top',\n  popperConfig: null,\n  sanitize: true,\n  sanitizeFn: null,\n  selector: false,\n  template: '<div class=\"tooltip\" role=\"tooltip\">' + '<div class=\"tooltip-arrow\"></div>' + '<div class=\"tooltip-inner\"></div>' + '</div>',\n  title: '',\n  trigger: 'hover focus'\n};\nconst DefaultType$3 = {\n  allowList: 'object',\n  animation: 'boolean',\n  boundary: '(string|element)',\n  container: '(string|element|boolean)',\n  customClass: '(string|function)',\n  delay: '(number|object)',\n  fallbackPlacements: 'array',\n  html: 'boolean',\n  offset: '(array|string|function)',\n  placement: '(string|function)',\n  popperConfig: '(null|object|function)',\n  sanitize: 'boolean',\n  sanitizeFn: '(null|function)',\n  selector: '(string|boolean)',\n  template: 'string',\n  title: '(string|element|function)',\n  trigger: 'string'\n};\n/**\n * Class definition\n */\n\nclass Tooltip extends BaseComponent {\n  constructor(element, config) {\n    if (typeof Popper === 'undefined') {\n      throw new TypeError('Bootstrap\\'s tooltips require Popper (https://popper.js.org)');\n    }\n\n    super(element, config); // Private\n\n    this._isEnabled = true;\n    this._timeout = 0;\n    this._isHovered = null;\n    this._activeTrigger = {};\n    this._popper = null;\n    this._templateFactory = null;\n    this._newContent = null; // Protected\n\n    this.tip = null;\n\n    this._setListeners();\n\n    if (!this._config.selector) {\n      this._fixTitle();\n    }\n  } // Getters\n\n\n  static get Default() {\n    return Default$3;\n  }\n\n  static get DefaultType() {\n    return DefaultType$3;\n  }\n\n  static get NAME() {\n    return NAME$4;\n  } // Public\n\n\n  enable() {\n    this._isEnabled = true;\n  }\n\n  disable() {\n    this._isEnabled = false;\n  }\n\n  toggleEnabled() {\n    this._isEnabled = !this._isEnabled;\n  }\n\n  toggle() {\n    if (!this._isEnabled) {\n      return;\n    }\n\n    this._activeTrigger.click = !this._activeTrigger.click;\n\n    if (this._isShown()) {\n      this._leave();\n\n      return;\n    }\n\n    this._enter();\n  }\n\n  dispose() {\n    clearTimeout(this._timeout);\n    EventHandler.off(this._element.closest(SELECTOR_MODAL), EVENT_MODAL_HIDE, this._hideModalHandler);\n\n    if (this._element.getAttribute('data-bs-original-title')) {\n      this._element.setAttribute('title', this._element.getAttribute('data-bs-original-title'));\n    }\n\n    this._disposePopper();\n\n    super.dispose();\n  }\n\n  show() {\n    if (this._element.style.display === 'none') {\n      throw new Error('Please use show on visible elements');\n    }\n\n    if (!(this._isWithContent() && this._isEnabled)) {\n      return;\n    }\n\n    const showEvent = EventHandler.trigger(this._element, this.constructor.eventName(EVENT_SHOW$2));\n    const shadowRoot = findShadowRoot(this._element);\n\n    const isInTheDom = (shadowRoot || this._element.ownerDocument.documentElement).contains(this._element);\n\n    if (showEvent.defaultPrevented || !isInTheDom) {\n      return;\n    } // todo v6 remove this OR make it optional\n\n\n    this._disposePopper();\n\n    const tip = this._getTipElement();\n\n    this._element.setAttribute('aria-describedby', tip.getAttribute('id'));\n\n    const {\n      container\n    } = this._config;\n\n    if (!this._element.ownerDocument.documentElement.contains(this.tip)) {\n      container.append(tip);\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_INSERTED));\n    }\n\n    this._popper = this._createPopper(tip);\n    tip.classList.add(CLASS_NAME_SHOW$2); // If this is a touch-enabled device we add extra\n    // empty mouseover listeners to the body's immediate children;\n    // only needed because of broken event delegation on iOS\n    // https://www.quirksmode.org/blog/archives/2014/02/mouse_event_bub.html\n\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.on(element, 'mouseover', noop);\n      }\n    }\n\n    const complete = () => {\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_SHOWN$2));\n\n      if (this._isHovered === false) {\n        this._leave();\n      }\n\n      this._isHovered = false;\n    };\n\n    this._queueCallback(complete, this.tip, this._isAnimated());\n  }\n\n  hide() {\n    if (!this._isShown()) {\n      return;\n    }\n\n    const hideEvent = EventHandler.trigger(this._element, this.constructor.eventName(EVENT_HIDE$2));\n\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n\n    const tip = this._getTipElement();\n\n    tip.classList.remove(CLASS_NAME_SHOW$2); // If this is a touch-enabled device we remove the extra\n    // empty mouseover listeners we added for iOS support\n\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.off(element, 'mouseover', noop);\n      }\n    }\n\n    this._activeTrigger[TRIGGER_CLICK] = false;\n    this._activeTrigger[TRIGGER_FOCUS] = false;\n    this._activeTrigger[TRIGGER_HOVER] = false;\n    this._isHovered = null; // it is a trick to support manual triggering\n\n    const complete = () => {\n      if (this._isWithActiveTrigger()) {\n        return;\n      }\n\n      if (!this._isHovered) {\n        this._disposePopper();\n      }\n\n      this._element.removeAttribute('aria-describedby');\n\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_HIDDEN$2));\n    };\n\n    this._queueCallback(complete, this.tip, this._isAnimated());\n  }\n\n  update() {\n    if (this._popper) {\n      this._popper.update();\n    }\n  } // Protected\n\n\n  _isWithContent() {\n    return Boolean(this._getTitle());\n  }\n\n  _getTipElement() {\n    if (!this.tip) {\n      this.tip = this._createTipElement(this._newContent || this._getContentForTemplate());\n    }\n\n    return this.tip;\n  }\n\n  _createTipElement(content) {\n    const tip = this._getTemplateFactory(content).toHtml(); // todo: remove this check on v6\n\n\n    if (!tip) {\n      return null;\n    }\n\n    tip.classList.remove(CLASS_NAME_FADE$2, CLASS_NAME_SHOW$2); // todo: on v6 the following can be achieved with CSS only\n\n    tip.classList.add(`bs-${this.constructor.NAME}-auto`);\n    const tipId = getUID(this.constructor.NAME).toString();\n    tip.setAttribute('id', tipId);\n\n    if (this._isAnimated()) {\n      tip.classList.add(CLASS_NAME_FADE$2);\n    }\n\n    return tip;\n  }\n\n  setContent(content) {\n    this._newContent = content;\n\n    if (this._isShown()) {\n      this._disposePopper();\n\n      this.show();\n    }\n  }\n\n  _getTemplateFactory(content) {\n    if (this._templateFactory) {\n      this._templateFactory.changeContent(content);\n    } else {\n      this._templateFactory = new TemplateFactory({ ...this._config,\n        // the `content` var has to be after `this._config`\n        // to override config.content in case of popover\n        content,\n        extraClass: this._resolvePossibleFunction(this._config.customClass)\n      });\n    }\n\n    return this._templateFactory;\n  }\n\n  _getContentForTemplate() {\n    return {\n      [SELECTOR_TOOLTIP_INNER]: this._getTitle()\n    };\n  }\n\n  _getTitle() {\n    return this._resolvePossibleFunction(this._config.title) || this._element.getAttribute('data-bs-original-title');\n  } // Private\n\n\n  _initializeOnDelegatedTarget(event) {\n    return this.constructor.getOrCreateInstance(event.delegateTarget, this._getDelegateConfig());\n  }\n\n  _isAnimated() {\n    return this._config.animation || this.tip && this.tip.classList.contains(CLASS_NAME_FADE$2);\n  }\n\n  _isShown() {\n    return this.tip && this.tip.classList.contains(CLASS_NAME_SHOW$2);\n  }\n\n  _createPopper(tip) {\n    const placement = typeof this._config.placement === 'function' ? this._config.placement.call(this, tip, this._element) : this._config.placement;\n    const attachment = AttachmentMap[placement.toUpperCase()];\n    return Popper.createPopper(this._element, tip, this._getPopperConfig(attachment));\n  }\n\n  _getOffset() {\n    const {\n      offset\n    } = this._config;\n\n    if (typeof offset === 'string') {\n      return offset.split(',').map(value => Number.parseInt(value, 10));\n    }\n\n    if (typeof offset === 'function') {\n      return popperData => offset(popperData, this._element);\n    }\n\n    return offset;\n  }\n\n  _resolvePossibleFunction(arg) {\n    return typeof arg === 'function' ? arg.call(this._element) : arg;\n  }\n\n  _getPopperConfig(attachment) {\n    const defaultBsPopperConfig = {\n      placement: attachment,\n      modifiers: [{\n        name: 'flip',\n        options: {\n          fallbackPlacements: this._config.fallbackPlacements\n        }\n      }, {\n        name: 'offset',\n        options: {\n          offset: this._getOffset()\n        }\n      }, {\n        name: 'preventOverflow',\n        options: {\n          boundary: this._config.boundary\n        }\n      }, {\n        name: 'arrow',\n        options: {\n          element: `.${this.constructor.NAME}-arrow`\n        }\n      }, {\n        name: 'preSetPlacement',\n        enabled: true,\n        phase: 'beforeMain',\n        fn: data => {\n          // Pre-set Popper's placement attribute in order to read the arrow sizes properly.\n          // Otherwise, Popper mixes up the width and height dimensions since the initial arrow style is for top placement\n          this._getTipElement().setAttribute('data-popper-placement', data.state.placement);\n        }\n      }]\n    };\n    return { ...defaultBsPopperConfig,\n      ...(typeof this._config.popperConfig === 'function' ? this._config.popperConfig(defaultBsPopperConfig) : this._config.popperConfig)\n    };\n  }\n\n  _setListeners() {\n    const triggers = this._config.trigger.split(' ');\n\n    for (const trigger of triggers) {\n      if (trigger === 'click') {\n        EventHandler.on(this._element, this.constructor.eventName(EVENT_CLICK$1), this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n\n          context.toggle();\n        });\n      } else if (trigger !== TRIGGER_MANUAL) {\n        const eventIn = trigger === TRIGGER_HOVER ? this.constructor.eventName(EVENT_MOUSEENTER) : this.constructor.eventName(EVENT_FOCUSIN$1);\n        const eventOut = trigger === TRIGGER_HOVER ? this.constructor.eventName(EVENT_MOUSELEAVE) : this.constructor.eventName(EVENT_FOCUSOUT$1);\n        EventHandler.on(this._element, eventIn, this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n\n          context._activeTrigger[event.type === 'focusin' ? TRIGGER_FOCUS : TRIGGER_HOVER] = true;\n\n          context._enter();\n        });\n        EventHandler.on(this._element, eventOut, this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n\n          context._activeTrigger[event.type === 'focusout' ? TRIGGER_FOCUS : TRIGGER_HOVER] = context._element.contains(event.relatedTarget);\n\n          context._leave();\n        });\n      }\n    }\n\n    this._hideModalHandler = () => {\n      if (this._element) {\n        this.hide();\n      }\n    };\n\n    EventHandler.on(this._element.closest(SELECTOR_MODAL), EVENT_MODAL_HIDE, this._hideModalHandler);\n  }\n\n  _fixTitle() {\n    const title = this._element.getAttribute('title');\n\n    if (!title) {\n      return;\n    }\n\n    if (!this._element.getAttribute('aria-label') && !this._element.textContent.trim()) {\n      this._element.setAttribute('aria-label', title);\n    }\n\n    this._element.setAttribute('data-bs-original-title', title); // DO NOT USE IT. Is only for backwards compatibility\n\n\n    this._element.removeAttribute('title');\n  }\n\n  _enter() {\n    if (this._isShown() || this._isHovered) {\n      this._isHovered = true;\n      return;\n    }\n\n    this._isHovered = true;\n\n    this._setTimeout(() => {\n      if (this._isHovered) {\n        this.show();\n      }\n    }, this._config.delay.show);\n  }\n\n  _leave() {\n    if (this._isWithActiveTrigger()) {\n      return;\n    }\n\n    this._isHovered = false;\n\n    this._setTimeout(() => {\n      if (!this._isHovered) {\n        this.hide();\n      }\n    }, this._config.delay.hide);\n  }\n\n  _setTimeout(handler, timeout) {\n    clearTimeout(this._timeout);\n    this._timeout = setTimeout(handler, timeout);\n  }\n\n  _isWithActiveTrigger() {\n    return Object.values(this._activeTrigger).includes(true);\n  }\n\n  _getConfig(config) {\n    const dataAttributes = Manipulator.getDataAttributes(this._element);\n\n    for (const dataAttribute of Object.keys(dataAttributes)) {\n      if (DISALLOWED_ATTRIBUTES.has(dataAttribute)) {\n        delete dataAttributes[dataAttribute];\n      }\n    }\n\n    config = { ...dataAttributes,\n      ...(typeof config === 'object' && config ? config : {})\n    };\n    config = this._mergeConfigObj(config);\n    config = this._configAfterMerge(config);\n\n    this._typeCheckConfig(config);\n\n    return config;\n  }\n\n  _configAfterMerge(config) {\n    config.container = config.container === false ? document.body : getElement(config.container);\n\n    if (typeof config.delay === 'number') {\n      config.delay = {\n        show: config.delay,\n        hide: config.delay\n      };\n    }\n\n    if (typeof config.title === 'number') {\n      config.title = config.title.toString();\n    }\n\n    if (typeof config.content === 'number') {\n      config.content = config.content.toString();\n    }\n\n    return config;\n  }\n\n  _getDelegateConfig() {\n    const config = {};\n\n    for (const key in this._config) {\n      if (this.constructor.Default[key] !== this._config[key]) {\n        config[key] = this._config[key];\n      }\n    }\n\n    config.selector = false;\n    config.trigger = 'manual'; // In the future can be replaced with:\n    // const keysWithDifferentValues = Object.entries(this._config).filter(entry => this.constructor.Default[entry[0]] !== this._config[entry[0]])\n    // `Object.fromEntries(keysWithDifferentValues)`\n\n    return config;\n  }\n\n  _disposePopper() {\n    if (this._popper) {\n      this._popper.destroy();\n\n      this._popper = null;\n    }\n\n    if (this.tip) {\n      this.tip.remove();\n      this.tip = null;\n    }\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Tooltip.getOrCreateInstance(this, config);\n\n      if (typeof config !== 'string') {\n        return;\n      }\n\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n\n      data[config]();\n    });\n  }\n\n}\n/**\n * jQuery\n */\n\n\ndefineJQueryPlugin(Tooltip);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): popover.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$3 = 'popover';\nconst SELECTOR_TITLE = '.popover-header';\nconst SELECTOR_CONTENT = '.popover-body';\nconst Default$2 = { ...Tooltip.Default,\n  content: '',\n  offset: [0, 8],\n  placement: 'right',\n  template: '<div class=\"popover\" role=\"tooltip\">' + '<div class=\"popover-arrow\"></div>' + '<h3 class=\"popover-header\"></h3>' + '<div class=\"popover-body\"></div>' + '</div>',\n  trigger: 'click'\n};\nconst DefaultType$2 = { ...Tooltip.DefaultType,\n  content: '(null|string|element|function)'\n};\n/**\n * Class definition\n */\n\nclass Popover extends Tooltip {\n  // Getters\n  static get Default() {\n    return Default$2;\n  }\n\n  static get DefaultType() {\n    return DefaultType$2;\n  }\n\n  static get NAME() {\n    return NAME$3;\n  } // Overrides\n\n\n  _isWithContent() {\n    return this._getTitle() || this._getContent();\n  } // Private\n\n\n  _getContentForTemplate() {\n    return {\n      [SELECTOR_TITLE]: this._getTitle(),\n      [SELECTOR_CONTENT]: this._getContent()\n    };\n  }\n\n  _getContent() {\n    return this._resolvePossibleFunction(this._config.content);\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Popover.getOrCreateInstance(this, config);\n\n      if (typeof config !== 'string') {\n        return;\n      }\n\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n\n      data[config]();\n    });\n  }\n\n}\n/**\n * jQuery\n */\n\n\ndefineJQueryPlugin(Popover);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): scrollspy.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$2 = 'scrollspy';\nconst DATA_KEY$2 = 'bs.scrollspy';\nconst EVENT_KEY$2 = `.${DATA_KEY$2}`;\nconst DATA_API_KEY = '.data-api';\nconst EVENT_ACTIVATE = `activate${EVENT_KEY$2}`;\nconst EVENT_CLICK = `click${EVENT_KEY$2}`;\nconst EVENT_LOAD_DATA_API$1 = `load${EVENT_KEY$2}${DATA_API_KEY}`;\nconst CLASS_NAME_DROPDOWN_ITEM = 'dropdown-item';\nconst CLASS_NAME_ACTIVE$1 = 'active';\nconst SELECTOR_DATA_SPY = '[data-bs-spy=\"scroll\"]';\nconst SELECTOR_TARGET_LINKS = '[href]';\nconst SELECTOR_NAV_LIST_GROUP = '.nav, .list-group';\nconst SELECTOR_NAV_LINKS = '.nav-link';\nconst SELECTOR_NAV_ITEMS = '.nav-item';\nconst SELECTOR_LIST_ITEMS = '.list-group-item';\nconst SELECTOR_LINK_ITEMS = `${SELECTOR_NAV_LINKS}, ${SELECTOR_NAV_ITEMS} > ${SELECTOR_NAV_LINKS}, ${SELECTOR_LIST_ITEMS}`;\nconst SELECTOR_DROPDOWN = '.dropdown';\nconst SELECTOR_DROPDOWN_TOGGLE$1 = '.dropdown-toggle';\nconst Default$1 = {\n  offset: null,\n  // TODO: v6 @deprecated, keep it for backwards compatibility reasons\n  rootMargin: '0px 0px -25%',\n  smoothScroll: false,\n  target: null,\n  threshold: [0.1, 0.5, 1]\n};\nconst DefaultType$1 = {\n  offset: '(number|null)',\n  // TODO v6 @deprecated, keep it for backwards compatibility reasons\n  rootMargin: 'string',\n  smoothScroll: 'boolean',\n  target: 'element',\n  threshold: 'array'\n};\n/**\n * Class definition\n */\n\nclass ScrollSpy extends BaseComponent {\n  constructor(element, config) {\n    super(element, config); // this._element is the observablesContainer and config.target the menu links wrapper\n\n    this._targetLinks = new Map();\n    this._observableSections = new Map();\n    this._rootElement = getComputedStyle(this._element).overflowY === 'visible' ? null : this._element;\n    this._activeTarget = null;\n    this._observer = null;\n    this._previousScrollData = {\n      visibleEntryTop: 0,\n      parentScrollTop: 0\n    };\n    this.refresh(); // initialize\n  } // Getters\n\n\n  static get Default() {\n    return Default$1;\n  }\n\n  static get DefaultType() {\n    return DefaultType$1;\n  }\n\n  static get NAME() {\n    return NAME$2;\n  } // Public\n\n\n  refresh() {\n    this._initializeTargetsAndObservables();\n\n    this._maybeEnableSmoothScroll();\n\n    if (this._observer) {\n      this._observer.disconnect();\n    } else {\n      this._observer = this._getNewObserver();\n    }\n\n    for (const section of this._observableSections.values()) {\n      this._observer.observe(section);\n    }\n  }\n\n  dispose() {\n    this._observer.disconnect();\n\n    super.dispose();\n  } // Private\n\n\n  _configAfterMerge(config) {\n    // TODO: on v6 target should be given explicitly & remove the {target: 'ss-target'} case\n    config.target = getElement(config.target) || document.body; // TODO: v6 Only for backwards compatibility reasons. Use rootMargin only\n\n    config.rootMargin = config.offset ? `${config.offset}px 0px -30%` : config.rootMargin;\n\n    if (typeof config.threshold === 'string') {\n      config.threshold = config.threshold.split(',').map(value => Number.parseFloat(value));\n    }\n\n    return config;\n  }\n\n  _maybeEnableSmoothScroll() {\n    if (!this._config.smoothScroll) {\n      return;\n    } // unregister any previous listeners\n\n\n    EventHandler.off(this._config.target, EVENT_CLICK);\n    EventHandler.on(this._config.target, EVENT_CLICK, SELECTOR_TARGET_LINKS, event => {\n      const observableSection = this._observableSections.get(event.target.hash);\n\n      if (observableSection) {\n        event.preventDefault();\n        const root = this._rootElement || window;\n        const height = observableSection.offsetTop - this._element.offsetTop;\n\n        if (root.scrollTo) {\n          root.scrollTo({\n            top: height,\n            behavior: 'smooth'\n          });\n          return;\n        } // Chrome 60 doesn't support `scrollTo`\n\n\n        root.scrollTop = height;\n      }\n    });\n  }\n\n  _getNewObserver() {\n    const options = {\n      root: this._rootElement,\n      threshold: this._config.threshold,\n      rootMargin: this._config.rootMargin\n    };\n    return new IntersectionObserver(entries => this._observerCallback(entries), options);\n  } // The logic of selection\n\n\n  _observerCallback(entries) {\n    const targetElement = entry => this._targetLinks.get(`#${entry.target.id}`);\n\n    const activate = entry => {\n      this._previousScrollData.visibleEntryTop = entry.target.offsetTop;\n\n      this._process(targetElement(entry));\n    };\n\n    const parentScrollTop = (this._rootElement || document.documentElement).scrollTop;\n    const userScrollsDown = parentScrollTop >= this._previousScrollData.parentScrollTop;\n    this._previousScrollData.parentScrollTop = parentScrollTop;\n\n    for (const entry of entries) {\n      if (!entry.isIntersecting) {\n        this._activeTarget = null;\n\n        this._clearActiveClass(targetElement(entry));\n\n        continue;\n      }\n\n      const entryIsLowerThanPrevious = entry.target.offsetTop >= this._previousScrollData.visibleEntryTop; // if we are scrolling down, pick the bigger offsetTop\n\n      if (userScrollsDown && entryIsLowerThanPrevious) {\n        activate(entry); // if parent isn't scrolled, let's keep the first visible item, breaking the iteration\n\n        if (!parentScrollTop) {\n          return;\n        }\n\n        continue;\n      } // if we are scrolling up, pick the smallest offsetTop\n\n\n      if (!userScrollsDown && !entryIsLowerThanPrevious) {\n        activate(entry);\n      }\n    }\n  }\n\n  _initializeTargetsAndObservables() {\n    this._targetLinks = new Map();\n    this._observableSections = new Map();\n    const targetLinks = SelectorEngine.find(SELECTOR_TARGET_LINKS, this._config.target);\n\n    for (const anchor of targetLinks) {\n      // ensure that the anchor has an id and is not disabled\n      if (!anchor.hash || isDisabled(anchor)) {\n        continue;\n      }\n\n      const observableSection = SelectorEngine.findOne(anchor.hash, this._element); // ensure that the observableSection exists & is visible\n\n      if (isVisible(observableSection)) {\n        this._targetLinks.set(anchor.hash, anchor);\n\n        this._observableSections.set(anchor.hash, observableSection);\n      }\n    }\n  }\n\n  _process(target) {\n    if (this._activeTarget === target) {\n      return;\n    }\n\n    this._clearActiveClass(this._config.target);\n\n    this._activeTarget = target;\n    target.classList.add(CLASS_NAME_ACTIVE$1);\n\n    this._activateParents(target);\n\n    EventHandler.trigger(this._element, EVENT_ACTIVATE, {\n      relatedTarget: target\n    });\n  }\n\n  _activateParents(target) {\n    // Activate dropdown parents\n    if (target.classList.contains(CLASS_NAME_DROPDOWN_ITEM)) {\n      SelectorEngine.findOne(SELECTOR_DROPDOWN_TOGGLE$1, target.closest(SELECTOR_DROPDOWN)).classList.add(CLASS_NAME_ACTIVE$1);\n      return;\n    }\n\n    for (const listGroup of SelectorEngine.parents(target, SELECTOR_NAV_LIST_GROUP)) {\n      // Set triggered links parents as active\n      // With both <ul> and <nav> markup a parent is the previous sibling of any nav ancestor\n      for (const item of SelectorEngine.prev(listGroup, SELECTOR_LINK_ITEMS)) {\n        item.classList.add(CLASS_NAME_ACTIVE$1);\n      }\n    }\n  }\n\n  _clearActiveClass(parent) {\n    parent.classList.remove(CLASS_NAME_ACTIVE$1);\n    const activeNodes = SelectorEngine.find(`${SELECTOR_TARGET_LINKS}.${CLASS_NAME_ACTIVE$1}`, parent);\n\n    for (const node of activeNodes) {\n      node.classList.remove(CLASS_NAME_ACTIVE$1);\n    }\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = ScrollSpy.getOrCreateInstance(this, config);\n\n      if (typeof config !== 'string') {\n        return;\n      }\n\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n\n      data[config]();\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nEventHandler.on(window, EVENT_LOAD_DATA_API$1, () => {\n  for (const spy of SelectorEngine.find(SELECTOR_DATA_SPY)) {\n    ScrollSpy.getOrCreateInstance(spy);\n  }\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(ScrollSpy);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): tab.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME$1 = 'tab';\nconst DATA_KEY$1 = 'bs.tab';\nconst EVENT_KEY$1 = `.${DATA_KEY$1}`;\nconst EVENT_HIDE$1 = `hide${EVENT_KEY$1}`;\nconst EVENT_HIDDEN$1 = `hidden${EVENT_KEY$1}`;\nconst EVENT_SHOW$1 = `show${EVENT_KEY$1}`;\nconst EVENT_SHOWN$1 = `shown${EVENT_KEY$1}`;\nconst EVENT_CLICK_DATA_API = `click${EVENT_KEY$1}`;\nconst EVENT_KEYDOWN = `keydown${EVENT_KEY$1}`;\nconst EVENT_LOAD_DATA_API = `load${EVENT_KEY$1}`;\nconst ARROW_LEFT_KEY = 'ArrowLeft';\nconst ARROW_RIGHT_KEY = 'ArrowRight';\nconst ARROW_UP_KEY = 'ArrowUp';\nconst ARROW_DOWN_KEY = 'ArrowDown';\nconst CLASS_NAME_ACTIVE = 'active';\nconst CLASS_NAME_FADE$1 = 'fade';\nconst CLASS_NAME_SHOW$1 = 'show';\nconst CLASS_DROPDOWN = 'dropdown';\nconst SELECTOR_DROPDOWN_TOGGLE = '.dropdown-toggle';\nconst SELECTOR_DROPDOWN_MENU = '.dropdown-menu';\nconst NOT_SELECTOR_DROPDOWN_TOGGLE = ':not(.dropdown-toggle)';\nconst SELECTOR_TAB_PANEL = '.list-group, .nav, [role=\"tablist\"]';\nconst SELECTOR_OUTER = '.nav-item, .list-group-item';\nconst SELECTOR_INNER = `.nav-link${NOT_SELECTOR_DROPDOWN_TOGGLE}, .list-group-item${NOT_SELECTOR_DROPDOWN_TOGGLE}, [role=\"tab\"]${NOT_SELECTOR_DROPDOWN_TOGGLE}`;\nconst SELECTOR_DATA_TOGGLE = '[data-bs-toggle=\"tab\"], [data-bs-toggle=\"pill\"], [data-bs-toggle=\"list\"]'; // todo:v6: could be only `tab`\n\nconst SELECTOR_INNER_ELEM = `${SELECTOR_INNER}, ${SELECTOR_DATA_TOGGLE}`;\nconst SELECTOR_DATA_TOGGLE_ACTIVE = `.${CLASS_NAME_ACTIVE}[data-bs-toggle=\"tab\"], .${CLASS_NAME_ACTIVE}[data-bs-toggle=\"pill\"], .${CLASS_NAME_ACTIVE}[data-bs-toggle=\"list\"]`;\n/**\n * Class definition\n */\n\nclass Tab extends BaseComponent {\n  constructor(element) {\n    super(element);\n    this._parent = this._element.closest(SELECTOR_TAB_PANEL);\n\n    if (!this._parent) {\n      return; // todo: should Throw exception on v6\n      // throw new TypeError(`${element.outerHTML} has not a valid parent ${SELECTOR_INNER_ELEM}`)\n    } // Set up initial aria attributes\n\n\n    this._setInitialAttributes(this._parent, this._getChildren());\n\n    EventHandler.on(this._element, EVENT_KEYDOWN, event => this._keydown(event));\n  } // Getters\n\n\n  static get NAME() {\n    return NAME$1;\n  } // Public\n\n\n  show() {\n    // Shows this elem and deactivate the active sibling if exists\n    const innerElem = this._element;\n\n    if (this._elemIsActive(innerElem)) {\n      return;\n    } // Search for active tab on same parent to deactivate it\n\n\n    const active = this._getActiveElem();\n\n    const hideEvent = active ? EventHandler.trigger(active, EVENT_HIDE$1, {\n      relatedTarget: innerElem\n    }) : null;\n    const showEvent = EventHandler.trigger(innerElem, EVENT_SHOW$1, {\n      relatedTarget: active\n    });\n\n    if (showEvent.defaultPrevented || hideEvent && hideEvent.defaultPrevented) {\n      return;\n    }\n\n    this._deactivate(active, innerElem);\n\n    this._activate(innerElem, active);\n  } // Private\n\n\n  _activate(element, relatedElem) {\n    if (!element) {\n      return;\n    }\n\n    element.classList.add(CLASS_NAME_ACTIVE);\n\n    this._activate(getElementFromSelector(element)); // Search and activate/show the proper section\n\n\n    const complete = () => {\n      if (element.getAttribute('role') !== 'tab') {\n        element.classList.add(CLASS_NAME_SHOW$1);\n        return;\n      }\n\n      element.removeAttribute('tabindex');\n      element.setAttribute('aria-selected', true);\n\n      this._toggleDropDown(element, true);\n\n      EventHandler.trigger(element, EVENT_SHOWN$1, {\n        relatedTarget: relatedElem\n      });\n    };\n\n    this._queueCallback(complete, element, element.classList.contains(CLASS_NAME_FADE$1));\n  }\n\n  _deactivate(element, relatedElem) {\n    if (!element) {\n      return;\n    }\n\n    element.classList.remove(CLASS_NAME_ACTIVE);\n    element.blur();\n\n    this._deactivate(getElementFromSelector(element)); // Search and deactivate the shown section too\n\n\n    const complete = () => {\n      if (element.getAttribute('role') !== 'tab') {\n        element.classList.remove(CLASS_NAME_SHOW$1);\n        return;\n      }\n\n      element.setAttribute('aria-selected', false);\n      element.setAttribute('tabindex', '-1');\n\n      this._toggleDropDown(element, false);\n\n      EventHandler.trigger(element, EVENT_HIDDEN$1, {\n        relatedTarget: relatedElem\n      });\n    };\n\n    this._queueCallback(complete, element, element.classList.contains(CLASS_NAME_FADE$1));\n  }\n\n  _keydown(event) {\n    if (![ARROW_LEFT_KEY, ARROW_RIGHT_KEY, ARROW_UP_KEY, ARROW_DOWN_KEY].includes(event.key)) {\n      return;\n    }\n\n    event.stopPropagation(); // stopPropagation/preventDefault both added to support up/down keys without scrolling the page\n\n    event.preventDefault();\n    const isNext = [ARROW_RIGHT_KEY, ARROW_DOWN_KEY].includes(event.key);\n    const nextActiveElement = getNextActiveElement(this._getChildren().filter(element => !isDisabled(element)), event.target, isNext, true);\n\n    if (nextActiveElement) {\n      nextActiveElement.focus({\n        preventScroll: true\n      });\n      Tab.getOrCreateInstance(nextActiveElement).show();\n    }\n  }\n\n  _getChildren() {\n    // collection of inner elements\n    return SelectorEngine.find(SELECTOR_INNER_ELEM, this._parent);\n  }\n\n  _getActiveElem() {\n    return this._getChildren().find(child => this._elemIsActive(child)) || null;\n  }\n\n  _setInitialAttributes(parent, children) {\n    this._setAttributeIfNotExists(parent, 'role', 'tablist');\n\n    for (const child of children) {\n      this._setInitialAttributesOnChild(child);\n    }\n  }\n\n  _setInitialAttributesOnChild(child) {\n    child = this._getInnerElement(child);\n\n    const isActive = this._elemIsActive(child);\n\n    const outerElem = this._getOuterElement(child);\n\n    child.setAttribute('aria-selected', isActive);\n\n    if (outerElem !== child) {\n      this._setAttributeIfNotExists(outerElem, 'role', 'presentation');\n    }\n\n    if (!isActive) {\n      child.setAttribute('tabindex', '-1');\n    }\n\n    this._setAttributeIfNotExists(child, 'role', 'tab'); // set attributes to the related panel too\n\n\n    this._setInitialAttributesOnTargetPanel(child);\n  }\n\n  _setInitialAttributesOnTargetPanel(child) {\n    const target = getElementFromSelector(child);\n\n    if (!target) {\n      return;\n    }\n\n    this._setAttributeIfNotExists(target, 'role', 'tabpanel');\n\n    if (child.id) {\n      this._setAttributeIfNotExists(target, 'aria-labelledby', `#${child.id}`);\n    }\n  }\n\n  _toggleDropDown(element, open) {\n    const outerElem = this._getOuterElement(element);\n\n    if (!outerElem.classList.contains(CLASS_DROPDOWN)) {\n      return;\n    }\n\n    const toggle = (selector, className) => {\n      const element = SelectorEngine.findOne(selector, outerElem);\n\n      if (element) {\n        element.classList.toggle(className, open);\n      }\n    };\n\n    toggle(SELECTOR_DROPDOWN_TOGGLE, CLASS_NAME_ACTIVE);\n    toggle(SELECTOR_DROPDOWN_MENU, CLASS_NAME_SHOW$1);\n    outerElem.setAttribute('aria-expanded', open);\n  }\n\n  _setAttributeIfNotExists(element, attribute, value) {\n    if (!element.hasAttribute(attribute)) {\n      element.setAttribute(attribute, value);\n    }\n  }\n\n  _elemIsActive(elem) {\n    return elem.classList.contains(CLASS_NAME_ACTIVE);\n  } // Try to get the inner element (usually the .nav-link)\n\n\n  _getInnerElement(elem) {\n    return elem.matches(SELECTOR_INNER_ELEM) ? elem : SelectorEngine.findOne(SELECTOR_INNER_ELEM, elem);\n  } // Try to get the outer element (usually the .nav-item)\n\n\n  _getOuterElement(elem) {\n    return elem.closest(SELECTOR_OUTER) || elem;\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Tab.getOrCreateInstance(this);\n\n      if (typeof config !== 'string') {\n        return;\n      }\n\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n\n      data[config]();\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nEventHandler.on(document, EVENT_CLICK_DATA_API, SELECTOR_DATA_TOGGLE, function (event) {\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n\n  if (isDisabled(this)) {\n    return;\n  }\n\n  Tab.getOrCreateInstance(this).show();\n});\n/**\n * Initialize on focus\n */\n\nEventHandler.on(window, EVENT_LOAD_DATA_API, () => {\n  for (const element of SelectorEngine.find(SELECTOR_DATA_TOGGLE_ACTIVE)) {\n    Tab.getOrCreateInstance(element);\n  }\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Tab);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap (v5.2.3): toast.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n/**\n * Constants\n */\n\nconst NAME = 'toast';\nconst DATA_KEY = 'bs.toast';\nconst EVENT_KEY = `.${DATA_KEY}`;\nconst EVENT_MOUSEOVER = `mouseover${EVENT_KEY}`;\nconst EVENT_MOUSEOUT = `mouseout${EVENT_KEY}`;\nconst EVENT_FOCUSIN = `focusin${EVENT_KEY}`;\nconst EVENT_FOCUSOUT = `focusout${EVENT_KEY}`;\nconst EVENT_HIDE = `hide${EVENT_KEY}`;\nconst EVENT_HIDDEN = `hidden${EVENT_KEY}`;\nconst EVENT_SHOW = `show${EVENT_KEY}`;\nconst EVENT_SHOWN = `shown${EVENT_KEY}`;\nconst CLASS_NAME_FADE = 'fade';\nconst CLASS_NAME_HIDE = 'hide'; // @deprecated - kept here only for backwards compatibility\n\nconst CLASS_NAME_SHOW = 'show';\nconst CLASS_NAME_SHOWING = 'showing';\nconst DefaultType = {\n  animation: 'boolean',\n  autohide: 'boolean',\n  delay: 'number'\n};\nconst Default = {\n  animation: true,\n  autohide: true,\n  delay: 5000\n};\n/**\n * Class definition\n */\n\nclass Toast extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._timeout = null;\n    this._hasMouseInteraction = false;\n    this._hasKeyboardInteraction = false;\n\n    this._setListeners();\n  } // Getters\n\n\n  static get Default() {\n    return Default;\n  }\n\n  static get DefaultType() {\n    return DefaultType;\n  }\n\n  static get NAME() {\n    return NAME;\n  } // Public\n\n\n  show() {\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW);\n\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n\n    this._clearTimeout();\n\n    if (this._config.animation) {\n      this._element.classList.add(CLASS_NAME_FADE);\n    }\n\n    const complete = () => {\n      this._element.classList.remove(CLASS_NAME_SHOWING);\n\n      EventHandler.trigger(this._element, EVENT_SHOWN);\n\n      this._maybeScheduleHide();\n    };\n\n    this._element.classList.remove(CLASS_NAME_HIDE); // @deprecated\n\n\n    reflow(this._element);\n\n    this._element.classList.add(CLASS_NAME_SHOW, CLASS_NAME_SHOWING);\n\n    this._queueCallback(complete, this._element, this._config.animation);\n  }\n\n  hide() {\n    if (!this.isShown()) {\n      return;\n    }\n\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE);\n\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n\n    const complete = () => {\n      this._element.classList.add(CLASS_NAME_HIDE); // @deprecated\n\n\n      this._element.classList.remove(CLASS_NAME_SHOWING, CLASS_NAME_SHOW);\n\n      EventHandler.trigger(this._element, EVENT_HIDDEN);\n    };\n\n    this._element.classList.add(CLASS_NAME_SHOWING);\n\n    this._queueCallback(complete, this._element, this._config.animation);\n  }\n\n  dispose() {\n    this._clearTimeout();\n\n    if (this.isShown()) {\n      this._element.classList.remove(CLASS_NAME_SHOW);\n    }\n\n    super.dispose();\n  }\n\n  isShown() {\n    return this._element.classList.contains(CLASS_NAME_SHOW);\n  } // Private\n\n\n  _maybeScheduleHide() {\n    if (!this._config.autohide) {\n      return;\n    }\n\n    if (this._hasMouseInteraction || this._hasKeyboardInteraction) {\n      return;\n    }\n\n    this._timeout = setTimeout(() => {\n      this.hide();\n    }, this._config.delay);\n  }\n\n  _onInteraction(event, isInteracting) {\n    switch (event.type) {\n      case 'mouseover':\n      case 'mouseout':\n        {\n          this._hasMouseInteraction = isInteracting;\n          break;\n        }\n\n      case 'focusin':\n      case 'focusout':\n        {\n          this._hasKeyboardInteraction = isInteracting;\n          break;\n        }\n    }\n\n    if (isInteracting) {\n      this._clearTimeout();\n\n      return;\n    }\n\n    const nextElement = event.relatedTarget;\n\n    if (this._element === nextElement || this._element.contains(nextElement)) {\n      return;\n    }\n\n    this._maybeScheduleHide();\n  }\n\n  _setListeners() {\n    EventHandler.on(this._element, EVENT_MOUSEOVER, event => this._onInteraction(event, true));\n    EventHandler.on(this._element, EVENT_MOUSEOUT, event => this._onInteraction(event, false));\n    EventHandler.on(this._element, EVENT_FOCUSIN, event => this._onInteraction(event, true));\n    EventHandler.on(this._element, EVENT_FOCUSOUT, event => this._onInteraction(event, false));\n  }\n\n  _clearTimeout() {\n    clearTimeout(this._timeout);\n    this._timeout = null;\n  } // Static\n\n\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Toast.getOrCreateInstance(this, config);\n\n      if (typeof config === 'string') {\n        if (typeof data[config] === 'undefined') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n\n        data[config](this);\n      }\n    });\n  }\n\n}\n/**\n * Data API implementation\n */\n\n\nenableDismissTrigger(Toast);\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Toast);\n\nexport { Alert, Button, Carousel, Collapse, Dropdown, Modal, Offcanvas, Popover, ScrollSpy, Tab, Toast, Tooltip };\n//# sourceMappingURL=bootstrap.esm.js.map\n","/* define several functions to replace jQuery methods\n * inspired by https://tobiasahlin.com/blog/move-from-jquery-to-vanilla-javascript/\n */\n\n/**\n * Execute a method if DOM has finished loading\n *\n * @param {function} callback the method to execute\n */\nexport function documentReady(callback) {\n  if (document.readyState != \"loading\") callback();\n  else document.addEventListener(\"DOMContentLoaded\", callback);\n}\n","// Import and setup functions to control Bootstrap's behavior.\nimport 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 if (latest_active.getBoundingClientRect().y > window.innerHeight * 0.5) {\n        let buffer = 0.25; // Buffer so we have some space above the scrolled item\n        sidebar.scrollTop = offset - sidebar.clientHeight * buffer;\n        console.log(\"[PST]: Scrolled sidebar using last active link...\");\n      }\n    }\n  }\n\n  // Store the sidebar scroll position\n  window.addEventListener(\"beforeunload\", () => {\n    sessionStorage.setItem(\"sidebar-scroll-top\", sidebar.scrollTop);\n  });\n}\n\n/*******************************************************************************\n * Search\n */\n\n/**\n * Find any search forms on the page and return their input element\n */\nvar findSearchInput = () => {\n  let forms = document.querySelectorAll(\"form.bd-search\");\n  if (!forms.length) {\n    // no search form found\n    return;\n  } else {\n    var form;\n    if (forms.length == 1) {\n      // there is exactly one search form (persistent or hidden)\n      form = forms[0];\n    } 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\ No newline at end of file
diff --git a/_static/scripts/sphinx-book-theme.js b/_static/scripts/sphinx-book-theme.js
new file mode 100644
index 00000000..be9fa44d
--- /dev/null
+++ b/_static/scripts/sphinx-book-theme.js
@@ -0,0 +1,2 @@
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diff --git a/_static/scripts/sphinx-book-theme.js.map b/_static/scripts/sphinx-book-theme.js.map
new file mode 100644
index 00000000..b8abcf86
--- /dev/null
+++ b/_static/scripts/sphinx-book-theme.js.map
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diff --git a/_static/searchtools.js b/_static/searchtools.js
new file mode 100644
index 00000000..0a44e858
--- /dev/null
+++ b/_static/searchtools.js
@@ -0,0 +1,525 @@
+/*
+ * searchtools.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * Sphinx JavaScript utilities for the full-text search.
+ *
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+if (!Scorer) {
+  /**
+   * Simple result scoring code.
+   */
+  var Scorer = {
+    // Implement the following function to further tweak the score for each result
+    // The function takes a result array [filename, title, anchor, descr, score]
+    // and returns the new score.
+    /*
+    score: function(result) {
+      return result[4];
+    },
+    */
+
+    // query matches the full name of an object
+    objNameMatch: 11,
+    // or matches in the last dotted part of the object name
+    objPartialMatch: 6,
+    // Additive scores depending on the priority of the object
+    objPrio: {0:  15,   // used to be importantResults
+              1:  5,   // used to be objectResults
+              2: -5},  // used to be unimportantResults
+    //  Used when the priority is not in the mapping.
+    objPrioDefault: 0,
+
+    // query found in title
+    title: 15,
+    partialTitle: 7,
+    // query found in terms
+    term: 5,
+    partialTerm: 2
+  };
+}
+
+if (!splitQuery) {
+  function splitQuery(query) {
+    return query.split(/\s+/);
+  }
+}
+
+/**
+ * Search Module
+ */
+var Search = {
+
+  _index : null,
+  _queued_query : null,
+  _pulse_status : -1,
+
+  htmlToText : function(htmlString) {
+      var virtualDocument = document.implementation.createHTMLDocument('virtual');
+      var htmlElement = $(htmlString, virtualDocument);
+      htmlElement.find('.headerlink').remove();
+      docContent = htmlElement.find('[role=main]')[0];
+      if(docContent === undefined) {
+          console.warn("Content block not found. Sphinx search tries to obtain it " +
+                       "via '[role=main]'. Could you check your theme or template.");
+          return "";
+      }
+      return docContent.textContent || docContent.innerText;
+  },
+
+  init : function() {
+      var params = $.getQueryParameters();
+      if (params.q) {
+          var query = params.q[0];
+          $('input[name="q"]')[0].value = query;
+          this.performSearch(query);
+      }
+  },
+
+  loadIndex : function(url) {
+    $.ajax({type: "GET", url: url, data: null,
+            dataType: "script", cache: true,
+            complete: function(jqxhr, textstatus) {
+              if (textstatus != "success") {
+                document.getElementById("searchindexloader").src = url;
+              }
+            }});
+  },
+
+  setIndex : function(index) {
+    var q;
+    this._index = index;
+    if ((q = this._queued_query) !== null) {
+      this._queued_query = null;
+      Search.query(q);
+    }
+  },
+
+  hasIndex : function() {
+      return this._index !== null;
+  },
+
+  deferQuery : function(query) {
+      this._queued_query = query;
+  },
+
+  stopPulse : function() {
+      this._pulse_status = 0;
+  },
+
+  startPulse : function() {
+    if (this._pulse_status >= 0)
+        return;
+    function pulse() {
+      var i;
+      Search._pulse_status = (Search._pulse_status + 1) % 4;
+      var dotString = '';
+      for (i = 0; i < Search._pulse_status; i++)
+        dotString += '.';
+      Search.dots.text(dotString);
+      if (Search._pulse_status > -1)
+        window.setTimeout(pulse, 500);
+    }
+    pulse();
+  },
+
+  /**
+   * perform a search for something (or wait until index is loaded)
+   */
+  performSearch : function(query) {
+    // create the required interface elements
+    this.out = $('#search-results');
+    this.title = $('<h2>' + _('Searching') + '</h2>').appendTo(this.out);
+    this.dots = $('<span></span>').appendTo(this.title);
+    this.status = $('<p class="search-summary">&nbsp;</p>').appendTo(this.out);
+    this.output = $('<ul class="search"/>').appendTo(this.out);
+
+    $('#search-progress').text(_('Preparing search...'));
+    this.startPulse();
+
+    // index already loaded, the browser was quick!
+    if (this.hasIndex())
+      this.query(query);
+    else
+      this.deferQuery(query);
+  },
+
+  /**
+   * execute search (requires search index to be loaded)
+   */
+  query : function(query) {
+    var i;
+
+    // stem the searchterms and add them to the correct list
+    var stemmer = new Stemmer();
+    var searchterms = [];
+    var excluded = [];
+    var hlterms = [];
+    var tmp = splitQuery(query);
+    var objectterms = [];
+    for (i = 0; i < tmp.length; i++) {
+      if (tmp[i] !== "") {
+          objectterms.push(tmp[i].toLowerCase());
+      }
+
+      if ($u.indexOf(stopwords, tmp[i].toLowerCase()) != -1 || tmp[i] === "") {
+        // skip this "word"
+        continue;
+      }
+      // stem the word
+      var word = stemmer.stemWord(tmp[i].toLowerCase());
+      var toAppend;
+      // select the correct list
+      if (word[0] == '-') {
+        toAppend = excluded;
+        word = word.substr(1);
+      }
+      else {
+        toAppend = searchterms;
+        hlterms.push(tmp[i].toLowerCase());
+      }
+      // only add if not already in the list
+      if (!$u.contains(toAppend, word))
+        toAppend.push(word);
+    }
+    var highlightstring = '?highlight=' + $.urlencode(hlterms.join(" "));
+
+    // console.debug('SEARCH: searching for:');
+    // console.info('required: ', searchterms);
+    // console.info('excluded: ', excluded);
+
+    // prepare search
+    var terms = this._index.terms;
+    var titleterms = this._index.titleterms;
+
+    // array of [filename, title, anchor, descr, score]
+    var results = [];
+    $('#search-progress').empty();
+
+    // lookup as object
+    for (i = 0; i < objectterms.length; i++) {
+      var others = [].concat(objectterms.slice(0, i),
+                             objectterms.slice(i+1, objectterms.length));
+      results = results.concat(this.performObjectSearch(objectterms[i], others));
+    }
+
+    // lookup as search terms in fulltext
+    results = results.concat(this.performTermsSearch(searchterms, excluded, terms, titleterms));
+
+    // let the scorer override scores with a custom scoring function
+    if (Scorer.score) {
+      for (i = 0; i < results.length; i++)
+        results[i][4] = Scorer.score(results[i]);
+    }
+
+    // now sort the results by score (in opposite order of appearance, since the
+    // display function below uses pop() to retrieve items) and then
+    // alphabetically
+    results.sort(function(a, b) {
+      var left = a[4];
+      var right = b[4];
+      if (left > right) {
+        return 1;
+      } else if (left < right) {
+        return -1;
+      } else {
+        // same score: sort alphabetically
+        left = a[1].toLowerCase();
+        right = b[1].toLowerCase();
+        return (left > right) ? -1 : ((left < right) ? 1 : 0);
+      }
+    });
+
+    // for debugging
+    //Search.lastresults = results.slice();  // a copy
+    //console.info('search results:', Search.lastresults);
+
+    // print the results
+    var resultCount = results.length;
+    function displayNextItem() {
+      // results left, load the summary and display it
+      if (results.length) {
+        var item = results.pop();
+        var listItem = $('<li></li>');
+        var requestUrl = "";
+        var linkUrl = "";
+        if (DOCUMENTATION_OPTIONS.BUILDER === 'dirhtml') {
+          // dirhtml builder
+          var dirname = item[0] + '/';
+          if (dirname.match(/\/index\/$/)) {
+            dirname = dirname.substring(0, dirname.length-6);
+          } else if (dirname == 'index/') {
+            dirname = '';
+          }
+          requestUrl = DOCUMENTATION_OPTIONS.URL_ROOT + dirname;
+          linkUrl = requestUrl;
+
+        } else {
+          // normal html builders
+          requestUrl = DOCUMENTATION_OPTIONS.URL_ROOT + item[0] + DOCUMENTATION_OPTIONS.FILE_SUFFIX;
+          linkUrl = item[0] + DOCUMENTATION_OPTIONS.LINK_SUFFIX;
+        }
+        listItem.append($('<a/>').attr('href',
+            linkUrl +
+            highlightstring + item[2]).html(item[1]));
+        if (item[3]) {
+          listItem.append($('<span> (' + item[3] + ')</span>'));
+          Search.output.append(listItem);
+          setTimeout(function() {
+            displayNextItem();
+          }, 5);
+        } else if (DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY) {
+          $.ajax({url: requestUrl,
+                  dataType: "text",
+                  complete: function(jqxhr, textstatus) {
+                    var data = jqxhr.responseText;
+                    if (data !== '' && data !== undefined) {
+                      var summary = Search.makeSearchSummary(data, searchterms, hlterms);
+                      if (summary) {
+                        listItem.append(summary);
+                      }
+                    }
+                    Search.output.append(listItem);
+                    setTimeout(function() {
+                      displayNextItem();
+                    }, 5);
+                  }});
+        } else {
+          // just display title
+          Search.output.append(listItem);
+          setTimeout(function() {
+            displayNextItem();
+          }, 5);
+        }
+      }
+      // search finished, update title and status message
+      else {
+        Search.stopPulse();
+        Search.title.text(_('Search Results'));
+        if (!resultCount)
+          Search.status.text(_('Your search did not match any documents. Please make sure that all words are spelled correctly and that you\'ve selected enough categories.'));
+        else
+            Search.status.text(_('Search finished, found %s page(s) matching the search query.').replace('%s', resultCount));
+        Search.status.fadeIn(500);
+      }
+    }
+    displayNextItem();
+  },
+
+  /**
+   * search for object names
+   */
+  performObjectSearch : function(object, otherterms) {
+    var filenames = this._index.filenames;
+    var docnames = this._index.docnames;
+    var objects = this._index.objects;
+    var objnames = this._index.objnames;
+    var titles = this._index.titles;
+
+    var i;
+    var results = [];
+
+    for (var prefix in objects) {
+      for (var iMatch = 0; iMatch != objects[prefix].length; ++iMatch) {
+        var match = objects[prefix][iMatch];
+        var name = match[4];
+        var fullname = (prefix ? prefix + '.' : '') + name;
+        var fullnameLower = fullname.toLowerCase()
+        if (fullnameLower.indexOf(object) > -1) {
+          var score = 0;
+          var parts = fullnameLower.split('.');
+          // check for different match types: exact matches of full name or
+          // "last name" (i.e. last dotted part)
+          if (fullnameLower == object || parts[parts.length - 1] == object) {
+            score += Scorer.objNameMatch;
+          // matches in last name
+          } else if (parts[parts.length - 1].indexOf(object) > -1) {
+            score += Scorer.objPartialMatch;
+          }
+          var objname = objnames[match[1]][2];
+          var title = titles[match[0]];
+          // If more than one term searched for, we require other words to be
+          // found in the name/title/description
+          if (otherterms.length > 0) {
+            var haystack = (prefix + ' ' + name + ' ' +
+                            objname + ' ' + title).toLowerCase();
+            var allfound = true;
+            for (i = 0; i < otherterms.length; i++) {
+              if (haystack.indexOf(otherterms[i]) == -1) {
+                allfound = false;
+                break;
+              }
+            }
+            if (!allfound) {
+              continue;
+            }
+          }
+          var descr = objname + _(', in ') + title;
+
+          var anchor = match[3];
+          if (anchor === '')
+            anchor = fullname;
+          else if (anchor == '-')
+            anchor = objnames[match[1]][1] + '-' + fullname;
+          // add custom score for some objects according to scorer
+          if (Scorer.objPrio.hasOwnProperty(match[2])) {
+            score += Scorer.objPrio[match[2]];
+          } else {
+            score += Scorer.objPrioDefault;
+          }
+          results.push([docnames[match[0]], fullname, '#'+anchor, descr, score, filenames[match[0]]]);
+        }
+      }
+    }
+
+    return results;
+  },
+
+  /**
+   * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions
+   */
+  escapeRegExp : function(string) {
+    return string.replace(/[.*+\-?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
+  },
+
+  /**
+   * search for full-text terms in the index
+   */
+  performTermsSearch : function(searchterms, excluded, terms, titleterms) {
+    var docnames = this._index.docnames;
+    var filenames = this._index.filenames;
+    var titles = this._index.titles;
+
+    var i, j, file;
+    var fileMap = {};
+    var scoreMap = {};
+    var results = [];
+
+    // perform the search on the required terms
+    for (i = 0; i < searchterms.length; i++) {
+      var word = searchterms[i];
+      var files = [];
+      var _o = [
+        {files: terms[word], score: Scorer.term},
+        {files: titleterms[word], score: Scorer.title}
+      ];
+      // add support for partial matches
+      if (word.length > 2) {
+        var word_regex = this.escapeRegExp(word);
+        for (var w in terms) {
+          if (w.match(word_regex) && !terms[word]) {
+            _o.push({files: terms[w], score: Scorer.partialTerm})
+          }
+        }
+        for (var w in titleterms) {
+          if (w.match(word_regex) && !titleterms[word]) {
+              _o.push({files: titleterms[w], score: Scorer.partialTitle})
+          }
+        }
+      }
+
+      // no match but word was a required one
+      if ($u.every(_o, function(o){return o.files === undefined;})) {
+        break;
+      }
+      // found search word in contents
+      $u.each(_o, function(o) {
+        var _files = o.files;
+        if (_files === undefined)
+          return
+
+        if (_files.length === undefined)
+          _files = [_files];
+        files = files.concat(_files);
+
+        // set score for the word in each file to Scorer.term
+        for (j = 0; j < _files.length; j++) {
+          file = _files[j];
+          if (!(file in scoreMap))
+            scoreMap[file] = {};
+          scoreMap[file][word] = o.score;
+        }
+      });
+
+      // create the mapping
+      for (j = 0; j < files.length; j++) {
+        file = files[j];
+        if (file in fileMap && fileMap[file].indexOf(word) === -1)
+          fileMap[file].push(word);
+        else
+          fileMap[file] = [word];
+      }
+    }
+
+    // now check if the files don't contain excluded terms
+    for (file in fileMap) {
+      var valid = true;
+
+      // check if all requirements are matched
+      var filteredTermCount = // as search terms with length < 3 are discarded: ignore
+        searchterms.filter(function(term){return term.length > 2}).length
+      if (
+        fileMap[file].length != searchterms.length &&
+        fileMap[file].length != filteredTermCount
+      ) continue;
+
+      // ensure that none of the excluded terms is in the search result
+      for (i = 0; i < excluded.length; i++) {
+        if (terms[excluded[i]] == file ||
+            titleterms[excluded[i]] == file ||
+            $u.contains(terms[excluded[i]] || [], file) ||
+            $u.contains(titleterms[excluded[i]] || [], file)) {
+          valid = false;
+          break;
+        }
+      }
+
+      // if we have still a valid result we can add it to the result list
+      if (valid) {
+        // select one (max) score for the file.
+        // for better ranking, we should calculate ranking by using words statistics like basic tf-idf...
+        var score = $u.max($u.map(fileMap[file], function(w){return scoreMap[file][w]}));
+        results.push([docnames[file], titles[file], '', null, score, filenames[file]]);
+      }
+    }
+    return results;
+  },
+
+  /**
+   * helper function to return a node containing the
+   * search summary for a given text. keywords is a list
+   * of stemmed words, hlwords is the list of normal, unstemmed
+   * words. the first one is used to find the occurrence, the
+   * latter for highlighting it.
+   */
+  makeSearchSummary : function(htmlText, keywords, hlwords) {
+    var text = Search.htmlToText(htmlText);
+    if (text == "") {
+      return null;
+    }
+    var textLower = text.toLowerCase();
+    var start = 0;
+    $.each(keywords, function() {
+      var i = textLower.indexOf(this.toLowerCase());
+      if (i > -1)
+        start = i;
+    });
+    start = Math.max(start - 120, 0);
+    var excerpt = ((start > 0) ? '...' : '') +
+      $.trim(text.substr(start, 240)) +
+      ((start + 240 - text.length) ? '...' : '');
+    var rv = $('<p class="context"></p>').text(excerpt);
+    $.each(hlwords, function() {
+      rv = rv.highlightText(this, 'highlighted');
+    });
+    return rv;
+  }
+};
+
+$(document).ready(function() {
+  Search.init();
+});
diff --git a/_static/sphinx-thebe.css b/_static/sphinx-thebe.css
new file mode 100644
index 00000000..1da27930
--- /dev/null
+++ b/_static/sphinx-thebe.css
@@ -0,0 +1,129 @@
+/* Thebelab Buttons */
+.thebelab-button {
+  z-index: 999;
+  display: inline-block;
+  padding: 0.35em 1.2em;
+  margin: 0px 1px;
+  border-radius: 0.12em;
+  box-sizing: border-box;
+  text-decoration: none;
+  font-family: "Roboto", sans-serif;
+  font-weight: 300;
+  text-align: center;
+  transition: all 0.2s;
+  background-color: #dddddd;
+  border: 0.05em solid white;
+  color: #000000;
+}
+
+.thebelab-button:hover {
+  border: 0.05em solid black;
+  background-color: #fcfcfc;
+}
+
+.thebe-launch-button {
+  height: 2.2em;
+  font-size: 0.8em;
+  border: 1px black solid;
+}
+
+/* Thebelab Cell */
+.thebelab-cell pre {
+  background: none;
+}
+
+.thebelab-cell .thebelab-input {
+  padding-left: 1em;
+  margin-bottom: 0.5em;
+  margin-top: 0.5em;
+}
+
+.thebelab-cell .jp-OutputArea {
+  margin-top: 0.5em;
+  margin-left: 1em;
+}
+
+button.thebelab-button.thebelab-run-button {
+  margin-left: 1.5em;
+  margin-bottom: 0.5em;
+}
+
+/* Loading button */
+button.thebe-launch-button div.spinner {
+  float: left;
+  margin-right: 1em;
+}
+
+/* Remove the spinner when thebelab is ready */
+.thebe-launch-button.thebe-status-ready .spinner {
+  display: none;
+}
+
+.thebe-launch-button span.status {
+  font-family: monospace;
+  font-weight: bold;
+}
+
+.thebe-launch-button.thebe-status-ready span.status {
+  color: green;
+}
+
+.spinner {
+  height: 2em;
+  text-align: center;
+  font-size: 0.7em;
+}
+
+.spinner > div {
+  background-color: #f37726;
+  height: 100%;
+  width: 6px;
+  display: inline-block;
+
+  -webkit-animation: sk-stretchdelay 1.2s infinite ease-in-out;
+  animation: sk-stretchdelay 1.2s infinite ease-in-out;
+}
+
+.spinner .rect2 {
+  -webkit-animation-delay: -1.1s;
+  animation-delay: -1.1s;
+}
+
+.spinner .rect3 {
+  -webkit-animation-delay: -1s;
+  animation-delay: -1s;
+}
+
+.spinner .rect4 {
+  -webkit-animation-delay: -0.9s;
+  animation-delay: -0.9s;
+}
+
+.spinner .rect5 {
+  -webkit-animation-delay: -0.8s;
+  animation-delay: -0.8s;
+}
+
+@-webkit-keyframes sk-stretchdelay {
+  0%,
+  40%,
+  100% {
+    -webkit-transform: scaleY(0.4);
+  }
+  20% {
+    -webkit-transform: scaleY(1);
+  }
+}
+
+@keyframes sk-stretchdelay {
+  0%,
+  40%,
+  100% {
+    transform: scaleY(0.4);
+    -webkit-transform: scaleY(0.4);
+  }
+  20% {
+    transform: scaleY(1);
+    -webkit-transform: scaleY(1);
+  }
+}
diff --git a/_static/sphinx-thebe.js b/_static/sphinx-thebe.js
new file mode 100644
index 00000000..7626dbb0
--- /dev/null
+++ b/_static/sphinx-thebe.js
@@ -0,0 +1,126 @@
+/**
+ * Add attributes to Thebe blocks to initialize thebe properly
+ */
+var configureThebe = () => {
+  // Load thebe config in case we want to update it as some point
+  console.log("[sphinx-thebe]: Loading thebe config...");
+  thebe_config = $('script[type="text/x-thebe-config"]')[0];
+
+  // If we already detect a Thebe cell, don't re-run
+  if (document.querySelectorAll("div.thebe-cell").length > 0) {
+    return;
+  }
+
+  // Update thebe buttons with loading message
+  $(".thebe-launch-button").each((ii, button) => {
+    button.innerHTML = `
+        <div class="spinner">
+            <div class="rect1"></div>
+            <div class="rect2"></div>
+            <div class="rect3"></div>
+            <div class="rect4"></div>
+        </div>
+        <span class="loading-text"></span>`;
+  });
+
+  // Set thebe event hooks
+  var thebeStatus;
+  thebelab.on("status", function (evt, data) {
+    console.log("Status changed:", data.status, data.message);
+
+    $(".thebe-launch-button ")
+      .removeClass("thebe-status-" + thebeStatus)
+      .addClass("thebe-status-" + data.status)
+      .find(".loading-text")
+      .html(
+        "<span class='launch_msg'>Launching from mybinder.org: </span><span class='status'>" +
+          data.status +
+          "</span>"
+      );
+
+    // Now update our thebe status
+    thebeStatus = data.status;
+
+    // Find any cells with an initialization tag and ask thebe to run them when ready
+    if (data.status === "ready") {
+      var thebeInitCells = document.querySelectorAll(
+        ".thebe-init, .tag_thebe-init"
+      );
+      thebeInitCells.forEach((cell) => {
+        console.log("Initializing Thebe with cell: " + cell.id);
+        cell.querySelector(".thebelab-run-button").click();
+      });
+    }
+  });
+};
+
+/**
+ * Update the page DOM to use Thebe elements
+ */
+var modifyDOMForThebe = () => {
+  // Find all code cells, replace with Thebe interactive code cells
+  const codeCells = document.querySelectorAll(thebe_selector);
+  codeCells.forEach((codeCell, index) => {
+    const codeCellId = (index) => `codecell${index}`;
+    codeCell.id = codeCellId(index);
+    codeCellText = codeCell.querySelector(thebe_selector_input);
+    codeCellOutput = codeCell.querySelector(thebe_selector_output);
+
+    // Clean up the language to make it work w/ CodeMirror and add it to the cell
+    dataLanguage = detectLanguage(kernelName);
+
+    // Re-arrange the cell and add metadata
+    if (codeCellText) {
+      codeCellText.setAttribute("data-language", dataLanguage);
+      codeCellText.setAttribute("data-executable", "true");
+
+      // If we had an output, insert it just after the `pre` cell
+      if (codeCellOutput) {
+        $(codeCellOutput).attr("data-output", "");
+        $(codeCellOutput).insertAfter(codeCellText);
+      }
+    }
+
+    // Remove sphinx-copybutton blocks, which are common in Sphinx
+    codeCell.querySelectorAll("button.copybtn").forEach((el) => {
+      el.remove();
+    });
+  });
+};
+
+var initThebe = () => {
+  // Load thebe dynamically if it's not already loaded
+  if (typeof thebelab === "undefined") {
+    console.log("[sphinx-thebe]: Loading thebe from CDN...");
+    $(".thebe-launch-button ").text("Loading thebe from CDN...");
+
+    const script = document.createElement("script");
+    script.src = `${THEBE_JS_URL}`;
+    document.head.appendChild(script);
+
+    // Runs once the script has finished loading
+    script.addEventListener("load", () => {
+      console.log("[sphinx-thebe]: Finished loading thebe from CDN...");
+      configureThebe();
+      modifyDOMForThebe();
+      thebelab.bootstrap();
+    });
+  } else {
+    console.log(
+      "[sphinx-thebe]: thebe already loaded, not loading from CDN..."
+    );
+    configureThebe();
+    modifyDOMForThebe();
+    thebelab.bootstrap();
+  }
+};
+
+// Helper function to munge the language name
+var detectLanguage = (language) => {
+  if (language.indexOf("python") > -1) {
+    language = "python";
+  } else if (language === "ir") {
+    language = "r";
+  }
+  return language;
+};
diff --git a/_static/styles/bootstrap.css b/_static/styles/bootstrap.css
new file mode 100644
index 00000000..0aa5f54f
--- /dev/null
+++ b/_static/styles/bootstrap.css
@@ -0,0 +1,6 @@
+/*!
+ * Bootstrap  v5.2.3 (https://getbootstrap.com/)
+ * Copyright 2011-2022 The Bootstrap Authors
+ * Copyright 2011-2022 Twitter, Inc.
+ * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+ */:root{--bs-blue:#0d6efd;--bs-indigo:#6610f2;--bs-purple:#6f42c1;--bs-pink:#d63384;--bs-red:#dc3545;--bs-orange:#fd7e14;--bs-yellow:#ffc107;--bs-green:#198754;--bs-teal:#20c997;--bs-cyan:#0dcaf0;--bs-black:#000;--bs-white:#fff;--bs-gray:#6c757d;--bs-gray-dark:#343a40;--bs-gray-100:#f8f9fa;--bs-gray-200:#e9ecef;--bs-gray-300:#dee2e6;--bs-gray-400:#ced4da;--bs-gray-500:#adb5bd;--bs-gray-600:#6c757d;--bs-gray-700:#495057;--bs-gray-800:#343a40;--bs-gray-900:#212529;--bs-primary:#0d6efd;--bs-secondary:#6c757d;--bs-success:#198754;--bs-info:#0dcaf0;--bs-warning:#ffc107;--bs-danger:#dc3545;--bs-light:#f8f9fa;--bs-dark:#212529;--bs-primary-rgb:13,110,253;--bs-secondary-rgb:108,117,125;--bs-success-rgb:25,135,84;--bs-info-rgb:13,202,240;--bs-warning-rgb:255,193,7;--bs-danger-rgb:220,53,69;--bs-light-rgb:248,249,250;--bs-dark-rgb:33,37,41;--bs-white-rgb:255,255,255;--bs-black-rgb:0,0,0;--bs-body-color-rgb:33,37,41;--bs-body-bg-rgb:255,255,255;--bs-font-sans-serif:system-ui,-apple-system,"Segoe UI",Roboto,"Helvetica Neue","Noto Sans","Liberation Sans",Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";--bs-font-monospace:SFMono-Regular,Menlo,Monaco,Consolas,"Liberation Mono","Courier 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diff --git a/_static/styles/pydata-sphinx-theme.css b/_static/styles/pydata-sphinx-theme.css
new file mode 100644
index 00000000..ca295d8a
--- /dev/null
+++ b/_static/styles/pydata-sphinx-theme.css
@@ -0,0 +1 @@
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diff --git a/_static/styles/sphinx-book-theme.css b/_static/styles/sphinx-book-theme.css
new file mode 100644
index 00000000..e514f24e
--- /dev/null
+++ b/_static/styles/sphinx-book-theme.css
@@ -0,0 +1,8 @@
+/*! sphinx-book-theme CSS
+ * BSD 3-Clause License
+ * Copyright (c) 2020, EBP
+ * All rights reserved.
+ *
+ * This follows the 7-1 pattern described here:
+ * https://sass-guidelin.es/#architecture
+ */html{--pst-font-size-base:none}html[data-theme=light]{--sbt-color-announcement:#616161}html[data-theme=dark],html[data-theme=light]{--pst-color-primary:#579aca}html{scroll-padding-top:4rem}.sbt-scroll-pixel-helper{position:absolute;width:0;height:0;top:0;left:0}.d-n,.onlyprint{display:none}@media print{.onlyprint{display:block!important}}@media print{.noprint{display:none!important}}.bd-article-container h1,.bd-article-container h2,.bd-article-container h3,.bd-article-container h4,.bd-article-container h5,.bd-article-container p.caption{color:var(--pst-color-muted)}.bd-article-container h1,.bd-article-container h2{font-weight:500}a.brackets:before{color:inherit;font-family:inherit;margin-right:0}table{position:relative}@media print{.bd-main .bd-content{margin-left:2rem;height:auto}.bd-main .bd-content #jb-print-docs-body{margin-left:0}.bd-main .bd-content #jb-print-docs-body h1{font-size:3em;text-align:center;margin-bottom:0}.bd-main .bd-content .bd-article{padding-top:0}.bd-main 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ease}.bd-sidebar-secondary.hide:not(:hover){max-height:3rem;overflow-y:hidden}.scrolled .bd-sidebar-secondary.hide:not(:hover){box-shadow:0 6px 6px -6px rgba(0,0,0,.3)}.bd-sidebar-secondary.hide:not(:hover) .onthispage:after{opacity:1;content:"\f107";font-family:Font Awesome\ 5 Free;font-weight:900;padding-left:.5em;transition:opacity .3s ease}.bd-sidebar-secondary.hide:not(:hover) nav.page-toc{opacity:0}}footer{font-size:var(--sbt-font-size-small-1)}footer.bd-footer-content{display:flex;flex-wrap:wrap;padding:15px;border-top:1px solid #ccc;font-size:87.5%}footer.bd-footer-content .bd-footer-content__inner{padding-left:0}footer.bd-footer-content .bd-footer-content__inner p{margin-bottom:0}.bd-footer-article{padding:0 1rem}@media (max-width:768px){.bd-footer-article{padding:0 .5rem}}.bd-sidebar-primary .navbar-icon-links{column-gap:.5rem}.bd-sidebar-primary .navbar-icon-links .nav-link i,.bd-sidebar-primary .navbar-icon-links .nav-link span{font-size:1.2rem}.bd-sidebar-primary .navbar-icon-links .nav-link img{font-size:.8rem}.navbar-brand{height:unset;max-height:unset;flex-direction:column;justify-content:center;gap:.25rem}.navbar-brand:hover{text-decoration:none}.navbar-brand .logo__title{font-size:1.25rem;white-space:normal;text-align:center}.navbar-brand .logo__image{height:unset}.bd-search-container{margin:2em}.bd-search-container #search-results h2:first-child{display:none}.search-button i{font-size:1.1rem}div#searchbox{padding-right:2rem;padding-left:2rem}@media (max-width:768px){div#searchbox{padding-right:1rem;padding-left:1rem}}@media (min-width:768px){div#searchbox p.highlight-link{margin-left:0}div#searchbox p.highlight-link a{font-size:1rem}}img{max-width:100%}img.align-center{margin-left:auto;margin-right:auto;display:block}img.align-left{clear:left;float:left;margin-right:1em}img.align-right{clear:right;float:right;margin-left:1em}div.figure{width:100%;margin-bottom:1em;text-align:center}div.figure.align-left{text-align:left}div.figure.align-left p.caption{margin-left:0}div.figure.align-right{text-align:right}div.figure.align-right p.caption{margin-right:0}div.figure p.caption{margin:.5em 10%}div.figure.margin-caption p.caption,div.figure.margin p.caption{margin:.5em 0}div.figure.margin-caption p.caption{text-align:left}div.figure span.caption-number{font-weight:700}div.figure span{font-size:.9rem}label.margin-toggle{margin-bottom:0}label.margin-toggle.marginnote-label{display:none}label.margin-toggle sup{user-select:none}@media (max-width:992px){label.margin-toggle{cursor:pointer;color:#0071bc}label.margin-toggle.marginnote-label{display:inline}label.margin-toggle.marginnote-label:after{content:"\2295"}}input.margin-toggle{display:none}@media (max-width:992px){input.margin-toggle:checked+.marginnote,input.margin-toggle:checked+.sidenote{display:block;float:left;left:1rem;clear:both;width:95%;margin:1rem 2.5%;position:relative}}span.marginnote,span.sidenote{z-index:2;position:relative;width:40%;float:right;background-color:unset;font-size:.9em;margin-left:.5rem;border-left:none}span.marginnote sup,span.sidenote sup{user-select:none}@media (min-width:992px),print{span.marginnote,span.sidenote{width:33%;margin:0 -36% 0 0;clear:right}span.marginnote p.sidebar-title,span.sidenote p.sidebar-title{margin-bottom:-1rem;border-bottom:none;padding-left:0}span.marginnote p.sidebar-title~*,span.sidenote p.sidebar-title~*{padding-left:0;padding-right:0}}@media (max-width:992px){span.marginnote,span.sidenote{display:none}}aside.sidebar .note{margin:1rem;padding:0 0 1rem}aside.sidebar .admonition-title{margin:0 -1rem 0 0}aside.sidebar.margin .sidebar-title:empty{display:none}aside.sidebar.margin .admonition{margin:.5rem;padding-left:0;padding-right:0}aside.sidebar.margin .admonition .admonition-title{margin-left:0;margin-right:0}@media (min-width:992px){aside.sidebar.margin{border:none}aside.sidebar.margin .admonition{margin:1rem 0;padding:0 0 1rem}}.cell.tag_margin,.cell.tag_popout,.margin.docutils.container,aside.margin,div.margin,figure.margin{z-index:2;position:relative;width:40%;float:right;background-color:unset;font-size:.9em;margin-left:.5rem}@media (min-width:992px),print{.cell.tag_margin,.cell.tag_popout,.margin.docutils.container,aside.margin,div.margin,figure.margin{width:33%;margin:0 -36% 0 0;clear:right}.cell.tag_margin p.sidebar-title,.cell.tag_popout p.sidebar-title,.margin.docutils.container p.sidebar-title,aside.margin p.sidebar-title,div.margin p.sidebar-title,figure.margin p.sidebar-title{margin-bottom:-1rem;border-bottom:none;padding-left:0}.cell.tag_margin p.sidebar-title~*,.cell.tag_popout p.sidebar-title~*,.margin.docutils.container p.sidebar-title~*,aside.margin p.sidebar-title~*,div.margin p.sidebar-title~*,figure.margin p.sidebar-title~*{padding-left:0;padding-right:0}}.cell.tag_margin div.cell.tag_margin .cell_output,.cell.tag_popout div.cell.tag_margin .cell_output,.margin.docutils.container div.cell.tag_margin .cell_output,aside.margin div.cell.tag_margin .cell_output,div.margin div.cell.tag_margin .cell_output,figure.margin div.cell.tag_margin .cell_output{padding-left:0}div.figure.margin-caption figcaption,div.figure.margin-caption p.caption,figure.margin-caption figcaption{z-index:2;position:relative;width:40%;float:right;background-color:unset;font-size:.9em;margin-left:.5rem}@media (min-width:992px),print{div.figure.margin-caption figcaption,div.figure.margin-caption p.caption,figure.margin-caption figcaption{width:33%;margin:0 -36% 0 0;clear:right}div.figure.margin-caption figcaption p.sidebar-title,div.figure.margin-caption p.caption p.sidebar-title,figure.margin-caption figcaption p.sidebar-title{margin-bottom:-1rem;border-bottom:none;padding-left:0}div.figure.margin-caption figcaption p.sidebar-title~*,div.figure.margin-caption p.caption p.sidebar-title~*,figure.margin-caption figcaption p.sidebar-title~*{padding-left:0;padding-right:0}}.margin-caption figcaption{text-align:left}div.cell.tag_full-width,div.cell.tag_full_width,div.full-width,div.full_width{z-index:2;position:relative}@media (min-width:992px){div.cell.tag_full-width,div.cell.tag_full_width,div.full-width,div.full_width{max-width:136%;width:136%}}blockquote.epigraph,blockquote.highlights,blockquote.pull-quote{font-size:1.25em;border-left:none;background-color:var(--pst-color-background)}blockquote div>p+p.attribution{font-style:normal;font-size:.9em;text-align:right;color:#6c757d;padding-right:2em}div[class*=highlight-],pre{clear:none}div.cell.tag_output_scroll div.cell_output,div.cell.tag_scroll-input div.cell_input,div.cell.tag_scroll-output div.cell_output{max-height:24em;overflow-y:auto}@media only print{div.utterances,hypothesis-sidebar{display:none}}.thebelab-cell{border:none!important;margin-right:.5em!important}.thebelab-cell .thebelab-input{padding-left:10px!important}.cell.docutils.container{padding-right:0!important}button.thebe-launch-button{height:2.5em;font-size:1em}
\ No newline at end of file
diff --git a/_static/styles/theme.css b/_static/styles/theme.css
new file mode 100644
index 00000000..4519dd91
--- /dev/null
+++ b/_static/styles/theme.css
@@ -0,0 +1,2 @@
+/* Provided by Sphinx's 'basic' theme, and included in the final set of assets */
+@import "../basic.css";
diff --git a/_static/togglebutton.css b/_static/togglebutton.css
new file mode 100644
index 00000000..54a67879
--- /dev/null
+++ b/_static/togglebutton.css
@@ -0,0 +1,160 @@
+/**
+ * Admonition-based toggles
+ */
+
+/* Visibility of the target */
+.admonition.toggle .admonition-title ~ * {
+    transition: opacity .3s, height .3s;
+}
+
+/* Toggle buttons inside admonitions so we see the title */
+.admonition.toggle {
+    position: relative;
+}
+
+/* Titles should cut off earlier to avoid overlapping w/ button */
+.admonition.toggle .admonition-title {
+    padding-right: 25%;
+    cursor: pointer;
+}
+
+/* Hovering will cause a slight shift in color to make it feel interactive */
+.admonition.toggle .admonition-title:hover {
+    box-shadow: inset 0 0 0px 20px rgb(0 0 0 / 1%);
+}
+
+/* Hovering will cause a slight shift in color to make it feel interactive */
+.admonition.toggle .admonition-title:active {
+    box-shadow: inset 0 0 0px 20px rgb(0 0 0 / 3%);
+}
+
+/* Remove extra whitespace below the admonition title when hidden */
+.admonition.toggle-hidden {
+    padding-bottom: 0;
+}
+
+.admonition.toggle-hidden .admonition-title {
+    margin-bottom: 0;
+}
+
+/* hides all the content of a page until de-toggled */
+.admonition.toggle-hidden .admonition-title ~ * {
+    height: 0;
+    margin: 0;
+    opacity: 0;
+    visibility: hidden;
+}
+
+/* General button style and position*/
+button.toggle-button {
+    /**
+     * Background and shape. By default there's no background
+     * but users can style as they wish
+     */  
+    background: none;
+    border: none;
+    outline: none;
+
+    /* Positioning just inside the admonition title */
+    position: absolute;
+    right: 0.5em;
+    padding: 0px;
+    border: none;
+    outline: none;
+}
+
+/* Display the toggle hint on wide screens */
+@media (min-width: 768px) {
+    button.toggle-button.toggle-button-hidden:before {
+        content: attr(data-toggle-hint);  /* This will be filled in by JS */
+        font-size: .8em;
+        align-self: center;
+    }
+}
+
+/* Icon behavior */
+.tb-icon {
+    transition: transform .2s ease-out;
+    height: 1.5em;
+    width: 1.5em;
+    stroke: currentColor;  /* So that we inherit the color of other text */
+}
+
+/* The icon should point right when closed, down when open. */
+/* Open */
+.admonition.toggle button .tb-icon {
+    transform: rotate(90deg);
+}
+
+/* Closed */
+.admonition.toggle button.toggle-button-hidden .tb-icon {
+    transform: rotate(0deg);
+}
+
+/* With details toggles, we don't rotate the icon so it points right */
+details.toggle-details .tb-icon {
+    height: 1.4em;
+    width: 1.4em;
+    margin-top: 0.1em;  /* To center the button vertically */
+}
+
+
+/**
+ * Details-based toggles.
+ * In this case, we wrap elements with `.toggle` in a details block.
+ */
+
+/* Details blocks */
+details.toggle-details {
+    margin: 1em 0;
+}
+
+
+details.toggle-details summary {
+    display: flex;
+    align-items: center;
+    cursor: pointer;
+    list-style: none;
+    border-radius: .2em;
+    border-left: 3px solid #1976d2;
+    background-color: rgb(204 204 204 / 10%);
+    padding: 0.2em 0.7em 0.3em 0.5em; /* Less padding on left because the SVG has left margin */
+    font-size: 0.9em;
+}
+
+details.toggle-details summary:hover {
+    background-color: rgb(204 204 204 / 20%);
+}
+
+details.toggle-details summary:active {
+    background: rgb(204 204 204 / 28%);
+}
+
+.toggle-details__summary-text {
+    margin-left: 0.2em;
+}
+
+details.toggle-details[open] summary {
+    margin-bottom: .5em;
+}
+
+details.toggle-details[open] summary .tb-icon {
+    transform: rotate(90deg);
+}
+
+details.toggle-details[open] summary ~ * {
+    animation: toggle-fade-in .3s ease-out;
+}
+
+@keyframes toggle-fade-in {
+  from {opacity: 0%;}
+  to {opacity: 100%;}
+}
+
+/* Print rules - we hide all toggle button elements at print */
+@media print {
+    /* Always hide the summary so the button doesn't show up */
+    details.toggle-details summary {
+        display: none;
+    }
+}
\ No newline at end of file
diff --git a/_static/togglebutton.js b/_static/togglebutton.js
new file mode 100644
index 00000000..215a7eef
--- /dev/null
+++ b/_static/togglebutton.js
@@ -0,0 +1,187 @@
+/**
+ * Add Toggle Buttons to elements
+ */
+
+let toggleChevron = `
+<svg xmlns="http://www.w3.org/2000/svg" class="tb-icon toggle-chevron-right" width="44" height="44" viewBox="0 0 24 24" stroke-width="1.5" stroke="#000000" fill="none" stroke-linecap="round" stroke-linejoin="round">
+<path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+<polyline points="9 6 15 12 9 18" />
+</svg>`;
+
+var initToggleItems = () => {
+  var itemsToToggle = document.querySelectorAll(togglebuttonSelector);
+  console.log(`[togglebutton]: Adding toggle buttons to ${itemsToToggle.length} items`)
+  // Add the button to each admonition and hook up a callback to toggle visibility
+  itemsToToggle.forEach((item, index) => {
+    if (item.classList.contains("admonition")) {
+      // If it's an admonition block, then we'll add a button inside
+      // Generate unique IDs for this item
+      var toggleID = `toggle-${index}`;
+      var buttonID = `button-${toggleID}`;
+
+      item.setAttribute('id', toggleID);
+      if (!item.classList.contains("toggle")){
+        item.classList.add("toggle");
+      }
+      // This is the button that will be added to each item to trigger the toggle
+      var collapseButton = `
+        <button type="button" id="${buttonID}" class="toggle-button" data-target="${toggleID}" data-button="${buttonID}" data-toggle-hint="${toggleHintShow}" aria-label="Toggle hidden content">
+            ${toggleChevron}
+        </button>`;
+
+      title = item.querySelector(".admonition-title")
+      title.insertAdjacentHTML("beforeend", collapseButton);
+      thisButton = document.getElementById(buttonID);
+
+      // Add click handlers for the button + admonition title (if admonition)
+      admonitionTitle = document.querySelector(`#${toggleID} > .admonition-title`)
+      if (admonitionTitle) {
+        // If an admonition, then make the whole title block clickable
+        admonitionTitle.addEventListener('click', toggleClickHandler);
+        admonitionTitle.dataset.target = toggleID
+        admonitionTitle.dataset.button = buttonID
+      } else {
+        // If not an admonition then we'll listen for the button click
+        thisButton.addEventListener('click', toggleClickHandler);
+      }
+
+      // Now hide the item for this toggle button unless explicitly noted to show
+      if (!item.classList.contains("toggle-shown")) {
+        toggleHidden(thisButton);
+      }
+    } else {
+      // If not an admonition, wrap the block in a <details> block
+      // Define the structure of the details block and insert it as a sibling
+      var detailsBlock = `
+        <details class="toggle-details">
+            <summary class="toggle-details__summary">
+              ${toggleChevron}
+              <span class="toggle-details__summary-text">${toggleHintShow}</span>
+            </summary>
+        </details>`;
+      item.insertAdjacentHTML("beforebegin", detailsBlock);
+
+      // Now move the toggle-able content inside of the details block
+      details = item.previousElementSibling
+      details.appendChild(item)
+      item.classList.add("toggle-details__container")
+
+      // Set up a click trigger to change the text as needed
+      details.addEventListener('click', (click) => {
+        let parent = click.target.parentElement;
+        if (parent.tagName.toLowerCase() == "details") {
+          summary = parent.querySelector("summary");
+          details = parent;
+        } else {
+          summary = parent;
+          details = parent.parentElement;
+        }
+        // Update the inner text for the proper hint
+        if (details.open) {
+          summary.querySelector("span.toggle-details__summary-text").innerText = toggleHintShow;
+        } else {
+          summary.querySelector("span.toggle-details__summary-text").innerText = toggleHintHide;
+        }
+        
+      });
+
+      // If we have a toggle-shown class, open details block should be open
+      if (item.classList.contains("toggle-shown")) {
+        details.click();
+      }
+    }
+  })
+};
+
+// This should simply add / remove the collapsed class and change the button text
+var toggleHidden = (button) => {
+  target = button.dataset['target']
+  var itemToToggle = document.getElementById(target);
+  if (itemToToggle.classList.contains("toggle-hidden")) {
+    itemToToggle.classList.remove("toggle-hidden");
+    button.classList.remove("toggle-button-hidden");
+  } else {
+    itemToToggle.classList.add("toggle-hidden");
+    button.classList.add("toggle-button-hidden");
+  }
+}
+
+var toggleClickHandler = (click) => {
+  // Be cause the admonition title is clickable and extends to the whole admonition
+  // We only look for a click event on this title to trigger the toggle.
+
+  if (click.target.classList.contains("admonition-title")) {
+    button = click.target.querySelector(".toggle-button");
+  } else if (click.target.classList.contains("tb-icon")) {
+    // We've clicked the icon and need to search up one parent for the button
+    button = click.target.parentElement;
+  } else if (click.target.tagName == "polyline") {
+    // We've clicked the SVG elements inside the button, need to up 2 layers
+    button = click.target.parentElement.parentElement;
+  } else if (click.target.classList.contains("toggle-button")) {
+    // We've clicked the button itself and so don't need to do anything
+    button = click.target;
+  } else {
+    console.log(`[togglebutton]: Couldn't find button for ${click.target}`)
+  }
+  target = document.getElementById(button.dataset['button']);
+  toggleHidden(target);
+}
+
+// If we want to blanket-add toggle classes to certain cells
+var addToggleToSelector = () => {
+  const selector = "";
+  if (selector.length > 0) {
+    document.querySelectorAll(selector).forEach((item) => {
+      item.classList.add("toggle");
+    })
+  }
+}
+
+// Helper function to run when the DOM is finished
+const sphinxToggleRunWhenDOMLoaded = cb => {
+  if (document.readyState != 'loading') {
+    cb()
+  } else if (document.addEventListener) {
+    document.addEventListener('DOMContentLoaded', cb)
+  } else {
+    document.attachEvent('onreadystatechange', function() {
+      if (document.readyState == 'complete') cb()
+    })
+  }
+}
+sphinxToggleRunWhenDOMLoaded(addToggleToSelector)
+sphinxToggleRunWhenDOMLoaded(initToggleItems)
+
+/** Toggle details blocks to be open when printing */
+if (toggleOpenOnPrint == "true") {
+  window.addEventListener("beforeprint", () => {
+    // Open the details
+    document.querySelectorAll("details.toggle-details").forEach((el) => {
+      el.dataset["togglestatus"] = el.open;
+      el.open = true;
+    });
+  
+    // Open the admonitions
+    document.querySelectorAll(".admonition.toggle.toggle-hidden").forEach((el) => {
+      console.log(el);
+      el.querySelector("button.toggle-button").click();
+      el.dataset["toggle_after_print"] = "true";
+    });
+  });
+  window.addEventListener("afterprint", () => {
+    // Re-close the details that were closed
+    document.querySelectorAll("details.toggle-details").forEach((el) => {
+      el.open = el.dataset["togglestatus"] == "true";
+      delete el.dataset["togglestatus"];
+    });
+  
+    // Re-close the admonition toggle buttons
+    document.querySelectorAll(".admonition.toggle").forEach((el) => {
+      if (el.dataset["toggle_after_print"] == "true") {
+        el.querySelector("button.toggle-button").click();
+        delete el.dataset["toggle_after_print"];
+      }
+    });
+  });
+}
diff --git a/_static/underscore-1.13.1.js b/_static/underscore-1.13.1.js
new file mode 100644
index 00000000..ffd77af9
--- /dev/null
+++ b/_static/underscore-1.13.1.js
@@ -0,0 +1,2042 @@
+(function (global, factory) {
+  typeof exports === 'object' && typeof module !== 'undefined' ? module.exports = factory() :
+  typeof define === 'function' && define.amd ? define('underscore', factory) :
+  (global = typeof globalThis !== 'undefined' ? globalThis : global || self, (function () {
+    var current = global._;
+    var exports = global._ = factory();
+    exports.noConflict = function () { global._ = current; return exports; };
+  }()));
+}(this, (function () {
+  //     Underscore.js 1.13.1
+  //     https://underscorejs.org
+  //     (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors
+  //     Underscore may be freely distributed under the MIT license.
+
+  // Current version.
+  var VERSION = '1.13.1';
+
+  // Establish the root object, `window` (`self`) in the browser, `global`
+  // on the server, or `this` in some virtual machines. We use `self`
+  // instead of `window` for `WebWorker` support.
+  var root = typeof self == 'object' && self.self === self && self ||
+            typeof global == 'object' && global.global === global && global ||
+            Function('return this')() ||
+            {};
+
+  // Save bytes in the minified (but not gzipped) version:
+  var ArrayProto = Array.prototype, ObjProto = Object.prototype;
+  var SymbolProto = typeof Symbol !== 'undefined' ? Symbol.prototype : null;
+
+  // Create quick reference variables for speed access to core prototypes.
+  var push = ArrayProto.push,
+      slice = ArrayProto.slice,
+      toString = ObjProto.toString,
+      hasOwnProperty = ObjProto.hasOwnProperty;
+
+  // Modern feature detection.
+  var supportsArrayBuffer = typeof ArrayBuffer !== 'undefined',
+      supportsDataView = typeof DataView !== 'undefined';
+
+  // All **ECMAScript 5+** native function implementations that we hope to use
+  // are declared here.
+  var nativeIsArray = Array.isArray,
+      nativeKeys = Object.keys,
+      nativeCreate = Object.create,
+      nativeIsView = supportsArrayBuffer && ArrayBuffer.isView;
+
+  // Create references to these builtin functions because we override them.
+  var _isNaN = isNaN,
+      _isFinite = isFinite;
+
+  // Keys in IE < 9 that won't be iterated by `for key in ...` and thus missed.
+  var hasEnumBug = !{toString: null}.propertyIsEnumerable('toString');
+  var nonEnumerableProps = ['valueOf', 'isPrototypeOf', 'toString',
+    'propertyIsEnumerable', 'hasOwnProperty', 'toLocaleString'];
+
+  // The largest integer that can be represented exactly.
+  var MAX_ARRAY_INDEX = Math.pow(2, 53) - 1;
+
+  // Some functions take a variable number of arguments, or a few expected
+  // arguments at the beginning and then a variable number of values to operate
+  // on. This helper accumulates all remaining arguments past the function’s
+  // argument length (or an explicit `startIndex`), into an array that becomes
+  // the last argument. Similar to ES6’s "rest parameter".
+  function restArguments(func, startIndex) {
+    startIndex = startIndex == null ? func.length - 1 : +startIndex;
+    return function() {
+      var length = Math.max(arguments.length - startIndex, 0),
+          rest = Array(length),
+          index = 0;
+      for (; index < length; index++) {
+        rest[index] = arguments[index + startIndex];
+      }
+      switch (startIndex) {
+        case 0: return func.call(this, rest);
+        case 1: return func.call(this, arguments[0], rest);
+        case 2: return func.call(this, arguments[0], arguments[1], rest);
+      }
+      var args = Array(startIndex + 1);
+      for (index = 0; index < startIndex; index++) {
+        args[index] = arguments[index];
+      }
+      args[startIndex] = rest;
+      return func.apply(this, args);
+    };
+  }
+
+  // Is a given variable an object?
+  function isObject(obj) {
+    var type = typeof obj;
+    return type === 'function' || type === 'object' && !!obj;
+  }
+
+  // Is a given value equal to null?
+  function isNull(obj) {
+    return obj === null;
+  }
+
+  // Is a given variable undefined?
+  function isUndefined(obj) {
+    return obj === void 0;
+  }
+
+  // Is a given value a boolean?
+  function isBoolean(obj) {
+    return obj === true || obj === false || toString.call(obj) === '[object Boolean]';
+  }
+
+  // Is a given value a DOM element?
+  function isElement(obj) {
+    return !!(obj && obj.nodeType === 1);
+  }
+
+  // Internal function for creating a `toString`-based type tester.
+  function tagTester(name) {
+    var tag = '[object ' + name + ']';
+    return function(obj) {
+      return toString.call(obj) === tag;
+    };
+  }
+
+  var isString = tagTester('String');
+
+  var isNumber = tagTester('Number');
+
+  var isDate = tagTester('Date');
+
+  var isRegExp = tagTester('RegExp');
+
+  var isError = tagTester('Error');
+
+  var isSymbol = tagTester('Symbol');
+
+  var isArrayBuffer = tagTester('ArrayBuffer');
+
+  var isFunction = tagTester('Function');
+
+  // Optimize `isFunction` if appropriate. Work around some `typeof` bugs in old
+  // v8, IE 11 (#1621), Safari 8 (#1929), and PhantomJS (#2236).
+  var nodelist = root.document && root.document.childNodes;
+  if (typeof /./ != 'function' && typeof Int8Array != 'object' && typeof nodelist != 'function') {
+    isFunction = function(obj) {
+      return typeof obj == 'function' || false;
+    };
+  }
+
+  var isFunction$1 = isFunction;
+
+  var hasObjectTag = tagTester('Object');
+
+  // In IE 10 - Edge 13, `DataView` has string tag `'[object Object]'`.
+  // In IE 11, the most common among them, this problem also applies to
+  // `Map`, `WeakMap` and `Set`.
+  var hasStringTagBug = (
+        supportsDataView && hasObjectTag(new DataView(new ArrayBuffer(8)))
+      ),
+      isIE11 = (typeof Map !== 'undefined' && hasObjectTag(new Map));
+
+  var isDataView = tagTester('DataView');
+
+  // In IE 10 - Edge 13, we need a different heuristic
+  // to determine whether an object is a `DataView`.
+  function ie10IsDataView(obj) {
+    return obj != null && isFunction$1(obj.getInt8) && isArrayBuffer(obj.buffer);
+  }
+
+  var isDataView$1 = (hasStringTagBug ? ie10IsDataView : isDataView);
+
+  // Is a given value an array?
+  // Delegates to ECMA5's native `Array.isArray`.
+  var isArray = nativeIsArray || tagTester('Array');
+
+  // Internal function to check whether `key` is an own property name of `obj`.
+  function has$1(obj, key) {
+    return obj != null && hasOwnProperty.call(obj, key);
+  }
+
+  var isArguments = tagTester('Arguments');
+
+  // Define a fallback version of the method in browsers (ahem, IE < 9), where
+  // there isn't any inspectable "Arguments" type.
+  (function() {
+    if (!isArguments(arguments)) {
+      isArguments = function(obj) {
+        return has$1(obj, 'callee');
+      };
+    }
+  }());
+
+  var isArguments$1 = isArguments;
+
+  // Is a given object a finite number?
+  function isFinite$1(obj) {
+    return !isSymbol(obj) && _isFinite(obj) && !isNaN(parseFloat(obj));
+  }
+
+  // Is the given value `NaN`?
+  function isNaN$1(obj) {
+    return isNumber(obj) && _isNaN(obj);
+  }
+
+  // Predicate-generating function. Often useful outside of Underscore.
+  function constant(value) {
+    return function() {
+      return value;
+    };
+  }
+
+  // Common internal logic for `isArrayLike` and `isBufferLike`.
+  function createSizePropertyCheck(getSizeProperty) {
+    return function(collection) {
+      var sizeProperty = getSizeProperty(collection);
+      return typeof sizeProperty == 'number' && sizeProperty >= 0 && sizeProperty <= MAX_ARRAY_INDEX;
+    }
+  }
+
+  // Internal helper to generate a function to obtain property `key` from `obj`.
+  function shallowProperty(key) {
+    return function(obj) {
+      return obj == null ? void 0 : obj[key];
+    };
+  }
+
+  // Internal helper to obtain the `byteLength` property of an object.
+  var getByteLength = shallowProperty('byteLength');
+
+  // Internal helper to determine whether we should spend extensive checks against
+  // `ArrayBuffer` et al.
+  var isBufferLike = createSizePropertyCheck(getByteLength);
+
+  // Is a given value a typed array?
+  var typedArrayPattern = /\[object ((I|Ui)nt(8|16|32)|Float(32|64)|Uint8Clamped|Big(I|Ui)nt64)Array\]/;
+  function isTypedArray(obj) {
+    // `ArrayBuffer.isView` is the most future-proof, so use it when available.
+    // Otherwise, fall back on the above regular expression.
+    return nativeIsView ? (nativeIsView(obj) && !isDataView$1(obj)) :
+                  isBufferLike(obj) && typedArrayPattern.test(toString.call(obj));
+  }
+
+  var isTypedArray$1 = supportsArrayBuffer ? isTypedArray : constant(false);
+
+  // Internal helper to obtain the `length` property of an object.
+  var getLength = shallowProperty('length');
+
+  // Internal helper to create a simple lookup structure.
+  // `collectNonEnumProps` used to depend on `_.contains`, but this led to
+  // circular imports. `emulatedSet` is a one-off solution that only works for
+  // arrays of strings.
+  function emulatedSet(keys) {
+    var hash = {};
+    for (var l = keys.length, i = 0; i < l; ++i) hash[keys[i]] = true;
+    return {
+      contains: function(key) { return hash[key]; },
+      push: function(key) {
+        hash[key] = true;
+        return keys.push(key);
+      }
+    };
+  }
+
+  // Internal helper. Checks `keys` for the presence of keys in IE < 9 that won't
+  // be iterated by `for key in ...` and thus missed. Extends `keys` in place if
+  // needed.
+  function collectNonEnumProps(obj, keys) {
+    keys = emulatedSet(keys);
+    var nonEnumIdx = nonEnumerableProps.length;
+    var constructor = obj.constructor;
+    var proto = isFunction$1(constructor) && constructor.prototype || ObjProto;
+
+    // Constructor is a special case.
+    var prop = 'constructor';
+    if (has$1(obj, prop) && !keys.contains(prop)) keys.push(prop);
+
+    while (nonEnumIdx--) {
+      prop = nonEnumerableProps[nonEnumIdx];
+      if (prop in obj && obj[prop] !== proto[prop] && !keys.contains(prop)) {
+        keys.push(prop);
+      }
+    }
+  }
+
+  // Retrieve the names of an object's own properties.
+  // Delegates to **ECMAScript 5**'s native `Object.keys`.
+  function keys(obj) {
+    if (!isObject(obj)) return [];
+    if (nativeKeys) return nativeKeys(obj);
+    var keys = [];
+    for (var key in obj) if (has$1(obj, key)) keys.push(key);
+    // Ahem, IE < 9.
+    if (hasEnumBug) collectNonEnumProps(obj, keys);
+    return keys;
+  }
+
+  // Is a given array, string, or object empty?
+  // An "empty" object has no enumerable own-properties.
+  function isEmpty(obj) {
+    if (obj == null) return true;
+    // Skip the more expensive `toString`-based type checks if `obj` has no
+    // `.length`.
+    var length = getLength(obj);
+    if (typeof length == 'number' && (
+      isArray(obj) || isString(obj) || isArguments$1(obj)
+    )) return length === 0;
+    return getLength(keys(obj)) === 0;
+  }
+
+  // Returns whether an object has a given set of `key:value` pairs.
+  function isMatch(object, attrs) {
+    var _keys = keys(attrs), length = _keys.length;
+    if (object == null) return !length;
+    var obj = Object(object);
+    for (var i = 0; i < length; i++) {
+      var key = _keys[i];
+      if (attrs[key] !== obj[key] || !(key in obj)) return false;
+    }
+    return true;
+  }
+
+  // If Underscore is called as a function, it returns a wrapped object that can
+  // be used OO-style. This wrapper holds altered versions of all functions added
+  // through `_.mixin`. Wrapped objects may be chained.
+  function _$1(obj) {
+    if (obj instanceof _$1) return obj;
+    if (!(this instanceof _$1)) return new _$1(obj);
+    this._wrapped = obj;
+  }
+
+  _$1.VERSION = VERSION;
+
+  // Extracts the result from a wrapped and chained object.
+  _$1.prototype.value = function() {
+    return this._wrapped;
+  };
+
+  // Provide unwrapping proxies for some methods used in engine operations
+  // such as arithmetic and JSON stringification.
+  _$1.prototype.valueOf = _$1.prototype.toJSON = _$1.prototype.value;
+
+  _$1.prototype.toString = function() {
+    return String(this._wrapped);
+  };
+
+  // Internal function to wrap or shallow-copy an ArrayBuffer,
+  // typed array or DataView to a new view, reusing the buffer.
+  function toBufferView(bufferSource) {
+    return new Uint8Array(
+      bufferSource.buffer || bufferSource,
+      bufferSource.byteOffset || 0,
+      getByteLength(bufferSource)
+    );
+  }
+
+  // We use this string twice, so give it a name for minification.
+  var tagDataView = '[object DataView]';
+
+  // Internal recursive comparison function for `_.isEqual`.
+  function eq(a, b, aStack, bStack) {
+    // Identical objects are equal. `0 === -0`, but they aren't identical.
+    // See the [Harmony `egal` proposal](https://wiki.ecmascript.org/doku.php?id=harmony:egal).
+    if (a === b) return a !== 0 || 1 / a === 1 / b;
+    // `null` or `undefined` only equal to itself (strict comparison).
+    if (a == null || b == null) return false;
+    // `NaN`s are equivalent, but non-reflexive.
+    if (a !== a) return b !== b;
+    // Exhaust primitive checks
+    var type = typeof a;
+    if (type !== 'function' && type !== 'object' && typeof b != 'object') return false;
+    return deepEq(a, b, aStack, bStack);
+  }
+
+  // Internal recursive comparison function for `_.isEqual`.
+  function deepEq(a, b, aStack, bStack) {
+    // Unwrap any wrapped objects.
+    if (a instanceof _$1) a = a._wrapped;
+    if (b instanceof _$1) b = b._wrapped;
+    // Compare `[[Class]]` names.
+    var className = toString.call(a);
+    if (className !== toString.call(b)) return false;
+    // Work around a bug in IE 10 - Edge 13.
+    if (hasStringTagBug && className == '[object Object]' && isDataView$1(a)) {
+      if (!isDataView$1(b)) return false;
+      className = tagDataView;
+    }
+    switch (className) {
+      // These types are compared by value.
+      case '[object RegExp]':
+        // RegExps are coerced to strings for comparison (Note: '' + /a/i === '/a/i')
+      case '[object String]':
+        // Primitives and their corresponding object wrappers are equivalent; thus, `"5"` is
+        // equivalent to `new String("5")`.
+        return '' + a === '' + b;
+      case '[object Number]':
+        // `NaN`s are equivalent, but non-reflexive.
+        // Object(NaN) is equivalent to NaN.
+        if (+a !== +a) return +b !== +b;
+        // An `egal` comparison is performed for other numeric values.
+        return +a === 0 ? 1 / +a === 1 / b : +a === +b;
+      case '[object Date]':
+      case '[object Boolean]':
+        // Coerce dates and booleans to numeric primitive values. Dates are compared by their
+        // millisecond representations. Note that invalid dates with millisecond representations
+        // of `NaN` are not equivalent.
+        return +a === +b;
+      case '[object Symbol]':
+        return SymbolProto.valueOf.call(a) === SymbolProto.valueOf.call(b);
+      case '[object ArrayBuffer]':
+      case tagDataView:
+        // Coerce to typed array so we can fall through.
+        return deepEq(toBufferView(a), toBufferView(b), aStack, bStack);
+    }
+
+    var areArrays = className === '[object Array]';
+    if (!areArrays && isTypedArray$1(a)) {
+        var byteLength = getByteLength(a);
+        if (byteLength !== getByteLength(b)) return false;
+        if (a.buffer === b.buffer && a.byteOffset === b.byteOffset) return true;
+        areArrays = true;
+    }
+    if (!areArrays) {
+      if (typeof a != 'object' || typeof b != 'object') return false;
+
+      // Objects with different constructors are not equivalent, but `Object`s or `Array`s
+      // from different frames are.
+      var aCtor = a.constructor, bCtor = b.constructor;
+      if (aCtor !== bCtor && !(isFunction$1(aCtor) && aCtor instanceof aCtor &&
+                               isFunction$1(bCtor) && bCtor instanceof bCtor)
+                          && ('constructor' in a && 'constructor' in b)) {
+        return false;
+      }
+    }
+    // Assume equality for cyclic structures. The algorithm for detecting cyclic
+    // structures is adapted from ES 5.1 section 15.12.3, abstract operation `JO`.
+
+    // Initializing stack of traversed objects.
+    // It's done here since we only need them for objects and arrays comparison.
+    aStack = aStack || [];
+    bStack = bStack || [];
+    var length = aStack.length;
+    while (length--) {
+      // Linear search. Performance is inversely proportional to the number of
+      // unique nested structures.
+      if (aStack[length] === a) return bStack[length] === b;
+    }
+
+    // Add the first object to the stack of traversed objects.
+    aStack.push(a);
+    bStack.push(b);
+
+    // Recursively compare objects and arrays.
+    if (areArrays) {
+      // Compare array lengths to determine if a deep comparison is necessary.
+      length = a.length;
+      if (length !== b.length) return false;
+      // Deep compare the contents, ignoring non-numeric properties.
+      while (length--) {
+        if (!eq(a[length], b[length], aStack, bStack)) return false;
+      }
+    } else {
+      // Deep compare objects.
+      var _keys = keys(a), key;
+      length = _keys.length;
+      // Ensure that both objects contain the same number of properties before comparing deep equality.
+      if (keys(b).length !== length) return false;
+      while (length--) {
+        // Deep compare each member
+        key = _keys[length];
+        if (!(has$1(b, key) && eq(a[key], b[key], aStack, bStack))) return false;
+      }
+    }
+    // Remove the first object from the stack of traversed objects.
+    aStack.pop();
+    bStack.pop();
+    return true;
+  }
+
+  // Perform a deep comparison to check if two objects are equal.
+  function isEqual(a, b) {
+    return eq(a, b);
+  }
+
+  // Retrieve all the enumerable property names of an object.
+  function allKeys(obj) {
+    if (!isObject(obj)) return [];
+    var keys = [];
+    for (var key in obj) keys.push(key);
+    // Ahem, IE < 9.
+    if (hasEnumBug) collectNonEnumProps(obj, keys);
+    return keys;
+  }
+
+  // Since the regular `Object.prototype.toString` type tests don't work for
+  // some types in IE 11, we use a fingerprinting heuristic instead, based
+  // on the methods. It's not great, but it's the best we got.
+  // The fingerprint method lists are defined below.
+  function ie11fingerprint(methods) {
+    var length = getLength(methods);
+    return function(obj) {
+      if (obj == null) return false;
+      // `Map`, `WeakMap` and `Set` have no enumerable keys.
+      var keys = allKeys(obj);
+      if (getLength(keys)) return false;
+      for (var i = 0; i < length; i++) {
+        if (!isFunction$1(obj[methods[i]])) return false;
+      }
+      // If we are testing against `WeakMap`, we need to ensure that
+      // `obj` doesn't have a `forEach` method in order to distinguish
+      // it from a regular `Map`.
+      return methods !== weakMapMethods || !isFunction$1(obj[forEachName]);
+    };
+  }
+
+  // In the interest of compact minification, we write
+  // each string in the fingerprints only once.
+  var forEachName = 'forEach',
+      hasName = 'has',
+      commonInit = ['clear', 'delete'],
+      mapTail = ['get', hasName, 'set'];
+
+  // `Map`, `WeakMap` and `Set` each have slightly different
+  // combinations of the above sublists.
+  var mapMethods = commonInit.concat(forEachName, mapTail),
+      weakMapMethods = commonInit.concat(mapTail),
+      setMethods = ['add'].concat(commonInit, forEachName, hasName);
+
+  var isMap = isIE11 ? ie11fingerprint(mapMethods) : tagTester('Map');
+
+  var isWeakMap = isIE11 ? ie11fingerprint(weakMapMethods) : tagTester('WeakMap');
+
+  var isSet = isIE11 ? ie11fingerprint(setMethods) : tagTester('Set');
+
+  var isWeakSet = tagTester('WeakSet');
+
+  // Retrieve the values of an object's properties.
+  function values(obj) {
+    var _keys = keys(obj);
+    var length = _keys.length;
+    var values = Array(length);
+    for (var i = 0; i < length; i++) {
+      values[i] = obj[_keys[i]];
+    }
+    return values;
+  }
+
+  // Convert an object into a list of `[key, value]` pairs.
+  // The opposite of `_.object` with one argument.
+  function pairs(obj) {
+    var _keys = keys(obj);
+    var length = _keys.length;
+    var pairs = Array(length);
+    for (var i = 0; i < length; i++) {
+      pairs[i] = [_keys[i], obj[_keys[i]]];
+    }
+    return pairs;
+  }
+
+  // Invert the keys and values of an object. The values must be serializable.
+  function invert(obj) {
+    var result = {};
+    var _keys = keys(obj);
+    for (var i = 0, length = _keys.length; i < length; i++) {
+      result[obj[_keys[i]]] = _keys[i];
+    }
+    return result;
+  }
+
+  // Return a sorted list of the function names available on the object.
+  function functions(obj) {
+    var names = [];
+    for (var key in obj) {
+      if (isFunction$1(obj[key])) names.push(key);
+    }
+    return names.sort();
+  }
+
+  // An internal function for creating assigner functions.
+  function createAssigner(keysFunc, defaults) {
+    return function(obj) {
+      var length = arguments.length;
+      if (defaults) obj = Object(obj);
+      if (length < 2 || obj == null) return obj;
+      for (var index = 1; index < length; index++) {
+        var source = arguments[index],
+            keys = keysFunc(source),
+            l = keys.length;
+        for (var i = 0; i < l; i++) {
+          var key = keys[i];
+          if (!defaults || obj[key] === void 0) obj[key] = source[key];
+        }
+      }
+      return obj;
+    };
+  }
+
+  // Extend a given object with all the properties in passed-in object(s).
+  var extend = createAssigner(allKeys);
+
+  // Assigns a given object with all the own properties in the passed-in
+  // object(s).
+  // (https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object/assign)
+  var extendOwn = createAssigner(keys);
+
+  // Fill in a given object with default properties.
+  var defaults = createAssigner(allKeys, true);
+
+  // Create a naked function reference for surrogate-prototype-swapping.
+  function ctor() {
+    return function(){};
+  }
+
+  // An internal function for creating a new object that inherits from another.
+  function baseCreate(prototype) {
+    if (!isObject(prototype)) return {};
+    if (nativeCreate) return nativeCreate(prototype);
+    var Ctor = ctor();
+    Ctor.prototype = prototype;
+    var result = new Ctor;
+    Ctor.prototype = null;
+    return result;
+  }
+
+  // Creates an object that inherits from the given prototype object.
+  // If additional properties are provided then they will be added to the
+  // created object.
+  function create(prototype, props) {
+    var result = baseCreate(prototype);
+    if (props) extendOwn(result, props);
+    return result;
+  }
+
+  // Create a (shallow-cloned) duplicate of an object.
+  function clone(obj) {
+    if (!isObject(obj)) return obj;
+    return isArray(obj) ? obj.slice() : extend({}, obj);
+  }
+
+  // Invokes `interceptor` with the `obj` and then returns `obj`.
+  // The primary purpose of this method is to "tap into" a method chain, in
+  // order to perform operations on intermediate results within the chain.
+  function tap(obj, interceptor) {
+    interceptor(obj);
+    return obj;
+  }
+
+  // Normalize a (deep) property `path` to array.
+  // Like `_.iteratee`, this function can be customized.
+  function toPath$1(path) {
+    return isArray(path) ? path : [path];
+  }
+  _$1.toPath = toPath$1;
+
+  // Internal wrapper for `_.toPath` to enable minification.
+  // Similar to `cb` for `_.iteratee`.
+  function toPath(path) {
+    return _$1.toPath(path);
+  }
+
+  // Internal function to obtain a nested property in `obj` along `path`.
+  function deepGet(obj, path) {
+    var length = path.length;
+    for (var i = 0; i < length; i++) {
+      if (obj == null) return void 0;
+      obj = obj[path[i]];
+    }
+    return length ? obj : void 0;
+  }
+
+  // Get the value of the (deep) property on `path` from `object`.
+  // If any property in `path` does not exist or if the value is
+  // `undefined`, return `defaultValue` instead.
+  // The `path` is normalized through `_.toPath`.
+  function get(object, path, defaultValue) {
+    var value = deepGet(object, toPath(path));
+    return isUndefined(value) ? defaultValue : value;
+  }
+
+  // Shortcut function for checking if an object has a given property directly on
+  // itself (in other words, not on a prototype). Unlike the internal `has`
+  // function, this public version can also traverse nested properties.
+  function has(obj, path) {
+    path = toPath(path);
+    var length = path.length;
+    for (var i = 0; i < length; i++) {
+      var key = path[i];
+      if (!has$1(obj, key)) return false;
+      obj = obj[key];
+    }
+    return !!length;
+  }
+
+  // Keep the identity function around for default iteratees.
+  function identity(value) {
+    return value;
+  }
+
+  // Returns a predicate for checking whether an object has a given set of
+  // `key:value` pairs.
+  function matcher(attrs) {
+    attrs = extendOwn({}, attrs);
+    return function(obj) {
+      return isMatch(obj, attrs);
+    };
+  }
+
+  // Creates a function that, when passed an object, will traverse that object’s
+  // properties down the given `path`, specified as an array of keys or indices.
+  function property(path) {
+    path = toPath(path);
+    return function(obj) {
+      return deepGet(obj, path);
+    };
+  }
+
+  // Internal function that returns an efficient (for current engines) version
+  // of the passed-in callback, to be repeatedly applied in other Underscore
+  // functions.
+  function optimizeCb(func, context, argCount) {
+    if (context === void 0) return func;
+    switch (argCount == null ? 3 : argCount) {
+      case 1: return function(value) {
+        return func.call(context, value);
+      };
+      // The 2-argument case is omitted because we’re not using it.
+      case 3: return function(value, index, collection) {
+        return func.call(context, value, index, collection);
+      };
+      case 4: return function(accumulator, value, index, collection) {
+        return func.call(context, accumulator, value, index, collection);
+      };
+    }
+    return function() {
+      return func.apply(context, arguments);
+    };
+  }
+
+  // An internal function to generate callbacks that can be applied to each
+  // element in a collection, returning the desired result — either `_.identity`,
+  // an arbitrary callback, a property matcher, or a property accessor.
+  function baseIteratee(value, context, argCount) {
+    if (value == null) return identity;
+    if (isFunction$1(value)) return optimizeCb(value, context, argCount);
+    if (isObject(value) && !isArray(value)) return matcher(value);
+    return property(value);
+  }
+
+  // External wrapper for our callback generator. Users may customize
+  // `_.iteratee` if they want additional predicate/iteratee shorthand styles.
+  // This abstraction hides the internal-only `argCount` argument.
+  function iteratee(value, context) {
+    return baseIteratee(value, context, Infinity);
+  }
+  _$1.iteratee = iteratee;
+
+  // The function we call internally to generate a callback. It invokes
+  // `_.iteratee` if overridden, otherwise `baseIteratee`.
+  function cb(value, context, argCount) {
+    if (_$1.iteratee !== iteratee) return _$1.iteratee(value, context);
+    return baseIteratee(value, context, argCount);
+  }
+
+  // Returns the results of applying the `iteratee` to each element of `obj`.
+  // In contrast to `_.map` it returns an object.
+  function mapObject(obj, iteratee, context) {
+    iteratee = cb(iteratee, context);
+    var _keys = keys(obj),
+        length = _keys.length,
+        results = {};
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys[index];
+      results[currentKey] = iteratee(obj[currentKey], currentKey, obj);
+    }
+    return results;
+  }
+
+  // Predicate-generating function. Often useful outside of Underscore.
+  function noop(){}
+
+  // Generates a function for a given object that returns a given property.
+  function propertyOf(obj) {
+    if (obj == null) return noop;
+    return function(path) {
+      return get(obj, path);
+    };
+  }
+
+  // Run a function **n** times.
+  function times(n, iteratee, context) {
+    var accum = Array(Math.max(0, n));
+    iteratee = optimizeCb(iteratee, context, 1);
+    for (var i = 0; i < n; i++) accum[i] = iteratee(i);
+    return accum;
+  }
+
+  // Return a random integer between `min` and `max` (inclusive).
+  function random(min, max) {
+    if (max == null) {
+      max = min;
+      min = 0;
+    }
+    return min + Math.floor(Math.random() * (max - min + 1));
+  }
+
+  // A (possibly faster) way to get the current timestamp as an integer.
+  var now = Date.now || function() {
+    return new Date().getTime();
+  };
+
+  // Internal helper to generate functions for escaping and unescaping strings
+  // to/from HTML interpolation.
+  function createEscaper(map) {
+    var escaper = function(match) {
+      return map[match];
+    };
+    // Regexes for identifying a key that needs to be escaped.
+    var source = '(?:' + keys(map).join('|') + ')';
+    var testRegexp = RegExp(source);
+    var replaceRegexp = RegExp(source, 'g');
+    return function(string) {
+      string = string == null ? '' : '' + string;
+      return testRegexp.test(string) ? string.replace(replaceRegexp, escaper) : string;
+    };
+  }
+
+  // Internal list of HTML entities for escaping.
+  var escapeMap = {
+    '&': '&amp;',
+    '<': '&lt;',
+    '>': '&gt;',
+    '"': '&quot;',
+    "'": '&#x27;',
+    '`': '&#x60;'
+  };
+
+  // Function for escaping strings to HTML interpolation.
+  var _escape = createEscaper(escapeMap);
+
+  // Internal list of HTML entities for unescaping.
+  var unescapeMap = invert(escapeMap);
+
+  // Function for unescaping strings from HTML interpolation.
+  var _unescape = createEscaper(unescapeMap);
+
+  // By default, Underscore uses ERB-style template delimiters. Change the
+  // following template settings to use alternative delimiters.
+  var templateSettings = _$1.templateSettings = {
+    evaluate: /<%([\s\S]+?)%>/g,
+    interpolate: /<%=([\s\S]+?)%>/g,
+    escape: /<%-([\s\S]+?)%>/g
+  };
+
+  // When customizing `_.templateSettings`, if you don't want to define an
+  // interpolation, evaluation or escaping regex, we need one that is
+  // guaranteed not to match.
+  var noMatch = /(.)^/;
+
+  // Certain characters need to be escaped so that they can be put into a
+  // string literal.
+  var escapes = {
+    "'": "'",
+    '\\': '\\',
+    '\r': 'r',
+    '\n': 'n',
+    '\u2028': 'u2028',
+    '\u2029': 'u2029'
+  };
+
+  var escapeRegExp = /\\|'|\r|\n|\u2028|\u2029/g;
+
+  function escapeChar(match) {
+    return '\\' + escapes[match];
+  }
+
+  // In order to prevent third-party code injection through
+  // `_.templateSettings.variable`, we test it against the following regular
+  // expression. It is intentionally a bit more liberal than just matching valid
+  // identifiers, but still prevents possible loopholes through defaults or
+  // destructuring assignment.
+  var bareIdentifier = /^\s*(\w|\$)+\s*$/;
+
+  // JavaScript micro-templating, similar to John Resig's implementation.
+  // Underscore templating handles arbitrary delimiters, preserves whitespace,
+  // and correctly escapes quotes within interpolated code.
+  // NB: `oldSettings` only exists for backwards compatibility.
+  function template(text, settings, oldSettings) {
+    if (!settings && oldSettings) settings = oldSettings;
+    settings = defaults({}, settings, _$1.templateSettings);
+
+    // Combine delimiters into one regular expression via alternation.
+    var matcher = RegExp([
+      (settings.escape || noMatch).source,
+      (settings.interpolate || noMatch).source,
+      (settings.evaluate || noMatch).source
+    ].join('|') + '|$', 'g');
+
+    // Compile the template source, escaping string literals appropriately.
+    var index = 0;
+    var source = "__p+='";
+    text.replace(matcher, function(match, escape, interpolate, evaluate, offset) {
+      source += text.slice(index, offset).replace(escapeRegExp, escapeChar);
+      index = offset + match.length;
+
+      if (escape) {
+        source += "'+\n((__t=(" + escape + "))==null?'':_.escape(__t))+\n'";
+      } else if (interpolate) {
+        source += "'+\n((__t=(" + interpolate + "))==null?'':__t)+\n'";
+      } else if (evaluate) {
+        source += "';\n" + evaluate + "\n__p+='";
+      }
+
+      // Adobe VMs need the match returned to produce the correct offset.
+      return match;
+    });
+    source += "';\n";
+
+    var argument = settings.variable;
+    if (argument) {
+      // Insure against third-party code injection. (CVE-2021-23358)
+      if (!bareIdentifier.test(argument)) throw new Error(
+        'variable is not a bare identifier: ' + argument
+      );
+    } else {
+      // If a variable is not specified, place data values in local scope.
+      source = 'with(obj||{}){\n' + source + '}\n';
+      argument = 'obj';
+    }
+
+    source = "var __t,__p='',__j=Array.prototype.join," +
+      "print=function(){__p+=__j.call(arguments,'');};\n" +
+      source + 'return __p;\n';
+
+    var render;
+    try {
+      render = new Function(argument, '_', source);
+    } catch (e) {
+      e.source = source;
+      throw e;
+    }
+
+    var template = function(data) {
+      return render.call(this, data, _$1);
+    };
+
+    // Provide the compiled source as a convenience for precompilation.
+    template.source = 'function(' + argument + '){\n' + source + '}';
+
+    return template;
+  }
+
+  // Traverses the children of `obj` along `path`. If a child is a function, it
+  // is invoked with its parent as context. Returns the value of the final
+  // child, or `fallback` if any child is undefined.
+  function result(obj, path, fallback) {
+    path = toPath(path);
+    var length = path.length;
+    if (!length) {
+      return isFunction$1(fallback) ? fallback.call(obj) : fallback;
+    }
+    for (var i = 0; i < length; i++) {
+      var prop = obj == null ? void 0 : obj[path[i]];
+      if (prop === void 0) {
+        prop = fallback;
+        i = length; // Ensure we don't continue iterating.
+      }
+      obj = isFunction$1(prop) ? prop.call(obj) : prop;
+    }
+    return obj;
+  }
+
+  // Generate a unique integer id (unique within the entire client session).
+  // Useful for temporary DOM ids.
+  var idCounter = 0;
+  function uniqueId(prefix) {
+    var id = ++idCounter + '';
+    return prefix ? prefix + id : id;
+  }
+
+  // Start chaining a wrapped Underscore object.
+  function chain(obj) {
+    var instance = _$1(obj);
+    instance._chain = true;
+    return instance;
+  }
+
+  // Internal function to execute `sourceFunc` bound to `context` with optional
+  // `args`. Determines whether to execute a function as a constructor or as a
+  // normal function.
+  function executeBound(sourceFunc, boundFunc, context, callingContext, args) {
+    if (!(callingContext instanceof boundFunc)) return sourceFunc.apply(context, args);
+    var self = baseCreate(sourceFunc.prototype);
+    var result = sourceFunc.apply(self, args);
+    if (isObject(result)) return result;
+    return self;
+  }
+
+  // Partially apply a function by creating a version that has had some of its
+  // arguments pre-filled, without changing its dynamic `this` context. `_` acts
+  // as a placeholder by default, allowing any combination of arguments to be
+  // pre-filled. Set `_.partial.placeholder` for a custom placeholder argument.
+  var partial = restArguments(function(func, boundArgs) {
+    var placeholder = partial.placeholder;
+    var bound = function() {
+      var position = 0, length = boundArgs.length;
+      var args = Array(length);
+      for (var i = 0; i < length; i++) {
+        args[i] = boundArgs[i] === placeholder ? arguments[position++] : boundArgs[i];
+      }
+      while (position < arguments.length) args.push(arguments[position++]);
+      return executeBound(func, bound, this, this, args);
+    };
+    return bound;
+  });
+
+  partial.placeholder = _$1;
+
+  // Create a function bound to a given object (assigning `this`, and arguments,
+  // optionally).
+  var bind = restArguments(function(func, context, args) {
+    if (!isFunction$1(func)) throw new TypeError('Bind must be called on a function');
+    var bound = restArguments(function(callArgs) {
+      return executeBound(func, bound, context, this, args.concat(callArgs));
+    });
+    return bound;
+  });
+
+  // Internal helper for collection methods to determine whether a collection
+  // should be iterated as an array or as an object.
+  // Related: https://people.mozilla.org/~jorendorff/es6-draft.html#sec-tolength
+  // Avoids a very nasty iOS 8 JIT bug on ARM-64. #2094
+  var isArrayLike = createSizePropertyCheck(getLength);
+
+  // Internal implementation of a recursive `flatten` function.
+  function flatten$1(input, depth, strict, output) {
+    output = output || [];
+    if (!depth && depth !== 0) {
+      depth = Infinity;
+    } else if (depth <= 0) {
+      return output.concat(input);
+    }
+    var idx = output.length;
+    for (var i = 0, length = getLength(input); i < length; i++) {
+      var value = input[i];
+      if (isArrayLike(value) && (isArray(value) || isArguments$1(value))) {
+        // Flatten current level of array or arguments object.
+        if (depth > 1) {
+          flatten$1(value, depth - 1, strict, output);
+          idx = output.length;
+        } else {
+          var j = 0, len = value.length;
+          while (j < len) output[idx++] = value[j++];
+        }
+      } else if (!strict) {
+        output[idx++] = value;
+      }
+    }
+    return output;
+  }
+
+  // Bind a number of an object's methods to that object. Remaining arguments
+  // are the method names to be bound. Useful for ensuring that all callbacks
+  // defined on an object belong to it.
+  var bindAll = restArguments(function(obj, keys) {
+    keys = flatten$1(keys, false, false);
+    var index = keys.length;
+    if (index < 1) throw new Error('bindAll must be passed function names');
+    while (index--) {
+      var key = keys[index];
+      obj[key] = bind(obj[key], obj);
+    }
+    return obj;
+  });
+
+  // Memoize an expensive function by storing its results.
+  function memoize(func, hasher) {
+    var memoize = function(key) {
+      var cache = memoize.cache;
+      var address = '' + (hasher ? hasher.apply(this, arguments) : key);
+      if (!has$1(cache, address)) cache[address] = func.apply(this, arguments);
+      return cache[address];
+    };
+    memoize.cache = {};
+    return memoize;
+  }
+
+  // Delays a function for the given number of milliseconds, and then calls
+  // it with the arguments supplied.
+  var delay = restArguments(function(func, wait, args) {
+    return setTimeout(function() {
+      return func.apply(null, args);
+    }, wait);
+  });
+
+  // Defers a function, scheduling it to run after the current call stack has
+  // cleared.
+  var defer = partial(delay, _$1, 1);
+
+  // Returns a function, that, when invoked, will only be triggered at most once
+  // during a given window of time. Normally, the throttled function will run
+  // as much as it can, without ever going more than once per `wait` duration;
+  // but if you'd like to disable the execution on the leading edge, pass
+  // `{leading: false}`. To disable execution on the trailing edge, ditto.
+  function throttle(func, wait, options) {
+    var timeout, context, args, result;
+    var previous = 0;
+    if (!options) options = {};
+
+    var later = function() {
+      previous = options.leading === false ? 0 : now();
+      timeout = null;
+      result = func.apply(context, args);
+      if (!timeout) context = args = null;
+    };
+
+    var throttled = function() {
+      var _now = now();
+      if (!previous && options.leading === false) previous = _now;
+      var remaining = wait - (_now - previous);
+      context = this;
+      args = arguments;
+      if (remaining <= 0 || remaining > wait) {
+        if (timeout) {
+          clearTimeout(timeout);
+          timeout = null;
+        }
+        previous = _now;
+        result = func.apply(context, args);
+        if (!timeout) context = args = null;
+      } else if (!timeout && options.trailing !== false) {
+        timeout = setTimeout(later, remaining);
+      }
+      return result;
+    };
+
+    throttled.cancel = function() {
+      clearTimeout(timeout);
+      previous = 0;
+      timeout = context = args = null;
+    };
+
+    return throttled;
+  }
+
+  // When a sequence of calls of the returned function ends, the argument
+  // function is triggered. The end of a sequence is defined by the `wait`
+  // parameter. If `immediate` is passed, the argument function will be
+  // triggered at the beginning of the sequence instead of at the end.
+  function debounce(func, wait, immediate) {
+    var timeout, previous, args, result, context;
+
+    var later = function() {
+      var passed = now() - previous;
+      if (wait > passed) {
+        timeout = setTimeout(later, wait - passed);
+      } else {
+        timeout = null;
+        if (!immediate) result = func.apply(context, args);
+        // This check is needed because `func` can recursively invoke `debounced`.
+        if (!timeout) args = context = null;
+      }
+    };
+
+    var debounced = restArguments(function(_args) {
+      context = this;
+      args = _args;
+      previous = now();
+      if (!timeout) {
+        timeout = setTimeout(later, wait);
+        if (immediate) result = func.apply(context, args);
+      }
+      return result;
+    });
+
+    debounced.cancel = function() {
+      clearTimeout(timeout);
+      timeout = args = context = null;
+    };
+
+    return debounced;
+  }
+
+  // Returns the first function passed as an argument to the second,
+  // allowing you to adjust arguments, run code before and after, and
+  // conditionally execute the original function.
+  function wrap(func, wrapper) {
+    return partial(wrapper, func);
+  }
+
+  // Returns a negated version of the passed-in predicate.
+  function negate(predicate) {
+    return function() {
+      return !predicate.apply(this, arguments);
+    };
+  }
+
+  // Returns a function that is the composition of a list of functions, each
+  // consuming the return value of the function that follows.
+  function compose() {
+    var args = arguments;
+    var start = args.length - 1;
+    return function() {
+      var i = start;
+      var result = args[start].apply(this, arguments);
+      while (i--) result = args[i].call(this, result);
+      return result;
+    };
+  }
+
+  // Returns a function that will only be executed on and after the Nth call.
+  function after(times, func) {
+    return function() {
+      if (--times < 1) {
+        return func.apply(this, arguments);
+      }
+    };
+  }
+
+  // Returns a function that will only be executed up to (but not including) the
+  // Nth call.
+  function before(times, func) {
+    var memo;
+    return function() {
+      if (--times > 0) {
+        memo = func.apply(this, arguments);
+      }
+      if (times <= 1) func = null;
+      return memo;
+    };
+  }
+
+  // Returns a function that will be executed at most one time, no matter how
+  // often you call it. Useful for lazy initialization.
+  var once = partial(before, 2);
+
+  // Returns the first key on an object that passes a truth test.
+  function findKey(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = keys(obj), key;
+    for (var i = 0, length = _keys.length; i < length; i++) {
+      key = _keys[i];
+      if (predicate(obj[key], key, obj)) return key;
+    }
+  }
+
+  // Internal function to generate `_.findIndex` and `_.findLastIndex`.
+  function createPredicateIndexFinder(dir) {
+    return function(array, predicate, context) {
+      predicate = cb(predicate, context);
+      var length = getLength(array);
+      var index = dir > 0 ? 0 : length - 1;
+      for (; index >= 0 && index < length; index += dir) {
+        if (predicate(array[index], index, array)) return index;
+      }
+      return -1;
+    };
+  }
+
+  // Returns the first index on an array-like that passes a truth test.
+  var findIndex = createPredicateIndexFinder(1);
+
+  // Returns the last index on an array-like that passes a truth test.
+  var findLastIndex = createPredicateIndexFinder(-1);
+
+  // Use a comparator function to figure out the smallest index at which
+  // an object should be inserted so as to maintain order. Uses binary search.
+  function sortedIndex(array, obj, iteratee, context) {
+    iteratee = cb(iteratee, context, 1);
+    var value = iteratee(obj);
+    var low = 0, high = getLength(array);
+    while (low < high) {
+      var mid = Math.floor((low + high) / 2);
+      if (iteratee(array[mid]) < value) low = mid + 1; else high = mid;
+    }
+    return low;
+  }
+
+  // Internal function to generate the `_.indexOf` and `_.lastIndexOf` functions.
+  function createIndexFinder(dir, predicateFind, sortedIndex) {
+    return function(array, item, idx) {
+      var i = 0, length = getLength(array);
+      if (typeof idx == 'number') {
+        if (dir > 0) {
+          i = idx >= 0 ? idx : Math.max(idx + length, i);
+        } else {
+          length = idx >= 0 ? Math.min(idx + 1, length) : idx + length + 1;
+        }
+      } else if (sortedIndex && idx && length) {
+        idx = sortedIndex(array, item);
+        return array[idx] === item ? idx : -1;
+      }
+      if (item !== item) {
+        idx = predicateFind(slice.call(array, i, length), isNaN$1);
+        return idx >= 0 ? idx + i : -1;
+      }
+      for (idx = dir > 0 ? i : length - 1; idx >= 0 && idx < length; idx += dir) {
+        if (array[idx] === item) return idx;
+      }
+      return -1;
+    };
+  }
+
+  // Return the position of the first occurrence of an item in an array,
+  // or -1 if the item is not included in the array.
+  // If the array is large and already in sort order, pass `true`
+  // for **isSorted** to use binary search.
+  var indexOf = createIndexFinder(1, findIndex, sortedIndex);
+
+  // Return the position of the last occurrence of an item in an array,
+  // or -1 if the item is not included in the array.
+  var lastIndexOf = createIndexFinder(-1, findLastIndex);
+
+  // Return the first value which passes a truth test.
+  function find(obj, predicate, context) {
+    var keyFinder = isArrayLike(obj) ? findIndex : findKey;
+    var key = keyFinder(obj, predicate, context);
+    if (key !== void 0 && key !== -1) return obj[key];
+  }
+
+  // Convenience version of a common use case of `_.find`: getting the first
+  // object containing specific `key:value` pairs.
+  function findWhere(obj, attrs) {
+    return find(obj, matcher(attrs));
+  }
+
+  // The cornerstone for collection functions, an `each`
+  // implementation, aka `forEach`.
+  // Handles raw objects in addition to array-likes. Treats all
+  // sparse array-likes as if they were dense.
+  function each(obj, iteratee, context) {
+    iteratee = optimizeCb(iteratee, context);
+    var i, length;
+    if (isArrayLike(obj)) {
+      for (i = 0, length = obj.length; i < length; i++) {
+        iteratee(obj[i], i, obj);
+      }
+    } else {
+      var _keys = keys(obj);
+      for (i = 0, length = _keys.length; i < length; i++) {
+        iteratee(obj[_keys[i]], _keys[i], obj);
+      }
+    }
+    return obj;
+  }
+
+  // Return the results of applying the iteratee to each element.
+  function map(obj, iteratee, context) {
+    iteratee = cb(iteratee, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length,
+        results = Array(length);
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      results[index] = iteratee(obj[currentKey], currentKey, obj);
+    }
+    return results;
+  }
+
+  // Internal helper to create a reducing function, iterating left or right.
+  function createReduce(dir) {
+    // Wrap code that reassigns argument variables in a separate function than
+    // the one that accesses `arguments.length` to avoid a perf hit. (#1991)
+    var reducer = function(obj, iteratee, memo, initial) {
+      var _keys = !isArrayLike(obj) && keys(obj),
+          length = (_keys || obj).length,
+          index = dir > 0 ? 0 : length - 1;
+      if (!initial) {
+        memo = obj[_keys ? _keys[index] : index];
+        index += dir;
+      }
+      for (; index >= 0 && index < length; index += dir) {
+        var currentKey = _keys ? _keys[index] : index;
+        memo = iteratee(memo, obj[currentKey], currentKey, obj);
+      }
+      return memo;
+    };
+
+    return function(obj, iteratee, memo, context) {
+      var initial = arguments.length >= 3;
+      return reducer(obj, optimizeCb(iteratee, context, 4), memo, initial);
+    };
+  }
+
+  // **Reduce** builds up a single result from a list of values, aka `inject`,
+  // or `foldl`.
+  var reduce = createReduce(1);
+
+  // The right-associative version of reduce, also known as `foldr`.
+  var reduceRight = createReduce(-1);
+
+  // Return all the elements that pass a truth test.
+  function filter(obj, predicate, context) {
+    var results = [];
+    predicate = cb(predicate, context);
+    each(obj, function(value, index, list) {
+      if (predicate(value, index, list)) results.push(value);
+    });
+    return results;
+  }
+
+  // Return all the elements for which a truth test fails.
+  function reject(obj, predicate, context) {
+    return filter(obj, negate(cb(predicate)), context);
+  }
+
+  // Determine whether all of the elements pass a truth test.
+  function every(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length;
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      if (!predicate(obj[currentKey], currentKey, obj)) return false;
+    }
+    return true;
+  }
+
+  // Determine if at least one element in the object passes a truth test.
+  function some(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length;
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      if (predicate(obj[currentKey], currentKey, obj)) return true;
+    }
+    return false;
+  }
+
+  // Determine if the array or object contains a given item (using `===`).
+  function contains(obj, item, fromIndex, guard) {
+    if (!isArrayLike(obj)) obj = values(obj);
+    if (typeof fromIndex != 'number' || guard) fromIndex = 0;
+    return indexOf(obj, item, fromIndex) >= 0;
+  }
+
+  // Invoke a method (with arguments) on every item in a collection.
+  var invoke = restArguments(function(obj, path, args) {
+    var contextPath, func;
+    if (isFunction$1(path)) {
+      func = path;
+    } else {
+      path = toPath(path);
+      contextPath = path.slice(0, -1);
+      path = path[path.length - 1];
+    }
+    return map(obj, function(context) {
+      var method = func;
+      if (!method) {
+        if (contextPath && contextPath.length) {
+          context = deepGet(context, contextPath);
+        }
+        if (context == null) return void 0;
+        method = context[path];
+      }
+      return method == null ? method : method.apply(context, args);
+    });
+  });
+
+  // Convenience version of a common use case of `_.map`: fetching a property.
+  function pluck(obj, key) {
+    return map(obj, property(key));
+  }
+
+  // Convenience version of a common use case of `_.filter`: selecting only
+  // objects containing specific `key:value` pairs.
+  function where(obj, attrs) {
+    return filter(obj, matcher(attrs));
+  }
+
+  // Return the maximum element (or element-based computation).
+  function max(obj, iteratee, context) {
+    var result = -Infinity, lastComputed = -Infinity,
+        value, computed;
+    if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) {
+      obj = isArrayLike(obj) ? obj : values(obj);
+      for (var i = 0, length = obj.length; i < length; i++) {
+        value = obj[i];
+        if (value != null && value > result) {
+          result = value;
+        }
+      }
+    } else {
+      iteratee = cb(iteratee, context);
+      each(obj, function(v, index, list) {
+        computed = iteratee(v, index, list);
+        if (computed > lastComputed || computed === -Infinity && result === -Infinity) {
+          result = v;
+          lastComputed = computed;
+        }
+      });
+    }
+    return result;
+  }
+
+  // Return the minimum element (or element-based computation).
+  function min(obj, iteratee, context) {
+    var result = Infinity, lastComputed = Infinity,
+        value, computed;
+    if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) {
+      obj = isArrayLike(obj) ? obj : values(obj);
+      for (var i = 0, length = obj.length; i < length; i++) {
+        value = obj[i];
+        if (value != null && value < result) {
+          result = value;
+        }
+      }
+    } else {
+      iteratee = cb(iteratee, context);
+      each(obj, function(v, index, list) {
+        computed = iteratee(v, index, list);
+        if (computed < lastComputed || computed === Infinity && result === Infinity) {
+          result = v;
+          lastComputed = computed;
+        }
+      });
+    }
+    return result;
+  }
+
+  // Sample **n** random values from a collection using the modern version of the
+  // [Fisher-Yates shuffle](https://en.wikipedia.org/wiki/Fisher–Yates_shuffle).
+  // If **n** is not specified, returns a single random element.
+  // The internal `guard` argument allows it to work with `_.map`.
+  function sample(obj, n, guard) {
+    if (n == null || guard) {
+      if (!isArrayLike(obj)) obj = values(obj);
+      return obj[random(obj.length - 1)];
+    }
+    var sample = isArrayLike(obj) ? clone(obj) : values(obj);
+    var length = getLength(sample);
+    n = Math.max(Math.min(n, length), 0);
+    var last = length - 1;
+    for (var index = 0; index < n; index++) {
+      var rand = random(index, last);
+      var temp = sample[index];
+      sample[index] = sample[rand];
+      sample[rand] = temp;
+    }
+    return sample.slice(0, n);
+  }
+
+  // Shuffle a collection.
+  function shuffle(obj) {
+    return sample(obj, Infinity);
+  }
+
+  // Sort the object's values by a criterion produced by an iteratee.
+  function sortBy(obj, iteratee, context) {
+    var index = 0;
+    iteratee = cb(iteratee, context);
+    return pluck(map(obj, function(value, key, list) {
+      return {
+        value: value,
+        index: index++,
+        criteria: iteratee(value, key, list)
+      };
+    }).sort(function(left, right) {
+      var a = left.criteria;
+      var b = right.criteria;
+      if (a !== b) {
+        if (a > b || a === void 0) return 1;
+        if (a < b || b === void 0) return -1;
+      }
+      return left.index - right.index;
+    }), 'value');
+  }
+
+  // An internal function used for aggregate "group by" operations.
+  function group(behavior, partition) {
+    return function(obj, iteratee, context) {
+      var result = partition ? [[], []] : {};
+      iteratee = cb(iteratee, context);
+      each(obj, function(value, index) {
+        var key = iteratee(value, index, obj);
+        behavior(result, value, key);
+      });
+      return result;
+    };
+  }
+
+  // Groups the object's values by a criterion. Pass either a string attribute
+  // to group by, or a function that returns the criterion.
+  var groupBy = group(function(result, value, key) {
+    if (has$1(result, key)) result[key].push(value); else result[key] = [value];
+  });
+
+  // Indexes the object's values by a criterion, similar to `_.groupBy`, but for
+  // when you know that your index values will be unique.
+  var indexBy = group(function(result, value, key) {
+    result[key] = value;
+  });
+
+  // Counts instances of an object that group by a certain criterion. Pass
+  // either a string attribute to count by, or a function that returns the
+  // criterion.
+  var countBy = group(function(result, value, key) {
+    if (has$1(result, key)) result[key]++; else result[key] = 1;
+  });
+
+  // Split a collection into two arrays: one whose elements all pass the given
+  // truth test, and one whose elements all do not pass the truth test.
+  var partition = group(function(result, value, pass) {
+    result[pass ? 0 : 1].push(value);
+  }, true);
+
+  // Safely create a real, live array from anything iterable.
+  var reStrSymbol = /[^\ud800-\udfff]|[\ud800-\udbff][\udc00-\udfff]|[\ud800-\udfff]/g;
+  function toArray(obj) {
+    if (!obj) return [];
+    if (isArray(obj)) return slice.call(obj);
+    if (isString(obj)) {
+      // Keep surrogate pair characters together.
+      return obj.match(reStrSymbol);
+    }
+    if (isArrayLike(obj)) return map(obj, identity);
+    return values(obj);
+  }
+
+  // Return the number of elements in a collection.
+  function size(obj) {
+    if (obj == null) return 0;
+    return isArrayLike(obj) ? obj.length : keys(obj).length;
+  }
+
+  // Internal `_.pick` helper function to determine whether `key` is an enumerable
+  // property name of `obj`.
+  function keyInObj(value, key, obj) {
+    return key in obj;
+  }
+
+  // Return a copy of the object only containing the allowed properties.
+  var pick = restArguments(function(obj, keys) {
+    var result = {}, iteratee = keys[0];
+    if (obj == null) return result;
+    if (isFunction$1(iteratee)) {
+      if (keys.length > 1) iteratee = optimizeCb(iteratee, keys[1]);
+      keys = allKeys(obj);
+    } else {
+      iteratee = keyInObj;
+      keys = flatten$1(keys, false, false);
+      obj = Object(obj);
+    }
+    for (var i = 0, length = keys.length; i < length; i++) {
+      var key = keys[i];
+      var value = obj[key];
+      if (iteratee(value, key, obj)) result[key] = value;
+    }
+    return result;
+  });
+
+  // Return a copy of the object without the disallowed properties.
+  var omit = restArguments(function(obj, keys) {
+    var iteratee = keys[0], context;
+    if (isFunction$1(iteratee)) {
+      iteratee = negate(iteratee);
+      if (keys.length > 1) context = keys[1];
+    } else {
+      keys = map(flatten$1(keys, false, false), String);
+      iteratee = function(value, key) {
+        return !contains(keys, key);
+      };
+    }
+    return pick(obj, iteratee, context);
+  });
+
+  // Returns everything but the last entry of the array. Especially useful on
+  // the arguments object. Passing **n** will return all the values in
+  // the array, excluding the last N.
+  function initial(array, n, guard) {
+    return slice.call(array, 0, Math.max(0, array.length - (n == null || guard ? 1 : n)));
+  }
+
+  // Get the first element of an array. Passing **n** will return the first N
+  // values in the array. The **guard** check allows it to work with `_.map`.
+  function first(array, n, guard) {
+    if (array == null || array.length < 1) return n == null || guard ? void 0 : [];
+    if (n == null || guard) return array[0];
+    return initial(array, array.length - n);
+  }
+
+  // Returns everything but the first entry of the `array`. Especially useful on
+  // the `arguments` object. Passing an **n** will return the rest N values in the
+  // `array`.
+  function rest(array, n, guard) {
+    return slice.call(array, n == null || guard ? 1 : n);
+  }
+
+  // Get the last element of an array. Passing **n** will return the last N
+  // values in the array.
+  function last(array, n, guard) {
+    if (array == null || array.length < 1) return n == null || guard ? void 0 : [];
+    if (n == null || guard) return array[array.length - 1];
+    return rest(array, Math.max(0, array.length - n));
+  }
+
+  // Trim out all falsy values from an array.
+  function compact(array) {
+    return filter(array, Boolean);
+  }
+
+  // Flatten out an array, either recursively (by default), or up to `depth`.
+  // Passing `true` or `false` as `depth` means `1` or `Infinity`, respectively.
+  function flatten(array, depth) {
+    return flatten$1(array, depth, false);
+  }
+
+  // Take the difference between one array and a number of other arrays.
+  // Only the elements present in just the first array will remain.
+  var difference = restArguments(function(array, rest) {
+    rest = flatten$1(rest, true, true);
+    return filter(array, function(value){
+      return !contains(rest, value);
+    });
+  });
+
+  // Return a version of the array that does not contain the specified value(s).
+  var without = restArguments(function(array, otherArrays) {
+    return difference(array, otherArrays);
+  });
+
+  // Produce a duplicate-free version of the array. If the array has already
+  // been sorted, you have the option of using a faster algorithm.
+  // The faster algorithm will not work with an iteratee if the iteratee
+  // is not a one-to-one function, so providing an iteratee will disable
+  // the faster algorithm.
+  function uniq(array, isSorted, iteratee, context) {
+    if (!isBoolean(isSorted)) {
+      context = iteratee;
+      iteratee = isSorted;
+      isSorted = false;
+    }
+    if (iteratee != null) iteratee = cb(iteratee, context);
+    var result = [];
+    var seen = [];
+    for (var i = 0, length = getLength(array); i < length; i++) {
+      var value = array[i],
+          computed = iteratee ? iteratee(value, i, array) : value;
+      if (isSorted && !iteratee) {
+        if (!i || seen !== computed) result.push(value);
+        seen = computed;
+      } else if (iteratee) {
+        if (!contains(seen, computed)) {
+          seen.push(computed);
+          result.push(value);
+        }
+      } else if (!contains(result, value)) {
+        result.push(value);
+      }
+    }
+    return result;
+  }
+
+  // Produce an array that contains the union: each distinct element from all of
+  // the passed-in arrays.
+  var union = restArguments(function(arrays) {
+    return uniq(flatten$1(arrays, true, true));
+  });
+
+  // Produce an array that contains every item shared between all the
+  // passed-in arrays.
+  function intersection(array) {
+    var result = [];
+    var argsLength = arguments.length;
+    for (var i = 0, length = getLength(array); i < length; i++) {
+      var item = array[i];
+      if (contains(result, item)) continue;
+      var j;
+      for (j = 1; j < argsLength; j++) {
+        if (!contains(arguments[j], item)) break;
+      }
+      if (j === argsLength) result.push(item);
+    }
+    return result;
+  }
+
+  // Complement of zip. Unzip accepts an array of arrays and groups
+  // each array's elements on shared indices.
+  function unzip(array) {
+    var length = array && max(array, getLength).length || 0;
+    var result = Array(length);
+
+    for (var index = 0; index < length; index++) {
+      result[index] = pluck(array, index);
+    }
+    return result;
+  }
+
+  // Zip together multiple lists into a single array -- elements that share
+  // an index go together.
+  var zip = restArguments(unzip);
+
+  // Converts lists into objects. Pass either a single array of `[key, value]`
+  // pairs, or two parallel arrays of the same length -- one of keys, and one of
+  // the corresponding values. Passing by pairs is the reverse of `_.pairs`.
+  function object(list, values) {
+    var result = {};
+    for (var i = 0, length = getLength(list); i < length; i++) {
+      if (values) {
+        result[list[i]] = values[i];
+      } else {
+        result[list[i][0]] = list[i][1];
+      }
+    }
+    return result;
+  }
+
+  // Generate an integer Array containing an arithmetic progression. A port of
+  // the native Python `range()` function. See
+  // [the Python documentation](https://docs.python.org/library/functions.html#range).
+  function range(start, stop, step) {
+    if (stop == null) {
+      stop = start || 0;
+      start = 0;
+    }
+    if (!step) {
+      step = stop < start ? -1 : 1;
+    }
+
+    var length = Math.max(Math.ceil((stop - start) / step), 0);
+    var range = Array(length);
+
+    for (var idx = 0; idx < length; idx++, start += step) {
+      range[idx] = start;
+    }
+
+    return range;
+  }
+
+  // Chunk a single array into multiple arrays, each containing `count` or fewer
+  // items.
+  function chunk(array, count) {
+    if (count == null || count < 1) return [];
+    var result = [];
+    var i = 0, length = array.length;
+    while (i < length) {
+      result.push(slice.call(array, i, i += count));
+    }
+    return result;
+  }
+
+  // Helper function to continue chaining intermediate results.
+  function chainResult(instance, obj) {
+    return instance._chain ? _$1(obj).chain() : obj;
+  }
+
+  // Add your own custom functions to the Underscore object.
+  function mixin(obj) {
+    each(functions(obj), function(name) {
+      var func = _$1[name] = obj[name];
+      _$1.prototype[name] = function() {
+        var args = [this._wrapped];
+        push.apply(args, arguments);
+        return chainResult(this, func.apply(_$1, args));
+      };
+    });
+    return _$1;
+  }
+
+  // Add all mutator `Array` functions to the wrapper.
+  each(['pop', 'push', 'reverse', 'shift', 'sort', 'splice', 'unshift'], function(name) {
+    var method = ArrayProto[name];
+    _$1.prototype[name] = function() {
+      var obj = this._wrapped;
+      if (obj != null) {
+        method.apply(obj, arguments);
+        if ((name === 'shift' || name === 'splice') && obj.length === 0) {
+          delete obj[0];
+        }
+      }
+      return chainResult(this, obj);
+    };
+  });
+
+  // Add all accessor `Array` functions to the wrapper.
+  each(['concat', 'join', 'slice'], function(name) {
+    var method = ArrayProto[name];
+    _$1.prototype[name] = function() {
+      var obj = this._wrapped;
+      if (obj != null) obj = method.apply(obj, arguments);
+      return chainResult(this, obj);
+    };
+  });
+
+  // Named Exports
+
+  var allExports = {
+    __proto__: null,
+    VERSION: VERSION,
+    restArguments: restArguments,
+    isObject: isObject,
+    isNull: isNull,
+    isUndefined: isUndefined,
+    isBoolean: isBoolean,
+    isElement: isElement,
+    isString: isString,
+    isNumber: isNumber,
+    isDate: isDate,
+    isRegExp: isRegExp,
+    isError: isError,
+    isSymbol: isSymbol,
+    isArrayBuffer: isArrayBuffer,
+    isDataView: isDataView$1,
+    isArray: isArray,
+    isFunction: isFunction$1,
+    isArguments: isArguments$1,
+    isFinite: isFinite$1,
+    isNaN: isNaN$1,
+    isTypedArray: isTypedArray$1,
+    isEmpty: isEmpty,
+    isMatch: isMatch,
+    isEqual: isEqual,
+    isMap: isMap,
+    isWeakMap: isWeakMap,
+    isSet: isSet,
+    isWeakSet: isWeakSet,
+    keys: keys,
+    allKeys: allKeys,
+    values: values,
+    pairs: pairs,
+    invert: invert,
+    functions: functions,
+    methods: functions,
+    extend: extend,
+    extendOwn: extendOwn,
+    assign: extendOwn,
+    defaults: defaults,
+    create: create,
+    clone: clone,
+    tap: tap,
+    get: get,
+    has: has,
+    mapObject: mapObject,
+    identity: identity,
+    constant: constant,
+    noop: noop,
+    toPath: toPath$1,
+    property: property,
+    propertyOf: propertyOf,
+    matcher: matcher,
+    matches: matcher,
+    times: times,
+    random: random,
+    now: now,
+    escape: _escape,
+    unescape: _unescape,
+    templateSettings: templateSettings,
+    template: template,
+    result: result,
+    uniqueId: uniqueId,
+    chain: chain,
+    iteratee: iteratee,
+    partial: partial,
+    bind: bind,
+    bindAll: bindAll,
+    memoize: memoize,
+    delay: delay,
+    defer: defer,
+    throttle: throttle,
+    debounce: debounce,
+    wrap: wrap,
+    negate: negate,
+    compose: compose,
+    after: after,
+    before: before,
+    once: once,
+    findKey: findKey,
+    findIndex: findIndex,
+    findLastIndex: findLastIndex,
+    sortedIndex: sortedIndex,
+    indexOf: indexOf,
+    lastIndexOf: lastIndexOf,
+    find: find,
+    detect: find,
+    findWhere: findWhere,
+    each: each,
+    forEach: each,
+    map: map,
+    collect: map,
+    reduce: reduce,
+    foldl: reduce,
+    inject: reduce,
+    reduceRight: reduceRight,
+    foldr: reduceRight,
+    filter: filter,
+    select: filter,
+    reject: reject,
+    every: every,
+    all: every,
+    some: some,
+    any: some,
+    contains: contains,
+    includes: contains,
+    include: contains,
+    invoke: invoke,
+    pluck: pluck,
+    where: where,
+    max: max,
+    min: min,
+    shuffle: shuffle,
+    sample: sample,
+    sortBy: sortBy,
+    groupBy: groupBy,
+    indexBy: indexBy,
+    countBy: countBy,
+    partition: partition,
+    toArray: toArray,
+    size: size,
+    pick: pick,
+    omit: omit,
+    first: first,
+    head: first,
+    take: first,
+    initial: initial,
+    last: last,
+    rest: rest,
+    tail: rest,
+    drop: rest,
+    compact: compact,
+    flatten: flatten,
+    without: without,
+    uniq: uniq,
+    unique: uniq,
+    union: union,
+    intersection: intersection,
+    difference: difference,
+    unzip: unzip,
+    transpose: unzip,
+    zip: zip,
+    object: object,
+    range: range,
+    chunk: chunk,
+    mixin: mixin,
+    'default': _$1
+  };
+
+  // Default Export
+
+  // Add all of the Underscore functions to the wrapper object.
+  var _ = mixin(allExports);
+  // Legacy Node.js API.
+  _._ = _;
+
+  return _;
+
+})));
+//# sourceMappingURL=underscore-umd.js.map
diff --git a/_static/underscore.js b/_static/underscore.js
new file mode 100644
index 00000000..cf177d42
--- /dev/null
+++ b/_static/underscore.js
@@ -0,0 +1,6 @@
+!function(n,r){"object"==typeof exports&&"undefined"!=typeof module?module.exports=r():"function"==typeof define&&define.amd?define("underscore",r):(n="undefined"!=typeof globalThis?globalThis:n||self,function(){var t=n._,e=n._=r();e.noConflict=function(){return n._=t,e}}())}(this,(function(){
+//     Underscore.js 1.13.1
+//     https://underscorejs.org
+//     (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors
+//     Underscore may be freely distributed under the MIT license.
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diff --git a/_static/vendor/fontawesome/6.1.2/LICENSE.txt b/_static/vendor/fontawesome/6.1.2/LICENSE.txt
new file mode 100644
index 00000000..cc557ece
--- /dev/null
+++ b/_static/vendor/fontawesome/6.1.2/LICENSE.txt
@@ -0,0 +1,165 @@
+Fonticons, Inc. (https://fontawesome.com)
+
+--------------------------------------------------------------------------------
+
+Font Awesome Free License
+
+Font Awesome Free is free, open source, and GPL friendly. You can use it for
+commercial projects, open source projects, or really almost whatever you want.
+Full Font Awesome Free license: https://fontawesome.com/license/free.
+
+--------------------------------------------------------------------------------
+
+# Icons: CC BY 4.0 License (https://creativecommons.org/licenses/by/4.0/)
+
+The Font Awesome Free download is licensed under a Creative Commons
+Attribution 4.0 International License and applies to all icons packaged
+as SVG and JS file types.
+
+--------------------------------------------------------------------------------
+
+# Fonts: SIL OFL 1.1 License
+
+In the Font Awesome Free download, the SIL OFL license applies to all icons
+packaged as web and desktop font files.
+
+Copyright (c) 2022 Fonticons, Inc. (https://fontawesome.com)
+with Reserved Font Name: "Font Awesome".
+
+This Font Software is licensed under the SIL Open Font License, Version 1.1.
+This license is copied below, and is also available with a FAQ at:
+http://scripts.sil.org/OFL
+
+SIL OPEN FONT LICENSE
+Version 1.1 - 26 February 2007
+
+PREAMBLE
+The goals of the Open Font License (OFL) are to stimulate worldwide
+development of collaborative font projects, to support the font creation
+efforts of academic and linguistic communities, and to provide a free and
+open framework in which fonts may be shared and improved in partnership
+with others.
+
+The OFL allows the licensed fonts to be used, studied, modified and
+redistributed freely as long as they are not sold by themselves. The
+fonts, including any derivative works, can be bundled, embedded,
+redistributed and/or sold with any software provided that any reserved
+names are not used by derivative works. The fonts and derivatives,
+however, cannot be released under any other type of license. The
+requirement for fonts to remain under this license does not apply
+to any document created using the fonts or their derivatives.
+
+DEFINITIONS
+"Font Software" refers to the set of files released by the Copyright
+Holder(s) under this license and clearly marked as such. This may
+include source files, build scripts and documentation.
+
+"Reserved Font Name" refers to any names specified as such after the
+copyright statement(s).
+
+"Original Version" refers to the collection of Font Software components as
+distributed by the Copyright Holder(s).
+
+"Modified Version" refers to any derivative made by adding to, deleting,
+or substituting — in part or in whole — any of the components of the
+Original Version, by changing formats or by porting the Font Software to a
+new environment.
+
+"Author" refers to any designer, engineer, programmer, technical
+writer or other person who contributed to the Font Software.
+
+PERMISSION & CONDITIONS
+Permission is hereby granted, free of charge, to any person obtaining
+a copy of the Font Software, to use, study, copy, merge, embed, modify,
+redistribute, and sell modified and unmodified copies of the Font
+Software, subject to the following conditions:
+
+1) Neither the Font Software nor any of its individual components,
+in Original or Modified Versions, may be sold by itself.
+
+2) Original or Modified Versions of the Font Software may be bundled,
+redistributed and/or sold with any software, provided that each copy
+contains the above copyright notice and this license. These can be
+included either as stand-alone text files, human-readable headers or
+in the appropriate machine-readable metadata fields within text or
+binary files as long as those fields can be easily viewed by the user.
+
+3) No Modified Version of the Font Software may use the Reserved Font
+Name(s) unless explicit written permission is granted by the corresponding
+Copyright Holder. This restriction only applies to the primary font name as
+presented to the users.
+
+4) The name(s) of the Copyright Holder(s) or the Author(s) of the Font
+Software shall not be used to promote, endorse or advertise any
+Modified Version, except to acknowledge the contribution(s) of the
+Copyright Holder(s) and the Author(s) or with their explicit written
+permission.
+
+5) The Font Software, modified or unmodified, in part or in whole,
+must be distributed entirely under this license, and must not be
+distributed under any other license. The requirement for fonts to
+remain under this license does not apply to any document created
+using the Font Software.
+
+TERMINATION
+This license becomes null and void if any of the above conditions are
+not met.
+
+DISCLAIMER
+THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
+OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL THE
+COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
+INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL
+DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM
+OTHER DEALINGS IN THE FONT SOFTWARE.
+
+--------------------------------------------------------------------------------
+
+# Code: MIT License (https://opensource.org/licenses/MIT)
+
+In the Font Awesome Free download, the MIT license applies to all non-font and
+non-icon files.
+
+Copyright 2022 Fonticons, Inc.
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in the
+Software without restriction, including without limitation the rights to use, copy,
+modify, merge, publish, distribute, sublicense, and/or sell copies of the Software,
+and to permit persons to whom the Software is furnished to do so, subject to the
+following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
+INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
+PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
+HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
+OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
+SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+--------------------------------------------------------------------------------
+
+# Attribution
+
+Attribution is required by MIT, SIL OFL, and CC BY licenses. Downloaded Font
+Awesome Free files already contain embedded comments with sufficient
+attribution, so you shouldn't need to do anything additional when using these
+files normally.
+
+We've kept attribution comments terse, so we ask that you do not actively work
+to remove them from files, especially code. They're a great way for folks to
+learn about Font Awesome.
+
+--------------------------------------------------------------------------------
+
+# Brand Icons
+
+All brand icons are trademarks of their respective owners. The use of these
+trademarks does not indicate endorsement of the trademark holder by Font
+Awesome, nor vice versa. **Please do not use brand logos for any purpose except
+to represent the company, product, or service to which they refer.**
diff --git a/_static/vendor/fontawesome/6.1.2/css/all.min.css b/_static/vendor/fontawesome/6.1.2/css/all.min.css
new file mode 100644
index 00000000..aad4b1d9
--- /dev/null
+++ b/_static/vendor/fontawesome/6.1.2/css/all.min.css
@@ -0,0 +1,5 @@
+/*!
+ * Font Awesome Free 6.1.2 by @fontawesome - https://fontawesome.com
+ * License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License)
+ * Copyright 2022 Fonticons, Inc.
+ */.fa{font-family:var(--fa-style-family,"Font Awesome 6 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\ No newline at end of file
diff --git a/_static/vendor/fontawesome/6.1.2/webfonts/fa-brands-400.ttf b/_static/vendor/fontawesome/6.1.2/webfonts/fa-brands-400.ttf
new file mode 100644
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+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
+  
+</a></div>
+        <div class="sidebar-primary-item"><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="notebooks/symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/closing_remarks.html">Outlook</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/citing.html">Citing this book</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">References</a></li>
+</ul>
+
+    </div>
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+    <h1>References</h1>
+    <!-- Table of contents -->
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+  <div class="tex2jax_ignore mathjax_ignore section" id="references">
+<h1>References<a class="headerlink" href="#references" title="Permalink to this headline">#</a></h1>
+<div class="docutils container" id="id1">
+<dl class="citation">
+<dt class="label" id="id18"><span class="brackets">AGM+18</span></dt>
+<dd><p>Julius Adebayo, Justin Gilmer, Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been Kim. Sanity checks for saliency maps. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, <em>Advances in Neural Information Processing Systems</em>, volume 31. Curran Associates, Inc., 2018. URL: <a class="reference external" href="https://proceedings.neurips.cc/paper_files/paper/2018/file/294a8ed24b1ad22ec2e7efea049b8737-Paper.pdf">https://proceedings.neurips.cc/paper_files/paper/2018/file/294a8ed24b1ad22ec2e7efea049b8737-Paper.pdf</a>.</p>
+</dd>
+<dt class="label" id="id19"><span class="brackets">ACOG18</span></dt>
+<dd><p>Marco Ancona, Enea Ceolini, Cengiz Öztireli, and Markus Gross. Towards better understanding of gradient-based attribution methods for deep neural networks. In <em>International Conference on Learning Representations</em>. 2018. URL: <a class="reference external" href="https://openreview.net/forum?id=Sy21R9JAW">https://openreview.net/forum?id=Sy21R9JAW</a>.</p>
+</dd>
+<dt class="label" id="id4"><span class="brackets">AMP13</span></dt>
+<dd><p>H. M. Arnold, I. M. Moroz, and T. N. Palmer. Stochastic parametrizations and model uncertainty in the lorenz '96 system. <em>Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences</em>, 371(1991):20110479, 2013. URL: <a class="reference external" href="https://doi.org/10.1098/rsta.2011.0479">https://doi.org/10.1098/rsta.2011.0479</a>, <a class="reference external" href="https://doi.org/10.1098/rsta.2011.0479">doi:10.1098/rsta.2011.0479</a>.</p>
+</dd>
+<dt class="label" id="id20"><span class="brackets">BBM+15</span></dt>
+<dd><p>Sebastian Bach, Alexander Binder, Grégoire Montavon, Frederick Klauschen, Klaus-Robert Müller, and Wojciech Samek. On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation. <em>PLOS ONE</em>, 10(7):1–46, 07 2015. URL: <a class="reference external" href="https://doi.org/10.1371/journal.pone.0130140">https://doi.org/10.1371/journal.pone.0130140</a>, <a class="reference external" href="https://doi.org/10.1371/journal.pone.0130140">doi:10.1371/journal.pone.0130140</a>.</p>
+</dd>
+<dt class="label" id="id17"><span class="brackets">BSH+10</span></dt>
+<dd><p>David Baehrens, Timon Schroeter, Stefan Harmeling, Motoaki Kawanabe, Katja Hansen, and Klaus-Robert Müller. How to explain individual classification decisions. <em>Journal of Machine Learning Research</em>, 11(61):1803–1831, 2010. URL: <a class="reference external" href="http://jmlr.org/papers/v11/baehrens10a.html">http://jmlr.org/papers/v11/baehrens10a.html</a>.</p>
+</dd>
+<dt class="label" id="id22"><span class="brackets">Bis06</span></dt>
+<dd><p>Christopher M. Bishop. <em>Pattern Recognition and Machine Learning (Information Science and Statistics)</em>. Springer-Verlag, Berlin, Heidelberg, 2006. ISBN 0387310738.</p>
+</dd>
+<dt class="label" id="id29"><span class="brackets">CB23</span></dt>
+<dd><p>William E Chapman and Judith Berner. Benefits of deterministic and stochastic tendency adjustments in a climate model. <em>arXiv preprint arXiv:2308.15295</em>, 2023.</p>
+</dd>
+<dt class="label" id="id13"><span class="brackets">CL15</span></dt>
+<dd><p>Alexandre J. Chorin and Fei Lu. Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics. <em>Proceedings of the National Academy of Sciences</em>, 112(32):9804–9809, July 2015. URL: <a class="reference external" href="https://doi.org/10.1073/pnas.1512080112">https://doi.org/10.1073/pnas.1512080112</a>, <a class="reference external" href="https://doi.org/10.1073/pnas.1512080112">doi:10.1073/pnas.1512080112</a>.</p>
+</dd>
+<dt class="label" id="id8"><span class="brackets">CVE08</span></dt>
+<dd><p>Daan Crommelin and Eric Vanden-Eijnden. Subgrid-scale parameterization with conditional markov chains. <em>Journal of the Atmospheric Sciences</em>, 65(8):2661–2675, August 2008. URL: <a class="reference external" href="https://doi.org/10.1175/2008jas2566.1">https://doi.org/10.1175/2008jas2566.1</a>, <a class="reference external" href="https://doi.org/10.1175/2008jas2566.1">doi:10.1175/2008jas2566.1</a>.</p>
+</dd>
+<dt class="label" id="id9"><span class="brackets">DCBBoing13</span></dt>
+<dd><p>J. Dorrestijn, D. T. Crommelin, J. A. Biello, and S. J. Böing. A data-driven multi-cloud model for stochastic parametrization of deep convection. <em>Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences</em>, 371(1991):20120374, May 2013. URL: <a class="reference external" href="https://doi.org/10.1098/rsta.2012.0374">https://doi.org/10.1098/rsta.2012.0374</a>, <a class="reference external" href="https://doi.org/10.1098/rsta.2012.0374">doi:10.1098/rsta.2012.0374</a>.</p>
+</dd>
+<dt class="label" id="id14"><span class="brackets">DB18</span></dt>
+<dd><p>Peter D. Dueben and Peter Bauer. Challenges and design choices for global weather and climate models based on machine learning. <em>Geoscientific Model Development</em>, 11(10):3999–4009, October 2018. URL: <a class="reference external" href="https://doi.org/10.5194/gmd-11-3999-2018">https://doi.org/10.5194/gmd-11-3999-2018</a>, <a class="reference external" href="https://doi.org/10.5194/gmd-11-3999-2018">doi:10.5194/gmd-11-3999-2018</a>.</p>
+</dd>
+<dt class="label" id="id32"><span class="brackets">FLSF+22</span></dt>
+<dd><p>Hugo Frezat, Julien Le Sommer, Ronan Fablet, Guillaume Balarac, and Redouane Lguensat. A posteriori learning for quasi-geostrophic turbulence parametrization. <em>Journal of Advances in Modeling Earth Systems</em>, 14(11):e2022MS003124, 2022. e2022MS003124 2022MS003124. URL: <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2022MS003124">https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2022MS003124</a>, <a class="reference external" href="https://arxiv.org/abs/https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2022MS003124">arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2022MS003124</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1029/2022MS003124">doi:https://doi.org/10.1029/2022MS003124</a>.</p>
+</dd>
+<dt class="label" id="id15"><span class="brackets">GCSM20</span></dt>
+<dd><p>David John Gagne, Hannah M. Christensen, Aneesh C. Subramanian, and Adam H. Monahan. Machine learning for stochastic parameterization: generative adversarial networks in the lorenz \textquotesingle 96 model. <em>Journal of Advances in Modeling Earth Systems</em>, March 2020. URL: <a class="reference external" href="https://doi.org/10.1029/2019ms001896">https://doi.org/10.1029/2019ms001896</a>, <a class="reference external" href="https://doi.org/10.1029/2019ms001896">doi:10.1029/2019ms001896</a>.</p>
+</dd>
+<dt class="label" id="id30"><span class="brackets">GBA+23</span></dt>
+<dd><p>William Gregory, Mitchell Bushuk, Alistair Adcroft, Yongfei Zhang, and Laure Zanna. Deep learning of systematic sea ice model errors from data assimilation increments. <em>Journal of Advances in Modeling Earth Systems</em>, 15(10):e2023MS003757, 2023.</p>
+</dd>
+<dt class="label" id="id31"><span class="brackets">GBZ+23</span></dt>
+<dd><p>William Gregory, Mitchell Bushuk, Yongfei Zhang, Alistair Adcroft, and Laure Zanna. Machine learning for online sea ice bias correction within global ice-ocean simulations. 2023. <a class="reference external" href="https://arxiv.org/abs/2310.02488">arXiv:2310.02488</a>.</p>
+</dd>
+<dt class="label" id="id28"><span class="brackets">GZ21</span></dt>
+<dd><p>Arthur P. Guillaumin and Laure Zanna. Stochastic-deep learning parameterization of ocean momentum forcing. <em>Journal of Advances in Modeling Earth Systems</em>, 13(9):e2021MS002534, 2021. e2021MS002534 2021MS002534. URL: <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021MS002534">https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021MS002534</a>, <a class="reference external" href="https://arxiv.org/abs/https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2021MS002534">arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2021MS002534</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1029/2021MS002534">doi:https://doi.org/10.1029/2021MS002534</a>.</p>
+</dd>
+<dt class="label" id="id11"><span class="brackets">HSPDuben17</span></dt>
+<dd><p>Sam Hatfield, Aneesh Subramanian, Tim Palmer, and Peter Düben. Improving weather forecast skill through reduced-precision data assimilation. <em>Monthly Weather Review</em>, 146(1):49–62, December 2017. URL: <a class="reference external" href="https://doi.org/10.1175/mwr-d-17-0132.1">https://doi.org/10.1175/mwr-d-17-0132.1</a>, <a class="reference external" href="https://doi.org/10.1175/mwr-d-17-0132.1">doi:10.1175/mwr-d-17-0132.1</a>.</p>
+</dd>
+<dt class="label" id="id12"><span class="brackets">Kwa12</span></dt>
+<dd><p>Frank Kwasniok. Data-based stochastic subgrid-scale parametrization: an approach using cluster-weighted modelling. <em>Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences</em>, 370(1962):1061–1086, March 2012. URL: <a class="reference external" href="https://doi.org/10.1098/rsta.2011.0384">https://doi.org/10.1098/rsta.2011.0384</a>, <a class="reference external" href="https://doi.org/10.1098/rsta.2011.0384">doi:10.1098/rsta.2011.0384</a>.</p>
+</dd>
+<dt class="label" id="id10"><span class="brackets">LSASS16</span></dt>
+<dd><p>K.J.H. Law, D. Sanz-Alonso, A. Shukla, and A.M. Stuart. Filter accuracy for the lorenz 96 model: fixed versus adaptive observation operators. <em>Physica D: Nonlinear Phenomena</em>, 325:1–13, June 2016. URL: <a class="reference external" href="https://doi.org/10.1016/j.physd.2015.12.008">https://doi.org/10.1016/j.physd.2015.12.008</a>, <a class="reference external" href="https://doi.org/10.1016/j.physd.2015.12.008">doi:10.1016/j.physd.2015.12.008</a>.</p>
+</dd>
+<dt class="label" id="id2"><span class="brackets">Lor95</span></dt>
+<dd><p>E.N. Lorenz. Predictability: a problem partly solved. <em>Seminar on Predictability</em>, 1:1–18, 1995. URL: <a class="reference external" href="https://www.ecmwf.int/node/10829">https://www.ecmwf.int/node/10829</a>.</p>
+</dd>
+<dt class="label" id="id34"><span class="brackets">OZB23</span></dt>
+<dd><p>Karl Otness, Laure Zanna, and Joan Bruna. Data-driven multiscale modeling of subgrid parameterizations in climate models. <em>arXiv preprint arXiv:2303.17496</em>, 2023.</p>
+</dd>
+<dt class="label" id="id33"><span class="brackets">PZBP23</span></dt>
+<dd><p>Christian Pedersen, Laure Zanna, Joan Bruna, and Pavel Perezhogin. Reliable coarse-grained turbulent simulations through combined offline learning and neural emulation. <em>arXiv preprint arXiv:2307.13144</em>, 2023.</p>
+</dd>
+<dt class="label" id="id26"><span class="brackets">PZFG23</span></dt>
+<dd><p>Pavel Perezhogin, Laure Zanna, and Carlos Fernandez-Granda. Generative data-driven approaches for stochastic subgrid parameterizations in an idealized ocean model. <em>arXiv preprint arXiv:2302.07984</em>, 2023.</p>
+</dd>
+<dt class="label" id="id21"><span class="brackets">RW05</span></dt>
+<dd><p>Carl Edward Rasmussen and Christopher K. I. Williams. <em>Gaussian Processes for Machine Learning</em>. The MIT Press, 2005. URL: <a class="reference external" href="http://dx.doi.org/10.7551/mitpress/3206.001.0001">http://dx.doi.org/10.7551/mitpress/3206.001.0001</a>, <a class="reference external" href="https://doi.org/10.7551/mitpress/3206.001.0001">doi:10.7551/mitpress/3206.001.0001</a>.</p>
+</dd>
+<dt class="label" id="id35"><span class="brackets">RLP+22</span></dt>
+<dd><p>Andrew Ross, Ziwei Li, Pavel Perezhogin, Carlos Fernandez-Granda, and Laure Zanna. Benchmarking of machine learning ocean subgrid parameterizations in an idealized model. <em>Journal of Advances in Modeling Earth Systems</em>, pages e2022MS003258, 2022. e2022MS003258 2022MS003258. URL: <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2022MS003258">https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2022MS003258</a>, <a class="reference external" href="https://arxiv.org/abs/https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2022MS003258">arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2022MS003258</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1029/2022MS003258">doi:https://doi.org/10.1029/2022MS003258</a>.</p>
+</dd>
+<dt class="label" id="id24"><span class="brackets">SRAZ23</span></dt>
+<dd><p>Aakash Sane, Brandon G Reichl, Alistair Adcroft, and Laure Zanna. Parameterizing vertical mixing coefficients in the ocean surface boundary layer using neural networks. <em>arXiv preprint arXiv:2306.09045</em>, 2023.</p>
+</dd>
+<dt class="label" id="id6"><span class="brackets">SLST17</span></dt>
+<dd><p>Tapio Schneider, Shiwei Lan, Andrew Stuart, and João Teixeira. Earth system modeling 2.0: a blueprint for models that learn from observations and targeted high-resolution simulations. <em>Geophysical Research Letters</em>, December 2017. URL: <a class="reference external" href="https://doi.org/10.1002/2017gl076101">https://doi.org/10.1002/2017gl076101</a>, <a class="reference external" href="https://doi.org/10.1002/2017gl076101">doi:10.1002/2017gl076101</a>.</p>
+</dd>
+<dt class="label" id="id23"><span class="brackets">SG23</span></dt>
+<dd><p><strong>missing journal in Shamekh_2023</strong></p>
+</dd>
+<dt class="label" id="id16"><span class="brackets">SVZ13</span></dt>
+<dd><p>Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: visualising image classification models and saliency maps. 2013. URL: <a class="reference external" href="https://arxiv.org/abs/1312.6034">https://arxiv.org/abs/1312.6034</a>, <a class="reference external" href="https://doi.org/10.48550/ARXIV.1312.6034">doi:10.48550/ARXIV.1312.6034</a>.</p>
+</dd>
+<dt class="label" id="id3"><span class="brackets">Wil05</span></dt>
+<dd><p>Daniel S. Wilks. Effects of stochastic parametrizations in the lorenz \textquotesingle 96 system. <em>Quarterly Journal of the Royal Meteorological Society</em>, 131(606):389–407, 2005. URL: <a class="reference external" href="https://doi.org/10.1256/qj.04.03">https://doi.org/10.1256/qj.04.03</a>, <a class="reference external" href="https://doi.org/10.1256/qj.04.03">doi:10.1256/qj.04.03</a>.</p>
+</dd>
+<dt class="label" id="id25"><span class="brackets">YOGorman23</span></dt>
+<dd><p>Janni Yuval and Paul A. O'Gorman. Neural-network parameterization of subgrid momentum transport in the atmosphere. <em>Journal of Advances in Modeling Earth Systems</em>, 15(4):e2023MS003606, 2023. e2023MS003606 2023MS003606. URL: <a class="reference external" href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2023MS003606">https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2023MS003606</a>, <a class="reference external" href="https://arxiv.org/abs/https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2023MS003606">arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2023MS003606</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1029/2023MS003606">doi:https://doi.org/10.1029/2023MS003606</a>.</p>
+</dd>
+<dt class="label" id="id27"><span class="brackets">ZAH+23</span></dt>
+<dd><p>Lorenzo Zampieri, Gabriele Arduini, Marika Holland, Sarah P. E. Keeley, Kristian Mogensen, Matthew D. Shupe, and Steffen Tietsche. A machine learning correction model of the winter clear-sky temperature bias over the arctic sea ice in atmospheric reanalyses. <em>Monthly Weather Review</em>, 151(6):1443 – 1458, 2023. URL: <a class="reference external" href="https://journals.ametsoc.org/view/journals/mwre/151/6/MWR-D-22-0130.1.xml">https://journals.ametsoc.org/view/journals/mwre/151/6/MWR-D-22-0130.1.xml</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1175/MWR-D-22-0130.1">doi:https://doi.org/10.1175/MWR-D-22-0130.1</a>.</p>
+</dd>
+<dt class="label" id="id36"><span class="brackets">ZPG+23</span></dt>
+<dd><p>Cheng Zhang, Pavel Perezhogin, Cem Gultekin, Alistair Adcroft, Carlos Fernandez-Granda, and Laure Zanna. Implementation and evaluation of a machine learned mesoscale eddy parameterization into a numerical ocean circulation model. <em>arXiv preprint arXiv:2303.00962</em>, 2023.</p>
+</dd>
+</dl>
+</div>
+</div>
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/constraints.html">Adding constraints to Neural Networks</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/constraints.html">Adding constraints to Neural Networks</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
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+<h1>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">#</a></h1>
+<p>The climate system is composed of many interacting components (e.g., ocean, atmosphere, ice) and described by complex nonlinear equations. To simulate, understand and predict climate, these equations are solved numerically under a number of simplifications, therefore leading to errors. The errors are the result of numerics used to solve the equations and the lack of appropriate representations of processes occurring below the resolution of the climate model grid (i.e., subgrid processes).</p>
+<p>The goal of this book is to conceptualize the problems associated with climate models within a simple and computationally accessible framework. We will introduce the readers to climate modeling by using a simple tool, the <span id="id1">[<a class="reference internal" href="bibliography.html#id2" title="E.N. Lorenz. Predictability: a problem partly solved. Seminar on Predictability, 1:1-18, 1995. URL: https://www.ecmwf.int/node/10829.">Lorenz, 1995</a>]</span> (L96) two-timescale model. We discuss the numerical aspects of the L96 model, the approximate representation of subgrid processes (known as parameterizations or closures), and simple data assimilation problems (a data-model fusion method). We will then use the L96 results to demonstrate how to learn subgrid parameterizations from data with machine learning, and then test the parameterizations offline (apriori) and online (aposteriori), with a focus on the interpretability of the results.</p>
+<p>The book was created by and as part of <a class="reference external" href="https://m2lines.github.io/">M<sup>2</sup>LInES</a>, an international collaboration supported by <a class="reference external" href="https://www.schmidtfutures.com/">Schmidt Futures</a>, to improve climate models with scientific machine learning. The goal for this book was for our team to work together and learn from each other; in particular, to get up to speed on the key scientific aspects of our collaboration (parameterizations, machine learning, data assimilation, uncertainty quantification) and to develop new ideas. Ultimately, we are happy to share these resources with the scientific community, to introduce our research ideas and foster the use of machine learning techniques for tackling climate science problems.</p>
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+        <div class="sidebar-primary-item"><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="closing_remarks.html">Outlook</a></li>
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+    <h1>Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#what-is-da-why-do-we-do-it">What is DA? Why do we do it?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#design-of-the-data-assimilation-experiments-in-l96">Design of the data assimilation experiments in L96</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-our-gcm-and-da-parameters-to-use-throughout-notebook">1. Define our “GCM” and DA parameters to use throughout notebook</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-truth-run-from-two-time-scale-l96-model">2. Generate <code class="docutils literal notranslate"><span class="pre">truth</span></code> run from two time-scale L96 model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-synthetic-observations">3. Generate synthetic observations</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#check-the-model-background-covariances-for-the-variable-x">Check the Model Background Covariances for the variable <code class="docutils literal notranslate"><span class="pre">X</span></code></a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#run-data-assimilation">4. Run Data Assimilation</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#post-processing-and-visualization">5. Post Processing and Visualization</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="data-assimilation-demo-in-the-lorenz-96-l96-two-time-scale-model">
+<h1>Data Assimilation demo in the Lorenz 96 (L96) two time-scale model<a class="headerlink" href="#data-assimilation-demo-in-the-lorenz-96-l96-two-time-scale-model" title="Permalink to this headline">#</a></h1>
+<div class="section" id="what-is-da-why-do-we-do-it">
+<h2>What is DA? Why do we do it?<a class="headerlink" href="#what-is-da-why-do-we-do-it" title="Permalink to this headline">#</a></h2>
+<p>Data assimilation is an approach for combining data (e.g. observations) with prior knowledge (e.g. from a physical model) to obtain an estimate of the true state of a system. We use data assimilation to generate accurate initial conditions for weather and climate predictions, and to produce reanalysis products (state estimates) of the atmosphere-ocean-sea ice-land system.</p>
+<p>Conceptually, data assimilation follows the Bayes’ Rule:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-0dc0e919-8d92-43ce-9fb2-83909695ae12">
+<span class="eqno">(18)<a class="headerlink" href="#equation-0dc0e919-8d92-43ce-9fb2-83909695ae12" title="Permalink to this equation">#</a></span>\[\begin{align}
+\ P(X|Obs)=\frac{P(Obs|X)P(X)}{P(Obs)}
+\end{align}\]</div>
+<p>where <span class="math notranslate nohighlight">\(X\)</span> is the state of the dynamic system, <span class="math notranslate nohighlight">\(P(X|Obs)\)</span> is the posterior distribution, or the analysis in DA terms, and <span class="math notranslate nohighlight">\(P(X)\)</span> is the prior distribution, or the forecast in DA terms. <span class="math notranslate nohighlight">\(P(Obs|X)\)</span> is the observation distribution given the prior. Simply speaking, we want to know the best estimate of the current state <span class="math notranslate nohighlight">\(X\)</span> of the system given the available observations. Below is a simple schematic representation of one DA step.</p>
+<div class="figure align-default">
+<img alt="../_images/DA_step.png" src="../_images/DA_step.png" />
+</div>
+</div>
+<div class="section" id="design-of-the-data-assimilation-experiments-in-l96">
+<h2>Design of the data assimilation experiments in L96<a class="headerlink" href="#design-of-the-data-assimilation-experiments-in-l96" title="Permalink to this headline">#</a></h2>
+<p>We use the 2-scale Lorenz-96 model as our truth, or our representation of the “real world”. We then use the 1-scale L96 model as our DA model, or our representation of the “GCM”. Detailed description of this setup can be find here (https://m2lines.github.io/L96_demo/notebooks/gcm-analogue.html). This is what we call an idealized biased-model framework in DA research, in the sense that our DA model is biased compared to the truth, similar to real-world situations where our GCMs are biased against the natural weather or climate systems.</p>
+<p>We will follow the follwing steps when doing DA in this idealized biased L96 framework:</p>
+<ol class="arabic simple">
+<li><p>Specify a biased “GCM” model for performing DA, and set the parameters for the “GCM”, observations, and DA method in <code class="docutils literal notranslate"><span class="pre">Config</span></code>.</p>
+<ul class="simple">
+<li><p>Parameters to experiment in <code class="docutils literal notranslate"><span class="pre">Config</span></code>: forcing <code class="docutils literal notranslate"><span class="pre">F</span></code>, seasonal cycle, experiment length,</p></li>
+</ul>
+</li>
+<li><p>Generate a “truth” signal from the 2-scale L96 model</p></li>
+<li><p>Collect sparse and noisy observations, sampled from “truth”</p>
+<ul class="simple">
+<li><p>Parameters to experiment in <code class="docutils literal notranslate"><span class="pre">Config</span></code>: observation density in space, frequency in time, observational error</p></li>
+</ul>
+</li>
+<li><p>Perform DA using a method of choice</p>
+<ul class="simple">
+<li><p>Parameters to experiment in <code class="docutils literal notranslate"><span class="pre">Config</span></code>: DA method (EnKF, 3DVAR, Hybrid EnKF), DA frequency, ensemble size (for EnKF), localization, inflation method and factor</p></li>
+<li><p>Optional: parameter estimation to recover the true forcing</p></li>
+</ul>
+</li>
+<li><p>Assess DA performance</p>
+<ul class="simple">
+<li><p>RMSE of DA experiment relative to truth</p></li>
+</ul>
+</li>
+<li><p>Analyze DA increments to assess GCM model error (optional)</p></li>
+</ol>
+<div class="section" id="define-our-gcm-and-da-parameters-to-use-throughout-notebook">
+<h3>1. Define our “GCM” and DA parameters to use throughout notebook<a class="headerlink" href="#define-our-gcm-and-da-parameters-to-use-throughout-notebook" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
+<span class="kn">import</span> <span class="nn">time</span>
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">RK4</span><span class="p">,</span> <span class="n">L96_2t_xdot_ydot</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span><span class="p">,</span> <span class="n">L96s</span>
+<span class="kn">import</span> <span class="nn">DA_methods</span>
+
+<span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">GCM</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;A toy `General Circulation Model` (GCM) that uses the single time-scale Lorenz 1996 model</span>
+<span class="sd">    with a parameterised coupling term to represent the interaction between the observed coarse</span>
+<span class="sd">    scale processes `X` and unobserved fine scale processes `Y` of the two time-scale model.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X_init: Initial conditions of X</span>
+<span class="sd">        F: Forcing term</span>
+<span class="sd">        dt: Sampling frequency of the model</span>
+<span class="sd">        nt: Number of timesteps for which to run the model</span>
+<span class="sd">        param: Weights to give to the coupling term</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        Model output for all variables of X at each timestep along with the corresponding time units</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+        <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_init</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+        <span class="n">X_init</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+    <span class="p">)</span>
+    <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">+</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">F</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">))</span>
+        <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Config</span><span class="p">:</span>
+    <span class="c1"># fmt: off</span>
+    <span class="n">K</span> <span class="o">=</span> <span class="mi">40</span>                              <span class="c1"># Dimension of L96 `X` variables</span>
+    <span class="n">J</span> <span class="o">=</span> <span class="mi">10</span>                              <span class="c1"># Dimension of L96 `Y` variables</span>
+    <span class="n">obs_freq</span> <span class="o">=</span> <span class="mi">10</span>                       <span class="c1"># Observation frequency (number of sampling intervals (si) per observation)</span>
+    <span class="n">F_truth</span> <span class="o">=</span> <span class="mi">10</span>                        <span class="c1"># F for truth signal</span>
+    <span class="n">F_fcst</span> <span class="o">=</span> <span class="mi">10</span>                         <span class="c1"># F for forecast (DA) model</span>
+    <span class="n">GCM_param</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>  <span class="c1"># Polynomial coefficicents for GCM parameterization</span>
+    <span class="n">ns_da</span> <span class="o">=</span> <span class="mi">2000</span>                        <span class="c1"># Number of time samples for DA</span>
+    <span class="n">ns</span> <span class="o">=</span> <span class="mi">2000</span>                           <span class="c1"># Number of time samples for truth signal</span>
+    <span class="n">ns_spinup</span> <span class="o">=</span> <span class="mi">200</span>                     <span class="c1"># Number of time samples for spin up</span>
+    <span class="n">dt</span> <span class="o">=</span> <span class="mf">0.005</span>                          <span class="c1"># Model timestep</span>
+    <span class="n">si</span> <span class="o">=</span> <span class="mf">0.005</span>                          <span class="c1"># Truth sampling interval</span>
+    <span class="n">seasonal</span> <span class="o">=</span> <span class="kc">False</span>                    <span class="c1"># Option for adding a seasonal cycle to the forcing in the L96 truth model</span>
+    <span class="n">B_loc</span> <span class="o">=</span> <span class="mi">5</span>                           <span class="c1"># Spatial localization radius for DA in terms of model grid points</span>
+    <span class="n">DA</span> <span class="o">=</span> <span class="s2">&quot;EnKF&quot;</span>                         <span class="c1"># DA method</span>
+    <span class="n">nens</span> <span class="o">=</span> <span class="mi">500</span>                          <span class="c1"># Number of ensemble members for DA</span>
+    <span class="n">inflate_opt</span> <span class="o">=</span> <span class="s2">&quot;relaxation&quot;</span>          <span class="c1"># Method for DA model covariance inflation</span>
+    <span class="n">inflate_factor</span> <span class="o">=</span> <span class="mf">0.65</span>                <span class="c1"># Inflation factor</span>
+    <span class="n">hybrid_factor</span> <span class="o">=</span> <span class="mf">0.1</span>                 <span class="c1"># Weighting factor for hybrid EnKF method (weight of static B matrix)</span>
+    <span class="n">param_DA</span> <span class="o">=</span> <span class="kc">False</span>                    <span class="c1"># Switch to turn on parameter estimation with DA</span>
+    <span class="n">param_sd</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.02</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]</span>   <span class="c1"># Polynomal parameter standard deviation</span>
+    <span class="n">param_inflate</span> <span class="o">=</span> <span class="s2">&quot;multiplicative&quot;</span>    <span class="c1"># Method for parameter variance inflation</span>
+    <span class="n">param_inf_factor</span> <span class="o">=</span> <span class="mf">0.02</span>             <span class="c1"># Parameter inflation factor</span>
+    <span class="n">obs_density</span> <span class="o">=</span> <span class="mf">0.2</span>                   <span class="c1"># Fraction of spatial gridpoints where observations are collected</span>
+    <span class="n">DA_freq</span> <span class="o">=</span> <span class="mi">10</span>                        <span class="c1"># Assimilation frequency (number of sampling intervals (si) per assimilation step)</span>
+    <span class="n">obs_sigma</span> <span class="o">=</span> <span class="mf">0.5</span>                     <span class="c1"># Observational error standard deviation</span>
+    <span class="n">save_fig</span> <span class="o">=</span> <span class="kc">False</span>                    <span class="c1"># Switch to save figure file</span>
+    <span class="n">save_data</span> <span class="o">=</span> <span class="kc">False</span>                   <span class="c1"># Switch to save</span>
+    <span class="c1"># fmt: on</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="generate-truth-run-from-two-time-scale-l96-model">
+<h3>2. Generate <code class="docutils literal notranslate"><span class="pre">truth</span></code> run from two time-scale L96 model<a class="headerlink" href="#generate-truth-run-from-two-time-scale-l96-model" title="Permalink to this headline">#</a></h3>
+<p>We initialise the L96 two time-scale model using a set of random normally distributed values for <span class="math notranslate nohighlight">\(X\)</span>, and zeros for <span class="math notranslate nohighlight">\(Y\)</span>, and run it forward for a period <code class="docutils literal notranslate"><span class="pre">ns_spinup</span></code> to allow the model to spinup. The <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(Y\)</span> components of the spinup are then used as initial conditions for another forward run for a time period <code class="docutils literal notranslate"><span class="pre">ns</span></code> to represent the unobserved <code class="docutils literal notranslate"><span class="pre">truth</span></code> field from which our observations will be derived.</p>
+<p><em>Note that in reality we do not observe the fine scale components (<span class="math notranslate nohighlight">\(Y\)</span>s) and so this is what we aim to represent in the parameterisation in the <code class="docutils literal notranslate"><span class="pre">GCM()</span></code> function.</em></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up the &quot;truth&quot; 2-scale L96 model</span>
+<span class="n">M_truth</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">F_truth</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">dt</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Generate the initial conditions of <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(Y\)</span></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M_truth</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">rng</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">((</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">)),</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
+
+<span class="c1"># The model runs for `ns_spinup` timesteps to spin-up</span>
+<span class="n">X_spinup</span><span class="p">,</span> <span class="n">Y_spinup</span><span class="p">,</span> <span class="n">t_spinup</span> <span class="o">=</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns_spinup</span><span class="p">)</span>
+
+<span class="c1"># The initial conditions for the first forecast (prior to DA)</span>
+<span class="c1"># are the last time sample after spinup</span>
+<span class="n">X_init</span> <span class="o">=</span> <span class="n">X_spinup</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">Y_init</span> <span class="o">=</span> <span class="n">Y_spinup</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Using the generated initial conditions above, run <strong>L96</strong> to generate the <code class="docutils literal notranslate"><span class="pre">truth</span></code>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M_truth</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">Y_init</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>L96: K=40 J=10 F=10 h=1 b=10 c=10 dt=0.005
+</pre></div>
+</div>
+</div>
+</div>
+<p>If given in <code class="docutils literal notranslate"><span class="pre">Config</span></code>, give <code class="docutils literal notranslate"><span class="pre">F</span></code> a <strong>seasonal cycle</strong> in the truth model</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">seasonal</span><span class="p">:</span>
+    <span class="n">ann_period</span> <span class="o">=</span> <span class="mi">2000</span>
+    <span class="n">mon_period</span> <span class="o">=</span> <span class="mi">100</span>
+    <span class="n">mon_per_ann</span> <span class="o">=</span> <span class="n">ann_period</span> <span class="o">/</span> <span class="n">mon_period</span>
+    <span class="n">X_truth</span><span class="p">,</span> <span class="n">Y_truth</span><span class="p">,</span> <span class="n">t_truth</span> <span class="o">=</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span> <span class="o">*</span> <span class="n">mon_period</span><span class="p">)</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns</span> <span class="o">/</span> <span class="n">mon_period</span><span class="p">)):</span>
+        <span class="n">M_truth</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">X_truth</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">...</span><span class="p">],</span> <span class="n">Y_truth</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">...</span><span class="p">])</span>
+        <span class="n">M_truth</span><span class="o">.</span><span class="n">set_param</span><span class="p">(</span><span class="n">F</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">F_truth</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">i</span> <span class="o">/</span> <span class="n">mon_per_ann</span><span class="p">))</span>
+        <span class="n">X_step</span><span class="p">,</span> <span class="n">Y_step</span><span class="p">,</span> <span class="n">t_step</span> <span class="o">=</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span> <span class="o">*</span> <span class="n">mon_period</span><span class="p">)</span>
+        <span class="n">X_truth</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">X_truth</span><span class="p">,</span> <span class="n">X_step</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="kc">None</span><span class="p">,</span> <span class="o">...</span><span class="p">]))</span>
+        <span class="n">Y_truth</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">Y_truth</span><span class="p">,</span> <span class="n">Y_step</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="kc">None</span><span class="p">,</span> <span class="o">...</span><span class="p">]))</span>
+        <span class="n">t_truth</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">t_truth</span><span class="p">,</span> <span class="n">t_truth</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">t_step</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="kc">None</span><span class="p">]))</span>
+<span class="k">else</span><span class="p">:</span>
+    <span class="n">X_truth</span><span class="p">,</span> <span class="n">Y_truth</span><span class="p">,</span> <span class="n">t_truth</span> <span class="o">=</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Now, we generate climatological background (temporal) covariance <code class="docutils literal notranslate"><span class="pre">B_clim2</span></code> for the 2-scale L96 model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">B_clim2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">X_truth</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Plotting values of <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(Y\)</span></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">221</span><span class="p">)</span>
+
+<span class="c1"># Snapshot of X[k]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">X_truth</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;X&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">j</span> <span class="o">/</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">J</span><span class="p">,</span> <span class="n">Y_truth</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Y&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$X,Y$ @ $t=N\Delta t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:],</span> <span class="s2">&quot;k:&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">j</span> <span class="o">/</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">J</span><span class="p">,</span> <span class="n">Y_truth</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:],</span> <span class="s2">&quot;k:&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">222</span><span class="p">)</span>
+
+<span class="c1"># Sample time-series X[0](t), Y[0](t)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">,</span> <span class="n">X_truth</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;X&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">,</span> <span class="n">Y_truth</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Y&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$X[0,t]$, $Y[0,t]$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">223</span><span class="p">)</span>
+
+<span class="c1"># Full model history of X</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">t_truth</span><span class="p">,</span> <span class="n">X_truth</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">orientation</span><span class="o">=</span><span class="s2">&quot;horizontal&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$X(k,t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">224</span><span class="p">)</span>
+
+<span class="c1"># Full model history of Y</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">j</span> <span class="o">/</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">J</span><span class="p">,</span> <span class="n">t_truth</span><span class="p">,</span> <span class="n">Y_truth</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">orientation</span><span class="o">=</span><span class="s2">&quot;horizontal&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$Y(k,t)$&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/740e8bac0bfac35e909f3e0127fd3ef3704b3c9373f1504320d73867901490a0.png" src="../_images/740e8bac0bfac35e909f3e0127fd3ef3704b3c9373f1504320d73867901490a0.png" />
+</div>
+</div>
+<p>Generating climatological background covariance <code class="docutils literal notranslate"><span class="pre">B_clim1</span></code> for 1-scale L96 model. This climatological covariance will be used by the 3-dimensional variational (3DVar) DA method, or by the hybrid 3DVar-EnKF method. Since we cannot observe every state variable at every possible location, when assimilting sparse observations into the model, we need to use each observation to update multiple model locations, usually around the observed location. The covariance are thus needed to project the model-observation misfits between different model locations in certain DA methods such as 3DVar and EnKF.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M_1s</span> <span class="o">=</span> <span class="n">L96s</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">F_truth</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="n">RK4</span><span class="p">)</span>
+<span class="n">M_1s</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">X_init</span><span class="p">)</span>
+<span class="n">X1_truth</span><span class="p">,</span> <span class="n">t1_truth</span> <span class="o">=</span> <span class="n">M_1s</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">si</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns</span><span class="p">)</span>
+<span class="n">B_clim1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">X1_truth</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="generate-synthetic-observations">
+<h3>3. Generate synthetic observations<a class="headerlink" href="#generate-synthetic-observations" title="Permalink to this headline">#</a></h3>
+<p>Here we generate a set of sparse observations by sampling from the <code class="docutils literal notranslate"><span class="pre">X_truth</span></code> run and adding some random Gaussian noise. The observations are stored as individual samples, each containing the time, location, and value.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Sample the &quot;truth&quot; to generate observations at certain times (t_obs) and locations (l_obs)</span>
+<span class="n">t_obs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="p">),</span>
+    <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="p">),</span> <span class="mi">1</span><span class="p">],</span>
+<span class="p">)</span><span class="o">.</span><span class="n">T</span>
+
+<span class="n">l_obs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">t_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="s2">&quot;int&quot;</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">l_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+    <span class="n">l_obs</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="p">),</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span>
+    <span class="p">)</span>
+
+<span class="n">X_obs</span> <span class="o">=</span> <span class="n">X_truth</span><span class="p">[</span><span class="n">t_obs</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">]</span> <span class="o">+</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span> <span class="o">*</span> <span class="n">rng</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">(</span><span class="n">l_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>The covariance matrix <code class="docutils literal notranslate"><span class="pre">R</span></code> for the observations (used to express the uncertainty in the observations during DA) is given as a diagonal matrix with entries defined by the square of the “observational error standard deviation”. <span class="math notranslate nohighlight">\(R\)</span> is a diagonal matrix because we assume that the errors of different observations are independent. The function <code class="docutils literal notranslate"><span class="pre">find_obs</span></code> is used here to find the locations and values of a set of observations given the observed times.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Calculated observation covariance matrix, assuming independent observations</span>
+<span class="n">R</span> <span class="o">=</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">1500</span><span class="p">),</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">1000</span><span class="p">:</span><span class="mi">1500</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
+    <span class="n">t_obs</span><span class="p">[</span><span class="mi">100</span><span class="p">:</span><span class="mi">150</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
+    <span class="n">DA_methods</span><span class="o">.</span><span class="n">find_obs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">X_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">t_obs</span><span class="p">[</span><span class="mi">150</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;obs&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/c13ac65f00c200689d59f8fc66be392f99887f9470e0953aeadb9da603c374b3.png" src="../_images/c13ac65f00c200689d59f8fc66be392f99887f9470e0953aeadb9da603c374b3.png" />
+</div>
+</div>
+<div class="section" id="check-the-model-background-covariances-for-the-variable-x">
+<h4>Check the Model Background Covariances for the variable <code class="docutils literal notranslate"><span class="pre">X</span></code><a class="headerlink" href="#check-the-model-background-covariances-for-the-variable-x" title="Permalink to this headline">#</a></h4>
+<p>The climatological covariances of both the “truth” (2-scale L96) and “GCM” (1-scale L96) are computed a-priori from a long run.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">B_corr1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B_clim1</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="n">B_corr2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B_clim2</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B_clim1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B_clim1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">B_corr1</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">B_clim1</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">B_clim1</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">B_clim1</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
+        <span class="n">B_corr2</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">B_clim2</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">B_clim2</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">B_clim2</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">B_corr1</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;bwr&quot;</span><span class="p">,</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">,</span> <span class="mi">20</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Background correlation matrix: 1-scale L96&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">B_corr2</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;bwr&quot;</span><span class="p">,</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">,</span> <span class="mi">20</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Background correlation matrix: 2-scale L96&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/3b7f92401785c5b58dd8d3410df0f3b52d2d8a6eb39293ddb55a5ef34fa79a77.png" src="../_images/3b7f92401785c5b58dd8d3410df0f3b52d2d8a6eb39293ddb55a5ef34fa79a77.png" />
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="run-data-assimilation">
+<h3>4. Run Data Assimilation<a class="headerlink" href="#run-data-assimilation" title="Permalink to this headline">#</a></h3>
+<p>During each DA cycle, we produce an updated state estimate for all <span class="math notranslate nohighlight">\(K\)</span> grid points, and all <code class="docutils literal notranslate"><span class="pre">n</span></code> number of ensemble members.</p>
+<p>The inputs are:</p>
+<ul class="simple">
+<li><p>prior estimate (the forecast at time t for all K grid points, and all n number of ensemble members)</p></li>
+<li><p>observations (N observations)</p></li>
+<li><p>operator matrix (N x K)</p></li>
+<li><p>covariance over obs (N x N)</p></li>
+<li><p>covariance over model (K x K)</p></li>
+</ul>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+
+<span class="c1"># Set up array to store DA increments</span>
+<span class="n">X_inc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">),</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">))</span>
+<span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;3DVar&quot;</span><span class="p">:</span>
+    <span class="n">X_inc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">X_inc</span><span class="p">)</span>
+
+<span class="n">t_DA</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Initialize ensemble with perturbations</span>
+<span class="n">i_t</span> <span class="o">=</span> <span class="mi">0</span>
+<span class="n">ensX</span> <span class="o">=</span> <span class="n">X_init</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">+</span> <span class="n">rng</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">))</span>
+<span class="n">X_post</span> <span class="o">=</span> <span class="n">ensX</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Localize the covariance matrix in advance to avoid the calculation at each DA step</span>
+<span class="c1"># The localization weight matrix is also saved for EnKF</span>
+
+<span class="n">B_loc</span><span class="p">,</span> <span class="n">W_clim</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">localize_covariance</span><span class="p">(</span><span class="n">B_clim1</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">B_loc</span><span class="p">)</span>
+
+<span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">param_DA</span><span class="p">:</span>
+    <span class="n">mean_param</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)))</span>
+    <span class="n">spread_param</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)))</span>
+    <span class="n">param_scale</span> <span class="o">=</span> <span class="n">Config</span><span class="o">.</span><span class="n">param_sd</span>
+    <span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">),</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)))</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">]</span> <span class="o">=</span> <span class="n">W_clim</span>
+<span class="k">else</span><span class="p">:</span>
+    <span class="n">W</span> <span class="o">=</span> <span class="n">W_clim</span>
+    <span class="n">param_scale</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ens_param</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">),</span> <span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)):</span>
+    <span class="n">ens_param</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span>
+        <span class="n">scale</span><span class="o">=</span><span class="n">param_scale</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">nens</span>
+    <span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">cycle</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">):</span>
+    <span class="c1"># Set up array to store model forecast for each DA cycle</span>
+    <span class="n">ensX_fcst</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">))</span>
+
+    <span class="c1"># Model forecast for next DA cycle</span>
+    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">):</span>
+        <span class="n">ensX_fcst</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span>
+            <span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">n</span><span class="p">],</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">F_fcst</span><span class="p">,</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">,</span>
+            <span class="n">ens_param</span><span class="p">[:,</span> <span class="n">n</span><span class="p">],</span>
+        <span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
+
+    <span class="n">i_t</span> <span class="o">=</span> <span class="n">i_t</span> <span class="o">+</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span>
+
+    <span class="c1"># Get prior/background from the forecast</span>
+    <span class="n">X_prior</span> <span class="o">=</span> <span class="n">ensX_fcst</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span>
+
+    <span class="c1"># Call DA</span>
+    <span class="n">t_DA</span><span class="p">[</span><span class="n">cycle</span><span class="p">]</span> <span class="o">=</span> <span class="n">t_truth</span><span class="p">[</span><span class="n">i_t</span><span class="p">]</span>
+    <span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;EnKF&quot;</span><span class="p">:</span>
+        <span class="n">H</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">observation_operator</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">i_t</span><span class="p">)</span>
+
+        <span class="c1"># Augment state vector with parameters when doing parameter estimation</span>
+        <span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">param_DA</span><span class="p">:</span>
+            <span class="n">H</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">(</span>
+                <span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">H</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)))),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span>
+            <span class="p">)</span>
+            <span class="n">X_prior</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">X_prior</span><span class="p">,</span> <span class="n">ens_param</span><span class="p">))</span>
+        <span class="n">B_ens</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">X_prior</span><span class="p">)</span>
+        <span class="n">B_ens_loc</span> <span class="o">=</span> <span class="n">B_ens</span> <span class="o">*</span> <span class="n">W</span>
+        <span class="n">X_post</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">EnKF</span><span class="p">(</span><span class="n">X_prior</span><span class="p">,</span> <span class="n">X_obs</span><span class="p">[</span><span class="n">t_obs</span> <span class="o">==</span> <span class="n">i_t</span><span class="p">],</span> <span class="n">H</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">B_ens_loc</span><span class="p">)</span>
+
+        <span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">ens_inflate</span><span class="p">(</span>
+            <span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="p">:],</span>
+            <span class="n">X_prior</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="p">:],</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">inflate_opt</span><span class="p">,</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">inflate_factor</span><span class="p">,</span>
+        <span class="p">)</span>
+
+        <span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">param_DA</span><span class="p">:</span>
+            <span class="n">X_post</span><span class="p">[</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)</span> <span class="p">:,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">ens_inflate</span><span class="p">(</span>
+                <span class="n">X_post</span><span class="p">[</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)</span> <span class="p">:,</span> <span class="p">:],</span>
+                <span class="n">X_prior</span><span class="p">[</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)</span> <span class="p">:,</span> <span class="p">:],</span>
+                <span class="n">Config</span><span class="o">.</span><span class="n">param_inflate</span><span class="p">,</span>
+                <span class="n">Config</span><span class="o">.</span><span class="n">param_inf_factor</span><span class="p">,</span>
+            <span class="p">)</span>
+            <span class="n">ens_param</span> <span class="o">=</span> <span class="n">X_post</span><span class="p">[</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">)</span> <span class="p">:,</span> <span class="p">:]</span>
+
+    <span class="k">elif</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;HyEnKF&quot;</span><span class="p">:</span>
+        <span class="n">H</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">observation_operator</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">i_t</span><span class="p">)</span>
+        <span class="n">B_ens</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">X_prior</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">Config</span><span class="o">.</span><span class="n">hybrid_factor</span><span class="p">)</span>
+            <span class="o">+</span> <span class="n">B_clim1</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">hybrid_factor</span>
+        <span class="p">)</span>
+        <span class="n">B_ens_loc</span> <span class="o">=</span> <span class="n">B_ens</span> <span class="o">*</span> <span class="n">W</span>
+        <span class="n">X_post</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">EnKF</span><span class="p">(</span><span class="n">X_prior</span><span class="p">,</span> <span class="n">X_obs</span><span class="p">[</span><span class="n">t_obs</span> <span class="o">==</span> <span class="n">i_t</span><span class="p">],</span> <span class="n">H</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">B_ens_loc</span><span class="p">)</span>
+        <span class="n">X_post</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">ens_inflate</span><span class="p">(</span>
+            <span class="n">X_post</span><span class="p">,</span> <span class="n">X_prior</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">inflate_opt</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">inflate_factor</span>
+        <span class="p">)</span>
+
+    <span class="k">elif</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;3DVar&quot;</span><span class="p">:</span>
+        <span class="n">X_prior</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">X_prior</span><span class="p">)</span>
+        <span class="n">H</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">observation_operator</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">i_t</span><span class="p">)</span>
+        <span class="n">X_post</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">Lin3dvar</span><span class="p">(</span><span class="n">X_prior</span><span class="p">,</span> <span class="n">X_obs</span><span class="p">[</span><span class="n">t_obs</span> <span class="o">==</span> <span class="n">i_t</span><span class="p">],</span> <span class="n">H</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">B_loc</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
+
+    <span class="k">elif</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;Replace&quot;</span><span class="p">:</span>
+        <span class="n">X_post</span> <span class="o">=</span> <span class="n">X_prior</span>
+        <span class="n">X_post</span><span class="p">[</span><span class="n">l_obs</span><span class="p">[</span><span class="n">t_obs</span> <span class="o">==</span> <span class="n">i_t</span><span class="p">]]</span> <span class="o">=</span> <span class="n">X_obs</span><span class="p">[</span><span class="n">t_obs</span> <span class="o">==</span> <span class="n">i_t</span><span class="p">]</span>
+
+    <span class="k">elif</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;None&quot;</span><span class="p">:</span>
+        <span class="n">X_post</span> <span class="o">=</span> <span class="n">X_prior</span>
+
+    <span class="k">if</span> <span class="ow">not</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;None&quot;</span><span class="p">:</span>
+        <span class="c1"># Get current increments</span>
+        <span class="n">X_inc</span><span class="p">[</span><span class="n">cycle</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="o">...</span><span class="p">])</span> <span class="o">-</span> <span class="n">X_prior</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span>
+        <span class="p">)</span>
+
+        <span class="c1"># Get posterior info about the estimated parameters</span>
+        <span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">param_DA</span><span class="p">:</span>
+            <span class="n">mean_param</span><span class="p">[</span><span class="n">cycle</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">ens_param</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+            <span class="n">spread_param</span><span class="p">[</span><span class="n">cycle</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">ens_param</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="c1"># Reset initial conditions for next DA cycle</span>
+    <span class="n">ensX_fcst</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">ensX</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">ensX</span><span class="p">,</span> <span class="n">ensX_fcst</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="kc">None</span><span class="p">,</span> <span class="o">...</span><span class="p">]))</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Time to complete DA: </span><span class="si">{</span><span class="nb">round</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">start_time</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="si">}</span><span class="s2"> (s)&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Time to complete DA: 34.19 (s)
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="post-processing-and-visualization">
+<h3>5. Post Processing and Visualization<a class="headerlink" href="#post-processing-and-visualization" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">mean_X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">ensX</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">clim</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">mean_X</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:]))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
+<span class="n">ch</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)],</span>
+    <span class="n">mean_X</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:],</span>
+    <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;bwr&quot;</span><span class="p">,</span>
+    <span class="n">vmin</span><span class="o">=-</span><span class="n">clim</span><span class="p">,</span>
+    <span class="n">vmax</span><span class="o">=</span><span class="n">clim</span><span class="p">,</span>
+    <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;neither&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">ch</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">orientation</span><span class="o">=</span><span class="s2">&quot;horizontal&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;s&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;X - X_truth&quot;</span><span class="p">)</span>
+
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(((</span><span class="n">mean_X</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;RMSE&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">ensX</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Spread&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)],</span>
+    <span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Obs error&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;RMSE (X - X_truth)&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">which</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(((</span><span class="n">mean_X</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;RMSE&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">X_inc_ave</span> <span class="o">=</span> <span class="n">X_inc</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">X_inc_ave</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Inc&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">DA_methods</span><span class="o">.</span><span class="n">running_average</span><span class="p">(</span><span class="n">X_inc_ave</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">7</span><span class="p">),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Inc Ave&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">F_fcst</span> <span class="o">-</span> <span class="n">Config</span><span class="o">.</span><span class="n">F_truth</span><span class="p">),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;F_bias&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">X_inc</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+    <span class="s2">&quot;k:&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ave Inc&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;s&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Increments&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">which</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+
+<span class="n">plot_start</span><span class="p">,</span> <span class="n">plot_end</span> <span class="o">=</span> <span class="mi">1000</span><span class="p">,</span> <span class="mi">1500</span>
+<span class="n">plot_start_DA</span><span class="p">,</span> <span class="n">plot_end_DA</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">plot_start</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span>
+    <span class="n">plot_end</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span>
+<span class="p">)</span>
+<span class="n">plot_x</span> <span class="o">=</span> <span class="mi">0</span>
+
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="n">plot_start</span><span class="p">:</span><span class="n">plot_end</span><span class="p">],</span> <span class="n">X_truth</span><span class="p">[</span><span class="n">plot_start</span><span class="p">:</span><span class="n">plot_end</span><span class="p">,</span> <span class="n">plot_x</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;truth&quot;</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="n">plot_start</span><span class="p">:</span><span class="n">plot_end</span><span class="p">],</span> <span class="n">mean_X</span><span class="p">[</span><span class="n">plot_start</span><span class="p">:</span><span class="n">plot_end</span><span class="p">,</span> <span class="n">plot_x</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;forecast&quot;</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
+    <span class="n">t_DA</span><span class="p">[</span><span class="n">plot_start_DA</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">:</span> <span class="n">plot_end_DA</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span>
+    <span class="n">DA_methods</span><span class="o">.</span><span class="n">find_obs</span><span class="p">(</span><span class="n">plot_x</span><span class="p">,</span> <span class="n">X_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="p">[</span><span class="n">plot_start</span><span class="p">,</span> <span class="n">plot_end</span><span class="p">]),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;obs&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">which</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;k=&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">plot_x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot; truth and forecast&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+
+<span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">param_DA</span><span class="p">:</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="s2">&quot;g&quot;</span><span class="p">,</span> <span class="s2">&quot;k&quot;</span><span class="p">]):</span>
+        <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+            <span class="n">t_DA</span><span class="p">,</span>
+            <span class="n">DA_methods</span><span class="o">.</span><span class="n">running_average</span><span class="p">(</span><span class="n">mean_param</span><span class="p">[:,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">100</span><span class="p">),</span>
+            <span class="n">c</span> <span class="o">+</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span>
+            <span class="n">label</span><span class="o">=</span><span class="s2">&quot;C</span><span class="si">{}</span><span class="s2"> </span><span class="si">{:3f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
+                <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">mean_param</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t_DA</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="p">:,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+            <span class="p">),</span>
+        <span class="p">)</span>
+        <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+            <span class="n">t_DA</span><span class="p">,</span>
+            <span class="n">DA_methods</span><span class="o">.</span><span class="n">running_average</span><span class="p">(</span>
+                <span class="n">mean_param</span><span class="p">[:,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">spread_param</span><span class="p">[:,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">100</span>
+            <span class="p">),</span>
+            <span class="n">c</span> <span class="o">+</span> <span class="s2">&quot;:&quot;</span><span class="p">,</span>
+            <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">&quot;SD </span><span class="si">{</span><span class="n">spread_param</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">t_DA</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">):,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s2">3f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">,</span>
+        <span class="p">)</span>
+        <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+            <span class="n">t_DA</span><span class="p">,</span>
+            <span class="n">DA_methods</span><span class="o">.</span><span class="n">running_average</span><span class="p">(</span>
+                <span class="n">mean_param</span><span class="p">[:,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">spread_param</span><span class="p">[:,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">100</span>
+            <span class="p">),</span>
+            <span class="n">c</span> <span class="o">+</span> <span class="s2">&quot;:&quot;</span><span class="p">,</span>
+        <span class="p">)</span>
+    <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+    <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">which</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+
+<span class="k">else</span><span class="p">:</span>
+    <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;off&quot;</span><span class="p">)</span>
+
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">text</span><span class="p">(</span>
+    <span class="mf">0.1</span><span class="p">,</span>
+    <span class="mf">0.2</span><span class="p">,</span>
+    <span class="p">(</span>
+        <span class="sa">f</span><span class="s2">&quot;RMSE=</span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(((</span><span class="n">mean_X</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="p">:])</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span><span class="si">:</span><span class="s2">3f</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;Spread=</span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">ensX</span><span class="p">,</span><span class="w"> </span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">))</span><span class="si">:</span><span class="s2">3f</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;GCM param=</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;DA=</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">DA</span><span class="si">}</span><span class="s2">, </span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;DA_freq=</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;B_loc=</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">B_loc</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;inflation=</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">inflate_opt</span><span class="si">}</span><span class="s2">, </span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">inflate_factor</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;obs_density=</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;obs_sigma=</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;obs_freq=</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="si">}</span><span class="se">\n</span><span class="s2">&quot;</span>
+    <span class="p">),</span>
+    <span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;off&quot;</span><span class="p">)</span>
+
+<span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">save_fig</span><span class="p">:</span>
+    <span class="n">output_dir</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="s2">&quot;DA_data&quot;</span><span class="p">)</span>
+    <span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">output_dir</span><span class="p">,</span> <span class="n">exist_ok</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+    <span class="n">exp_number</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
+    <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">output_dir</span><span class="p">,</span> <span class="sa">f</span><span class="s2">&quot;config_</span><span class="si">{</span><span class="n">exp_number</span><span class="si">}</span><span class="s2">.txt&quot;</span><span class="p">),</span> <span class="s2">&quot;w&quot;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
+        <span class="n">f</span><span class="o">.</span><span class="n">write</span><span class="p">(</span>
+            <span class="n">output_dir</span>
+            <span class="o">+</span> <span class="nb">str</span><span class="p">({</span><span class="n">k</span><span class="p">:</span> <span class="n">v</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">Config</span><span class="o">.</span><span class="vm">__dict__</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">k</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s2">&quot;_&quot;</span><span class="p">)})</span>
+        <span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">output_dir</span><span class="p">,</span> <span class="sa">f</span><span class="s2">&quot;fig_</span><span class="si">{</span><span class="n">exp_number</span><span class="si">}</span><span class="s2">.png&quot;</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/5efcebdb340a3a6ab21b115115866079ae6e109dc255409c98a0c2254de9ace0.png" src="../_images/5efcebdb340a3a6ab21b115115866079ae6e109dc255409c98a0c2254de9ace0.png" />
+</div>
+</div>
+<p>Save DA output for further analysis</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">save_data</span><span class="p">:</span>
+    <span class="n">output_filename</span> <span class="o">=</span> <span class="p">(</span>
+        <span class="sa">f</span><span class="s2">&quot;K_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;J_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">J</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;obs_freq_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;F_truth_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">F_truth</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;F_fcst_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">F_fcst</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;ns_da_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;ns_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">ns</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;ns_spinup_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_spinup</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;dt_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">dt</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;si_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;B_loc_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">B_loc</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;DA_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">DA</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;nens_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;inflate_opt_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">inflate_opt</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;inflate_factor_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">inflate_factor</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;hybrid_factor_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">hybrid_factor</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;obs_density_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;DA_freq_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="si">}</span><span class="s2">_&quot;</span>
+        <span class="sa">f</span><span class="s2">&quot;obs_sigma_</span><span class="si">{</span><span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span><span class="si">}</span><span class="s2">.npz&quot;</span>
+    <span class="p">)</span>
+    <span class="n">output_dir</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="s2">&quot;DA_data&quot;</span><span class="p">)</span>
+    <span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">output_dir</span><span class="p">,</span> <span class="n">exist_ok</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">savez</span><span class="p">(</span>
+        <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">output_dir</span><span class="p">,</span> <span class="n">output_filename</span><span class="p">),</span>
+        <span class="n">mean_X</span><span class="o">=</span><span class="n">mean_X</span><span class="p">,</span>
+        <span class="n">X_truth</span><span class="o">=</span><span class="n">X_truth</span><span class="p">,</span>
+        <span class="n">X_inc_ave</span><span class="o">=</span><span class="n">X_inc_ave</span><span class="p">,</span>
+    <span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+</div>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
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+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#what-is-da-why-do-we-do-it">What is DA? Why do we do it?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#design-of-the-data-assimilation-experiments-in-l96">Design of the data assimilation experiments in L96</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-our-gcm-and-da-parameters-to-use-throughout-notebook">1. Define our “GCM” and DA parameters to use throughout notebook</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-truth-run-from-two-time-scale-l96-model">2. Generate <code class="docutils literal notranslate"><span class="pre">truth</span></code> run from two time-scale L96 model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-synthetic-observations">3. Generate synthetic observations</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#check-the-model-background-covariances-for-the-variable-x">Check the Model Background Covariances for the variable <code class="docutils literal notranslate"><span class="pre">X</span></code></a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#run-data-assimilation">4. Run Data Assimilation</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#post-processing-and-visualization">5. Post Processing and Visualization</a></li>
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+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
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+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
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+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
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+    <h1>Gaussian Processes</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#recap-lorenz-96">Recap: Lorenz 96</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#recap-gaussian-processes-gps">Recap: Gaussian Processes (GPs)</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exact-inference">Exact inference</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#variational-inference">Variational inference</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sparse-gps">Sparse GPs</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learn-the-sub-grid-tendencies-offline">Learn the sub-grid tendencies offline</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#local-mappings-with-a-linear-kernel">Local mappings with a linear kernel</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#local-mappings-with-a-non-linear-kernel">Local mappings with a non-linear kernel</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#implement-the-linear-gp-parameterisation-in-an-online-setting">Implement the linear GP parameterisation in an online setting</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#use-multi-task-gps-to-exploit-non-local-features">Use multi-task GPs to exploit non-local features</a></li>
+</ul>
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+                
+<div id="searchbox"></div>
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+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="gaussian-processes">
+<h1>Gaussian Processes<a class="headerlink" href="#gaussian-processes" title="Permalink to this headline">#</a></h1>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
+<span class="kn">import</span> <span class="nn">gpflow</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span><span class="p">,</span> <span class="n">integrate_L96_2t</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">EulerFwd</span><span class="p">,</span> <span class="n">RK2</span><span class="p">,</span> <span class="n">RK4</span>
+
+<span class="n">time_method</span> <span class="o">=</span> <span class="n">RK4</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2024-01-28 13:14:45.061788: I external/local_tsl/tsl/cuda/cudart_stub.cc:31] Could not find cuda drivers on your machine, GPU will not be used.
+2024-01-28 13:14:45.094002: E external/local_xla/xla/stream_executor/cuda/cuda_dnn.cc:9261] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered
+2024-01-28 13:14:45.094033: E external/local_xla/xla/stream_executor/cuda/cuda_fft.cc:607] Unable to register cuFFT factory: Attempting to register factory for plugin cuFFT when one has already been registered
+2024-01-28 13:14:45.095237: E external/local_xla/xla/stream_executor/cuda/cuda_blas.cc:1515] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered
+2024-01-28 13:14:45.100944: I external/local_tsl/tsl/cuda/cudart_stub.cc:31] Could not find cuda drivers on your machine, GPU will not be used.
+2024-01-28 13:14:45.101385: I tensorflow/core/platform/cpu_feature_guard.cc:182] This TensorFlow binary is optimized to use available CPU instructions in performance-critical operations.
+To enable the following instructions: AVX2 FMA, in other operations, rebuild TensorFlow with the appropriate compiler flags.
+2024-01-28 13:14:45.820674: W tensorflow/compiler/tf2tensorrt/utils/py_utils.cc:38] TF-TRT Warning: Could not find TensorRT
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="recap-lorenz-96">
+<h2>Recap: Lorenz 96<a class="headerlink" href="#recap-lorenz-96" title="Permalink to this headline">#</a></h2>
+<div class="figure align-default" id="l96-equation-figure">
+<a class="reference internal image-reference" href="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png"><img alt="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" src="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" style="width: 400px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 9 </span><span class="caption-text"><em>Visualisation of a two-scale Lorenz ‘96 system with J = 8 and K = 6. Global-scale variables (<span class="math notranslate nohighlight">\(X_k\)</span>) are updated based on neighbouring variables and on the local-scale variables (<span class="math notranslate nohighlight">\(Y_{j,k}\)</span>) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from <a class="reference external" href="https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436">Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.</a></em>.</span><a class="headerlink" href="#l96-equation-figure" title="Permalink to this image">#</a></p>
+</div>
+<p><span id="id1">[<a class="reference internal" href="../bibliography.html#id2" title="E.N. Lorenz. Predictability: a problem partly solved. Seminar on Predictability, 1:1-18, 1995. URL: https://www.ecmwf.int/node/10829.">Lorenz, 1995</a>]</span> describes a “two time-scale” model in two equations (2 and 3) which are:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-c55c3e2c-6fe7-4ef5-a1a9-d597ebbb0267">
+<span class="eqno">(21)<a class="headerlink" href="#equation-c55c3e2c-6fe7-4ef5-a1a9-d597ebbb0267" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}
+\\
+\frac{d}{dt} Y_{j,k}
+&amp;= - cbY_{j+1,k} \left( Y_{j+2,k} - X_{j-1,k} \right) - c Y_{j,k} + \frac{hc}{b} X_k
+\end{align}\]</div>
+</div>
+<div class="section" id="recap-gaussian-processes-gps">
+<h2>Recap: Gaussian Processes (GPs)<a class="headerlink" href="#recap-gaussian-processes-gps" title="Permalink to this headline">#</a></h2>
+<div class="section" id="exact-inference">
+<h3>Exact inference<a class="headerlink" href="#exact-inference" title="Permalink to this headline">#</a></h3>
+<p>Consider the following case where we have a dataset <span class="math notranslate nohighlight">\(\mathcal{D}\)</span> comprising <span class="math notranslate nohighlight">\(n\)</span> pairs of observations <span class="math notranslate nohighlight">\(\mathcal{D} = \{(x_1,y_1),(x_2,y_2),...,(x_n,y_n)\}\)</span>, where <span class="math notranslate nohighlight">\(x_i\)</span> could e.g., be the coarse-grid term at time <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(y_i\)</span> the corresponding sub-grid term. In the standard supervised learning problem, we wish to recover the function <span class="math notranslate nohighlight">\(f\)</span> which maps a given input <span class="math notranslate nohighlight">\(x_i\)</span> to a given output <span class="math notranslate nohighlight">\(y_i\)</span>, where we assume that the outputs have been corrupted by some random noise <span class="math notranslate nohighlight">\(\varepsilon_i\)</span>:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-c0e055b9-a1a0-4ad4-a3f8-2229793e365f">
+<span class="eqno">(22)<a class="headerlink" href="#equation-c0e055b9-a1a0-4ad4-a3f8-2229793e365f" title="Permalink to this equation">#</a></span>\[\begin{equation}
+y_i = f(x_i) + \varepsilon_i,
+\end{equation}\]</div>
+<p>or in vector format:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-1fa7c1a6-43a4-404b-a066-c32a701033e8">
+<span class="eqno">(23)<a class="headerlink" href="#equation-1fa7c1a6-43a4-404b-a066-c32a701033e8" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\textbf{y} = f(\textbf{x}) + \pmb{\varepsilon}. 
+\end{equation}\]</div>
+<p>Then after learning the function <span class="math notranslate nohighlight">\(f\)</span>, we wish to make reliable predictions at un-observed test input locations <span class="math notranslate nohighlight">\(\textbf{x}_* \in \mathbb{R}^{n_*}\)</span>, such that:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-2102cdd3-7492-4657-ab6a-7cb9c3588b9f">
+<span class="eqno">(24)<a class="headerlink" href="#equation-2102cdd3-7492-4657-ab6a-7cb9c3588b9f" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\textbf{f}_* = f(\textbf{x}_*). 
+\end{equation}\]</div>
+<p>In the GP framework, we assume that the noise term <span class="math notranslate nohighlight">\(\pmb{\varepsilon}\)</span> represents independent and identically distributed zero-mean Gaussian noise, with variance <span class="math notranslate nohighlight">\(\sigma^2\)</span>, such that <span class="math notranslate nohighlight">\(\varepsilon_i \sim \mathcal{N}(0,\sigma^2)\)</span>, and that the function <span class="math notranslate nohighlight">\(f\)</span> itself is a GP:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-4db7f621-0f6c-40f6-beef-7b30f5b0d08d">
+<span class="eqno">(25)<a class="headerlink" href="#equation-4db7f621-0f6c-40f6-beef-7b30f5b0d08d" title="Permalink to this equation">#</a></span>\[\begin{equation}
+f(\pmb{\phi}) \sim \mathcal{GP}(m(\pmb{\phi}),k(\pmb{\phi},\pmb{\phi})),
+\end{equation}\]</div>
+<p>where <span class="math notranslate nohighlight">\(m(\pmb{\phi})\)</span> and <span class="math notranslate nohighlight">\(k(\pmb{\phi},\pmb{\phi})\)</span> are the mean and covariance (or kernel) of the GP respectively, and <span class="math notranslate nohighlight">\(\pmb{\phi}\)</span> represents arbitrary function inputs - either  <span class="math notranslate nohighlight">\(\textbf{x}\)</span> or <span class="math notranslate nohighlight">\(\textbf{x}_*\)</span>. Through the assumption that <span class="math notranslate nohighlight">\(f\)</span> is a GP, we effectively place prior probabilities over all conceviable functions <span class="math notranslate nohighlight">\(f_i\)</span>, where <span class="math notranslate nohighlight">\(i = \{1,2,...,\infty\}\)</span>, but assign higher probabilities to specific types of functions (e.g., linear, periodic, anisotropic, non-stationary etc) through our choice of <span class="math notranslate nohighlight">\(m(\pmb{\phi})\)</span> and <span class="math notranslate nohighlight">\(k(\pmb{\phi},\pmb{\phi})\)</span>. After which, we use the observations <span class="math notranslate nohighlight">\(\textbf{y}\)</span> to update these prior probabilities in the form of a posterior distribution over functions. Let’s see how we do this in practice:</p>
+<p>Continuing with the notation that <span class="math notranslate nohighlight">\(\textbf{x} = \{x_1,x_2,...,x_n\}\)</span> represents the set of training inputs, and <span class="math notranslate nohighlight">\(\textbf{x}_* = \{x_{*_1},x_{*_2},...,x_{*_{n_*}}\}\)</span> the set of test inputs, we wish to train a model which is able to make accurate predictions at the test locations: <span class="math notranslate nohighlight">\(\textbf{f}_* = \{f(x_{*_1}),f(x_{*_2}),...,f(x_{*_{n_*}})\}\)</span>. We start by defining a series of covariance functions associated with the training and test inputs:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-076fee2d-1984-4640-8540-52e9fb8cdecc">
+<span class="eqno">(26)<a class="headerlink" href="#equation-076fee2d-1984-4640-8540-52e9fb8cdecc" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\begin{aligned}
+\textbf{K} &amp;= k(\textbf{x},\textbf{x}) + \sigma^2\pmb{I} \\
+\textbf{k}_{*} &amp;= k(\textbf{x},\textbf{x}_*) \\
+\pmb{k}_* &amp;= k(\textbf{x}_*,\textbf{x}_*).
+\end{aligned}
+\end{equation}\]</div>
+<p>The covariance function of the GP prior is a particularly important concept, as this is what allows us to model the different kinds of functions described above, so consideration should go into choosing the most appropriate function (there are data driven ways to make this choice. We could e.g., use cross-validation, or more commonly in Bayesian statistics; compare models via the <em>log marginal likelihood</em> - GPflow is setup to do the latter). Now, because a GP can ultimately be described as a collection of random Gaussian variables, under our GP prior (defined by <span class="math notranslate nohighlight">\(m(\pmb{\phi})\)</span> and <span class="math notranslate nohighlight">\(k(\pmb{\phi},\pmb{\phi})\)</span>), the function values <span class="math notranslate nohighlight">\(\textbf{f}_*\)</span> and the training observations <span class="math notranslate nohighlight">\(\textbf{y}\)</span> are considered to form a joint Gaussian distribution:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-2edeb304-a503-48c3-9776-71ce1d29f307">
+<span class="eqno">(27)<a class="headerlink" href="#equation-2edeb304-a503-48c3-9776-71ce1d29f307" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\begin{bmatrix}
+\textbf{f}_* \\
+\textbf{y}
+\end{bmatrix} \sim \mathcal{N}\Bigg(
+\begin{bmatrix}
+m(\textbf{x}_*) \\
+m(\textbf{x})
+\end{bmatrix},
+\begin{bmatrix}
+\pmb{k}_* &amp; \textbf{k}_*^\text{T} \\
+\textbf{k}_* &amp; \textbf{K}
+\end{bmatrix} \Bigg),
+\end{equation}\]</div>
+<p>where we see that the marginal distributions (under the prior) of both <span class="math notranslate nohighlight">\(\textbf{f}_*\)</span> and <span class="math notranslate nohighlight">\(\textbf{y}\)</span> are given by:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-d4e508db-3be5-4821-8776-4fcf717359ce">
+<span class="eqno">(28)<a class="headerlink" href="#equation-d4e508db-3be5-4821-8776-4fcf717359ce" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\textbf{f}_* \sim \mathcal{N}\big(m(\textbf{x}_*),\pmb{k}_*\big) \quad , \quad \textbf{y} \sim \mathcal{N}\big(m(\textbf{x}),\textbf{K}\big).
+\label{marginalsGP}
+\end{equation}\]</div>
+<p>In reality however, we are interested in the conditional distribution of the function values, given the training data - <span class="math notranslate nohighlight">\(p(\textbf{f}_* | \textbf{y},\textbf{x},\textbf{x}_*)\)</span>, which is also Gaussian such that <span class="math notranslate nohighlight">\(\textbf{f}_* | \textbf{y},\textbf{x},\textbf{x}_* \sim \mathcal{N}(\bar{\textbf{f}}_*,\sigma^2_{\textbf{f}_*})\)</span>. The mean and covariance of this distribution then tells us everything we need about the posterior distribution of the function values, and can be derived analytically through a bit of linear algebra (see <span id="id2">[<a class="reference internal" href="../bibliography.html#id21" title="Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning. The MIT Press, 2005. URL: http://dx.doi.org/10.7551/mitpress/3206.001.0001, doi:10.7551/mitpress/3206.001.0001.">Rasmussen and Williams, 2005</a>]</span> or <span id="id3">[<a class="reference internal" href="../bibliography.html#id22" title="Christopher M. Bishop. Pattern Recognition and Machine Learning (Information Science and Statistics). Springer-Verlag, Berlin, Heidelberg, 2006. ISBN 0387310738.">Bishop, 2006</a>]</span> for details). Here is the final solution:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-ee2abc29-27cc-490a-a87a-ad02b34fd010">
+<span class="eqno">(29)<a class="headerlink" href="#equation-ee2abc29-27cc-490a-a87a-ad02b34fd010" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\begin{aligned}
+\bar{\textbf{f}}_* &amp;= m(\textbf{x}_*) + \textbf{k}_*^\text{T}\textbf{K}^{-1}\big(\textbf{y} - m(\textbf{x})\big) \\
+\sigma^2_{\textbf{f}_*} &amp;= \pmb{k}_* - \textbf{k}_*^\text{T}\textbf{K}^{-1}\textbf{k}_*.
+\end{aligned}
+\end{equation}\]</div>
+</div>
+<div class="section" id="variational-inference">
+<h3>Variational inference<a class="headerlink" href="#variational-inference" title="Permalink to this headline">#</a></h3>
+<p>The analytical solution in the above description relies on the assumption that the data are Gaussian distributed, which makes the maths straightfoward because the product of 2 Gaussians (the prior and likelihood) results in another Gaussian (the posterior). In the case where the data are not Gaussian distributed, the resultant posterior will no longer have a closed-form solution, so instead we need to approximate the posterior as a multivarite Gaussian (or other) using variational techniques. To do this, GPflow implements the approach outlined in <a class="reference external" href="http://www0.cs.ucl.ac.uk/staff/c.archambeau/publ/neco_mo09_web.pdf">Opper and Archambeau, (2009)</a>, and we give a brief summary here to follow on from the section above.</p>
+<p>Recall the posterior predictive distribution from before: <span class="math notranslate nohighlight">\(p(\textbf{f}_* | \textbf{y})\)</span>, where we have removed the dependence on the inputs <span class="math notranslate nohighlight">\(\textbf{x}\)</span> and <span class="math notranslate nohighlight">\(\textbf{x}_*\)</span> to keep the notation uncluttered. In the following discussion we assume that this distribution is no longer Gaussian, because e.g., the observations <span class="math notranslate nohighlight">\(\textbf{y}\)</span> are no longer Gaussian, and hence we no longer have a Gaussian likelihood. Therefore instead, we wish to approximate <span class="math notranslate nohighlight">\(p(\textbf{f}_* | \textbf{y})\)</span> by a multivariate Gaussian <span class="math notranslate nohighlight">\(q(\textbf{f}_*) \sim \mathcal{N}(\textbf{m},\textbf{S})\)</span>. This is achieved by choosing the values of <span class="math notranslate nohighlight">\(\textbf{m}\)</span> and <span class="math notranslate nohighlight">\(\textbf{S}\)</span> which minimise the <em>variational free energy</em> cost function:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-928c1ad0-2e43-49f8-a12b-0a110ba9d92b">
+<span class="eqno">(30)<a class="headerlink" href="#equation-928c1ad0-2e43-49f8-a12b-0a110ba9d92b" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\mathcal{J}\big(q(\textbf{f}_*)\big) = - \ln p(\textbf{y}) + \mathcal{K}\mathcal{L}[q(\textbf{f}_*) \vert\vert p(\textbf{f}_* | \textbf{y})], 
+\end{equation}\]</div>
+<p>where</p>
+<div class="amsmath math notranslate nohighlight" id="equation-7eb9f48c-443d-4527-8f78-3e1173dc6f33">
+<span class="eqno">(31)<a class="headerlink" href="#equation-7eb9f48c-443d-4527-8f78-3e1173dc6f33" title="Permalink to this equation">#</a></span>\[\begin{equation}
+p(\textbf{y}) = \int p(\textbf{y} | \textbf{f}_*)p(\textbf{f}_*) \ \ d\textbf{f}_*
+\end{equation}\]</div>
+<p>is the marginal likelihood (the probability of the observations under the model prior; commonly referred to as the <em>evidence</em>), and</p>
+<div class="amsmath math notranslate nohighlight" id="equation-4e4a5e86-c3fc-4a82-8a7d-80a65a89cc88">
+<span class="eqno">(32)<a class="headerlink" href="#equation-4e4a5e86-c3fc-4a82-8a7d-80a65a89cc88" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\mathcal{K}\mathcal{L}[q(\textbf{f}_*) \vert\vert p(\textbf{f}_* | \textbf{y})] = \int q(\textbf{f}_*)\ln \dfrac{q(\textbf{f}_*)}{p(\textbf{f}_* | \textbf{y})} \ \ d\textbf{f}_*
+\end{equation}\]</div>
+<p>is the Kullback-Leibler divergence which compares the similarity of two probability distributions (in this case, <span class="math notranslate nohighlight">\(q(\textbf{f}_*)\)</span> and <span class="math notranslate nohighlight">\(p(\textbf{f}_* | \textbf{y}\)</span>)).</p>
+<p>The variational free energy <span class="math notranslate nohighlight">\(\mathcal{J}\big(q(\textbf{f}_*)\big)\)</span> represents an upper bound on the negative (log) evidence <span class="math notranslate nohighlight">\(-\ln p(\textbf{y})\)</span>, and is the quantity that GPflow minimises in the variational setting. It is however common practice to track the maximisation of the negative variational free energy, which then represents a lower bound on the (log) evidence, known as the <em>Evidence Lower BOund</em> (ELBO):</p>
+<div class="amsmath math notranslate nohighlight" id="equation-e2a4d99d-879c-4007-92b7-205360d3ef94">
+<span class="eqno">(33)<a class="headerlink" href="#equation-e2a4d99d-879c-4007-92b7-205360d3ef94" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\ln p(\textbf{y}) = \text{ELBO} - \mathcal{K}\mathcal{L}[q(\textbf{f}_*) \vert\vert p(\textbf{f}_* | \textbf{y})] \geq \text{ELBO}.
+\end{equation}\]</div>
+<p>Once the values of <span class="math notranslate nohighlight">\(\textbf{m}\)</span> and <span class="math notranslate nohighlight">\(\textbf{S}\)</span> have been optimised, the Gaussian approximation for the mean and covariance of the posterior can be computed as:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-2eabecd8-0928-46fc-be95-88cefdffc305">
+<span class="eqno">(34)<a class="headerlink" href="#equation-2eabecd8-0928-46fc-be95-88cefdffc305" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\begin{aligned}
+\bar{\textbf{f}}_* &amp;= m(\textbf{x}_*) + \textbf{k}_*^\text{T}\textbf{K}^{-1}\big(\textbf{m} - m(\textbf{x})\big) \\
+\sigma^2_{\textbf{f}_*} &amp;= \pmb{k}_* + \textbf{k}_*^\text{T}\textbf{K}^{-1}\big(\textbf{S} - \textbf{K}\big)\textbf{K}^{-1}\textbf{k}_*.
+\end{aligned}
+\end{equation}\]</div>
+<p>Note that in practice there is a fair bit of maths involved in computing <span class="math notranslate nohighlight">\(\mathcal{J}\big(q(\textbf{f}_*)\big)\)</span> which we do not show here, however the reader is referred to the <a class="reference external" href="https://gpflow.github.io/GPflow/2.4.0/notebooks/theory/vgp_notes.html">GPflow docs</a> and the <a class="reference external" href="http://www0.cs.ucl.ac.uk/staff/c.archambeau/publ/neco_mo09_web.pdf">Opper and Archambeau, (2009)</a> paper for more information. See also this nice <a class="reference external" href="https://github.com/sotakao/moap-bml-workshop/blob/main/2_sparse_gp.ipynb">Jupyter notebook</a> workthrough by So Takao from the University College London Centre for AI, which includes the derivations of the KL divergence term in the Appendix.</p>
+</div>
+<div class="section" id="sparse-gps">
+<h3>Sparse GPs<a class="headerlink" href="#sparse-gps" title="Permalink to this headline">#</a></h3>
+<p>Due to the need to invert the kernel matrix <span class="math notranslate nohighlight">\(\textbf{K}\)</span> (which has size <span class="math notranslate nohighlight">\(n \times n\)</span>) in both the exact and variational settings, GP problems scale with complexity <span class="math notranslate nohighlight">\(\mathcal{O}(n^3)\)</span>. Therefore for data sets with more than a few thousand training points, the standard GP implementations above become problematic. A solution to this is to consider sparse GPs which rely on a set of <em>inducing variables</em> to learn the function <span class="math notranslate nohighlight">\(f\)</span>, as outlined in <a class="reference external" href="https://proceedings.mlr.press/v5/titsias09a/titsias09a.pdf">Titsias, (2009)</a>. Note that GPflow has 2 commonly used classes for implementation of the inducing points method, namely ‘SGPR’ and ‘SVGP’. Although at first it perhaps may seem that ‘SVGP’ is for variational inference and ‘SGPR’ is for exact inference, this is not actually the case. Sparse learning requires variational inference in either case, it is just that ‘SVGP’ is used for non-Gaussian data (as described above), while ‘SGPR’ is used for Gaussian data. Let’s introduce the inducing points method now:</p>
+<p>Consider the set of points <span class="math notranslate nohighlight">\(\textbf{z} = \{z_1,z_2,...,z_m\}\)</span> where <span class="math notranslate nohighlight">\(m &lt;&lt; n\)</span>, which could perhaps correspond to a subset of the training inputs, or a hold-out set etc, however the important point is that they exist in the same space as the inputs. These points are known as the <em>inducing points</em>, and subsequently their corresponding latent function values <span class="math notranslate nohighlight">\(\textbf{u} = \{f(z_1),f(z_2),...,f(z_m)\}\)</span> are known as the <em>inducing variables</em>. If we then assume that the posterior predictive function values at the input locations <span class="math notranslate nohighlight">\(\textbf{x}\)</span> and <span class="math notranslate nohighlight">\(\textbf{x}_*\)</span> (given by <span class="math notranslate nohighlight">\(\textbf{f}\)</span> and <span class="math notranslate nohighlight">\(\textbf{f}_*\)</span> respectively) are conditionally independent given <span class="math notranslate nohighlight">\(\textbf{u}\)</span>, we can describe the posterior <span class="math notranslate nohighlight">\(p(\textbf{f}_* | \textbf{y})\)</span> as a function of <span class="math notranslate nohighlight">\(\textbf{u}\)</span> as follows:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-c86a51bd-377c-42fc-bb6b-95eeb9d0577d">
+<span class="eqno">(35)<a class="headerlink" href="#equation-c86a51bd-377c-42fc-bb6b-95eeb9d0577d" title="Permalink to this equation">#</a></span>\[\begin{equation}
+p(\textbf{f}_* | \textbf{y}) = \int p(\textbf{f}_* | \textbf{u})p(\textbf{u} | \textbf{y}) \ \ d\textbf{u}.
+\end{equation}\]</div>
+<p>We can subsequently derive an analytical solution for <span class="math notranslate nohighlight">\(p(\textbf{f}_* | \textbf{u})\)</span> above by recalling in the first section on exact inference how <span class="math notranslate nohighlight">\(\textbf{y}\)</span> and <span class="math notranslate nohighlight">\(\textbf{f}_*\)</span> formed a joint Gaussian under the GP prior. The same principles apply here to <span class="math notranslate nohighlight">\(\textbf{u}\)</span> and <span class="math notranslate nohighlight">\(\textbf{f}_*\)</span>, hence the mean and covariance of <span class="math notranslate nohighlight">\(p(\textbf{f}_* | \textbf{u})\)</span> follows the same form as those defined by <span class="math notranslate nohighlight">\(\bar{\textbf{f}}_*\)</span> and <span class="math notranslate nohighlight">\(\sigma_{\textbf{f}_*}\)</span> previously, except this time we replace all occurrences of <span class="math notranslate nohighlight">\(\textbf{x}\)</span> with <span class="math notranslate nohighlight">\(\textbf{z}\)</span>, as well as re-defining the covariance matrices:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-07c9429c-783a-4ca6-bb7e-998470c64f75">
+<span class="eqno">(36)<a class="headerlink" href="#equation-07c9429c-783a-4ca6-bb7e-998470c64f75" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\begin{aligned}
+\textbf{K} = k(\textbf{z},\textbf{z}) \\
+\textbf{k}_* = k(\textbf{z},\textbf{x}_*) \\
+\pmb{k}_* = k(\textbf{z},\textbf{z}).
+\end{aligned}
+\end{equation}\]</div>
+<p>The second distribution above, <span class="math notranslate nohighlight">\(p(\textbf{u} | \textbf{y})\)</span>, does not have a closed-form solution however, so we need to rely on variational inference to approximate it as a multivariate Gaussian <span class="math notranslate nohighlight">\(q(\textbf{u}) \sim \mathcal{N}(\textbf{m},\textbf{S})\)</span>. Similar to what we saw in the variational inference discussion above, both <span class="math notranslate nohighlight">\(\textbf{m}\)</span> and <span class="math notranslate nohighlight">\(\textbf{S}\)</span> (as well as <span class="math notranslate nohighlight">\(\textbf{z}\)</span>) can be optimised by minimising the variational free energy cost function <span class="math notranslate nohighlight">\(\mathcal{J}\big(q(\textbf{f}_*)\big)\)</span>. After which, we can produce an analytical solution for the approximate posterior mean and covariance as we have already seen, by once again replacing all occurrences of <span class="math notranslate nohighlight">\(\textbf{x}\)</span> with <span class="math notranslate nohighlight">\(\textbf{z}\)</span>, as well as re-defining the covariance matrices. This results in much cheaper inference than before as the covariance matrix <span class="math notranslate nohighlight">\(\textbf{K}\)</span> is now of size <span class="math notranslate nohighlight">\(m \times m\)</span>, hence the computational cost of the matrix inverse is now <span class="math notranslate nohighlight">\(\mathcal{O}(m^3)\)</span> as opposed to <span class="math notranslate nohighlight">\(\mathcal{O}(n^3)\)</span>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># - a GCM class without any parameterization</span>
+<span class="k">class</span> <span class="nc">GCM_no_param</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">time_method</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+        <span class="c1"># X0 - initial conditions, dt - time increment, nt - number of forward steps to take</span>
+        <span class="c1"># param - parameters of our closure</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+
+
+<span class="c1"># - a GCM class including a linear parameterization in rhs of equation for tendency</span>
+
+
+<span class="k">class</span> <span class="nc">GCM</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">parameterization</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">time_method</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">parameterization</span> <span class="o">=</span> <span class="n">parameterization</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">parameterization</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+        <span class="c1"># X0 - initial conditions, dt - time increment, nt - number of forward steps to take</span>
+        <span class="c1"># param - parameters of our closure</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+
+
+<span class="c1"># - a GCM class including a GP parameterization in rhs of equation for tendency</span>
+
+
+<span class="k">class</span> <span class="nc">GCM_GP</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">GP</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">time_method</span><span class="p">,</span> <span class="n">stochastic</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">local</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">GP</span> <span class="o">=</span> <span class="n">GP</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">stochastic</span> <span class="o">=</span> <span class="n">stochastic</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">local</span> <span class="o">=</span> <span class="n">local</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">local</span><span class="p">:</span>  <span class="c1"># i.e. outputs are shape n*K</span>
+            <span class="n">X_ten</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>  <span class="c1"># i.e. outputs are shape n x K</span>
+            <span class="n">X_augment</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">))</span><span class="o">.</span><span class="n">T</span><span class="p">))</span>
+            <span class="n">X_ten</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span><span class="n">X_augment</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">stochastic</span><span class="p">:</span>
+            <span class="n">f_mean</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">GP</span><span class="o">.</span><span class="n">predict_f_samples</span><span class="p">(</span>
+                <span class="n">X_ten</span><span class="p">,</span> <span class="mi">1</span>
+            <span class="p">)</span>  <span class="c1"># produce 1 random sample from the posterior over functions</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">f_mean</span><span class="p">,</span> <span class="n">f_variance</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">GP</span><span class="o">.</span><span class="n">predict_f</span><span class="p">(</span><span class="n">X_ten</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">f_mean</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+        <span class="c1"># X0 - initial conditions, dt - time increment, nt - number of forward steps to take</span>
+        <span class="c1"># param - parameters of our closure</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+
+
+<span class="c1"># Define the function which will be used to track the optimisation of the models. Much of the code in the &#39;GP&#39;</span>
+<span class="c1"># function is borrowed from the example on the GPflow website:</span>
+<span class="c1"># https://gpflow.github.io/GPflow/2.4.0/notebooks/advanced/gps_for_big_data.html</span>
+
+
+<span class="k">def</span> <span class="nf">trainGP</span><span class="p">(</span>
+    <span class="n">model</span><span class="p">,</span>
+    <span class="n">training_data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+    <span class="n">validation_data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+    <span class="n">batch_size</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+    <span class="n">iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span>
+    <span class="n">verbose</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
+<span class="p">):</span>
+    <span class="n">tf</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">set_seed</span><span class="p">(</span><span class="mi">766</span><span class="p">)</span>
+
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">training_data</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+    <span class="k">if</span> <span class="n">batch_size</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="n">batch_size</span> <span class="o">=</span> <span class="n">n</span>
+    <span class="n">train_dataset</span> <span class="o">=</span> <span class="p">(</span>
+        <span class="n">tf</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">Dataset</span><span class="o">.</span><span class="n">from_tensor_slices</span><span class="p">(</span><span class="n">training_data</span><span class="p">)</span><span class="o">.</span><span class="n">repeat</span><span class="p">()</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
+    <span class="p">)</span>
+    <span class="n">train_iter</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">train_dataset</span><span class="o">.</span><span class="n">batch</span><span class="p">(</span><span class="n">batch_size</span><span class="p">))</span>
+    <span class="n">training_loss</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">training_loss_closure</span><span class="p">(</span><span class="n">train_iter</span><span class="p">,</span> <span class="nb">compile</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+    <span class="n">optimizer</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">Adam</span><span class="p">()</span>
+
+    <span class="nd">@tf</span><span class="o">.</span><span class="n">function</span>
+    <span class="k">def</span> <span class="nf">optimization_step</span><span class="p">(</span><span class="n">trainable_variables</span><span class="p">):</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">training_loss</span><span class="p">,</span> <span class="n">trainable_variables</span><span class="p">)</span>
+
+    <span class="n">elbo_log</span> <span class="o">=</span> <span class="p">[]</span>
+    <span class="n">iters</span> <span class="o">=</span> <span class="p">[]</span>
+    <span class="n">pred_error</span> <span class="o">=</span> <span class="p">[]</span>
+    <span class="k">for</span> <span class="n">step</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">iterations</span><span class="p">):</span>
+        <span class="n">optimization_step</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">trainable_variables</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">step</span> <span class="o">%</span> <span class="mi">10</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="o">-</span><span class="n">training_loss</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+            <span class="n">prediction</span><span class="p">,</span> <span class="n">uncertainty</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict_f</span><span class="p">(</span><span class="n">validation_data</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+            <span class="n">pred_error</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">prediction</span> <span class="o">-</span> <span class="n">validation_data</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span>
+            <span class="n">elbo_log</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">loss</span><span class="p">)</span>
+            <span class="n">iters</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">step</span><span class="p">)</span>
+        <span class="k">if</span> <span class="n">verbose</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="k">if</span> <span class="n">step</span> <span class="o">&lt;</span> <span class="mi">1000</span><span class="p">:</span>
+                <span class="k">if</span> <span class="n">step</span> <span class="o">%</span> <span class="mi">100</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+                    <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;ELBO at step </span><span class="si">{</span><span class="n">step</span><span class="si">}</span><span class="s2">: </span><span class="si">{</span><span class="n">loss</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="k">if</span> <span class="n">step</span> <span class="o">%</span> <span class="mi">1000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+                    <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;ELBO at step </span><span class="si">{</span><span class="n">step</span><span class="si">}</span><span class="s2">: </span><span class="si">{</span><span class="n">loss</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">model</span><span class="p">,</span> <span class="n">iters</span><span class="p">,</span> <span class="n">elbo_log</span><span class="p">,</span> <span class="n">pred_error</span>
+
+
+<span class="k">def</span> <span class="nf">plot_loss</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">train</span><span class="p">,</span> <span class="n">valid</span><span class="p">,</span> <span class="n">labels</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;ELBO&quot;</span><span class="p">,</span> <span class="s2">&quot;MSE (Validation)&quot;</span><span class="p">]):</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">17</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+    <span class="n">plots</span> <span class="o">=</span> <span class="p">[</span><span class="n">train</span><span class="p">,</span> <span class="n">valid</span><span class="p">]</span>
+    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
+        <span class="n">ax</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+        <span class="n">ax</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+        <span class="n">ax</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">plots</span><span class="p">[</span><span class="n">c</span><span class="p">],</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+        <span class="n">ax</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;iteration&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+        <span class="n">ax</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="n">labels</span><span class="p">[</span><span class="n">c</span><span class="p">],</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+        <span class="n">ax</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">wspace</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="learn-the-sub-grid-tendencies-offline">
+<h2>Learn the sub-grid tendencies offline<a class="headerlink" href="#learn-the-sub-grid-tendencies-offline" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">time_steps</span> <span class="o">=</span> <span class="mi">20000</span>
+<span class="n">Forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span> <span class="o">*</span> <span class="n">time_steps</span>
+
+<span class="c1"># Create a &quot;real world&quot; with K=8 and J=32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Forcing</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Get training and validation data for the GP model</span>
+<span class="n">val_size</span> <span class="o">=</span> <span class="mi">4000</span>
+
+<span class="c1"># - Run the true state and output subgrid tendencies (the effect of Y on X is xytrue):</span>
+<span class="n">Xtrue</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">xytrue</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM_no_param</span><span class="p">(</span><span class="n">Forcing</span><span class="p">)</span>
+
+<span class="c1"># train:</span>
+<span class="n">X_train</span> <span class="o">=</span> <span class="n">Xtrue</span><span class="p">[:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">Y_train</span> <span class="o">=</span> <span class="n">xytrue</span><span class="p">[:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="c1"># test:</span>
+<span class="n">X_valid</span> <span class="o">=</span> <span class="n">Xtrue</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+<span class="n">Y_valid</span> <span class="o">=</span> <span class="n">xytrue</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+
+<span class="n">n</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="n">Y_train</span><span class="o">.</span><span class="n">shape</span>
+<span class="n">ns</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_valid</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="local-mappings-with-a-linear-kernel">
+<h3>Local mappings with a linear kernel<a class="headerlink" href="#local-mappings-with-a-linear-kernel" title="Permalink to this headline">#</a></h3>
+<p>The linear GP kernel is equivalent to:</p>
+<p><span class="math notranslate nohighlight">\(k(\textbf{x},\textbf{x}') = \theta^2 \big(\textbf{x}\textbf{x}'^\text{T}\big)\)</span>, where <span class="math notranslate nohighlight">\(\theta\)</span> is a tunable hyperparameter.</p>
+<p>First we will build a GP with a linear kernel while assuming the outputs (i.e., each of the Ks) are independent.  The extension to learning non-linear functions is trivial. We simply define a new kernel, e.g., gpflow.kernels.Matern32(). Incorporating non-local information (i.e., from other ks (we have 8 ks in total)) is a
+little more tricky, but we will tackle this later.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="n">x_train</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="n">y_train</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span><span class="n">Y_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="n">x_valid</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="n">y_valid</span> <span class="o">=</span> <span class="n">Y_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">)</span>
+
+<span class="n">M</span> <span class="o">=</span> <span class="mi">50</span>  <span class="c1"># number of inducing variables</span>
+<span class="n">Z</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">x_train</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">x_train</span><span class="p">),</span> <span class="n">M</span><span class="p">))</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span>
+<span class="p">)</span>
+
+<span class="n">kernel</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">kernels</span><span class="o">.</span><span class="n">Linear</span><span class="p">()</span>
+<span class="n">lik</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">likelihoods</span><span class="o">.</span><span class="n">Gaussian</span><span class="p">()</span>
+<span class="n">gp_linear</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">models</span><span class="o">.</span><span class="n">SVGP</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">kernel</span><span class="p">,</span> <span class="n">likelihood</span><span class="o">=</span><span class="n">lik</span><span class="p">,</span> <span class="n">inducing_variable</span><span class="o">=</span><span class="n">Z</span><span class="p">)</span>
+<span class="n">gp_linear</span><span class="o">.</span><span class="n">likelihood</span><span class="o">.</span><span class="n">variance</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_train</span><span class="p">))</span>
+<span class="n">gp_linear</span><span class="p">,</span> <span class="n">iters</span><span class="p">,</span> <span class="n">elbo</span><span class="p">,</span> <span class="n">pred_error</span> <span class="o">=</span> <span class="n">trainGP</span><span class="p">(</span>
+    <span class="n">gp_linear</span><span class="p">,</span>
+    <span class="n">training_data</span><span class="o">=</span><span class="p">(</span><span class="n">x_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">),</span>
+    <span class="n">validation_data</span><span class="o">=</span><span class="p">(</span><span class="n">x_valid</span><span class="p">,</span> <span class="n">y_valid</span><span class="p">),</span>
+    <span class="n">iterations</span><span class="o">=</span><span class="mi">5000</span><span class="p">,</span>
+    <span class="n">batch_size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
+    <span class="n">verbose</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plot_loss</span><span class="p">(</span><span class="n">iters</span><span class="p">,</span> <span class="n">elbo</span><span class="p">,</span> <span class="n">pred_error</span><span class="p">)</span>
+<span class="n">gp_linear</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/1c7fcf648f20f599597d2fec4229eaa05c2913bbac1227aff02be70c3cdeb710.png" src="../_images/1c7fcf648f20f599597d2fec4229eaa05c2913bbac1227aff02be70c3cdeb710.png" />
+<div class="output text_html">&lt;gpflow.models.svgp.SVGP object at 0x7f1724f91970&gt;
+<table>
+<thead>
+<tr><th>name                    </th><th>class    </th><th>transform       </th><th>prior  </th><th>trainable  </th><th>shape      </th><th>dtype  </th><th>value                                </th></tr>
+</thead>
+<tbody>
+<tr><td>SVGP.kernel.variance    </td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>0.97062                              </td></tr>
+<tr><td>SVGP.likelihood.variance</td><td>Parameter</td><td>Softplus + Shift</td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>15.3553                              </td></tr>
+<tr><td>SVGP.inducing_variable.Z</td><td>Parameter</td><td>Identity        </td><td>       </td><td>True       </td><td>(50, 1)    </td><td>float64</td><td>[[-33.72872...                       </td></tr>
+<tr><td>SVGP.q_mu               </td><td>Parameter</td><td>Identity        </td><td>       </td><td>True       </td><td>(50, 1)    </td><td>float64</td><td>[[9.3964e-01...                      </td></tr>
+<tr><td>SVGP.q_sqrt             </td><td>Parameter</td><td>FillTriangular  </td><td>       </td><td>True       </td><td>(1, 50, 50)</td><td>float64</td><td>[[[6.895e-02, 0.000e+00, 0.000e+00...</td></tr>
+</tbody>
+</table></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">prediction_linGP</span><span class="p">,</span> <span class="n">uncertainty_linGP</span> <span class="o">=</span> <span class="n">gp_linear</span><span class="o">.</span><span class="n">predict_f</span><span class="p">(</span><span class="n">x_valid</span><span class="p">)</span>
+<span class="n">prediction_linGP</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">prediction_linGP</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+<span class="n">uncertainty_linGP</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">uncertainty_linGP</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+
+<span class="n">n_std</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">start</span> <span class="o">=</span> <span class="mi">31000</span>
+<span class="n">end</span> <span class="o">=</span> <span class="n">start</span> <span class="o">+</span> <span class="mi">1000</span>
+<span class="n">prediction_linGPplot</span> <span class="o">=</span> <span class="n">prediction_linGP</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">]</span>
+<span class="n">uncertainty_linGPplot</span> <span class="o">=</span> <span class="n">n_std</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">uncertainty_linGP</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">])</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">17</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">prediction_linGPplot</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted values&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1000</span><span class="p">),</span>
+    <span class="n">prediction_linGPplot</span> <span class="o">+</span> <span class="n">uncertainty_linGPplot</span><span class="p">,</span>
+    <span class="n">prediction_linGPplot</span> <span class="o">-</span> <span class="n">uncertainty_linGPplot</span><span class="p">,</span>
+    <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">y_valid</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="nb">sorted</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">prediction_linGP</span><span class="p">)))</span>
+<span class="n">a</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span>
+    <span class="o">*</span><span class="nb">sorted</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">n_std</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">uncertainty_linGP</span><span class="p">)))</span>
+<span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
+<span class="n">c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
+    <span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span>
+    <span class="n">Y_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span>
+    <span class="n">s</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">prediction_linGP</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">-</span> <span class="n">c</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/e8026a46d837d87d213ba1065089be0f83288ac10e4c2f47e834177e9d581bea.png" src="../_images/e8026a46d837d87d213ba1065089be0f83288ac10e4c2f47e834177e9d581bea.png" />
+</div>
+</div>
+</div>
+<div class="section" id="local-mappings-with-a-non-linear-kernel">
+<h3>Local mappings with a non-linear kernel<a class="headerlink" href="#local-mappings-with-a-non-linear-kernel" title="Permalink to this headline">#</a></h3>
+<p>Here we are going to opt for a combination of 2 kernels: The Rational Quadratic kernel:</p>
+<p><span class="math notranslate nohighlight">\(k_1(\textbf{x},\textbf{x}') = \theta_1^2 \bigg(1 + \dfrac{\Vert \textbf{x} - \textbf{x}' \Vert^2}{2\theta_2^2\theta_3}\bigg)^{-\theta_3}\)</span></p>
+<p>and the linear kernel:</p>
+<p><span class="math notranslate nohighlight">\(k_2(\textbf{x},\textbf{x}') = \theta_4^2 \big(\textbf{x}\textbf{x}^\text{T}\big)\)</span></p>
+<p>such that our kernel becomes <span class="math notranslate nohighlight">\(k(\textbf{x},\textbf{x}') = k_1(\textbf{x},\textbf{x}') + k_2(\textbf{x},\textbf{x}')\)</span></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="n">x_train</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="n">y_train</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span><span class="n">Y_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="n">x_valid</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
+<span class="n">y_valid</span> <span class="o">=</span> <span class="n">Y_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">)</span>
+
+<span class="n">M</span> <span class="o">=</span> <span class="mi">50</span>  <span class="c1"># number of inducing variables</span>
+<span class="n">Z</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">x_train</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">x_train</span><span class="p">),</span> <span class="n">M</span><span class="p">))</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span>
+<span class="p">)</span>
+
+<span class="n">kernel</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">kernels</span><span class="o">.</span><span class="n">RationalQuadratic</span><span class="p">()</span> <span class="o">+</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">kernels</span><span class="o">.</span><span class="n">Linear</span><span class="p">()</span>
+<span class="n">lik</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">likelihoods</span><span class="o">.</span><span class="n">Gaussian</span><span class="p">()</span>
+<span class="n">gp_nonlinear</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">models</span><span class="o">.</span><span class="n">SVGP</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">kernel</span><span class="p">,</span> <span class="n">likelihood</span><span class="o">=</span><span class="n">lik</span><span class="p">,</span> <span class="n">inducing_variable</span><span class="o">=</span><span class="n">Z</span><span class="p">)</span>
+<span class="n">gp_nonlinear</span><span class="o">.</span><span class="n">likelihood</span><span class="o">.</span><span class="n">variance</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_train</span><span class="p">))</span>
+<span class="n">gp_nonlinear</span><span class="p">,</span> <span class="n">iters</span><span class="p">,</span> <span class="n">elbo</span><span class="p">,</span> <span class="n">pred_error</span> <span class="o">=</span> <span class="n">trainGP</span><span class="p">(</span>
+    <span class="n">gp_nonlinear</span><span class="p">,</span>
+    <span class="n">training_data</span><span class="o">=</span><span class="p">(</span><span class="n">x_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">),</span>
+    <span class="n">validation_data</span><span class="o">=</span><span class="p">(</span><span class="n">x_valid</span><span class="p">,</span> <span class="n">y_valid</span><span class="p">),</span>
+    <span class="n">iterations</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span>
+    <span class="n">batch_size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
+    <span class="n">verbose</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plot_loss</span><span class="p">(</span><span class="n">iters</span><span class="p">,</span> <span class="n">elbo</span><span class="p">,</span> <span class="n">pred_error</span><span class="p">)</span>
+<span class="n">gp_nonlinear</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/2f0af083ad45a1695326b046a40d1cf355ae008a9e92dee1be2a50763c778e8d.png" src="../_images/2f0af083ad45a1695326b046a40d1cf355ae008a9e92dee1be2a50763c778e8d.png" />
+<div class="output text_html">&lt;gpflow.models.svgp.SVGP object at 0x7f168bcd7610&gt;
+<table>
+<thead>
+<tr><th>name                               </th><th>class    </th><th>transform       </th><th>prior  </th><th>trainable  </th><th>shape      </th><th>dtype  </th><th>value                                   </th></tr>
+</thead>
+<tbody>
+<tr><td>SVGP.kernel.kernels[0].variance    </td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>1.83679                                 </td></tr>
+<tr><td>SVGP.kernel.kernels[0].lengthscales</td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>3.72028                                 </td></tr>
+<tr><td>SVGP.kernel.kernels[0].alpha       </td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>3.19889                                 </td></tr>
+<tr><td>SVGP.kernel.kernels[1].variance    </td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>0.82989                                 </td></tr>
+<tr><td>SVGP.likelihood.variance           </td><td>Parameter</td><td>Softplus + Shift</td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>9.70904                                 </td></tr>
+<tr><td>SVGP.inducing_variable.Z           </td><td>Parameter</td><td>Identity        </td><td>       </td><td>True       </td><td>(50, 1)    </td><td>float64</td><td>[[-3.334647e+01...                      </td></tr>
+<tr><td>SVGP.q_mu                          </td><td>Parameter</td><td>Identity        </td><td>       </td><td>True       </td><td>(50, 1)    </td><td>float64</td><td>[[8.9104e-01...                         </td></tr>
+<tr><td>SVGP.q_sqrt                        </td><td>Parameter</td><td>FillTriangular  </td><td>       </td><td>True       </td><td>(1, 50, 50)</td><td>float64</td><td>[[[1.4250e-01, 0.0000e+00, 0.0000e+00...</td></tr>
+</tbody>
+</table></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">prediction_nonlinGP</span><span class="p">,</span> <span class="n">uncertainty_nonlinGP</span> <span class="o">=</span> <span class="n">gp_nonlinear</span><span class="o">.</span><span class="n">predict_f</span><span class="p">(</span><span class="n">x_valid</span><span class="p">)</span>
+<span class="n">prediction_nonlinGP</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">prediction_nonlinGP</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+<span class="n">uncertainty_nonlinGP</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">uncertainty_nonlinGP</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+
+<span class="n">n_std</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">start</span> <span class="o">=</span> <span class="mi">31000</span>
+<span class="n">end</span> <span class="o">=</span> <span class="n">start</span> <span class="o">+</span> <span class="mi">1000</span>
+<span class="n">prediction_nonlinGPplot</span> <span class="o">=</span> <span class="n">prediction_nonlinGP</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">]</span>
+<span class="n">uncertainty_nonlinGPplot</span> <span class="o">=</span> <span class="n">n_std</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">uncertainty_nonlinGP</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">])</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">17</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">prediction_nonlinGPplot</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted values&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1000</span><span class="p">),</span>
+    <span class="n">prediction_nonlinGPplot</span> <span class="o">+</span> <span class="n">uncertainty_nonlinGPplot</span><span class="p">,</span>
+    <span class="n">prediction_nonlinGPplot</span> <span class="o">-</span> <span class="n">uncertainty_nonlinGPplot</span><span class="p">,</span>
+    <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">y_valid</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="nb">sorted</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">prediction_nonlinGP</span><span class="p">)))</span>
+<span class="n">a</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span>
+    <span class="o">*</span><span class="nb">sorted</span><span class="p">(</span>
+        <span class="nb">zip</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">n_std</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">uncertainty_nonlinGP</span><span class="p">))</span>
+    <span class="p">)</span>
+<span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
+<span class="n">c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
+    <span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span>
+    <span class="n">Y_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span>
+    <span class="n">s</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">-</span> <span class="n">c</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/1885d2d31683446e0e68520bee9fc1f8f6ccead12f24f79c4b7564dfe328c201.png" src="../_images/1885d2d31683446e0e68520bee9fc1f8f6ccead12f24f79c4b7564dfe328c201.png" />
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="implement-the-linear-gp-parameterisation-in-an-online-setting">
+<h2>Implement the linear GP parameterisation in an online setting<a class="headerlink" href="#implement-the-linear-gp-parameterisation-in-an-online-setting" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">T_test</span> <span class="o">=</span> <span class="mi">10</span>
+
+<span class="n">X_full</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T_test</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>  <span class="c1"># Full model</span>
+
+<span class="n">init_cond</span> <span class="o">=</span> <span class="n">Xtrue</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="n">gcm_gp_det</span> <span class="o">=</span> <span class="n">GCM_GP</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">gp_nonlinear</span><span class="p">,</span> <span class="n">stochastic</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">Xgp_det</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_gp_det</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">gp_nonlinear</span><span class="p">)</span>
+
+<span class="n">gcm_gp_sto</span> <span class="o">=</span> <span class="n">GCM_GP</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">gp_nonlinear</span><span class="p">,</span> <span class="n">stochastic</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">Xgp_sto</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_gp_sto</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">gp_nonlinear</span><span class="p">)</span>
+
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM_no_param</span><span class="p">(</span><span class="n">Forcing</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+
+<span class="n">naive_parameterization</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+<span class="n">X_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mf">0.85439536</span><span class="p">,</span> <span class="mf">1.75218026</span><span class="p">])</span>
+
+<span class="n">time_i</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">X_full</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Full L96&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span>
+    <span class="n">Xgp_det</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GP (nonlinear kernel det.)&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;darkorange&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">Xgp_sto</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GP (nonlinear kernel stoch.)&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">X_no_param</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;no param&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;darkviolet&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">X_param</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;polynomial&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;crimson&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper left&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/bed63cc2ccd257eb22673e5c6ffd8b62831dad7601b1020c26af24fac4597867.png" src="../_images/bed63cc2ccd257eb22673e5c6ffd8b62831dad7601b1020c26af24fac4597867.png" />
+</div>
+</div>
+</div>
+<div class="section" id="use-multi-task-gps-to-exploit-non-local-features">
+<h2>Use multi-task GPs to exploit non-local features<a class="headerlink" href="#use-multi-task-gps-to-exploit-non-local-features" title="Permalink to this headline">#</a></h2>
+<p>In the GP literature, it is common to see the multi-task learning problem framed the same set of inputs are used to learn each of the functional mappings to the various outputs:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-2111e37c-afe8-4dd1-9939-54801385a830">
+<span class="eqno">(37)<a class="headerlink" href="#equation-2111e37c-afe8-4dd1-9939-54801385a830" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\textbf{y}_k = f_k(\textbf{x}) + \pmb{\varepsilon}_k,
+\end{equation}\]</div>
+<p>however it is also possible to define a unique set of inputs which map to the corresponding <span class="math notranslate nohighlight">\(k\)</span>, i.e:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-6c4a3119-670f-4d94-aa40-9709dfb7c5a7">
+<span class="eqno">(38)<a class="headerlink" href="#equation-6c4a3119-670f-4d94-aa40-9709dfb7c5a7" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\textbf{y}_k = f_k(\textbf{x}_k) + \pmb{\varepsilon}_k.
+\end{equation}\]</div>
+<p>This is more applicable to our case as we have 8 coarse grid terms, each of which map to their own respective sub-grid terms. We then want to model for any potential correlation between the outputs <span class="math notranslate nohighlight">\(\textbf{y}_k\)</span>, and hence the various functions <span class="math notranslate nohighlight">\(f_k(\textbf{x}_k)\)</span>. One way to achieve this is through a concept known as <em>Linear models for coregionalisation</em> (see e.g., <a class="reference external" href="https://arxiv.org/pdf/1106.6251.pdf">Álvarez et al, 2012</a>), where we assume that the function <span class="math notranslate nohighlight">\(f(\textbf{x})\)</span> is drawn from the linear combination of some latent function <span class="math notranslate nohighlight">\(g(\textbf{x})\)</span>, such that:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-165c47ec-d268-4189-809f-9ae3733290d7">
+<span class="eqno">(39)<a class="headerlink" href="#equation-165c47ec-d268-4189-809f-9ae3733290d7" title="Permalink to this equation">#</a></span>\[\begin{equation}
+f(\textbf{x}) = \textbf{W}g(\textbf{x}),
+\end{equation}\]</div>
+<p>where <span class="math notranslate nohighlight">\(f(\textbf{x}) \in \mathbb{R}^K\)</span>, <span class="math notranslate nohighlight">\(g(\textbf{x}) \in \mathbb{R}^L\)</span>, and <span class="math notranslate nohighlight">\(\textbf{W} \in \mathbb{R}^{K \times L}\)</span>, where <span class="math notranslate nohighlight">\(L \leq K\)</span>. The principle is that the outputs of <span class="math notranslate nohighlight">\(g(\textbf{x})\)</span> are uncorrelated, but through combining them with <span class="math notranslate nohighlight">\(\textbf{W}\)</span>, they become correlated. In GPflow, the only possibility for incorporating the unique input-output task pairing is via a simpler approach known as the <em>Intrinsic model for Coregionalisation</em> (see the <a class="reference external" href="https://gpflow.github.io/GPflow/2.4.0/notebooks/advanced/coregionalisation.html">docs</a>). In this approach we assume the covariance of the latent function values between two tasks is given by a linear combination of the input kernel:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-d11b272f-5c7e-4a8c-acfb-d9cc68ffc721">
+<span class="eqno">(40)<a class="headerlink" href="#equation-d11b272f-5c7e-4a8c-acfb-d9cc68ffc721" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\text{cov}\big(f_i(\textbf{x}),f_j(\textbf{x}')\big) = k(\textbf{x},\textbf{x}')\cdot \text{B}_{ij},
+\end{equation}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\textbf{B} = \textbf{W}\textbf{W}^\text{T} + \text{diag}(\kappa)\)</span>, and <span class="math notranslate nohighlight">\(\textbf{W} \in \mathbb{R}^{K \times L}\)</span>. The values of <span class="math notranslate nohighlight">\(\textbf{W}\)</span> and <span class="math notranslate nohighlight">\(\kappa\)</span> are optimised along with the kernel hyperparameters with the ELBO.</p>
+<p>We will use the same nonlinear kernel (Rational Quadratic + Linear) as before.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">gpflow.kernels</span> <span class="kn">import</span> <span class="n">Coregion</span>
+
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="n">heterosked</span> <span class="o">=</span> <span class="kc">False</span>  <span class="c1"># separate noise variance for each task</span>
+
+<span class="c1"># in this case we use the idea of &#39;coregionalisation&#39; to model the dependence between outputs</span>
+<span class="n">task_id_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span><span class="o">.</span><span class="n">T</span>
+<span class="n">task_id_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">ns</span><span class="p">)))</span><span class="o">.</span><span class="n">T</span>
+
+<span class="c1"># augment the inputs with an index for which output each row belongs to</span>
+<span class="n">x_train</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">X_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">task_id_train</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span>
+<span class="p">)</span>
+<span class="k">if</span> <span class="n">heterosked</span><span class="p">:</span>
+    <span class="n">y_train</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">Y_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">task_id_train</span><span class="p">)),</span>
+        <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">,</span>
+    <span class="p">)</span>
+    <span class="n">lik</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">likelihoods</span><span class="o">.</span><span class="n">SwitchedLikelihood</span><span class="p">(</span>
+        <span class="p">[</span><span class="n">gpflow</span><span class="o">.</span><span class="n">likelihoods</span><span class="o">.</span><span class="n">Gaussian</span><span class="p">()</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">K</span><span class="p">)]</span>
+    <span class="p">)</span>
+<span class="k">else</span><span class="p">:</span>
+    <span class="n">y_train</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span>
+        <span class="n">Y_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span>
+    <span class="p">)</span>
+    <span class="n">lik</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">likelihoods</span><span class="o">.</span><span class="n">Gaussian</span><span class="p">()</span>
+<span class="n">x_valid</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">task_id_test</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span>
+<span class="p">)</span>
+<span class="n">y_valid</span> <span class="o">=</span> <span class="n">Y_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">)</span>
+
+<span class="n">M</span> <span class="o">=</span> <span class="mi">56</span>
+<span class="n">Z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
+    <span class="p">[</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">X_train</span><span class="p">[:,</span> <span class="n">k</span><span class="p">]),</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">X_train</span><span class="p">[:,</span> <span class="n">k</span><span class="p">]),</span> <span class="n">M</span> <span class="o">//</span> <span class="n">K</span><span class="p">)</span>
+        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>
+    <span class="p">]</span>
+<span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">M</span> <span class="o">//</span> <span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">)</span>
+<span class="n">Z</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">convert_to_tensor</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">Z</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">kron</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">M</span> <span class="o">//</span> <span class="n">K</span><span class="p">)))</span><span class="o">.</span><span class="n">T</span><span class="p">)),</span>
+    <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">rank</span> <span class="o">=</span> <span class="n">K</span>  <span class="c1"># for larger data sets it may be quicker to set a rank lower than full-rank.</span>
+<span class="n">coreg</span> <span class="o">=</span> <span class="n">Coregion</span><span class="p">(</span>
+    <span class="n">output_dim</span><span class="o">=</span><span class="n">K</span><span class="p">,</span> <span class="n">rank</span><span class="o">=</span><span class="n">rank</span><span class="p">,</span> <span class="n">active_dims</span><span class="o">=</span><span class="p">[</span><span class="n">x_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
+<span class="p">)</span>  <span class="c1"># the last column of the input data</span>
+<span class="c1"># contains the indices for each task</span>
+
+<span class="n">dims</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">x_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">k1</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">kernels</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">active_dims</span><span class="o">=</span><span class="n">dims</span><span class="p">)</span>
+<span class="n">k2</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">kernels</span><span class="o">.</span><span class="n">RationalQuadratic</span><span class="p">(</span><span class="n">active_dims</span><span class="o">=</span><span class="n">dims</span><span class="p">)</span>
+<span class="n">kernel</span> <span class="o">=</span> <span class="p">(</span><span class="n">k1</span> <span class="o">+</span> <span class="n">k2</span><span class="p">)</span> <span class="o">*</span> <span class="n">coreg</span>
+
+<span class="n">gp_nonlocal</span> <span class="o">=</span> <span class="n">gpflow</span><span class="o">.</span><span class="n">models</span><span class="o">.</span><span class="n">SVGP</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">kernel</span><span class="p">,</span> <span class="n">likelihood</span><span class="o">=</span><span class="n">lik</span><span class="p">,</span> <span class="n">inducing_variable</span><span class="o">=</span><span class="n">Z</span><span class="p">)</span>
+<span class="n">gp_nonlocal</span><span class="p">,</span> <span class="n">iters</span><span class="p">,</span> <span class="n">elbo</span><span class="p">,</span> <span class="n">pred_error</span> <span class="o">=</span> <span class="n">trainGP</span><span class="p">(</span>
+    <span class="n">gp_nonlocal</span><span class="p">,</span>
+    <span class="n">training_data</span><span class="o">=</span><span class="p">(</span><span class="n">x_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">),</span>
+    <span class="n">validation_data</span><span class="o">=</span><span class="p">(</span><span class="n">x_valid</span><span class="p">,</span> <span class="n">y_valid</span><span class="p">),</span>
+    <span class="n">iterations</span><span class="o">=</span><span class="mi">5000</span><span class="p">,</span>
+    <span class="n">batch_size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
+    <span class="n">verbose</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plot_loss</span><span class="p">(</span><span class="n">iters</span><span class="p">,</span> <span class="n">elbo</span><span class="p">,</span> <span class="n">pred_error</span><span class="p">)</span>
+<span class="n">gp_nonlocal</span>  <span class="c1"># print the model</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/824d0d11706f019795b818807ff030f12ec7f816000b4270fb9d065798780d41.png" src="../_images/824d0d11706f019795b818807ff030f12ec7f816000b4270fb9d065798780d41.png" />
+<div class="output text_html">&lt;gpflow.models.svgp.SVGP object at 0x7f1676e74520&gt;
+<table>
+<thead>
+<tr><th>name                                          </th><th>class    </th><th>transform       </th><th>prior  </th><th>trainable  </th><th>shape      </th><th>dtype  </th><th>value                                   </th></tr>
+</thead>
+<tbody>
+<tr><td>SVGP.kernel.kernels[0].kernels[0].variance    </td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>0.40711                                 </td></tr>
+<tr><td>SVGP.kernel.kernels[0].kernels[1].variance    </td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>1.56768                                 </td></tr>
+<tr><td>SVGP.kernel.kernels[0].kernels[1].lengthscales</td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>3.13803                                 </td></tr>
+<tr><td>SVGP.kernel.kernels[0].kernels[1].alpha       </td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>0.45191                                 </td></tr>
+<tr><td>SVGP.kernel.kernels[1].W                      </td><td>Parameter</td><td>Identity        </td><td>       </td><td>True       </td><td>(8, 8)     </td><td>float64</td><td>[[0.52882, 0.52882, 0.52882...          </td></tr>
+<tr><td>SVGP.kernel.kernels[1].kappa                  </td><td>Parameter</td><td>Softplus        </td><td>       </td><td>True       </td><td>(8,)       </td><td>float64</td><td>[0.77143, 0.62198, 0.50819...           </td></tr>
+<tr><td>SVGP.likelihood.variance                      </td><td>Parameter</td><td>Softplus + Shift</td><td>       </td><td>True       </td><td>()         </td><td>float64</td><td>1.89604                                 </td></tr>
+<tr><td>SVGP.inducing_variable.Z                      </td><td>Parameter</td><td>Identity        </td><td>       </td><td>True       </td><td>(56, 2)    </td><td>float64</td><td>[[-32.94938, 0....                      </td></tr>
+<tr><td>SVGP.q_mu                                     </td><td>Parameter</td><td>Identity        </td><td>       </td><td>True       </td><td>(56, 1)    </td><td>float64</td><td>[[0.57454...                            </td></tr>
+<tr><td>SVGP.q_sqrt                                   </td><td>Parameter</td><td>FillTriangular  </td><td>       </td><td>True       </td><td>(1, 56, 56)</td><td>float64</td><td>[[[2.5483e-01, 0.0000e+00, 0.0000e+00...</td></tr>
+</tbody>
+</table></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">prediction_nonlocGP</span><span class="p">,</span> <span class="n">uncertainty_nonlocGP</span> <span class="o">=</span> <span class="n">gp_nonlocal</span><span class="o">.</span><span class="n">predict_f</span><span class="p">(</span><span class="n">x_valid</span><span class="p">)</span>
+<span class="n">prediction_nonlocGP</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">prediction_nonlocGP</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+<span class="n">uncertainty_nonlocGP</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">uncertainty_nonlocGP</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+
+<span class="n">n_std</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">start</span> <span class="o">=</span> <span class="mi">31000</span>
+<span class="n">end</span> <span class="o">=</span> <span class="n">start</span> <span class="o">+</span> <span class="mi">1000</span>
+<span class="n">prediction_nonlocGPplot</span> <span class="o">=</span> <span class="n">prediction_nonlocGP</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">]</span>
+<span class="n">uncertainty_nonlocGPplot</span> <span class="o">=</span> <span class="n">n_std</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">uncertainty_nonlocGP</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">])</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">17</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">prediction_nonlocGPplot</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted values&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1000</span><span class="p">),</span>
+    <span class="n">prediction_nonlocGPplot</span> <span class="o">+</span> <span class="n">uncertainty_nonlocGPplot</span><span class="p">,</span>
+    <span class="n">prediction_nonlocGPplot</span> <span class="o">-</span> <span class="n">uncertainty_nonlocGPplot</span><span class="p">,</span>
+    <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">y_valid</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="nb">sorted</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">prediction_nonlocGP</span><span class="p">)))</span>
+<span class="n">a</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span>
+    <span class="o">*</span><span class="nb">sorted</span><span class="p">(</span>
+        <span class="nb">zip</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span> <span class="n">n_std</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">uncertainty_nonlocGP</span><span class="p">))</span>
+    <span class="p">)</span>
+<span class="p">)</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
+<span class="n">c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
+    <span class="n">X_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span>
+    <span class="n">Y_valid</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ns</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s2">&quot;F&quot;</span><span class="p">),</span>
+    <span class="n">s</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">-</span> <span class="n">c</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
+<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/31d0ae4625a7c80c4ea094ae9dda890040a155cedd81cfe7a7fdec45eb77e52b.png" src="../_images/31d0ae4625a7c80c4ea094ae9dda890040a155cedd81cfe7a7fdec45eb77e52b.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">T_test</span> <span class="o">=</span> <span class="mi">10</span>
+
+<span class="n">X_full</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T_test</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>  <span class="c1"># Full model</span>
+
+<span class="n">init_cond</span> <span class="o">=</span> <span class="n">Xtrue</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="n">gcm_gp_nonlinear</span> <span class="o">=</span> <span class="n">GCM_GP</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">gp_nonlinear</span><span class="p">,</span> <span class="n">stochastic</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">Xgp_nonlinear</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_gp_nonlinear</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">gp_nonlinear</span><span class="p">)</span>
+
+<span class="n">gcm_gp_nonlocal</span> <span class="o">=</span> <span class="n">GCM_GP</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">gp_nonlocal</span><span class="p">,</span> <span class="n">stochastic</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">local</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">Xgp_nonlocal</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_gp_nonlocal</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">gp_nonlocal</span><span class="p">)</span>
+
+<span class="n">gcm_gp_nonlocal_det</span> <span class="o">=</span> <span class="n">GCM_GP</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">gp_nonlocal</span><span class="p">,</span> <span class="n">stochastic</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">local</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">Xgp_nonlocal_det</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_gp_nonlocal_det</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">gp_nonlocal</span><span class="p">)</span>
+
+<span class="n">time_i</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;top&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s2">&quot;right&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">X_full</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Full L96&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span>
+    <span class="n">Xgp_nonlinear</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GP (nonlinear local kernel)&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;teal&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span>
+    <span class="n">Xgp_nonlocal</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GP (nonlinear nonlocal kernel sto)&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;crimson&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span>
+    <span class="n">Xgp_nonlocal_det</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GP (nonlinear nonlocal kernel det)&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;darkviolet&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#recap-lorenz-96">Recap: Lorenz 96</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#recap-gaussian-processes-gps">Recap: Gaussian Processes (GPs)</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exact-inference">Exact inference</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#variational-inference">Variational inference</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sparse-gps">Sparse GPs</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learn-the-sub-grid-tendencies-offline">Learn the sub-grid tendencies offline</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#local-mappings-with-a-linear-kernel">Local mappings with a linear kernel</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#local-mappings-with-a-non-linear-kernel">Local mappings with a non-linear kernel</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#implement-the-linear-gp-parameterisation-in-an-online-setting">Implement the linear GP parameterisation in an online setting</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#use-multi-task-gps-to-exploit-non-local-features">Use multi-task GPs to exploit non-local features</a></li>
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+    <script>document.write(`<img src="../_static/newlogo.png" class="logo__image only-dark" alt="Logo image"/>`);</script>
+  
+  
+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
+  
+</a></div>
+        <div class="sidebar-primary-item"><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
+</ul>
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+    <h1>Improving Performance of Neural Networks</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-dataset-model-and-the-training-code">Setting up the Dataset, Model, and the Training Code</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#create-the-dataset">Create the Dataset</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#create-the-neural-network">Create the Neural Network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-and-evaluate-the-network">Train and Evaluate the Network</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regularization-and-overfitting">Regularization and Overfitting</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regularization-intuition">Regularization Intuition</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regularization-of-neural-networks">Regularization of Neural Networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#weight-decay-l2-norm">Weight decay (L2 norm)</a><ul class="nav section-nav flex-column">
+<li class="toc-h5 nav-item toc-entry"><a class="reference internal nav-link" href="#using-weight-decay">Using Weight Decay</a></li>
+</ul>
+</li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#dropout">Dropout</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#choosing-the-learning-rate">Choosing the Learning Rate</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-the-optimal-learning-rate">Finding the Optimal Learning Rate</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#recommended-reading">Recommended Reading</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#batch-normalization">Batch Normalization</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cyclic-learning-rate">Cyclic learning rate</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="improving-performance-of-neural-networks">
+<h1>Improving Performance of Neural Networks<a class="headerlink" href="#improving-performance-of-neural-networks" title="Permalink to this headline">#</a></h1>
+<p>In this notebook, we will briefly introduce several techniques that are generally used to improve the accuracy of neural networks.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="kn">import</span> <span class="nn">time</span>
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+<span class="kn">from</span> <span class="nn">torch.utils.data</span> <span class="kn">import</span> <span class="n">TensorDataset</span><span class="p">,</span> <span class="n">DataLoader</span>
+
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Ensuring reproducibility</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">14</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="setting-up-the-dataset-model-and-the-training-code">
+<h2>Setting up the Dataset, Model, and the Training Code<a class="headerlink" href="#setting-up-the-dataset-model-and-the-training-code" title="Permalink to this headline">#</a></h2>
+<p>First, we setup all the necessary code that’s required to build the dataset, create our neural network and train the network on the dataset.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The dataset, the neural network, and the training and evaluation functions that we use in this notebook are same as the one defined in <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a> and <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/L96_offline_NN.html">Using Neural Networks for L96 Parameterization</a>.</p>
+</div>
+<div class="section" id="create-the-dataset">
+<h3>Create the Dataset<a class="headerlink" href="#create-the-dataset" title="Permalink to this headline">#</a></h3>
+<div class="cell tag_hide-cell docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell content</span>
+<span class="expanded">Hide code cell content</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Generating the Ground Truth</span>
+<span class="c1"># ---------------------------</span>
+
+<span class="n">time_steps</span> <span class="o">=</span> <span class="mi">2000</span>
+<span class="n">forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span> <span class="o">*</span> <span class="n">time_steps</span>
+
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">forcing</span><span class="p">)</span>
+
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">xy_true</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">xy_true</span> <span class="o">=</span> <span class="n">X_true</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">),</span> <span class="n">xy_true</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+
+
+<span class="c1"># Splitting the into Training and Test Dataset</span>
+<span class="c1"># --------------------------------------------</span>
+
+<span class="n">val_size</span> <span class="o">=</span> <span class="mi">400</span>
+
+<span class="c1"># Training Data</span>
+<span class="n">X_true_train</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">subgrid_tend_train</span> <span class="o">=</span> <span class="n">xy_true</span><span class="p">[:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="c1"># Test Data</span>
+<span class="n">X_true_test</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+<span class="n">subgrid_tend_test</span> <span class="o">=</span> <span class="n">xy_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+
+
+<span class="c1"># Building the Dataset and the Dataloaders</span>
+<span class="c1"># ----------------------------------------</span>
+
+<span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">1024</span>
+
+<span class="c1"># Training Data</span>
+<span class="n">nlocal_data</span> <span class="o">=</span> <span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_train</span><span class="p">),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_tend_train</span><span class="p">),</span>
+<span class="p">)</span>
+<span class="n">loader_train</span> <span class="o">=</span> <span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">nlocal_data</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+
+<span class="c1"># Test Data</span>
+<span class="n">nlocal_data_test</span> <span class="o">=</span> <span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">)</span>
+<span class="p">)</span>
+<span class="n">loader_test</span> <span class="o">=</span> <span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">nlocal_data_test</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</details>
+</div>
+</div>
+<div class="section" id="create-the-neural-network">
+<h3>Create the Neural Network<a class="headerlink" href="#create-the-neural-network" title="Permalink to this headline">#</a></h3>
+<p>We build a 3 layer neural network consisting of two hidden layers and an output layer. Each hidden layer is followed by the <code class="docutils literal notranslate"><span class="pre">ReLU</span></code> activation function.</p>
+<div class="cell tag_hide-cell docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell content</span>
+<span class="expanded">Hide code cell content</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">NetANN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</details>
+</div>
+</div>
+<div class="section" id="train-and-evaluate-the-network">
+<h3>Train and Evaluate the Network<a class="headerlink" href="#train-and-evaluate-the-network" title="Permalink to this headline">#</a></h3>
+<div class="cell tag_hide-cell docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell content</span>
+<span class="expanded">Hide code cell content</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train the network for one epoch&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
+
+    <span class="n">train_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+        <span class="c1"># Get predictions</span>
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="c1"># This if block is needed to add a dummy dimension if our inputs are 1D</span>
+            <span class="c1"># (where each number is a different sample)</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">network</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>
+
+        <span class="c1"># Compute the loss</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span>
+        <span class="n">train_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Clear the gradients</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>
+
+        <span class="c1"># Backpropagation to compute the gradients and update the weights</span>
+        <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>
+
+    <span class="k">return</span> <span class="n">train_loss</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+
+
+<span class="k">def</span> <span class="nf">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Test the network&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>  <span class="c1"># Evaluation mode (important when having dropout layers)</span>
+
+    <span class="n">test_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+            <span class="c1"># Get predictions</span>
+            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="c1"># This if block is needed to add a dummy dimension if our inputs are 1D</span>
+                <span class="c1"># (where each number is a different sample)</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">network</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>
+
+            <span class="c1"># Compute the loss</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span>
+            <span class="n">test_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Get an average loss for the entire dataset</span>
+        <span class="n">test_loss</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">test_loss</span>
+
+
+<span class="k">def</span> <span class="nf">fit_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">val_loader</span><span class="p">,</span> <span class="n">n_epochs</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train and validate the network&quot;&quot;&quot;</span>
+    <span class="n">train_losses</span><span class="p">,</span> <span class="n">val_losses</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
+    <span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+        <span class="n">train_loss</span> <span class="o">=</span> <span class="n">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+        <span class="n">val_loss</span> <span class="o">=</span> <span class="n">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">val_loader</span><span class="p">)</span>
+        <span class="n">train_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">train_loss</span><span class="p">)</span>
+        <span class="n">val_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">val_loss</span><span class="p">)</span>
+    <span class="n">end_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Training completed in </span><span class="si">{</span><span class="nb">int</span><span class="p">(</span><span class="n">end_time</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">start_time</span><span class="p">)</span><span class="si">}</span><span class="s2"> seconds.&quot;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">train_losses</span><span class="p">,</span> <span class="n">val_losses</span>
+</pre></div>
+</div>
+</div>
+</details>
+</div>
+</div>
+</div>
+<div class="section" id="regularization-and-overfitting">
+<h2>Regularization and Overfitting<a class="headerlink" href="#regularization-and-overfitting" title="Permalink to this headline">#</a></h2>
+<p>One of the most common issues that happen while training a neural network is when the model memorizes the training dataset. It causes the model to perform very accurately on the training set but shows very poor performance on the validation set. This phenomenon is termed overfitting. One of the ways to prevent overfitting is to add regularization to our model as described below.</p>
+<div class="figure align-default" id="overfitting-curve">
+<a class="reference internal image-reference" href="../_images/overfitting.png"><img alt="../_images/overfitting.png" src="../_images/overfitting.png" style="width: 700px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 3 </span><span class="caption-text">Curves showing different types of model fits. The image is taken from the Python Machine Learning book by Sebastian Raschka.</span><a class="headerlink" href="#overfitting-curve" title="Permalink to this image">#</a></p>
+</div>
+<p>The curve on the far right of the plot above predicts perfectly on the given set, yet it’s not the best choice. This is because if you were to gather some new data points, they most likely would not be on that curve. Instead, those new points would be closer to the curve in the middle graph since it generalizes better to the dataset.</p>
+<div class="section" id="regularization-intuition">
+<h3>Regularization Intuition<a class="headerlink" href="#regularization-intuition" title="Permalink to this headline">#</a></h3>
+<p>Regularization can be thought of as <strong>putting constraints on the model</strong> to obtain better generalizability i.e. <em>avoiding remembering</em> the training data.</p>
+<p>One of the ways to achieve this can be by adding a term to the loss function such that:</p>
+<blockquote>
+<div><p>Loss = Training Loss + Regularization</p>
+</div></blockquote>
+<p>This puts a penalty for making the model more complex.</p>
+<p>Very braodly speaking (just to gain intuition) - if we want to reduce the training loss (reduce bias) we should try using a more complex model (if we have enough data) and if we want to reduce overfitting (reduce variace) we should simplify or constraint the model (increase regularization).</p>
+</div>
+<div class="section" id="regularization-of-neural-networks">
+<h3>Regularization of Neural Networks<a class="headerlink" href="#regularization-of-neural-networks" title="Permalink to this headline">#</a></h3>
+<p>Some of the ways to add regularization in neural networks are:</p>
+<ul class="simple">
+<li><p>Dropout (added in the definition of the network).</p></li>
+<li><p>Early stopping</p></li>
+<li><p>Weight decay (added in the optimizer part - see <code class="docutils literal notranslate"><span class="pre">optim.Adam</span></code> in PyTorch)</p></li>
+<li><p>Data augmentation (usually for images)</p></li>
+</ul>
+<p>Below we show how some of these regularizations can be applied to the L96 parameterization problem. However, this particular problem is not very plagued by the issue of overfitting, and so the advantages of these processes are not clear in all their glory.</p>
+<div class="section" id="weight-decay-l2-norm">
+<h4>Weight decay (L2 norm)<a class="headerlink" href="#weight-decay-l2-norm" title="Permalink to this headline">#</a></h4>
+<p>Weight decay is usually defined as a term that’s added directly to the update rule.
+Namely, to update a certain weight <span class="math notranslate nohighlight">\(w\)</span> in the <span class="math notranslate nohighlight">\(i+1\)</span> iteration, we would use a modified rule:</p>
+<p><span class="math notranslate nohighlight">\(w_{i+1} = w_{i} - \gamma ( \frac{\partial L}{\partial w} + A w_{i})\)</span></p>
+<p>In practice, this is almost identical to L2 regularization, though there is some difference (e.g., see <a class="reference external" href="https://bbabenko.github.io/weight-decay/">here</a>)</p>
+<p>Weight decay is one of the parameters of the optimizer - see <code class="docutils literal notranslate"><span class="pre">torch.optim.SGD</span></code></p>
+<div class="section" id="using-weight-decay">
+<h5>Using Weight Decay<a class="headerlink" href="#using-weight-decay" title="Permalink to this headline">#</a></h5>
+<p>Now we try to train our <code class="docutils literal notranslate"><span class="pre">NetANN</span></code> model again but this time by adding a weight decay to it.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_3l_decay</span> <span class="o">=</span> <span class="n">NetANN</span><span class="p">()</span>
+
+<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">80</span>
+<span class="n">criterion</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>
+<span class="n">optimizer_decay</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_3l_decay</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.003</span><span class="p">,</span> <span class="n">weight_decay</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">train_loss_decay</span><span class="p">,</span> <span class="n">val_loss_decay</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">nn_3l_decay</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer_decay</span><span class="p">,</span> <span class="n">loader_train</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">n_epochs</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 0 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+<p>Plotting the training and validation loss curves</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_decay</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Training loss, with weight decay&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">val_loss_decay</span><span class="p">,</span> <span class="s2">&quot;r--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Validation loss, with weight decay&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/bc0d6dfca4ec5cfcb32557a551a5eac5ad5158a5fad49440890b827eeb5564ce.png" src="../_images/bc0d6dfca4ec5cfcb32557a551a5eac5ad5158a5fad49440890b827eeb5564ce.png" />
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="dropout">
+<h4>Dropout<a class="headerlink" href="#dropout" title="Permalink to this headline">#</a></h4>
+<p>Dropout means randomly deactivating or temporarily removing some units from a layer of the network while training, along with all its incoming and outgoing connections. See more details <a class="reference external" href="http://jmlr.org/papers/v15/srivastava14a.html">here</a>.
+It is usually the most useful regularization that we can do in fully connected layers.</p>
+<p>In convolutional layers dropout makes less sense - see more discussion <a class="reference external" href="https://www.kdnuggets.com/2018/09/dropout-convolutional-networks.html">here</a></p>
+<div class="figure align-default" id="dropout-layer">
+<a class="reference internal image-reference" href="../_images/Dropout_layer.png"><img alt="../_images/Dropout_layer.png" src="../_images/Dropout_layer.png" style="width: 700px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 4 </span><span class="caption-text">Dropout Neural Net Model. Left: A standard neural net with 2 hidden layers. Right: An example of a thinned net produced by applying dropout to the network on the left. Crossed units have been dropped. Image taken from <a class="reference external" href="http://www.jmlr.org/papers/volume15/srivastava14a/srivastava14a.pdf">here</a>.</span><a class="headerlink" href="#dropout-layer" title="Permalink to this image">#</a></p>
+</div>
+<p>In the network defined below, we add dropout to with a probability of 20% to each layer. This means that during each training step, random 20% of the units within each layer will be deactivated.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">NetANNDropout</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dropout_rate</span><span class="o">=</span><span class="mf">0.2</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">dropout</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Dropout</span><span class="p">(</span><span class="n">dropout_rate</span><span class="p">)</span>  <span class="c1"># Dropout regularization</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dropout</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dropout</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Network with very high dropout</span>
+<span class="n">nn_3l_drop</span> <span class="o">=</span> <span class="n">NetANNDropout</span><span class="p">(</span><span class="n">dropout_rate</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
+
+<span class="n">criterion</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>
+<span class="n">optimizer_drop</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_3l_drop</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.003</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">train_loss_drop</span><span class="p">,</span> <span class="n">val_loss_drop</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">nn_3l_drop</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer_drop</span><span class="p">,</span> <span class="n">loader_train</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">n_epochs</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 0 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+<p>Plotting the training and validation loss curves.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_drop</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Training loss, with dropout&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">val_loss_drop</span><span class="p">,</span> <span class="s2">&quot;r--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Validation loss, with dropout&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/88737b958659020b17697275e287862b1727d707c07594ba52af674142b1d1e7.png" src="../_images/88737b958659020b17697275e287862b1727d707c07594ba52af674142b1d1e7.png" />
+</div>
+</div>
+<p>In this particular problem drop out does not seem to provide much advantage. While the model with drop out never seems to reach a state of overfitting (validation loss&gt;training loss), it also does not perform as well as the model without any dropout.</p>
+</div>
+</div>
+</div>
+<div class="section" id="choosing-the-learning-rate">
+<h2>Choosing the Learning Rate<a class="headerlink" href="#choosing-the-learning-rate" title="Permalink to this headline">#</a></h2>
+<p>While training a neural network <strong>selecting a good learning rate (LR) is essential for both fast convergence and a lower error</strong>. A high learning rate can cause the training loss to never converge while a too small learning rate will cause the model to converge extremely slowly.</p>
+<div class="section" id="finding-the-optimal-learning-rate">
+<h3>Finding the Optimal Learning Rate<a class="headerlink" href="#finding-the-optimal-learning-rate" title="Permalink to this headline">#</a></h3>
+<p>To choose the optimal learning rate for our network, we can use an LR finding algorithm. The objective of a LR Finder is to find the highest LR which still minimises the loss and does not make the loss diverge/explode. This is done by first starting with an extremely small LR and then increasing the LR after each batch until the corresponding loss starts to explode. To read more about learning rate finders, read <a class="reference external" href="https://towardsdatascience.com/speeding-up-neural-net-training-with-lr-finder-c3b401a116d0">this blog</a>.</p>
+<p>For our use case, we use the LR finder from the <code class="docutils literal notranslate"><span class="pre">torch-lr-finder</span></code> package to find the best learning rate for our neural network.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">torch_lr_finder</span> <span class="kn">import</span> <span class="n">LRFinder</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/usr/share/miniconda/envs/L96M2lines/lib/python3.9/site-packages/torch_lr_finder/lr_finder.py:5: TqdmExperimentalWarning: Using `tqdm.autonotebook.tqdm` in notebook mode. Use `tqdm.tqdm` instead to force console mode (e.g. in jupyter console)
+  from tqdm.autonotebook import tqdm
+</pre></div>
+</div>
+</div>
+</div>
+<p>Define the model and the optimizer. The optimizer is <strong>initialized with a very small learning rate</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_3l_lr</span> <span class="o">=</span> <span class="n">NetANN</span><span class="p">()</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_3l_lr</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">1e-7</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Now we setup the LR finder and make it run for 200 iterations during which the learning rate varies from 1e-7 to 100.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">lr_finder</span> <span class="o">=</span> <span class="n">LRFinder</span><span class="p">(</span><span class="n">nn_3l_lr</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">criterion</span><span class="p">)</span>
+<span class="n">lr_finder</span><span class="o">.</span><span class="n">range_test</span><span class="p">(</span><span class="n">loader_train</span><span class="p">,</span> <span class="n">end_lr</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">num_iter</span><span class="o">=</span><span class="mi">200</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<script type="application/vnd.jupyter.widget-view+json">{"version_major": 2, "version_minor": 0, "model_id": "5cbb622a7d6343c2b26a45b01f0b902f"}</script><div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Stopping early, the loss has diverged
+Learning rate search finished. See the graph with {finder_name}.plot()
+</pre></div>
+</div>
+</div>
+</div>
+<p>Now we plot the LR vs the loss curve to find the best learning rate.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot the lr vs the loss curve</span>
+<span class="n">lr_finder</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
+
+<span class="c1"># Reset the model and optimizer to their initial state</span>
+<span class="n">lr_finder</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>LR suggestion: steepest gradient
+Suggested LR: 3.29E-02
+</pre></div>
+</div>
+<img alt="../_images/db04845ee402a409b80c7962908e50196407f4f39d6a1513d6bff21616f7b3b2.png" src="../_images/db04845ee402a409b80c7962908e50196407f4f39d6a1513d6bff21616f7b3b2.png" />
+</div>
+</div>
+<p>From the curve, we see that at the learning of approximately 0.01 we get the steepest gradient. So we choose 0.01 as the learning rate for our neural network.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">20</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_3l_lr</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+
+<span class="n">train_loss</span><span class="p">,</span> <span class="n">val_loss</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">nn_3l_lr</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">loader_train</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">n_epochs</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 0 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+<p>Plotting the training and validation loss curves.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Training loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">val_loss</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Validation loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/88910c75b2b3080c6db0a73cfb377a524387fcd6a82156466659e50d3240fb75.png" src="../_images/88910c75b2b3080c6db0a73cfb377a524387fcd6a82156466659e50d3240fb75.png" />
+</div>
+</div>
+<p>From the loss curves we can see that <strong>the loss has converged much faster</strong> than before.</p>
+</div>
+</div>
+<div class="section" id="recommended-reading">
+<h2>Recommended Reading<a class="headerlink" href="#recommended-reading" title="Permalink to this headline">#</a></h2>
+<div class="section" id="batch-normalization">
+<h3>Batch Normalization<a class="headerlink" href="#batch-normalization" title="Permalink to this headline">#</a></h3>
+<p>Normalize the activation values such that the hidden representation don’t vary drastically and also helps to get improvement in the training speed.</p>
+</div>
+<div class="section" id="cyclic-learning-rate">
+<h3>Cyclic learning rate<a class="headerlink" href="#cyclic-learning-rate" title="Permalink to this headline">#</a></h3>
+<p>The cyclic learning rate policy, introduced in <a class="reference external" href="https://arxiv.org/abs/1506.01186">Cyclical Learning Rates for Training Neural Networks</a>, cycles the learning rate between two boundaries with a constant frequency in a triangular fashion. To read more about the cyclic learning rates and the one cycle policy, read <a class="reference external" href="https://sgugger.github.io/the-1cycle-policy.html">here</a>.</p>
+<p>In PyTorch, cyclic learning rate can be used from <code class="docutils literal notranslate"><span class="pre">optim.lr_scheduler.CyclicLR</span></code>.</p>
+<div class="figure align-default" id="cyclic-lr">
+<a class="reference internal image-reference" href="../_images/cyclic_lr.png"><img alt="../_images/cyclic_lr.png" src="../_images/cyclic_lr.png" style="width: 500px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 5 </span><span class="caption-text">Cyclical learning rates oscillate back and forth between two bounds when training, slowly increasing the learning rate after every batch update. Image taken from <a class="reference external" href="https://pyimagesearch.com/2019/07/29/cyclical-learning-rates-with-keras-and-deep-learning/">here</a>.</span><a class="headerlink" href="#cyclic-lr" title="Permalink to this image">#</a></p>
+</div>
+</div>
+</div>
+</div>
+
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-dataset-model-and-the-training-code">Setting up the Dataset, Model, and the Training Code</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#create-the-dataset">Create the Dataset</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#create-the-neural-network">Create the Neural Network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-and-evaluate-the-network">Train and Evaluate the Network</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regularization-and-overfitting">Regularization and Overfitting</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regularization-intuition">Regularization Intuition</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#regularization-of-neural-networks">Regularization of Neural Networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#weight-decay-l2-norm">Weight decay (L2 norm)</a><ul class="nav section-nav flex-column">
+<li class="toc-h5 nav-item toc-entry"><a class="reference internal nav-link" href="#using-weight-decay">Using Weight Decay</a></li>
+</ul>
+</li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#dropout">Dropout</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#choosing-the-learning-rate">Choosing the Learning Rate</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-the-optimal-learning-rate">Finding the Optimal Learning Rate</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#recommended-reading">Recommended Reading</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#batch-normalization">Batch Normalization</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cyclic-learning-rate">Cyclic learning rate</a></li>
+</ul>
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+</ul>
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+  <head>
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+    <title>The Lorenz-96 Two-Timescale System &#8212; Learning Machine Learning with Lorenz-96</title>
+  
+  
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+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
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+    <h1>The Lorenz-96 Two-Timescale System</h1>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#code-to-numerically-integrate-the-two-time-scale-l96-model">Code to numerically integrate the two time-scale L96 model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-class-for-numerical-integration-of-the-two-time-scale-l96">A class for numerical integration of the two time-scale L96</a></li>
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+  <div class="tex2jax_ignore mathjax_ignore section" id="the-lorenz-96-two-timescale-system">
+<h1>The Lorenz-96 Two-Timescale System<a class="headerlink" href="#the-lorenz-96-two-timescale-system" title="Permalink to this headline">#</a></h1>
+<p>In 1996, Lorenz proposed a two-time scale dynamical system, referred to as the Lorenz-96 model (L96), whose behavior mimics the non-linear dynamics of the extratropical atmosphere with simplified representation of multiscale interactions and nonlinear advection <span id="id1">[<a class="reference internal" href="../bibliography.html#id2" title="E.N. Lorenz. Predictability: a problem partly solved. Seminar on Predictability, 1:1-18, 1995. URL: https://www.ecmwf.int/node/10829.">Lorenz, 1995</a>]</span>. The L96 model consists of two sets of equations coupling two sets of variables (<span class="math notranslate nohighlight">\(X_k\)</span> and <span class="math notranslate nohighlight">\(Y_{j,k}\)</span>), which evolve over two (slow and fast) timescales and are arranged around a latitude circle as shown in Fig. 1. The equations comprising L96 are:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-b8654deb-d7b3-40a3-8bb2-3f708e40a7c6">
+<span class="eqno">(1)<a class="headerlink" href="#equation-b8654deb-d7b3-40a3-8bb2-3f708e40a7c6" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k},
+\\
+\frac{d}{dt} Y_{j,k}
+&amp;= - cbY_{j+1,k} \left( Y_{j+2,k} - Y_{j-1,k} \right) - c Y_{j,k} + \frac{hc}{b} X_k
+\end{align}\]</div>
+<p>where <span class="math notranslate nohighlight">\(X_k\)</span>, <span class="math notranslate nohighlight">\(k=1,\ldots,K\)</span>, denotes <span class="math notranslate nohighlight">\(K\)</span> slow (or low-frequency) variables, and <span class="math notranslate nohighlight">\(Y_{j,k}\)</span> , <span class="math notranslate nohighlight">\(j=1,\ldots,J\)</span> denotes <span class="math notranslate nohighlight">\(J*K\)</span> fast (or high-frequency) variables.
+The slow equations are coupled to the fast equations via a coupling term, <span class="math notranslate nohighlight">\(\sum_{j=0}^{J-1} Y_{j,k}\)</span>, which sums over the <span class="math notranslate nohighlight">\(J\)</span> fast variables corresponding to a particular <span class="math notranslate nohighlight">\(k\)</span>. On the other hand, each fast equation is forced by a coupling term, <span class="math notranslate nohighlight">\(\frac{hc}{b} X_k\)</span>, that depends on the slow variable corresponding to that particular <span class="math notranslate nohighlight">\(k\)</span>.</p>
+<p>The evolution of this two-time scale system depend on three key parameters: <span class="math notranslate nohighlight">\(b\)</span>, <span class="math notranslate nohighlight">\(c\)</span> and <span class="math notranslate nohighlight">\(h\)</span>. Here <span class="math notranslate nohighlight">\(b\)</span> determines the magnitude of the nonlinear interactions among the fast variables, <span class="math notranslate nohighlight">\(c\)</span> controls how rapidly the fast variables fluctuate compared to the slow variables and, <span class="math notranslate nohighlight">\(h\)</span> governs the strength of the coupling between the slow and fast variables. Moreover, the slow time-scale equation is forced by the parameter <span class="math notranslate nohighlight">\(F\)</span>, whose value determines the chaotic behaviour of the system. E.g. <span id="id2">[<a class="reference internal" href="../bibliography.html#id3" title="Daniel S. Wilks. Effects of stochastic parametrizations in the lorenz \textquotesingle 96 system. Quarterly Journal of the Royal Meteorological Society, 131(606):389–407, 2005. URL: https://doi.org/10.1256/qj.04.03, doi:10.1256/qj.04.03.">Wilks, 2005</a>]</span>.</p>
+<p>The chaotic dynamical system L96 is very useful for testing different numerical methods in atmospheric modeling thanks to its transparency, low computational cost and simplicity compared to Global Climate Models (GCM). The interaction between variables of different scales makes the L96 model of particular interest when evaluating new parameterization methodologies. As such, it was used in assessing different techniques that were later incorporated into GCMs (<span id="id3">[<a class="reference internal" href="../bibliography.html#id8" title="Daan Crommelin and Eric Vanden-Eijnden. Subgrid-scale parameterization with conditional markov chains. Journal of the Atmospheric Sciences, 65(8):2661–2675, August 2008. URL: https://doi.org/10.1175/2008jas2566.1, doi:10.1175/2008jas2566.1.">Crommelin and Vanden-Eijnden, 2008</a>]</span>, <span id="id4">[<a class="reference internal" href="../bibliography.html#id9" title="J. Dorrestijn, D. T. Crommelin, J. A. Biello, and S. J. Böing. A data-driven multi-cloud model for stochastic parametrization of deep convection. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1991):20120374, May 2013. URL: https://doi.org/10.1098/rsta.2012.0374, doi:10.1098/rsta.2012.0374.">Dorrestijn <em>et al.</em>, 2013</a>]</span>).</p>
+<p>The L96 model has been extensively used as a test bed in various studies including data assimilation approaches (<span id="id5">[<a class="reference internal" href="../bibliography.html#id10" title="K.J.H. Law, D. Sanz-Alonso, A. Shukla, and A.M. Stuart. Filter accuracy for the lorenz 96 model: fixed versus adaptive observation operators. Physica D: Nonlinear Phenomena, 325:1–13, June 2016. URL: https://doi.org/10.1016/j.physd.2015.12.008, doi:10.1016/j.physd.2015.12.008.">Law <em>et al.</em>, 2016</a>]</span>, <span id="id6">[<a class="reference internal" href="../bibliography.html#id11" title="Sam Hatfield, Aneesh Subramanian, Tim Palmer, and Peter Düben. Improving weather forecast skill through reduced-precision data assimilation. Monthly Weather Review, 146(1):49–62, December 2017. URL: https://doi.org/10.1175/mwr-d-17-0132.1, doi:10.1175/mwr-d-17-0132.1.">Hatfield <em>et al.</em>, 2017</a>]</span>), stochastic parameterization schemes (<span id="id7">[<a class="reference internal" href="../bibliography.html#id12" title="Frank Kwasniok. Data-based stochastic subgrid-scale parametrization: an approach using cluster-weighted modelling. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370(1962):1061–1086, March 2012. URL: https://doi.org/10.1098/rsta.2011.0384, doi:10.1098/rsta.2011.0384.">Kwasniok, 2012</a>]</span>, <span id="id8">[<a class="reference internal" href="../bibliography.html#id4" title="H. M. Arnold, I. M. Moroz, and T. N. Palmer. Stochastic parametrizations and model uncertainty in the lorenz '96 system. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1991):20110479, 2013. URL: https://doi.org/10.1098/rsta.2011.0479, doi:10.1098/rsta.2011.0479.">Arnold <em>et al.</em>, 2013</a>]</span>, <span id="id9">[<a class="reference internal" href="../bibliography.html#id13" title="Alexandre J. Chorin and Fei Lu. Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics. Proceedings of the National Academy of Sciences, 112(32):9804–9809, July 2015. URL: https://doi.org/10.1073/pnas.1512080112, doi:10.1073/pnas.1512080112.">Chorin and Lu, 2015</a>]</span>)  and Machine Learning-based parameterization methodologies (<span id="id10">[<a class="reference internal" href="../bibliography.html#id6" title="Tapio Schneider, Shiwei Lan, Andrew Stuart, and João Teixeira. Earth system modeling 2.0: a blueprint for models that learn from observations and targeted high-resolution simulations. Geophysical Research Letters, December 2017. URL: https://doi.org/10.1002/2017gl076101, doi:10.1002/2017gl076101.">Schneider <em>et al.</em>, 2017</a>]</span>, <span id="id11">[<a class="reference internal" href="../bibliography.html#id14" title="Peter D. Dueben and Peter Bauer. Challenges and design choices for global weather and climate models based on machine learning. Geoscientific Model Development, 11(10):3999–4009, October 2018. URL: https://doi.org/10.5194/gmd-11-3999-2018, doi:10.5194/gmd-11-3999-2018.">Dueben and Bauer, 2018</a>]</span> , <span id="id12">[<a class="reference internal" href="../bibliography.html#id15" title="David John Gagne, Hannah M. Christensen, Aneesh C. Subramanian, and Adam H. Monahan. Machine learning for stochastic parameterization: generative adversarial networks in the lorenz \textquotesingle 96 model. Journal of Advances in Modeling Earth Systems, March 2020. URL: https://doi.org/10.1029/2019ms001896, doi:10.1029/2019ms001896.">Gagne <em>et al.</em>, 2020</a>]</span>).</p>
+<div class="figure align-default" id="l96-equation-figure">
+<a class="reference internal image-reference" href="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png"><img alt="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" src="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" style="width: 400px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 1 </span><span class="caption-text"><em>Visualisation of a two-scale Lorenz ‘96 system with J = 8 and K = 6. Global-scale variables (<span class="math notranslate nohighlight">\(X_k\)</span>) are updated based on neighbouring variables and on the local-scale variables (<span class="math notranslate nohighlight">\(Y_{j,k}\)</span>) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from <a class="reference external" href="https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436">Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.</a></em>.</span><a class="headerlink" href="#l96-equation-figure" title="Permalink to this image">#</a></p>
+</div>
+<div class="section" id="code-to-numerically-integrate-the-two-time-scale-l96-model">
+<h2>Code to numerically integrate the two time-scale L96 model<a class="headerlink" href="#code-to-numerically-integrate-the-two-time-scale-l96-model" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">display_source</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>In the <code class="docutils literal notranslate"><span class="pre">L96_model</span></code> module we provide the function <code class="docutils literal notranslate"><span class="pre">L96_2t_xdot_ydot</span></code> (shown next) that returns the tendencies (sum of the RHS) for the X and Y equations, as well as the tendency corresponding to the coupling in the X equation, <span class="math notranslate nohighlight">\(-hc/b \sum_{j=0}^{J-1} Y_{j,k}\)</span>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96_2t_xdot_ydot</span>
+
+<span class="n">display_source</span><span class="p">(</span><span class="n">L96_2t_xdot_ydot</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
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+.highlight .vm { color: #19177C } /* Name.Variable.Magic */
+.highlight .il { color: #666666 } /* Literal.Number.Integer.Long */</style><div class="highlight"><pre><span></span><span class="nd">@njit</span>
+<span class="k">def</span> <span class="nf">L96_2t_xdot_ydot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Calculate the time rate of change for the X and Y variables for the Lorenz &#39;96, two time-scale</span>
+<span class="sd">    model, equations 2 and 3:</span>
+<span class="sd">        d/dt X[k] =     -X[k-1] ( X[k-2] - X[k+1] )   - X[k] + F - h.c/b sum_j Y[j,k]</span>
+<span class="sd">        d/dt Y[j] = -b c Y[j+1] ( Y[j+2] - Y[j-1] ) - c Y[j]     + h.c/b X[k]</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X : Values of X variables at the current time step</span>
+<span class="sd">        Y : Values of Y variables at the current time step</span>
+<span class="sd">        F : Forcing term</span>
+<span class="sd">        h : coupling coefficient</span>
+<span class="sd">        b : ratio of amplitudes</span>
+<span class="sd">        c : time-scale ratio</span>
+<span class="sd">    Returns:</span>
+<span class="sd">        dXdt, dYdt, C : Arrays of X and Y time tendencies, and the coupling term -hc/b*sum(Y,j)</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">JK</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Y</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+    <span class="n">J</span> <span class="o">=</span> <span class="n">JK</span> <span class="o">//</span> <span class="n">K</span>
+    <span class="k">assert</span> <span class="n">JK</span> <span class="o">==</span> <span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">,</span> <span class="s2">&quot;X and Y have incompatible shapes&quot;</span>
+    <span class="n">Xdot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>
+    <span class="n">hcb</span> <span class="o">=</span> <span class="p">(</span><span class="n">h</span> <span class="o">*</span> <span class="n">c</span><span class="p">)</span> <span class="o">/</span> <span class="n">b</span>
+
+    <span class="n">Ysummed</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+
+    <span class="n">Xdot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">-</span> <span class="n">X</span> <span class="o">+</span> <span class="n">F</span> <span class="o">-</span> <span class="n">hcb</span> <span class="o">*</span> <span class="n">Ysummed</span>
+    <span class="n">Ydot</span> <span class="o">=</span> <span class="p">(</span>
+        <span class="o">-</span><span class="n">c</span> <span class="o">*</span> <span class="n">b</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+        <span class="o">-</span> <span class="n">c</span> <span class="o">*</span> <span class="n">Y</span>
+        <span class="o">+</span> <span class="n">hcb</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">J</span><span class="p">)</span>
+    <span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">Xdot</span><span class="p">,</span> <span class="n">Ydot</span><span class="p">,</span> <span class="o">-</span><span class="n">hcb</span> <span class="o">*</span> <span class="n">Ysummed</span>
+</pre></div>
+</div></div>
+</div>
+<p>These tendencies describe the continuous evolution of the L96 model at a particular time. However, to obtain a discrete solution we must integrate numerically in time. In the <code class="docutils literal notranslate"><span class="pre">L96_model</span></code> module, we provide the function <code class="docutils literal notranslate"><span class="pre">integrate_L96_2t</span></code> (shown next) that uses fourth-order Runge-Kutta integration. Starting from the initial conditions <code class="docutils literal notranslate"><span class="pre">X0,Y0</span></code>, the function returns the trajectories of <code class="docutils literal notranslate"><span class="pre">X,Y</span></code> sampled at an interval <code class="docutils literal notranslate"><span class="pre">si</span></code>. There is a related function, <code class="docutils literal notranslate"><span class="pre">integrate_L96_2t_with_coupling</span></code>, that in addition to the trajectories, also returns the coupling term, <span class="math notranslate nohighlight">\(-hc/b \sum_{j=0}^{J-1} Y_{j,k}\)</span> at each point in the trajectory.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">integrate_L96_2t</span>
+
+<span class="n">display_source</span><span class="p">(</span><span class="n">integrate_L96_2t</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><style>pre { line-height: 125%; }
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+.highlight .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */
+.highlight .kp { color: #008000 } /* Keyword.Pseudo */
+.highlight .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */
+.highlight .kt { color: #B00040 } /* Keyword.Type */
+.highlight .m { color: #666666 } /* Literal.Number */
+.highlight .s { color: #BA2121 } /* Literal.String */
+.highlight .na { color: #687822 } /* Name.Attribute */
+.highlight .nb { color: #008000 } /* Name.Builtin */
+.highlight .nc { color: #0000FF; font-weight: bold } /* Name.Class */
+.highlight .no { color: #880000 } /* Name.Constant */
+.highlight .nd { color: #AA22FF } /* Name.Decorator */
+.highlight .ni { color: #717171; font-weight: bold } /* Name.Entity */
+.highlight .ne { color: #CB3F38; font-weight: bold } /* Name.Exception */
+.highlight .nf { color: #0000FF } /* Name.Function */
+.highlight .nl { color: #767600 } /* Name.Label */
+.highlight .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */
+.highlight .nt { color: #008000; font-weight: bold } /* Name.Tag */
+.highlight .nv { color: #19177C } /* Name.Variable */
+.highlight .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */
+.highlight .w { color: #bbbbbb } /* Text.Whitespace */
+.highlight .mb { color: #666666 } /* Literal.Number.Bin */
+.highlight .mf { color: #666666 } /* Literal.Number.Float */
+.highlight .mh { color: #666666 } /* Literal.Number.Hex */
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+.highlight .sa { color: #BA2121 } /* Literal.String.Affix */
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+.highlight .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */
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+.highlight .se { color: #AA5D1F; font-weight: bold } /* Literal.String.Escape */
+.highlight .sh { color: #BA2121 } /* Literal.String.Heredoc */
+.highlight .si { color: #A45A77; font-weight: bold } /* Literal.String.Interpol */
+.highlight .sx { color: #008000 } /* Literal.String.Other */
+.highlight .sr { color: #A45A77 } /* Literal.String.Regex */
+.highlight .s1 { color: #BA2121 } /* Literal.String.Single */
+.highlight .ss { color: #19177C } /* Literal.String.Symbol */
+.highlight .bp { color: #008000 } /* Name.Builtin.Pseudo */
+.highlight .fm { color: #0000FF } /* Name.Function.Magic */
+.highlight .vc { color: #19177C } /* Name.Variable.Class */
+.highlight .vg { color: #19177C } /* Name.Variable.Global */
+.highlight .vi { color: #19177C } /* Name.Variable.Instance */
+.highlight .vm { color: #19177C } /* Name.Variable.Magic */
+.highlight .il { color: #666666 } /* Literal.Number.Integer.Long */</style><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">integrate_L96_2t</span><span class="p">(</span><span class="n">X0</span><span class="p">,</span> <span class="n">Y0</span><span class="p">,</span> <span class="n">si</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">t0</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="mf">0.001</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Integrates forward-in-time the two time-scale Lorenz 1996 model, using the RK4 integration method.</span>
+<span class="sd">    Returns the full history with nt+1 values starting with initial conditions, X[:,0]=X0 and Y[:,0]=Y0,</span>
+<span class="sd">    and ending with the final state, X[:,nt+1] and Y[:,nt+1] at time t0+nt*si.</span>
+
+<span class="sd">    Note the model is intergrated</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X0 : Values of X variables at the current time</span>
+<span class="sd">        Y0 : Values of Y variables at the current time</span>
+<span class="sd">        si : Sampling time interval</span>
+<span class="sd">        nt : Number of sample segments (results in nt+1 samples incl. initial state)</span>
+<span class="sd">        F  : Forcing term</span>
+<span class="sd">        h  : coupling coefficient</span>
+<span class="sd">        b  : ratio of amplitudes</span>
+<span class="sd">        c  : time-scale ratio</span>
+<span class="sd">        t0 : Initial time (defaults to 0)</span>
+<span class="sd">        dt : The actual time step. If dt&lt;si, then si is used. Otherwise si/dt must be a whole number. Default 0.001.</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        X[:,:], Y[:,:], time[:] : the full history X[n,k] and Y[n,k] at times t[n]</span>
+
+<span class="sd">    Example usage:</span>
+<span class="sd">        X,Y,t = integrate_L96_2t(5+5*np.random.rand(8), np.random.rand(8*4), 0.01, 500, 18, 1, 10, 10)</span>
+<span class="sd">        plt.plot( t, X);</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">xhist</span><span class="p">,</span> <span class="n">yhist</span><span class="p">,</span> <span class="n">time</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">integrate_L96_2t_with_coupling</span><span class="p">(</span>
+        <span class="n">X0</span><span class="p">,</span> <span class="n">Y0</span><span class="p">,</span> <span class="n">si</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">t0</span><span class="o">=</span><span class="n">t0</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span>
+    <span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">xhist</span><span class="p">,</span> <span class="n">yhist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div></div>
+</div>
+<p>An example of how to use <code class="docutils literal notranslate"><span class="pre">integrate_L96_2t</span></code> is shown below. Here we choose <span class="math notranslate nohighlight">\(K=36\)</span> and <span class="math notranslate nohighlight">\(J=10\)</span> (i.e. 36 values of <span class="math notranslate nohighlight">\(X\)</span> and 10 values of <span class="math notranslate nohighlight">\(Y\)</span> per value of <span class="math notranslate nohighlight">\(X\)</span>). Also, in accordance with previous studies, we set <span class="math notranslate nohighlight">\(h=1\)</span>, <span class="math notranslate nohighlight">\(c=10\)</span>, and <span class="math notranslate nohighlight">\(b=10\)</span>. The initial condition for <span class="math notranslate nohighlight">\(X\)</span> is setup as <span class="math notranslate nohighlight">\(X(s)=s(s-1)(s+1)\)</span>, for <span class="math notranslate nohighlight">\(s=-1\ldots 1\)</span>, and the <span class="math notranslate nohighlight">\(Y\)</span> is initialized with zeros. The value of <span class="math notranslate nohighlight">\(F\)</span> is set to <span class="math notranslate nohighlight">\(10\)</span>, which is sufficient to obtain chaotic behavior.</p>
+<p><em>Note: if you increase <span class="math notranslate nohighlight">\(F\)</span>, you may need to reduce <span class="math notranslate nohighlight">\(dt\)</span> for numerical stability.</em></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">K</span> <span class="o">=</span> <span class="mi">36</span>  <span class="c1"># Number of globa-scale variables X</span>
+<span class="n">J</span> <span class="o">=</span> <span class="mi">10</span>  <span class="c1"># Number of local-scale Y variables per single global-scale X variable</span>
+<span class="n">nt</span> <span class="o">=</span> <span class="mi">1000</span>  <span class="c1"># Number of time steps</span>
+<span class="n">si</span> <span class="o">=</span> <span class="mf">0.005</span>  <span class="c1"># Sampling time interval</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.005</span>  <span class="c1"># Time step</span>
+<span class="n">F</span> <span class="o">=</span> <span class="mf">10.0</span>  <span class="c1"># Focring</span>
+<span class="n">h</span> <span class="o">=</span> <span class="mf">1.0</span>  <span class="c1"># Coupling coefficient</span>
+<span class="n">b</span> <span class="o">=</span> <span class="mf">10.0</span>  <span class="c1"># ratio of amplitudes</span>
+<span class="n">c</span> <span class="o">=</span> <span class="mf">10.0</span>  <span class="c1"># time-scale ratio</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;A non-dimension coordinate from -1..+1 corresponding to k=0..K&quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="n">K</span> <span class="o">-</span> <span class="mi">1</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">k</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>  <span class="c1"># For coordinate in plots</span>
+<span class="n">j</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">)</span>  <span class="c1"># For coordinate in plots</span>
+
+<span class="c1"># Initial conditions</span>
+<span class="n">X_init</span> <span class="o">=</span> <span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">Y_init</span> <span class="o">=</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">s</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+
+<span class="c1"># &quot;Run&quot; model</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">integrate_L96_2t</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">Y_init</span><span class="p">,</span> <span class="n">si</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>After running the model, we plot the results.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">221</span><span class="p">)</span>
+
+<span class="c1"># Snapshot of X[k]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">X</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$X_k(t=n_t dt)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">j</span> <span class="o">/</span> <span class="n">J</span><span class="p">,</span> <span class="n">Y</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$Y_{j,k}(t=n_t dt)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">X_init</span><span class="p">,</span> <span class="s2">&quot;k:&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$X_k(t=0)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">j</span> <span class="o">/</span> <span class="n">J</span><span class="p">,</span> <span class="n">Y_init</span><span class="p">,</span> <span class="s2">&quot;k:&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$Y_{j,k}(t=0)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k, k + j/J&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$X_k, Y_{j,k}$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">222</span><span class="p">)</span>
+
+<span class="c1"># Sample time-series X[0](t), Y[0](t)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$X_0(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">Y</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$Y_{0,0}(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">223</span><span class="p">)</span>
+
+<span class="c1"># Full model history of X</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">orientation</span><span class="o">=</span><span class="s2">&quot;horizontal&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$X_k(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">224</span><span class="p">)</span>
+
+<span class="c1"># Full model history of Y</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">j</span> <span class="o">/</span> <span class="n">J</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">orientation</span><span class="o">=</span><span class="s2">&quot;horizontal&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k + j/J&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$Y_{j,k}(t)$&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/e10ecaf0247573745362e20554abe003f66cbb1b36b34c4a24808899880ff865.png" src="../_images/e10ecaf0247573745362e20554abe003f66cbb1b36b34c4a24808899880ff865.png" />
+</div>
+</div>
+</div>
+<div class="section" id="a-class-for-numerical-integration-of-the-two-time-scale-l96">
+<h2>A class for numerical integration of the two time-scale L96<a class="headerlink" href="#a-class-for-numerical-integration-of-the-two-time-scale-l96" title="Permalink to this headline">#</a></h2>
+<p>For convenience, we also provide a class named <code class="docutils literal notranslate"><span class="pre">L96</span></code> in <code class="docutils literal notranslate"><span class="pre">L96_model.py</span></code> which helps to reduce the amount of code when using the <code class="docutils literal notranslate"><span class="pre">L96_2t_xdot_ydot</span></code> function.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span>
+
+<span class="n">help</span><span class="p">(</span><span class="n">L96</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Help on class L96 in module L96_model:
+
+class L96(builtins.object)
+ |  L96(K, J, F=18, h=1, b=10, c=10, t=0, dt=0.001)
+ |  
+ |  Class for two time-scale Lorenz 1996 model
+ |  
+ |  Methods defined here:
+ |  
+ |  __init__(self, K, J, F=18, h=1, b=10, c=10, t=0, dt=0.001)
+ |      Construct a two time-scale model with parameters:
+ |      K  : Number of X values
+ |      J  : Number of Y values per X value
+ |      F  : Forcing term (default 18.)
+ |      h  : coupling coefficient (default 1.)
+ |      b  : ratio of amplitudes (default 10.)
+ |      c  : time-scale ratio (default 10.)
+ |      t  : Initial time (default 0.)
+ |      dt : Time step (default 0.001)
+ |  
+ |  __repr__(self)
+ |      Return repr(self).
+ |  
+ |  __str__(self)
+ |      Return str(self).
+ |  
+ |  copy(self)
+ |  
+ |  print(self)
+ |  
+ |  randomize_IC(self)
+ |      Randomize the initial conditions (or current state)
+ |  
+ |  run(self, si, T, store=False, return_coupling=False)
+ |      Run model for a total time of T, sampling at intervals of si.
+ |      If store=Ture, then stores the final state as the initial conditions for the next segment.
+ |      If return_coupling=True, returns C in addition to X,Y,t.
+ |      Returns sampled history: X[:,:],Y[:,:],t[:],C[:,:].
+ |  
+ |  set_param(self, dt=None, F=None, h=None, b=None, c=None, t=None)
+ |      Set a model parameter, e.g. .set_param(si=0.01, dt=0.002)
+ |  
+ |  set_state(self, X, Y, t=None)
+ |      Set initial conditions (or current state), e.g. .set_state(X,Y)
+ |  
+ |  ----------------------------------------------------------------------
+ |  Data descriptors defined here:
+ |  
+ |  __dict__
+ |      dictionary for instance variables (if defined)
+ |  
+ |  __weakref__
+ |      list of weak references to the object (if defined)
+ |  
+ |  ----------------------------------------------------------------------
+ |  Data and other attributes defined here:
+ |  
+ |  F = &#39;Forcing&#39;
+ |  
+ |  X = &#39;Current X state or initial conditions&#39;
+ |  
+ |  Y = &#39;Current Y state or initial conditions&#39;
+ |  
+ |  b = &#39;Ratio of timescales&#39;
+ |  
+ |  c = &#39;Ratio of amplitudes&#39;
+ |  
+ |  dt = &#39;Time step&#39;
+ |  
+ |  h = &#39;Coupling coefficient&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<p>For example, to setup the same model as above with <span class="math notranslate nohighlight">\(K=36\)</span>, <span class="math notranslate nohighlight">\(J=10\)</span> and <span class="math notranslate nohighlight">\(F=10\)</span>, use</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create an instance of the L96 model, default parameters except those given</span>
+<span class="n">M</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">36</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="mf">0.005</span><span class="p">)</span>
+<span class="n">M</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>L96: K=36 J=10 F=10 h=1 b=10 c=10 dt=0.005
+</pre></div>
+</div>
+</div>
+</div>
+<p>Note that parameters <span class="math notranslate nohighlight">\(F\)</span>, <span class="math notranslate nohighlight">\(h\)</span>, <span class="math notranslate nohighlight">\(b\)</span>, <span class="math notranslate nohighlight">\(c\)</span> have defaults since in many of the applications in later notebooks these will not change. The initial conditions are automatically randomized to <span class="math notranslate nohighlight">\(X = b {\cal N}\)</span>, <span class="math notranslate nohighlight">\(Y = {\cal N}\)</span> where <span class="math notranslate nohighlight">\({\cal N}\)</span> is a random number drawn from a Guassian distribution. To set the same initial condition as used above, use</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set the initial conditions (here X is the same cubic as above, Y=0)</span>
+<span class="n">M</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">K</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">M</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>L96: K=36 J=10 F=10 h=1 b=10 c=10 dt=0.005
+ X=[ 0.05326217  0.14641204  0.22258659  0.28281464  0.328125    0.35954647
+  0.37810785  0.38483796  0.3807656   0.36691958  0.3443287   0.31402178
+  0.27702761  0.234375    0.18709276  0.13620971  0.08275463  0.02775634
+ -0.02775634 -0.08275463 -0.13620971 -0.18709276 -0.234375   -0.27702761
+ -0.31402178 -0.3443287  -0.36691958 -0.3807656  -0.38483796 -0.37810785
+ -0.35954647 -0.328125   -0.28281464 -0.22258659 -0.14641204 -0.05326217]
+ Y=[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+ t=0
+</pre></div>
+</div>
+</div>
+</div>
+<p>To sample the same tracjectories as above use the <code class="docutils literal notranslate"><span class="pre">.run()</span></code> method.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Run the model for 1000 sample intervals, or for time 1000*0.005 = 5</span>
+<span class="n">X2</span><span class="p">,</span> <span class="n">Y2</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mf">0.005</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Mean absolute difference =&quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X2</span> <span class="o">-</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">Y2</span> <span class="o">-</span> <span class="n">Y</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Mean absolute difference = 0.0
+</pre></div>
+</div>
+</div>
+</div>
+<p>Re-running the above cell gives the same answer each time because the same initial conditions are used for each invocation or <code class="docutils literal notranslate"><span class="pre">.run()</span></code>. To allow sequences of run, using the end state of one as the starting point for the next, add the parameter <code class="docutils literal notranslate"><span class="pre">store=True</span></code> which updates the initial condition stored in the object (<code class="docutils literal notranslate"><span class="pre">M</span></code>) with the final state of the run.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Run the model as above but use the end state as an initial condition for the next run.</span>
+<span class="c1"># (Re-running this cell will give different answers after the first run)</span>
+<span class="n">X2</span><span class="p">,</span> <span class="n">Y2</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mf">0.005</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span>
+    <span class="s2">&quot;Mean absolute difference 1st run=&quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X2</span> <span class="o">-</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">Y2</span> <span class="o">-</span> <span class="n">Y</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+<span class="p">)</span>
+
+<span class="n">X2</span><span class="p">,</span> <span class="n">Y2</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mf">0.005</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span>
+    <span class="s2">&quot;Mean absolute difference 2nd run=&quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X2</span> <span class="o">-</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">Y2</span> <span class="o">-</span> <span class="n">Y</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Mean absolute difference 1st run= 0.0
+Mean absolute difference 2nd run= 4.761055522025149
+</pre></div>
+</div>
+</div>
+</div>
+<p>The entire model creation and execution process demonstrated above can also be implemented in a one-liner version as shown below.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X3</span><span class="p">,</span> <span class="n">Y3</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="p">(</span>
+    <span class="n">L96</span><span class="p">(</span><span class="mi">36</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="mf">0.005</span><span class="p">)</span>
+    <span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">K</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">M</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
+    <span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mf">0.005</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-class-for-numerical-integration-of-the-two-time-scale-l96">A class for numerical integration of the two time-scale L96</a></li>
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+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
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+<li class="toctree-l1 current active"><a class="current reference internal" href="#">ResNet and Recurrent Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
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+    <h1>ResNet and Recurrent Neural Networks</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generating-data-from-l96">Generating data from L96:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#janni-s-fully-connected-3-layer-artificial-neural-network-ann">Janni’s fully connected, 3-layer Artificial Neural Network (ANN)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resnet">ResNet</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#main-points">Main points</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#recurrent-neural-networks-rnn">Recurrent Neural Networks (RNN)</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="resnet-and-recurrent-neural-networks">
+<h1>ResNet and Recurrent Neural Networks<a class="headerlink" href="#resnet-and-recurrent-neural-networks" title="Permalink to this headline">#</a></h1>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="kn">from</span> <span class="nn">matplotlib.animation</span> <span class="kn">import</span> <span class="n">FuncAnimation</span>
+<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">HTML</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">math</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch.autograd</span> <span class="kn">import</span> <span class="n">Variable</span>
+<span class="kn">import</span> <span class="nn">torch.nn.functional</span> <span class="k">as</span> <span class="nn">F</span>
+<span class="kn">import</span> <span class="nn">torch.utils.data</span> <span class="k">as</span> <span class="nn">Data</span>
+<span class="kn">import</span> <span class="nn">torchvision</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+
+<span class="c1"># from torch_lr_finder import LRFinder</span>
+<span class="kn">import</span> <span class="nn">torch.nn.functional</span> <span class="k">as</span> <span class="nn">F</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span><span class="p">,</span> <span class="n">integrate_L96_2t</span><span class="p">,</span> <span class="n">EulerFwd</span><span class="p">,</span> <span class="n">RK2</span><span class="p">,</span> <span class="n">RK4</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="figure align-default">
+<a class="reference internal image-reference" href="https://www.asimovinstitute.org/wp-content/uploads/2019/04/NeuralNetworkZo19High.png"><img alt="https://www.asimovinstitute.org/wp-content/uploads/2019/04/NeuralNetworkZo19High.png" src="https://www.asimovinstitute.org/wp-content/uploads/2019/04/NeuralNetworkZo19High.png" style="width: 600px;" /></a>
+</div>
+<p><a class="reference external" href="https://www.ecmwf.int/en/elibrary/10829-predictability-problem-partly-solved">Lorenz (1996)</a> describes a “two time-scale” model in two equations (2 and 3) which are:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-00ce264c-955e-43a1-8379-85d83b8945c8">
+<span class="eqno">(51)<a class="headerlink" href="#equation-00ce264c-955e-43a1-8379-85d83b8945c8" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}
+\\
+\frac{d}{dt} Y_{j,k}
+&amp;= - cbY_{j+1,k} \left( Y_{j+2,k} - Y_{j-1,k} \right) - c Y_{j,k} + \frac{hc}{b} X_k
+\end{align}\]</div>
+<div class="figure align-default" id="id1">
+<a class="reference internal image-reference" href="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png"><img alt="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" src="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" style="width: 400px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 10 </span><span class="caption-text"><em>Visualisation of a two-scale Lorenz ‘96 system with J = 8 and K = 6. Global-scale variables (<span class="math notranslate nohighlight">\(X_k\)</span>) are updated based on neighbouring variables and on the local-scale variables (<span class="math notranslate nohighlight">\(Y_{j,k}\)</span>) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from <a class="reference external" href="https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436">Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.</a></em>.</span><a class="headerlink" href="#id1" title="Permalink to this image">#</a></p>
+</div>
+<div class="section" id="generating-data-from-l96">
+<h2>Generating data from L96:<a class="headerlink" href="#generating-data-from-l96" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">time_steps</span> <span class="o">=</span> <span class="mi">20000</span>
+<span class="n">Forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span> <span class="o">*</span> <span class="n">time_steps</span>
+
+<span class="c1"># Create a &quot;real world&quot; with K=8 and J=32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Forcing</span><span class="p">)</span>
+
+<span class="c1"># Get training data for the neural network.</span>
+<span class="c1"># - Run the true state and output subgrid tendencies (the effect of Y on X is xytrue):</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">xy_true</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">val_size</span> <span class="o">=</span> <span class="mi">4000</span>  <span class="c1"># number of time steps for validation</span>
+<span class="c1"># train:</span>
+<span class="n">X_true_train</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span>
+    <span class="p">:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:</span>
+<span class="p">]</span>  <span class="c1"># Flatten because we first use single input as a sample</span>
+<span class="n">subgrid_tend_train</span> <span class="o">=</span> <span class="n">xy_true</span><span class="p">[:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="c1"># test:</span>
+<span class="n">X_true_test</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+<span class="n">subgrid_tend_test</span> <span class="o">=</span> <span class="n">xy_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create non local training data</span>
+<span class="c1"># Define a data loader (8 inputs, 8 outputs)</span>
+
+<span class="c1"># Define our X,Y pairs (state, subgrid tendency) for the linear regression local network.local_torch_dataset = Data.TensorDataset(</span>
+<span class="n">torch_dataset</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_train</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_tend_train</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+<span class="p">)</span>
+
+<span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">1024</span>  <span class="c1"># Number of sample in each batch</span>
+
+<span class="n">loader</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">torch_dataset</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># Define a test dataloader (8 inputs, 8 outputs)</span>
+
+<span class="n">torch_dataset_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="p">)</span>
+
+<span class="n">loader_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">torch_dataset_test</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="amsmath math notranslate nohighlight" id="equation-4c0fb91b-32c6-4164-b0ce-4dbaa9e64782">
+<span class="eqno">(52)<a class="headerlink" href="#equation-4c0fb91b-32c6-4164-b0ce-4dbaa9e64782" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}
+\\
+\frac{d}{dt} Y_{j,k}
+&amp;= - cbY_{j+1,k} \left( Y_{j+2,k} - Y_{j-1,k} \right) - c Y_{j,k} + \frac{hc}{b} X_k
+\end{align}\]</div>
+<p>Goal: Use <span class="math notranslate nohighlight">\(X_k\)</span> to predict subgrid terms,  <span class="math notranslate nohighlight">\(- \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}\)</span></p>
+</div>
+<div class="section" id="janni-s-fully-connected-3-layer-artificial-neural-network-ann">
+<h2>Janni’s fully connected, 3-layer Artificial Neural Network (ANN)<a class="headerlink" href="#janni-s-fully-connected-3-layer-artificial-neural-network-ann" title="Permalink to this headline">#</a></h2>
+<div class="figure align-default">
+<a class="reference internal image-reference" href="https://miro.medium.com/max/720/1*VHOUViL8dHGfvxCsswPv-Q.png"><img alt="https://miro.medium.com/max/720/1*VHOUViL8dHGfvxCsswPv-Q.png" src="https://miro.medium.com/max/720/1*VHOUViL8dHGfvxCsswPv-Q.png" style="width: 400px;" /></a>
+</div>
+<p>K=8 and J=32</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Net_ANN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_ANN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs, 16 neurons for first hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 16 neurons for second hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+    <span class="c1">#         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train_model</span><span class="p">(</span><span class="n">net</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">trainloader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+    <span class="n">net</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
+    <span class="n">test_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">for</span> <span class="n">step</span><span class="p">,</span> <span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">trainloader</span><span class="p">):</span>  <span class="c1"># for each training step</span>
+        <span class="n">b_x</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>  <span class="c1"># Inputs</span>
+        <span class="n">b_y</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">batch_y</span><span class="p">)</span>  <span class="c1"># outputs</span>
+        <span class="k">if</span> <span class="p">(</span>
+            <span class="nb">len</span><span class="p">(</span><span class="n">b_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
+        <span class="p">):</span>  <span class="c1"># If is needed to add a dummy dimension if our inputs are 1D (where each number is a different sample)</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span>
+                <span class="n">net</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">b_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+            <span class="p">)</span>  <span class="c1"># input x and predict based on x</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">net</span><span class="p">(</span><span class="n">b_x</span><span class="p">)</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">b_y</span><span class="p">)</span>  <span class="c1"># Calculating loss</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>  <span class="c1"># clear gradients for next train</span>
+        <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>  <span class="c1"># backpropagation, compute gradients</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>  <span class="c1"># apply gradients to update weights</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">test_model</span><span class="p">(</span><span class="n">net</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">trainloader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">text</span><span class="o">=</span><span class="s2">&quot;validation&quot;</span><span class="p">):</span>
+    <span class="n">net</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>  <span class="c1"># Evaluation mode (important when having dropout layers)</span>
+    <span class="n">test_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="k">for</span> <span class="n">step</span><span class="p">,</span> <span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span>
+            <span class="n">trainloader</span>
+        <span class="p">):</span>  <span class="c1"># for each training step</span>
+            <span class="n">b_x</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>  <span class="c1"># Inputs</span>
+            <span class="n">b_y</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">batch_y</span><span class="p">)</span>  <span class="c1"># outputs</span>
+            <span class="k">if</span> <span class="p">(</span>
+                <span class="nb">len</span><span class="p">(</span><span class="n">b_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
+            <span class="p">):</span>  <span class="c1"># If is needed to add a dummy dimension if our inputs are 1D (where each number is a different sample)</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span>
+                    <span class="n">net</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">b_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+                <span class="p">)</span>  <span class="c1"># input x and predict based on x</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">net</span><span class="p">(</span><span class="n">b_x</span><span class="p">)</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">b_y</span><span class="p">)</span>  <span class="c1"># Calculating loss</span>
+            <span class="n">test_loss</span> <span class="o">=</span> <span class="n">test_loss</span> <span class="o">+</span> <span class="n">loss</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>  <span class="c1"># Keep track of the loss</span>
+        <span class="n">test_loss</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">trainloader</span><span class="p">)</span>  <span class="c1"># dividing by the number of batches</span>
+    <span class="c1">#         print(len(trainloader))</span>
+    <span class="c1"># disabling prints to only show graphs</span>
+    <span class="c1"># print(text + &#39; loss:&#39;,test_loss)</span>
+    <span class="k">return</span> <span class="n">test_loss</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">criterion</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>  <span class="c1"># MSE loss function</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_3l</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">()</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="c1"># optimizer = optim.Adam(nn_3l.parameters(), lr=1e-7)</span>
+<span class="c1"># lr_finder = LRFinder(nn_3l, optimizer, criterion)</span>
+<span class="c1"># lr_finder.range_test(loader, end_lr=100, num_iter=200)</span>
+<span class="c1"># lr_finder.plot() # to inspect the loss-learning rate graph</span>
+<span class="c1"># lr_finder.reset() # to reset the model and optimizer to their initial state</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">20</span>  <span class="c1"># Number of epocs</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_3l</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">validation_loss</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
+<span class="n">train_loss</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
+<span class="c1"># time0 = time()</span>
+<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+    <span class="n">train_model</span><span class="p">(</span><span class="n">nn_3l</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+    <span class="n">train_loss</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_model</span><span class="p">(</span><span class="n">nn_3l</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="s2">&quot;train&quot;</span><span class="p">))</span>
+    <span class="n">validation_loss</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_model</span><span class="p">(</span><span class="n">nn_3l</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;training loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">validation_loss</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;validation loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epochs&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">final_losses</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
+<span class="n">final_losses</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">validation_loss</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/c17ea4a24949a4db439f81358d3731880f20bec38bf8d4a8579bf2c5cd124a9a.png" src="../_images/c17ea4a24949a4db439f81358d3731880f20bec38bf8d4a8579bf2c5cd124a9a.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">preds22</span> <span class="o">=</span> <span class="n">nn_3l</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="p">:])</span><span class="o">.</span><span class="n">double</span><span class="p">())</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds22</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;NN Predicted values&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">[:</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/976a727af220692d5da9f0315062ba1747b4814f696636e0fed76ef950f681ab.png" src="../_images/976a727af220692d5da9f0315062ba1747b4814f696636e0fed76ef950f681ab.png" />
+</div>
+</div>
+<p>3-layer ANN is (too) good!</p>
+</div>
+<div class="section" id="resnet">
+<h2>ResNet<a class="headerlink" href="#resnet" title="Permalink to this headline">#</a></h2>
+<p>ResNet is based on residual blocks, depicted below<br />
+Residual blocks are based on skip connections<br />
+Skip connections add output from one layer to that of a deeper layer</p>
+<div class="figure align-default">
+<a class="reference internal image-reference" href="https://d1m75rqqgidzqn.cloudfront.net/wp-data/2020/10/22125000/Image-recreation-Sep-15-1-1-640x420.jpg"><img alt="https://d1m75rqqgidzqn.cloudfront.net/wp-data/2020/10/22125000/Image-recreation-Sep-15-1-1-640x420.jpg" src="https://d1m75rqqgidzqn.cloudfront.net/wp-data/2020/10/22125000/Image-recreation-Sep-15-1-1-640x420.jpg" style="width: 600px;" /></a>
+</div>
+<p>ResNet is famously applied to <em>image recognition</em> using NN that are <em>very deep</em></p>
+<blockquote>
+<div><p>He, Kaiming, et al. “Deep Residual Learning for Image Recognition.” ArXiv.org, 2015, arxiv.org/abs/1512.03385. Accessed 5 Oct. 2021.</p>
+</div></blockquote>
+<p>Pretrained image recognition ResNet models are available with tens or &gt;100 layers<br />
+This is not our use case for Lorenz ‘96</p>
+<blockquote>
+<div><p>However, adaptation for dynamic systems do exist, e.g. https://towardsdatascience.com/neural-odes-breakdown-of-another-deep-learning-breakthrough-3e78c7213795\</p>
+</div></blockquote>
+<p>Here, we’ll look at skip connections and residual blocks, rather than ResNet<br />
+First, need to ‘break’ Janni’s model by adding too many layers</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Net_deepNN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_deepNN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs, 16 neurons for first hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 16 neurons for second hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear4</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear5</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear6</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+    <span class="c1">#         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear4</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear5</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear6</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_deep</span> <span class="o">=</span> <span class="n">Net_deepNN</span><span class="p">()</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="c1"># optimizer = optim.Adam(nn_deep.parameters(), lr=1e-7)</span>
+<span class="c1"># lr_finder = LRFinder(nn_deep, optimizer, criterion)</span>
+<span class="c1"># lr_finder.range_test(loader, end_lr=100, num_iter=200)</span>
+<span class="c1"># lr_finder.plot() # to inspect the loss-learning rate graph</span>
+<span class="c1"># lr_finder.reset() # to reset the model and optimizer to their initial state</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">40</span>  <span class="c1"># Number of epocs</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_deep</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">validation_loss</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
+<span class="n">train_loss</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
+<span class="c1"># time0 = time()</span>
+<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+    <span class="n">train_model</span><span class="p">(</span><span class="n">nn_deep</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+    <span class="n">train_loss</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_model</span><span class="p">(</span><span class="n">nn_deep</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="s2">&quot;train&quot;</span><span class="p">))</span>
+    <span class="n">validation_loss</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_model</span><span class="p">(</span><span class="n">nn_deep</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;training loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">validation_loss</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;validation loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epochs&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">final_losses</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">validation_loss</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/72f29ea394fc357824959899b713c6c4094ab4bc9df7b27dd25e63142edd3354.png" src="../_images/72f29ea394fc357824959899b713c6c4094ab4bc9df7b27dd25e63142edd3354.png" />
+</div>
+</div>
+<p>Worse loss with twice the training epochs</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">preds_deep</span> <span class="o">=</span> <span class="n">nn_deep</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="p">:])</span><span class="o">.</span><span class="n">double</span><span class="p">())</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds22</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;3 layer Predictions&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">[:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_deep</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;deepNN Predictions&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/2a9ec6e755706008b244e144e7d759464fa47b934303991f09406ba69c3bd00f.png" src="../_images/2a9ec6e755706008b244e144e7d759464fa47b934303991f09406ba69c3bd00f.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Net_deepNN_withSkips</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_deepNN_withSkips</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs, 16 neurons for first hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 16 neurons for second hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear4</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear5</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear6</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+    <span class="c1">#         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x1</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x2</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x1</span><span class="p">))</span>
+        <span class="n">x3</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x2</span><span class="p">))</span> <span class="o">+</span> <span class="n">x1</span>
+        <span class="n">x4</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear4</span><span class="p">(</span><span class="n">x3</span><span class="p">))</span>
+        <span class="n">x5</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear5</span><span class="p">(</span><span class="n">x4</span><span class="p">))</span> <span class="o">+</span> <span class="n">x3</span>
+        <span class="n">x6</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear6</span><span class="p">(</span><span class="n">x5</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x6</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_deep_skips</span> <span class="o">=</span> <span class="n">Net_deepNN_withSkips</span><span class="p">()</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">20</span>  <span class="c1"># Number of epocs</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_deep_skips</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">validation_loss</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
+<span class="n">train_loss</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
+<span class="c1"># time0 = time()</span>
+<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+    <span class="n">train_model</span><span class="p">(</span><span class="n">nn_deep_skips</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+    <span class="n">train_loss</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_model</span><span class="p">(</span><span class="n">nn_deep_skips</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="s2">&quot;train&quot;</span><span class="p">))</span>
+    <span class="n">validation_loss</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_model</span><span class="p">(</span><span class="n">nn_deep_skips</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;training loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">validation_loss</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;validation loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epochs&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="c1"># final_losses=np.append(final_losses,validation_loss[-1])</span>
+<span class="n">final_losses</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">validation_loss</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/2cd7d2726e03d3c9005fa27b25ece861da8cb8883859bb9609c83bbabc9df633.png" src="../_images/2cd7d2726e03d3c9005fa27b25ece861da8cb8883859bb9609c83bbabc9df633.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">preds_deep_skips</span> <span class="o">=</span> <span class="n">nn_deep_skips</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="p">:])</span><span class="o">.</span><span class="n">double</span><span class="p">())</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds22</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;3 layer Predictions&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">[:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_deep</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;deepNN Predictions&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">preds_deep_skips</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;deepNN w/ skips Predictions&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/fb37ae9a498e44eec8b5f0b9a1c872bccd3c5fb8f38e9d546c0473cdda218e0f.png" src="../_images/fb37ae9a498e44eec8b5f0b9a1c872bccd3c5fb8f38e9d546c0473cdda218e0f.png" />
+</div>
+</div>
+<p>Adding skips led to two improvements</p>
+<ul class="simple">
+<li><p>Improved loss, lower than that of deep network without skips and the original 3-layer NN</p></li>
+<li><p>Faster training</p></li>
+</ul>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Net_3L_withSkip</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_3L_withSkip</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs, 16 neurons for first hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 16 neurons for second hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+    <span class="c1">#         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">+</span> <span class="n">x</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_3l_skip</span> <span class="o">=</span> <span class="n">Net_3L_withSkip</span><span class="p">()</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="c1"># optimizer = optim.Adam(nn_3l_res.parameters(), lr=1e-7)</span>
+<span class="c1"># lr_finder = LRFinder(nn_3l_res, optimizer, criterion)</span>
+<span class="c1"># lr_finder.range_test(loader, end_lr=100, num_iter=200)</span>
+<span class="c1"># lr_finder.plot() # to inspect the loss-learning rate graph</span>
+<span class="c1"># lr_finder.reset() # to reset the model and optimizer to their initial state</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">20</span>  <span class="c1"># Number of epocs</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_3l_skip</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">validation_loss</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
+<span class="n">train_loss</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
+<span class="c1"># time0 = time()</span>
+<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+    <span class="n">train_model</span><span class="p">(</span><span class="n">nn_3l_skip</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+    <span class="n">train_loss</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_model</span><span class="p">(</span><span class="n">nn_3l_skip</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="s2">&quot;train&quot;</span><span class="p">))</span>
+    <span class="n">validation_loss</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_model</span><span class="p">(</span><span class="n">nn_3l_skip</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;training loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">validation_loss</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;validation loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epochs&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">final_losses</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">validation_loss</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/b4d0fba179fadf2e0ed8d3e7aa576b623cc185a006220c9da465f089f4e70916.png" src="../_images/b4d0fba179fadf2e0ed8d3e7aa576b623cc185a006220c9da465f089f4e70916.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">preds_3l_skip</span> <span class="o">=</span> <span class="n">nn_3l_skip</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="p">:])</span><span class="o">.</span><span class="n">double</span><span class="p">())</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds22</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;3 layer Predictions&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">[:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_deep</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;deepNN Predictions&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">preds_deep_skips</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;deepNN w/ skips Predictions&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_3l_skip</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;3 layer w/ skip Predictions&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/daaf3a955a9a46aacc9e8bffb7df75a4b9f33525b839972cbe0b6356d9020ed5.png" src="../_images/daaf3a955a9a46aacc9e8bffb7df75a4b9f33525b839972cbe0b6356d9020ed5.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model_names</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;3 layer&quot;</span><span class="p">,</span> <span class="s2">&quot;deep&quot;</span><span class="p">,</span> <span class="s2">&quot;deep w/ skips&quot;</span><span class="p">,</span> <span class="s2">&quot;3 layer w/ skip&quot;</span><span class="p">]</span>
+<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">final_losses</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="n">model_names</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">])</span>
+<span class="n">df</span><span class="p">[</span><span class="s2">&quot;epochs&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">20</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">]</span>
+<span class="n">df</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><div>
+<style scoped>
+    .dataframe tbody tr th:only-of-type {
+        vertical-align: middle;
+    }
+
+    .dataframe tbody tr th {
+        vertical-align: top;
+    }
+
+    .dataframe thead th {
+        text-align: right;
+    }
+</style>
+<table border="1" class="dataframe">
+  <thead>
+    <tr style="text-align: right;">
+      <th></th>
+      <th>loss</th>
+      <th>epochs</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <th>3 layer</th>
+      <td>2.563390</td>
+      <td>20</td>
+    </tr>
+    <tr>
+      <th>deep</th>
+      <td>5.650750</td>
+      <td>40</td>
+    </tr>
+    <tr>
+      <th>deep w/ skips</th>
+      <td>2.264253</td>
+      <td>20</td>
+    </tr>
+    <tr>
+      <th>3 layer w/ skip</th>
+      <td>2.497974</td>
+      <td>20</td>
+    </tr>
+  </tbody>
+</table>
+</div></div></div>
+</div>
+<div class="section" id="main-points">
+<h3>Main points<a class="headerlink" href="#main-points" title="Permalink to this headline">#</a></h3>
+<ul class="simple">
+<li><p>Skip connections are easy to program in pytorch</p></li>
+<li><p>Deeper isn’t always better</p></li>
+<li><p>Residual structure can lower loss, and lead to faster training</p></li>
+<li><p>Even shallow NN can benefit from using a residual block</p></li>
+</ul>
+</div>
+</div>
+<div class="section" id="recurrent-neural-networks-rnn">
+<h2>Recurrent Neural Networks (RNN)<a class="headerlink" href="#recurrent-neural-networks-rnn" title="Permalink to this headline">#</a></h2>
+<p>#What is RNN? &gt;  Recurrent Neural Network.
+The hidden state (hidden layer) is used in the next time step. Useful in operations which involve recurrence.
+In the image below, <span class="math notranslate nohighlight">\(x_t\)</span> is the vector of <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(o_t\)</span> is the vector of tendency terms. The network uses <span class="math notranslate nohighlight">\(x_t\)</span> and hidden state <span class="math notranslate nohighlight">\(s_{t-1}\)</span> as inputs to obtain the tendency term <span class="math notranslate nohighlight">\(o_t\)</span>. A normal fully connected ANN will only use <span class="math notranslate nohighlight">\(x_t\)</span> to obtain <span class="math notranslate nohighlight">\(o_t\)</span> but RNN uses <span class="math notranslate nohighlight">\(s_{t-1}\)</span> in addition to <span class="math notranslate nohighlight">\(x_t\)</span> to get the output, <span class="math notranslate nohighlight">\(o_t\)</span>.</p>
+<div class="figure align-default">
+<a class="reference internal image-reference" href="https://upload.wikimedia.org/wikipedia/commons/0/05/RNN.png"><img alt="https://upload.wikimedia.org/wikipedia/commons/0/05/RNN.png" src="https://upload.wikimedia.org/wikipedia/commons/0/05/RNN.png" style="width: 600px;" /></a>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">### Applying RNN to L96 model:</span>
+
+<span class="c1"># preparing data for training</span>
+
+<span class="n">x1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">(</span><span class="n">X_true</span><span class="p">)</span>  <span class="c1"># xk&#39;s w.r.t to time</span>
+<span class="n">x2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">(</span><span class="n">xy_true</span><span class="p">)</span>  <span class="c1"># tendency terms</span>
+
+
+<span class="k">class</span> <span class="nc">L96_RNN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">in_n</span><span class="p">,</span> <span class="n">hid</span><span class="p">,</span> <span class="n">out_n</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">L96_RNN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">h_size</span> <span class="o">=</span> <span class="n">hid</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span>
+            <span class="n">in_n</span> <span class="o">+</span> <span class="n">hid</span><span class="p">,</span> <span class="n">hid</span>
+        <span class="p">)</span>  <span class="c1"># note that X_k + hidden state is the input</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">hid</span><span class="p">,</span> <span class="n">out_n</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">hidden</span><span class="p">):</span>
+        <span class="c1"># forward code for RNN is different than ANN. Hidden state is also given as input to forward function</span>
+        <span class="n">x2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">hidden</span><span class="p">),</span> <span class="mi">1</span><span class="p">))</span>  <span class="c1"># input layer to 1st layer</span>
+        <span class="n">x3</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span>
+            <span class="n">x2</span>
+        <span class="p">)</span>  <span class="c1"># x2 is the hidden state which will be used as input in the next iteration</span>
+        <span class="n">x4</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x3</span><span class="p">)</span>  <span class="c1"># x4 is the output layer, tendency terms in L96 case.</span>
+
+        <span class="k">return</span> <span class="p">(</span>
+            <span class="n">x4</span><span class="p">,</span>
+            <span class="n">x3</span><span class="p">,</span>
+        <span class="p">)</span>  <span class="c1"># x4 is the output layer, and x3 is the hidden layer returned for using in the next iteration</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">### deletes the model if it already exists</span>
+<span class="k">try</span><span class="p">:</span>
+    <span class="k">del</span> <span class="n">model</span>
+<span class="k">except</span><span class="p">:</span>
+    <span class="k">pass</span>
+<span class="c1">###</span>
+
+
+<span class="n">m</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c1"># no. of time steps to use in RNN. untested, code might break if changed.</span>
+
+<span class="n">in_n</span><span class="p">,</span> <span class="n">hid</span><span class="p">,</span> <span class="n">out_n</span> <span class="o">=</span> <span class="p">(</span>
+    <span class="mi">8</span><span class="p">,</span>
+    <span class="mi">8</span><span class="p">,</span>
+    <span class="mi">8</span><span class="p">,</span>
+<span class="p">)</span>  <span class="c1"># X_k, k=1 to 8 and corresponding tendency terms. hid layer nodes can be chaned</span>
+
+<span class="n">model</span> <span class="o">=</span> <span class="n">L96_RNN</span><span class="p">(</span><span class="n">in_n</span><span class="p">,</span> <span class="n">hid</span><span class="p">,</span> <span class="n">out_n</span><span class="p">)</span>  <span class="c1"># model initialize</span>
+
+<span class="n">l_r</span> <span class="o">=</span> <span class="mf">1e-03</span>  <span class="c1"># learning rate</span>
+
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="n">l_r</span><span class="p">)</span>  <span class="c1"># using Adam optimiser</span>
+<span class="n">loss_fn</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">(</span><span class="n">reduction</span><span class="o">=</span><span class="s2">&quot;mean&quot;</span><span class="p">)</span>  <span class="c1"># loss function</span>
+
+<span class="n">epochs</span> <span class="o">=</span> <span class="mi">20</span>  <span class="c1"># epochs</span>
+
+<span class="n">loss_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="n">epochs</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>  <span class="c1"># to store loss for plotting purposes</span>
+
+<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">epochs</span><span class="p">):</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">time_steps</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+        <span class="n">hidd</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">h_size</span><span class="p">)))</span>
+        <span class="n">inp</span> <span class="o">=</span> <span class="n">x1</span><span class="p">[</span><span class="n">i</span> <span class="p">:</span> <span class="n">i</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>
+        <span class="n">outp</span> <span class="o">=</span> <span class="n">x2</span><span class="p">[</span><span class="n">i</span> <span class="p">:</span> <span class="n">i</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>
+        <span class="n">pred</span><span class="p">,</span> <span class="n">hidd</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">forward</span><span class="p">(</span><span class="n">inp</span><span class="p">,</span> <span class="n">hidd</span><span class="p">)</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">loss_fn</span><span class="p">(</span><span class="n">pred</span><span class="p">,</span> <span class="n">outp</span><span class="p">)</span>
+        <span class="c1"># loss.backward()</span>
+        <span class="n">model</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>
+        <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+        <span class="c1"># loss.backward(retain_graph=True)</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>
+
+    <span class="n">loss_array</span><span class="p">[</span><span class="n">p</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span>
+    <span class="n">loss_array</span><span class="p">[</span><span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">loss</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+    <span class="c1"># print(&#39;epoch no. &gt;&#39;,p, &#39;loss &gt;&#39;,loss.detach().numpy())</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">loss_array</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epochs&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/fb085bc3408cc7325c740add41f3deb3c25b00b8826de3d8d9c3dce4844b9a5f.png" src="../_images/fb085bc3408cc7325c740add41f3deb3c25b00b8826de3d8d9c3dce4844b9a5f.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># checking tendency term output from L96 RNN:</span>
+
+<span class="n">hidd</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">h_size</span><span class="p">))</span>
+<span class="p">)</span>  <span class="c1"># hidden state needs to be initialized for the 1st iteration</span>
+
+<span class="n">pred</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="n">time_steps</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>  <span class="c1"># tendency prediction</span>
+
+<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">time_steps</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+    <span class="n">inp</span> <span class="o">=</span> <span class="n">x1</span><span class="p">[</span><span class="n">j</span> <span class="p">:</span> <span class="n">j</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>
+    <span class="n">pred</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="p">:],</span> <span class="n">hidd</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">forward</span><span class="p">(</span><span class="n">inp</span><span class="p">,</span> <span class="n">hidd</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">l</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c1"># k value. l=k. l goes from 1 to 8 for this example</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">x2</span><span class="p">[:,</span> <span class="n">l</span><span class="p">],</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;actual&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">pred</span><span class="p">[:,</span> <span class="n">l</span><span class="p">]</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">(),</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;RNN&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>  <span class="c1"># looking only till t=10</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;tendency&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/f4164bd788faf6e0ea54371b190a3d89dc1d3b9dcc40c34c46586060f18abd8f.png" src="../_images/f4164bd788faf6e0ea54371b190a3d89dc1d3b9dcc40c34c46586060f18abd8f.png" />
+</div>
+</div>
+</div>
+</div>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
+        codeMirrorConfig: {
+            theme: "abcdef",
+            mode: "python"
+        },
+        kernelOptions: {
+            name: "python3",
+            path: "./notebooks"
+        },
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
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+    <h1>Using Neural Networks for L96 Parameterization: Offline Training</h1>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generating-the-data">Generating the data</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#build-the-real-world-to-generate-the-ground-truth-dataset">Build the <em>Real World</em> to Generate the Ground Truth Dataset</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#run-the-simulation-to-generate-data">Run the simulation to generate data</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#split-the-data-into-training-and-testing-set">Split the Data into Training and Testing Set</a></li>
+</ul>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-dataloaders">The Dataloaders</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-local-ml-models">The <em>local</em> ML models</a><ul class="nav section-nav flex-column">
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fully-connected-neural-network-fcnn">Fully connected neural network (FCNN)</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-loss-function">The Loss Function</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-optimizer">The Optimizer</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#combining-it-all-together-training-the-network">Combining it all Together: Training the network</a><ul class="nav section-nav flex-column">
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#compare-predictions-with-ground-truth">Compare Predictions with Ground Truth</a></li>
+</ul>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-non-local-ml-models">The <em>non-local</em> ML models</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#save-the-networks">Save the networks</a></li>
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+<h1>Using Neural Networks for L96 Parameterization: Offline Training<a class="headerlink" href="#using-neural-networks-for-l96-parameterization-offline-training" title="Permalink to this headline">#</a></h1>
+<div class="section" id="outline">
+<h2>Outline:<a class="headerlink" href="#outline" title="Permalink to this headline">#</a></h2>
+<p>In this notebook, we’ll apply the concepts learned in the <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/gradient_decent.html">previous notebook</a> to the parameterization task in L96 model, which was highlighted <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/gcm-analogue.ipynb">here</a> and <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/gcm-parameterization-problem.ipynb">here</a>.</p>
+<p>As a succinct reminder, the parameterization task requires us to represent the impact of the fast scales on the slow scales as a function of the slow scales. Machine learning provides us a way to learn complex non-linear functional dependecies from a given data set. In this notebook we will show :</p>
+<ul class="simple">
+<li><p>how the data to train a machine learning model is generated.</p></li>
+<li><p>how a ML model can be trained to learn functional dependence in this data.</p></li>
+</ul>
+</div>
+<div class="section" id="generating-the-data">
+<h2>Generating the data<a class="headerlink" href="#generating-the-data" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="kn">import</span> <span class="nn">time</span>
+<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">import</span> <span class="nn">torch.utils.data</span> <span class="k">as</span> <span class="nn">Data</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">RK2</span><span class="p">,</span> <span class="n">RK4</span><span class="p">,</span> <span class="n">EulerFwd</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span><span class="p">,</span> <span class="n">integrate_L96_2t</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Ensuring reproducibility</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">14</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="build-the-real-world-to-generate-the-ground-truth-dataset">
+<h3>Build the <em>Real World</em> to Generate the Ground Truth Dataset<a class="headerlink" href="#build-the-real-world-to-generate-the-ground-truth-dataset" title="Permalink to this headline">#</a></h3>
+<p>Remeber from the <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/gcm-analogue.ipynb">notebook</a> that described the parameterization problem, the impact of the fast time scales on the slow time scales is captured by the term:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-1d6d716f-2e4d-4456-a87b-84435a1df5a1">
+<span class="eqno">(8)<a class="headerlink" href="#equation-1d6d716f-2e4d-4456-a87b-84435a1df5a1" title="Permalink to this equation">#</a></span>\[\begin{align}
+\underbrace{ 
+\left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}
+}_{=U_k}.
+\end{align}\]</div>
+<p>The goal of the parameterization is to express <span class="math notranslate nohighlight">\(U_k\)</span> purely as a function of the large scale variables <span class="math notranslate nohighlight">\(P(X_k)\)</span>. In the previous <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/gcm-analogue.ipynb">notebook</a> this was done using simple polynomial functions with specified form, but here we will estimate this function using a more general function - neural network.</p>
+<p>First we need to run a two time-scale model and generate data for <span class="math notranslate nohighlight">\(U_k\)</span> and <span class="math notranslate nohighlight">\(X_k\)</span>.
+We initialise the L96 two time-scale model using <span class="math notranslate nohighlight">\(K\)</span> (set to 8) values of <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(J\)</span> (set to 32) values of <span class="math notranslate nohighlight">\(Y\)</span> for each <span class="math notranslate nohighlight">\(X\)</span>. The model is run for 20,000 timesteps to generate the dataset for the neural network.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">time_steps</span> <span class="o">=</span> <span class="mi">20000</span>
+<span class="n">forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span> <span class="o">*</span> <span class="n">time_steps</span>
+
+<span class="c1"># Create a &quot;real world&quot; with K=8 and J=32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">forcing</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="run-the-simulation-to-generate-data">
+<h3>Run the simulation to generate data<a class="headerlink" href="#run-the-simulation-to-generate-data" title="Permalink to this headline">#</a></h3>
+<p>Using the <em>real world</em> model created above we generate the training data (input and output pairs) for the neural network by running the simulation and writing out the large scale state and subgrid tendencies.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># The effect of Y on X is `xy_true`</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">xy_true</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># Change the data type to `float32` in order to avoid doing type conversions later on</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">xy_true</span> <span class="o">=</span> <span class="n">X_true</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">),</span> <span class="n">xy_true</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="split-the-data-into-training-and-testing-set">
+<h3>Split the Data into Training and Testing Set<a class="headerlink" href="#split-the-data-into-training-and-testing-set" title="Permalink to this headline">#</a></h3>
+<p>We split the data into 80% for training and 20% for testing. In more conventional ML tasks one may split the data into training, validation and testing sets, but for simplicity we do not do this here.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Number of time steps for validation</span>
+<span class="n">val_size</span> <span class="o">=</span> <span class="mi">4000</span>
+
+<span class="c1"># Training Data</span>
+<span class="n">X_true_train</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span>
+    <span class="p">:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:</span>
+<span class="p">]</span>  <span class="c1"># Flatten because we first use single input as a sample</span>
+<span class="n">subgrid_tend_train</span> <span class="o">=</span> <span class="n">xy_true</span><span class="p">[:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="c1"># Test Data</span>
+<span class="n">X_true_test</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+<span class="n">subgrid_tend_test</span> <span class="o">=</span> <span class="n">xy_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_true_train</span><span class="o">.</span><span class="n">shape</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(16001, 8)
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="the-dataloaders">
+<h2>The Dataloaders<a class="headerlink" href="#the-dataloaders" title="Permalink to this headline">#</a></h2>
+<p>In ML work flows it is common practice to create ways to access the data that allow for easily iterating over the whole data set. This is done because usually the datasets are very large, and breaking the data up into smaller chunks (batches) that can be loaded into memory (GPU or CPU) one at a time allow us to train using the whole dataset (see more details <a class="reference external" href="https://machinelearningmastery.com/gentle-introduction-mini-batch-gradient-descent-configure-batch-size/">here</a> and <a class="reference external" href="https://pytorch.org/tutorials/beginner/data_loading_tutorial.html">here</a>).</p>
+<p><code class="docutils literal notranslate"><span class="pre">Dataset</span></code> and <code class="docutils literal notranslate"><span class="pre">Dataloader</span></code> classes in PyTorch provide a very convenient way of iterating over a dataset while training a deep learning model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Number of sample in each batch</span>
+<span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">2000</span>
+<span class="c1"># this batch size would result in 2 test batches and 8 training batches.</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Define the X (state), Y (subgrid tendency) pairs for the linear regression local network.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">local_dataset</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true_train</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">subgrid_tend_train</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span>
+<span class="p">)</span>
+
+<span class="n">local_loader</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">local_dataset</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Define the dataloader for the test set.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">local_dataset_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span>
+<span class="p">)</span>
+
+<span class="n">local_loader_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">local_dataset_test</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Display a batch of samples from the dataset.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Iterating over the data to get one batch</span>
+<span class="n">data_iterator</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">local_loader</span><span class="p">)</span>
+<span class="n">X_iter</span><span class="p">,</span> <span class="n">subgrid_tend_iter</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">data_iterator</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;X (State):</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">,</span> <span class="n">X_iter</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">Y (Subgrid Tendency):</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">,</span> <span class="n">subgrid_tend_iter</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_iter</span><span class="p">,</span> <span class="n">subgrid_tend_iter</span><span class="p">,</span> <span class="s2">&quot;.&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;State - X&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Subgrid tendency - U&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">16</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>X (State):
+ tensor([12.9715,  5.3797,  0.8659,  ...,  8.3222, -2.6947,  4.2560])
+
+Y (Subgrid Tendency):
+ tensor([-7.5589, -6.6147, -0.8452,  ..., -7.2965,  4.5814, -7.6241])
+</pre></div>
+</div>
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(-12.0, 16.0)
+</pre></div>
+</div>
+<img alt="../_images/7ebe103e74036aece384205d5ebdce085cc88785a43ade68338481c6ce49ffa5.png" src="../_images/7ebe103e74036aece384205d5ebdce085cc88785a43ade68338481c6ce49ffa5.png" />
+</div>
+</div>
+<p>Clearly there is a relationship between <span class="math notranslate nohighlight">\(U_k\)</span> and <span class="math notranslate nohighlight">\(X_k\)</span>, we plan to learn this with the help of ML models.</p>
+</div>
+<div class="section" id="the-local-ml-models">
+<h2>The <em>local</em> ML models<a class="headerlink" href="#the-local-ml-models" title="Permalink to this headline">#</a></h2>
+<p>Here we show how a few different machine learning models can be setup. These models will predict the <span class="math notranslate nohighlight">\(U_k\)</span> at a specific <span class="math notranslate nohighlight">\(k\)</span> using the <span class="math notranslate nohighlight">\(X_k\)</span> as the same <span class="math notranslate nohighlight">\(k\)</span>, and so may be considered local. Later we will consider non-local models, which predict <span class="math notranslate nohighlight">\(U_k\)</span> at specific <span class="math notranslate nohighlight">\(k\)</span> using <span class="math notranslate nohighlight">\(X_k\)</span> at multiple locations (values of <span class="math notranslate nohighlight">\(k\)</span>).</p>
+<p>We will consider a linear regression network and a non-linear fully connected neural network (FCNN) here.</p>
+<div class="section" id="linear-regression-network">
+<h3>Linear regression network<a class="headerlink" href="#linear-regression-network" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">LinearRegression</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># A single input and a single output</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="c1"># This method is automatically executed when</span>
+        <span class="c1"># we call a object of this class</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linear_network</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
+<span class="n">linear_network</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>LinearRegression(
+  (linear1): Linear(in_features=1, out_features=1, bias=True)
+)
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="fully-connected-neural-network-fcnn">
+<h3>Fully connected neural network (FCNN)<a class="headerlink" href="#fully-connected-neural-network-fcnn" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">FCNN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fcnn_network</span> <span class="o">=</span> <span class="n">FCNN</span><span class="p">()</span>
+<span class="n">fcnn_network</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>FCNN(
+  (linear1): Linear(in_features=1, out_features=16, bias=True)
+  (linear2): Linear(in_features=16, out_features=16, bias=True)
+  (linear3): Linear(in_features=16, out_features=1, bias=True)
+  (relu): ReLU()
+)
+</pre></div>
+</div>
+</div>
+</div>
+<p>We do not expect the linear regression to do well in this problem, as the dependence seenin the figure is clearly more complex than a simple line. We include it here as a baseline.</p>
+<p>For the FCNN the non-linearity is introduced by adding the activation function, which is chosen to be ReLU in this case. The number of layers and width of layers are hyperparameters, which the practitioner needs to set based on some trial and error (or systematic hyperparameter sweeps). The goal is to make the network as complex as needed but no more.</p>
+<div class="admonition-need-for-activation-functions admonition">
+<p class="admonition-title">Need for Activation Functions</p>
+<p><strong>If layers of a neural network contain only fully-connected layers (matrix multiplications), everything would be linear.</strong></p>
+<p>For example, if we have an input <span class="math notranslate nohighlight">\(x\)</span> along with 2 layers of weight matrices <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> then the neural network would compute the output as <span class="math notranslate nohighlight">\(A(Bx)\)</span>, which is linear (in <span class="math notranslate nohighlight">\(x\)</span>). Thus, in order to introduce some non-linearity we use activation functions.</p>
+<p>Now the same neural network as above with an activation function <span class="math notranslate nohighlight">\(\phi\)</span> would compute the output as <span class="math notranslate nohighlight">\(A(\phi(Bx))\)</span>.</p>
+</div>
+<div class="admonition-relu-activation-function admonition">
+<p class="admonition-title">ReLU Activation Function</p>
+<div class="figure align-default" id="relu-activation">
+<img alt="../_images/relu_activation_function.png" src="../_images/relu_activation_function.png" />
+</div>
+<p>The <code class="docutils literal notranslate"><span class="pre">ReLU</span></code> activation function is just a <span class="math notranslate nohighlight">\(max(0,X)\)</span> function (The image is referenced from <a class="reference external" href="https://analyticsindiamag.com/most-common-activation-functions-in-neural-networks-and-rationale-behind-it/">this blog</a>). Even a function as simple as this enables a typical NN to be a nonlinear function of the inputs!</p>
+</div>
+<p><strong>Test forward function</strong><br />
+Below we test to see if the models can return an output given an input. Since at this point the models have not been trained (parameters have not been optimized) the resulting output is just a random number.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">net_input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">out_linear</span> <span class="o">=</span> <span class="n">linear_network</span><span class="p">(</span><span class="n">net_input</span><span class="p">)</span>
+<span class="n">out_fcnn</span> <span class="o">=</span> <span class="n">fcnn_network</span><span class="p">(</span><span class="n">net_input</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span>
+    <span class="sa">f</span><span class="s2">&quot;The output of the random input from the linear network is: </span><span class="si">{</span><span class="n">out_linear</span><span class="o">.</span><span class="n">item</span><span class="p">()</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span>
+<span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The output of the random input from the fcnn is: </span><span class="si">{</span><span class="n">out_fcnn</span><span class="o">.</span><span class="n">item</span><span class="p">()</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The output of the random input from the linear network is: 0.6956
+The output of the random input from the fcnn is: -0.0326
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="the-loss-function">
+<h2>The Loss Function<a class="headerlink" href="#the-loss-function" title="Permalink to this headline">#</a></h2>
+<p>Apart from the architecutre of the model, the other major component of any machine learning task is the loss function. This is basically some quantitative way to test the skill of the model - whether the model does a good job or not at estimating the output. During the optimization of the weights, the weights are changed such that the model skill is improved.</p>
+<p>For our task, we choose the <em>Mean Squared Error</em> metric as our loss function. However, other loss functions may be used or need to be adapted to the problem.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># MSE loss function</span>
+<span class="n">loss_fn</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Load the input and output pair from the data loader</span>
+<span class="n">X_tmp</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="nb">iter</span><span class="p">(</span><span class="n">local_loader</span><span class="p">))</span>
+
+<span class="c1"># Predict the output</span>
+<span class="n">y_tmp_linear</span> <span class="o">=</span> <span class="n">linear_network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_tmp</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">))</span>
+<span class="n">y_tmp_fcnn</span> <span class="o">=</span> <span class="n">fcnn_network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_tmp</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">))</span>
+
+<span class="c1"># Calculate the MSE loss</span>
+<span class="n">loss_linear</span> <span class="o">=</span> <span class="n">loss_fn</span><span class="p">(</span><span class="n">y_tmp_linear</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_tmp</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Linear network MSE Loss: </span><span class="si">{</span><span class="n">loss_linear</span><span class="o">.</span><span class="n">item</span><span class="p">()</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+
+<span class="n">loss_fcnn</span> <span class="o">=</span> <span class="n">loss_fn</span><span class="p">(</span><span class="n">y_tmp_fcnn</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_tmp</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;FCNN MSE Loss: </span><span class="si">{</span><span class="n">loss_fcnn</span><span class="o">.</span><span class="n">item</span><span class="p">()</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Linear network MSE Loss: 97.5671
+FCNN MSE Loss: 31.5754
+</pre></div>
+</div>
+</div>
+</div>
+<p>Since at this point neither of the networks have been trained, the resultant loss is just arbitrary. If one model is doing better than the other, it is just a result of random chance.</p>
+</div>
+<div class="section" id="the-optimizer">
+<h2>The Optimizer<a class="headerlink" href="#the-optimizer" title="Permalink to this headline">#</a></h2>
+<p>Now in order to make the network learn (optimize the parameters), we need an algorithm that will update its weights depending on the loss function. This is achieved by using an optimizer. The implementation of many optimizers can be found in PyTorch. The choice of which optimizer we choose might be very important as it will determine how fast the network will be able to learn.</p>
+<p>In the example below, we show one of the popular optimizers <code class="docutils literal notranslate"><span class="pre">Adam</span></code>. The main hyperparameter in this optimizer is the <strong>learning rate</strong>.</p>
+<p>Another popular optimizer that may be useful in some situations is Stochastic Gradient Descent or <code class="docutils literal notranslate"><span class="pre">SGD</span></code>.
+The SGD optimizer consists of two major hyperparameters called the <strong>learning rate</strong> and <strong>momentum</strong>. The <strong>learning rate</strong> determines the magnitude with which the weights of the network update thus making it crucial to choose the correct learning rate (<span class="math notranslate nohighlight">\(LR\)</span>) otherwise the network will either fail to train, or take much longer to converge. <strong>Momentum</strong> determines how much of the loss function gradients are determined based on the past iteration vs the latest iteration.
+To read about momentum, check out this <a class="reference external" href="https://towardsdatascience.com/stochastic-gradient-descent-with-momentum-a84097641a5d">blog post</a>.</p>
+<p>The primary task of the optimizer is to update the parameters in little steps such that the loss function is reduced. In equations, the  effective value of the gradient <span class="math notranslate nohighlight">\(V\)</span> (direction in which loss function reduces) at step <span class="math notranslate nohighlight">\(t\)</span> in SGD with momentum (<span class="math notranslate nohighlight">\(\beta\)</span>) is determined by</p>
+<div class="amsmath math notranslate nohighlight" id="equation-7e5ee783-6067-426c-ad2e-638a46403d4b">
+<span class="eqno">(9)<a class="headerlink" href="#equation-7e5ee783-6067-426c-ad2e-638a46403d4b" title="Permalink to this equation">#</a></span>\[\begin{equation}
+V_t = \beta V_{t-1} + (1-\beta) \nabla_w L(W,X,y)
+\end{equation}\]</div>
+<p>and the updates to the parameters will be</p>
+<div class="amsmath math notranslate nohighlight" id="equation-cabaefa7-a3a1-448a-a975-64e35007209e">
+<span class="eqno">(10)<a class="headerlink" href="#equation-cabaefa7-a3a1-448a-a975-64e35007209e" title="Permalink to this equation">#</a></span>\[\begin{equation}
+w^{new} = w^{old} - LR * V_t
+\end{equation}\]</div>
+<p>See in the code sample below how taking a step with the optimizer changes the model parameters. In what comes next we will take these steps to reach a parameter state such that the loss function is as small as possible.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">learning_rate</span> <span class="o">=</span> <span class="mf">0.003</span>
+<span class="n">momentum</span> <span class="o">=</span> <span class="mf">0.1</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># switch the commenting out below to try a different optimizer.</span>
+<span class="c1"># optimizer_linear = optim.SGD(linear_network.parameters(), lr=learning_rate, momentum=momentum)</span>
+<span class="n">optimizer_linear</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">linear_network</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="n">learning_rate</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Before backward pass: </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">linear_network</span><span class="o">.</span><span class="n">parameters</span><span class="p">())[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+
+<span class="n">loss_linear</span><span class="o">.</span><span class="n">backward</span><span class="p">(</span><span class="n">retain_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">optimizer_linear</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">After backward pass: </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">linear_network</span><span class="o">.</span><span class="n">parameters</span><span class="p">())[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Before backward pass: 
+ [[0.6589745]]
+
+After backward pass: 
+ [[0.6559745]]
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># optimizer_fcnn = optim.SGD(fcnn_network.parameters(), lr=learning_rate, momentum=momentum)</span>
+<span class="n">optimizer_fcnn</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">fcnn_network</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="n">learning_rate</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="combining-it-all-together-training-the-network">
+<h2>Combining it all Together: Training the network<a class="headerlink" href="#combining-it-all-together-training-the-network" title="Permalink to this headline">#</a></h2>
+<div class="section" id="define-the-training-and-test-functions">
+<h3>Define the Training and Test Functions<a class="headerlink" href="#define-the-training-and-test-functions" title="Permalink to this headline">#</a></h3>
+<p>The <code class="docutils literal notranslate"><span class="pre">train_model</span></code> function update the network weight such that the loss is reduced. A single call to this function iterates over all the batches in the dataset, and this single iteration is called an epoch. To do its job this function needs access to the model that needs to be trained, the criterion or the loss function that determines the skill of the model, the data loader that allows the model to access the data, and the optimizer that appropriately adjusts the weights to reduce the loss. Also notice that the model weights are adjusted multiple times in a single epoch - everytime a batch is processed.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train the network for one epoch&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
+
+    <span class="n">train_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+        <span class="c1"># Get predictions</span>
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="c1"># This if block is needed to add a dummy dimension if our inputs are 1D</span>
+            <span class="c1"># (where each number is a different sample)</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">network</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>
+
+        <span class="c1"># Compute the loss</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span>
+        <span class="n">train_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Clear the gradients</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>
+
+        <span class="c1"># Backpropagation to compute the gradients and update the weights</span>
+        <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>
+
+    <span class="k">return</span> <span class="n">train_loss</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>The <code class="docutils literal notranslate"><span class="pre">test_model</span></code> function is meant to test the skill of the model, but does not update the weights to further improve it.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Test the network&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>  <span class="c1"># Evaluation mode (important when having dropout layers)</span>
+
+    <span class="n">test_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+            <span class="c1"># Get predictions</span>
+            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="c1"># This if block is needed to add a dummy dimension if our inputs are 1D</span>
+                <span class="c1"># (where each number is a different sample)</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">network</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>
+
+            <span class="c1"># Compute the loss</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span>
+            <span class="n">test_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Get an average loss for the entire dataset</span>
+        <span class="n">test_loss</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">test_loss</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>The <code class="docutils literal notranslate"><span class="pre">fit_model</span></code> iterates the <code class="docutils literal notranslate"><span class="pre">train_model</span></code> function over multiple epochs, such that the loss is reduced.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">fit_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">test_loader</span><span class="p">,</span> <span class="n">n_epochs</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train and validate the network&quot;&quot;&quot;</span>
+    <span class="n">train_losses</span><span class="p">,</span> <span class="n">test_losses</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
+    <span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+        <span class="n">train_loss</span> <span class="o">=</span> <span class="n">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+        <span class="n">test_loss</span> <span class="o">=</span> <span class="n">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">test_loader</span><span class="p">)</span>
+        <span class="n">train_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">train_loss</span><span class="p">)</span>
+        <span class="n">test_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_loss</span><span class="p">)</span>
+    <span class="n">end_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Training completed in </span><span class="si">{</span><span class="nb">int</span><span class="p">(</span><span class="n">end_time</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">start_time</span><span class="p">)</span><span class="si">}</span><span class="s2"> seconds.&quot;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">train_losses</span><span class="p">,</span> <span class="n">test_losses</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="train-the-networks">
+<h3>Train the networks<a class="headerlink" href="#train-the-networks" title="Permalink to this headline">#</a></h3>
+<p>Finally we have all the components ready and we can train our models.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Epochs refer to the number of times we iterate over the entire training data during training.</span>
+<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">15</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">train_loss_linear</span><span class="p">,</span> <span class="n">test_loss_linear</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">linear_network</span><span class="p">,</span> <span class="n">loss_fn</span><span class="p">,</span> <span class="n">optimizer_linear</span><span class="p">,</span> <span class="n">local_loader</span><span class="p">,</span> <span class="n">local_loader_test</span><span class="p">,</span> <span class="n">n_epochs</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 14 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">train_loss_fcnn</span><span class="p">,</span> <span class="n">test_loss_fcnn</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">fcnn_network</span><span class="p">,</span> <span class="n">loss_fn</span><span class="p">,</span> <span class="n">optimizer_fcnn</span><span class="p">,</span> <span class="n">local_loader</span><span class="p">,</span> <span class="n">local_loader_test</span><span class="p">,</span> <span class="n">n_epochs</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 15 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_linear</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Linear train loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">test_loss_linear</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Linear test loss&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_fcnn</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;FCNN train loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">test_loss_fcnn</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;FCNN test loss&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Iteration&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Loss vs Iteration&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/d229e9791f1ae649b95ba0fb6539035ed9f85603926960e70b3cd4d4da7ec157.png" src="../_images/d229e9791f1ae649b95ba0fb6539035ed9f85603926960e70b3cd4d4da7ec157.png" />
+</div>
+</div>
+</div>
+<div class="section" id="compare-predictions-with-ground-truth">
+<h3>Compare Predictions with Ground Truth<a class="headerlink" href="#compare-predictions-with-ground-truth" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">predictions_linear</span> <span class="o">=</span> <span class="n">linear_network</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">1</span><span class="p">)</span>
+<span class="p">)</span>
+<span class="n">predictions_fcnn</span> <span class="o">=</span> <span class="n">fcnn_network</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">1</span><span class="p">)</span>
+<span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">predictions_linear</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted from linear model&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">predictions_fcnn</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted from FCNN model&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">[:</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True Values&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/0ef505f397f52fcffe105be5d948c3a4039842efcb63a681814903a4b5b5b6d6.png" src="../_images/0ef505f397f52fcffe105be5d948c3a4039842efcb63a681814903a4b5b5b6d6.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_points</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">31</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span>
+
+<span class="n">linear_pred</span> <span class="o">=</span> <span class="n">linear_network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_points</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+
+<span class="n">fcnn_pred</span> <span class="o">=</span> <span class="n">fcnn_network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_points</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xy_true</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">bins</span><span class="o">=</span><span class="mi">91</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;Reds&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_points</span><span class="p">,</span> <span class="n">linear_pred</span><span class="p">,</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Linear predictions&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_points</span><span class="p">,</span> <span class="n">fcnn_pred</span><span class="p">,</span> <span class="s2">&quot;-&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;FCNN predictions&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;g&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">16</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;State - X&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Subgrid tendency - U&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/f6776f2d7c25b40143730973d5213ea990b3df28f2e54f3b6b5f3d49dd2e2c8e.png" src="../_images/f6776f2d7c25b40143730973d5213ea990b3df28f2e54f3b6b5f3d49dd2e2c8e.png" />
+</div>
+</div>
+<p>As seen above, the FCNN does a better job at learning the relationship between the <span class="math notranslate nohighlight">\(X_k\)</span> and <span class="math notranslate nohighlight">\(U_k\)</span>, when compared to the linear model.
+Can we do even better?</p>
+</div>
+</div>
+<div class="section" id="the-non-local-ml-models">
+<h2>The <em>non-local</em> ML models<a class="headerlink" href="#the-non-local-ml-models" title="Permalink to this headline">#</a></h2>
+<p>It is possible that sub-grid tendency at <span class="math notranslate nohighlight">\(k\)</span> depends on the neighbouring points (as shown in figure below), or points farther away.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">131</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xy_true</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span>
+    <span class="n">bins</span><span class="o">=</span><span class="mi">91</span><span class="p">,</span>
+    <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;Reds&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">16</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;State - $X_{k-1}$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Subgrid tendency - $U_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">132</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xy_true</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">bins</span><span class="o">=</span><span class="mi">91</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;Reds&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">16</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;State - $X_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Subgrid tendency - $U_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">133</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xy_true</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span>
+    <span class="n">bins</span><span class="o">=</span><span class="mi">91</span><span class="p">,</span>
+    <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;Reds&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">16</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;State - $X_{k+1}$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Subgrid tendency - $U_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/e5920a101289f5fe2b91d01168903234482de51ec3662fa47592fe0f6e92d2d4.png" src="../_images/e5920a101289f5fe2b91d01168903234482de51ec3662fa47592fe0f6e92d2d4.png" />
+</div>
+</div>
+<p>So to make more accurate model, one may want to build a model that learns relationship between not only the specific k point but also other k points.</p>
+<p>We will do this with the help of a non-local model, which uses all <span class="math notranslate nohighlight">\(X_k\)</span>s to predict the <span class="math notranslate nohighlight">\(U_k\)</span> at all 8 <span class="math notranslate nohighlight">\(k\)</span>s.</p>
+<p>To build this model, we need to change two things:</p>
+<ul class="simple">
+<li><p>The data sets need to be slightly altered, such that each sample for the ML model will be composed of all eight k points.</p></li>
+<li><p>The architecture of the neural network will have to be altered to take 8 inputs and produce 8 outputs.</p></li>
+</ul>
+<p>but apart from these changes, the rest of the training process is the same.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Training Dataset</span>
+<span class="c1"># ----------------</span>
+<span class="n">nlocal_data_train</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_train</span><span class="p">),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_tend_train</span><span class="p">),</span>
+<span class="p">)</span>
+<span class="n">nlocal_loader_train</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">nlocal_data_train</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span>
+<span class="p">)</span>
+
+
+<span class="c1"># Test Dataset</span>
+<span class="c1"># ------------</span>
+<span class="n">nlocal_data_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">)</span>
+<span class="p">)</span>
+<span class="n">nlocal_loader_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">nlocal_data_test</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">NonLocal_FCNN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nonlocal_fcnn_network</span> <span class="o">=</span> <span class="n">NonLocal_FCNN</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># optimizer_nonlocal_fcnn = optim.SGD(nonlocal_fcnn_network.parameters(),</span>
+<span class="c1">#                                    lr=learning_rate, momentum=momentum)</span>
+<span class="n">optimizer_nonlocal_fcnn</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span>
+    <span class="n">nonlocal_fcnn_network</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="n">learning_rate</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">120</span>
+<span class="n">train_loss_nonlocal</span><span class="p">,</span> <span class="n">test_loss_nonlocal</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">nonlocal_fcnn_network</span><span class="p">,</span>
+    <span class="n">loss_fn</span><span class="p">,</span>
+    <span class="n">optimizer_nonlocal_fcnn</span><span class="p">,</span>
+    <span class="n">nlocal_loader_train</span><span class="p">,</span>
+    <span class="n">nlocal_loader_test</span><span class="p">,</span>
+    <span class="n">n_epochs</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 14 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_nonlocal</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Non-local model train loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">test_loss_nonlocal</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Non-local model test loss&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_fcnn</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local FCNN train loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">test_loss_fcnn</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local FCNN test loss&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Epochs&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Loss vs Epochs&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/349b9651104a0d38271d4f642559a2a85003812916cbac695b15f8784fbff2bb.png" src="../_images/349b9651104a0d38271d4f642559a2a85003812916cbac695b15f8784fbff2bb.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">k_loc</span> <span class="o">=</span> <span class="mi">1</span>
+<span class="n">predictions_linear</span> <span class="o">=</span> <span class="n">linear_network</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="n">k_loc</span><span class="p">],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">1</span><span class="p">)</span>
+<span class="p">)</span>
+<span class="n">predictions_fcnn</span> <span class="o">=</span> <span class="n">fcnn_network</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="n">k_loc</span><span class="p">],</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">1</span><span class="p">)</span>
+<span class="p">)</span>
+
+<span class="n">predictions_nonlocal_fcnn</span> <span class="o">=</span> <span class="n">nonlocal_fcnn_network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">))</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">predictions_linear</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted from linear model&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">predictions_fcnn</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted from local FCNN model&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">predictions_nonlocal_fcnn</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[</span><span class="mi">0</span><span class="p">:</span><span class="mi">1000</span><span class="p">,</span> <span class="n">k_loc</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted from non-local FCNN model&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">subgrid_tend_test</span><span class="p">[:</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True Values&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/ecdd771fa1377dd16da8fc3477dafddfee8e4b294f02e81de51afc682bb5d282.png" src="../_images/ecdd771fa1377dd16da8fc3477dafddfee8e4b294f02e81de51afc682bb5d282.png" />
+</div>
+</div>
+<p>As seen above, the non-local model does about as good as the local model. It may be possible to improve the skill of the model by adjusting the hyperparameters associated with the training or model architecture, but we did not pursue this much more in this notebook.</p>
+</div>
+<div class="section" id="save-the-networks">
+<h2>Save the networks<a class="headerlink" href="#save-the-networks" title="Permalink to this headline">#</a></h2>
+<p>Now we save the weights that were learnt here, so we can use them directly later without having to retrain the neural network.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Save network</span>
+<span class="n">save_path</span> <span class="o">=</span> <span class="s2">&quot;./networks/non_local_FCNN.pth&quot;</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="n">nonlocal_fcnn_network</span><span class="o">.</span><span class="n">state_dict</span><span class="p">(),</span> <span class="n">save_path</span><span class="p">)</span>
+
+<span class="n">save_path</span> <span class="o">=</span> <span class="s2">&quot;./networks/local_FCNN.pth&quot;</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="n">fcnn_network</span><span class="o">.</span><span class="n">state_dict</span><span class="p">(),</span> <span class="n">save_path</span><span class="p">)</span>
+
+<span class="n">save_path</span> <span class="o">=</span> <span class="s2">&quot;./networks/linear.pth&quot;</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="n">linear_network</span><span class="o">.</span><span class="n">state_dict</span><span class="p">(),</span> <span class="n">save_path</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline">#</a></h2>
+<p>In this notebook we showed how machine learning based models, particularly fully connected neural networks, can be trained to learn the relationship between the sub-grid fluxes and the slow variables. This pattern learning exercise is sometimes referred to as offline training, as the skill of the ML model in an actual simulation of the slow variables (gcm) has not been evaluated. In the next notebook we show how these models trained offline can be incorporated into the gcm version of the L96 model, and how its skill can be evaluated.</p>
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generating-the-data">Generating the data</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#build-the-real-world-to-generate-the-ground-truth-dataset">Build the <em>Real World</em> to Generate the Ground Truth Dataset</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#run-the-simulation-to-generate-data">Run the simulation to generate data</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#split-the-data-into-training-and-testing-set">Split the Data into Training and Testing Set</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-dataloaders">The Dataloaders</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-local-ml-models">The <em>local</em> ML models</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#linear-regression-network">Linear regression network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fully-connected-neural-network-fcnn">Fully connected neural network (FCNN)</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-loss-function">The Loss Function</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-optimizer">The Optimizer</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#combining-it-all-together-training-the-network">Combining it all Together: Training the network</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-training-and-test-functions">Define the Training and Test Functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-the-networks">Train the networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#compare-predictions-with-ground-truth">Compare Predictions with Ground Truth</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-non-local-ml-models">The <em>non-local</em> ML models</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#save-the-networks">Save the networks</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
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+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
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+        <div class="sidebar-primary-item"><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="citing.html">Citing this book</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../bibliography.html">References</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Using Neural Networks for L96 Parameterization: Online Testing</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#load-in-the-pre-trained-parameterization-networks">Load in the pre-trained parameterization networks</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adding-parameterizations-to-gcm">Adding Parameterizations to GCM</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#gcm-without-neural-network-parameterization">GCM <em>Without</em> Neural Network Parameterization</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#gcm-with-neural-network-parameterization">GCM <em>With</em> Neural Network Parameterization</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#comparing-results">Comparing Results</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#checking-over-many-different-initial-conditions">Checking over many different initial conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="using-neural-networks-for-l96-parameterization-online-testing">
+<h1>Using Neural Networks for L96 Parameterization: Online Testing<a class="headerlink" href="#using-neural-networks-for-l96-parameterization-online-testing" title="Permalink to this headline">#</a></h1>
+<div class="section" id="outline">
+<h2>Outline:<a class="headerlink" href="#outline" title="Permalink to this headline">#</a></h2>
+<p>In this notebook, we’ll extend upon the concepts learned in the <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/L96_offline_nn.html">previous</a> notebook by implementing the neural networks as a parameterization in the single time scale Lorenz 96, and testing its impact in a simulation. Testing the impacts of parameterizations in a simulation is sometimes also referred to as online testing.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="kn">import</span> <span class="nn">time</span>
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+<span class="kn">from</span> <span class="nn">torch.utils.data</span> <span class="kn">import</span> <span class="n">TensorDataset</span><span class="p">,</span> <span class="n">DataLoader</span>
+
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">RK4</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Ensuring reproducibility</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">14</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="load-in-the-pre-trained-parameterization-networks">
+<h2>Load in the pre-trained parameterization networks<a class="headerlink" href="#load-in-the-pre-trained-parameterization-networks" title="Permalink to this headline">#</a></h2>
+<p>We first load in the networks that were trained in the previous <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/L96_offline_NN.html">notebook</a>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linear_weights</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="s2">&quot;./networks/linear.pth&quot;</span><span class="p">)</span>
+<span class="n">local_FCNN_weights</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="s2">&quot;./networks/local_FCNN.pth&quot;</span><span class="p">)</span>
+<span class="n">nonlocal_FCNN_weights</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="s2">&quot;./networks/non_local_FCNN.pth&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>We need to define the architecture that each model has before the weights can be assigned.</p>
+<div class="cell tag_hide-cell docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell content</span>
+<span class="expanded">Hide code cell content</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># The model architectures</span>
+<span class="c1"># ---------------------------</span>
+
+
+<span class="k">class</span> <span class="nc">LinearRegression</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># A single input and a single output</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="c1"># This method is automatically executed when</span>
+        <span class="c1"># we call a object of this class</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+
+
+<span class="k">class</span> <span class="nc">FCNN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+
+
+<span class="k">class</span> <span class="nc">NonLocal_FCNN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</details>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Initialize network instances and assign pre-trained weights.</span>
+<span class="n">linear_network</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
+<span class="n">linear_network</span><span class="o">.</span><span class="n">load_state_dict</span><span class="p">(</span><span class="n">linear_weights</span><span class="p">)</span>
+
+<span class="n">local_fcnn_network</span> <span class="o">=</span> <span class="n">FCNN</span><span class="p">()</span>
+<span class="n">local_fcnn_network</span><span class="o">.</span><span class="n">load_state_dict</span><span class="p">(</span><span class="n">local_FCNN_weights</span><span class="p">)</span>
+
+<span class="n">nonlocal_fcnn_network</span> <span class="o">=</span> <span class="n">NonLocal_FCNN</span><span class="p">()</span>
+<span class="n">nonlocal_fcnn_network</span><span class="o">.</span><span class="n">load_state_dict</span><span class="p">(</span><span class="n">nonlocal_FCNN_weights</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;All keys matched successfully&gt;
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="adding-parameterizations-to-gcm">
+<h2>Adding Parameterizations to GCM<a class="headerlink" href="#adding-parameterizations-to-gcm" title="Permalink to this headline">#</a></h2>
+<p>To recap, <span id="id1">[<a class="reference internal" href="../bibliography.html#id2" title="E.N. Lorenz. Predictability: a problem partly solved. Seminar on Predictability, 1:1-18, 1995. URL: https://www.ecmwf.int/node/10829.">Lorenz, 1995</a>]</span> describes a two-time scale dynamical system using two equations which are:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{gather*}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}
+\end{gather*}\]</div>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{gather*}
+\frac{d}{dt} Y_{j,k}
+&amp;= - cbY_{j+1,k} \left( Y_{j+2,k} - Y_{j-1,k} \right) - c Y_{j,k} + \frac{hc}{b} X_k
+\end{gather*}\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>All the GCM networks used in this notebook have been introduced earlier in notebooks <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/gcm-parameterization-problem.html">Key aspects of GCMs parameterizations</a> and <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/estimating-gcm-parameters.html">Tuning GCM Parameterizations</a>. To see the definition of those networks, expand the cells in the respective GCM sections below.</p>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">T_test</span> <span class="o">=</span> <span class="mi">10</span>
+<span class="n">forcing</span> <span class="o">=</span> <span class="mi">18</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.01</span>
+
+<span class="n">k</span> <span class="o">=</span> <span class="mi">8</span>
+<span class="n">j</span> <span class="o">=</span> <span class="mi">32</span>
+
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">forcing</span><span class="p">)</span>
+
+<span class="c1"># Full L96 model (two time scale model)</span>
+<span class="n">X_full</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T_test</span><span class="p">)</span>
+<span class="n">X_full</span> <span class="o">=</span> <span class="n">X_full</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+
+<span class="n">init_conditions</span> <span class="o">=</span> <span class="n">X_full</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="gcm-without-neural-network-parameterization">
+<h3>GCM <em>Without</em> Neural Network Parameterization<a class="headerlink" href="#gcm-without-neural-network-parameterization" title="Permalink to this headline">#</a></h3>
+<div class="cell tag_hide-cell docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell content</span>
+<span class="expanded">Hide code cell content</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GCM_without_parameterization</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;GCM without parameterization</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        F: Forcing term</span>
+<span class="sd">        time_stepping: Time stepping method</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">_</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Compute right hand side of the the GCM equations&quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Run GCM</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            X0: Initial conditions of X</span>
+<span class="sd">            dt: Time increment</span>
+<span class="sd">            nt: Number of forward steps to take</span>
+<span class="sd">            param: Parameters of closure</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Model output for all variables of X at each timestep</span>
+<span class="sd">            along with the corresponding time units</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</details>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM_without_parameterization</span><span class="p">(</span><span class="n">forcing</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="gcm-with-neural-network-parameterization">
+<h3>GCM <em>With</em> Neural Network Parameterization<a class="headerlink" href="#gcm-with-neural-network-parameterization" title="Permalink to this headline">#</a></h3>
+<div class="cell tag_hide-cell docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell content</span>
+<span class="expanded">Hide code cell content</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GCM_network</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;GCM with neural network parameterization</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        F: Forcing term</span>
+<span class="sd">        network: Neural network</span>
+<span class="sd">        time_stepping: Time stepping method</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">network</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">network</span> <span class="o">=</span> <span class="n">network</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">_</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Compute right hand side of the the GCM equations&quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="o">.</span><span class="n">linear1</span><span class="o">.</span><span class="n">in_features</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_torch</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
+
+        <span class="c1"># Adding NN parameterization</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="p">(</span><span class="n">X_torch</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Run GCM</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            X0: Initial conditions of X</span>
+<span class="sd">            dt: Time increment</span>
+<span class="sd">            nt: Number of forward steps to take</span>
+<span class="sd">            param: Parameters of closure</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Model output for all variables of X at each timestep</span>
+<span class="sd">            along with the corresponding time units</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</details>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Evaluate with linear network</span>
+<span class="n">gcm_linear_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">linear_network</span><span class="p">)</span>
+<span class="n">Xnn_linear</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_linear_net</span><span class="p">(</span><span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">linear_network</span><span class="p">)</span>
+
+<span class="c1"># Evaluate with local FCNN</span>
+<span class="n">gcm_local_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">local_fcnn_network</span><span class="p">)</span>
+<span class="n">Xnn_local</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_local_net</span><span class="p">(</span><span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">local_fcnn_network</span><span class="p">)</span>
+
+<span class="c1"># Evaluate with nonlocal FCNN</span>
+<span class="n">gcm_nonlocal_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">nonlocal_fcnn_network</span><span class="p">)</span>
+<span class="n">Xnn_nonlocal</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_nonlocal_net</span><span class="p">(</span>
+    <span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">nonlocal_fcnn_network</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="comparing-results">
+<h3>Comparing Results<a class="headerlink" href="#comparing-results" title="Permalink to this headline">#</a></h3>
+<p>Comparing the predictions of GCM with different parameterizations.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">time_i</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">X_full</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Full L96&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">X_no_param</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;No parameterization&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">Xnn_linear</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;linear parameterization&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">Xnn_local</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local NN&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">Xnn_nonlocal</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;nonlocal NN&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper left&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/182380f0ec2cd4008c052f74a5317eabfe91c6c004df7d4af8b5d78acc2d2a96.png" src="../_images/182380f0ec2cd4008c052f74a5317eabfe91c6c004df7d4af8b5d78acc2d2a96.png" />
+</div>
+</div>
+</div>
+<div class="section" id="checking-over-many-different-initial-conditions">
+<h3>Checking over many different initial conditions<a class="headerlink" href="#checking-over-many-different-initial-conditions" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">err_linear</span><span class="p">,</span> <span class="n">err_local</span><span class="p">,</span> <span class="n">err_nonlocal</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]</span>
+<span class="n">T_test</span> <span class="o">=</span> <span class="mi">1</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">90</span><span class="p">):</span>
+    <span class="n">init_conditions_i</span> <span class="o">=</span> <span class="n">X_full</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">10</span><span class="p">,</span> <span class="p">:]</span>
+
+    <span class="c1"># Evaluate with linear network</span>
+    <span class="n">gcm_linear_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">linear_network</span><span class="p">)</span>
+    <span class="n">Xnn_linear</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_linear_net</span><span class="p">(</span>
+        <span class="n">init_conditions_i</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">linear_network</span>
+    <span class="p">)</span>
+
+    <span class="c1"># Evaluate with local FCNN</span>
+    <span class="n">gcm_local_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">local_fcnn_network</span><span class="p">)</span>
+    <span class="n">Xnn_local</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_local_net</span><span class="p">(</span>
+        <span class="n">init_conditions_i</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">local_fcnn_network</span>
+    <span class="p">)</span>
+
+    <span class="c1"># Evaluate with nonlocal FCNN</span>
+    <span class="n">gcm_nonlocal_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">nonlocal_fcnn_network</span><span class="p">)</span>
+    <span class="n">Xnn_nonlocal</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_nonlocal_net</span><span class="p">(</span>
+        <span class="n">init_conditions_i</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">nonlocal_fcnn_network</span>
+    <span class="p">)</span>
+
+    <span class="c1"># GCM parameterized by the global 3-layer network</span>
+    <span class="c1"># gcm_net_3layers = GCM_network(forcing, nn_3l)</span>
+    <span class="c1"># Xnn_3layer_i, t = gcm_net_3layers(init_conditions_i, dt, int(T_test / dt), nn_3l)</span>
+
+    <span class="c1"># GCM parameterized by the linear network</span>
+    <span class="c1"># gcm_net_1layers = GCM_network(forcing, linear_network)</span>
+    <span class="c1"># Xnn_1layer_i, t = gcm_net_1layers(init_conditions_i, dt, int(T_test / dt), linear_network)</span>
+
+    <span class="n">err_linear</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X_full</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="p">:</span> <span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">+</span> <span class="n">T_test</span> <span class="o">*</span> <span class="mi">100</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">Xnn_linear</span><span class="p">))</span>
+    <span class="p">)</span>
+
+    <span class="n">err_local</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X_full</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="p">:</span> <span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">+</span> <span class="n">T_test</span> <span class="o">*</span> <span class="mi">100</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">Xnn_local</span><span class="p">))</span>
+    <span class="p">)</span>
+
+    <span class="n">err_nonlocal</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X_full</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="p">:</span> <span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">+</span> <span class="n">T_test</span> <span class="o">*</span> <span class="mi">100</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">Xnn_nonlocal</span><span class="p">))</span>
+    <span class="p">)</span>
+
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Sum of errors for linear: </span><span class="si">{</span><span class="nb">sum</span><span class="p">(</span><span class="n">err_linear</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Sum of errors for local neural network: </span><span class="si">{</span><span class="nb">sum</span><span class="p">(</span><span class="n">err_local</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Sum of errors for non-local neural network: </span><span class="si">{</span><span class="nb">sum</span><span class="p">(</span><span class="n">err_nonlocal</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Sum of errors for linear: 53703.23
+Sum of errors for local neural network: 31679.62
+Sum of errors for non-local neural network: 40778.95
+</pre></div>
+</div>
+</div>
+</div>
+<p>In this evaluation we see that the neural networks perform much better than the linear model. However, incorporating non-locality did not improve model performance over the local model. It may be possible to have improved this by building a better non-local model by a larger parameter sweep.</p>
+</div>
+</div>
+<div class="section" id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline">#</a></h2>
+<p>In this notebook we showed how neural networks can be added as a parameterization to the single timescale L96 model (the gcm analogue of the two timescale L96 model). The neural network parameterizations performed better than the linear parameterization, as shown by evaluating the model performance over many initial conditions.</p>
+<p>In these notebook we have followed the strategy of :</p>
+<ul class="simple">
+<li><p>running a realistic (more time scales resolved) simulation.</p></li>
+<li><p>evaluating the impact of the fast scales (which could not be resolved if the simulation resolved fewer time scales) on the slow scales in the realistic simulation.</p></li>
+<li><p>learning a functional relationship between the impact of the fast scales on the slow scale and the slow scales.</p></li>
+<li><p>incoporating this relationship (ML based parameterization) in the model of the slow time scales, and evaluating the success (if any).</p></li>
+</ul>
+<p>This is a strategy of offline training and online testing.</p>
+<p>A different strategy to have would have been to not necessarily learn the impact of or the patterns in the sub-grid scales. But rather optimize a model such that when it is added to the slow time scale model, it leads to the time evolution of the slow time scale model being closer to the two time scale model (which is our ultimate goal). This strategy is sometimes called online training, and discussed in the next notebook. A short primer to this was presented in <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/estimating-gcm-parameters.html">the notebook on gcm tuning</a>, where we tuned parameters to match the time evolution of the more realistic model.</p>
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#load-in-the-pre-trained-parameterization-networks">Load in the pre-trained parameterization networks</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adding-parameterizations-to-gcm">Adding Parameterizations to GCM</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#gcm-without-neural-network-parameterization">GCM <em>Without</em> Neural Network Parameterization</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#gcm-with-neural-network-parameterization">GCM <em>With</em> Neural Network Parameterization</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#comparing-results">Comparing Results</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#checking-over-many-different-initial-conditions">Checking over many different initial conditions</a></li>
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+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
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+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="closing_remarks.html">Outlook</a></li>
+<li class="toctree-l1"><a class="reference internal" href="citing.html">Citing this book</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../bibliography.html">References</a></li>
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+    <h1>Using Neural Networks for L96 Parameterization: Online Training</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#online-vs-offline-training">Online vs offline training</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-the-ground-truth-dataset-from-the-real-world">Generate the Ground Truth Dataset from the <em>Real World</em></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#create-a-differentiable-1-time-scale-l96-model">Create a differentiable 1 time-scale L96 model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-a-neural-network-for-the-parameterization">Define a Neural Network for the parameterization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#loss-function-and-optimizer-for-online-training">Loss function and optimizer for online training</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#training">Training</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#test-in-a-simulation-online-testing">Test in a simulation: Online testing</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="using-neural-networks-for-l96-parameterization-online-training">
+<h1>Using Neural Networks for L96 Parameterization: Online Training<a class="headerlink" href="#using-neural-networks-for-l96-parameterization-online-training" title="Permalink to this headline">#</a></h1>
+<div class="section" id="outline">
+<h2>Outline:<a class="headerlink" href="#outline" title="Permalink to this headline">#</a></h2>
+<p>In the previous two notebooks we showed how a parameterization can be developed for the L96 setup using neural networks using a methodology known as offline training. In this notebook we dive into an alternative training philosophy known as online training.</p>
+</div>
+<div class="section" id="online-vs-offline-training">
+<h2>Online vs offline training<a class="headerlink" href="#online-vs-offline-training" title="Permalink to this headline">#</a></h2>
+<p>The type of ML training done in the previous couple of notebooks, where we estimate the sub-grid terms explicitly as the terms corresponding to the missing scales from a full simulations, is known as offline training. In this training procedure it is assumed that know exactly how to estimate the effect of the missing scales. However, this might not be true in general. For the L96 case there can be differences in the numerics of the one time scale and two time scale models, which will not be accounted for in the sub-grid terms we estimate from the two time-scale more. In more general simulations, e.g. turbulent flows, a clear scale separation into scales is not available, and it is not obvious what the right way to filter the high resolution simulation might be. Additionally, the parameterization learnt using the offline procedure may turn out to be unstable because numerical stability of the parameterized model was never imposed as a constraint during the traing procedure.</p>
+<p>An alternate way to train machine learning models is called online training. In this procedure, rather than training a model that best fits a pre-defined sub-grid tendency, a model is trained which tries to produce a solution of the parameterized low resolution model (<span class="math notranslate nohighlight">\(X^{param-LR}\)</span>) that is closest to the high resolution model (<span class="math notranslate nohighlight">\(X^{HR}\)</span>, where this <span class="math notranslate nohighlight">\(X\)</span> has been appropriately downscaled to match the resolution of the low resolution model). For the L96 simulation, this would imply that the parameterized one time-scale model match the evolution of the slow variables in the two-time scale model.</p>
+<p>A corresponding loss function may be defined as:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-b1607528-f579-4110-8539-b4627f613540">
+<span class="eqno">(11)<a class="headerlink" href="#equation-b1607528-f579-4110-8539-b4627f613540" title="Permalink to this equation">#</a></span>\[\begin{equation}
+L^{Online} = \frac{1}{T}\int_0^T |X^{param-LR} - X^{HR}|^2 dt
+\end{equation}\]</div>
+<p>Contrast this to the loss function offline training:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-e9aedc11-00ed-498e-83f1-60b7caa4d4a1">
+<span class="eqno">(12)<a class="headerlink" href="#equation-e9aedc11-00ed-498e-83f1-60b7caa4d4a1" title="Permalink to this equation">#</a></span>\[\begin{equation}
+L^{Offline} = |P(X_k^{LR}) - U_k|^2,
+\end{equation}\]</div>
+<p>where <span class="math notranslate nohighlight">\(P(X)\)</span> is the parameterization of the sub-grid terms (coupling terms) <span class="math notranslate nohighlight">\(U\)</span>. <em>Note that in both of the loss functions above appropriate averages over available samples are taken.</em></p>
+<p>So, let’s see how this can be done in practice, and what technical changes we need to make.</p>
+<p>Note: Much of this notebook has been inspired by Pavel Perezhogin’s <a class="reference external" href="https://github.com/Pperezhogin/L96-predictability">repo</a>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="kn">import</span> <span class="nn">time</span>
+<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">import</span> <span class="nn">torch.utils.data</span> <span class="k">as</span> <span class="nn">Data</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">RK2</span><span class="p">,</span> <span class="n">RK4</span><span class="p">,</span> <span class="n">EulerFwd</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span><span class="p">,</span> <span class="n">integrate_L96_2t</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Ensuring reproducibility</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">14</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># device = torch.device(&#39;cuda&#39; if torch.cuda.is_available() else &#39;cpu&#39;) # You can try this if you have a GPU. However, speed up on GPU is only guaranteed after overhead cost of moving data between CPU and GPU is accounted for.</span>
+<span class="n">device</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">device</span><span class="p">(</span><span class="s2">&quot;cpu&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="generate-the-ground-truth-dataset-from-the-real-world">
+<h2>Generate the Ground Truth Dataset from the <em>Real World</em><a class="headerlink" href="#generate-the-ground-truth-dataset-from-the-real-world" title="Permalink to this headline">#</a></h2>
+<p>Same as the past notebooks, we first generate some instance from the two-time scale (also called our “real world”) simulation.</p>
+<p>We initialise the L96 two time-scale model using <span class="math notranslate nohighlight">\(K\)</span> (set to 8) values of <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(J\)</span> (set to 32) values of <span class="math notranslate nohighlight">\(Y\)</span> for each <span class="math notranslate nohighlight">\(X\)</span>. The model is run for many timesteps to generate the dataset for training later.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Setup the two time-scale model</span>
+<span class="n">time_steps</span> <span class="o">=</span> <span class="mi">32000</span>
+<span class="n">forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span> <span class="o">*</span> <span class="n">time_steps</span>
+
+<span class="c1"># Create a &quot;real world&quot; with K=8 and J=32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">forcing</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Run the two time-scale model</span>
+
+<span class="c1"># The effect of Y on X is `xy_true`</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># Change the data type to `float32` in order to avoid doing type conversions later on</span>
+<span class="n">X_true</span> <span class="o">=</span> <span class="n">X_true</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+
+<span class="c1"># Notice that we only output the model trajectory here, as we don&#39;t need the coupling terms for training here.</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>We now need to set the number of time steps that the training process will use for every sample (how long of a trajectory are we trying to match).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Tstep_train</span> <span class="o">=</span> <span class="mi">128</span>
+
+<span class="c1"># We split the simulation into ensembles members without any overlap,</span>
+<span class="c1"># and each of these will be used as a separate sample.</span>
+<span class="n">N_ens</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">time_steps</span> <span class="o">/</span> <span class="n">Tstep_train</span><span class="p">)</span>
+<span class="n">X_true_ens</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">N_ens</span> <span class="o">*</span> <span class="n">Tstep_train</span><span class="p">,</span> <span class="p">:],</span> <span class="p">(</span><span class="n">N_ens</span><span class="p">,</span> <span class="n">Tstep_train</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Split the data into training and test</span>
+
+<span class="c1"># Set the number of time series that will be part of test ensemble.</span>
+<span class="n">test_ens</span> <span class="o">=</span> <span class="mi">32</span>
+
+<span class="c1"># Training Data</span>
+<span class="n">X_true_train</span> <span class="o">=</span> <span class="n">X_true_ens</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="o">-</span><span class="n">test_ens</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span>
+
+<span class="c1"># Test Data</span>
+<span class="n">X_true_test</span> <span class="o">=</span> <span class="n">X_true_ens</span><span class="p">[</span><span class="o">-</span><span class="n">test_ens</span><span class="p">:,</span> <span class="p">:,</span> <span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_true_train</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">X_true_test</span><span class="o">.</span><span class="n">shape</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>((218, 128, 8), (32, 128, 8))
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Number of sample in each batch</span>
+<span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">32</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Notice that in the training and testing datasets defined below the input to the model is the initial condition, and the output that the model will be evaluated against is a time series from the simulation.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Training Dataset</span>
+<span class="c1"># ----------------</span>
+<span class="n">nlocal_data_train</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_train</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">:]),</span>  <span class="c1"># expected input is an initial condition</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_train</span><span class="p">),</span>  <span class="c1"># expected output as a time series</span>
+<span class="p">)</span>
+
+<span class="n">nlocal_loader_train</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">nlocal_data_train</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+
+<span class="c1"># Test Dataset</span>
+<span class="c1"># ------------</span>
+<span class="n">nlocal_data_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">:]),</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">)</span>
+<span class="p">)</span>
+<span class="n">nlocal_loader_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">nlocal_data_test</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="create-a-differentiable-1-time-scale-l96-model">
+<h2>Create a differentiable 1 time-scale L96 model<a class="headerlink" href="#create-a-differentiable-1-time-scale-l96-model" title="Permalink to this headline">#</a></h2>
+<p>One of the key components needed for online training is a differentiable solver. This can be seen by the presence of <span class="math notranslate nohighlight">\(X^{param-LR}\)</span> in the loss function, which indicates that derivatives should be able to pass through a function that not only corresponds to the extra terms that are added, but also produces the full solution.</p>
+<p>This relatively to easy to do with modern machine learning frameworks like PyTorch or JAX, as long as the model is simple enough to be rewritten using these frameworks.</p>
+<p>Here we will write a differentiable solver for :</p>
+<div class="amsmath math notranslate nohighlight" id="equation-a36e3eb0-47cb-4200-83eb-4ead630ff90b">
+<span class="eqno">(13)<a class="headerlink" href="#equation-a36e3eb0-47cb-4200-83eb-4ead630ff90b" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F.
+\end{align}\]</div>
+<p>Notice below that in this case this task was as simple as using the word <code class="docutils literal notranslate"><span class="pre">torch</span></code> instead of <code class="docutils literal notranslate"><span class="pre">np</span></code>, basically swapping out numpy calls with PyTorch calls.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">L96_eq1_xdot_torch</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="mi">20</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Compared to older function works on batches,</span>
+<span class="sd">    i.e. dimension is Nbatch x K (single time step for many batches is input)</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="p">(</span>
+        <span class="n">torch</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">shifts</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">dims</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="o">*</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">shifts</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">dims</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">shifts</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">dims</span><span class="o">=-</span><span class="mi">1</span><span class="p">))</span>
+        <span class="o">-</span> <span class="n">X</span>
+        <span class="o">+</span> <span class="n">F</span>
+    <span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">RK4</span><span class="p">(</span><span class="n">fn</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="o">*</span><span class="n">kw</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Calculate the new state X(n+1) for d/dt X = fn(X,t,...) using the fourth order Runge-Kutta method.</span>
+<span class="sd">    Args:</span>
+<span class="sd">        fn     : The function returning the time rate of change of model variables X</span>
+<span class="sd">        dt     : The time step</span>
+<span class="sd">        X      : Values of X variables at the current time, t</span>
+<span class="sd">        kw     : All other arguments that should be passed to fn, i.e. fn(X, t, *kw)</span>
+<span class="sd">    Returns:</span>
+<span class="sd">        X at t+dt</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">Xdot1</span> <span class="o">=</span> <span class="n">fn</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="o">*</span><span class="n">kw</span><span class="p">)</span>
+    <span class="n">Xdot2</span> <span class="o">=</span> <span class="n">fn</span><span class="p">(</span><span class="n">X</span> <span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">Xdot1</span><span class="p">,</span> <span class="o">*</span><span class="n">kw</span><span class="p">)</span>
+    <span class="n">Xdot3</span> <span class="o">=</span> <span class="n">fn</span><span class="p">(</span><span class="n">X</span> <span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">Xdot2</span><span class="p">,</span> <span class="o">*</span><span class="n">kw</span><span class="p">)</span>
+    <span class="n">Xdot4</span> <span class="o">=</span> <span class="n">fn</span><span class="p">(</span><span class="n">X</span> <span class="o">+</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">Xdot3</span><span class="p">,</span> <span class="o">*</span><span class="n">kw</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">X</span> <span class="o">+</span> <span class="p">(</span><span class="n">dt</span> <span class="o">/</span> <span class="mf">6.0</span><span class="p">)</span> <span class="o">*</span> <span class="p">((</span><span class="n">Xdot1</span> <span class="o">+</span> <span class="n">Xdot4</span><span class="p">)</span> <span class="o">+</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="p">(</span><span class="n">Xdot2</span> <span class="o">+</span> <span class="n">Xdot3</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="define-a-neural-network-for-the-parameterization">
+<h2>Define a Neural Network for the parameterization<a class="headerlink" href="#define-a-neural-network-for-the-parameterization" title="Permalink to this headline">#</a></h2>
+<p>Here we use a neural network architecture that is exactly the same as the non-local architecture that was used in the previous notebooks.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">FCNN_nonlocal</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fcnn_online_network</span> <span class="o">=</span> <span class="n">FCNN_nonlocal</span><span class="p">()</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="loss-function-and-optimizer-for-online-training">
+<h2>Loss function and optimizer for online training<a class="headerlink" href="#loss-function-and-optimizer-for-online-training" title="Permalink to this headline">#</a></h2>
+<p>The target of the loss function in online training is not simply to match the sub-grid fluxes, but instead to also track the solution of the two time-scale model using the single time-scale model.</p>
+<p>The loss function below is where the magic happens. Notice two key elements: (1) the neural network is combined with the numerical solver that produces the tendency at every time step, (2) a time integration is performed over the number some number of time steps.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">compute_loss</span><span class="p">(</span><span class="n">NN</span><span class="p">,</span> <span class="n">x_in</span><span class="p">,</span> <span class="n">y_true</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    NN: the neural network that parameterizes the missing terms</span>
+<span class="sd">    x_in: initial conditions, which are the input. Shape (batch size X K)</span>
+<span class="sd">    y_true: full solution from the two time scale model. Shape (batch size X Nt X K)</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">full_xdot</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">NN</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">L96_eq1_xdot_torch</span><span class="p">(</span>
+        <span class="n">x</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">forcing</span>
+    <span class="p">)</span>  <span class="c1"># make a function that returns tendency with NN as param.</span>
+
+    <span class="n">y_pred</span> <span class="o">=</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">y_true</span>  <span class="c1"># Use this to store the model prediction we make</span>
+
+    <span class="n">y_pred</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">x_in</span>  <span class="c1"># intiailize IC (which is the only input to model).</span>
+
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">y_true</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>  <span class="c1"># loop over time steps</span>
+        <span class="n">y_pred</span><span class="p">[:,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">RK4</span><span class="p">(</span>
+            <span class="n">full_xdot</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">[:,</span> <span class="n">j</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">clone</span><span class="p">()</span>
+        <span class="p">)</span>  <span class="c1"># time step forward.</span>
+
+    <span class="k">return</span> <span class="p">((</span><span class="n">y_pred</span> <span class="o">-</span> <span class="n">y_true</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Since the only free parameters correspond to the weights of the neural network, they are passed to the optimizer. Notice that even though the loss function is much more complex than the offline training case, the parameters that are being optimized are the same.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># setup optimizer.</span>
+
+<span class="n">optimizer_fcnn</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">fcnn_online_network</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="training">
+<h2>Training<a class="headerlink" href="#training" title="Permalink to this headline">#</a></h2>
+<p>First we define the helper functions for training, testing and fitting, same as we did in previous notebooks</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train the network for one epoch&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
+
+    <span class="n">train_loss</span> <span class="o">=</span> <span class="mi">0</span>
+
+    <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+        <span class="c1"># Compute the loss (now the predictions are done directly in the loss function).</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">batch_x</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">),</span> <span class="n">batch_y</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">))</span>
+        <span class="n">train_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Clear the gradients</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>
+
+        <span class="c1"># Backpropagation to compute the gradients and update the weights</span>
+        <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>
+
+    <span class="k">return</span> <span class="n">train_loss</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+
+
+<span class="k">def</span> <span class="nf">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Test the network&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>  <span class="c1"># Evaluation mode (important when having dropout layers)</span>
+
+    <span class="n">test_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+            <span class="c1"># Compute the loss</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">batch_x</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">),</span> <span class="n">batch_y</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">device</span><span class="p">))</span>
+            <span class="n">test_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Get an average loss for the entire dataset</span>
+        <span class="n">test_loss</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">test_loss</span>
+
+
+<span class="k">def</span> <span class="nf">fit_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">test_loader</span><span class="p">,</span> <span class="n">n_epochs</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train and validate the network&quot;&quot;&quot;</span>
+    <span class="n">train_losses</span><span class="p">,</span> <span class="n">test_losses</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
+    <span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+        <span class="n">train_loss</span> <span class="o">=</span> <span class="n">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+        <span class="n">test_loss</span> <span class="o">=</span> <span class="n">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">test_loader</span><span class="p">)</span>
+        <span class="n">train_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">train_loss</span><span class="p">)</span>
+        <span class="n">test_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test_loss</span><span class="p">)</span>
+    <span class="n">end_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Training completed in </span><span class="si">{</span><span class="nb">int</span><span class="p">(</span><span class="n">end_time</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">start_time</span><span class="p">)</span><span class="si">}</span><span class="s2"> seconds.&quot;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">train_losses</span><span class="p">,</span> <span class="n">test_losses</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Epochs refer to the number of times we iterate over the entire training data during training.</span>
+<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">100</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Train the model.</span>
+<span class="n">train_loss_online</span><span class="p">,</span> <span class="n">test_loss_online</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">fcnn_online_network</span><span class="p">,</span>
+    <span class="n">compute_loss</span><span class="p">,</span>
+    <span class="n">optimizer_fcnn</span><span class="p">,</span>
+    <span class="n">nlocal_loader_train</span><span class="p">,</span>
+    <span class="n">nlocal_loader_test</span><span class="p">,</span>
+    <span class="n">n_epochs</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 105 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">train_loss_online</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Training loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">test_loss_online</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Test loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;MSE&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Epochs&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0.5, 0, &#39;Epochs&#39;)
+</pre></div>
+</div>
+<img alt="../_images/bd0060e0df1c0abe4a50d89cd347e4be2f5d97e1791d6ee28888efb3f5dd0a0f.png" src="../_images/bd0060e0df1c0abe4a50d89cd347e4be2f5d97e1791d6ee28888efb3f5dd0a0f.png" />
+</div>
+</div>
+<p>The loss curve above shows that online model is training well, and parameters have been optimized in some sense. Let us check below how this online trained model compares against the offline trained model from the previous notebook.</p>
+</div>
+<div class="section" id="test-in-a-simulation-online-testing">
+<h2>Test in a simulation: Online testing<a class="headerlink" href="#test-in-a-simulation-online-testing" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Load in the previously trained non-local offline network.</span>
+<span class="n">nonlocal_FCNN_weights</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="s2">&quot;./networks/non_local_FCNN.pth&quot;</span><span class="p">)</span>
+
+<span class="n">fcnn_offline_network</span> <span class="o">=</span> <span class="n">FCNN_nonlocal</span><span class="p">()</span>
+<span class="n">fcnn_offline_network</span><span class="o">.</span><span class="n">load_state_dict</span><span class="p">(</span><span class="n">nonlocal_FCNN_weights</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;All keys matched successfully&gt;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># The different GCM classes</span>
+<span class="c1"># ---------------------------</span>
+
+
+<span class="k">class</span> <span class="nc">GCM_without_parameterization</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;GCM without parameterization</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        F: Forcing term</span>
+<span class="sd">        time_stepping: Time stepping method</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">_</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Compute right hand side of the the GCM equations&quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Run GCM</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            X0: Initial conditions of X</span>
+<span class="sd">            dt: Time increment</span>
+<span class="sd">            nt: Number of forward steps to take</span>
+<span class="sd">            param: Parameters of closure</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Model output for all variables of X at each timestep</span>
+<span class="sd">            along with the corresponding time units</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+
+
+<span class="k">class</span> <span class="nc">GCM_network</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;GCM with neural network parameterization</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        F: Forcing term</span>
+<span class="sd">        network: Neural network</span>
+<span class="sd">        time_stepping: Time stepping method</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">network</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">network</span> <span class="o">=</span> <span class="n">network</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">_</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Compute right hand side of the the GCM equations&quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="o">.</span><span class="n">linear1</span><span class="o">.</span><span class="n">in_features</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_torch</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
+
+        <span class="c1"># Adding NN parameterization</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="p">(</span><span class="n">X_torch</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;Run GCM</span>
+
+<span class="sd">        Args:</span>
+<span class="sd">            X0: Initial conditions of X</span>
+<span class="sd">            dt: Time increment</span>
+<span class="sd">            nt: Number of forward steps to take</span>
+<span class="sd">            param: Parameters of closure</span>
+
+<span class="sd">        Returns:</span>
+<span class="sd">            Model output for all variables of X at each timestep</span>
+<span class="sd">            along with the corresponding time units</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Now let us pick a random point in the simulation as our initial condition, and compare if there is some drastic difference between offline and online parameterization that can be seen visually.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">start</span> <span class="o">=</span> <span class="mi">2100</span>
+<span class="n">init_conditions</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="n">start</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">T_test</span> <span class="o">=</span> <span class="mi">10</span>
+<span class="n">forcing</span> <span class="o">=</span> <span class="mi">18</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.01</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM_without_parameterization</span><span class="p">(</span><span class="n">forcing</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Evaluate with nonlocal offline FCNN</span>
+<span class="n">gcm_nonlocal_offline_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">fcnn_offline_network</span><span class="p">)</span>
+<span class="n">Xnn_nonlocal_offline</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_nonlocal_offline_net</span><span class="p">(</span>
+    <span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">fcnn_offline_network</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Evaluate with nonlocal offline FCNN</span>
+<span class="n">gcm_nonlocal_online_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">fcnn_online_network</span><span class="p">)</span>
+<span class="n">Xnn_nonlocal_online</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_nonlocal_online_net</span><span class="p">(</span>
+    <span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">fcnn_online_network</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">time_i</span> <span class="o">=</span> <span class="mi">300</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">X_true</span><span class="p">[</span><span class="n">start</span> <span class="p">:</span> <span class="n">start</span> <span class="o">+</span> <span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Full L96&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">X_no_param</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;No parameterization&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">Xnn_nonlocal_offline</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Offline parameterization&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">time_i</span><span class="p">],</span> <span class="n">Xnn_nonlocal_online</span><span class="p">[:</span><span class="n">time_i</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Online parameterization&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">hlines</span><span class="p">(</span>
+    <span class="mi">0</span><span class="p">,</span>
+    <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
+    <span class="n">t</span><span class="p">[</span><span class="n">Tstep_train</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;time range of online training&quot;</span><span class="p">,</span>
+    <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;r&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;lower right&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/91ad94acaed53c78536427a711bdf743c2f06080c31d0844043de57b9534e791.png" src="../_images/91ad94acaed53c78536427a711bdf743c2f06080c31d0844043de57b9534e791.png" />
+</div>
+</div>
+<p>The above plot shows that both offline and online trained models perform much better than the simulation without any parameterization. However, it is unclear if there is any signficant gain in the online case. To be more precise, we compare the different cases below over many simulation.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">err_noparam</span><span class="p">,</span> <span class="n">err_offline</span><span class="p">,</span> <span class="n">err_online</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]</span>
+<span class="n">T_test</span> <span class="o">=</span> <span class="mi">1</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">):</span>
+    <span class="n">init_conditions_i</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">10</span><span class="p">,</span> <span class="p">:]</span>
+
+    <span class="c1"># Evaluate with no parameterization</span>
+    <span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM_without_parameterization</span><span class="p">(</span><span class="n">forcing</span><span class="p">)</span>
+    <span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+
+    <span class="c1"># Evaluate with local FCNN</span>
+    <span class="n">gcm_nonlocal_offline_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">fcnn_offline_network</span><span class="p">)</span>
+    <span class="n">Xnn_nonlocal_offline</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_nonlocal_offline_net</span><span class="p">(</span>
+        <span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">fcnn_offline_network</span>
+    <span class="p">)</span>
+
+    <span class="c1"># Evaluate with nonlocal FCNN</span>
+    <span class="n">gcm_nonlocal_online_net</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">forcing</span><span class="p">,</span> <span class="n">fcnn_online_network</span><span class="p">)</span>
+    <span class="n">Xnn_nonlocal_online</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_nonlocal_online_net</span><span class="p">(</span>
+        <span class="n">init_conditions</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">fcnn_online_network</span>
+    <span class="p">)</span>
+
+    <span class="n">err_noparam</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="p">:</span> <span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">+</span> <span class="n">T_test</span> <span class="o">*</span> <span class="mi">100</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_no_param</span><span class="p">))</span>
+    <span class="p">)</span>
+
+    <span class="n">err_offline</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="p">:</span> <span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">+</span> <span class="n">T_test</span> <span class="o">*</span> <span class="mi">100</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">Xnn_nonlocal_offline</span><span class="p">)</span>
+        <span class="p">)</span>
+    <span class="p">)</span>
+
+    <span class="n">err_online</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="p">:</span> <span class="n">i</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">+</span> <span class="n">T_test</span> <span class="o">*</span> <span class="mi">100</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">Xnn_nonlocal_online</span><span class="p">))</span>
+    <span class="p">)</span>
+
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Sum of errors for no parameterization: </span><span class="si">{</span><span class="nb">sum</span><span class="p">(</span><span class="n">err_noparam</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Sum of errors for offline parameterization: </span><span class="si">{</span><span class="nb">sum</span><span class="p">(</span><span class="n">err_offline</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Sum of errors for online parameterization: </span><span class="si">{</span><span class="nb">sum</span><span class="p">(</span><span class="n">err_online</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Sum of errors for no parameterization: 526830.43
+Sum of errors for offline parameterization: 432909.76
+Sum of errors for online parameterization: 428407.07
+</pre></div>
+</div>
+</div>
+</div>
+<p>This assessment shows that the online parameterization performs about the same as offline parameterzation. However, atleast for the L96 model the gains (if any), which come at the cost of signfiant complexity, are not drastic.</p>
+</div>
+<div class="section" id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline">#</a></h2>
+<p>In this notebook we described how online training differs from offline training, and provided an example of how online training can be done for the L96 model. While for the L96 model the online training procedure did not produce significant improvements, the gains for other models may be much greater. If interested, you may look at this <a class="reference external" href="https://raspstephan.github.io/blog/lorenz-96-is-too-easy/">blog post</a> arguing that L96 might be too simple a test case, which is why different training procedures do not result in very significant differences.</p>
+<p>In the next few notebooks we show a few tricks to potentially improve performance of neural networks, how to interpret neural networks, and how physics constraints can be added in their architecture.</p>
+</div>
+</div>
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+    <i class="fa-solid fa-list"></i> Contents
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+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#online-vs-offline-training">Online vs offline training</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-the-ground-truth-dataset-from-the-real-world">Generate the Ground Truth Dataset from the <em>Real World</em></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#create-a-differentiable-1-time-scale-l96-model">Create a differentiable 1 time-scale L96 model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-a-neural-network-for-the-parameterization">Define a Neural Network for the parameterization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#loss-function-and-optimizer-for-online-training">Loss function and optimizer for online training</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#training">Training</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#test-in-a-simulation-online-testing">Test in a simulation: Online testing</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
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+    <h1>Learning Data Assimilation Increments</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-model-its-parameters-and-other-utility-functions">Defining the Model, its Parameters, and Other Utility Functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-general-circulation-model-gcm">Defining the General Circulation Model (GCM)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-utility-functions">Defining the Utility Functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initializing-the-lorenz-1996-model-parameters">Initializing the Lorenz 1996 Model Parameters</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#suggestions-for-modifying-the-l96-model-paramters">Suggestions for Modifying the L96 Model Paramters</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-truth-run-from-two-time-scale-l96-model">Generate <code class="docutils literal notranslate"><span class="pre">Truth</span></code> Run from Two Time-Scale L96 Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-synthetic-observations">Generate Synthetic Observations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#apply-localization-to-the-background-model-covariance">Apply Localization to the Background Model Covariance</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#run-data-assimilation">Run Data Assimilation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#post-processing-and-visualization">Post Processing and Visualization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examining-the-relationship-between-the-members-and-their-increments">Examining the Relationship between the Members and their Increments</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#individual-ensemble-members">Individual Ensemble Members</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#mean-over-ensemble-members">Mean over Ensemble Members</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-the-da-increments">Learning the DA Increments</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#observing-the-dataset">Observing the Dataset</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#creating-the-dataset-split">Creating the Dataset Split</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#building-the-dataset">Building the Dataset</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-model">Define the Model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-training-and-test-functions">Define the Training and Test Functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-the-network">Train the Network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-results">Visualizing the Results</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="learning-data-assimilation-increments">
+<h1>Learning Data Assimilation Increments<a class="headerlink" href="#learning-data-assimilation-increments" title="Permalink to this headline">#</a></h1>
+<p>This notebook is derived from the previous notebooks: <span class="xref std std-doc">Neural_network_for_Lorenz96</span> and <a class="reference internal" href="DA_demo_L96.html"><span class="doc">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</span></a>.</p>
+<p>We’ve restricted it to only using the 3-layer network (not the linear regression model)</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+<span class="kn">from</span> <span class="nn">torch.optim</span> <span class="kn">import</span> <span class="n">Adam</span>
+<span class="kn">from</span> <span class="nn">torch.utils.data</span> <span class="kn">import</span> <span class="n">DataLoader</span><span class="p">,</span> <span class="n">TensorDataset</span>
+
+<span class="kn">import</span> <span class="nn">DA_methods</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Ensuring reproducibility</span>
+<span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">()</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">42</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="defining-the-model-its-parameters-and-other-utility-functions">
+<h2>Defining the Model, its Parameters, and Other Utility Functions<a class="headerlink" href="#defining-the-model-its-parameters-and-other-utility-functions" title="Permalink to this headline">#</a></h2>
+<div class="section" id="defining-the-general-circulation-model-gcm">
+<h3>Defining the General Circulation Model (GCM)<a class="headerlink" href="#defining-the-general-circulation-model-gcm" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">GCM</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;A toy `General Circulation Model` (GCM) that uses the single</span>
+<span class="sd">    time-scale Lorenz 1996 model with a parameterised coupling term</span>
+<span class="sd">    to represent the interaction between the observed coarse scale</span>
+<span class="sd">    processes `X` and unobserved fine scale processes `Y` of the two</span>
+<span class="sd">    time-scale model.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X_init: Initial conditions of X</span>
+<span class="sd">        F: Forcing term</span>
+<span class="sd">        dt: Sampling frequency of the model</span>
+<span class="sd">        nt: Number of timesteps for which to run the model</span>
+<span class="sd">        param: Weights to give to the coupling term</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        Model output for all variables of X at each timestep along with</span>
+<span class="sd">        the corresponding time units</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+        <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_init</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+        <span class="n">X_init</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+    <span class="p">)</span>
+    <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">+</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">F</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">))</span>
+        <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="defining-the-utility-functions">
+<h3>Defining the Utility Functions<a class="headerlink" href="#defining-the-utility-functions" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;A non-dimension coordinate from -1..+1 corresponding to k=0..K&quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="n">K</span> <span class="o">-</span> <span class="mi">1</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">get_dist</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Compute the absolute distance between two element indices</span>
+<span class="sd">    within a square matrix of size (K x K)</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        i: the ith row index</span>
+<span class="sd">        j: the jth column index</span>
+<span class="sd">        K: shape of square array</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        Distance</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">)</span> <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">K</span> <span class="o">/</span> <span class="mi">2</span> <span class="k">else</span> <span class="n">K</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">observation_operator</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">i_t</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Observation operator to map between model and observation space,</span>
+<span class="sd">    assuming linearity and model space observations.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        K: spatial dimension of the model</span>
+<span class="sd">        l_obs: spatial positions of observations on model grid</span>
+<span class="sd">        t_obs: time positions of observations</span>
+<span class="sd">        i_t: the timestep of the current DA cycle</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        Operator matrix (K * observation_density, K)</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">n</span> <span class="o">=</span> <span class="n">l_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="n">H</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
+    <span class="n">H</span><span class="p">[</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">l_obs</span><span class="p">[</span><span class="n">t_obs</span> <span class="o">==</span> <span class="n">i_t</span><span class="p">]]</span> <span class="o">=</span> <span class="mi">1</span>
+    <span class="k">return</span> <span class="n">H</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">gaspari_cohn</span><span class="p">(</span><span class="n">distance</span><span class="p">,</span> <span class="n">radius</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Compute the appropriate distance dependent weighting of a</span>
+<span class="sd">    covariance matrix, after Gaspari &amp; Cohn, 1999 (https://doi.org/10.1002/qj.49712555417)</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        distance: the distance between array elements</span>
+<span class="sd">        radius: localization radius for DA</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        distance dependent weight of the (i,j) index of a covariance matrix</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="n">distance</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="n">weight</span> <span class="o">=</span> <span class="mf">1.0</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">if</span> <span class="n">radius</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">weight</span> <span class="o">=</span> <span class="mf">0.0</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">ratio</span> <span class="o">=</span> <span class="n">distance</span> <span class="o">/</span> <span class="n">radius</span>
+            <span class="n">weight</span> <span class="o">=</span> <span class="mf">0.0</span>
+            <span class="k">if</span> <span class="n">ratio</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="n">weight</span> <span class="o">=</span> <span class="p">(</span>
+                    <span class="o">-</span><span class="p">(</span><span class="n">ratio</span><span class="o">**</span><span class="mi">5</span><span class="p">)</span> <span class="o">/</span> <span class="mi">4</span>
+                    <span class="o">+</span> <span class="n">ratio</span><span class="o">**</span><span class="mi">4</span> <span class="o">/</span> <span class="mi">2</span>
+                    <span class="o">+</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">ratio</span><span class="o">**</span><span class="mi">3</span> <span class="o">/</span> <span class="mi">8</span>
+                    <span class="o">-</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">ratio</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="mi">3</span>
+                    <span class="o">+</span> <span class="mi">1</span>
+                <span class="p">)</span>
+            <span class="k">elif</span> <span class="n">ratio</span> <span class="o">&lt;=</span> <span class="mi">2</span><span class="p">:</span>
+                <span class="n">weight</span> <span class="o">=</span> <span class="p">(</span>
+                    <span class="n">ratio</span><span class="o">**</span><span class="mi">5</span> <span class="o">/</span> <span class="mi">12</span>
+                    <span class="o">-</span> <span class="n">ratio</span><span class="o">**</span><span class="mi">4</span> <span class="o">/</span> <span class="mi">2</span>
+                    <span class="o">+</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">ratio</span><span class="o">**</span><span class="mi">3</span> <span class="o">/</span> <span class="mi">8</span>
+                    <span class="o">+</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">ratio</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="mi">3</span>
+                    <span class="o">-</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">ratio</span>
+                    <span class="o">+</span> <span class="mi">4</span>
+                    <span class="o">-</span> <span class="mi">2</span> <span class="o">/</span> <span class="mi">3</span> <span class="o">/</span> <span class="n">ratio</span>
+                <span class="p">)</span>
+    <span class="k">return</span> <span class="n">weight</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">localize_covariance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Localize the model climatology covariance matrix, based on</span>
+<span class="sd">    the Gaspari-Cohn function.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        B: Covariance matrix over a long model run &#39;M_truth&#39; (K, K)</span>
+<span class="sd">        loc: spatial localization radius for DA</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        Covariance matrix scaled to zero outside distance &#39;loc&#39; from diagonal and</span>
+<span class="sd">        the matrix of weights which are used to scale covariance matrix</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">M</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">shape</span>
+    <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ix_</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">M</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N</span><span class="p">))</span>
+    <span class="n">dist</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">get_dist</span><span class="p">)(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span>
+    <span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">gaspari_cohn</span><span class="p">)(</span><span class="n">dist</span><span class="p">,</span> <span class="n">loc</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">B</span> <span class="o">*</span> <span class="n">W</span><span class="p">,</span> <span class="n">W</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">running_average</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Compute running mean over a user-specified window.</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X: Input vector of arbitrary length &#39;n&#39;</span>
+<span class="sd">        N: Size of window over which to compute mean</span>
+
+<span class="sd">    Returns:</span>
+<span class="sd">        X averaged over window N</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="k">if</span> <span class="n">N</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="n">N1</span><span class="p">,</span> <span class="n">N2</span> <span class="o">=</span> <span class="o">-</span><span class="n">N</span> <span class="o">/</span> <span class="mi">2</span><span class="p">,</span> <span class="n">N</span> <span class="o">/</span> <span class="mi">2</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">N1</span><span class="p">,</span> <span class="n">N2</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
+    <span class="n">X_sum</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">N1</span><span class="p">,</span> <span class="n">N2</span><span class="p">):</span>
+        <span class="n">X_sum</span> <span class="o">=</span> <span class="n">X_sum</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">X_sum</span> <span class="o">/</span> <span class="n">N</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">find_obs</span><span class="p">(</span><span class="n">loc</span><span class="p">,</span> <span class="n">obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="n">period</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;NOTE: This function is for plotting purposes only.&quot;&quot;&quot;</span>
+    <span class="n">t_period</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">((</span><span class="n">t_obs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">period</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">&amp;</span> <span class="p">(</span><span class="n">t_obs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">period</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
+    <span class="n">obs_period</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">t_period</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+    <span class="n">obs_period</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">obs_period</span><span class="p">)):</span>
+        <span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">l_obs</span><span class="p">[</span><span class="n">t_period</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]]</span> <span class="o">==</span> <span class="n">loc</span><span class="p">):</span>
+            <span class="n">obs_period</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">obs</span><span class="p">[</span><span class="n">t_period</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]][</span><span class="n">l_obs</span><span class="p">[</span><span class="n">t_period</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]]</span> <span class="o">==</span> <span class="n">loc</span><span class="p">]</span>
+    <span class="k">return</span> <span class="n">obs_period</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="initializing-the-lorenz-1996-model-parameters">
+<h3>Initializing the Lorenz 1996 Model Parameters<a class="headerlink" href="#initializing-the-lorenz-1996-model-parameters" title="Permalink to this headline">#</a></h3>
+<p>Let’s define the parameters of the Lorenz 1996 model that match those of Wilks, 2005; K=8, J=32, F=18.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Config</span><span class="p">:</span>
+    <span class="c1"># fmt: off</span>
+    <span class="n">K</span> <span class="o">=</span> <span class="mi">8</span>                               <span class="c1"># Dimension of L96 `X` variables</span>
+    <span class="n">J</span> <span class="o">=</span> <span class="mi">32</span>                              <span class="c1"># Dimension of L96 `Y` variables</span>
+    <span class="n">obs_freq</span> <span class="o">=</span> <span class="mi">10</span>                       <span class="c1"># Observation frequency (number of sampling intervals (si) per observation)</span>
+    <span class="n">F_truth</span> <span class="o">=</span> <span class="mi">18</span>                        <span class="c1"># F for truth signal</span>
+    <span class="n">F_fcst</span> <span class="o">=</span> <span class="mi">18</span>                         <span class="c1"># F for forecast (DA) model</span>
+    <span class="n">GCM_param</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>  <span class="c1"># Polynomial coefficicents for GCM parameterization</span>
+    <span class="n">ns_da</span> <span class="o">=</span> <span class="mi">4000</span>                        <span class="c1"># Number of time samples for DA</span>
+    <span class="n">ns</span> <span class="o">=</span> <span class="mi">4000</span>                           <span class="c1"># Number of time samples for truth signal</span>
+    <span class="n">ns_spinup</span> <span class="o">=</span> <span class="mi">200</span>                     <span class="c1"># Number of time samples for spin up</span>
+    <span class="n">dt</span> <span class="o">=</span> <span class="mf">0.005</span>                          <span class="c1"># Model timestep</span>
+    <span class="n">si</span> <span class="o">=</span> <span class="mf">0.005</span>                          <span class="c1"># Truth sampling interval</span>
+    <span class="n">B_loc</span> <span class="o">=</span> <span class="mf">0.0</span>                         <span class="c1"># Spatial localization radius for DA</span>
+    <span class="n">DA</span> <span class="o">=</span> <span class="s2">&quot;EnKF&quot;</span>                         <span class="c1"># DA method</span>
+    <span class="n">nens</span> <span class="o">=</span> <span class="mi">50</span>                           <span class="c1"># Number of ensemble members for DA</span>
+    <span class="n">inflate_opt</span> <span class="o">=</span> <span class="s2">&quot;relaxation&quot;</span>          <span class="c1"># Method for DA model covariance inflation</span>
+    <span class="n">inflate_factor</span> <span class="o">=</span> <span class="mf">0.86</span>               <span class="c1"># Inflation factor</span>
+    <span class="n">obs_density</span> <span class="o">=</span> <span class="mf">1.0</span>                   <span class="c1"># Fraction of spatial gridpoints where observations are collected</span>
+    <span class="n">DA_freq</span> <span class="o">=</span> <span class="mi">10</span>                        <span class="c1"># Assimilation frequency (number of sampling intervals (si) per assimilation step)</span>
+    <span class="n">obs_sigma</span> <span class="o">=</span> <span class="mf">0.1</span>                     <span class="c1"># Observational error standard deviation</span>
+    <span class="n">initial_spread</span> <span class="o">=</span> <span class="mf">0.1</span>                <span class="c1"># Initial spread added to initial conditions</span>
+    <span class="c1"># fmt: on</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="suggestions-for-modifying-the-l96-model-paramters">
+<h4>Suggestions for Modifying the L96 Model Paramters<a class="headerlink" href="#suggestions-for-modifying-the-l96-model-paramters" title="Permalink to this headline">#</a></h4>
+<p>If you want to modify the default model parameters given above, you can use the suggestions below for alternate values depending upon the desired behaviour.</p>
+<p><strong>Less certain observations</strong></p>
+<ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">obs_sigma</span></code>: 1.0</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">initial_spread</span></code>: 1.0</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">inflate_factor</span></code>: 0.5</p></li>
+</ul>
+<p><strong>Less frequent observations</strong></p>
+<ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">obs_freq</span></code>: 50</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">DA_freq</span></code>: 50</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">inflate_factor</span></code>: 0.4</p></li>
+</ul>
+<p><strong>Very infrequent observations</strong></p>
+<ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">obs_freq</span></code>: 200</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">DA_freq</span></code>: 200</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">inflate_factor</span></code>: 0.5</p></li>
+</ul>
+<p><strong>More frequent observations</strong></p>
+<ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">obs_freq</span></code>: 5</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">DA_freq</span></code>: 5</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">inflate_factor</span></code>: 0.9</p></li>
+</ul>
+<p><strong>Very frequent observations</strong></p>
+<ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">obs_freq</span></code>: 1</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">DA_freq</span></code>: 1</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">inflate_factor</span></code>: 0.98</p></li>
+</ul>
+<p><strong>Very frequent observations but less accurate</strong></p>
+<ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">obs_freq</span></code>: 1</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">DA_freq</span></code>: 1</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">obs_sigma</span></code>: 1.0</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">initial_spread</span></code>: 1.0</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">inflate_factor</span></code>: 0.9</p></li>
+</ul>
+<p><strong>Different time-scale</strong></p>
+<ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">F_fcst</span></code>: 16</p></li>
+</ul>
+</div>
+</div>
+</div>
+<div class="section" id="generate-truth-run-from-two-time-scale-l96-model">
+<h2>Generate <code class="docutils literal notranslate"><span class="pre">Truth</span></code> Run from Two Time-Scale L96 Model<a class="headerlink" href="#generate-truth-run-from-two-time-scale-l96-model" title="Permalink to this headline">#</a></h2>
+<p>The L96 two time-scale model acts as the <strong>real world</strong> using which we obtain the unobserved <code class="docutils literal notranslate"><span class="pre">truth</span></code> field from which our observations will be derived.</p>
+<p>We begin by spinning-up a state and then record a series of <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(Y\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> in arrays <code class="docutils literal notranslate"><span class="pre">X_truth</span></code>, <code class="docutils literal notranslate"><span class="pre">Y_truth</span></code> and <code class="docutils literal notranslate"><span class="pre">t_truth</span></code> respectively. The initial state <span class="math notranslate nohighlight">\(X(t=0\)</span> is recorded in <code class="docutils literal notranslate"><span class="pre">X_init</span></code> (and equal to <code class="docutils literal notranslate"><span class="pre">X_truth[0]</span></code>).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up the &quot;truth&quot; 2-scale L96 model</span>
+<span class="n">M_truth</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">F_truth</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">dt</span><span class="p">)</span>
+<span class="n">M_truth</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">rng</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">((</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">)),</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
+
+<span class="c1"># The model runs for `ns_spinup` timesteps to spin-up</span>
+<span class="n">X_spinup</span><span class="p">,</span> <span class="n">Y_spinup</span><span class="p">,</span> <span class="n">t_spinup</span> <span class="o">=</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns_spinup</span><span class="p">)</span>
+
+<span class="c1"># Generate the initial conditions of X and Y</span>
+<span class="n">X_init</span> <span class="o">=</span> <span class="n">X_spinup</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">Y_init</span> <span class="o">=</span> <span class="n">Y_spinup</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="c1"># Using the initial conditions, generate the truth</span>
+<span class="n">M_truth</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">Y_init</span><span class="p">)</span>
+<span class="n">X_truth</span><span class="p">,</span> <span class="n">Y_truth</span><span class="p">,</span> <span class="n">t_truth</span> <span class="o">=</span> <span class="n">M_truth</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="generate-synthetic-observations">
+<h2>Generate Synthetic Observations<a class="headerlink" href="#generate-synthetic-observations" title="Permalink to this headline">#</a></h2>
+<p>Now we create some <strong>observations</strong> of the <strong>real world</strong> by sampling at <code class="docutils literal notranslate"><span class="pre">obs_freq</span></code> intervals and adding some noise (observational error).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Sample the &quot;truth&quot; to generate observations at certain times (t_obs) and locations (l_obs)</span>
+<span class="n">t_obs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">tile</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="p">),</span>
+    <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="p">),</span> <span class="mi">1</span><span class="p">],</span>
+<span class="p">)</span><span class="o">.</span><span class="n">T</span>
+
+<span class="n">l_obs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">t_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="s2">&quot;int&quot;</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">l_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+    <span class="n">l_obs</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="p">),</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span>
+    <span class="p">)</span>
+
+<span class="n">X_obs</span> <span class="o">=</span> <span class="n">X_truth</span><span class="p">[</span><span class="n">t_obs</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">]</span> <span class="o">+</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span> <span class="o">*</span> <span class="n">rng</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">(</span><span class="n">l_obs</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Calculated observation covariance matrix, assuming independent observations</span>
+<span class="n">R</span> <span class="o">=</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span> <span class="o">*</span> <span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">[:],</span> <span class="n">X_truth</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="mi">1</span><span class="p">:,</span> <span class="mi">0</span><span class="p">]],</span>
+    <span class="n">find_obs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">X_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">t_obs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;obs&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;X(t)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Observations at k=0&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/750326bacd741c98a6bcb4ba3e205817fc829de7e7319a009dd3a427166770c3.png" src="../_images/750326bacd741c98a6bcb4ba3e205817fc829de7e7319a009dd3a427166770c3.png" />
+</div>
+</div>
+</div>
+<div class="section" id="apply-localization-to-the-background-model-covariance">
+<h2>Apply Localization to the Background Model Covariance<a class="headerlink" href="#apply-localization-to-the-background-model-covariance" title="Permalink to this headline">#</a></h2>
+<p>We run the model in forward mode for 5000 steps to calculate the background covariance. The model is the <strong>GCM</strong> function defined above which integrates forward.</p>
+<div class="amsmath math notranslate nohighlight" id="equation-2f69e48d-813d-46f3-b1bd-15feff11cc37">
+<span class="eqno">(19)<a class="headerlink" href="#equation-2f69e48d-813d-46f3-b1bd-15feff11cc37" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F
+\end{align}\]</div>
+<p>The absence of the coupling term to the <span class="math notranslate nohighlight">\(Y\)</span> equations makes this a model with <strong>missing physics</strong> that we hope the Ensemble Kalman Filter will correct.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Generate climatological background covariance for 1-scale L96 model</span>
+<span class="n">X1_clim</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">F_fcst</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="mi">5000</span><span class="p">)</span>
+<span class="n">B_clim1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">X1_clim</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Load pre-calculated climatological background covariance matrix from a long simulation</span>
+<span class="n">B_loc</span><span class="p">,</span> <span class="n">W_clim</span> <span class="o">=</span> <span class="n">localize_covariance</span><span class="p">(</span><span class="n">B_clim1</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="n">Config</span><span class="o">.</span><span class="n">B_loc</span><span class="p">)</span>
+
+<span class="n">B_corr1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">B_clim1</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B_clim1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B_clim1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">B_corr1</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">B_clim1</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">B_clim1</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">B_clim1</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">131</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">B_corr1</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;bwr&quot;</span><span class="p">,</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">,</span> <span class="mi">20</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Background correlation matrix: 1-scale L96&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">132</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">B_loc</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;B_loc&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">133</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">W_clim</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;W_clim&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/5c65ea268088d2309cbe9994a3a35a2dfea8eff1bac9ad19b29891184e46feb3.png" src="../_images/5c65ea268088d2309cbe9994a3a35a2dfea8eff1bac9ad19b29891184e46feb3.png" />
+</div>
+</div>
+</div>
+<div class="section" id="run-data-assimilation">
+<h2>Run Data Assimilation<a class="headerlink" href="#run-data-assimilation" title="Permalink to this headline">#</a></h2>
+<p>The algorithms steps through segments of time (<strong>DA cycles</strong>), launching an ensemble of short forecasts from the posterior estimate of the preceding segment, each perturbed by noise in their initial condition (inflation).</p>
+<p>Each ensemble trajectory is stored in <code class="docutils literal notranslate"><span class="pre">ensX</span></code>. The increment added to correct the prior is in <code class="docutils literal notranslate"><span class="pre">X_inc</span></code>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up array to store DA increments</span>
+<span class="n">X_inc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">),</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">))</span>
+<span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;3DVar&quot;</span><span class="p">:</span>
+    <span class="n">X_inc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">X_inc</span><span class="p">)</span>
+<span class="n">t_DA</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Initialize ensemble with perturbations</span>
+<span class="n">i_t</span> <span class="o">=</span> <span class="mi">0</span>
+<span class="n">ensX</span> <span class="o">=</span> <span class="n">X_init</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">+</span> <span class="n">rng</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">))</span>
+<span class="n">X_post</span> <span class="o">=</span> <span class="n">ensX</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">W</span> <span class="o">=</span> <span class="n">W_clim</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">cycle</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">):</span>
+    <span class="c1"># Set up array to store model forecast for each DA cycle</span>
+    <span class="n">ensX_fcst</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">))</span>
+
+    <span class="c1"># Model forecast for next DA cycle</span>
+    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">):</span>
+        <span class="n">ensX_fcst</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span>
+            <span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">n</span><span class="p">],</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">F_fcst</span><span class="p">,</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">,</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">,</span>
+        <span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
+
+    <span class="n">i_t</span> <span class="o">=</span> <span class="n">i_t</span> <span class="o">+</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span>
+
+    <span class="c1"># Get prior/background from the forecast</span>
+    <span class="n">X_prior</span> <span class="o">=</span> <span class="n">ensX_fcst</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span>
+
+    <span class="c1"># Call DA</span>
+    <span class="n">t_DA</span><span class="p">[</span><span class="n">cycle</span><span class="p">]</span> <span class="o">=</span> <span class="n">t_truth</span><span class="p">[</span><span class="n">i_t</span><span class="p">]</span>
+    <span class="k">if</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;EnKF&quot;</span><span class="p">:</span>
+        <span class="n">H</span> <span class="o">=</span> <span class="n">observation_operator</span><span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">i_t</span><span class="p">)</span>
+
+        <span class="c1"># Augment state vector with parameters when doing parameter estimation</span>
+        <span class="n">B_ens</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">X_prior</span><span class="p">)</span>
+        <span class="n">B_ens_loc</span> <span class="o">=</span> <span class="n">B_ens</span> <span class="o">*</span> <span class="n">W</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">]</span>
+        <span class="n">X_post</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">EnKF</span><span class="p">(</span><span class="n">X_prior</span><span class="p">,</span> <span class="n">X_obs</span><span class="p">[</span><span class="n">t_obs</span> <span class="o">==</span> <span class="n">i_t</span><span class="p">],</span> <span class="n">H</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">B_ens_loc</span><span class="p">)</span>
+        <span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">DA_methods</span><span class="o">.</span><span class="n">ens_inflate</span><span class="p">(</span>
+            <span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="p">:],</span>
+            <span class="n">X_prior</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="p">:],</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">inflate_opt</span><span class="p">,</span>
+            <span class="n">Config</span><span class="o">.</span><span class="n">inflate_factor</span><span class="p">,</span>
+        <span class="p">)</span>
+    <span class="k">elif</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;None&quot;</span><span class="p">:</span>
+        <span class="n">X_post</span> <span class="o">=</span> <span class="n">X_prior</span>
+
+    <span class="k">if</span> <span class="ow">not</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA</span> <span class="o">==</span> <span class="s2">&quot;None&quot;</span><span class="p">:</span>
+        <span class="c1"># Get current increments</span>
+        <span class="n">X_inc</span><span class="p">[</span><span class="n">cycle</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="o">...</span><span class="p">])</span> <span class="o">-</span> <span class="n">X_prior</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="o">...</span><span class="p">]</span>
+        <span class="p">)</span>
+
+    <span class="c1"># Reset initial conditions for next DA cycle</span>
+    <span class="n">ensX_fcst</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">X_post</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="n">Config</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">ensX</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">ensX</span><span class="p">,</span> <span class="n">ensX_fcst</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="kc">None</span><span class="p">,</span> <span class="o">...</span><span class="p">]))</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="post-processing-and-visualization">
+<h2>Post Processing and Visualization<a class="headerlink" href="#post-processing-and-visualization" title="Permalink to this headline">#</a></h2>
+<p><code class="docutils literal notranslate"><span class="pre">meanX</span></code> is the ensemble mean forecast, averaging over all the ensemble members. It has discontinuities due to the increment addition between each DA segment.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">meanX</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">ensX</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">clim</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">meanX</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:]))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
+<span class="n">ch</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)],</span>
+    <span class="n">meanX</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:],</span>
+    <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;bwr&quot;</span><span class="p">,</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mf">6.5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">ch</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">orientation</span><span class="o">=</span><span class="s2">&quot;horizontal&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;s&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;X - X_truth&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(((</span><span class="n">meanX</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;RMSE&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">ensX</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Spread&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)],</span>
+    <span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Obs error&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;RMSE (X - X_truth)&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">which</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(((</span><span class="n">meanX</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;RMSE&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">X_inc_ave</span> <span class="o">=</span> <span class="n">X_inc</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">X_inc_ave</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Inc&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">running_average</span><span class="p">(</span><span class="n">X_inc_ave</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="mi">7</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Inc Ave&quot;</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">F_fcst</span> <span class="o">-</span> <span class="n">Config</span><span class="o">.</span><span class="n">F_truth</span><span class="p">),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;F_bias&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">M_truth</span><span class="o">.</span><span class="n">k</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">X_inc</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">si</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span>
+    <span class="s2">&quot;k:&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ave Inc&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;s&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Increments&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">which</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+
+<span class="n">plot_start</span><span class="p">,</span> <span class="n">plot_end</span> <span class="o">=</span> <span class="mi">200</span><span class="p">,</span> <span class="mi">800</span>
+<span class="n">plot_start_DA</span><span class="p">,</span> <span class="n">plot_end_DA</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">plot_start</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span>
+    <span class="n">plot_end</span> <span class="o">/</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span>
+<span class="p">)</span>
+<span class="n">plot_x</span> <span class="o">=</span> <span class="mi">0</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="n">plot_start</span><span class="p">:</span><span class="n">plot_end</span><span class="p">],</span> <span class="n">X_truth</span><span class="p">[</span><span class="n">plot_start</span><span class="p">:</span><span class="n">plot_end</span><span class="p">,</span> <span class="n">plot_x</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;truth&quot;</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="n">plot_start</span><span class="p">:</span><span class="n">plot_end</span><span class="p">],</span> <span class="n">meanX</span><span class="p">[</span><span class="n">plot_start</span><span class="p">:</span><span class="n">plot_end</span><span class="p">,</span> <span class="n">plot_x</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;forecast&quot;</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
+    <span class="n">t_DA</span><span class="p">[</span><span class="n">plot_start_DA</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">:</span> <span class="n">plot_end_DA</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span>
+    <span class="n">find_obs</span><span class="p">(</span><span class="n">plot_x</span><span class="p">,</span> <span class="n">X_obs</span><span class="p">,</span> <span class="n">t_obs</span><span class="p">,</span> <span class="n">l_obs</span><span class="p">,</span> <span class="p">[</span><span class="n">plot_start</span><span class="p">,</span> <span class="n">plot_end</span><span class="p">]),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;obs&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">which</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;k=&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">plot_x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot; truth and forecast&quot;</span><span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;off&quot;</span><span class="p">)</span>
+
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">text</span><span class="p">(</span>
+    <span class="mf">0.1</span><span class="p">,</span>
+    <span class="mf">0.1</span><span class="p">,</span>
+    <span class="s2">&quot;RMSE=</span><span class="si">{:3f}</span><span class="se">\n</span><span class="s2">Spread=</span><span class="si">{:3f}</span><span class="se">\n</span><span class="s2">GCM param=</span><span class="si">{}</span><span class="se">\n</span><span class="s2">DA=</span><span class="si">{}</span><span class="s2">,</span><span class="si">{}</span><span class="se">\n</span><span class="s2">DA_freq=</span><span class="si">{}</span><span class="se">\n</span><span class="s2">B_loc=</span><span class="si">{}</span><span class="se">\n</span><span class="s2">inflation=</span><span class="si">{}</span><span class="s2">,</span><span class="si">{}</span><span class="se">\n</span><span class="s2">obs_density=</span><span class="si">{}</span><span class="se">\n</span><span class="s2">obs_sigma=</span><span class="si">{}</span><span class="se">\n</span><span class="s2">obs_freq=</span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(((</span><span class="n">meanX</span> <span class="o">-</span> <span class="n">X_truth</span><span class="p">[</span><span class="mi">0</span> <span class="p">:</span> <span class="p">(</span><span class="n">Config</span><span class="o">.</span><span class="n">ns_da</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">:])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()),</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">ensX</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)),</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">DA</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">GCM_param</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">nens</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">B_loc</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">inflate_opt</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">inflate_factor</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">obs_density</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">obs_sigma</span><span class="p">,</span>
+        <span class="n">Config</span><span class="o">.</span><span class="n">obs_freq</span><span class="p">,</span>
+    <span class="p">),</span>
+    <span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;off&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/23488f54e098afc5bacf9ac884680a11197baaa835ebfd096f8c71fd34d1b99f.png" src="../_images/23488f54e098afc5bacf9ac884680a11197baaa835ebfd096f8c71fd34d1b99f.png" />
+</div>
+</div>
+</div>
+<div class="section" id="examining-the-relationship-between-the-members-and-their-increments">
+<h2>Examining the Relationship between the Members and their Increments<a class="headerlink" href="#examining-the-relationship-between-the-members-and-their-increments" title="Permalink to this headline">#</a></h2>
+<p>Converting the increment <code class="docutils literal notranslate"><span class="pre">X_inc</span></code> into a tendency, we can examine the relationship between <span class="math notranslate nohighlight">\(\dot{X}\)</span> (due to the missing physics) and the state of the model at the beginning of each DA segment.</p>
+<p>If we are properly correcting the absence of the coupling term then this structure should look like the parameterization of the coupling term, as done in Wilks, 2005.</p>
+<div class="section" id="individual-ensemble-members">
+<h3>Individual Ensemble Members<a class="headerlink" href="#individual-ensemble-members" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">jj</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X_inc_ave</span><span class="p">[</span><span class="mi">0</span><span class="p">:,</span> <span class="p">:]</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span> <span class="o">&gt;</span> <span class="o">-</span><span class="mf">1e-7</span>
+
+<span class="c1"># The offset by DA_freq looks at the previous posterior</span>
+<span class="n">x_input</span> <span class="o">=</span> <span class="n">ensX</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="mi">0</span><span class="p">:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="n">jj</span><span class="p">]</span>
+
+<span class="c1"># Mid-point of trajectory</span>
+<span class="n">x_input</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">x_input</span> <span class="o">+</span> <span class="n">ensX</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="mi">0</span><span class="p">:,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">:]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="n">jj</span><span class="p">])</span>
+
+<span class="n">xinc_output</span> <span class="o">=</span> <span class="n">X_inc_ave</span><span class="p">[</span><span class="mi">0</span><span class="p">:,</span> <span class="p">:]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="n">jj</span><span class="p">]</span>
+
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
+<span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">x_input</span><span class="p">,</span> <span class="n">xinc_output</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
+<span class="n">p18</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.000707</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0130</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0190</span><span class="p">,</span> <span class="mf">1.59</span><span class="p">,</span> <span class="mf">0.275</span><span class="p">]</span>  <span class="c1"># Polynomial from Wilks, 2005</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;All time, all individial k, and all ensemble members&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_input</span><span class="p">,</span> <span class="n">xinc_output</span><span class="p">,</span> <span class="s2">&quot;k.&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p18</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ - Wilks, 2005&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Ensemble member $X_i(k,t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Ensemble member increment $\dot</span><span class="si">{X}</span><span class="s2">$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span>
+    <span class="n">x_input</span><span class="p">,</span> <span class="n">xinc_output</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">50</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">150</span><span class="p">))</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p18</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ - Wilks, 2005&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Ensemble member $X_i(k,t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Ensemble member increment $\dot</span><span class="si">{X}</span><span class="s2">$&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/69d7008e0ce0f1d4e02e7b34b1258ccab514eb6340cae4d7fc2a1295becac678.png" src="../_images/69d7008e0ce0f1d4e02e7b34b1258ccab514eb6340cae4d7fc2a1295becac678.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">xl</span> <span class="o">=</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">10</span>
+<span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
+<span class="n">e</span> <span class="o">=</span> <span class="mi">19</span>
+<span class="n">si</span> <span class="o">=</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span>
+
+<span class="n">l</span> <span class="o">=</span> <span class="p">(</span><span class="n">l_obs</span> <span class="o">==</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;Ensemble member </span><span class="si">%i</span><span class="s2"> , k = </span><span class="si">%i</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">,</span> <span class="n">X_truth</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">,</span> <span class="n">ensX</span><span class="p">[:,</span> <span class="n">k</span><span class="p">,</span> <span class="n">e</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ensemble member forecast&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="n">l</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
+    <span class="n">ensX</span><span class="p">[</span><span class="n">si</span><span class="p">::</span><span class="n">si</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">e</span><span class="p">][</span><span class="n">l</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_inc</span><span class="p">[:,</span> <span class="p">:,</span> <span class="n">e</span><span class="p">][</span><span class="n">l</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span>
+    <span class="s2">&quot;.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ensemble member prior&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="n">l</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="n">ensX</span><span class="p">[</span><span class="n">si</span><span class="p">::</span><span class="n">si</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">e</span><span class="p">][</span><span class="n">l</span><span class="p">],</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ensemble member post&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">xl</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time, t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/bf661d2456b68fdc5b5def47d9c5152fb1c35e8b613d54adfc18c0c24b664024.png" src="../_images/bf661d2456b68fdc5b5def47d9c5152fb1c35e8b613d54adfc18c0c24b664024.png" />
+</div>
+</div>
+</div>
+<div class="section" id="mean-over-ensemble-members">
+<h3>Mean over Ensemble Members<a class="headerlink" href="#mean-over-ensemble-members" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">jj</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X_inc_ave</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">flatten</span><span class="p">())</span> <span class="o">&gt;</span> <span class="o">-</span><span class="mf">1e-7</span>
+
+<span class="c1"># The offset by DA_freq looks at the previous posterior</span>
+<span class="n">x_input</span> <span class="o">=</span> <span class="n">meanX</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="mi">0</span><span class="p">:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span><span class="p">]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="n">jj</span><span class="p">]</span>
+
+<span class="c1"># Mid-point of trajectory</span>
+<span class="n">x_input</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">x_input</span> <span class="o">+</span> <span class="n">meanX</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="mi">0</span><span class="p">:,</span> <span class="mi">0</span><span class="p">]]</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="n">jj</span><span class="p">])</span>
+
+<span class="n">xinc_output</span> <span class="o">=</span> <span class="n">X_inc_ave</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="n">jj</span><span class="p">]</span>
+
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
+<span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">x_input</span><span class="p">,</span> <span class="n">xinc_output</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
+<span class="n">p18</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.000707</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0130</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0190</span><span class="p">,</span> <span class="mf">1.59</span><span class="p">,</span> <span class="mf">0.275</span><span class="p">]</span>  <span class="c1"># Polynomial from Wilks, 2005</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;All time, all individial k, mean over ensemble members&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_input</span><span class="p">,</span> <span class="n">xinc_output</span><span class="p">,</span> <span class="s2">&quot;k.&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p18</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ - Wilks, 2005&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Ensemble member $X_i(k,t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Ensemble member increment $\dot</span><span class="si">{X}</span><span class="s2">$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span>
+    <span class="n">x_input</span><span class="p">,</span> <span class="n">xinc_output</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">50</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">150</span><span class="p">))</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p18</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ - Wilks, 2005&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Ensemble member $X_i(k,t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Ensemble member increment $\dot</span><span class="si">{X}</span><span class="s2">$&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/289f4fb04d58ed75d8a6d3670e8add57cacf146681e015b9de873916f3a5cd8c.png" src="../_images/289f4fb04d58ed75d8a6d3670e8add57cacf146681e015b9de873916f3a5cd8c.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">xl</span> <span class="o">=</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">10</span>
+<span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
+<span class="n">si</span> <span class="o">=</span> <span class="n">Config</span><span class="o">.</span><span class="n">DA_freq</span>
+
+<span class="n">l</span> <span class="o">=</span> <span class="p">(</span><span class="n">l_obs</span> <span class="o">==</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;Ensemble mean, k = </span><span class="si">%i</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">,</span> <span class="n">X_truth</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">,</span>
+    <span class="n">meanX</span><span class="p">[:,</span> <span class="n">k</span><span class="p">]</span> <span class="o">-</span> <span class="n">ensX</span><span class="p">[:,</span> <span class="n">k</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span>
+    <span class="n">meanX</span><span class="p">[:,</span> <span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">ensX</span><span class="p">[:,</span> <span class="n">k</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;grey&quot;</span><span class="p">,</span>
+    <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ensemble spread&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">,</span> <span class="n">meanX</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ensemble mean forecast&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_truth</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="n">l</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
+    <span class="n">meanX</span><span class="p">[</span><span class="n">si</span><span class="p">::</span><span class="n">si</span><span class="p">,</span> <span class="n">k</span><span class="p">][</span><span class="n">l</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_inc</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)[</span><span class="n">l</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span>
+    <span class="s2">&quot;.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ensemble mean prior&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_truth</span><span class="p">[</span><span class="n">t_obs</span><span class="p">[</span><span class="n">l</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="n">meanX</span><span class="p">[</span><span class="n">si</span><span class="p">::</span><span class="n">si</span><span class="p">,</span> <span class="n">k</span><span class="p">][</span><span class="n">l</span><span class="p">],</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ensemble mean post&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">xl</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time, t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/3955805ac255a4697bcd823a416b62a9b21d8eb49ed4e5816fcfe37d602b4cda.png" src="../_images/3955805ac255a4697bcd823a416b62a9b21d8eb49ed4e5816fcfe37d602b4cda.png" />
+</div>
+</div>
+<p>With this, we have successfully <strong>created the DA increments</strong>.</p>
+</div>
+</div>
+<div class="section" id="learning-the-da-increments">
+<h2>Learning the DA Increments<a class="headerlink" href="#learning-the-da-increments" title="Permalink to this headline">#</a></h2>
+<p>With the dataset for DA Increments created above, we now build a neural network which will try to model that dataset so that we can use it for unseen observations.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">t_inc</span> <span class="o">=</span> <span class="n">t_truth</span><span class="p">[</span>
+    <span class="n">t_obs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
+<span class="p">]</span>  <span class="c1"># These are the times in the &quot;real world&quot; when observations were made</span>
+<span class="n">dt_inc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t_inc</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>  <span class="c1"># Time interval between increments</span>
+
+<span class="n">dt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t_truth</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>  <span class="c1"># Time-step of &quot;real world&quot; model</span>
+<span class="n">da_interval</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">dt_inc</span> <span class="o">/</span> <span class="n">dt</span><span class="p">)</span>
+
+<span class="c1"># Data from DA system (increments for individual ensemble members)</span>
+<span class="n">x_input</span> <span class="o">=</span> <span class="n">ensX</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span><span class="n">da_interval</span><span class="p">]</span>
+<span class="n">X_tend</span> <span class="o">=</span> <span class="n">X_inc</span> <span class="o">/</span> <span class="n">dt_inc</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="observing-the-dataset">
+<h3>Observing the Dataset<a class="headerlink" href="#observing-the-dataset" title="Permalink to this headline">#</a></h3>
+<p>As a sanity check, we look at the data for obvious structure. A polyfit to the data will compare well to Wilks 2005, if the data is similar in distribution. We show Wilks 2005 and the 4th order polyfit for reference but neither are used or needed for training the neural network.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># A simple scatter plot of x_tend against x_input</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
+<span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span><span class="n">x_input</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="n">X_tend</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="mi">4</span><span class="p">)</span>
+<span class="n">p18</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.000707</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0130</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.0190</span><span class="p">,</span> <span class="mf">1.59</span><span class="p">,</span> <span class="mf">0.275</span><span class="p">]</span>  <span class="c1"># Polynomial from Wilks, 2005</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_input</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="n">X_tend</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="s2">&quot;k.&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p18</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ - Wilks, 2005&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/9facc671a81c9b5655296f1f7c70eb592b9ef3c456da22959826b997844868f8.png" src="../_images/9facc671a81c9b5655296f1f7c70eb592b9ef3c456da22959826b997844868f8.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># A PDf of x_tend against x_input</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span>
+    <span class="n">x_input</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span>
+    <span class="n">X_tend</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span>
+    <span class="n">bins</span><span class="o">=</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">50</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">150</span><span class="p">)),</span>
+    <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Greys</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p18</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="s2">&quot;k--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ - Wilks, 2005&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ lin. regr.&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;X$_k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Missing X$_k$ tendency&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Conventional linear regression&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/c4ad60668fc0fb989a4274b581872a1958e5650b65e2f5dabd576c259174cfb3.png" src="../_images/c4ad60668fc0fb989a4274b581872a1958e5650b65e2f5dabd576c259174cfb3.png" />
+</div>
+</div>
+</div>
+<div class="section" id="creating-the-dataset-split">
+<h3>Creating the Dataset Split<a class="headerlink" href="#creating-the-dataset-split" title="Permalink to this headline">#</a></h3>
+<p>We partition the dataset into <strong>training</strong> (seen by the network during optimization of the weights) and <strong>validation</strong> (used for evaluation) sets.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">train_size</span> <span class="o">=</span> <span class="n">x_input</span><span class="o">.</span><span class="n">size</span> <span class="o">//</span> <span class="mi">2</span>
+<span class="n">train_size</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">0.7</span> <span class="o">*</span> <span class="n">x_input</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Training set size =&quot;</span><span class="p">,</span> <span class="n">train_size</span><span class="p">,</span> <span class="s2">&quot;out of&quot;</span><span class="p">,</span> <span class="n">x_input</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
+
+<span class="c1"># Convert the data to type float32</span>
+<span class="n">x_input</span><span class="p">,</span> <span class="n">X_tend</span> <span class="o">=</span> <span class="n">x_input</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">),</span> <span class="n">X_tend</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+
+<span class="n">X_train</span> <span class="o">=</span> <span class="n">x_input</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[:</span><span class="n">train_size</span><span class="p">]</span>
+<span class="n">Y_train</span> <span class="o">=</span> <span class="n">X_tend</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[:</span><span class="n">train_size</span><span class="p">]</span>
+<span class="n">X_val</span> <span class="o">=</span> <span class="n">x_input</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="n">train_size</span><span class="p">:]</span>
+<span class="n">Y_val</span> <span class="o">=</span> <span class="n">X_tend</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="n">train_size</span><span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training set size = 112000 out of 160000
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="building-the-dataset">
+<h3>Building the Dataset<a class="headerlink" href="#building-the-dataset" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">1024</span>
+
+<span class="c1"># Training Dataset</span>
+<span class="n">train_dataset</span> <span class="o">=</span> <span class="n">TensorDataset</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">Y_train</span><span class="p">))</span>
+<span class="n">train_loader</span> <span class="o">=</span> <span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">train_dataset</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># Validation Dataset</span>
+<span class="n">val_dataset</span> <span class="o">=</span> <span class="n">TensorDataset</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_val</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">Y_val</span><span class="p">))</span>
+<span class="n">val_loader</span> <span class="o">=</span> <span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">val_dataset</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="define-the-model">
+<h3>Define the Model<a class="headerlink" href="#define-the-model" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">NetANN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">W</span><span class="o">=</span><span class="mi">16</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">W</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">W</span><span class="p">,</span> <span class="n">W</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">W</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="define-the-training-and-test-functions">
+<h3>Define the Training and Test Functions<a class="headerlink" href="#define-the-training-and-test-functions" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train the network for one epoch&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
+
+    <span class="n">train_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+        <span class="c1"># Get predictions</span>
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="c1"># This if block is needed to add a dummy dimension if our inputs are 1D</span>
+            <span class="c1"># (where each number is a different sample)</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">network</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>
+
+        <span class="c1"># Compute the loss</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span>
+        <span class="n">train_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Clear the gradients</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>
+
+        <span class="c1"># Backpropagation to compute the gradients and update the weights</span>
+        <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>
+
+    <span class="k">return</span> <span class="n">train_loss</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Test the network&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>  <span class="c1"># Evaluation mode (important when having dropout layers)</span>
+
+    <span class="n">test_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+            <span class="c1"># Get predictions</span>
+            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="c1"># This if block is needed to add a dummy dimension if our inputs are 1D</span>
+                <span class="c1"># (where each number is a different sample)</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">network</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>
+
+            <span class="c1"># Compute the loss</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span>
+            <span class="n">test_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Get an average loss for the entire dataset</span>
+        <span class="n">test_loss</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">test_loss</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">fit_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">val_loader</span><span class="p">,</span> <span class="n">n_epochs</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train and validate the network&quot;&quot;&quot;</span>
+    <span class="n">train_losses</span><span class="p">,</span> <span class="n">val_losses</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
+    <span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+        <span class="n">train_loss</span> <span class="o">=</span> <span class="n">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+        <span class="n">val_loss</span> <span class="o">=</span> <span class="n">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">val_loader</span><span class="p">)</span>
+        <span class="n">train_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">train_loss</span><span class="p">)</span>
+        <span class="n">val_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">val_loss</span><span class="p">)</span>
+    <span class="n">end_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Training completed in </span><span class="si">{</span><span class="nb">int</span><span class="p">(</span><span class="n">end_time</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">start_time</span><span class="p">)</span><span class="si">}</span><span class="s2"> seconds.&quot;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">train_losses</span><span class="p">,</span> <span class="n">val_losses</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="train-the-network">
+<h3>Train the Network<a class="headerlink" href="#train-the-network" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Initialize the network</span>
+<span class="n">nn_3l</span> <span class="o">=</span> <span class="n">NetANN</span><span class="p">(</span><span class="n">W</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+
+<span class="c1"># MSE loss function</span>
+<span class="n">criterion</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>
+
+<span class="c1"># Adam optimizer</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">Adam</span><span class="p">(</span><span class="n">nn_3l</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.003</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">24</span>
+<span class="n">train_loss</span><span class="p">,</span> <span class="n">val_loss</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">nn_3l</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">val_loader</span><span class="p">,</span> <span class="n">n_epochs</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 21 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="visualizing-the-results">
+<h3>Visualizing the Results<a class="headerlink" href="#visualizing-the-results" title="Permalink to this headline">#</a></h3>
+<p>Comparing the training and validation loss curves</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Training loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">val_loss</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Validation loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/76454e3245c1605c48f1add1948c7f100bfd8efb8252b1f386eb748ff7cc603c.png" src="../_images/76454e3245c1605c48f1add1948c7f100bfd8efb8252b1f386eb748ff7cc603c.png" />
+</div>
+</div>
+<p>Since the NN has one input and one output, we can plot it as a function <span class="math notranslate nohighlight">\(nn(X)\)</span> (orange), and compare it to the polyfit (blue) and Wilks 2005 polynomial (black dashed).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span>
+    <span class="n">x_input</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span>
+    <span class="n">X_tend</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span>
+    <span class="n">bins</span><span class="o">=</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">50</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">150</span><span class="p">)),</span>
+    <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Greys</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p18</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="s2">&quot;k--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ - Wilks, 2005&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ lin. regr.&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">x</span><span class="p">,</span>
+    <span class="p">(</span><span class="n">nn_3l</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)),</span> <span class="mi">1</span><span class="p">)))</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">(),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;NN-3L&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;X$_k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Missing X$_k$ tendency&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;NN fit&quot;</span><span class="p">);</span>
+</pre></div>
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-model-its-parameters-and-other-utility-functions">Defining the Model, its Parameters, and Other Utility Functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-general-circulation-model-gcm">Defining the General Circulation Model (GCM)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-utility-functions">Defining the Utility Functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initializing-the-lorenz-1996-model-parameters">Initializing the Lorenz 1996 Model Parameters</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#suggestions-for-modifying-the-l96-model-paramters">Suggestions for Modifying the L96 Model Paramters</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-truth-run-from-two-time-scale-l96-model">Generate <code class="docutils literal notranslate"><span class="pre">Truth</span></code> Run from Two Time-Scale L96 Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-synthetic-observations">Generate Synthetic Observations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#apply-localization-to-the-background-model-covariance">Apply Localization to the Background Model Covariance</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#run-data-assimilation">Run Data Assimilation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#post-processing-and-visualization">Post Processing and Visualization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examining-the-relationship-between-the-members-and-their-increments">Examining the Relationship between the Members and their Increments</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#individual-ensemble-members">Individual Ensemble Members</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#mean-over-ensemble-members">Mean over Ensemble Members</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-the-da-increments">Learning the DA Increments</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#observing-the-dataset">Observing the Dataset</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#creating-the-dataset-split">Creating the Dataset Split</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#building-the-dataset">Building the Dataset</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-model">Define the Model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-training-and-test-functions">Define the Training and Test Functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-the-network">Train the Network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-results">Visualizing the Results</a></li>
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+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
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+        <div class="sidebar-primary-item"><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="citing.html">Citing this book</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../bibliography.html">References</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Interpreting Neural Networks</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#set-up-the-model-and-networks">Set up the model and networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-l96-data">Generate L96 data</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#load-a-pretrained-neural-network">Load a pretrained neural network</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#feature-importance">Feature importance</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#saliency-maps">Saliency maps</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-saliency-maps-using-input-gradients">Generate saliency maps using input gradients</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#full-jacobian-method">Full Jacobian method</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#create-and-fit-a-linear-regression-model">Create and fit a Linear Regression model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#approximate-jacobian-method">Approximate Jacobian method</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-saliency-maps-layerwise-relevance-propagation-lrp">Generate saliency maps layerwise relevance propagation (LRP)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparing-all-of-the-methods">Comparing all of the methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="interpreting-neural-networks">
+<h1>Interpreting Neural Networks<a class="headerlink" href="#interpreting-neural-networks" title="Permalink to this headline">#</a></h1>
+<div class="section" id="outline">
+<h2>Outline<a class="headerlink" href="#outline" title="Permalink to this headline">#</a></h2>
+<p>Neural networks are great at approximating complex functions, but usually provide little direct insight or interpretation into the functional relationships that are approximated. In this notebook we present some techniques that can be used to probe neural networks and indirectly infer what the network has learnt.</p>
+<p>We present two class of approaches: feature importance and saliency maps.</p>
+<div class="section" id="set-up-the-model-and-networks">
+<h3>Set up the model and networks<a class="headerlink" href="#set-up-the-model-and-networks" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">math</span>
+
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch.autograd</span> <span class="kn">import</span> <span class="n">Variable</span><span class="p">,</span> <span class="n">grad</span>
+<span class="kn">from</span> <span class="nn">torch.autograd.functional</span> <span class="kn">import</span> <span class="n">jacobian</span>
+<span class="kn">import</span> <span class="nn">torch.nn.functional</span> <span class="k">as</span> <span class="nn">F</span>
+<span class="kn">import</span> <span class="nn">torch.utils.data</span> <span class="k">as</span> <span class="nn">Data</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+
+<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">r2_score</span>
+<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
+
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="p">(</span>
+    <span class="n">L96</span><span class="p">,</span>
+    <span class="n">L96_eq1_xdot</span><span class="p">,</span>
+    <span class="n">integrate_L96_2t</span><span class="p">,</span>
+    <span class="n">EulerFwd</span><span class="p">,</span>
+    <span class="n">RK2</span><span class="p">,</span>
+    <span class="n">RK4</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="o">%</span><span class="k">matplotlib</span> inline
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>  <span class="c1"># For reproducibility</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>  <span class="c1"># For reproducibility</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;torch._C.Generator at 0x7fd0101f0a90&gt;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create a confusion matrix like figure</span>
+<span class="k">def</span> <span class="nf">imshow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">colorbar_pct</span><span class="o">=</span><span class="mf">97.5</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;seismic&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">vlim</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kw</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">vlim</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="n">vlim</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">percentile</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">colorbar_pct</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Input dimension&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Output dimension&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">))</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">))</span>
+    <span class="n">im</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=-</span><span class="n">vlim</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="n">vlim</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">cmap</span><span class="p">,</span> <span class="o">**</span><span class="n">kw</span><span class="p">)</span>
+    <span class="n">cb</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+    <span class="k">if</span> <span class="n">label</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="n">cb</span><span class="o">.</span><span class="n">set_label</span><span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">im</span>
+
+
+<span class="k">def</span> <span class="nf">plot_feature_importance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">feature_index</span><span class="p">):</span>
+    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
+    <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_axes</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Shift in score&quot;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Column&quot;</span><span class="p">)</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Permutation Feature Importance for k=&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">feature_index</span><span class="p">))</span>
+
+    <span class="n">predictors</span> <span class="o">=</span> <span class="n">result</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
+    <span class="n">scores</span> <span class="o">=</span> <span class="n">result</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span>
+    <span class="n">y_pos</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">predictors</span><span class="p">))</span>
+
+    <span class="n">ax</span><span class="o">.</span><span class="n">barh</span><span class="p">(</span><span class="n">y_pos</span><span class="p">,</span> <span class="n">scores</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">y_pos</span><span class="p">,</span> <span class="n">predictors</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="generate-l96-data">
+<h3>Generate L96 data<a class="headerlink" href="#generate-l96-data" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">time_steps</span> <span class="o">=</span> <span class="mi">20000</span>
+<span class="n">Forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span> <span class="o">*</span> <span class="n">time_steps</span>
+
+<span class="c1"># Create a &quot;synthetic world&quot; with K=8 and J=32</span>
+<span class="n">K</span> <span class="o">=</span> <span class="mi">8</span>
+<span class="n">J</span> <span class="o">=</span> <span class="mi">32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Forcing</span><span class="p">)</span>
+<span class="c1"># Get training data for the neural network.</span>
+
+<span class="c1"># - Run the true state and output subgrid tendencies (the effect of Y on X is xytrue):</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">xy_true</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="load-a-pretrained-neural-network">
+<h3>Load a pretrained neural network<a class="headerlink" href="#load-a-pretrained-neural-network" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Specify a path</span>
+<span class="n">PATH</span> <span class="o">=</span> <span class="s2">&quot;networks/network_3_layers_100_epoches.pth&quot;</span>
+<span class="c1"># Load</span>
+<span class="n">weights</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">PATH</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Net_ANN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_ANN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs, 16 neurons for first hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 16 neurons for second hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="o">.</span><span class="n">weight</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">()</span>
+<span class="n">model</span><span class="o">.</span><span class="n">load_state_dict</span><span class="p">(</span><span class="n">weights</span><span class="p">)</span>
+<span class="n">model</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Net_ANN(
+  (linear1): Linear(in_features=8, out_features=16, bias=True)
+  (linear2): Linear(in_features=16, out_features=16, bias=True)
+  (linear3): Linear(in_features=16, out_features=8, bias=True)
+)
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="feature-importance">
+<h2>Feature importance<a class="headerlink" href="#feature-importance" title="Permalink to this headline">#</a></h2>
+<p>Feature importance is a simple technique that allows us to identify the contribution that each input feature has towards bringing a neural network’s output close to the target (reducing the loss or performing well on some skill metric).</p>
+<p>In this process the importance of each input feature is evaluate by scrambling its value and evaluating how much the network output changed in response. This changed output is compared to the target, to see whether the particular input has small or large effect on the network output.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># First we compute the baeline score (how the model does with original data)</span>
+<span class="n">feature_index</span> <span class="o">=</span> <span class="mi">3</span>  <span class="c1"># which k do we focus on for output.</span>
+<span class="n">baseline_r2</span> <span class="o">=</span> <span class="n">r2_score</span><span class="p">(</span>
+    <span class="n">model</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">X_true</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[:,</span> <span class="n">feature_index</span><span class="p">],</span>
+    <span class="n">xy_true</span><span class="p">[:,</span> <span class="n">feature_index</span><span class="p">],</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">scrambled_r2</span> <span class="o">=</span> <span class="p">[]</span>
+<span class="k">for</span> <span class="n">column</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+    <span class="c1"># Create a copy of X_test</span>
+    <span class="n">X_copy</span> <span class="o">=</span> <span class="n">X_true</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+    <span class="c1"># Scramble the values of the given predictor</span>
+    <span class="n">shuffle_col</span> <span class="o">=</span> <span class="n">X_copy</span><span class="p">[:,</span> <span class="n">column</span><span class="p">]</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">shuffle_col</span><span class="p">)</span>
+    <span class="n">X_copy</span><span class="p">[:,</span> <span class="n">column</span><span class="p">]</span> <span class="o">=</span> <span class="n">shuffle_col</span>
+
+    <span class="c1"># Calculate the new R2</span>
+    <span class="n">score</span> <span class="o">=</span> <span class="n">r2_score</span><span class="p">(</span>
+        <span class="n">model</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">X_copy</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()[:,</span> <span class="n">feature_index</span><span class="p">],</span>
+        <span class="n">xy_true</span><span class="p">[:,</span> <span class="n">feature_index</span><span class="p">],</span>
+    <span class="p">)</span>
+
+    <span class="c1"># Append the increase in R2 to the list of results</span>
+    <span class="n">scrambled_r2</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">column</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">score</span> <span class="o">-</span> <span class="n">baseline_r2</span><span class="p">)])</span>
+
+<span class="c1"># Put the results into a pandas dataframe and rank the predictors by score</span>
+<span class="n">scrambled_r2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">scrambled_r2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plot_feature_importance</span><span class="p">(</span><span class="n">scrambled_r2</span><span class="p">,</span> <span class="n">feature_index</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/9ce19cfb313f251e04c2548f77b1645f31f16d4ff080cd2d3237d9e6563913d4.png" src="../_images/9ce19cfb313f251e04c2548f77b1645f31f16d4ff080cd2d3237d9e6563913d4.png" />
+</div>
+</div>
+<p>Notice above that that we are considering the impact that shuffling the inputs has on a particular output feature ( feature index). The plot below shows that for this network the largest deterioration in model skill takes place when the input feature at the same k as the output k is varied, suggesting the local behavior of the dependence between the inputs and outputs.</p>
+</div>
+<div class="section" id="saliency-maps">
+<h2>Saliency maps<a class="headerlink" href="#saliency-maps" title="Permalink to this headline">#</a></h2>
+<p>Saliency maps are popular visualization techniques for gaining insights on why neural networks made a particular decision on the given input data. They are usually rendered as a heatmap, where hotness corresponds to regions that have a big impact on the model’s final decision. They are helpful, for example, when you are frustrated by your model incorrectly classifying a certain datapoint, because you can look at the input features that led to that decision <span id="id1">[<a class="reference internal" href="../bibliography.html#id16" title="Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: visualising image classification models and saliency maps. 2013. URL: https://arxiv.org/abs/1312.6034, doi:10.48550/ARXIV.1312.6034.">Simonyan <em>et al.</em>, 2013</a>]</span>, <span id="id2">[<a class="reference internal" href="../bibliography.html#id17" title="David Baehrens, Timon Schroeter, Stefan Harmeling, Motoaki Kawanabe, Katja Hansen, and Klaus-Robert Müller. How to explain individual classification decisions. Journal of Machine Learning Research, 11(61):1803–1831, 2010. URL: http://jmlr.org/papers/v11/baehrens10a.html.">Baehrens <em>et al.</em>, 2010</a>]</span>, <span id="id3">[<a class="reference internal" href="../bibliography.html#id18" title="Julius Adebayo, Justin Gilmer, Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been Kim. Sanity checks for saliency maps. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL: https://proceedings.neurips.cc/paper_files/paper/2018/file/294a8ed24b1ad22ec2e7efea049b8737-Paper.pdf.">Adebayo <em>et al.</em>, 2018</a>]</span>.</p>
+<p>In this notebook, we will explore the different ways one could render saliency maps for the L96 data and visualize them.</p>
+</div>
+<div class="section" id="generate-saliency-maps-using-input-gradients">
+<h2>Generate saliency maps using input gradients<a class="headerlink" href="#generate-saliency-maps-using-input-gradients" title="Permalink to this headline">#</a></h2>
+<p>Since neural networks are differentiable, the simplest way of generating saliency maps (i.e. a quantification of the sensitivity of the output to the input) is to just take its first derivative with respect to its inputs (or Jacobian, for networks with multiple inputs and outputs).</p>
+<p>We can do this easily in Pytorch using <code class="docutils literal notranslate"><span class="pre">torch.autograd.functional.jacobian</span></code>.</p>
+<div class="section" id="full-jacobian-method">
+<h3>Full Jacobian method<a class="headerlink" href="#full-jacobian-method" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%%time</span>
+<span class="n">jacobians</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
+    <span class="p">[</span>
+        <span class="n">jacobian</span><span class="p">(</span>
+            <span class="n">model</span><span class="p">,</span>
+            <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">single</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]),</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">),</span>
+            <span class="n">create_graph</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+        <span class="p">)</span>
+        <span class="o">.</span><span class="n">detach</span><span class="p">()</span>
+        <span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">200</span><span class="p">)</span>
+    <span class="p">]</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>CPU times: user 240 ms, sys: 3.74 ms, total: 244 ms
+Wall time: 206 ms
+</pre></div>
+</div>
+</div>
+</div>
+<p>This gives us an array of 8x8 gradients, one for each of 200 input samples. Let’s visualize their average value, as well as their standard deviation:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span>
+    <span class="s2">&quot;Mean and standard deviation of NN input gradients across the dataset&quot;</span><span class="p">,</span>
+    <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">,</span>
+    <span class="n">y</span><span class="o">=</span><span class="mf">1.025</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Average value&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">imshow</span><span class="p">(</span><span class="n">jacobians</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Average input derivative&quot;</span><span class="p">,</span> <span class="n">vlim</span><span class="o">=</span><span class="mf">0.85</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Standard deviation&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">imshow</span><span class="p">(</span><span class="n">jacobians</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Standard deviation&quot;</span><span class="p">,</span> <span class="n">vlim</span><span class="o">=</span><span class="mf">0.85</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/381a4153528383c6d1e79ec17985197010841832732ce37af443c79b4121a3f4.png" src="../_images/381a4153528383c6d1e79ec17985197010841832732ce37af443c79b4121a3f4.png" />
+</div>
+</div>
+<p>The dominant term in the average gradient is close to -1 along the main diagonal, but there are significant off-diagonal elements, and also significant deviation across samples. It’s interesting to compare this to the behavior of a linear regression model:</p>
+</div>
+<div class="section" id="create-and-fit-a-linear-regression-model">
+<h3>Create and fit a Linear Regression model<a class="headerlink" href="#create-and-fit-a-linear-regression-model" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">lr</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
+<span class="n">lr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">xy_true</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Comparing to linear model&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">imshow</span><span class="p">(</span><span class="n">lr</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Linear regression weight&quot;</span><span class="p">,</span> <span class="n">vlim</span><span class="o">=</span><span class="mf">0.85</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/81380d6e67290d31cc5b4e1c8fce97df6fb0cb7b059ac3427541424bc8b32bf9.png" src="../_images/81380d6e67290d31cc5b4e1c8fce97df6fb0cb7b059ac3427541424bc8b32bf9.png" />
+</div>
+</div>
+<p>We see that the weights of a linear regression model generally match the average input gradients of the neural network, especially down the main diagonal. This makes some sense given that, at each point, input gradients represent the best <em>local</em> linear model that approximates the nonlinear neural network.</p>
+<p>Although computing full Jacobians works fine for a small example, it can become expensive for large networks and input/output dimensions, so we can also approximate it using finite differences:</p>
+</div>
+<div class="section" id="approximate-jacobian-method">
+<h3>Approximate Jacobian method<a class="headerlink" href="#approximate-jacobian-method" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%%time</span>
+<span class="n">epsilon</span> <span class="o">=</span> <span class="mf">1e-2</span>
+<span class="n">approx_jacobians</span> <span class="o">=</span> <span class="p">[]</span>
+
+<span class="k">for</span> <span class="n">case</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">200</span><span class="p">):</span>
+    <span class="n">inputs</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="n">case</span><span class="p">,</span> <span class="p">:]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+    <span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">single</span><span class="p">(</span><span class="n">inputs</span><span class="p">),</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+    <span class="n">pred</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
+    <span class="n">Js</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">pred</span><span class="p">)))</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)):</span>
+        <span class="n">perturb</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
+        <span class="n">perturb</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">epsilon</span>
+        <span class="n">inputs_perturbed</span> <span class="o">=</span> <span class="p">(</span><span class="n">inputs</span> <span class="o">+</span> <span class="n">perturb</span><span class="p">)</span><span class="o">.</span><span class="n">clone</span><span class="p">()</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
+        <span class="n">Js</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">model</span><span class="p">(</span><span class="n">inputs_perturbed</span><span class="p">)</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span> <span class="o">-</span> <span class="n">pred</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+        <span class="p">)</span> <span class="o">/</span> <span class="n">epsilon</span>
+    <span class="n">approx_jacobians</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Js</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
+
+<span class="n">approx_jacobians</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">approx_jacobians</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>CPU times: user 232 ms, sys: 0 ns, total: 232 ms
+Wall time: 232 ms
+</pre></div>
+</div>
+</div>
+</div>
+<p>(Technically this is slightly slower than the previous example, but it can be more performant for large networks.)</p>
+<p>Let’s see how that looks:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Interpreting the network with finite differences&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">imshow</span><span class="p">(</span><span class="n">approx_jacobians</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Average finite difference&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/d28615143968924da88b9cc3758ea7cb8e649ad8f2d45d795a6ffd4cdc307dad.png" src="../_images/d28615143968924da88b9cc3758ea7cb8e649ad8f2d45d795a6ffd4cdc307dad.png" />
+</div>
+</div>
+<p>As expected, it’s fairly similar to the full Jacobian method.</p>
+</div>
+</div>
+<div class="section" id="generate-saliency-maps-layerwise-relevance-propagation-lrp">
+<h2>Generate saliency maps layerwise relevance propagation (LRP)<a class="headerlink" href="#generate-saliency-maps-layerwise-relevance-propagation-lrp" title="Permalink to this headline">#</a></h2>
+<p>Another proposed method for generating saliency maps is <span id="id4">[<a class="reference internal" href="../bibliography.html#id20" title="Sebastian Bach, Alexander Binder, Grégoire Montavon, Frederick Klauschen, Klaus-Robert Müller, and Wojciech Samek. On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation. PLOS ONE, 10(7):1-46, 07 2015. URL: https://doi.org/10.1371/journal.pone.0130140, doi:10.1371/journal.pone.0130140.">Bach <em>et al.</em>, 2015</a>]</span>. With a baseline of 0, this method is akin to multiplying the input gradients by the input itself (per <span id="id5">[<a class="reference internal" href="../bibliography.html#id19" title="Marco Ancona, Enea Ceolini, Cengiz Öztireli, and Markus Gross. Towards better understanding of gradient-based attribution methods for deep neural networks. In International Conference on Learning Representations. 2018. URL: https://openreview.net/forum?id=Sy21R9JAW.">Ancona <em>et al.</em>, 2018</a>]</span>), but it has become popular in the climate community and does support alternative baselines.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">weights</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
+    <span class="nb">print</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>linear1.weight
+linear1.bias
+linear2.weight
+linear2.bias
+linear3.weight
+linear3.bias
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">epsilon</span> <span class="o">=</span> <span class="mf">0.0</span>  <span class="c1"># filtering small values</span>
+<span class="n">gamma</span> <span class="o">=</span> <span class="mf">0.0</span>  <span class="c1"># give more weights to positive values</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">## get the weight and bias of the NN</span>
+<span class="k">def</span> <span class="nf">get_weight</span><span class="p">(</span><span class="n">weightsname</span><span class="p">):</span>
+    <span class="n">Ws</span> <span class="o">=</span> <span class="p">[]</span>
+    <span class="n">Bs</span> <span class="o">=</span> <span class="p">[]</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">name</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">weights</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
+        <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">Ws</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">weights</span><span class="p">[</span><span class="n">name</span><span class="p">]))</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">Bs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">weights</span><span class="p">[</span><span class="n">name</span><span class="p">]))</span>
+    <span class="k">return</span> <span class="n">Ws</span><span class="p">,</span> <span class="n">Bs</span>  <span class="c1"># weights and biases</span>
+
+
+<span class="c1"># forward pass to calculate the output of each layer</span>
+<span class="k">def</span> <span class="nf">forward_pass</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">Ws</span><span class="p">,</span> <span class="n">Bs</span><span class="p">):</span>
+    <span class="n">L</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Ws</span><span class="p">)</span>
+    <span class="n">forward</span> <span class="o">=</span> <span class="p">[</span><span class="n">data</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">L</span>
+
+    <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">L</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+        <span class="n">forward</span><span class="p">[</span><span class="n">l</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">Ws</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">forward</span><span class="p">[</span><span class="n">l</span><span class="p">]))</span> <span class="o">+</span> <span class="n">Bs</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>  <span class="c1"># ativation ReLu</span>
+
+    <span class="c1">## for last layer that does not have activation function</span>
+    <span class="n">forward</span><span class="p">[</span><span class="n">L</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ws</span><span class="p">[</span><span class="n">L</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">forward</span><span class="p">[</span><span class="n">L</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="n">Bs</span><span class="p">[</span><span class="n">L</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>  <span class="c1"># linear last layer</span>
+    <span class="k">return</span> <span class="n">forward</span>
+
+
+<span class="k">def</span> <span class="nf">rho</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">l</span><span class="p">):</span>
+    <span class="n">w_intermediate</span> <span class="o">=</span> <span class="n">w</span> <span class="o">+</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">][</span><span class="n">l</span><span class="p">]</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">w_intermediate</span> <span class="o">+</span> <span class="n">gamma</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">w_intermediate</span><span class="p">)</span>
+
+
+<span class="k">def</span> <span class="nf">incr</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">l</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">z</span> <span class="o">+</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">][</span><span class="n">l</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">**</span> <span class="mf">0.5</span> <span class="o">+</span> <span class="mf">1e-9</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">## backward pass to compute the LRP of each layer. Same rule applied to the first layer (input layer)</span>
+<span class="k">def</span> <span class="nf">onelayer_LRP</span><span class="p">(</span><span class="n">W</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">forward</span><span class="p">,</span> <span class="n">nz</span><span class="p">,</span> <span class="n">zz</span><span class="p">):</span>
+    <span class="n">mask</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nz</span><span class="p">))</span>
+    <span class="n">mask</span><span class="p">[</span><span class="n">zz</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
+    <span class="n">L</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">W</span><span class="p">)</span>
+    <span class="n">R</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">L</span> <span class="o">+</span> <span class="p">[</span><span class="n">forward</span><span class="p">[</span><span class="n">L</span><span class="p">]</span> <span class="o">*</span> <span class="n">mask</span><span class="p">]</span>  <span class="c1"># start from last layer Relevance</span>
+
+    <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">L</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
+        <span class="n">w</span> <span class="o">=</span> <span class="n">rho</span><span class="p">(</span><span class="n">W</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">l</span><span class="p">)</span>
+        <span class="n">b</span> <span class="o">=</span> <span class="n">rho</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">l</span><span class="p">)</span>
+        <span class="n">z</span> <span class="o">=</span> <span class="n">incr</span><span class="p">(</span><span class="n">w</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">forward</span><span class="p">[</span><span class="n">l</span><span class="p">])</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">epsilon</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>  <span class="c1"># step 1 - forward pass</span>
+        <span class="n">s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">l</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>  <span class="c1"># step 2 - element-wise division</span>
+        <span class="n">c</span> <span class="o">=</span> <span class="n">w</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>  <span class="c1"># step 3 - backward pass</span>
+        <span class="n">R</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">forward</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">*</span> <span class="n">c</span>  <span class="c1"># step 4 - element-wise product</span>
+    <span class="k">return</span> <span class="n">R</span>
+
+
+<span class="k">def</span> <span class="nf">LRP_alllayer</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">weights</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;inputs:</span>
+<span class="sd">        data: for single sample, with the right asix, the shape is (nz,naxis)</span>
+<span class="sd">        weights: dictionary of weights and biases</span>
+<span class="sd">    output:</span>
+<span class="sd">        LRP, shape: (nx,L+1) that each of the column consist of L+1 array</span>
+<span class="sd">        Relevance of fisrt layer&#39;s pixels&quot;&quot;&quot;</span>
+    <span class="n">nx</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+    <span class="c1">## step 1: get the wieghts</span>
+    <span class="n">Ws</span><span class="p">,</span> <span class="n">Bs</span> <span class="o">=</span> <span class="n">get_weight</span><span class="p">(</span><span class="n">weights</span><span class="p">)</span>
+
+    <span class="c1">## step 2: call the forward pass to get the intermediate layers output</span>
+    <span class="n">inter_layer</span> <span class="o">=</span> <span class="n">forward_pass</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">Ws</span><span class="p">,</span> <span class="n">Bs</span><span class="p">)</span>
+
+    <span class="c1">## loop over all z and get the LRP of each layer</span>
+    <span class="n">R_all</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">nx</span>
+    <span class="n">relevance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="n">nx</span><span class="p">))</span>
+    <span class="k">for</span> <span class="n">xx</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nx</span><span class="p">):</span>
+        <span class="n">R_all</span><span class="p">[</span><span class="n">xx</span><span class="p">]</span> <span class="o">=</span> <span class="n">onelayer_LRP</span><span class="p">(</span><span class="n">Ws</span><span class="p">,</span> <span class="n">Bs</span><span class="p">,</span> <span class="n">inter_layer</span><span class="p">,</span> <span class="n">nx</span><span class="p">,</span> <span class="n">xx</span><span class="p">)</span>
+        <span class="n">relevance</span><span class="p">[</span><span class="n">xx</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">R_all</span><span class="p">[</span><span class="n">xx</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
+
+    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">R_all</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">),</span> <span class="n">relevance</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%%time</span>
+<span class="n">R_many</span> <span class="o">=</span> <span class="p">[]</span>
+<span class="k">for</span> <span class="n">case</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">200</span><span class="p">):</span>
+    <span class="n">inputs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">copy</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">case</span><span class="p">,</span> <span class="p">:])</span>
+    <span class="n">_</span><span class="p">,</span> <span class="n">Rs</span> <span class="o">=</span> <span class="n">LRP_alllayer</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">weights</span><span class="p">)</span>
+    <span class="n">R_many</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Rs</span><span class="p">)</span>
+<span class="n">LRP</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="n">R_many</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>CPU times: user 145 ms, sys: 79 µs, total: 145 ms
+Wall time: 144 ms
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Interpreting the network with</span><span class="se">\n</span><span class="s2">layerwise relevance propagation&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">imshow</span><span class="p">(</span><span class="n">LRP</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Average LRP&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/41e34cf57dc43bc73b5d468bd5140914de293287344dd7e35641073fec7c6d40.png" src="../_images/41e34cf57dc43bc73b5d468bd5140914de293287344dd7e35641073fec7c6d40.png" />
+</div>
+</div>
+<p>LRP outputs something qualitatively different from the gradient-based methods, and in fact each element should be interpreted more as an attribution score (i.e. the actual contribution of an input to the output) than as a sensitivity score (i.e. how much an output changes with an input). Below, we’ll see that this approximately reduces to multiplying the gradient by the input.</p>
+</div>
+<div class="section" id="comparing-all-of-the-methods">
+<h2>Comparing all of the methods<a class="headerlink" href="#comparing-all-of-the-methods" title="Permalink to this headline">#</a></h2>
+<p>Let’s look at the output of these methods side-by-side:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">comparisons</span> <span class="o">=</span> <span class="p">[</span>
+    <span class="p">(</span><span class="n">jacobians</span><span class="p">,</span> <span class="s2">&quot;Jacobian (exact)&quot;</span><span class="p">),</span>
+    <span class="p">(</span><span class="n">approx_jacobians</span><span class="p">,</span> <span class="s2">&quot;Jacobian (finite diff)&quot;</span><span class="p">),</span>
+    <span class="p">(</span><span class="n">LRP</span><span class="p">,</span> <span class="s2">&quot;LRP&quot;</span><span class="p">),</span>
+    <span class="p">(</span><span class="n">jacobians</span> <span class="o">*</span> <span class="n">X_true</span><span class="p">[:</span><span class="mi">200</span><span class="p">][:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:],</span> <span class="s2">&quot;Jacobian * input&quot;</span><span class="p">),</span>
+<span class="p">]</span>
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span>
+    <span class="s2">&quot;Comparing average outputs of saliency methods&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s2">&quot;bottom&quot;</span>
+<span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">saliency</span><span class="p">,</span> <span class="n">label</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">comparisons</span><span class="p">):</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+    <span class="n">imshow</span><span class="p">(</span><span class="n">saliency</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/17508470a9d04959beee538fb629d9535416609512c702ec8f635271176753e9.png" src="../_images/17508470a9d04959beee538fb629d9535416609512c702ec8f635271176753e9.png" />
+</div>
+</div>
+<p>As expected, LRP and Jacobian * input produce very similar outputs. For more information on the intricacies of saliency maps, see <span id="id6">[<a class="reference internal" href="../bibliography.html#id19" title="Marco Ancona, Enea Ceolini, Cengiz Öztireli, and Markus Gross. Towards better understanding of gradient-based attribution methods for deep neural networks. In International Conference on Learning Representations. 2018. URL: https://openreview.net/forum?id=Sy21R9JAW.">Ancona <em>et al.</em>, 2018</a>]</span> and <span id="id7">[<a class="reference internal" href="../bibliography.html#id20" title="Sebastian Bach, Alexander Binder, Grégoire Montavon, Frederick Klauschen, Klaus-Robert Müller, and Wojciech Samek. On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation. PLOS ONE, 10(7):1-46, 07 2015. URL: https://doi.org/10.1371/journal.pone.0130140, doi:10.1371/journal.pone.0130140.">Bach <em>et al.</em>, 2015</a>]</span>.</p>
+</div>
+<div class="section" id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline">#</a></h2>
+<p>In this notebook we introduced some approaches that can be used to better understand how neural networks estimate the output from a given input. This is an important area of research, as we would want to test that the neural networks are producing the right answers for the right reasons (physically reasonable models). This becomes particularly important when we want our networks to produce accurate results (extrapolate) for inputs that were not observed in training data.</p>
+<p>In the next notebook we show how physics based constraints can be incorporated into neural networks, in an attempt to further guide the neural network to learn more appropriate relationships between inputs and outputs.</p>
+</div>
+</div>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#set-up-the-model-and-networks">Set up the model and networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-l96-data">Generate L96 data</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#load-a-pretrained-neural-network">Load a pretrained neural network</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#feature-importance">Feature importance</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#saliency-maps">Saliency maps</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-saliency-maps-using-input-gradients">Generate saliency maps using input gradients</a><ul class="nav section-nav flex-column">
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#approximate-jacobian-method">Approximate Jacobian method</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-saliency-maps-layerwise-relevance-propagation-lrp">Generate saliency maps layerwise relevance propagation (LRP)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparing-all-of-the-methods">Comparing all of the methods</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
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+<p>If you found this work useful and would like to cite it, please use :
+Balwada, D., Abernathey, R., Acharya, S., Adcroft, A., Brener, J., Balaji, V., … &amp; Zanna, L. (2023). Learning Machine Learning with Lorenz-96. Authorea Preprints.</p>
+<p>DOI: 10.22541/essoar.170365239.95851488/v1
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
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+<h1>Outlook<a class="headerlink" href="#outlook" title="Permalink to this headline">#</a></h1>
+<p>With this JupyterBook, we have designed an introduction to the problem of machine learning for improving climate models. The Lorenz 96 model is a perfect benchmark to illustrate several ideas and concepts, but remains too simple to present a real test case for complex problems in climate physics and modeling.</p>
+<p>M2LInES has been pushing the boundaries of climate model development through the innovative use of scientific machine learning models to explore:</p>
+<ul class="simple">
+<li><p>New subgrid physics parameterizations in the ocean <span id="id1">[<a class="reference internal" href="../bibliography.html#id28" title="Arthur P. Guillaumin and Laure Zanna. Stochastic-deep learning parameterization of ocean momentum forcing. Journal of Advances in Modeling Earth Systems, 13(9):e2021MS002534, 2021. e2021MS002534 2021MS002534. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021MS002534, arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2021MS002534, doi:https://doi.org/10.1029/2021MS002534.">Guillaumin and Zanna, 2021</a>, <a class="reference internal" href="../bibliography.html#id26" title="Pavel Perezhogin, Laure Zanna, and Carlos Fernandez-Granda. Generative data-driven approaches for stochastic subgrid parameterizations in an idealized ocean model. arXiv preprint arXiv:2302.07984, 2023.">Perezhogin <em>et al.</em>, 2023</a>, <a class="reference internal" href="../bibliography.html#id24" title="Aakash Sane, Brandon G Reichl, Alistair Adcroft, and Laure Zanna. Parameterizing vertical mixing coefficients in the ocean surface boundary layer using neural networks. arXiv preprint arXiv:2306.09045, 2023.">Sane <em>et al.</em>, 2023</a>]</span>, sea-ice <span id="id2">[<a class="reference internal" href="../bibliography.html#id27" title="Lorenzo Zampieri, Gabriele Arduini, Marika Holland, Sarah P. E. Keeley, Kristian Mogensen, Matthew D. Shupe, and Steffen Tietsche. A machine learning correction model of the winter clear-sky temperature bias over the arctic sea ice in atmospheric reanalyses. Monthly Weather Review, 151(6):1443 - 1458, 2023. URL: https://journals.ametsoc.org/view/journals/mwre/151/6/MWR-D-22-0130.1.xml, doi:https://doi.org/10.1175/MWR-D-22-0130.1.">Zampieri <em>et al.</em>, 2023</a>]</span>, and atmosphere <span id="id3">[<a class="reference internal" href="../bibliography.html#id23">Shamekh and Gentine, 2023</a>, <a class="reference internal" href="../bibliography.html#id25" title="Janni Yuval and Paul A. O'Gorman. Neural-network parameterization of subgrid momentum transport in the atmosphere. Journal of Advances in Modeling Earth Systems, 15(4):e2023MS003606, 2023. e2023MS003606 2023MS003606. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2023MS003606, arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2023MS003606, doi:https://doi.org/10.1029/2023MS003606.">Yuval and O'Gorman, 2023</a>]</span>.</p></li>
+<li><p>New corrections from data assimilation for climate model components <span id="id4">[<a class="reference internal" href="../bibliography.html#id29" title="William E Chapman and Judith Berner. Benefits of deterministic and stochastic tendency adjustments in a climate model. arXiv preprint arXiv:2308.15295, 2023.">Chapman and Berner, 2023</a>, <a class="reference internal" href="../bibliography.html#id30" title="William Gregory, Mitchell Bushuk, Alistair Adcroft, Yongfei Zhang, and Laure Zanna. Deep learning of systematic sea ice model errors from data assimilation increments. Journal of Advances in Modeling Earth Systems, 15(10):e2023MS003757, 2023.">Gregory <em>et al.</em>, 2023</a>, <a class="reference internal" href="../bibliography.html#id31" title="William Gregory, Mitchell Bushuk, Yongfei Zhang, Alistair Adcroft, and Laure Zanna. Machine learning for online sea ice bias correction within global ice-ocean simulations. 2023. arXiv:2310.02488.">Gregory <em>et al.</em>, 2023</a>]</span>.</p></li>
+<li><p>Development of innovative architecture and algorithms for multiscale machine learning related to  turbulence in oceans and atmosphere <span id="id5">[<a class="reference internal" href="../bibliography.html#id32" title="Hugo Frezat, Julien Le Sommer, Ronan Fablet, Guillaume Balarac, and Redouane Lguensat. A posteriori learning for quasi-geostrophic turbulence parametrization. Journal of Advances in Modeling Earth Systems, 14(11):e2022MS003124, 2022. e2022MS003124 2022MS003124. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2022MS003124, arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2022MS003124, doi:https://doi.org/10.1029/2022MS003124.">Frezat <em>et al.</em>, 2022</a>, <a class="reference internal" href="../bibliography.html#id34" title="Karl Otness, Laure Zanna, and Joan Bruna. Data-driven multiscale modeling of subgrid parameterizations in climate models. arXiv preprint arXiv:2303.17496, 2023.">Otness <em>et al.</em>, 2023</a>, <a class="reference internal" href="../bibliography.html#id33" title="Christian Pedersen, Laure Zanna, Joan Bruna, and Pavel Perezhogin. Reliable coarse-grained turbulent simulations through combined offline learning and neural emulation. arXiv preprint arXiv:2307.13144, 2023.">Pedersen <em>et al.</em>, 2023</a>, <a class="reference internal" href="../bibliography.html#id35" title="Andrew Ross, Ziwei Li, Pavel Perezhogin, Carlos Fernandez-Granda, and Laure Zanna. Benchmarking of machine learning ocean subgrid parameterizations in an idealized model. Journal of Advances in Modeling Earth Systems, pages e2022MS003258, 2022. e2022MS003258 2022MS003258. URL: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2022MS003258, arXiv:https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2022MS003258, doi:https://doi.org/10.1029/2022MS003258.">Ross <em>et al.</em>, 2022</a>]</span>.</p></li>
+</ul>
+<p>These novel techniques and data-driven models are now making their way into global climate models <span id="id6">[<a class="reference internal" href="../bibliography.html#id36" title="Cheng Zhang, Pavel Perezhogin, Cem Gultekin, Alistair Adcroft, Carlos Fernandez-Granda, and Laure Zanna. Implementation and evaluation of a machine learned mesoscale eddy parameterization into a numerical ocean circulation model. arXiv preprint arXiv:2303.00962, 2023.">Zhang <em>et al.</em>, 2023</a>]</span>, illustrating the importance of bringing climate physics, scientific computing, and modern machine learning techniques together.</p>
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+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
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+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Adding constraints to Neural Networks</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Adding constraints to Neural Networks</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#physical-constraints-in-l96">Physical constraints in L96</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#l96-model-with-momentum-conserving-term">L96 model with momentum conserving term</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#skill-of-different-ml-models-in-prediction">Skill of different ML models in prediction</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="adding-constraints-to-neural-networks">
+<h1>Adding constraints to Neural Networks<a class="headerlink" href="#adding-constraints-to-neural-networks" title="Permalink to this headline">#</a></h1>
+<div class="section" id="outline">
+<h2>Outline<a class="headerlink" href="#outline" title="Permalink to this headline">#</a></h2>
+<p>The goal of this notebook is to show how a physical or mathematical constraint can be built in to the learning process to try and nudge the ML model towards more consistent solutions.</p>
+</div>
+<div class="section" id="physical-constraints-in-l96">
+<h2>Physical constraints in L96<a class="headerlink" href="#physical-constraints-in-l96" title="Permalink to this headline">#</a></h2>
+<p>The <strong><span class="math notranslate nohighlight">\(1^{st}\)</span> order equation L96 model</strong>,</p>
+<div class="amsmath math notranslate nohighlight" id="equation-f640bbdd-181e-4f86-9525-4715d1cf38e3">
+<span class="eqno">(14)<a class="headerlink" href="#equation-f640bbdd-181e-4f86-9525-4715d1cf38e3" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\frac{d}{dt} X_k
+= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F 
+\end{equation}\]</div>
+<p>conserves energy,
+$<span class="math notranslate nohighlight">\(E_X = \frac{1}{2}&lt;X_K^2&gt;,\)</span>$
+in the unforced and undamped form (last 2 terms on the RHS are zero), which results due to the form of the non-linearity.</p>
+<p>However, the <strong><span class="math notranslate nohighlight">\(2^{nd}\)</span> order equation L96 model</strong></p>
+<div class="amsmath math notranslate nohighlight" id="equation-fb2860bd-11e8-4b9a-904a-fc442956277a">
+<span class="eqno">(15)<a class="headerlink" href="#equation-fb2860bd-11e8-4b9a-904a-fc442956277a" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}
+\\
+\frac{d}{dt} Y_{j,k}
+&amp;= - cbY_{j+1,k} \left( Y_{j+2,k} - Y_{j-1,k} \right) - c Y_{j,k} + \frac{hc}{b} X_k
+\end{align}\]</div>
+<p>does not have the same conservation property. Instead, for the unforced and undamped system the total energy,
+$<span class="math notranslate nohighlight">\(E = E_X + E_Y = \frac{1}{2}(&lt;X_K^2&gt; + &lt;Y_j^2&gt;),\)</span><span class="math notranslate nohighlight">\( 
+is conserved, with the &quot;subgrid tendencies&quot; being the exchange terms (\)</span>&lt;X_kY_{j,k}&gt;<span class="math notranslate nohighlight">\() between the large scale (\)</span>\frac{1}{2}&lt;X_K^2&gt;<span class="math notranslate nohighlight">\() and small scale energy reservoirs \)</span>\frac{1}{2}&lt;Y_j^2&gt;$.</p>
+<p>Using energetic arguments to contrain a parameterization is non-trivial, but can prove to be very beneficial. For example the Gent-McWilliams scheme guarantees that the sub-grid parameterization will reduce APE. The GEOMETRIC parameterization keeps track of sub-grid energy and the exchange between the resolved and sub-grid energy, in an attempt to have total energy conservation.</p>
+</div>
+<div class="section" id="l96-model-with-momentum-conserving-term">
+<h2>L96 model with momentum conserving term<a class="headerlink" href="#l96-model-with-momentum-conserving-term" title="Permalink to this headline">#</a></h2>
+<p>As shown above, the L96 model conserves total energy that is contained over both X and Y variables. However, for demonstration purposes we created a modified L96, where the coupling term is momentum conserving.
+This is a bit silly to do, as the L96 model does not conserve momentum (but no one can stop you!). This also changes the energetics of the original L96. The main reason for this choice was because it was the easiest to implement.</p>
+<p>The equations of the new model are:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-1392ad8c-e72f-4301-843f-dd37b2989361">
+<span class="eqno">(16)<a class="headerlink" href="#equation-1392ad8c-e72f-4301-843f-dd37b2989361" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \left( \frac{hc}{b} \right) \left(\sum_{j=0}^{J-1} Y_{j,k-1} - \sum_{j=0}^{J-1} Y_{j,k+1}\right)
+\\
+\frac{d}{dt} Y_{j,k}
+&amp;= - cbY_{j+1,k} \left( Y_{j+2,k} - Y_{j-1,k} \right) - c Y_{j,k} + \frac{hc}{b} X_k
+\end{align}\]</div>
+<p>The small scale forcing from the small scale to the large scale is:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-0cbea324-b614-42d7-93b1-be57b5406d55">
+<span class="eqno">(17)<a class="headerlink" href="#equation-0cbea324-b614-42d7-93b1-be57b5406d55" title="Permalink to this equation">#</a></span>\[\begin{equation}
+G_K = S_{k-1} - S_{k+1} = - \left( \frac{hc}{b} \right) \left(\sum_{j=0}^{J-1} Y_{j,k-1} - \sum_{j=0}^{J-1} Y_{j,k+1}\right)
+\end{equation}\]</div>
+<p>It is clear that summing <span class="math notranslate nohighlight">\(G_k\)</span> over the full domain is identically zero, since it is in a flux form. This term does not input any net momentum into the system, <span class="math notranslate nohighlight">\(&lt;G_k&gt;=0\)</span>. We will use this as a contrain in this exercise.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">xarray</span> <span class="k">as</span> <span class="nn">xr</span>
+
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch.autograd</span> <span class="kn">import</span> <span class="n">Variable</span>
+<span class="kn">import</span> <span class="nn">torch.nn.functional</span> <span class="k">as</span> <span class="nn">F</span>
+<span class="kn">import</span> <span class="nn">torch.utils.data</span> <span class="k">as</span> <span class="nn">Data</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+<span class="kn">from</span> <span class="nn">tqdm</span> <span class="kn">import</span> <span class="n">tqdm</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Load modules for the model</span>
+<span class="kn">from</span> <span class="nn">L96_conservative_model</span> <span class="kn">import</span> <span class="p">(</span>
+    <span class="n">L96</span><span class="p">,</span>
+    <span class="n">L96_eq1_xdot</span><span class="p">,</span>
+    <span class="n">integrate_L96_2t</span><span class="p">,</span>
+<span class="p">)</span>  <span class="c1"># L96_model_XYtend Adds the option to ouptput the subgrid tendencies (effect of Y on X)</span>
+<span class="kn">from</span> <span class="nn">L96_conservative_model</span> <span class="kn">import</span> <span class="n">EulerFwd</span><span class="p">,</span> <span class="n">RK2</span><span class="p">,</span> <span class="n">RK4</span>
+
+<span class="c1"># Note: We are using a slightly different L96 model here.</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/home/runner/work/L96_demo/L96_demo/notebooks/L96_conservative_model.py:17: NumbaDeprecationWarning: <span class=" -Color -Color-Bold">The &#39;nopython&#39; keyword argument was not supplied to the &#39;numba.jit&#39; decorator. The implicit default value for this argument is currently False, but it will be changed to True in Numba 0.59.0. See https://numba.readthedocs.io/en/stable/reference/deprecation.html#deprecation-of-object-mode-fall-back-behaviour-when-using-jit for details.</span>
+  def L96_eq1_xdot(X, F):
+/home/runner/work/L96_demo/L96_demo/notebooks/L96_conservative_model.py:39: NumbaDeprecationWarning: <span class=" -Color -Color-Bold">The &#39;nopython&#39; keyword argument was not supplied to the &#39;numba.jit&#39; decorator. The implicit default value for this argument is currently False, but it will be changed to True in Numba 0.59.0. See https://numba.readthedocs.io/en/stable/reference/deprecation.html#deprecation-of-object-mode-fall-back-behaviour-when-using-jit for details.</span>
+  def L96_2t_xdot_ydot(X, Y, F, h, b, c):
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s2">&quot;seaborn-v0_8&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s2">&quot;font.size&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">16</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Simulate the world</span>
+
+<span class="n">time_steps</span> <span class="o">=</span> <span class="mi">20000</span>
+<span class="n">Forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">12</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span> <span class="o">*</span> <span class="n">time_steps</span>
+
+<span class="c1"># Create a &quot;real world&quot; with K=8 and J=32</span>
+<span class="n">K</span> <span class="o">=</span> <span class="mi">8</span>
+<span class="n">J</span> <span class="o">=</span> <span class="mi">32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Forcing</span><span class="p">)</span>
+
+<span class="c1"># Get training data for the neural network.</span>
+<span class="c1"># - Run the true state and output subgrid tendencies (the effect of Y on X is xytrue):</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">T_true</span><span class="p">,</span> <span class="n">S_true</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># G_k = S_(k-1) - S_(k+1)</span>
+<span class="n">G_true</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">S_true</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">S_true</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># convert to xarray for convenience</span>
+
+<span class="n">X_true_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">X_true</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">T_true</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;X&quot;</span><span class="p">)</span>
+<span class="n">Y_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">y</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;JK&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">T_true</span><span class="p">,</span> <span class="s2">&quot;JK&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span> <span class="o">*</span> <span class="n">J</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;Y&quot;</span><span class="p">)</span>
+<span class="n">S_true_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">S_true</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">T_true</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;S&quot;</span><span class="p">)</span>
+<span class="n">G_true_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">G_true</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">T_true</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;G&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">141</span><span class="p">)</span>
+<span class="n">X_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">30</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">vmin</span><span class="o">=-</span><span class="mi">16</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="mi">21</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">142</span><span class="p">)</span>
+<span class="n">Y_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">30</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">vmin</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="mi">21</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">143</span><span class="p">)</span>
+<span class="n">G_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">30</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">levels</span><span class="o">=</span><span class="mi">21</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">144</span><span class="p">)</span>
+<span class="n">S_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">30</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">levels</span><span class="o">=</span><span class="mi">21</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/5ae128218660729dc5de566e3015b3c00cf9f48b23357b10ad8249e8a89daada.png" src="../_images/5ae128218660729dc5de566e3015b3c00cf9f48b23357b10ad8249e8a89daada.png" />
+</div>
+</div>
+<p>The modified system looks fairly organized and chaotic. So the essence of the original L96 still remains, which is what we wanted.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">131</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">0</span><span class="p">),</span> <span class="n">S_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$X_{k-1}$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$S_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">132</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">S_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$X_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$S_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">133</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">2</span><span class="p">),</span> <span class="n">S_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$X_{k+1}$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$S_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/52aff1cd24a85fdc97b64d6afbcae7c5a6004c151109dd64239ed8f0dfb89262.png" src="../_images/52aff1cd24a85fdc97b64d6afbcae7c5a6004c151109dd64239ed8f0dfb89262.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">131</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">0</span><span class="p">),</span> <span class="n">G_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$X_{k-1}$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$G_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">132</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">G_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$X_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$G_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">133</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">2</span><span class="p">),</span> <span class="n">G_true_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$X_{k+1}$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$G_</span><span class="si">{k}</span><span class="s2">$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/432a453aa52b50a02143ea9348d2be794e578fb05bdf019ef1550242cb0c9f8f.png" src="../_images/432a453aa52b50a02143ea9348d2be794e578fb05bdf019ef1550242cb0c9f8f.png" />
+</div>
+</div>
+<p>Since the dependence of Y on X remains the same, G now depends on the neighbouring points (and not the K itself).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">S_true_xr</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="s2">&quot;K&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$&lt;S&gt;$&quot;</span><span class="p">)</span>
+<span class="c1"># plt.title(&#39;Averaged subgrid term&#39;)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">G_true_xr</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="s2">&quot;K&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$&lt;G&gt;$&quot;</span><span class="p">)</span>
+<span class="c1"># plt.title(&#39;Averaged subgrid term&#39;)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/507fa542cd95c1044710e79f7a3a13016f08603648d0d52621eb64ace81ac4d0.png" src="../_images/507fa542cd95c1044710e79f7a3a13016f08603648d0d52621eb64ace81ac4d0.png" />
+</div>
+</div>
+<p>The momentum input from the subgrid term is essentially zero (to numerical precision), as should be.</p>
+<p><strong>What should we learn using ML?</strong><br />
+We will do three types of learning (all fully non-local):<br />
+a) Learn <span class="math notranslate nohighlight">\(G_k\)</span> directly as a function of <span class="math notranslate nohighlight">\(X_k\)</span>s by minimizing a MSE loss.<br />
+b) Learn <span class="math notranslate nohighlight">\(G_k\)</span> directly as a function of <span class="math notranslate nohighlight">\(X_k\)</span>s by minimizing a MSE loss + minimizing <span class="math notranslate nohighlight">\(&lt;G_k&gt;\)</span>.<br />
+c) Learn <span class="math notranslate nohighlight">\(S_k\)</span> as a function of <span class="math notranslate nohighlight">\(X_k\)</span>s, and then <span class="math notranslate nohighlight">\(&lt;G_k&gt;\)</span> will be zero by design.<br />
+d) Learn <span class="math notranslate nohighlight">\(S_k\)</span> but with <span class="math notranslate nohighlight">\(G_k\)</span> being the goal. (This will introduce the Gauge problem).</p>
+<p><em>We chose MSE as a loss function because as shown earlier, the learning for L96 is not sensitive to the choice of loss function.</em></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">### Split into Training and testing data</span>
+
+<span class="n">val_size</span> <span class="o">=</span> <span class="mi">4000</span>  <span class="c1"># number of time steps for validation</span>
+
+<span class="c1"># train:</span>
+<span class="n">X_true_train</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span>
+    <span class="p">:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:</span>
+<span class="p">]</span>  <span class="c1"># Flatten because we first use single input as a sample</span>
+<span class="n">subgrid_G_tend_train</span> <span class="o">=</span> <span class="n">G_true</span><span class="p">[:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">subgrid_S_tend_train</span> <span class="o">=</span> <span class="n">S_true</span><span class="p">[:</span><span class="o">-</span><span class="n">val_size</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="c1"># test:</span>
+<span class="n">X_true_test</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+<span class="n">subgrid_G_tend_test</span> <span class="o">=</span> <span class="n">G_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+<span class="n">subgrid_S_tend_test</span> <span class="o">=</span> <span class="n">S_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:,</span> <span class="p">:]</span>
+
+<span class="c1"># Create non local training data</span>
+<span class="c1"># Define a data loader (8 inputs, 8 outputs)</span>
+
+<span class="c1">### For the cases where will learn G_k</span>
+
+<span class="c1"># Define our X,Y pairs (state, subgrid tendency) for the linear regression local network.local_torch_dataset = Data.TensorDataset(</span>
+<span class="n">torch_dataset</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_train</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_G_tend_train</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+<span class="p">)</span>
+
+<span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">1024</span>  <span class="c1"># Number of sample in each batch</span>
+
+<span class="n">loader</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">torch_dataset</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># Define a test dataloader (8 inputs, 8 outputs)</span>
+
+<span class="n">torch_dataset_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_G_tend_test</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+<span class="p">)</span>
+
+<span class="n">loader_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">torch_dataset_test</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span>
+<span class="p">)</span>
+
+<span class="c1"># For the cases where will learn S_k</span>
+
+<span class="c1"># Define our X,Y pairs (state, subgrid tendency) for the linear regression local network.local_torch_dataset = Data.TensorDataset(</span>
+<span class="n">torch_dataset</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_train</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_S_tend_train</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+<span class="p">)</span>
+
+<span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">1024</span>  <span class="c1"># Number of sample in each batch</span>
+
+<span class="n">loader_S</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">torch_dataset</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># Define a test dataloader (8 inputs, 8 outputs)</span>
+
+<span class="n">torch_dataset_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">subgrid_S_tend_test</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span>
+<span class="p">)</span>
+
+<span class="n">loader_S_test</span> <span class="o">=</span> <span class="n">Data</span><span class="o">.</span><span class="n">DataLoader</span><span class="p">(</span>
+    <span class="n">dataset</span><span class="o">=</span><span class="n">torch_dataset_test</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Class for the neural network architecture</span>
+<span class="k">class</span> <span class="nc">Net_ANN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s2">&quot;Default&quot;</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_ANN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs, 16 neurons for first hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 16 neurons for second hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
+
+    <span class="c1">#         self.lin_drop = nn.Dropout(0.1) #regularization method to prevent overfitting.</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Setup function to take one step in the learning</span>
+<span class="k">def</span> <span class="nf">train_model</span><span class="p">(</span><span class="n">net</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">trainloader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+    <span class="c1"># set model in to training mode</span>
+    <span class="n">net</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
+
+    <span class="c1"># Loop through all the training subsets (notice we batched in size of 1024)</span>
+    <span class="k">for</span> <span class="n">step</span><span class="p">,</span> <span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">trainloader</span><span class="p">):</span>  <span class="c1"># for each training step</span>
+        <span class="n">b_x</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>  <span class="c1"># Inputs</span>
+        <span class="n">b_y</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">batch_y</span><span class="p">)</span>  <span class="c1"># outputs</span>
+
+        <span class="k">if</span> <span class="p">(</span>
+            <span class="nb">len</span><span class="p">(</span><span class="n">b_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
+        <span class="p">):</span>  <span class="c1"># If is needed to add a dummy dimension if our inputs are 1D (where each number is a different sample)</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span>
+                <span class="n">net</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">b_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+            <span class="p">)</span>  <span class="c1"># input x and predict based on x</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">net</span><span class="p">(</span><span class="n">b_x</span><span class="p">)</span>
+
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">b_y</span><span class="p">)</span>  <span class="c1"># Calculating loss</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>  <span class="c1"># clear gradients for next train</span>
+        <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>  <span class="c1"># backpropagation, compute gradients</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>  <span class="c1"># apply gradients to update weights</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Setup function that evaluates a number of metrics at each training step</span>
+<span class="k">def</span> <span class="nf">test_model</span><span class="p">(</span>
+    <span class="n">net</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">metric_fun</span><span class="p">,</span> <span class="n">trainloader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">text</span><span class="o">=</span><span class="s2">&quot;validation&quot;</span><span class="p">,</span> <span class="n">print_flag</span><span class="o">=</span><span class="mi">0</span>
+<span class="p">):</span>
+    <span class="n">net</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>  <span class="c1"># Evaluation mode (important when having dropout layers)</span>
+
+    <span class="n">test_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="n">test_metric</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="n">test_metric_2</span> <span class="o">=</span> <span class="mi">0</span>
+
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="k">for</span> <span class="n">step</span><span class="p">,</span> <span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span>
+            <span class="n">trainloader</span>
+        <span class="p">):</span>  <span class="c1"># for each training step</span>
+            <span class="n">b_x</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>  <span class="c1"># Inputs</span>
+            <span class="n">b_y</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">batch_y</span><span class="p">)</span>  <span class="c1"># outputs</span>
+            <span class="k">if</span> <span class="p">(</span>
+                <span class="nb">len</span><span class="p">(</span><span class="n">b_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
+            <span class="p">):</span>  <span class="c1"># If is needed to add a dummy dimension if our inputs are 1D (where each number is a different sample)</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span>
+                    <span class="n">net</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">b_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+                <span class="p">)</span>  <span class="c1"># input x and predict based on x</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">net</span><span class="p">(</span><span class="n">b_x</span><span class="p">)</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">b_y</span><span class="p">)</span>  <span class="c1"># Calculating loss</span>
+            <span class="n">metric</span> <span class="o">=</span> <span class="n">metric_fun</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">b_y</span><span class="p">)</span>
+
+            <span class="n">test_loss</span> <span class="o">=</span> <span class="n">test_loss</span> <span class="o">+</span> <span class="n">loss</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>  <span class="c1"># Keep track of the loss</span>
+            <span class="n">test_metric</span> <span class="o">=</span> <span class="n">test_metric</span> <span class="o">+</span> <span class="n">metric</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>  <span class="c1"># Keep track of the loss</span>
+            <span class="n">test_metric_2</span> <span class="o">=</span> <span class="p">(</span>
+                <span class="n">test_metric_2</span>
+                <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)))</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+            <span class="p">)</span>
+
+        <span class="n">test_loss</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">trainloader</span><span class="p">)</span>  <span class="c1"># dividing by the number of batches</span>
+        <span class="n">test_metric</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">trainloader</span><span class="p">)</span>  <span class="c1"># dividing by the number of batches</span>
+        <span class="n">test_metric_2</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">trainloader</span><span class="p">)</span>
+    <span class="c1">#         print(len(trainloader))</span>
+    <span class="c1"># print(test_loss)</span>
+
+    <span class="k">if</span> <span class="n">print_flag</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+        <span class="nb">print</span><span class="p">(</span>
+            <span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">net</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s2"> </span><span class="si">{</span><span class="n">text</span><span class="si">}</span><span class="s2"> loss: </span><span class="si">{</span><span class="n">test_loss</span><span class="si">}</span><span class="s2">, metric_1: </span><span class="si">{</span><span class="n">test_metric</span><span class="si">}</span><span class="s2">, metric_2: </span><span class="si">{</span><span class="n">test_metric_2</span><span class="si">}</span><span class="s2">&quot;</span>
+        <span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">test_loss</span><span class="p">,</span> <span class="n">test_metric</span><span class="p">,</span> <span class="n">test_metric_2</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># The ML models</span>
+
+<span class="n">lr</span> <span class="o">=</span> <span class="mf">0.001</span>  <span class="c1"># the learning rate</span>
+
+<span class="c1"># (a) Learn G_k using MSE</span>
+<span class="n">criterion_a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>
+<span class="n">nn_a</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">(</span><span class="s2">&quot;Learn Gk&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="n">optimizer_a</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_a</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="n">lr</span><span class="p">)</span>
+
+<span class="c1"># (b) learn G_k using MSE + condition of reducing momentum forcing</span>
+
+<span class="n">alpha</span> <span class="o">=</span> <span class="mi">100</span>  <span class="c1"># This is the regularization parameter</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">custom_loss</span><span class="p">(</span><span class="n">Ypred</span><span class="p">,</span> <span class="n">Ytrue</span><span class="p">):</span>
+    <span class="c1"># loss = MSE + net momentum forcing penalty</span>
+    <span class="n">loss</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">Ypred</span> <span class="o">-</span> <span class="n">Ytrue</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span>
+        <span class="n">torch</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">Ypred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
+    <span class="p">)</span>
+    <span class="k">return</span> <span class="n">loss</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">criterion_b</span> <span class="o">=</span> <span class="n">custom_loss</span>
+<span class="n">nn_b</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">(</span><span class="s2">&quot;Learn Gk w/ constrain&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="n">optimizer_b</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_b</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="n">lr</span><span class="p">)</span>
+
+<span class="c1"># (c) Learn S_k using MSE</span>
+<span class="n">criterion_c</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>
+<span class="n">nn_c</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">(</span><span class="s2">&quot;Learn Sk&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="n">optimizer_c</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_c</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="n">lr</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># (d) Learn S_k but with G_k as the objective</span>
+<span class="k">def</span> <span class="nf">custom_loss2</span><span class="p">(</span><span class="n">Spred</span><span class="p">,</span> <span class="n">Gtrue</span><span class="p">):</span>
+    <span class="c1"># Gtrue = np.roll(Strue, 1, axis=1) - np.roll(Strue, -1, axis=1)</span>
+    <span class="n">Gpred</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">Spred</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">Spred</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">loss</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">Gpred</span> <span class="o">-</span> <span class="n">Gtrue</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">loss</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">criterion_d</span> <span class="o">=</span> <span class="n">custom_loss2</span>
+<span class="n">nn_d</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">(</span><span class="s2">&quot;Learn Sk from Gk&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="n">optimizer_d</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_d</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="n">lr</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>The custom loss function:
+$<span class="math notranslate nohighlight">\(
+loss = (G_{pred} - G_{true})^2 + \alpha |&lt;G_k&gt;|
+\)</span>$</p>
+<p>In model (b) we have to pick a value for the degree of regularization <span class="math notranslate nohighlight">\(\alpha\)</span>. Here we use a constant value, but this value itself needed to be determined. We determined it using some rough testing with the L-curve criterion (read about it more <a class="reference external" href="https://www.sintef.no/globalassets/project/evitameeting/2005/lcurve.pdf">here</a>).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Do the training</span>
+
+<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">100</span>
+
+<span class="c1"># Containers</span>
+<span class="n">validation_stat_a</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;loss&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric2&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+<span class="p">}</span>
+<span class="n">train_stat_a</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;loss&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric2&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+<span class="p">}</span>
+
+<span class="n">validation_stat_b</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;loss&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric2&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+<span class="p">}</span>
+<span class="n">train_stat_b</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;loss&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric2&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+<span class="p">}</span>
+
+<span class="n">validation_stat_c</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;loss&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric2&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+<span class="p">}</span>
+<span class="n">train_stat_c</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;loss&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric2&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+<span class="p">}</span>
+
+<span class="n">validation_stat_d</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;loss&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric2&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+<span class="p">}</span>
+<span class="n">train_stat_d</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;loss&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+    <span class="s2">&quot;metric2&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_epochs</span><span class="p">)),</span>
+<span class="p">}</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">helper_fun</span><span class="p">(</span><span class="n">nn_sel</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">metric</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+    <span class="n">train_model</span><span class="p">(</span><span class="n">nn_sel</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">test_model</span><span class="p">(</span>
+        <span class="n">nn_sel</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">metric</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">text</span><span class="o">=</span><span class="s2">&quot;train&quot;</span>
+    <span class="p">),</span> <span class="n">test_model</span><span class="p">(</span><span class="n">nn_sel</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">metric</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">print_flag</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="n">tqdm</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">)):</span>
+    <span class="c1"># print(&quot;--------------&quot;)</span>
+    <span class="c1"># print(&quot;Epoch num : &quot; + str(epoch))</span>
+
+    <span class="c1"># (a)</span>
+    <span class="p">(</span>
+        <span class="n">validation_stat_a</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">validation_stat_a</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">validation_stat_a</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+    <span class="p">),</span> <span class="p">(</span>
+        <span class="n">train_stat_a</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">train_stat_a</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">train_stat_a</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+    <span class="p">)</span> <span class="o">=</span> <span class="n">helper_fun</span><span class="p">(</span>
+        <span class="n">nn_a</span><span class="p">,</span> <span class="n">criterion_a</span><span class="p">,</span> <span class="n">criterion_a</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer_a</span>
+    <span class="p">)</span>
+
+    <span class="c1"># (b)</span>
+    <span class="p">(</span>
+        <span class="n">validation_stat_b</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">validation_stat_b</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">validation_stat_b</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+    <span class="p">),</span> <span class="p">(</span>
+        <span class="n">train_stat_b</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">train_stat_b</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">train_stat_b</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+    <span class="p">)</span> <span class="o">=</span> <span class="n">helper_fun</span><span class="p">(</span>
+        <span class="n">nn_b</span><span class="p">,</span> <span class="n">criterion_b</span><span class="p">,</span> <span class="n">criterion_a</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer_b</span>
+    <span class="p">)</span>
+
+    <span class="c1"># (c)</span>
+    <span class="p">(</span>
+        <span class="n">validation_stat_c</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">validation_stat_c</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">validation_stat_c</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+    <span class="p">),</span> <span class="p">(</span>
+        <span class="n">train_stat_c</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">train_stat_c</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">train_stat_c</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+    <span class="p">)</span> <span class="o">=</span> <span class="n">helper_fun</span><span class="p">(</span>
+        <span class="n">nn_c</span><span class="p">,</span> <span class="n">criterion_c</span><span class="p">,</span> <span class="n">criterion_a</span><span class="p">,</span> <span class="n">loader_S</span><span class="p">,</span> <span class="n">loader_S_test</span><span class="p">,</span> <span class="n">optimizer_c</span>
+    <span class="p">)</span>
+
+    <span class="c1"># (d)</span>
+    <span class="c1"># notice that this learns Sk (what is predicted), but the MSE is relative to Gk.</span>
+    <span class="p">(</span>
+        <span class="n">validation_stat_d</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">validation_stat_d</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">validation_stat_d</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+    <span class="p">),</span> <span class="p">(</span>
+        <span class="n">train_stat_d</span><span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">train_stat_d</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+        <span class="n">train_stat_d</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">][</span><span class="n">epoch</span><span class="p">],</span>
+    <span class="p">)</span> <span class="o">=</span> <span class="n">helper_fun</span><span class="p">(</span>
+        <span class="n">nn_d</span><span class="p">,</span> <span class="n">criterion_d</span><span class="p">,</span> <span class="n">criterion_d</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">optimizer_d</span>
+    <span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>100%|██████████| 100/100 [01:14&lt;00:00,  1.34it/s]
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">train_stat_a</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">train_stat_b</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">train_stat_c</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">train_stat_d</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_d</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="c1"># ax.set_xlim([0, 20])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;MSE in $G$ or $S$&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Epoch number&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Training&quot;</span><span class="p">)</span>
+
+
+<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">validation_stat_a</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">validation_stat_b</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">validation_stat_c</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">validation_stat_d</span><span class="p">[</span><span class="s2">&quot;metric&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_d</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Epoch number&quot;</span><span class="p">)</span>
+<span class="c1"># ax.set_xlim([0, 20])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Validation&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/8fcd085d52fd67fb7fe6e17f67d7f0741a6645d32c39582056e62165feb7adef.png" src="../_images/8fcd085d52fd67fb7fe6e17f67d7f0741a6645d32c39582056e62165feb7adef.png" />
+</div>
+</div>
+<p>Looking at the plots above we see that the training went well for all the cases, as the loss function is reduced. Note tht we should not compare the MSE of the cases where G_k is learnt with the cases where S_k is learnt, since the learning objective was different.</p>
+<p>We also see that adding a constrain to minimize net momentum input when learning G_k, made the MSE for the model a bit higher, as it is trying to reduce two things at once.</p>
+<p>It is interesting to note that even the model without the constraint is trying to reduce the momentum input over training epochs, which shows that it is something that the learning is trying to learn without help. (there might be situations with limited data where this help is important, but not super helpful for L96).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># How do the models that directly learn Gk do at mom cons of that term.</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">train_stat_a</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span> <span class="s2">&quot; Train&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">train_stat_b</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span> <span class="s2">&quot; Train&quot;</span><span class="p">)</span>
+
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">validation_stat_a</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span> <span class="s2">&quot; Test&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">),</span> <span class="n">validation_stat_b</span><span class="p">[</span><span class="s2">&quot;metric2&quot;</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span> <span class="s2">&quot; Test&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="c1"># ax.set_xlim([0, 20])</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$&lt;G_k&gt;$&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Epoch number&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Training&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/1f3341473c788328163cccafa47767ae8f72beb4594e75d3e9cceb1722278fa2.png" src="../_images/1f3341473c788328163cccafa47767ae8f72beb4594e75d3e9cceb1722278fa2.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Look at predictions</span>
+<span class="n">nn_a</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>
+<span class="n">nn_b</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>
+<span class="n">nn_c</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>
+<span class="n">nn_d</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>
+
+<span class="n">pred_G_a</span> <span class="o">=</span> <span class="n">nn_a</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+<span class="n">pred_G_b</span> <span class="o">=</span> <span class="n">nn_b</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+<span class="n">pred_S_c</span> <span class="o">=</span> <span class="n">nn_c</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+<span class="n">pred_G_c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">pred_S_c</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span>
+    <span class="n">pred_S_c</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span>
+<span class="p">)</span>  <span class="c1"># Since the last model predicts the S, we need to calculate G.</span>
+<span class="n">pred_S_d</span> <span class="o">=</span> <span class="n">nn_d</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X_true_test</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+<span class="n">pred_G_d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">pred_S_d</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span>
+    <span class="n">pred_S_d</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span>
+<span class="p">)</span>  <span class="c1"># Since the last model predicts the S, we need to calculate G.</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="o">=</span> <span class="mi">500</span>
+<span class="n">k</span> <span class="o">=</span> <span class="mi">2</span>
+
+<span class="n">T_test</span> <span class="o">=</span> <span class="n">T_true</span><span class="p">[</span><span class="o">-</span><span class="n">val_size</span><span class="p">:]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">[:</span><span class="n">n</span><span class="p">],</span> <span class="n">subgrid_G_tend_test</span><span class="p">[:</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">[:</span><span class="n">n</span><span class="p">],</span> <span class="n">pred_G_a</span><span class="p">[:</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">[:</span><span class="n">n</span><span class="p">],</span> <span class="n">pred_G_b</span><span class="p">[:</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">[:</span><span class="n">n</span><span class="p">],</span> <span class="n">pred_G_c</span><span class="p">[:</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">[:</span><span class="n">n</span><span class="p">],</span> <span class="n">pred_G_d</span><span class="p">[:</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$G_k$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="c1"># plt.plot(Xtrue_test[:,k],  subgrid_tend_test[:,k], &#39;.&#39;, markersize=1)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">subgrid_G_tend_test</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">pred_G_a</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mf">2.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">subgrid_G_tend_test</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">pred_G_b</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mf">2.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">subgrid_G_tend_test</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">pred_G_c</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mf">2.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">subgrid_G_tend_test</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">pred_G_d</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mf">2.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span>
+<span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;True G&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Predicted G&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="c1"># plt.plot(Xtrue_test[:,k],  pred_2.detach().numpy()[:,k], &#39;.&#39;, markersize=1.5)</span>
+<span class="c1"># plt.plot(Xtrue_test[:,k],  pred_3.detach().numpy()[:,k], &#39;.&#39;, markersize=1.5)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/3688c1b52ce927ce55a0334fbf31e1f6b11ec8612ce19ac8cf7ae6e0560049e1.png" src="../_images/3688c1b52ce927ce55a0334fbf31e1f6b11ec8612ce19ac8cf7ae6e0560049e1.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">MSE_a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">subgrid_G_tend_test</span> <span class="o">-</span> <span class="n">pred_G_a</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
+<span class="n">MSE_b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">subgrid_G_tend_test</span> <span class="o">-</span> <span class="n">pred_G_b</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
+<span class="n">MSE_c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">subgrid_G_tend_test</span> <span class="o">-</span> <span class="n">pred_G_c</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
+<span class="n">MSE_d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">subgrid_G_tend_test</span> <span class="o">-</span> <span class="n">pred_G_d</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
+
+<span class="n">MSE_S_c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">subgrid_S_tend_test</span> <span class="o">-</span> <span class="n">pred_S_c</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
+<span class="n">MSE_S_d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">subgrid_S_tend_test</span> <span class="o">-</span> <span class="n">pred_S_d</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_names</span> <span class="o">=</span> <span class="p">[</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">nn_b</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">nn_c</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">nn_d</span><span class="o">.</span><span class="n">name</span><span class="p">]</span>
+<span class="n">MSEs</span> <span class="o">=</span> <span class="p">[</span><span class="n">MSE_a</span><span class="p">,</span> <span class="n">MSE_b</span><span class="p">,</span> <span class="n">MSE_c</span><span class="p">,</span> <span class="n">MSE_d</span><span class="p">]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">nn_names</span><span class="p">,</span> <span class="n">MSEs</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">setp</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">get_xticklabels</span><span class="p">(),</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">45</span><span class="p">,</span> <span class="n">horizontalalignment</span><span class="o">=</span><span class="s2">&quot;right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;MSE in $G_k$&quot;</span><span class="p">);</span>
+<span class="c1"># plt.xlabel(&#39;Model&#39;)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/0a72b7c3057ad953cd0fe06ad19dd4c6effcee5273dfc9655f7c7494a811085a.png" src="../_images/0a72b7c3057ad953cd0fe06ad19dd4c6effcee5273dfc9655f7c7494a811085a.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_names</span> <span class="o">=</span> <span class="p">[</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">nn_d</span><span class="o">.</span><span class="n">name</span><span class="p">]</span>
+<span class="n">MSEs</span> <span class="o">=</span> <span class="p">[</span><span class="n">MSE_S_c</span><span class="p">,</span> <span class="n">MSE_S_d</span><span class="p">]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">nn_names</span><span class="p">,</span> <span class="n">MSEs</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">setp</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">get_xticklabels</span><span class="p">(),</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">45</span><span class="p">,</span> <span class="n">horizontalalignment</span><span class="o">=</span><span class="s2">&quot;right&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;MSE in $S_k$&quot;</span><span class="p">);</span>
+<span class="c1"># plt.xlabel(&#39;Model&#39;)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/f0e96450b1fcc484fcebba3d26238fbec0d25b0cb952ffb8672346f2963da56c.png" src="../_images/f0e96450b1fcc484fcebba3d26238fbec0d25b0cb952ffb8672346f2963da56c.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">subgrid_G_tend_test</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_G_a</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_G_b</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_G_c</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;-.&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_G_d</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;-.&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="mi">160</span><span class="p">,</span> <span class="mi">175</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$&lt;G_k&gt;$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time step&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_G_a</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_a</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_G_b</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_b</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$&lt;G_k&gt;$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;PDF&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">])</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/b395173e482d641136ecbde66dfe9a805a1b836b3ca6c768519202fcecf3b66e.png" src="../_images/b395173e482d641136ecbde66dfe9a805a1b836b3ca6c768519202fcecf3b66e.png" />
+</div>
+</div>
+<ul class="simple">
+<li><p>The models do quite a good job (visually) at capturing the patterns in the subgrid forcing.</p></li>
+<li><p>The model with the constrain of momentum conservation does a better job of it (pdf width is much smaller), BUT some bias is introduced (median is shifted from 0).</p></li>
+<li><p>From our experimenting this bias is not always the same, and can vary if trainined multiple times.</p></li>
+</ul>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">subgrid_S_tend_test</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_S_c</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;-.&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_S_d</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;-.&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_d</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="mi">160</span><span class="p">,</span> <span class="mi">175</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$&lt;S_k&gt;$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time step&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_S_c</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_c</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pred_S_d</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">nn_d</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$&lt;S_k&gt;$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;PDF&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/a52313f2422adbaaa2712f6036538deb826ce00dd828b28e61e4e5264c6eb0cc.png" src="../_images/a52313f2422adbaaa2712f6036538deb826ce00dd828b28e61e4e5264c6eb0cc.png" />
+</div>
+</div>
+<p><strong>When the model learns Sk from Gk, without knowing anything about Gk. It produces the right Gk, but a Sk that looks very different from the real Sk.</strong></p>
+<div class="section" id="skill-of-different-ml-models-in-prediction">
+<h3>Skill of different ML models in prediction<a class="headerlink" href="#skill-of-different-ml-models-in-prediction" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># - a GCM class including a neural network parameterization in rhs of equation for tendency</span>
+
+<span class="n">time_method</span> <span class="o">=</span> <span class="n">RK4</span>
+
+<span class="c1"># - a GCM class without any parameterization</span>
+
+
+<span class="k">class</span> <span class="nc">GCM_no_param</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">time_method</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+        <span class="c1"># X0 - initial conditions, dt - time increment, nt - number of forward steps to take</span>
+        <span class="c1"># param - parameters of our closure</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GCM_network</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">network</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">time_method</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">network</span> <span class="o">=</span> <span class="n">network</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="o">.</span><span class="n">linear1</span><span class="o">.</span><span class="n">in_features</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_torch</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="p">(</span><span class="n">X_torch</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+        <span class="p">)</span>  <span class="c1"># Adding NN parameterization</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+        <span class="c1"># X0 - initial conditions, dt - time increment, nt - number of forward steps to take</span>
+        <span class="c1"># param - parameters of our closure</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">histX</span><span class="p">,</span> <span class="n">histB</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">histX</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+        <span class="n">histB</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.0</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="c1"># this next if statement is being called twice</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="o">.</span><span class="n">linear1</span><span class="o">.</span><span class="n">in_features</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+                <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_torch</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+
+            <span class="n">histB</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="p">(</span><span class="n">X_torch</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
+
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">histX</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">histX</span><span class="p">,</span> <span class="n">time</span><span class="p">,</span> <span class="n">histB</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GCM_network_S</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">network</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">time_method</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">network</span> <span class="o">=</span> <span class="n">network</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">):</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="o">.</span><span class="n">linear1</span><span class="o">.</span><span class="n">in_features</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_torch</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+
+            <span class="n">predS</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="p">(</span><span class="n">X_torch</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+            <span class="n">predG</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">predS</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">predS</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        <span class="c1"># return L96_eq1_xdot(X, self.F) + np.squeeze(self.network(X_torch).data.numpy()) # Adding NN parameterization</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">predG</span><span class="p">)</span>  <span class="c1"># Adding NN parameterization</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+        <span class="c1"># X0 - initial conditions, dt - time increment, nt - number of forward steps to take</span>
+        <span class="c1"># param - parameters of our closure</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">histX</span><span class="p">,</span> <span class="n">histB</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">histX</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+        <span class="n">histB</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.0</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="c1"># this next if statement is being called twice</span>
+            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="o">.</span><span class="n">linear1</span><span class="o">.</span><span class="n">in_features</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+                <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_torch</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+
+            <span class="n">predS</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="p">(</span><span class="n">X_torch</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+            <span class="n">predG</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">predS</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">predS</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+            <span class="n">histB</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">predG</span>  <span class="c1"># np.squeeze(self.network(X_torch).data.numpy())</span>
+
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">histX</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">histX</span><span class="p">,</span> <span class="n">time</span><span class="p">,</span> <span class="n">histB</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">T_test</span> <span class="o">=</span> <span class="mi">20</span>
+
+<span class="n">X_full</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">S_full</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T_test</span><span class="p">)</span>  <span class="c1"># Full model</span>
+<span class="n">G_full</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">S_full</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">S_full</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">init_cond</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="n">gcm_a</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">nn_a</span><span class="p">)</span>
+<span class="n">X_a</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">G_a</span> <span class="o">=</span> <span class="n">gcm_a</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+
+<span class="n">gcm_b</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">nn_b</span><span class="p">)</span>
+<span class="n">X_b</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">G_b</span> <span class="o">=</span> <span class="n">gcm_b</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+
+<span class="n">gcm_c</span> <span class="o">=</span> <span class="n">GCM_network_S</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">nn_c</span><span class="p">)</span>
+<span class="n">X_c</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">G_c</span> <span class="o">=</span> <span class="n">gcm_c</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+
+<span class="n">gcm_d</span> <span class="o">=</span> <span class="n">GCM_network_S</span><span class="p">(</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">nn_d</span><span class="p">)</span>
+<span class="n">X_d</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">G_d</span> <span class="o">=</span> <span class="n">gcm_d</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+
+
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM_no_param</span><span class="p">(</span><span class="n">Forcing</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T_test</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># convert to xarray for convenience</span>
+<span class="n">X_full_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">X_full</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;X_true&quot;</span><span class="p">)</span>
+<span class="n">G_full_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">G_full</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;G_true&quot;</span><span class="p">)</span>
+
+<span class="n">X_a_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">X_a</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;X_a&quot;</span><span class="p">)</span>
+<span class="n">G_a_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">G_a</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;G_a&quot;</span><span class="p">)</span>
+
+<span class="n">X_b_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">X_b</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;X_b&quot;</span><span class="p">)</span>
+<span class="n">G_b_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">G_b</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;G_b&quot;</span><span class="p">)</span>
+
+<span class="n">X_c_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">X_c</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;X_c&quot;</span><span class="p">)</span>
+<span class="n">G_c_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">G_c</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;G_c&quot;</span><span class="p">)</span>
+
+<span class="n">X_d_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">X_d</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;X_d&quot;</span><span class="p">)</span>
+<span class="n">G_d_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">G_d</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;G_d&quot;</span><span class="p">)</span>
+
+<span class="n">X_noparam_xr</span> <span class="o">=</span> <span class="n">xr</span><span class="o">.</span><span class="n">DataArray</span><span class="p">(</span>
+    <span class="n">X_no_param</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;time&quot;</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">],</span> <span class="n">coords</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;time&quot;</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span> <span class="s2">&quot;K&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)}</span>
+<span class="p">)</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="s2">&quot;X_noparam&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">161</span><span class="p">)</span>
+<span class="n">X_full_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Full model&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">162</span><span class="p">)</span>
+<span class="n">X_a_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Learn Gk&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">163</span><span class="p">)</span>
+<span class="n">X_b_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Lrn Gk w/ con&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">164</span><span class="p">)</span>
+<span class="n">X_c_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Lrn Sk&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">165</span><span class="p">)</span>
+<span class="n">X_d_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Lrn Sk f/ Gk&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">166</span><span class="p">)</span>
+<span class="n">X_noparam_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">15</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span>
+    <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span>
+    <span class="n">cbar_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;aspect&quot;</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span> <span class="s2">&quot;shrink&quot;</span><span class="p">:</span> <span class="mf">0.2</span><span class="p">},</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;No param&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;X&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/7046a9edf14555e2d7325900c8c86bc04a730036a9f3e5cd3935b9088d166fc4.png" src="../_images/7046a9edf14555e2d7325900c8c86bc04a730036a9f3e5cd3935b9088d166fc4.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">161</span><span class="p">)</span>
+<span class="n">G_full_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Full model&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">162</span><span class="p">)</span>
+<span class="n">G_a_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Learn Gk&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">163</span><span class="p">)</span>
+<span class="n">G_b_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Lrn Gk w/ con&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">164</span><span class="p">)</span>
+<span class="n">G_c_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span> <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Lrn Sk&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">165</span><span class="p">)</span>
+<span class="n">G_d_xr</span><span class="o">.</span><span class="n">sel</span><span class="p">(</span><span class="n">time</span><span class="o">=</span><span class="nb">slice</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span>
+    <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span>
+    <span class="n">extend</span><span class="o">=</span><span class="s2">&quot;both&quot;</span><span class="p">,</span>
+    <span class="n">add_colorbar</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+    <span class="n">cbar_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;aspect&quot;</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span> <span class="s2">&quot;shrink&quot;</span><span class="p">:</span> <span class="mf">0.2</span><span class="p">},</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Lrn Sk&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">166</span><span class="p">)</span>
+<span class="c1"># X_noparam_xr.sel(time=slice(0, 15)).plot.contourf(levels=np.linspace(-15, 15, 9), extend=&#39;both&#39;, cbar_kwargs={&#39;aspect&#39;:5})</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;No param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;G&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/febfa9eacdd5243ebd4b05d403d5ca784ccd8005946eff6bc30bbd5b8efe74ce.png" src="../_images/febfa9eacdd5243ebd4b05d403d5ca784ccd8005946eff6bc30bbd5b8efe74ce.png" />
+</div>
+</div>
+<p>Given the simplicity of the parameterization problem in L96 model, all the different ML parameterizations performed relatively well. We expect to see much larger gains by adding constraints in more complex realistic problems, particularly the ones that are more data limited.</p>
+</div>
+</div>
+<div class="section" id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline">#</a></h2>
+<p>In this notebook we learnt how to add constraints to the learning process, and studied the impacts that this has on the trained models.</p>
+<p>With this notebook we wrap up the section on using neural networks to parameterize missing effects in the L96 model. In the next section we will be learning a different approach to improving models - data assimilation.</p>
+</div>
+</div>
+
+    <script type="text/x-thebe-config">
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+            repo: "binder-examples/jupyter-stacks-datascience",
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+            theme: "abcdef",
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#physical-constraints-in-l96">Physical constraints in L96</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#l96-model-with-momentum-conserving-term">L96 model with momentum conserving term</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#skill-of-different-ml-models-in-prediction">Skill of different ML models in prediction</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
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+    <title>Tuning GCM Parameterizations &#8212; Learning Machine Learning with Lorenz-96</title>
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="closing_remarks.html">Outlook</a></li>
+<li class="toctree-l1"><a class="reference internal" href="citing.html">Citing this book</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../bibliography.html">References</a></li>
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+    <h1>Tuning GCM Parameterizations</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-gcm-parameterization-problem">The GCM Parameterization Problem</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#variational-estimation-of-optimal-parameters-for-a-predefined-closure">Variational estimation of optimal parameters for a predefined closure</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#estimating-parameters-based-on-one-initial-condition-and-one-time-step">Estimating parameters based on one initial condition and one time step</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#cost-function">Cost function</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#minimization">Minimization</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#let-s-test-the-closure">Let’s test the closure.</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#results">Results</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#estimating-parameters-which-optimize-longer-trajectories">Estimating parameters which optimize longer trajectories</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Let’s test the closure.</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Results</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discussion-and-possible-next-steps">Discussion and possible next steps:</a></li>
+</ul>
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+              
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+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="tuning-gcm-parameterizations">
+<h1>Tuning GCM Parameterizations<a class="headerlink" href="#tuning-gcm-parameterizations" title="Permalink to this headline">#</a></h1>
+<p>The objective of this notebook is to show how GCM closures can be tuned in practice.
+We will assume a specific formulation of a closure and estimate its parameters through a standard optimisation procedure using a procedure similar to Data Assimilation (DA).</p>
+<p><strong>Resources</strong> : We have used material from Emmanuel Cosme’s nice GitHub <a class="reference external" href="https://github.com/ecosme38/Data-Assimilation-Notebooks">repository</a>.</p>
+<div class="section" id="the-gcm-parameterization-problem">
+<h2>The GCM Parameterization Problem<a class="headerlink" href="#the-gcm-parameterization-problem" title="Permalink to this headline">#</a></h2>
+<p>Here we quickly reintroduce the problem we are trying to solve, our starting point here is <a class="reference internal" href="gcm-parameterization-problem.html"><span class="doc">GCM parameterizations, skill metrics, and other sources of uncertainity</span></a>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">scipy.optimize</span> <span class="k">as</span> <span class="nn">opt</span>
+
+<span class="c1"># L96 provides the &quot;real world&quot;, L96_eq1_xdot is the beginning of rhs of X tendency</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">RK2</span><span class="p">,</span> <span class="n">RK4</span><span class="p">,</span> <span class="n">EulerFwd</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Setting the seed gives us reproducible results</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">13</span><span class="p">)</span>
+
+<span class="c1"># Create a &quot;real world&quot; with K=8 and J=32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Run “real world” for 3 days to forget initial conditons.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># (store=True save the final state as an initial condition for the next run)</span>
+<span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>From here on we can use <code class="docutils literal notranslate"><span class="pre">W.X</span></code> as perfect initial conditions for a model and sample the real world using <code class="docutils literal notranslate"><span class="pre">W.run(dt,T)</span></code></p>
+<p>The class defined below is just a sophisticated version of the single time-scale L96 gcm analogue that was defined in ‘The Lorenz-96 GCM Analog’ notebook.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GCM</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">parameterization</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">EulerFwd</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        GCM with paramerization</span>
+<span class="sd">        Args:</span>
+<span class="sd">            F: forcing</span>
+<span class="sd">            parameterization: function that takes parameters and returns a tendency</span>
+<span class="sd">            time_stepping: time stepping method</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">parameterization</span> <span class="o">=</span> <span class="n">parameterization</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            X: state vector</span>
+<span class="sd">            param: parameters of our closure</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">parameterization</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            X0: initial conditions</span>
+<span class="sd">            dt: time increment</span>
+<span class="sd">            nt: number of forward steps to take</span>
+<span class="sd">            param: parameters of our closure</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>We illustrate the concepts here with the help of the simple linear regression parameterization, which was also introduced in the <a class="reference internal" href="gcm-analogue.html"><span class="doc">The Lorenz-96 and its GCM Analog</span></a>. Remember that this was not a very good parameterization, but it is used to show how imperfect parameterizations can be tuned in different ways. Here we use parameters that were estimated by fitting the linear regression to the true sub-grid tendencies. Such parameter estimates are sometimes also referred to as “offline fitting” estimates.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">naive_parameterization</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.005</span><span class="p">,</span> <span class="mf">5.0</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+<span class="c1"># We pass the parameters that were obtained by fitting the linear in &#39;The Lorenz-96 GCM Analog&#39; notebook.</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mf">0.85439536</span><span class="p">,</span> <span class="mf">0.75218026</span><span class="p">])</span>
+<span class="c1"># We also simulate a case with no parameterization, to show that even the imperfect parameterization is helpful.</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Now we compare the model with the parameterization and trajectories from the “real world” truth.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># This samples the real world with the same time interval as &quot;dt&quot; used by the model</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Model with linear param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_no_param</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Model with no param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_4(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/1e353ac1bf7c67d6a0c92d8a4b8a61719b3bc2d8b575de710a40cd81f3bc9dd8.png" src="../_images/1e353ac1bf7c67d6a0c92d8a4b8a61719b3bc2d8b575de710a40cd81f3bc9dd8.png" />
+</div>
+</div>
+<p>The figure above show that the model with linear parameterization outperforms the model with no parameterization. Next we discuss if these parameter estimates can be made even better.</p>
+</div>
+<div class="section" id="variational-estimation-of-optimal-parameters-for-a-predefined-closure">
+<h2>Variational estimation of optimal parameters for a predefined closure<a class="headerlink" href="#variational-estimation-of-optimal-parameters-for-a-predefined-closure" title="Permalink to this headline">#</a></h2>
+<p>We will try here to estimate the parameters of <code class="docutils literal notranslate"><span class="pre">naive_parameterization</span></code> with a variational approach. This is different from the way the parameters used above were estimated, which had used an offline fitting approach.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># - assuming the formulation of the parameterization</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="estimating-parameters-based-on-one-initial-condition-and-one-time-step">
+<h3>Estimating parameters based on one initial condition and one time step<a class="headerlink" href="#estimating-parameters-based-on-one-initial-condition-and-one-time-step" title="Permalink to this headline">#</a></h3>
+<div class="section" id="cost-function">
+<h4>Cost function<a class="headerlink" href="#cost-function" title="Permalink to this headline">#</a></h4>
+<p>What we will be doing here is very close to what is done with classical variational data assimilation, where we try to estimate the state of the parameters of a model through the minimization of a cost function <span class="math notranslate nohighlight">\(J\)</span>. This is also very close to what is done when parameterizations are encoded as neural networks.</p>
+<p>We introduce a cost function <span class="math notranslate nohighlight">\(J(p)\)</span> which depends on the parameters of the closure.</p>
+<div class="math notranslate nohighlight">
+\[J(p) = ||X_p - X_{true}||_{d}\]</div>
+<p>where <span class="math notranslate nohighlight">\(p=[p1,p2]\)</span>, <span class="math notranslate nohighlight">\(X_p\)</span> is GCM solution computed with with parameters <span class="math notranslate nohighlight">\(p\)</span> and <span class="math notranslate nohighlight">\(||\cdot ||_{d}\)</span> is one of the distances above.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">norm_initial_tendency</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">):</span>
+    <span class="n">T1</span> <span class="o">=</span> <span class="n">X1</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">X1</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">T2</span> <span class="o">=</span> <span class="n">X2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">X2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">((</span><span class="n">T1</span> <span class="o">-</span> <span class="n">T2</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">cost_function</span><span class="p">(</span><span class="n">param</span><span class="p">):</span>
+    <span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span>
+    <span class="n">X_gcm</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">param</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">norm_initial_tendency</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_gcm</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="minimization">
+<h4>Minimization<a class="headerlink" href="#minimization" title="Permalink to this headline">#</a></h4>
+<p>Since the problem dimension is only two ( <span class="math notranslate nohighlight">\(p=[p1,p2]\)</span> ), we can use efficient derivative-free optimization methods.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">prior</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.85439536</span><span class="p">,</span> <span class="mf">0.75218026</span><span class="p">])</span>  <span class="c1">#  prior</span>
+<span class="n">res</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">cost_function</span><span class="p">,</span> <span class="n">prior</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s2">&quot;Powell&quot;</span><span class="p">)</span>
+<span class="n">opt_param</span> <span class="o">=</span> <span class="n">res</span><span class="p">[</span><span class="s2">&quot;x&quot;</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Optimized parameters = </span><span class="si">{</span><span class="n">opt_param</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimized parameters = [1.78084896 3.64518369]
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="let-s-test-the-closure">
+<h4>Let’s test the closure.<a class="headerlink" href="#let-s-test-the-closure" title="Permalink to this headline">#</a></h4>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">100.0</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+<span class="n">X_optimized</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">opt_param</span><span class="p">)</span>
+<span class="n">X_prior</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">prior</span><span class="p">)</span>
+
+<span class="c1"># - ... the true state</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="results">
+<h4>Results<a class="headerlink" href="#results" title="Permalink to this headline">#</a></h4>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_true</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_prior</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Initial GCM&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_optimized</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Optimized GCM&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_4(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/14e8227bbb6b7ee6861b5f340279ac345c63a9966e3abeff1fda06a37d38b1c6.png" src="../_images/14e8227bbb6b7ee6861b5f340279ac345c63a9966e3abeff1fda06a37d38b1c6.png" />
+</div>
+</div>
+<p>The results are better but not great. This problem is related to the question of <em>a priori</em> versus <em>a posteriori</em> skill in Large Eddy Simulation (LES) closures.</p>
+</div>
+</div>
+<div class="section" id="estimating-parameters-which-optimize-longer-trajectories">
+<h3>Estimating parameters which optimize longer trajectories<a class="headerlink" href="#estimating-parameters-which-optimize-longer-trajectories" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">5.0</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_gcm</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/usr/share/miniconda/envs/L96M2lines/lib/python3.9/site-packages/numpy/lib/polynomial.py:779: RuntimeWarning: invalid value encountered in multiply
+  y = y * x + pv
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># - pointwise distance :</span>
+<span class="k">def</span> <span class="nf">pointwise</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">L</span><span class="o">=</span><span class="mf">1.0</span><span class="p">):</span>
+    <span class="c1"># computed over some window t&lt;L.</span>
+    <span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="n">X1</span> <span class="o">-</span> <span class="n">X2</span><span class="p">)[</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">t</span> <span class="o">&lt;</span> <span class="n">L</span><span class="p">)]</span>
+    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">D</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">cost_function</span><span class="p">(</span><span class="n">param</span><span class="p">):</span>
+    <span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mi">5</span>
+    <span class="n">X_gcm</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">param</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">pointwise</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_gcm</span><span class="p">,</span> <span class="n">L</span><span class="o">=</span><span class="mf">5.0</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">prior</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.85439536</span><span class="p">,</span> <span class="mf">0.75218026</span><span class="p">])</span>  <span class="c1">#  prior</span>
+<span class="n">res</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">cost_function</span><span class="p">,</span> <span class="n">prior</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s2">&quot;Powell&quot;</span><span class="p">)</span>
+<span class="n">opt_param</span> <span class="o">=</span> <span class="n">res</span><span class="p">[</span><span class="s2">&quot;x&quot;</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/tmp/ipykernel_2862/3672341191.py:5: RuntimeWarning: overflow encountered in square
+  return np.sqrt(D**2).mean(axis=0)
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Optimized parameters = </span><span class="si">{</span><span class="n">opt_param</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Optimized parameters = [0.88644175 0.74928667]
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="id1">
+<h4>Let’s test the closure.<a class="headerlink" href="#id1" title="Permalink to this headline">#</a></h4>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">100.0</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+<span class="n">X_optimized</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">opt_param</span><span class="p">)</span>
+<span class="n">X_prior</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">prior</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="id2">
+<h4>Results<a class="headerlink" href="#id2" title="Permalink to this headline">#</a></h4>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_true</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_prior</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Initial GCM&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_optimized</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Optimized GCM&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_4(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/713f67431308f30416e05a2d7bd6cfa2f338b9e6170d76c0a233db209ee9293c.png" src="../_images/713f67431308f30416e05a2d7bd6cfa2f338b9e6170d76c0a233db209ee9293c.png" />
+</div>
+</div>
+<p>The closure produces better results but it is not clear how this would generalize to unseen initial conditions.</p>
+</div>
+</div>
+</div>
+<div class="section" id="discussion-and-possible-next-steps">
+<h2>Discussion and possible next steps:<a class="headerlink" href="#discussion-and-possible-next-steps" title="Permalink to this headline">#</a></h2>
+<ul class="simple">
+<li><p>Estimating parameters over one time-step but with an ensemble of initial conditions.</p></li>
+<li><p>Use neural networks to encode closures.</p></li>
+<li><p>Explore the feasability of differentiable GCMs.</p></li>
+</ul>
+</div>
+</div>
+
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+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-gcm-parameterization-problem">The GCM Parameterization Problem</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#variational-estimation-of-optimal-parameters-for-a-predefined-closure">Variational estimation of optimal parameters for a predefined closure</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#estimating-parameters-based-on-one-initial-condition-and-one-time-step">Estimating parameters based on one initial condition and one time step</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#cost-function">Cost function</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#minimization">Minimization</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#let-s-test-the-closure">Let’s test the closure.</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#results">Results</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#estimating-parameters-which-optimize-longer-trajectories">Estimating parameters which optimize longer trajectories</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Let’s test the closure.</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Results</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discussion-and-possible-next-steps">Discussion and possible next steps:</a></li>
+</ul>
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+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
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+    <h1>The Lorenz-96 and its GCM Analog</h1>
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+                <h2> Contents </h2>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-two-time-scale-model-analog-for-the-real-atmosphere">The two time-scale model: Analog for the real atmosphere</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-single-time-scale-model-analog-for-a-general-circulation-model-gcm">The single time-scale model: Analog for a general circulation model (GCM)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adding-a-parametization-to-the-single-time-scale-model">Adding a parametization to the single time-scale model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#testing-the-effect-of-parameterizations-in-the-gcm-model">Testing the effect of parameterizations in the “GCM” model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
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+  <div class="tex2jax_ignore mathjax_ignore section" id="the-lorenz-96-and-its-gcm-analog">
+<h1>The Lorenz-96 and its GCM Analog<a class="headerlink" href="#the-lorenz-96-and-its-gcm-analog" title="Permalink to this headline">#</a></h1>
+<p>The physical equations of motion of the system describing the atmosphere dynamics are known. Also in the real atmosphere there is variability over all scales, from the smallest molecular scales to the largest planetary scales.
+Thus, it is not feasible to explicitly solve the known physical equations of motion within Global Climate Models (GCMs), as computational resources limit the range of scales that can be resolved. To make the behavior of the GCM better match the real world, we need to effectively parameterize the effects that the unresolved scales have on the resolved (large) scales.</p>
+<p>The two time-scale L96 model, described in the the <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/L96-two-scale-description.html#">previous notebook</a>, can be considered as a simplified analog for real world atmosphere, where variability resides at only two discrete scales (a simplification of the continuum in the real world). In this analogy, a GCM would be a model that can only solve the equations for the slow time scale, where the effects of the fast time-scale variables would be missing. We introduce the single time-scale version of the L96 model below. To make the single time scale model match the two time scale model, the effects of the fast time scales need to parameterized into the single time scale model.</p>
+<div class="section" id="the-two-time-scale-model-analog-for-the-real-atmosphere">
+<h2>The two time-scale model: Analog for the real atmosphere<a class="headerlink" href="#the-two-time-scale-model-analog-for-the-real-atmosphere" title="Permalink to this headline">#</a></h2>
+<p>We will first describe a simulation with the two time-scale model from the <a class="reference internal" href="L96-two-scale-description.html"><span class="doc">The Lorenz-96 Two-Timescale System</span></a>, which is taken as the control simulation that we hope to replicate with the single time-scale model.</p>
+<p>The forcing and resolution parameters, <span class="math notranslate nohighlight">\(F\)</span>, <span class="math notranslate nohighlight">\(J\)</span> and <span class="math notranslate nohighlight">\(K\)</span>, for the two time-scale model are fixed based on <span id="id1">[<a class="reference internal" href="../bibliography.html#id3" title="Daniel S. Wilks. Effects of stochastic parametrizations in the lorenz \textquotesingle 96 system. Quarterly Journal of the Royal Meteorological Society, 131(606):389–407, 2005. URL: https://doi.org/10.1256/qj.04.03, doi:10.1256/qj.04.03.">Wilks, 2005</a>]</span>, as <span class="math notranslate nohighlight">\(F=18\)</span> or <span class="math notranslate nohighlight">\(20\)</span>, <span class="math notranslate nohighlight">\(K=8\)</span>, and <span class="math notranslate nohighlight">\(J=32\)</span>. Here, the value chosen for the parameter <span class="math notranslate nohighlight">\(F\)</span> is set large enough to ensure chaotic behavior. We also use the reference values for the <span class="math notranslate nohighlight">\(h\)</span>, <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(c\)</span> parameters to be, <span class="math notranslate nohighlight">\(h=1\)</span>, <span class="math notranslate nohighlight">\(b=10\)</span>, and <span class="math notranslate nohighlight">\(c=10\)</span>.
+With this particular choice of parameter values, one model time unit (MTU) is approximately equivalent to five atmospheric days. This estimate is obtained by comparing error-doubling times in the Lorenz-96 model and the real atmosphere <span id="id2">[<a class="reference internal" href="../bibliography.html#id2" title="E.N. Lorenz. Predictability: a problem partly solved. Seminar on Predictability, 1:1-18, 1995. URL: https://www.ecmwf.int/node/10829.">Lorenz, 1995</a>]</span>.</p>
+<p>In the following code, we solve the L96 system using an accurate time-stepping scheme (RK4 with sufficiently small <span class="math notranslate nohighlight">\(\Delta t\)</span>), and plot the time evolution for the <span class="math notranslate nohighlight">\(X_k\)</span> and <span class="math notranslate nohighlight">\(Y_{j,k}\)</span> variables.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">23</span><span class="p">)</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">)</span>
+<span class="n">T</span> <span class="o">=</span> <span class="mf">5.0</span>
+<span class="o">%</span><span class="k">time</span> X_true,Y_true,t = W.run(0.05, T)
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>CPU times: user 2.52 s, sys: 45.1 ms, total: 2.57 s
+Wall time: 2.57 s
+</pre></div>
+</div>
+</div>
+</div>
+<p>Here is the what the solution looks like:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$X_k(t)$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">j</span> <span class="o">/</span> <span class="n">W</span><span class="o">.</span><span class="n">J</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">Y_true</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k+j/J&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$Y_{j,k}(t)$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/8e6fb16b88b45800be7e1ddcb232672d663071c8b0b17df5eae198b2453fb025.png" src="../_images/8e6fb16b88b45800be7e1ddcb232672d663071c8b0b17df5eae198b2453fb025.png" />
+</div>
+</div>
+</div>
+<div class="section" id="the-single-time-scale-model-analog-for-a-general-circulation-model-gcm">
+<h2>The single time-scale model: Analog for a general circulation model (GCM)<a class="headerlink" href="#the-single-time-scale-model-analog-for-a-general-circulation-model-gcm" title="Permalink to this headline">#</a></h2>
+<p>The two time-scale model discussed above solves a set of equations for the slow and fast variables, where the equations for the slow variables are:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-6b6a63f5-c263-4d96-b7e1-269259bf3e29">
+<span class="eqno">(2)<a class="headerlink" href="#equation-6b6a63f5-c263-4d96-b7e1-269259bf3e29" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \underbrace{ 
+\left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}
+}_{=U_k} \ .
+\end{align}\]</div>
+<p>Here the effects of the fast scales on the slow scales are represented by the last term on the RHS, denoted as <span class="math notranslate nohighlight">\(U_k\)</span>, <span class="math notranslate nohighlight">\(k=1,\ldots,k\)</span>.</p>
+<p>In the single time-scale model the explicit equations for the fast scales are not known, and so we do not know what <span class="math notranslate nohighlight">\(Y_{j,k}\)</span> or subsequently <span class="math notranslate nohighlight">\(U_k\)</span> are. So, in the single time-scale model the effects of the fast scales are either missing, <span class="math notranslate nohighlight">\(U_k=0\)</span>, or they need to be specified/parameterized purely in terms of the known slow time-scale variables, <span class="math notranslate nohighlight">\(U_k \approx P (X_k)\)</span>.</p>
+<p>In the following code, we show how the single time scale model can be solved. We use <code class="docutils literal notranslate"><span class="pre">L96_eq1_xdot</span></code> (code shown below), which returns the tendency (RHS) corresponding to the following equation, where there are no coupling or parameterization terms,</p>
+<div class="amsmath math notranslate nohighlight" id="equation-b57cbfe8-96ed-42fc-98df-1406233d052f">
+<span class="eqno">(3)<a class="headerlink" href="#equation-b57cbfe8-96ed-42fc-98df-1406233d052f" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F.
+\end{align}\]</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96_eq1_xdot</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">display_source</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">display_source</span><span class="p">(</span><span class="n">L96_eq1_xdot</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><style>pre { line-height: 125%; }
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+.highlight .vc { color: #19177C } /* Name.Variable.Class */
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+.highlight .vm { color: #19177C } /* Name.Variable.Magic */
+.highlight .il { color: #666666 } /* Literal.Number.Integer.Long */</style><div class="highlight"><pre><span></span><span class="nd">@njit</span>
+<span class="k">def</span> <span class="nf">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">advect</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    Calculate the time rate of change for the X variables for the Lorenz &#39;96, equation 1:</span>
+<span class="sd">        d/dt X[k] = -X[k-2] X[k-1] + X[k-1] X[k+1] - X[k] + F</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X : Values of X variables at the current time step</span>
+<span class="sd">        F : Forcing term</span>
+<span class="sd">    Returns:</span>
+<span class="sd">        dXdt : Array of X time tendencies</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="n">K</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+    <span class="n">Xdot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>
+
+    <span class="k">if</span> <span class="n">advect</span><span class="p">:</span>
+        <span class="n">Xdot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">-</span> <span class="n">X</span> <span class="o">+</span> <span class="n">F</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">Xdot</span> <span class="o">=</span> <span class="o">-</span><span class="n">X</span> <span class="o">+</span> <span class="n">F</span>
+    <span class="c1">#     for k in range(K):</span>
+    <span class="c1">#         Xdot[k] = ( X[(k+1)%K] - X[k-2] ) * X[k-1] - X[k] + F</span>
+    <span class="k">return</span> <span class="n">Xdot</span>
+</pre></div>
+</div></div>
+</div>
+<p>Now we define <code class="docutils literal notranslate"><span class="pre">GCM</span></code>, which solves for the temporal evolution of <span class="math notranslate nohighlight">\(X\)</span>s using a simple Euler integration.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#  We define the GCM which solves for X in time and returns its time series</span>
+<span class="k">def</span> <span class="nf">GCM</span><span class="p">(</span><span class="n">X0</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+    <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span> <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+
+    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">+</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">F</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">))</span>
+        <span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mf">1e3</span><span class="p">:</span>
+            <span class="k">break</span>
+        <span class="n">hist</span><span class="p">[</span><span class="n">n</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Notice that we have added the possibility of adding a parameterization that may take the form a polynomial function, this will be discussed futher below.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_init</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">F_mod</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mf">0.002</span><span class="p">,</span> <span class="n">W</span><span class="o">.</span><span class="n">F</span>
+<span class="c1"># set the initial condition to be the same</span>
+<span class="c1"># we also set the dt to be smaller as the Euler scheme is less accurate</span>
+
+<span class="c1"># no parameterization</span>
+<span class="n">X_gcm_no_param</span><span class="p">,</span> <span class="n">T_gcm_no_param</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">F_mod</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">T_gcm_no_param</span><span class="p">,</span> <span class="n">X_gcm_no_param</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;$X_k(t)$&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/4ff447a44162aad66fe5cf9a70bed8be64046d13bc8fd8a677b2bd0b61099983.png" src="../_images/4ff447a44162aad66fe5cf9a70bed8be64046d13bc8fd8a677b2bd0b61099983.png" />
+</div>
+</div>
+<p>Clearly, the evolution the single time scale system does not even qualitatively match the evolution of the two time scale system. For example, the leftward propagation is completely missing.</p>
+</div>
+<div class="section" id="adding-a-parametization-to-the-single-time-scale-model">
+<h2>Adding a parametization to the single time-scale model<a class="headerlink" href="#adding-a-parametization-to-the-single-time-scale-model" title="Permalink to this headline">#</a></h2>
+<p>The single time-scale model does not behave the same as the two time-scale model, since the effects of the fast time-scales (or “unresolved scales” or “sub-grid effects” or “sub-grid tendencies”) are missing. To obtain a closer match to the two time-scale model, we need to introduce these missing effects into the single time-scale model. Adding these missing effects is known as parameterization, where the effect of the missing scales are represented purely in terms of the variables that are known in the gcm (<span class="math notranslate nohighlight">\(X\)</span>’s only in the single time-scale model).</p>
+<p>The one time-scale equation of the Lorenz-96 model with a parameterization for the effects of the fast time-scale , can be written as:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-dafb9a8a-6464-4528-a5a9-d353b3645264">
+<span class="eqno">(4)<a class="headerlink" href="#equation-dafb9a8a-6464-4528-a5a9-d353b3645264" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - P\left( X_k \right) \ .
+\end{align}\]</div>
+<p>Here, we consider a particular functional form for modeling the unknown parameterization, using a polynomial approximation.
+Example of such a parameterization include the fourth-order polynomial form proposed by
+<span id="id3">[<a class="reference internal" href="../bibliography.html#id3" title="Daniel S. Wilks. Effects of stochastic parametrizations in the lorenz \textquotesingle 96 system. Quarterly Journal of the Royal Meteorological Society, 131(606):389–407, 2005. URL: https://doi.org/10.1256/qj.04.03, doi:10.1256/qj.04.03.">Wilks, 2005</a>]</span>:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+P(X_k)=b_0 + b_1 X_k + b_2 X_k^2 + b_3 X_k^3 + b_4 X_k^4 + e_k.
+\end{equation*}\]</div>
+<p>Alternatively, we also consider a first-order polynomial as suggested by
+<span id="id4">[<a class="reference internal" href="../bibliography.html#id4" title="H. M. Arnold, I. M. Moroz, and T. N. Palmer. Stochastic parametrizations and model uncertainty in the lorenz '96 system. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1991):20110479, 2013. URL: https://doi.org/10.1098/rsta.2011.0479, doi:10.1098/rsta.2011.0479.">Arnold <em>et al.</em>, 2013</a>]</span>:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+P(X_k)=b_0 + b_1 X_k + e_k.
+\end{equation*}\]</div>
+<p>In both these forms <span class="math notranslate nohighlight">\(e_k\)</span> is a stochastic component, which will not be discussed further till the next notebook.</p>
+<p>All parameterizations have some unknown parameters, which need to be determined in some way. These parameters may be guessed based on some intuition about the physics, or determined from data collected in the real system (two time-scale model here), or optimized to make the evolution of the reduced (single time-scale) model match the evolution of the real world or full (two time-scale model) model. In this notebook we will use the second approach, and the last approach will be discussed in <a class="reference internal" href="estimating-gcm-parameters.html"><span class="doc">Tuning GCM Parameterizations</span></a>.</p>
+<p>In summary: with the “real atmosphere” or two time-scale system in hand, we can “observe” the effect of the sub-grid forcing on the large scale (<span class="math notranslate nohighlight">\(U_k\)</span>) and test the skill of the polynomial function, <span class="math notranslate nohighlight">\(P(X_k)\)</span>, models/parameterizations.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">time</span> X, Y, t = W.run(0.05, 200.) # We first run the 2 time-scale model again for longer time, to generate more data.
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>CPU times: user 8.06 s, sys: 10.9 ms, total: 8.07 s
+Wall time: 8.06 s
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_copy</span> <span class="o">=</span> <span class="n">X</span>
+<span class="c1"># Generate U_k samples from 2 time-scale model</span>
+<span class="c1"># using the longer time series, we now generate samples of U_k that are used to</span>
+<span class="c1"># estimate the unknown parameters.</span>
+<span class="n">U_sample</span> <span class="o">=</span> <span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">h</span> <span class="o">*</span> <span class="n">W</span><span class="o">.</span><span class="n">c</span> <span class="o">/</span> <span class="n">W</span><span class="o">.</span><span class="n">b</span><span class="p">)</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">Y</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">W</span><span class="o">.</span><span class="n">K</span><span class="p">,</span> <span class="n">W</span><span class="o">.</span><span class="n">J</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Fit polynomial of order 1.</span>
+<span class="n">p1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">polyfit</span><span class="p">(</span>
+    <span class="n">X_copy</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="n">U_sample</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="mi">1</span>
+<span class="p">)</span>  <span class="c1"># Fit a linear curve through the data.</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Linear Poly coeffs:&quot;</span><span class="p">,</span> <span class="n">p1</span><span class="p">)</span>
+
+<span class="c1"># We use the parameters from from Wilks, 2005</span>
+<span class="n">p4</span> <span class="o">=</span> <span class="p">[</span>
+    <span class="mf">0.000707</span><span class="p">,</span>
+    <span class="o">-</span><span class="mf">0.0130</span><span class="p">,</span>
+    <span class="o">-</span><span class="mf">0.0190</span><span class="p">,</span>
+    <span class="mf">1.59</span><span class="p">,</span>
+    <span class="mf">0.275</span><span class="p">,</span>
+<span class="p">]</span>
+<span class="c1"># We could have just as easily have fit the parameters using the data as well.</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Linear Poly coeffs: [0.85439536 0.75218026]
+</pre></div>
+</div>
+</div>
+</div>
+<p>We now compare the results obtained with a linear polynomial approximation, <span id="id5">Wilks [<a class="reference internal" href="../bibliography.html#id3" title="Daniel S. Wilks. Effects of stochastic parametrizations in the lorenz \textquotesingle 96 system. Quarterly Journal of the Royal Meteorological Society, 131(606):389–407, 2005. URL: https://doi.org/10.1256/qj.04.03, doi:10.1256/qj.04.03.">2005</a>]</span> polynomial parameterization and the “truth” values of the coupling terms.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="c1"># 2D histogram of X vs U</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">X_copy</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="n">U_sample</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="n">bins</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$X_k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$U_k = \frac</span><span class="si">{hc}{b}</span><span class="s2">\sum_j Y_{j,k}$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s2">&quot;PDF&quot;</span><span class="p">)</span>
+
+<span class="c1"># Fits from polynomials</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p4</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_4(X_k)$ - Wilks, 2005&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$P_1(X_k)$&quot;</span><span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/c20856426db6571b7afff12caee3a80fd0e2ee35f918c1995f9cfc0162d4584a.png" src="../_images/c20856426db6571b7afff12caee3a80fd0e2ee35f918c1995f9cfc0162d4584a.png" />
+</div>
+</div>
+<p id="sub-grid-hist-label">The figure above shows that the the relationship between the slow variables (<span class="math notranslate nohighlight">\(X_k\)</span>) and the observed coupling term (<span class="math notranslate nohighlight">\(U_k\)</span>) is non-linear. The higher order polynomlial, since it is more flexible, does a better job at capturing the overall pattern, relative to the linear fit.</p>
+<p>We had already setup the code in <code class="docutils literal notranslate"><span class="pre">GCM</span></code> to accept polynomial parameterizations, which can be turned on by passing the parameters. We will use this in the next section to test the effect that the parameterization has on the single time-scale model.</p>
+</div>
+<div class="section" id="testing-the-effect-of-parameterizations-in-the-gcm-model">
+<h2>Testing the effect of parameterizations in the “GCM” model<a class="headerlink" href="#testing-the-effect-of-parameterizations-in-the-gcm-model" title="Permalink to this headline">#</a></h2>
+<p>Now that we have a couple of different candidate parameterizations that can roughly predict the relationship between the slow variables and sub-grid forcing, we test their impact in a GCM simulation where the parameterization is required. We compare four simulations:</p>
+<ul class="simple">
+<li><p>“Real world”: corresponding to the “truth” model goverened by the full two time-scale Lorenz-96 system.</p></li>
+<li><p>GCM without parameterization: corresponding to the one time-scale Lorenz-96 system without any the coupling term. We use a forward-Euler scheme to integrate the model forward.</p></li>
+<li><p>GCM with our parameterization: corresponding to the one time-scale Lorenz-96 system with the linear polynomial approximation of the coupling terms as obtained above.</p></li>
+<li><p>GCM with <span id="id6">[<a class="reference internal" href="../bibliography.html#id3" title="Daniel S. Wilks. Effects of stochastic parametrizations in the lorenz \textquotesingle 96 system. Quarterly Journal of the Royal Meteorological Society, 131(606):389–407, 2005. URL: https://doi.org/10.1256/qj.04.03, doi:10.1256/qj.04.03.">Wilks, 2005</a>]</span> parameterization: corresponding to the one time-scale Lorenz-96 system with a third-order polynomial approximation of the coupling terms.</p></li>
+</ul>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">13</span><span class="p">)</span>
+<span class="n">T</span> <span class="o">=</span> <span class="mi">5</span>
+<span class="c1"># Real world</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">Y_true</span><span class="p">,</span> <span class="n">T_true</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">randomize_IC</span><span class="p">()</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+<span class="n">X_init</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">F_mod</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mf">0.0</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">K</span><span class="p">),</span> <span class="mf">0.002</span><span class="p">,</span> <span class="n">W</span><span class="o">.</span><span class="n">F</span> <span class="o">+</span> <span class="mf">0.0</span>
+<span class="c1"># The reason for adding the zero terms to X and F will become clear below, where the amplitude will be increased.</span>
+
+<span class="c1"># no parameterization</span>
+<span class="n">X_gcm1</span><span class="p">,</span> <span class="n">T_gcm1</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">F_mod</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+<span class="c1"># Linear parameterization</span>
+<span class="n">X_gcm2</span><span class="p">,</span> <span class="n">T_gcm2</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">F_mod</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">p1</span><span class="p">)</span>
+<span class="c1"># Wilks parameterization - 4th order polynomial.</span>
+<span class="n">X_gcm3</span><span class="p">,</span> <span class="n">T_gcm3</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">F_mod</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">p4</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Now we look at Hovmoeller plots of the different solutions.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">17</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
+<span class="n">lims</span> <span class="o">=</span> <span class="p">{</span><span class="s2">&quot;vmin&quot;</span><span class="p">:</span> <span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="s2">&quot;vmax&quot;</span><span class="p">:</span> <span class="mi">12</span><span class="p">,</span> <span class="s2">&quot;levels&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">),</span> <span class="s2">&quot;extend&quot;</span><span class="p">:</span> <span class="s2">&quot;both&quot;</span><span class="p">}</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">141</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">T_true</span> <span class="o">+</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">X_true</span><span class="p">,</span> <span class="o">**</span><span class="n">lims</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Real world&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">142</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">T_gcm1</span> <span class="o">+</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">X_gcm1</span><span class="p">,</span> <span class="o">**</span><span class="n">lims</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;GCM without parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">143</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">T_gcm3</span> <span class="o">+</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">X_gcm2</span><span class="p">,</span> <span class="o">**</span><span class="n">lims</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;GCM with linear parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">144</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="n">T_gcm2</span> <span class="o">+</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">X_gcm3</span><span class="p">,</span> <span class="o">**</span><span class="n">lims</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;GCM with Wilks&#39; parameterization&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/f72abcd2b030ce1152a4aa1e0cf3e7e2cc412457a41511880743bc9a595b4411.png" src="../_images/f72abcd2b030ce1152a4aa1e0cf3e7e2cc412457a41511880743bc9a595b4411.png" />
+</div>
+</div>
+<p>The next plot shows the temporal evolution of the variable <span class="math notranslate nohighlight">\(X_3\)</span> obtained with the four models listed above.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">k</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_true</span> <span class="o">+</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Real world&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_gcm1</span> <span class="o">+</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">X_gcm1</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="s2">&quot;:&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_gcm1</span> <span class="o">+</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">X_gcm2</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with our parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T_gcm1</span> <span class="o">+</span> <span class="n">t</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">X_gcm3</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with Wilks parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_3(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">25</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/d8e6ca354b9326e866c7705704354e0856e80cc51ccfc27d777a382cc016429c.png" src="../_images/d8e6ca354b9326e866c7705704354e0856e80cc51ccfc27d777a382cc016429c.png" />
+</div>
+</div>
+<p>As seen above, all the simulation diverge at long times,and the unparameterized simulation diverges very rapidly. On the other hand, the parameterized GCMs track the “real world” solution better. The Wilks parameterization does better than the linear fit.</p>
+</div>
+<div class="section" id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline">#</a></h2>
+<p>In this chapter:</p>
+<ul class="simple">
+<li><p>We argued that the single time-scale L96 model is a reduced representation of the two time-scale L96 model, which can be considered analogous to the way GCMs are a reduced repseration of the real world.</p></li>
+<li><p>We showed that missing effects need to be parameterized, if the reduced representation is required behave like the full model/real world.</p></li>
+<li><p>We used the two time-scale L96 model to generate a “truth” dataset for the effects of unresolved scales.</p></li>
+<li><p>We built a “GCM” with a polynomial parameterization of coupling to unresolved processes (<span class="math notranslate nohighlight">\(\left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}\)</span>)</p></li>
+<li><p>Finally we compared the solution from the two time-scale, single time-scale and parameterized single time-scale models, showing that the paramaterized version evolves more similarly to the two time-scale model.</p></li>
+</ul>
+<p>In the next chapter we will further explore some key properties of parameterizations, and also discuss the other sources of error that can lead to a difference between the reduced and full models (or GCMs and real world).</p>
+</div>
+</div>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-two-time-scale-model-analog-for-the-real-atmosphere">The two time-scale model: Analog for the real atmosphere</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-single-time-scale-model-analog-for-a-general-circulation-model-gcm">The single time-scale model: Analog for a general circulation model (GCM)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adding-a-parametization-to-the-single-time-scale-model">Adding a parametization to the single time-scale model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#testing-the-effect-of-parameterizations-in-the-gcm-model">Testing the effect of parameterizations in the “GCM” model</a></li>
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+<li class="toctree-l1 current active"><a class="current reference internal" href="#">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
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+    <h1>GCM parameterizations, skill metrics, and other sources of uncertainity</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-need-for-gcm-parameterizations">The need for GCM parameterizations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#should-parameterizations-be-deterministic-or-stochastic">Should parameterizations  be deterministic or stochastic ?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#how-to-measure-parameterization-skill">How to measure parameterization skill ?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sources-of-model-error">Sources of model error</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="gcm-parameterizations-skill-metrics-and-other-sources-of-uncertainity">
+<h1>GCM parameterizations, skill metrics, and other sources of uncertainity<a class="headerlink" href="#gcm-parameterizations-skill-metrics-and-other-sources-of-uncertainity" title="Permalink to this headline">#</a></h1>
+<p>In the last notebook we provided a very quick overview of the problem we run into when trying to simulate a real system (e.g. the atmosphere) on a computer using a General Circulation Model (GCM), even when the exact equations to be solved are known in principle. The problem of limited computational resources translates into an inability to resolve all scales of motion in a GCM, and the unresolved scales need to be parameterized.</p>
+<p>The objective of this notebook is to introduce some more of the key aspects of parameterizations in GCMs, illustrating the deterministic vs stochastic approaches, the interplay with numerical errors, and how to measure the skill of a parameterization. We also provide a comprehensive discussion of the different sources of errors that are present in GCMs.</p>
+<div class="section" id="the-need-for-gcm-parameterizations">
+<h2>The need for GCM parameterizations<a class="headerlink" href="#the-need-for-gcm-parameterizations" title="Permalink to this headline">#</a></h2>
+<p>Let’s first quickly review some concepts from the last notebook, using a slightly modified framing that might benefit some readers.
+We will assume from now on that the readers are familiar  with the <span id="id1">[<a class="reference internal" href="../bibliography.html#id2" title="E.N. Lorenz. Predictability: a problem partly solved. Seminar on Predictability, 1:1-18, 1995. URL: https://www.ecmwf.int/node/10829.">Lorenz, 1995</a>]</span> two-time scale model and its numerical implementation in the <code class="docutils literal notranslate"><span class="pre">L96_model</span></code> module, which was discussed in <a class="reference internal" href="L96-two-scale-description.html"><span class="doc">The Lorenz-96 Two-Timescale System</span></a>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">numba</span> <span class="kn">import</span> <span class="n">jit</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">RK2</span><span class="p">,</span> <span class="n">RK4</span><span class="p">,</span> <span class="n">EulerFwd</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Here, <code class="docutils literal notranslate"><span class="pre">L96</span></code> serves as the <strong>“real world”</strong> or two time-scale model, whereas <code class="docutils literal notranslate"><span class="pre">L96_eq1_xdot</span></code> represents the <strong>beginning of rhs of X tendency</strong> or the tendency in the single time-scale model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Setting the seed gives us reproducible results</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">13</span><span class="p">)</span>
+
+<span class="c1"># Create a &quot;real world&quot; with K = 8 and J = 32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Since we start the model with a random initial condition, there is no reason to expect that these initial conditions are an actual solution to the model. These arbitrary states can result in initial shocks to the system, which will are unrealistic features but get dissipated after some time.
+So we <em>run the “real world” for 3 days in order to forget the initial conditons</em>, and settle into a state that is an actual solution to the model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># `store=True` saves the final state as an initial condition for the next run.</span>
+<span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>From here on we can use <code class="docutils literal notranslate"><span class="pre">W.X</span></code> as perfect initial conditions for a model and sample the “real world” using <code class="docutils literal notranslate"><span class="pre">W.run(dt,</span> <span class="pre">T)</span></code>.</p>
+<p><strong>How to think about the real world vs models:</strong></p>
+<p>Let’s call <span class="math notranslate nohighlight">\(Z(t)\)</span> the trajectory of the full complexity physical system (say planet earth). Because in practice, for computational or observational reasons, we cannot afford describing and predicting <span class="math notranslate nohighlight">\(Z(t)\)</span>, we will only focus on a projection of <span class="math notranslate nohighlight">\(Z(t)\)</span> in some lower dimension space. Let’s call this reduced dimension state <span class="math notranslate nohighlight">\(X(t)\)</span>.</p>
+<p>In our L96 toy model (analog to the real world),  <span class="math notranslate nohighlight">\(Z(t)=(X(t),Y(t))\)</span> is the full complexity physical system, while <span class="math notranslate nohighlight">\(X(t)\)</span> is the lower dimension reduction (single time-scale models is analog to the GCM). In real world situations or more complex models (e.g. actual atmosphere or ocean models), the lower dimension representation of the real system may be thought of as a coarse-grained or a subsampled description of the full-scale system.</p>
+<p>Now, a GCM is simply a numerical machine which intends to predict the trajectory <span class="math notranslate nohighlight">\(X(t)\)</span> from knowledge of <span class="math notranslate nohighlight">\(X(t=0)\)</span> only. A GCM is generally built from first principle physical laws, by discretizing partial differential equations.</p>
+<p>In what follows, we therefore assume that we know a fraction of the terms that govern the evolution of <span class="math notranslate nohighlight">\(X\)</span>. We also assume that we do not know what governs the evolution of <span class="math notranslate nohighlight">\(Y\)</span> nor how <span class="math notranslate nohighlight">\(Y\)</span> may affect <span class="math notranslate nohighlight">\(X\)</span>.</p>
+<p><strong>GCM without parameterization</strong></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">GCM_no_parameterization</span><span class="p">(</span><span class="n">X0</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;GCM without parameterization</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        X0: initial conditions</span>
+<span class="sd">        dt: time increment</span>
+<span class="sd">        nt: number of forward steps to take</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+        <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+        <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+    <span class="p">)</span>
+    <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">+</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">F</span><span class="p">))</span>
+        <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>This GCM is unstable due to Euler forward time stepping scheme, so we don’t integrate it for too long and compare it to the real world with the same time interval as <code class="docutils literal notranslate"><span class="pre">dt</span></code> used by the model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">3.0</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">GCM_no_parameterization</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Compare with the real world</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth (L96)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Model (GCM)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_4(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/2a53176adb69bff5f76f533f664689079b4038a966b1e9136561c5fd07c79d6e.png" src="../_images/2a53176adb69bff5f76f533f664689079b4038a966b1e9136561c5fd07c79d6e.png" />
+</div>
+</div>
+<p>There are several reasons why the above model (i.e. single time-scale model) differs from truth (i.e. L96 two time-scale model), which will be discussed below in <a class="reference internal" href="#other-sources-affect-gcm"><span class="std std-ref">Sources of model error</span></a>. One of these reasons is missing physics.</p>
+<p>One way, discussed in the previous notebook, to reduce the differences between the Model and the Truth, is to add a <em>parameterization</em>: an extra term to the rhs of the Model evolution operator in order to reduce the Model error as compared to the Truth. It may account for missing processes that are present in the truth, but are not included in the reduced model. The missing processed may be a result of unresolved scales (sub-grid processes) or due to physical processes that could not be encoded into the full equations.
+<em>Parameterizations</em> are also commonly refered to as <em>closures</em>, in particular when they encode explicit physical assumptions on how non-represented variables (e.g. <span class="math notranslate nohighlight">\(Y\)</span>) impact represented variables (e.g. <span class="math notranslate nohighlight">\(X\)</span>).</p>
+<p>Parameterizations usually involve free parameters that need to be adjusted. The form of the parameterization may be dictated by physical laws, but generally it is unknown as well.</p>
+<p><strong>GCM with parameterization</strong></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Here we introduce a class to solve for the one time-scale problem,</span>
+<span class="c1"># which can take arbitrary parameterizations and time-stepping schemes as input.</span>
+<span class="k">class</span> <span class="nc">GCM</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;GCM with parameterization in rhs of equation for tendency&quot;&quot;&quot;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">parameterization</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">EulerFwd</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">parameterization</span> <span class="o">=</span> <span class="n">parameterization</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">):</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">parameterization</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            X0: initial conditions</span>
+<span class="sd">            dt: time increment</span>
+<span class="sd">            nt: number of forward steps to take</span>
+<span class="sd">            param: parameters of our closure</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>As a first step, we illustrate introducing a polynomial parameterization to GCM and then compare the model to the true trajectories obtained from the real world with the same time interval as <code class="docutils literal notranslate"><span class="pre">dt</span></code> used by the model. This is the same as what was done in the previous notebook, but is shown again for completeness.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">naive_parameterization</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+<span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">5.0</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mf">0.85439536</span><span class="p">,</span> <span class="mf">0.75218026</span><span class="p">])</span>
+<span class="c1"># we use the parameters for the linear polynomial parameterization that were learnt in the last notebook.</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Compare with the real world</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Model&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_4(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/c3c3bb667a65afa1f323ab5220a0dd100ae43389d6df40b0431cd81153458afd.png" src="../_images/c3c3bb667a65afa1f323ab5220a0dd100ae43389d6df40b0431cd81153458afd.png" />
+</div>
+</div>
+<p>While the GCM with parameterization is better than the GCM without parameterization, it is still not very good at reproducing the true evolution of the full system. It also remains to find the most appropriate coefficients of the polynomial parameterization to make the Model as close as possible to the Truth.</p>
+<p>In summary, the parameterization problem boils down to defining the functional form and finding the best parameters in order to minimize the distance between the true trajectory and the model trajectory.
+<strong>With M2LINES, we are approaching this problem as a Machine Learning problem. We want to learn parameterizations from objective measures of their skills through an optimization procedure.</strong>
+But we are not only interested in learning the parameters of existing functional forms. More generally, we would like to learn the functional forms too.</p>
+</div>
+<div class="section" id="should-parameterizations-be-deterministic-or-stochastic">
+<h2>Should parameterizations  be deterministic or stochastic ?<a class="headerlink" href="#should-parameterizations-be-deterministic-or-stochastic" title="Permalink to this headline">#</a></h2>
+<p>The <code class="docutils literal notranslate"><span class="pre">naive_parameterization</span></code> above has no particular physical nor mathematical justification. Most importantly, it relies on a very strong assumption, that the time rate of change of <span class="math notranslate nohighlight">\(X\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> is a function of <span class="math notranslate nohighlight">\(X(t)\)</span>. This assumption implies that the future evolution of the reduced dimension system <span class="math notranslate nohighlight">\(X(t)\)</span> is <em>deterministically</em> related to the initial reduced dimension state <span class="math notranslate nohighlight">\(X(0)\)</span>.</p>
+<p>But this is not a good assumption because the two identical reduced dimension states (<span class="math notranslate nohighlight">\(X\)</span>, macro-state) can be associated with very different fine scale states (<span class="math notranslate nohighlight">\(Y\)</span>, micro-state). This can also be seen visually by considering the <a class="reference internal" href="gcm-analogue.html#sub-grid-hist-label"><span class="std std-ref">plot</span></a> in the previous notebook, which shows that the for each value of <span class="math notranslate nohighlight">\(X\)</span> there is a range of possible values for the sub-grid effects.
+Given the non-linearity of the evolution equation for <span class="math notranslate nohighlight">\(Z\)</span>, the two large scale trajectories will diverge at some point due to these small differences in the un-observed states. Let’s illustrate that with L96 alone.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Randomising the initial Ys</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">13</span><span class="p">)</span>
+
+<span class="c1"># Duplicating L96 to create perturbed versions that include random perturbations in Y</span>
+<span class="n">Wp1</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+<span class="n">Yp1</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">Y</span><span class="o">.</span><span class="n">std</span><span class="p">()</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">Wp1</span><span class="o">.</span><span class="n">Y</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
+<span class="n">Wp1</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">Yp1</span><span class="p">)</span>
+
+<span class="n">Wp2</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+<span class="n">Yp2</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">Y</span> <span class="o">+</span> <span class="mf">0.0001</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">Wp2</span><span class="o">.</span><span class="n">Y</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
+<span class="n">Wp2</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">Yp2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>L96: K=8 J=32 F=18 h=1 b=10 c=10 dt=0.001
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Running L96 and perturbed versions to compare results</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+<span class="n">X_pert1</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">Wp1</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+<span class="n">X_pert2</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">Wp2</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_pert1</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Perturbed 1&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_pert2</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Perturbed 2&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_4(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/08e1e31ba58751e82d182d48ce994e9989cf3ec336fd6bdd7bbee8f7823bf355.png" src="../_images/08e1e31ba58751e82d182d48ce994e9989cf3ec336fd6bdd7bbee8f7823bf355.png" />
+</div>
+</div>
+<p>So even very small uncertainties in the micro-state (<span class="math notranslate nohighlight">\(Y\)</span>) of L96 can lead to large scale changes (i.e. of the variable <span class="math notranslate nohighlight">\(X\)</span>) over short time.</p>
+<p>In a Model that does not know anything about micro-state <span class="math notranslate nohighlight">\(Y\)</span>, it is possible to introduce this uncertainty through a stochastic form in the parameterization.</p>
+<p>In addition, with this illustration, we also highlight that there is a horizon after which pointwise comparisons of the model with the truth are meaningless, hence there is some needed discussion on how to measure the skill of a parameterization.</p>
+</div>
+<div class="section" id="how-to-measure-parameterization-skill">
+<h2>How to measure parameterization skill ?<a class="headerlink" href="#how-to-measure-parameterization-skill" title="Permalink to this headline">#</a></h2>
+<p>We would like to build our closures by systematically measuring their skills, so that we can compare different fomulations using these “skill scores”.</p>
+<p>Since we are interested in matching the evolution of the “real world” using a GCM, we define skill scores that measure the distance between the evolution of the true state <span class="math notranslate nohighlight">\(X_{true}\)</span> and the simulated state <span class="math notranslate nohighlight">\(X_{gcm}\)</span>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Defining our GCM</span>
+<span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.005</span><span class="p">,</span> <span class="mf">1000.0</span>
+<span class="c1"># Use Runge-Kutta time stepping scheme because forward Euler would lead to instabilities</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">)</span>
+
+<span class="c1"># Evaluate the GCMs</span>
+<span class="n">X_gcm</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mf">0.85439536</span><span class="p">,</span> <span class="mf">0.75218026</span><span class="p">])</span>
+<span class="n">X_gcm_no_param</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
+
+<span class="c1"># Evaluate the true state</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plotting the results</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_true</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_gcm_no_param</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X_gcm</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with param&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_4(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span>
+    <span class="p">(</span><span class="n">X_true</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_gcm_no_param</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[:</span><span class="mi">500</span><span class="p">],</span>
+    <span class="p">(</span><span class="n">X_true</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_gcm</span><span class="p">[:</span><span class="mi">500</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with param&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_4^</span><span class="si">{true}</span><span class="s2">(t) - X_4^</span><span class="si">{gcm}</span><span class="s2">(t)$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/8e4c6a663e7a3a3cc04be93fd9c822691ce775c9edaa4ec2efba3de1740d8f0e.png" src="../_images/8e4c6a663e7a3a3cc04be93fd9c822691ce775c9edaa4ec2efba3de1740d8f0e.png" />
+</div>
+</div>
+<p><strong>Error metric based on model evolution</strong>:</p>
+<p>Clearly, the absolute error between the true and gcm trajectory grows with time, and we would like measure how the error cumulates over time.
+A simple error metric is the point-wise root mean square error, which is averaged over time:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-cfc10c99-7a28-4355-8445-02153488318c">
+<span class="eqno">(5)<a class="headerlink" href="#equation-cfc10c99-7a28-4355-8445-02153488318c" title="Permalink to this equation">#</a></span>\[\begin{equation}
+E (t) = \frac{1}{t} \int_0^t |X_{true}(t) - X_{gcm}(t)| dt
+\end{equation}\]</div>
+<p>This can be computed for each <span class="math notranslate nohighlight">\(X_k\)</span> separately, or averaged over all <span class="math notranslate nohighlight">\(k\)</span>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">error_model_evolution</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">L</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Model evolution error computed over some window t &lt; L.&quot;&quot;&quot;</span>
+    <span class="n">D</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span>
+        <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">((</span><span class="n">X1</span><span class="p">[:</span> <span class="nb">int</span><span class="p">(</span><span class="n">L</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">X2</span><span class="p">[:</span> <span class="nb">int</span><span class="p">(</span><span class="n">L</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="p">:])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span>
+    <span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">L</span> <span class="o">/</span> <span class="n">dt</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
+    <span class="k">return</span> <span class="n">D</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">dist</span> <span class="o">=</span> <span class="n">error_model_evolution</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_gcm</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+<span class="n">dist_no_param</span> <span class="o">=</span> <span class="n">error_model_evolution</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_gcm_no_param</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plotting how this distance grows with the length of the window</span>
+<span class="c1"># for all the components of X</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dist_no_param</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dist</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with param&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span>
+<span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$||X_</span><span class="si">{true}</span><span class="s2">-X_</span><span class="si">{gcm}</span><span class="s2">||$&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/698819cf269a9dcd2f431b2e63e4f79e55591e0efd7d7a648d804c5de33401f6.png" src="../_images/698819cf269a9dcd2f431b2e63e4f79e55591e0efd7d7a648d804c5de33401f6.png" />
+</div>
+</div>
+<p>The error grows with time, but saturates to some a constant after the truth and GCM have gotten completely decorrelated. This constant is equal to the sum of the variance of the truth and GCM states. Also, as expected, the error grows much more rapidly without a parameterization, showing that adding the parameterization has resulted in a quantitative improvement in the solution.</p>
+<p><strong>Climatology based error metric:</strong></p>
+<p>Knowing from the above discussion that the system is not predictable after some time, we may as well have decided to measure how well the model captures the mean state. With this we expect to measure the “climate” of the system instead of the “weather”.</p>
+<div class="amsmath math notranslate nohighlight" id="equation-0dd65573-d515-4eaf-afed-f8ab3d0ffbb0">
+<span class="eqno">(6)<a class="headerlink" href="#equation-0dd65573-d515-4eaf-afed-f8ab3d0ffbb0" title="Permalink to this equation">#</a></span>\[\begin{equation}
+E_{clim}  = | \overline{X_{true}}  - \overline{X_{gcm}}|
+\end{equation}\]</div>
+<p>where the <span class="math notranslate nohighlight">\(\overline{(.)}\)</span> is some suitably defined average that may be variable in time (like a seasonal climatology).</p>
+<p>This metric will likely converge to some non-trivial values which are indicative of how well our model captures the “climate” of the system, and the difference indicating the bias in the model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">211</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_gcm_no_param</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
+    <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span>
+    <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_gcm</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
+    <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span>
+    <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with param&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$\overline{X_</span><span class="si">{k}</span><span class="s2">}$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">212</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_gcm_no_param</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">),</span>
+    <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span>
+    <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_gcm</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">),</span>
+    <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span>
+    <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with param&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$E_</span><span class="si">{clim}</span><span class="s2">$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/ceca613f6a0425b2c46fe9d2da7a3379a8def72999cb8e94f2be87367c9792ac.png" src="../_images/ceca613f6a0425b2c46fe9d2da7a3379a8def72999cb8e94f2be87367c9792ac.png" />
+</div>
+</div>
+<p>As seen in the first figure, the long term mean state of the true solution is different from the two GCM solutions. Surprisingly, the GCM without parameterization has a climatology that is slightly closer to the true climatology than the climatology of the GCM with parameterization. In other words, the error <span class="math notranslate nohighlight">\(E_{clim}\)</span> is larger for the GCM with parameterization. This is shown in the second figure.</p>
+<p>The above result is counter-intuitive, and suggests that <span class="math notranslate nohighlight">\(E_{clim}\)</span> is not a very powerful metric. This metric takes into account only the means of the underlying probability distributions. Let’s look at the full probability distributions next!</p>
+<p><strong>Distribution based error metric</strong>:</p>
+<p>We want to measure differences between the empirical distributions of the true solution vs. GCM solution. There are several options how to define distributional differences, but here we consider the earth mover’s distance or Wasserstein distance:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-3c4563a4-5ff5-4cca-8b27-5a151088717b">
+<span class="eqno">(7)<a class="headerlink" href="#equation-3c4563a4-5ff5-4cca-8b27-5a151088717b" title="Permalink to this equation">#</a></span>\[\begin{align}
+E_{distr}  &amp; = \int_{-\infty}^\infty |P(X^{true}\leq x) - P(X^{GCM}\leq x) \, | dx, \\
+\end{align}\]</div>
+<p>where <span class="math notranslate nohighlight">\(P(X\leq x)\)</span> is the cumulative distribution function of <span class="math notranslate nohighlight">\(X\)</span>.</p>
+<p>Before we compute the differences between the distribution <span class="math notranslate nohighlight">\(E_{distr}\)</span>, let’s first look at the empirical distributions themselves. For simplicity, we show the probability density functions (rather than the cumulative distribution functions, where the latter are used in the computation of <span class="math notranslate nohighlight">\(E_{distr}\)</span>).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">kdeplot</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;_none&quot;</span><span class="p">)</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">kdeplot</span><span class="p">(</span>
+    <span class="n">X_gcm_no_param</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">X_gcm_no_param</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;_none&quot;</span><span class="p">)</span>
+
+<span class="n">sns</span><span class="o">.</span><span class="n">kdeplot</span><span class="p">(</span><span class="n">X_gcm</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">X_gcm</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;_none&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$X$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7f0030482070&gt;
+</pre></div>
+</div>
+<img alt="../_images/14a5ff0814487e1fcfc9b95a3741eeeb429ad913299b436f859ad570e7b5879d.png" src="../_images/14a5ff0814487e1fcfc9b95a3741eeeb429ad913299b436f859ad570e7b5879d.png" />
+</div>
+</div>
+<p>The vertical lines in the figure above indicate the means of the respective empirical distributions. As we saw earlier, the mean of the GCM without parameterization is slightly closer to the real mean than the mean of the GCM with parameterization. However, if we take into account the entire distribution, the GCM with parameterization seems to be a closer match to the true distribution. Our eyeball estimate is quantitatively confirmed by the next figure, which computes the distributional error, <span class="math notranslate nohighlight">\(E_{distr}\)</span>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">wasserstein_distance</span>
+
+
+<span class="k">def</span> <span class="nf">error_distribution</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Distribution error measured by the Wasserstein distance.&quot;&quot;&quot;</span>
+
+    <span class="n">diff</span> <span class="o">=</span> <span class="n">wasserstein_distance</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">diff</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">E_distr_no_param</span> <span class="o">=</span> <span class="p">[</span><span class="n">error_distribution</span><span class="p">(</span><span class="n">X_true</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">X_gcm_no_param</span><span class="p">[:,</span> <span class="n">k</span><span class="p">])</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">]</span>
+<span class="n">E_distr_param</span> <span class="o">=</span> <span class="p">[</span><span class="n">error_distribution</span><span class="p">(</span><span class="n">X_true</span><span class="p">[:,</span> <span class="n">k</span><span class="p">],</span> <span class="n">X_gcm</span><span class="p">[:,</span> <span class="n">k</span><span class="p">])</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">E_distr_no_param</span><span class="p">,</span>
+    <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span><span class="p">,</span>
+    <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">W</span><span class="o">.</span><span class="n">k</span><span class="p">,</span>
+    <span class="n">E_distr_param</span><span class="p">,</span>
+    <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with param&quot;</span><span class="p">,</span>
+    <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$E_</span><span class="si">{distr}</span><span class="s2">$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7f00304b9f10&gt;
+</pre></div>
+</div>
+<img alt="../_images/0df982713de18b7004700e7913fdb51d768a90e5ba6ccb01fd8b979693a4e31d.png" src="../_images/0df982713de18b7004700e7913fdb51d768a90e5ba6ccb01fd8b979693a4e31d.png" />
+</div>
+</div>
+<p><strong>Tendency based error metric</strong>:</p>
+<p>It is also very common to formulate closures based on databases of initial tendencies.  In the Large Eddy Simulation  community, this is sometimes refered to as <em>a priori</em> skill, because you don’t need to run the full model to compute it.</p>
+<p>This is the sort of game that several of us have been playing, trying for instance to estimate subgrid fluxes from knowledge of the large scale quantities  $<span class="math notranslate nohighlight">\( \nabla\cdot \mathbf{s} =   \nabla\cdot\big(\overline{\mathbf{u}\,\Phi} - \overline{\mathbf{u}}\,\overline{\Phi}\big) \simeq f(\overline{\mathbf{u}},\overline{\Phi})\)</span>$</p>
+<p><em>Note that this is not exactly the same problem as the <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">priori</span></code> LES problem, because of the interplay with time-discretization. Let’s neglect that for the moment.</em></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">norm_initial_tendency</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">):</span>
+    <span class="n">T1</span> <span class="o">=</span> <span class="n">X1</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">X1</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">T2</span> <span class="o">=</span> <span class="n">X2</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">X2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">((</span><span class="n">T1</span> <span class="o">-</span> <span class="n">T2</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Because this metric is cheap to evaluate, as we do not need to integrate the GCM more than 1 time-step, we can start a sensitivity analysis in order to identify good optimal parameters for the specific formulation <code class="docutils literal notranslate"><span class="pre">naive_parameterization</span></code></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span>
+
+<span class="c1"># Let&#39;s define again the true state</span>
+<span class="c1"># But only run for 1 time step</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
+
+<span class="c1"># and an ensemble of trajectories</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+
+<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
+
+<span class="n">_p1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">_p2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">xp1</span><span class="p">,</span> <span class="n">yp2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">_p1</span><span class="p">,</span> <span class="n">_p2</span><span class="p">)</span>
+
+<span class="n">score</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
+
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">p1</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">_p1</span><span class="p">):</span>
+    <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">p2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">_p2</span><span class="p">):</span>
+        <span class="n">X_gcm</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">])</span>  <span class="c1"># run gcm for 1 time step</span>
+        <span class="n">score</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">norm_initial_tendency</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_gcm</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">xp1</span><span class="p">,</span> <span class="n">yp2</span><span class="p">,</span> <span class="n">score</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$p_2$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$p_1$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Score&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/df24707e7e69056417cb8386bc8462baca171a35a924364689a7ccd5178b3022.png" src="../_images/df24707e7e69056417cb8386bc8462baca171a35a924364689a7ccd5178b3022.png" />
+</div>
+</div>
+<p>From this analysis, we see that the optimisation problem is probably well posed as the cost function appears pretty smooth. One can also see that the parameter <span class="math notranslate nohighlight">\(p_1\)</span> is more important than <span class="math notranslate nohighlight">\(p_2\)</span>.</p>
+</div>
+<div class="section" id="sources-of-model-error">
+<span id="other-sources-affect-gcm"></span><h2>Sources of model error<a class="headerlink" href="#sources-of-model-error" title="Permalink to this headline">#</a></h2>
+<p>Missing physics is not the only source of error in GCMs, and many other factors can contribute to the GCM state diverging from the real world.
+Here, we consider the following five sources of error:</p>
+<ol class="arabic">
+<li><p>Missing physics: which is modeled with a GCM without parameterization that corresponds to the one time-scale Lorenz-96 system without any the coupling term.</p></li>
+<li><p>Poorly parameterized unresolved physics: which is studied by considering a first-order and third-order polynomial approximations of the coupling terms:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+    P_4 \rightarrow P_1
+    \end{equation*}\]</div>
+</li>
+<li><p>Unknown forcing: which is modeled by adding an error to the forcing term:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+    F \rightarrow F + error
+    \end{equation*}\]</div>
+</li>
+<li><p>Numerical approximation: which is studied by increasing the time-step:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+    \Delta t \rightarrow 10 \Delta t
+    \end{equation*}\]</div>
+</li>
+<li><p>Initialization error: which is modeled by adding an error to the initial condition:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+    X(t=0) \rightarrow X(t=0) + error
+    \end{equation*}\]</div>
+</li>
+</ol>
+<p>The first two sources have already been discussed in some detail previously, and included here to contrast against other error sources.</p>
+<p>The next code estimates these sources of error and the figure shows their relative contributions. For reference, we also plot the error of the GCM using <span id="id2">[<a class="reference internal" href="../bibliography.html#id3" title="Daniel S. Wilks. Effects of stochastic parametrizations in the lorenz \textquotesingle 96 system. Quarterly Journal of the Royal Meteorological Society, 131(606):389–407, 2005. URL: https://doi.org/10.1256/qj.04.03, doi:10.1256/qj.04.03.">Wilks, 2005</a>]</span> polynomial coupling term and without any of the sources of error listed above. All errors are evaluated by comparing the GCMs to the “truth” model goverened by the full two time-scale Lorenz-96 system.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Generic GCM parameters</span>
+<span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.002</span><span class="p">,</span> <span class="mf">40.0</span>
+<span class="n">X0</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">X</span>
+
+<span class="c1"># Remember the optimized parameters from previous notebook</span>
+<span class="n">p1</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.85439536</span><span class="p">,</span> <span class="mf">0.75218026</span><span class="p">]</span>
+<span class="n">p4</span> <span class="o">=</span> <span class="p">[</span>
+    <span class="mf">0.000707</span><span class="p">,</span>
+    <span class="o">-</span><span class="mf">0.0130</span><span class="p">,</span>
+    <span class="o">-</span><span class="mf">0.0190</span><span class="p">,</span>
+    <span class="mf">1.59</span><span class="p">,</span>
+    <span class="mf">0.275</span><span class="p">,</span>
+<span class="p">]</span>
+
+<span class="c1"># Sampling real world over a longer period of time</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
+
+<span class="c1"># GCM with different parameterizations</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>  <span class="c1"># Missing physics</span>
+<span class="n">X_p1</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">p1</span><span class="p">)</span>  <span class="c1"># Simpler but poorer parameterization</span>
+<span class="n">X_p4</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">p4</span><span class="p">)</span>  <span class="c1"># More complex parameterization</span>
+
+<span class="c1"># GCM with perturbed forcing</span>
+<span class="n">gcm_pert_F</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">F</span> <span class="o">+</span> <span class="mf">1.0</span><span class="p">,</span> <span class="n">naive_parameterization</span><span class="p">)</span>
+<span class="n">X_frc</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm_pert_F</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">p4</span><span class="p">)</span>  <span class="c1"># Perturbed forcing</span>
+
+<span class="c1"># GCM with perturbed IC</span>
+<span class="n">X_IC</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">dt</span><span class="p">),</span> <span class="n">param</span><span class="o">=</span><span class="n">p4</span><span class="p">)</span>  <span class="c1"># Perturbed IC</span>
+
+<span class="c1"># GCM with changed dt</span>
+<span class="n">X_dt</span><span class="p">,</span> <span class="n">t_dt</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">dt</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">dt</span><span class="p">)),</span> <span class="n">param</span><span class="o">=</span><span class="n">p4</span><span class="p">)</span>  <span class="c1"># Larged dt</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_no_param</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_p1</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with linear param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_p4</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with 4th order param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_frc</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with perturbed forcing&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_IC</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with pertured IC&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_dt</span><span class="p">,</span> <span class="n">X_dt</span><span class="p">[:,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with increased dt&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$X_</span><span class="si">{4}</span><span class="s2">(t)$&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/3de102cded80cc8765838dbaf843d5c91c3d2e57ebbac3987ccf4ae292cdc273.png" src="../_images/3de102cded80cc8765838dbaf843d5c91c3d2e57ebbac3987ccf4ae292cdc273.png" />
+</div>
+</div>
+<p>Clearly perturbing the gcm in different ways results in different solutions. Below we quantify these effects using the <code class="docutils literal notranslate"><span class="pre">error_model_evolution</span> <span class="pre">metric</span></code> that was defined earlier. We present the results as an average over all k.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">dist</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">error_model_evolution</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">dt</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">dist_no_param</span> <span class="o">=</span> <span class="n">dist</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_no_param</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
+<span class="n">dist_p1</span> <span class="o">=</span> <span class="n">dist</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_p1</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
+<span class="n">dist_p4</span> <span class="o">=</span> <span class="n">dist</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_p4</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
+<span class="n">dist_frc</span> <span class="o">=</span> <span class="n">dist</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_frc</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
+<span class="n">dist_IC</span> <span class="o">=</span> <span class="n">dist</span><span class="p">(</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_IC</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
+<span class="n">dist_dt</span> <span class="o">=</span> <span class="n">dist</span><span class="p">(</span><span class="n">X_true</span><span class="p">[::</span><span class="mi">5</span><span class="p">],</span> <span class="n">X_dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">dt</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">dist_no_param</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">dist_p1</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with linear param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">dist_p4</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with 4th order param&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">dist_frc</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with perturbed forcing&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">dist_IC</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with pertured IC&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_dt</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">dist_dt</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with increased dt&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$t$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$E(t)$&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/f02d520723e155426ad3f5582b5c3f764ca738f52843aa02c386200f57e43644.png" src="../_images/f02d520723e155426ad3f5582b5c3f764ca738f52843aa02c386200f57e43644.png" />
+</div>
+</div>
+<p>Under the perturbations considered above, the lack of missing physics contributes the most error to the GCM.<br />
+This can be fixed by adding parameterizations, which have different contributions to error at different times. Specifically, the error grows more rapidly for the higher order parameterization relative to the linear parameterization, but saturates to smaller error at longer time. This might partly be because we optimized the parameters for the linear parameterization, but used parameter estimates for the higher order parameterization from the literature.<br />
+The second largest error source is the numerical approximation (or changed dt), suggesting that we need to be careful about the design and choice of the numerical schemes.
+The errors due to the forcing and initial condition reult in smaller errors, but will likely grow as the perturbations to these change.</p>
+</div>
+<div class="section" id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline">#</a></h2>
+<p>In this chapter:</p>
+<ul class="simple">
+<li><p>We first reintroduced the GCM parameterization problem, discussing it in terms of limited observability of GCMs.</p></li>
+<li><p>Then we discussed the need for stochastic parameterizations, which arises because of the possible differences in the unresolved or hidden states.</p></li>
+<li><p>Then we discussed how skill can be measured, and defined four different metrics.</p></li>
+<li><p>Finally we discussed the different sources of error that may be presend in numerical GCMs.</p></li>
+</ul>
+<p>This chapter has made it clear that even models with some degree of tuning will diverge from the true solution, as there are multiple sources of errors. In the next chapter we discuss if and how GCMs and their parameterizations can be further tuned to enhance their range of predictability (which is known to not be in finite).</p>
+</div>
+</div>
+
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-need-for-gcm-parameterizations">The need for GCM parameterizations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#should-parameterizations-be-deterministic-or-stochastic">Should parameterizations  be deterministic or stochastic ?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#how-to-measure-parameterization-skill">How to measure parameterization skill ?</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sources-of-model-error">Sources of model error</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="closing_remarks.html">Outlook</a></li>
+<li class="toctree-l1"><a class="reference internal" href="citing.html">Citing this book</a></li>
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+    <h1>Introduction to Machine Learning and Neural Networks</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-machine-learning">Introduction to machine learning</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#pytorch">PyTorch</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#linear-regression">Linear regression</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#data-generation-for-linear-regression">Data generation for linear regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#loss-function">Loss function</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-descent-and-optimization">Gradient descent and optimization</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#neural-networks">Neural Networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-some-data">Generate some data</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-neural-network">Define the neural network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#updating-the-weights-using-an-optimizer">Updating the Weights using an Optimizer</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#adam-optimizer">Adam Optimizer</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Loss function</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#training-the-network">Training the network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#compare-predictions-with-ground-truth">Compare Predictions with Ground Truth</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
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+
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+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="introduction-to-machine-learning-and-neural-networks">
+<h1>Introduction to Machine Learning and Neural Networks<a class="headerlink" href="#introduction-to-machine-learning-and-neural-networks" title="Permalink to this headline">#</a></h1>
+<div class="section" id="outline">
+<h2>Outline:<a class="headerlink" href="#outline" title="Permalink to this headline">#</a></h2>
+<p>In this notebook we introduce the basics of machine learning, show how to optimize parameters in simple models (linear regression), and finally demonstrate a basic neural network training case. There are many excellent tutorials on machine learning available on the internet, and our goal here is not to be comprehensive but rather provide a gentle introduction that seems necessary to what comes later in the book.</p>
+</div>
+<div class="section" id="introduction-to-machine-learning">
+<h2>Introduction to machine learning<a class="headerlink" href="#introduction-to-machine-learning" title="Permalink to this headline">#</a></h2>
+<p>Machine learning can be defined as : “Machine learning is a way for computers to learn and improve from experience without being explicitly programmed.” This definition was generated with the help of an AI agent called <a class="reference external" href="https://chat.openai.com/share/4cb73dc4-e372-445c-8aca-94c6199540eb">ChatGPT</a>.</p>
+<p>It has proven to be a useful tool to the scientific community by being able to make advancements in a range of problems, which had traditionally been very challenging or impossible. In our project, <a class="reference external" href="https://m2lines.github.io/">M2LInES</a>, we are improving simulations of the climate system by targeting the parameterization or sub-grid closure problem with the help of ML.</p>
+<p>In simpler terms, ML for our context can be thought of as a sophisticated way to approximate or estimate function. This can be done in a way where the structure of the functions is prescribed and constrained to very limited space of functions (e.g. linear regression) or in a way where the the structure of the function is left relatively free (e.g. neural networks). Regardless of this choice, all ML tasks can be broken down into a few general pieces:</p>
+<ul class="simple">
+<li><p>Data : All ML tasks need data to learn patterns.</p></li>
+<li><p>Model : Some choice needs to be made about the model architecture (e.g. neural network, gaussian process, etc).</p></li>
+<li><p>Loss functions : The way a ML model gets closer to the true function is by minimizing some appropriately chosen loss function, a measure of error between truth and ML prediction.</p></li>
+<li><p>Optimizers : To minize the error or loss function some method is needed to estimate optimal parameters.</p></li>
+</ul>
+<p>Since all this is being done on a computer, some software architecture is needed. Possible, one of the major reasons for large scale adoption of machine learning has been the availability of easy to use, well documented, open source and general purpose machine learning libraries. Many such library exist, e.g. <a class="reference external" href="https://www.tensorflow.org/">Tensorflow</a>, <a class="reference external" href="https://pytorch.org/">PyTorch</a>, <a class="reference external" href="https://jax.readthedocs.io/en/latest/index.html">JAX</a>, etc. In this book we will primarily use PyTorch.</p>
+<div class="section" id="pytorch">
+<h3><a class="reference external" href="https://github.com/pytorch/pytorch">PyTorch</a><a class="headerlink" href="#pytorch" title="Permalink to this headline">#</a></h3>
+<p>In short, PyTorch is a Python-based machine learning framework that can be used for scientific machine learning tasks. There are many good reasons to use PyTorch, including that it can be used as a replacement for NumPy if we want to do computations on GPUs, it is extremely flexible and fast, but primarily that it has a large community and excellent tutorials to learn from.</p>
+<p>To understand and install PyTorch, we recommend going through the <a class="reference external" href="https://pytorch.org/tutorials/">tutorials</a>, <a class="reference external" href="https://pytorch.org/blog/">blogs</a>, or one of the many online tutorial (e.g. <a class="reference external" href="https://pytorch.org/tutorials/beginner/deep_learning_60min_blitz.html">Deep Learning With Pytorch: A 60 Minute Blitz</a>, <a class="reference external" href="https://pytorch.org/tutorials/beginner/blitz/neural_networks_tutorial.html">Neural Networks</a> etc).</p>
+<p>To follow along a single blogpost that goes through most concepts, we recommend <a class="reference external" href="https://blog.paperspace.com/pytorch-101-understanding-graphs-and-automatic-differentiation/">this</a>.</p>
+<hr class="docutils" />
+<p>Having understood how the tools work, we can now use them to model our data. To train any machine learning model, there is a general recipe followed. This recipe can be summarized in three steps:</p>
+<ol class="arabic simple">
+<li><p>Define a model representation.</p></li>
+<li><p>Choose a suitable loss function that tells us how far apart our model predictions are from the real data</p></li>
+<li><p>Update the parameters in model representation using an optimization algorithm to minimize the loss function.</p></li>
+</ol>
+<p>In this notebook, we will be working a linear regression model and a neural network. We will measure loss using the Mean Squared Error (MSE) function and will try to optimize or reduce the loss value using a popular optimization algorithm, gradient descent.</p>
+<p>To read further on how to train machine learning models, refer to Prof. Fund’s <a class="reference external" href="https://github.com/ffund/intro-ml-tss21">lecture notes</a> or Prof. Hegde’s <a class="reference external" href="https://chinmayhegde.github.io/dl-notes/notes/lecture03/">blogposts</a>.</p>
+</div>
+</div>
+<div class="section" id="linear-regression">
+<h2>Linear regression<a class="headerlink" href="#linear-regression" title="Permalink to this headline">#</a></h2>
+<p>Probably, the simplest “machine learning” model is <a class="reference external" href="https://en.wikipedia.org/wiki/Linear_regression">linear regression</a>. This model models the relationship between the input and output data as a linear function, which for a single input and single output dimension is equivalent to fitting a line through the data:
+$<span class="math notranslate nohighlight">\(
+y = w_1x + w_0,
+\)</span><span class="math notranslate nohighlight">\(
+where \)</span>w_1<span class="math notranslate nohighlight">\( (slope or kernel) and \)</span>w_0$ (bias) are parameters that need to best estimated.</p>
+<p>For multiple inputs and single output the model can be visually represented as follows:</p>
+<a class="reference internal image-reference" href="https://miro.medium.com/v2/resize:fit:1032/1*WswH2fPx0bf_JFRMm8V-HA.gif"><img alt="nn-output-computation" src="https://miro.medium.com/v2/resize:fit:1032/1*WswH2fPx0bf_JFRMm8V-HA.gif" style="width: 500px;" /></a>
+<p><em>Image is taken from <a class="reference external" href="https://blog.insightdatascience.com/a-quick-introduction-to-vanilla-neural-networks-b0998c6216a1">here</a>.</em> Similarly, this linear regression could be modified to have multiple inputs and multiple outputs.</p>
+<p>There are many ways to estimate optimal parameters for linear regression, and here we will show how this can be done using PyTorch and gradient descent.</p>
+<p>The best prediction is defined by minimizing some cost or loss function. For linear regression, generally the mean sequare error between the expected output <span class="math notranslate nohighlight">\(y\)</span> and the prediction (<span class="math notranslate nohighlight">\(\tilde{y}\)</span>), defined as mean of <span class="math notranslate nohighlight">\((y-\tilde{y})^2\)</span> over all output dimensions and samples, is minimized.</p>
+<p><em>Most of this lesson follows a tutorial from Jeremy Howard’s fast.ai <a class="reference external" href="https://www.youtube.com/watch?v=ACU-T9L4_lI">lesson zero</a></em>.</p>
+<p>We will learn the parameters <span class="math notranslate nohighlight">\(w_0\)</span> and <span class="math notranslate nohighlight">\(w_1\)</span> of a line using PyTorch.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Import machine-learning packages</span>
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
+
+<span class="c1"># Import plotting packages</span>
+<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span><span class="p">,</span> <span class="n">HTML</span>
+<span class="kn">from</span> <span class="nn">matplotlib.animation</span> <span class="kn">import</span> <span class="n">FuncAnimation</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">time</span>
+<span class="kn">import</span> <span class="nn">base64</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="o">%</span><span class="k">matplotlib</span> inline
+</pre></div>
+</div>
+</div>
+</div>
+<p>Set random seed for reproducibility</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="data-generation-for-linear-regression">
+<h3>Data generation for linear regression<a class="headerlink" href="#data-generation-for-linear-regression" title="Permalink to this headline">#</a></h3>
+<p>Since this is a toy problem, we first choose the true parameters so that we can later easily verify the success of our approach.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">w</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">as_tensor</span><span class="p">([</span><span class="mf">3.0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
+<span class="n">w</span>
+<span class="c1"># where 3 is the interesection point (sometimes also called bias) and 2 is the slope.</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>tensor([3., 2.])
+</pre></div>
+</div>
+</div>
+</div>
+<p>Notice that we are using tensors here, which is a data structure used by PyTorch to easily deploy computations to CPUs or GPUs.</p>
+<p>Then we create some data points x and y which are related to each other through the linear relationship and parameters chosen above.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># We create 100 data points.</span>
+<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+
+<span class="c1"># uniformly sample x points between -1 and 1.</span>
+<span class="c1"># Underscore functions in pytorch means replace the value (update)</span>
+<span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">uniform_</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">[:</span><span class="mi">6</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>tensor([[ 1.0000, -0.0428],
+        [ 1.0000, -0.2131],
+        [ 1.0000, -0.1181],
+        [ 1.0000, -0.2475],
+        [ 1.0000, -0.4128],
+        [ 1.0000, -0.3115]])
+</pre></div>
+</div>
+</div>
+</div>
+<p>Note above that for each sample we have created two numbers.
+The first is 1, to be used with <span class="math notranslate nohighlight">\(w_0\)</span>, and the second is x, to be used with <span class="math notranslate nohighlight">\(w_1\)</span>.</p>
+<p>Next we generate output data using the equation for linear regression, but also add some random noise to be in the more real world setting where there is always some noise or if linear regression is not the perfect model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Generate output data.</span>
+<span class="n">y</span> <span class="o">=</span> <span class="n">x</span> <span class="o">@</span> <span class="n">w</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>  <span class="c1"># @ is a matrix product (similar to matmul)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/1f6586dec843683ca074b53e805dd990c1181b755fea9dfa7a70d0611ca293e3.png" src="../_images/1f6586dec843683ca074b53e805dd990c1181b755fea9dfa7a70d0611ca293e3.png" />
+</div>
+</div>
+</div>
+<div class="section" id="loss-function">
+<h3>Loss function<a class="headerlink" href="#loss-function" title="Permalink to this headline">#</a></h3>
+<p>We define our loss function <span class="math notranslate nohighlight">\(L\)</span> as Mean Square Error loss as:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+L_{MSE} = \frac{1}{n} \cdot \sum_{i=1}^{n} (y_i - \tilde y_i)^2
+\end{equation*}\]</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">mse</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">):</span>
+    <span class="k">return</span> <span class="p">((</span><span class="n">y_true</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Written in terms of <span class="math notranslate nohighlight">\(w_0\)</span> and <span class="math notranslate nohighlight">\(w_1\)</span>, our <strong>loss function</strong> <span class="math notranslate nohighlight">\(L\)</span> is:</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+L_{MSE} = \frac{1}{n} \cdot \sum_{i=1}^{n} (y_i - (w_0 + \mathbf{w_1} \cdot x_i))^2
+\end{equation*}\]</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Initialize parameters with some randomly guessed values.</span>
+<span class="n">w_real</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">as_tensor</span><span class="p">([</span><span class="o">-</span><span class="mf">3.0</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">])</span>
+<span class="n">y_hat</span> <span class="o">=</span> <span class="n">x</span> <span class="o">@</span> <span class="n">w_real</span>
+<span class="c1"># Initial mean-squared error</span>
+<span class="n">mse</span><span class="p">(</span><span class="n">y_hat</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>tensor(66.1959)
+</pre></div>
+</div>
+</div>
+</div>
+<p>Here we chose randomly guessed initial parameter choices. While this choice is not consequential here, in more complex problems these choice prior parameter choice can be extremely important.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">y_hat</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$</span><span class="se">\\</span><span class="s2">tilde</span><span class="si">{y}</span><span class="s2">$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Prediction of model using non-optimized model&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/e095a0e1d6dd8884f85ae8ee38d1af5a3ed9b80d10b3f8567b72e41579e1a1d9.png" src="../_images/e095a0e1d6dd8884f85ae8ee38d1af5a3ed9b80d10b3f8567b72e41579e1a1d9.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Initialize parameter obect that will be optimized.</span>
+<span class="n">w</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">w_real</span><span class="p">)</span>
+<span class="n">w</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Parameter containing:
+tensor([-3., -5.], requires_grad=True)
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="gradient-descent-and-optimization">
+<h3>Gradient descent and optimization<a class="headerlink" href="#gradient-descent-and-optimization" title="Permalink to this headline">#</a></h3>
+<p>So far, we have specified the <em>model</em> (linear regression) and the <em>evaluation criteria</em> (or <em>loss function</em>). Now we need to handle <em>optimization</em>; that is, how do we find the best values for weights (<span class="math notranslate nohighlight">\(w_0\)</span>, <span class="math notranslate nohighlight">\(w_1\)</span>) such that they best fit the linear regression parameters?</p>
+<p>This optimization is done using gradient descent. In this process the gradient of the loss function is estimated relative to the unknown parameters, and then the parameters are updated in a direction that reduces the loss function.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Load the image file</span>
+<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s2">&quot;figs/Gradient_descent2.png&quot;</span><span class="p">,</span> <span class="s2">&quot;rb&quot;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
+    <span class="n">image_data</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">read</span><span class="p">()</span>
+
+<span class="c1"># Create the HTML code to display the image</span>
+<span class="n">html</span> <span class="o">=</span> <span class="sa">f</span><span class="s1">&#39;&lt;div style=&quot;text-align:center&quot;&gt;&lt;img src=&quot;data:image/png;base64,</span><span class="si">{</span><span class="n">base64</span><span class="o">.</span><span class="n">b64encode</span><span class="p">(</span><span class="n">image_data</span><span class="p">)</span><span class="o">.</span><span class="n">decode</span><span class="p">()</span><span class="si">}</span><span class="s1">&quot; style=&quot;max-width:700px;&quot;/&gt;&lt;/div&gt;&#39;</span>
+
+<span class="c1"># Display the HTML code in the notebook</span>
+<span class="n">display</span><span class="p">(</span><span class="n">HTML</span><span class="p">(</span><span class="n">html</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><div style="text-align:center"><img 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PfcHDt2xo0xppsWUENIAoS6EAIVJFDvvZe9tft9nneuub/17SJtgewg8Szp+9Zac836++TFGPPnd86cBhyWHLW1tcZPTk6Of2K6ziIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAs0J0KmkVEvzDEkK89TX11teXp4VFhY2y5eTFjbz58+3uXPnWt++fa19+/aWm5vrjbASfnjPD48odWIDzWpWggiIwDVCgE630eu2MqYcpPNz9R58l/Hgu+1SB/NeLJ/X5eI7uyZvIWkHFbRC7FKss+v0u1hn6lH2eC7j97nY8L9B11Jd0qUIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIXFMEOEcYvUldXZ3LmDhvyPm5OEcXB33ixAnbuHGj3Xrrrfb3f//3MbnxnCVsVq5caWvXrrXhw4dbt27dLL8g33JzMtKGDcXG4nVLjTbWrgsREIGrnABn6NsyS38ZQuA7SKTpizO+51rqalvyep5E2GTRaypWWpA2eI23kXlKkWU1gl+sSTthPG37jRqQrSVnwyZymrTTEh+liYAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiMD3iQDn3poGv3D8nKPjJ7oURtbs27fPli5d6g7mueeea4YpS9isW7fOduzYYSNHjrSePXtaQUGBN8QKY+WcTKTEiY3E9GY1K0EEROAaIdAWgdA2GXCNAGkUInwPtnTwPenxM3jsjqOp6MgqxofphLbwDq2mS4WGWupNTGPurBLxQdbZu9pKNgmbLFS6EQEREAEREAEREAEREAEREAEREAEREAEREAEnEH0Jbyhm0t6Ezyh0GIGzc+dOmzdvnvXq1cuefPLJZvSaCZtdu3bZuHHjrEePHo0N8YKVspF4ne5AunHPoC8REIFriMClBEKc3Y/n7/bQ+e6KR3ynxfuWzun8LT1vPS1wS96azbKF9JCMt2tbXEqzOjwhVbT18XDMYdzp8bSYP8GT3b9U0+kHqWRdioAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiMD3mUB0JtGX8Ex5w4N71vCgsJkzZ477lxkzZnha+quZsNm9e7eNHz/eC8SKY0Ppib543eKEX7oFXYuACFzlBDhDf6lZ+ihB4vkqH3IL3Y/vvBYetZAUwh0z3CKXcOayY5mjLXwzuZtetS3qJTYYz01ryb7P7l945iUv9c8guxrdiYAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiMD3igDnEKOkidd0KIyw4T0DZmbNmmXdu3e3mTNnNmOTJWzWr19vUdiwQFrYxArTNUjWpGnoWgSuVQJtEQpRBMTztcricsaV5ha5hLN7j5jkMuybm5C2CRv2mw02NnrRgbQobL55Fy/alh6KgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIwNVOIP6fvaNT4XiYFu95zQ/9SxQ2l4yw+eKLL9zwcEk07mETK/WLpIGYFhvjvcQNKegQgWuVQFo8tDbGpiKg6X1r5a719MiuCY8mtyESJybGcxvZJEtVZqJ5WirXUp0tpaEskltyM425W3rYUpNKEwEREAEREAEREAEREAEREAEREAEREAEREIHvAYEoYzhUupL0HjZ8FoNh+IxLolHYcEuap556qhmdrAibKGzGjh3rwiauq9bUAsUORFETz81qV4IIiMA1QCBKh7YOhVP7jdP7bS10jea7HHaRWTxfDpLLaYf1XuQ3wqOWnExjr1p6eDldVV4REAEREAEREAEREAEREAEREAEREAEREAERuMYIpJ0JhU2UNpQ1UdgwjcJm9uzZ7l8uuSTahg0bPCRn9OjRXiBdESvjEdN4zU7wkLBxDPoSgWuUwDeRAUTROMV/jXK5+LBysFbZ5b8byexyuV3u78N+xzbiOXssjUuioWqMwh+GnrWcP7u07kRABERABERABERABERABERABERABERABETg+0mAzoRzgnV1dX7mfXQqdCzcw2bevHkeYfPjH/+4GaSsCJsobMaMGeMFohViKVaWvm9WkxJEQASuUQKXKwSaTuo3vb9GMaWGxXdliFO5lLRJs0lfpypr0+Xl/EYttZNKSy6Dpkkax00qR5t6pEwiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIi8H0lQGnDT9qpRGEzd+5c9y9PPvlkMzytCxvsYcMJulhh/H+Kh4nI7KiamNasdiWIgAhcAwQuRwY0HW6c5o/nps+vjvv0C7YtPQ7vxLZwi1ziuS21N83Tlnaalknfx7bjOaimdI54nckRU3QWAREQAREQAREQAREQAREQAREQAREQAREQAREggdbmEKM/YdQNI2zmzJnjwmbGjBnNwDUTNnv27DHuYcNNb9IGqCVhExtqVqsSREAEriEC30YIxCn+eP5uYYnvtYv1Kr7nYt54f7Ey4VmGWyzbvEzkkvOtlpjk8ms8Wm8ntsx24nU8x4QQrklTH2qLz8OZuVA6O1F3IiACIiACIiACIiACIiACIiACIiACIiACIiACzeblOIeYnquL97t37/Y9bLp3725tjrAZN26csQAPn5zkHF0ygxcnK9lYvPaM+hIBEbhGCaSn79PXrQwXWcK+J0EO4FWBI30dyv1nvz/SL8w4kpiW7lu8ju88H04YVLN3oI8bGUIZzxlYhMvYTJNzkCDpdppkuORt24VNrCpb3MTfKD5t3MMmJuAsYZOCoUsREAEREAEREAEREAEREAEREAEREAEREAERaEIgPbcYr2MWzv0x7bKEzfr16z0kh8KGETZxAjFWxsrTabExnUVABK5dAvHlEv63T/OQ2AecgphpeeyUADGaI143wASwvrjRFtN5xHzh7uLfsT8xV3wnxfum56b5mz6P5bmGZMzLtPhpmp95mDeOgc9juXgd6wwD43qVQXjweeMz3lzGkW6DxZKgmlBxY4QNnyTtuQdKiZnU7+X9SZ6zRKYMfh+/88QQUYN8njXchQf6FgEREAEREAEREAEREAEREAEREAEREAEREAERaJFAnMeL5zjPGPewmT17dtuWRJOwaZGvEkXge00g/WLhxL5/UpP/bYXDeigKOP0fZUe4R0owAlniw3PGB21tpA35ophJixNeM53rSLJvsX+xD6HvmajCyCTdXLo+T0/6HvKibKI9+Ixt8YjtUHw11GdvQuYZWvtyjvGhl07Y8UFG0gS+2Wn8CT0FzONvEv6jEepjTxt/ECYxPyriTxTyJT9WyK5vERABERABERABERABERABERABERABERABERCBFIEw58a50Mx8X5x/5B42EjYpWLoUARG4PAJ8sfAIQoLXfNkwJaTzqvnBSf3wUqJACEcsl45eSZ6F7F4mydx4Cu023n7ri0sJG7aXl5eXjDc0F1+ysS88xzTm4H18Fu95dlqRHxNwgIKLIQqcKGw8PeZrUpcXaukr+Q3Cr5CKDmoULoF/+P3S/WOJwL2hIZSL/fAu8BH70JiHfYZgQrHcnNyscbbULaWJgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIwPeZQJw35DnOG/LMeUkJm+/zvwyNXQSuAIEw4R+kRKwupHHiP0z+433TeMT8TKAAoBTgkdM42R9eTuGFFQUAhQJzsb7Wj7QUaT3XxZ+k+xdzsl6mp5/FlynzNJU8MV/sT8wbI2c8PanTh5UCFMuyXj7zEScvb09L5eV96wdLZniRNQ8vjmuKofSR6RuY44H3Axfsa+hT+A0Q6MOnnu7SBnlYN69D11hahwiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIQEsEONcWP3HekGcJm5ZoKU0EROCyCITJ/DCxHwuGNM7s8xMn8sNTvnjqfdafk/4UMrEUz7yJgoB15iZpvOZztwW8aPFwEdLik7Ylxhclz+m64ouTtcQ86TSOKZZp6XnMy2fxwHY9XiYOKc0htM0B4+UNVrHuWE+so/VzZB9ysNm0GPN6vS+Z3yb0m5yTdhMGFDH1njcwiSMI+UL9zJO59x+q9a7piQiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAh8jwnE+cOmc34SNt/jfxQaughcKQJ8sfDITNgnIsJNBJ9lpADzcR8YfliOS23F5bb4QuKRk5PXmMZ7Vs8PPUJaasR2mSd9pPuRTr/YdayL59i3WE+6n+wrj5g/1sl7flgmXvMZ79OfxvwUMchPcVVXX+fJcTmxdH7mQUbPC4wk2VhfrKvlM7nzEw7WU1tby6osPz8fdWSPgW2y9rgEmrfLFE8PeVk2ZMv0gfn4Cb9hiMxhPTpEQAREQAREQAREQAREQAREQAREQAREQAREQARaJhDn1HhOzwVK2LTMS6kiIAKXQYAvFh5xcp/XIY3p/HCC31P9ntEadbV1VllZZRcuVLq0iC+p4uIiKy0twx4x+Y3ShtXzeZA7GRkQ22XNLR3p/sTnFysT+8BzdXU1+ldpNTU13jZflkVFRVZeXm6FhYWxOn+WLsd8/LBt7nMTZVTsC8/MzzxVVdVWUXXB26qvq3eRUoI2ioqL/ZploUO8rRxCQFkffYCZaJHwPHSI8qSxa7iIN4E/22TbrJcHr+MRrzneiooK9K3KHzFLQUG+/yZFRYXhN+YYGuv2X9fT2bswzsxvFOvXWQREQAREQAREQAREQAREQAREQAREQAREQAREIBDgXFz8cD4tfjh/pz1s9K9EBETgWxGIk/1hsj5UFdIoBPgJwgA7vfhEP5f4qqqqsYMHD9rOnTvtzJkzLkYoOHr27GUDBw6yTp06uRjhkmjBK2SETWzvUp1O9yfmba0s0/0PI14Q/XPs2DHbtm2bHT9+vFHAdO/e3W688Ubr3Lmzv0RZJkoQL497Co8LFy54HRQ7JSUlEB4FLknquVdP4kgohA4dOWJ79u6zY2ijBvKmvF0769evn/Xp08fKSktd2riVgf+gAsG7m99xKMlVZBySWx5f5E9JE+phZE/IG34fRtYw7fTpU7Zzx047dPgQRBJlVb2Pd8iQoda9e7fGsXA5N47FGSddyjFES3knM31s7KwuREAEREAEREAEREAEREAEREAEREAEREAEREAEnECcS+SZ82vxI2GjfyAiIALfmkCUBFGQxPtgJ4IQyMkJQoTGoBZC5Mzps7Zy5Up77733bMuWLXb+/HkrK2tnEydOtGnT7rebb77ZRUEDzECoj/VEaZGIglTPY9uppBYvM33LPI5pPFPWMLrkiy++sDfeeMPWr1/vS4lRvEyaNMmefvppGz58eGPkDCUMBRRfplxyjFE5RyBiTp48aR06dLAePXo0RuUwT1xu7RzG++maz2zBwoW2bt068DhjgwYOtKlT77E77rgT5bq77MnNhfygBEHfeOCqEYJfuwEKz/gojoXXmSMjbJjGqihiYl/CPkI56H+Nbd26zWbPnoXfZhXGcMLzUlI98sh0Gz16dBgLooAo10Jd4Xclf1/SzXsYeuYZ9CUCIiACIiACIiACIiACIiACIiACIiACIiACIpBFgHN48RNlDc8SNlmYdCMCInAlCGSkAUUCPxQG2cLm9OnTtmjhIvvDH/4AYbEeUTZnrX379nbXXXfZzJlP2bhx46wbIlooQ+qwXBgFQ9Ml0aKkiedL9T3Tr+Y54wsyCpsVK1bYb3/7W+OZ0TAUNlOmTLH/+l//q40cObJR2LAco2ooeRiVQ/m0adMmjxqidOI4GJnD8szLly4PCp2PFn9i77zzH7Z69WqXI0OHDrMnnnjCHnroQevVq5dH2eTmheXLyJEaJIw1CJFLC5socjICJcoaRtN4nYnBZ784zi+//NJeeeUV+/DDD+3o0aMubMaNG2s//cuf2q233erCpgjL1uVwWTVUgV+28f8BQLGU7hnHqUMEREAEREAEREAEREAEREAEREAEREAEREAERCCbQJyL5JnzffEjYZPNSXciIAJXiABfNkHW8MyXDq1x2EOlDnLgzJnT9sEHC+zf//3fbe1nn2MprjOQAe0RYTLVfvKTn9iECROsa9duLjlqamohOup8L5UY2cFuxhcZry91hP5k5wryA71EX+OHwobiYvny5fYv//IvtnTpUo+aKca+MuzbL3/5SxszZkyjsOFLlEugnTp1ykXNggULbNmyZR5tM336dESmPOLLnHHvG7YX2zl69JgtWPShvfX227YK0SwnTpywIUOH2lNPPWWPPvqIC5vSMiyLRjECdvibjJdjSGuRyDkztjBWFuAzHL5+WSjD/obIGsMYwj47zELGVdhPZ+PGjfbiiy8ax0Fhw7YmTpxgf/d3f2eTJ0+2dhgHl3jDUDAW1pFrebmItvHqJWzIUocIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIXIxAnCPkOc5x8ixhczFqeiYCIvCNCTRKA4+widVEYVNnp09B2Cz4AMLmd/b52i8SYVNu99xzjz377HM2fvx4j0zhcl0xwiYP0SYUBDziiyzWnD6HtjMpUS5kUsIV6+CRfkFyWTNGy6xatcp+85vfuLCpqKiw0rIym4q+/V//8A82ChE2+fn53ge+RBkt9PXXX9vixYtt/vz5Lm66du3q8oURM3379sVyb2VOogH5WebEiZP24Ucf21vvvIMIm0/tJIXNsKE2c8ZMm47lx3r17GGlpSU+3ly3IRQwiRcJdoQ9T64SMeOjiV9Mi+kcZ2asbD9Km/z8PG+D9xRPXAIuCpsjR47iGYXNRPsv/+VvIWxuy0TYcG8hb5+SJnTMfxNvJ7QVe6KzCIiACIiACIiACIiACIiACIiACIiACIiACIhAhkB6PjLOc/LMebtdu3Zhy4LZvtXCjBkzMoWSqxwUjrN+PpnHAlzmh3szxAlPnmO2dFqz2pQgAiJwjRJofE34+MJbIy0Nkms84B42jEj5AFEcv/vd8xA2IcKGS6JNmUJh82yWsGFdlAO5iBjxTe1dtDBqB+8dtMZl0/wI3gDvIoghJFAbxPcRI3RYT0iDAmEd+HgdobP+DqtDvmoIm82bN/teLuvXb0CEzQUrKS21sWPHecTM0KFDrACig7XxvXf8+HEInpU2Z84cW4SomV27dtvAQQMRKfQX9sTjj1vffv2srF0ZO+ovXXbkzNmztvbzL2zxkqX21Vdf2zncU+z86Ec/stuw9Fjnzp2tuAhLj6GPHDP++kEO/pdjTPrNZ8zHPzxCesKkMQ/rYLSOGXFRTPE/ABRP+RBhIbKoyjZC2LzwwgsQTx/Y4SPHLAfPGoXN7ZN92bqipF/eEW+R1bJ+/05SdBIBERABERABERABERABERABERABERABERABEWiJAOfv4sfn1Divho+ETUu0lCYCInCZBCgH+JJhsUQUJDWEvWuYjPREHjBi5tTpKGx+hyXRQoRNhw7tEWEz1Z57DhE2E8ZbNyyJhjcVXlShbG4uhQ0rZnQHjyAo6uqwzw3kQ9jjBkIC0oXtMjoExa0e7dVAUDC6JRfSIjcv37i0Wk4OP6EOFyHBhHj5M+jfvv377djxE1guDMuxFRZBonSxXr17WQeIpQKUD0u81dvxY8dtyZIl9t5779kSLKG2f/9BGzhwkD2DcTz6+GPWv38/K2/XLultPVYqa7Dqmmo7iroPHj7i0UXsI/N069YNS8F1tdLiEvQxRhMRQ+gnu1jXUOfChZKFqdznJh/9ycV4yJ8v9rAMGvf/CXny8zlucoOcQR21+CIPRtjkE2oDBE5djW3CkmjP//55mztvvh04dNTQW5swcRKWRPuZ3XHHndYev1FhQYguYrGkV/wxsq49QV8iIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIALNCERZwzPn/eJHwqYZKiWIgAhcPgHqk2CFm5alOOHBl09bhA33iXFhM36Cywsu+hWEDWUL9IEbAqR5dYg+SZZIq09ebuxHfX0tW7QcChv2C2KC4oJKgVIjF3uuMNrEJUjyQkwLG0bC1NRWW0VFpVVeqILcqLe8/AIrKiq2IoibwsJC7NtCyQExgvq5/wz3uqGwWbz4E9u374ANGDTInn762SBsBkDYlLeDKqFq4hEibS5UoY0L1RBCNc4nHyKJ0StFqL8AkS9xbF6k8QvjSyJ1+AInV5dQ4JCWJxneHDP3mkFPHR6I4FyHfW3IhXz8OX6nhloKmw0ZYXP4GNrKsfEUNj//ud1JYdO+HMIm7GHDcuk209eN3dWFCIiACIiACIiACIiACIiACIiACIiACIiACIhAFgHO3cVPlDU8S9hkYdKNCIjANyPw5xE2iC1xSQJ/AgFDWWJWUFDoUSLeA7zoaiAdaqopWShBGGmSiCK88AryCyEbipL9ZxCNQuGAdH7SwqYBESyUGZRAjK6phR3KY1QOJEqQPmGZspCvzoXNiuXLsYTaHPvo48W2d+8+6z9woM2c+bRNf/QR6z+gP0RHO0Sy5LroyWN7qLuOH47FxUuQHyFKCPmQJyz/lhkCrnzMFEXsW11dbYi0YXn/hBc9x+PjxgAZQVOAiJiw/w8HjHFDWEUR1gC5FZaaQxGw+3LDegib39u89+fbwSPHJWwCcX2LgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIwBUjEGUNz3F+kmcJmyuGWBWJwPeZwJ9e2DRgmbP6+hq7UFlhZ8+et6rqGpc17dt3sJKSUhcYVVUX7MyZU3b02BE/874eUoMqhJExHTt2QtROd+uEc0lxqe/P4lEs8BjhSKJXUKbqQqWdO19hFdi/phbSJg+yh/vYlJWV+d4yjJSpqqq08+fP2YEDB7CHzSpbtHChrVy52vYfOIj9aPrZI489btPun2a9+/a2jh06WFlJibUrK0UUTSHGUofonWo7izbO40O5VIxl0MrQRnFxsUfYUPBQKoU9esDYRYy5lKqsRP/OnfW9cM6cPu11+L40iCSqx4eihpE6jNjhEm5dunS2ci5nVlSSCBuOgNqGi6TxqLM6iK6NFDbPPw9hgz1sjkLY4KkibByQvkRABERABERABERABERABERABERABERABETgihCQsLkiGFWJCIhAywT+9MKmBnu+nD972nbv2WVff7XFjp84CfHSxW686WbfW+bU6ZMQJ/sR3bLHDh8+iD1yTkDuVAZhA9FBcdGxQyfr1au39evb3wYMGGi9e/e2Muwb49EneSHShiKFbR06dMA2bvzS9uzZb1y6rBDLoQ0YOMhG3nKL9eje3aohgw4dPGg7d2yzTZs223qIjo0bNtq2bdvt5KnT1rlLV+zDM9FGjh6FPWm64NPZ+vfra9dfd5317tkTdV6wPXv321dbttrefftQX7V1x/411w0fboMHDfb9bBgZE44QOcNl3c6cOWMHDx203bv3oI8c52k7dfIkJNY5RBZV+9JvlD95iKIpKirwvXA6depg3dDnbt17YA+evhBI/axdeXtE1iCSB+POw95A8PdWWxWEzQsvvGDvf7AA++toSbSW/70rVQREQAREQAREQAREQAREQAREQAREQAREQAS+OQEJm2/OTiVFQAQuSeBPL2wqKhDJsm+PLVnyic2eM9e2b98J6TLIHn30cevXr599veVrW40oly83bXRZU4HIl2pEjFDAMDCFS5pxD5puXbrZ8OHX2+Tb77C77rrLyxZC5hQU5kFc5GJ5MgqbKvvi88/tpRdfsuUrVyGK5Twia9rZD3/4I/vpT39qN95wg0uSLz7/zD766EP75JNPbPeuXXb23Dm7wD1vautRX6GVI/qHQqiktARyqJdNnDjBHnvkERs7ZrQxKmbJsuX23pw5tubTz+wsRMywYcPskenT7V7s49OzRw+PtAnouS9PPSJqztm27dtsxcqVthxLsG2F7GHa+YrzLqdqEHXEpdAYNcQoG/gYK0X0EaN6OnXqaD179bHRYyfY3ah/yJChVlSM/XjycxHNwyXYDJFEVVgSbYO99NJLNh/CZt/Bw+DR2h42Ye+bEJyU/r7kPxZlEAEREAEREAEREAEREAEREAEREAEREAEREIHvNQEJm+/1z6/Bi8CfmsA3FzbP/+55W7v2czt16ox1wJJdUyETnnvuJzZ+/HgsX9atca8VLj22d/dOW7Bgvv3xj2/ajp27IEH62qRJt7pM2bptq+3cud1OHEdUCJb38l1puNcLBQwiT7iEGteBLCkqxdJoHe2mm0bYlCn3QNzcbkOGDnFhQ7HDl2Ut9nJZ8+lq++1vfmufLFkGEVNhxVjObMo9U+2//cM/2KiRI+3kyRO2ds2nthDLoC1e/LHt2b0b4qQCkTI1kBwNlo+9ddqVl1tpKZZQKylyYTNp0kR7/PHHbPyYMRjvKft48Sf21jvvuoA5fuyYDUU/Zjw5w6VN3z69XdhQ1HCpsyNHjtiGjRtt9erVtuazNYjs2WEnEFnD8RVgbx3KKJ657w3LVCFaphoflqXEYbROEZaBGzJsuP3w7il2222TIa6GI+qo3IoKsK8NI2wguLiHzYt/+INH2Bw4dBT77DQRNhhTYWEBWKaFTfj3JW3zp/7fmeoXAREQAREQAREQAREQAREQAREQAREQARG4FghI2FwLv6LGIALfWQLfTNgsWLDAfof9UtZ+1gZhg/1adu3abnPnzLaXXn4V17sRNdIFETL96SPsyNEjvhxYeXkZ0hHZUlaCSBNEjUBYnDt71o4dOw5JctoqsF9MXV2D9cCyZJMn326PPfaYS5t27binDTQPJE8dhM3qVSvtX/7l/9iST5ZBxFRC6BTb1HvvtX/+53/2CJmziJDZhGieFStW2CpEvGzbtg1S5agvh1YNQVRS2g5Lj/X1dsrbleHcw0aOuNmm3H03InSuR19O2ocfL7Y333zHliFa5siRwzZo0EB7+qmn7InHn/DIH+5nU1NT41E0mzZtsnnz5kEgfWJbGFkDgVWMCJlu3br6Em1dsARbGSJpChBJVFdXj0ifSkQBYT+fo0ftOD6nEcFTieifzogwGj/pB3b/Aw/aXT/6kfXt0wt78kDAQNjUYE8e7mHzhxf/YPPnL7ADWUui/czuuONOa58Im1zYGX6CpOHvz+twF1M9UV8iIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAJZBCRssnDoRgRE4MoS+NMLm7Nnz9j2bV/bnNmz7JVXXrMdO3Z5FEsxokYKCgqsvLydXQ8RMgbLjd1ww3Dr0qWzyxeKiyOHD9vmr77ypce+2vy1VVZe8IiUwYOH2HQsUfbQww9Z/wH9EQlTbNU1F7A0WLWthoT51a9+ZcuWrvA9bIpLyuweRP/84z/+o40eNcqXHDt96oSLFka7fPbZZ7YSy6dtwD42x46fwPJjvexuRORQCvXo0Q377HSyLvj07tXTOrRvj+iY47bow4/tjTffhvRZaUdd2AyyGTOetMchkQb0H2DtsJxaJfbh2Yc9brjs2ttvv21frPvCl16jGuESanfdfZeNgAhiNFJxYRHGheXd8gsQVVSFPW4O2ZcelfOprV+/Af06bnn5hTZg0BB74MGHsJzcIzZk8CBImDLLR5BNTfUFj7AJS6IttP2HIMHw006YOMn+7ueJsGmPCBtE8jBaKUgb9kTC5sr+70m1iYAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIXMsEJGyu5V9XYxOB/3QCfw5hc9q2bf3aI2xeffV138MGK49hb5oC694d0SujbsEeM3fahAnjrE/f3saoFi4XVl11AXvanEb+HbZ0yVJEzCyx3bv3IkKlAiKlpz3wwAP2KATJTTffaB06toewqbb6WggbRNj8v4iwWbp0GaJVahDNUmr33DvV/vsvf4kl0Ua7rKhDPgqhgwcPepTNB4gYWoZ9afYfOGQDBg60H2N5s4cgRvr26wNJUx6WJcNyYvl5uXbixAlb9NHH9uZb79iK5Svs8OFDHmEzc8ZMj/rph+gcChvubfPll19iT5kPbNasWbZ161bfE2fIkCFY1uw2FzbDhg31vIUQNYWQVxQ2dXW1dhoRRZRJiyF73p//gX319RYfd9fuPW0axv3UzBke7dO5YwcXNlwSbeOGdfbiiy/6HjYHDnNJNAibSRA2P/s5ImzusPaQTUHYMMImSJv4z08RNpGEziIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiLQOgEJm9bZ6IkIiMC3JvDnEDaMsNli8+bOtdcgbLZBwNRhr5ZCLFU24pYRNn36IzZt2jSPrrEc7l1T63u51ENccE+X05A2K1euRoTOHJcwe/bus65du9s990yxRxBlM3bcOOuGSJj6Oux/01Bnq1atsP/1v/4fWwLJU3mh2qN57rvvPvuf//N/2phRoz2ohPu4cPk0ypZVq1b7kmUfQsLsRd2DBg+2p599zh57FNEyA/pBqJSBMjhhyTWu4cYl0T76+BN7E1Ezy7Ek2iFInwEDBtjTTz+NfW4eRyROLysrKcUSa6dcBs3FcmiLFy9G3XutL4QURdODDz5oNyG6hhE7HGMelkPLxzpwjH5hUzlopwIROsshhF5+9TUXN3v3HbCOnbvYfdPu87bGjBpp3RCNlANmdYiw4ZJoL7zwAgTPAjt4FPsBtSJsfDk0NENpwxibcMSreI7pOouACIiACIiACIiACIiACIiACIiACIiACIiACEQCEjaRhM4iIAJ/AgKXKWwgF05BRFzOHjZhSbStibB5zbZu2w4jkWMdO3SyH911lz3zzDOQLmOxFFonjK8OAqPOo0xqEDHDF2AV9m9hhMnCBYtsLqTPpk2brQPKTp482R6ePt1uv3Oy9enTx8vUQ8KsXLnc/vf//n8RlbMsCJvCQrv3Xgqb/9vGjB4d9mvhfjd1NVhy7QiWNVuO6J+5kCpLbO/+/S5snoGwYfROfwib8rIylAmcKGxOnjxhCxd9aH988y1b6UuiHbGBgwYaI2y4h03/fv2stKTE957ZsmWLrf18LZZb24DooN2IcmlnU6bcg4gXLLeGvXG4DFo9zEoeZE1uDjbu8QNt4e8FCBsuh/berNk2b/58ZxCEzTTsl/M0ooVGuLDJzW1I9rBZZy+/9BIibMKSaBQ247kk2s+wJNqdd1qH8vYeKdS4h42ETcJbJxEQAREQAREQAREQAREQAREQAREQAREQARFoGwEJm7ZxUi4REIFvRCCICHiI5Gi8oFPB0UBH4WdeMDLmFJbryhY2pyFQ2ttU7BPz3HM/sfHjx/u+LA3QHJQRFDY7tgdh8yr2sKGwycfyX3379reHHnrInvvJczZ06FArKMRmLFaPQJY6q0V0Dfdy4QuwDpEz+/btt8UfL7E333wTe858bmVl7WwcImseQKTKlKlTbODAgS5gamuqIFGW27/8Hy6JthzSo9oKi4pt6n332v/45/8BYTPGl0QzROJw6bFjR496lMxsRO98svgTCJsDkC+D7VmM47HHH7N+/fr6Em1EEaXNiRPHIWwW2R/fgLDBfjmsYwj21JkxY4YvicY9bEpLS7GHTYUdwbMDBw7459Chg5YLMTMKkTHXXX+9lUDq5OVxzCBF2ODMsTLipq62zirOnbOvvvraFiyEqHr/fdu0+Svr1KWr3Xvf/RA2MxuFDSNsaqoqfQ+bl19+ORE2h/FbNYQ9bCBsbk+ETWFBvo+fzfHDI5z4HT+erC8REAEREAEREAEREAEREAEREAEREAEREAEREIEmBCRsmgDRrQiIwJUkEIRMnLx3a9CsepgEiBP+oVAIwmah/e53v7O1az/3+7SwmTBhggsbRnikhQ2jY6Kwadeu3G666WZ76OGHITke9aiUvHxGmdAOQVhAqNTW1EDeQEZAXhw8eMiWYU+a119/w1avXu3Lqd08YoRxqbN7p91r3BfGI3MgbLgk2q9+/esgbLAkWklpmU1FhM0//eIXNnrUKMtzW0ExVG/Hjx3zfWhmz55tiz78yHbv2Yv9aAbbT/7iL+3xJx6HVOrjS6KF5cNC3yhsuHwa97BZvmyZHTt+3IYPG25PPvlje+ThR1zylCEqhy/vmtpaq66uthqM5cKFCy6JiooKsa9OMQRVoeViTxyKGvLleJmnqqoKsqfSTp44aZs3b/bIn0+WLrWdu/ZYl249IGym2VNPzbCxo0da1y4d4Z7ACkuibdq43l5EhM37H3xg+w8eAUMImwmIsPn5z+127GHDCJsobBhl4wd5Ix8WSIPAYYRPfBAe61sEREAEREAEREAEREAEREAEREAEREAEREAERCBDQMImw0JXIiACV5gAXzA8PMIjmIMWWgiT+i5sIE9OnT4TImx+9zyEzdo2CZv0HjZbt+2wDh07IdJkNPavedgefvhB692nN2QNZEEibChfarFkWQOsTy2kx6FDhyFslttrr/3RVq3+FEt7FUL43AR5cZ/v6TJ0KIVNLaRHla1etdJ+/et/dWFTieXUShCNc++997qwGTUyCBvuYdOAZdFc2CxbbrMSYbOHwgZ72Dz3k2xhE2JPGAnTYCdOnAjCBkuicQ+b48cgbK4b7kuicU+dvliejRE2UX646gJnLvFWUVGBz3k7f/68nUUEzQksr3YaS8xx2bcLlRfsPJ55Hjw/c+aMi6odO3bZrl277dSZs9irp5ePeeZMCptbIGw6hYgkjHvzxg0QNi/avPcpbA41Rtj87c9+jiXY7rD2EjYt/NtWkgiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAi0nYCETdtZKacIiMBlEvhzCpu5c+bY6xAu27bvtE6dOtvYsWNt+iPT7cGHHrBevXqi50GiUKTUc48Z7EfD6A9G2GQLG0bYUNiMcBFz37T7bIgLm5pWhE0qwqZR2BBUEDaMkpn13ixbhKiZIGyGQNj8hT3++OOIlumD5dfKvG+MSqFTYoTNokUf2RtYnm3FihV2PImwefrpp31JtD69eyOCpsTLsAAZM8qGS8MdQ0TP0aNH7OChQ7Z3314sFbfDz+cgYyhqzp+DzIG0YYRNNSJtqmtqfXk0MrCcvGxhg6XVunbFvj9c3o3C5suN9tKLL2L5tPm298DBZsKm6R42JEABFQ4qKUXYJDB0EgEREAEREAEREAEREAEREAEREAEREAEREIEWCUjYtIhFiSIgAleCwJ9P2GAPmzlzESHzum1DhE2nzl1s/ITxLmweeOA+69mzp4uNBsgHiht+aiFs2L90hM2rr76OJc8+xb40RXbzzdlLotUhIidE2KxoEmEDYTM1LIk2auRIy8M+MmFJsHo7gegYFzazZnnUDJdEGzBwIPawCcKmf/9+viRa3L+GwuYkImwWLvoQwuZt38PmBITNsGFDbebMmfboo49av759rbSk1EdB8XQOkTR79+61jRs32udffI49fLbakcNHEGVzzs4hkuYC5AyXTOO+NVxyjnvrUJ9wf5u8vHzIK4PAueDyplOXbr68G/ewGTtmlHXv2hmMsO9NbbVt+hIRNn8IwmbfgUO+HN2ESZMsRNjcaR3al2NJtIKwh433TsLmSvxvSHWIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAh8fwhI2Hx/fmuNVAT+7AT+XMJmx7Ztxj1sKGy2bt1unbp0sYkTJkLYPGzTECHTq2cPKARG2FDY8OB+OTWQDvUQNjHCZpk1FTbTpk3DEmH32uAhg110cC+X1aubCpuwJNov/vEXlhY2OYiwcWGDZc3eQ4TNRx9/bLsobAZA2Dz7nO9hQ2FT3q6dR9ZEkeTCBvvdvPHGWxA2qxBhc8y4JNtTTz1lj0PY9KWwwZJolDBnz57D3jM7bc2aNVjKbZWtX78e++TswXJnpxulDPeyKYaAKi0pgxxqh08prkusCGl1dYgCOn7S9u3bb8ewp01HiC4uAxeEzWjrBmEDW4MIm2rbvGmj/eEPf/AIm/0HD19C2ATGirDxf2z6EgEREAEREAEREAEREAEREAEREAEREAEREIE2EZCwaRMmZRIBEfgmBK6csOmAKJap9txzz2Gj+wnWrVs3jwypR3gIlwLbuX27CxsKly1bt1nnzp1tIqI/pk+fbvdBuFDYcP8aFzYML6Gw8QibRNhAQCxZEoUNlkSDzBgx4hbIniBsBlHYIH9dDYVNy3vYNBc22I8G0TGMsHkPETYfYUm03Xv22cBBg+yZZyBsHn/M+g/oh71f2iVoGfkDyXM87GHDCJvly1dgibPDNmTIEEieZ+zHj//Y+vTpgyXRiu3c2bO2fcd2W7J0qc2bN88jbE5jX5o67M+Tl5eLZd0KfOm0Du3bW/ce3a1/v/42aOAgCJ8+WOqsK/bpKbBD2Ivm09VrbPGSpVhKjqKrayJsnsIeNqOsS5eO8DW1HmHz1eYv2yxsQoRRiGQKg4vLoTn8ZLw6iYAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIpAlI2KRp6FoEROCKEvhzCZvtWAZs3tx59trrf0SEzTZfEm0Shc3D0yEgpmJJtCBsKETi8mNhSbTmwmblqtUefdKSsKmlsFm1wv71X39jS5cut8oLVVZSxgib++wX//iPiLAZhSXRchqXBXNhgwgb38Pmww9tz14Im8GD7emnn/X9aAYO7G/lFDZYmo2HR+VA2HC/mzfffgfCZrkdO3oUETZDEfXyVNjDBsKmBBEyRw4fto8RtTPv/Xn22Wef2YGDByw/v8DHOghtDManR48e1rFjByxX1gH7+nSEyOqC6/Zennv57NyxE/vlfGhz5kD4bNpsnSFypmIsT2H5tbGjuYdNiLCpRzQS97BhhM087GGzD4KrDrKMUqylJdGCsOGIwrg4ssyH6TpEQAREQAREQAREQAREQAREQAREQAREQAREQASaEpCwaUpE9yIgAleMwJUVNvcmETbjm0XYbHNhM9def/0NFzYUE5N+8AN7ePrDkClB2ITluRpc2FBWtCxsXsMyZBQ2xTbilqYRNtXYw4bCJomwWbbcLlyotlIsMxaETVgSLRcb0VBYRPmywpdEe88WYpkzCpvBg7G82dPP2GNY3mzgoAG+JBo5sX+USZQ8iz5a7MKGZY8dO4YIm6EuUbiHTYyw2bFjh73yystYbu1dO3TooFVVV1sP7NUzbuw4u23ybTZq9Ggb0J9CqNwKIHJ8zxrsr8N9crg8XHVVFfal2WSzZs2xd999zzZ8+aUviTZ16r02Y+YMGzdmtHXv1tVyLe5hE4XNB8Y9bOrgYiZMnGh/9/Of2x13ZO9hkxE28Z9SjKyJ55iuswiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIQCQgYRNJ6CwCInDFCVx5YfMTLIk2rm3CBtEfD2FJNBc2vXo0ShTXFU2EzUEsDeZLor3yuq1ChE3jkmj3T8OSavdZXBKNwuZTRNj86tf/aksbhU052rgXETYUNqO8nVzKF0gYyhcXNlgSbSEiWaKweRp72Dz22GPYz4Z72JQF7ugT+3by5En7cPEn9uZbWBJt6TJE0hzBHjpDEJXztD2OMr169fL8X0Kw/Pu//3+IjpltlZWV1h6RM6MgWVjv5MmTEWXU2crKyiBr8tGnHMgg/KGtcWfSYJUVFb4c2ptvvmULFi607dt3WvsOHe2ue+6BHJphE8aPx1Jy3ZAdcgt792zauMFeeulFRNgsCMIG3Z0wCcLmZxA2d2YLm8ZmGv9FRVETz40PdCECIiACIiACIiACIiACIiACIiACIiACIiACIpAQkLDRPwUREIE/GYHLFzb1dur0aVuwYIH97nfP29q1a+3UqdPWoQP3sGGEDYUNI2y6Zu1hs33bNuzjMtdeey21JNrESR5hMxUypWcibChSuCxaOsKmprbODiFipLU9bO6blhE2dbUhwuZXv/61L4nWeoRNRtgsSyJsPsQyZxQ2gwYlS6JhD5uBA5Il0ZI+cWm0U6dO2UeJsFkGYXP4yBEbgqicuCRaT0TR1NXV2RdffG6//e1vsXfPHBcxPSFyJt8+2aOQbr3tNo+oyc3Lcz/TUE8ZZFiuLdflTU1NDfbGOYol1T6yPyIq6dM1a9DOUStv39HunjLFZmL5tUkTJ1hv7H1jiLDJCJuXbN58RNjsD0uiMcLmbxFhc+edd1iH8va+b06ILoIXynIz8SaevTv6EgEREAEREAEREAEREAEREAEREAEREAEREAERSBGQsEnB0KUIiMCVJdA2YYM2ISqoUuqwwf03ETY7tm93YfPqq6+HPWw6dfb9VR5O9rDhUmGUNS4TcjLCph4io5bCBhE2SyFHWH7Vqk89wubmm0cYZQ0/jHDhPi51tdW+h82vfvVrFzxR2HDfl1/8IiyJlgdTEaRFiLBZtnQpli2bZR99vNiFzYCBA12IMBJm8OBBiIwpD+OvryMIFzYfIu+bb4U9bChWKGxmzHjSHnnkEUS99IKwqcW+NWvtt//2W3t/3vvYuybf+vbra7ffcbs98+wzNhEihdE0ebkQNjjXYLk0mpvCwkLLy8m1s2fP2pYtW2z27Nn2zttv2XYsr8axlCPC5u4pU7FkG4UNI2yw9099rQubzV8ywuZlRNh8YHv3Y0k07GEThM3PkiXRIGwKGM1j/mGDjdKGP67fSNiAhA4REAEREAEREAEREAEREAEREAEREAEREAERaJGAhE2LWJQoAiJwJQhcvrCpg7A5dckIm65du7rgqYc0OHv2jDUVNr6HDZZEe5hLot031br36AF9QGET9oqphxyh9GD/ampqE2GzPBE2YUk0FzZYDs2FzdDBEDYoU1uFZcRW2P+BsFm6ZLlVXqiykjLuYYMl0VzYjIIQyRY2y5ctwz4xs2zRhx/b7t17rHefPliq7RGXLzfffJN16dLJo2CCTDJfEu2jxUt8DxvKniOHDwdhM3OmPfroI9a7V2+XMOvXr7d/+7d/c+nCiJny9u2wb80oe3LGDPvhD39oHTt2stKSEstFVA3HmwtRw6Pi/HnbhoikxYs/to8+/BCROl/Y8RMnrBab0rTv0Ml+dNfdEEozPcKmDyKTuCRaQ20N9rvZYC+88HubM+8D23/wMOoMe9j87c+jsCmHsClwWUM3E5ZgS/4VSdgkIHQSAREQAREQAREQAREQAREQAREQAREQAREQgdYJSNi0zkZPREAErgABvmR87xRXLJy5b+FAMmNs6uqwJBqWBFu4MCyJ9tlnn6WWRJvqS6KNG8c9bCBsGqBgUDeFzfYYYfPa65AR243CZiKWRJv+yMO+h00PCBselCINEBD1iBqhgOFddXWNHcSSaJkIm9VWVFxkIcJmmk1DhM0QRNjUUfLUVNkqCJtfU9gspbC5YCWlZRYjbEZjD5s8NOLCAn07efKErVq50mYhkmXBgoW2Y+cu79ukST+wOyFVRoy42ffjKS0t9nMH7ENz6tRJW/zJEnvr7XfQxlI7fAjCZshgRNjMtMcefdT6QPgUQIwwQuZ5LBv37nvvejt1MCj9Bw6we7Ck2e133GGDBw2CDOriUTUcex3E1Nlz51DfQUiadfYhZM1G7EvDPXMuQDyxPJdEu+OOO+3JJ5+02ybfav36QA7xl0F00ZcQNs8//7zNnTffDh46giXpcjzC5mcubO6w9o1LonH8bmicA9s25A3/BvxOXyIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAi0QkLBpAYqSREAE/twEgsjh3iwnT1LYLLTf//55W7PmMxcK3MNmCkTEs88+a2PHjrWu3bpBACBiBMXOMMJmx3Z7//152MPmdZc3nTp3snHjxtvDDz8M4TLNemF/l1xfkgsrfFG84NOA6By+AC9Auhw4eAARM0vtlVdetdWrV1sJIlNuuWUEyt5v96P8kKFDfd+Y6uoqWwkB8+t//bUtx940LFsMuXPPPVPtn/7pn2zM6NG+dww7xn1yKJ8YwTIPy5Zxr5mtW7bheT7ESHvr3r279e7dx6UKhcx9iOYZhfLnzp31vrz9ztu2fNlyO4q9ZYKwmeFRORQ2paWltn/fPnvjjTfs3Xff8/Gfxt4/RcXFRjl1ww3X2+hRo20Qllwrh0jhcez4Mdu2dZtt2brFdu3caYcPH0V0UY33t7KywioqKq2srAx7BE1EJM+jWBrtbizZNhic6jzfxi832gu//73Nnz/f97shOy6J9vOf/z0kz+3Wrl25FRUVIaKHiicclDRcBC18+K1DBERABERABERABERABERABERABERABERABESgNQISNq2RUboIiMCfnQCFDSXHggULPJqDETaMAKGwueeee+yZZ56xMWPGWNeuEDZY6gtWxPdj2bVrp73/wXwIm9cgRbZap04dIR4m2EMUNhAhFDZB8CC+BmWCtMFSX5A2VVUX7NAhRNhg6bKXsUfL6tWrXIjcfPPNiM65zx64/34bCmHD/W6qsRfMylWr7F8hbJYtX2YXKiux302hTYWw+eV//6VLEi5BRjdUD2Hjy7Xt2Gkff/SRvfXWW7Z+/QaP6OGzwoJCCJJ2aKvMbrzxBvurv/5rl1JcsoyRNe/8xzsuh45jDxuKkyefpLCZ7hE2JSWldvLEcfT1U1u0aKHLox3Yh6YS/cnLy7MuWDJuQP/+LoUonzjm06fP2MGDByFqDnkfunTpan379oFwKnHJtRMSh3vhcKxc4u2JJ56wUaNG+bgpdjZs3OgSjcKG++rwoLD52c9+ZrfffruVQ9gUgAXHz8NlTSLJqG1Cqj/SlwiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIQAsEJGxagKIkERCB/xwCFDaMFKGweeGFF2zt2rUucKKweeqpp2w0olC41BcFDPewOYdlvvbu22MLsIza6xA2XCqsY0cIG8iEhx56yEVPz549fUB84TH8w5dfq8WyaJAwlBFHICBWrlhhL79MYYMIG0Sw3HD99Xb33VM8wmbosKEud2pQ5tNPV9tvfvsbRL8sswoIkqLCQrQxBRE2v7SRI0e6qODSa5QyFYhcOXnipH265lOXSatWrXZxUgPxg4XiPC+l0fVo6x/+23+z6Q9Ph3SpgDxaau9hqbM1a9bYiePHbeCAQRAoj3vEUO/evT2ShXLmFGQWGb377rvo16eIfDkCiXTBo3t8CbJEmPiw8UVm+fl5iOzpjSXjJiBaaRyWTCuC9FlkS5Ysgbyqcjn2gx/8wP7mb/7G7rzzTu9jLcbtwgZ72Cz44AM7duyY1zVhwnj7a+S77bbbfEm0/IJ8mJpcj7LhnjmhD2E/Gwmb/5z/TalVERABERABERABERABERABERABERABERCBq4eAhM3V81uppyJwzROgsDl79qxLkzlz5thXX33l9+Xl5RAMEz3yg3KjU6dOzoJ73pyvOO+RI4yMoXjYs3cv5EG5jbjlFps8ebKNHz/eunTqzG1UfAk0bq/CiBOKGt+XhlE9EB8UEh/M/8A2bdpkxVjaa+DAgV520qSJHonCPV4oLjZt2mzvYLmy9evX2wVE5xRjGTL27amZT9mw4cOw9BqFRa5LE0bkUNrs37/f941Zh+XRtm3bhuXJjlst9pTJyy+gtrG+/fr6/jSTJ9/ufdqE5ccYZbN582YInlPWu2dv+9Fdd9md2JumG5ZS4zJs7AuXZDsCSbPl669d3Kxbt872YvxkSEmDqrGnTq5H3XC5NEbVDB06BBE9N3rUDkXW+fMV9hEigFZAWJ09e86jbG644QZfFm3SpEkuhzj2bdu3YVm3uf7bMAqKMuZG5KMUuwWiqh2WU8vH3jqMfOLyc9kRNkHaXPP/gDVAERABERABERABERABERABERABERABERABEfgWBCRsvgU8FRUBEbiyBBjxUlFRYVyei/KBS3iFfWKKsY/LEBsxYoRHh3CvFb68mP8CokJOnTpp27EkGAXHiRMnrATLfPXr18+GXzfcBkG8MD+VgUfY4IrRLxQmvpcN6uEyZPsgVShrDh446FEo3bDs2qBBA/EZhIidDt4WxQU6LhmDAABAAElEQVT3u1n72VpE9ex1aVKICJshg4cgomeC7x9DWUNpE/tXU1tjldgf5jiWMNu9a7eLDy4pRmFUkF/ogoMCahSkx9Bhw1x0HDp0EGJnqx04cMAqzp33qBdKluuuu87aY/+bAiynxjghCqu6ulpIl/O2Y/t2W79hg+3ZvdtOQqhQfrEP7A/zl5WVYom0Hojmuc6GDRtunTt3dhlDIUTWlGPn0Bbr5D4448aNs+HDh7uQovg5fPiwL+m2Y+cO/42QZL0QqcNl0/r17Ye6CiGg8n3s8DU+DnwhF8nzE649QV8iIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIALNCHA+L37i/yGaZ86D7tq1y2bPnu1zdzNmzGhWNgcF+f/h9oP/b3MW4CQfJ/tYCQ+eY7Z0mj/UlwiIgAg0IeASBpEjlBCUGryndGAkSymWKqMgie8Vvlv4nCKF0S6VkD0UFYzyKEC0B8uU4JOXx6W60BCXQ2MZCJv4XmLzrKMWbXFJMLbJfNzPhRKC9eRijTMPWMH7jM8plWpquKwZ62zAPjZFxigg7ksTkvyJX7Otuto6o7hhWUbdsB32k/3iJ/a1CMuTcayUMBwP99ehWGJaGcbOpdpcfiTvV28AXxwL66YYYkRPFfhV455ROHyWj0gejqWoqNjP5MIP97thnkoso0Yxxj5xmTmmU3Jx/xtekyfHy9+EfWeEEscdRFCZ/ybsSw7XgsPBNvkbhWXRJGscir5EQAREQAREQAREQAREQAREQAREQAREQARE4BIEOK8WP5xfix/OX9K/SNhcAqAei4AIXDkCfBnx5cMzD57jSyme0y+s2DJzU4zEsozmoDyIy5NFWeN1urVB3fzD+r2S8M3LoGZwbpQisU/ZQoI1xCP0DXu2JAl8EiNKvF+IWmF+ig/mZT/54Z4ylDG5uXmxqsY6DGu3sX/pMfFhXG4sFiCq0FW2zqibusYPy/JgGxQv/MT8bJN9iRw8I74obLyvOXlezvOgUAMlFzPhi2V4+Lh54WlsB31gncjvz3AdmMQzM+sQAREQAREQAREQAREQAREQAREQAREQAREQARFoiUCc+8yaX8McG+f5JGxaIqY0ERCBPxkBvogoHHjOzaMASab7OfGfiIAoMDzyA2k8GPERX2YUJEylkKCoiEcUE14nMsT8zBvbYRqrDLKBefAQX6wzPAv9YJ2hfJAUvgQaNUbSj9AGa6YACeNhnewPzzxYNz+hrVBvKE8xkhEeIW+D1SLqpgEyhZE/bA+l+ahJm57kaewDpVVsLzxJvtk2/7C//idhwK6FalGOfcXSbvhTn/wmeRRL6D95UByxFMdH/myH++V4Gd6zqWSsrJT90SECIiACIiACIiACIiACIiACIiACIiACIiACItA6gTi/yHN63lDCpnVmeiICIvAnJBBfSmyiqWzgPZ/Hw+8pFDwqhAWSJ56n8cYlQswbUsN3rCvmZOnwMvSrrLZYeaN/SPKxGdabTmdJHqyHL9LwnC0EZUEBwjTex6GE+1CGZfm4wTvlZiWrH57MTDi8btaDP+kj9CumNO9fHDdzsGS8j/1g38KBM/7682SsTA/5PSHJx2yxDJKaXGY9ayyhCxEQAREQAREQAREQAREQAREQAREQAREQAREQgTQBzrvFj8/9+dyjImzSjHQtAiLwrQhEmZCaxb9IfUEGZDLwxdRSWszhESA0FGmhkAgML4d0LtXFyI9wsD9uIcItLmPPstsJL0eXDUmGjGBBUW8y1OX+JdTW+M26+GG7jEbxfV+8X6yM/WHWUHEcY2P7nsyRsf5wsB8xHyrOpIYqYras9NAHtILGYh9D0Uy/eR9aClEwzMvDW/BsSQNJgp+SDFRFrJh/eIQxgjfriA0iPanB8+hLBERABERABERABERABERABERABERABERABESgZQJxTjHMK4b5QM7XKcKmZV5KFQERuCwCnN7nJ07Zx3PrlcQlz5gjvYRYTE8vcxZraZQIrB432aKDreNBSiB4OZiKzIuP5RJt4UYjvgyTFpI6MwIkSWfNXi+XDUsvZcalwZiHz1l1iLbhfeh/kFBRjsSz50uWX4t72PAZl0CL7SSj8XFyabIsvEl/Qptsl0u5RWHlHfK+MM3rS/oWR0MCZBX7w3Rk9TSWaVxujkulIYEfHszPT/yN0svVeQZ9iYAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIXJJAnHPjOc65xXk37WFzSXzKIAIicHECnNAPU/4hX5AGFysTJUA8xxdTay8r1uWqJbiDxqqZlj68ZZoMHI29aMwSL+I5lKSPSIqEBHy3nJZe/qyprGHRKDcoNkKkT3zpNlacXMRxu7DhGJIuOYemmVu85+hie8wQZIqPw6tjhUzjWNBv5MVdIxS/ZrGY1zMyLxNDNvaFR2NfmR7TYn2eI5Oe3OokAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiLQCgHOt8VPnBflWRE2rQBTsgiIwOUQ4Cx/EAShVJjob0sNUQbwHF9OMYKD5dPRN6EVfOOvt4CXWCjPBKTwUVJPdBPMRwkR8yW+AalBWVD4hGeUG/yw1VAmXof7pi/RIGxYN/OxjnAd6uF1LMd647Vf4Cu2FbJBqLAAPpGBV+aZQ9l0X2IdbJP7+sQyMU9sO96zbv7hiJk3XrOeMPZQo/cFX5RNXjZ0zh/GfLGtpmWZrkMEREAEREAEREAEREAEREAEREAEREAEREAERODSBMJcIucT03N7EjaXJqccIiACbSDAmf3M7H4UBi5FLjGRH0VAGxphdTgS+ZFqL5PKZ+hHpitBPCQJ7Fe6O9ltZ2QN62t6NM/bXFCE+rPTs8s1rRX3qezsdo5/ZSWnMvElzjHEQuGetWbSeJdUwkscIb4mCJuQkvmmvGk8cBlKBrETWolSLOTieLLbatp2Y226EAEREAEREAEREAEREAEREAEREAEREAEREAERaIEA59jih3Nt8aMImxZgKUkEROByCXCaPzPxHyQFp/vDy+ZStYX82bmiFGj2rFFWsMXQZhQLjTXQaqSOHLcgqYTkMlP3xWVNLBlkSbyL0iSeY3rzc6ad5s/wNm4hkeRaPlhXZNNyjuap5MQ/rdaZcIwn/w9EY+5sYdO8dgmblpgoTQREQAREQAREQAREQAREQAREQAREQAREQARaI8A5vviJsoZnCZvWiCldBETgMgh8O2HDhtJSIy0k0uneoZTgoITgEURESkf8iYSNN9b4FduL5+wxNGa7xEUDiocaMvW4/EqcU5pFS1VFPhfLF3RN+I2gpppVk+GYeZbO1wRns/LphIv1I51P1yIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiLwfSUQZQ3PnE+LHwmb7+u/CI1bBK4ogWxhk6k6IwBCWriPL6JMvotfRSnhuVoRNtmCIbs/qSLNGgp186XY7FELCc0zXY7MaKFCKCf0tUnj3gqHgONSAiTNpmnepArUgivvaCbFK2/6hX6kR9h8bOmnTQsHYRX/49L8qVJEQAREQAREQAREQAREQAREQAREQAREQAREQARIgHN68RPn03iWsNG/DxEQgStAgCLgEjIgUQGeK8naVDBcrCMuJpqIjUxkCKNUokwIL7tMfy4tYygmmlTdSldiG/EcXq6tZL5kciOxTHVexm/byCgtbFg4zZRVkFGoL9x5A61+ZbPK1B07GM8tV8D88T8wLedQqgiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIAOfR4ifOp/EsYaN/GyIgAleAQNtkAMWIKwSc44vochr34qkCV0rYpKq8xGUUFvF8aWHDcbZ2BGpNR0X5hCNJvmj5ADSr+nT+WD/rw040jfkyIiYksZpMNzPSJpMvjiGew+8XK4z5eP4mv2usR2cREAEREAEREAEREAEREAEREAEREAEREAER+D4Q4Dxa/MT5NJ4lbL4Pv77GKAJ/cgJBDVyqmegXgiCgGMgIgEuVjc8z2iETPdK0liAQYs6MgIh1fPNzbCmeLy5s2jK+QC72NfQsUzv6fpHOhnFmZ0i3GetmHW0XNqwvwyy0ke5FfBbSQs/5HxiWyhY5rEmHCIiACIiACIiACIiACIiACIiACIiACIiACIhANoEoa3jmfF78SNhkc9KdCIjANyIQ1MClikbBEIRN7rcWNm4JkkZzcoI64G1sJzyKgiHJ2JYTqorRO2l5EZQEK0gLjLZU2Hqe2OvYXqg5u/7su0xd2eMM6d9E2GRqjFcpZgmLBnQCqUkGnuN1hlVIzaTH2nQWAREQAREQAREQAREQAREQAREQAREQAREQARHIEOC8XvxEWcOzhE2Gka5EQAS+MQFqh6geWq8kIxiSqf3LiLDxsk3z0/wkR1uFTboPTavzqhJBEesN5ygwooyI5+xcF7vLtJvKlerAxYSNs0W/0jIm1tK03nQe0mG9gXaGVXaZOLZYY+qcYkFhwyNIm1BjSAnfmXYun026Hl2LgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIwLVOgPNz8SNhc63/2hqfCPzZCQQ10JZmgyygJEhP7KevW66lUTI0lnObgMxBRDQmMyUlclwxpKrPPGtZVGSeN+1Hus+pCptma+G+9TqRGR2PsoZFQ82Z+qMI4TCzmYWG0nU3fR7IsE5exbtsPi5gMs2FSvmN7Ol+eWnP5yWQIV0ojIEp8Smr0CECIiACIiACIiACIiACIiACIiACIiACIiACItCcAOf04kfCpjkfpYiACHwrAq0JAUqO5hXTp2TSY4Z4bp6fKVFMZKREeKm5WaAmSBWPeUNNrT3LTo+thrLhWXY9zBEb4fN4HUtmzsEXueLIJDa58rpZBevxrAFKqDVTd1uFTUv9yfQAUTbeqUxKdnfQHrsR+5F+iDQXN3zu6Rg7z41ffpE8YXLm3hP1JQIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIikEWAc4PxI2GThUY3IiAC354Ap/LDdD7ryhYdQX603kZ6gj99nV0i1pkRE/WpdrLbiHlDDa09y06PrYWymWdN64p18pzpSywdzi2KjyRLrM/PcbiJ/PFbXIfk8H0pYZPdcvZd6he5hLAJvx77FPvn6iX2D9V6Xal776Xf8ys8CFfhOrsnuhMBERABERABERABERABERABERABERABERABEYgE4jwczxI2kYrOIiACV4jAxYQNm8gIkJYabIgbpCQPMyIkKAf/DpeNkqTB6mmGYgl3BlEVZKRDpjXW2ZpIybQXZVN2fzP1xRbCOXiWmJZpyyNSYteS5PgSrqurs9raGqupqbG6+nrLyc21/PwCKyjIxznfcnNyPegmU1usAEP0ppq31yxvpgiuQkdihE24w3dj/4Kkqa2ts6rqaqvGh5zy8vLQpwIrLCzAdb7zZZHYuvclfCWtpURTUnfob/L4Iiey4YZq5BbGT/6xpaYFY8dbe940v+5FQAREQAREQAREQAREQAREQAREQAREQARE4LtFIM4V8sx5sPjhHNmuXbts9uzZ1qNHD5sxY0azjmOeL0xz8sn69eu9wLhx47xAnFTjOWZLpzWrTQkiIALXGIFMVEb83z4HGN4afOEECcL7nERGhOsMBryH/P2RmxtESXhRxTrwkIdX5BfBGqA+TvDHaXtO47syQIKnJc/5kmMK62a9oW3mCJKAfc7NzUv6yfpDTY0V+z3Tkvr9KnxlWo+9iO9BPmFHQjsNlBFouKa6yk6fPm2HDh20ffv225lz5624tMx69+5tffv0tc6dO1lRUWHStA8gVIHm2G8eySlVP58xNd1mwgBp/BP+eo/wgPnYH3IJ5Shpjh07Ydu377C9ew9YXV29tW/fwfr06WP9+/ezLl27glEum8dBhqFsLscXh+4XHD9qZ3/wN/yeSbtJ2aQzuGNeQ956l1eVlZV+LiktseKSEstDewk+L+nXSR0sh18NX42N+xN9iYAIiIAIiIAIiIAIiIAIiIAIiIAIiIAIiMDVQCDMU3KeDXN1mOOKHwmbq+HXUx9F4DtOIAgDzp9jEp4z9fgbDr5wcIsXj0+y+wR7epKdLyNM2tc3FvD7UAHTWBsFAw6viPU3OfgQiSFfeLmFTCFnVt+8H0wPhcILMTeROaFe7yKfe6OoN5ET4Tb0N8mZ9IVPksw4h/aSNpjMNl3Y1FvF+fO2f/9eW7fuC/v00zV26Mgx69ilm40ZM9bGjhljgwcNsnbtylgLC/pftpVdZ2jPW0RnOe6YJ/Qi9DFcx2esI2EDYWNWhwRKlyCSzp8/57JmyZLltmbN51ZVVW09e/a20aNH24SJE2zIkCFWgOifMEy02IDyaDf0ga3j8O6yP7neFr4QLYN09DH0hZfMxPYhdfCbM6rn7NkzdvDQITt8+DBrsQGDBrrA8sgeSiLP749CeVz6WCRsAhR9i4AIiIAIiIAIiIAIiIAIiIAIiIAIiIAIXHUEON8XP1HW8Cxhc9X9lOqwCHy3CcQXDXtJ+cEPj8y8exAHIR+fB8EQ75nXJ/Yx9c+0WAen+fHE62FURnyReXJ4iKdsEClJYzzF9j1f8hWibhgBwigO1hmibVg4pKGKJC+FTZA2oeoYJZTcNebL3KOk/01qQN0UNuzT6dOnbPPmL23+/Pn23nvv2Y7de6xv/4F2/7QHbNq0aTZq1Ejr3KkjB+DD8HrQfowSysuLUSUUI6HF+M3ukgsfRKYhlzeN5uOPQVFTh/pZA+uus5MnT9gXX6yzN9542+bOnQ+JUmGDBg22++67zx586GEXN8XFBSmWYTwYWeint530KIfRSolo4djx4UF5w37xYFpNda2dh8DavnOHrV692rZu3WplZWV219132ZixY620tDRIIq+DfWb58ME37tBGY+t8qkMEREAEREAEREAEREAEREAEREAEREAEREAErg4CnB+LnzjPybOEzdXx+6mXIvCdJpB+uYSOhkl6twepnnPuPgiPKEkyE/qZFxPL4uN/mwsb7vvCFxf3OonLdAVxwNl8/uVkPqsIdTMv+xfy8hnrj5KI/QnXSRGXAnzOXOHgVUZMhLaQ5PIB0SRJf3IgKrwNvlhd8iQ1sB9JntOnTtqXmzba+/Pm2X+8867t2LXb+g8cbA9Nn24P3H+/3XLLzdapY6fQOsqxiYwgCn1lMBKjUzLP2P8wZo6Th9/6ddIHr5FkWCHSPEIGHHGuq6uBsDkJYbPeXn3ldZs1ay5ESiWiaoba1HvvtenTH7Hx48dbcXEhxufVozyYcg8h58K28eE4vR1IMG+H36F99it0EaWwhw8jeA4ePIgoo09txYqVtgWyhvLmuuuusx8/+aTdeuutWBquyPfRCW2x5mRcqJN3XBLNxVDSJZ1EQAREQAREQAREQAREQAREQAREQAREQARE4GohEOdTw7wZ5rowecaPhM3V8guqnyLwHSYQXzDsIifm+aEvYDon9TmhH9PCNaffg3jgS4hH3O8k5A8T/UxnOZcMqIPVMT+lTVrYhHxs2GvnyQ/mra2t9X6k96mJbaVfiEkRP3nfk4RQV+gPVUHoH/tFacCXaIPv+cL68/Ly+KBR2Pj48ZxRLBQPp8+ctq+/+soWLVpoc+fMs1179lnfAQNsGmTNPVOm2PXXX28dO3TwNhpQhnvE5GMpshD1w3pDWzyzfT6PL3O2691jaQ7AtUbot1Nh1AvzeKbQn7r6WquFsDkFYbNu/UZ75aXXEPkz2yoqLtjgwUNsyj332COPPGoTJkywkpIi8yq8Bv5mSdSLU2DbibBpCJEvoV+hydAfSB6wqK2tsXPYu2fTpi/t9T/+0d5//33sn3McS8GV222TJ9tPf/pXdvvtt1thIQURlQ/G4JFDGekTxpNE8qAnOkRABERABERABERABERABERABERABERABETgaiLA+bL4ifN7PEvYXE2/ovoqAt9hAmFSnh1kREuYqM/ImLj0WBAuLhw40e8CIZ7DsmS+3Blq4fJffByWP2N93BuF93iZYeI/tpd+ocVr9oJqggf7EF5+vAsCg/fMG0QIBUNomzm8HaYkY+ByaIwKqcOHlbJfFCXeOS4zFseBNP+TPGO7dXW1kEtYfgyV5SE8pfJChe3ds8fWrl2LZcA+tcNHj1nnbj1s7Nhx2MNmtA0cNNDalZa5oGC7rJ19bOwnx86+8YOvGMkT5FUitFCOET0+PlbAA+xYhw+A8iOHsiWIJPbx9KlTEDbr7ZWXX7N3352NaJcKGzBgkN0NifTEE0/YxIkTraAQe9ig0TA8wsF1Mnby8N/Dl13LQ81omFlw4ofP6tFO+B0grk6fRkTPF/bKK6/YBx8ssLPnzllHRBZNhrD5i7/4S5sMYRMibMKYOARKG2fCdjEeRjShdn+kLxEQAREQAREQAREQAREQAREQAREQAREQARG4mgiEebIwxxnnNHmWsLmafkX1VQS+8wQwMU8hQJuQTNT7xD0iJfjC4TVci0/cc7KfB24hE8I1pYjLBtwHScG8QS6wQpb1AjiFdigEUBPz+yR+0g6e82Ct7ArrcOHCNOR3uYCnQQoFYRPS+Dw36SOvWZaCqM5flqwwROew4lA3rjxjkAhBMiEBj4MEouxhJ/LQR0aXnMKyaAcOHLD9+JzD0mNFpe2sV6/e1qd3b+vcpZMVFRSgPSoPignW5F9JM5RWZMxbnjMv9Rht4/1Ferhn06yHjCBcQinUGJgGLrXepy8+X2evvvpHLIk2B3vYnIewGegRNo899rgLm6KiUD50h0LIO8Fu+MF+5lAKRZGS9NEZosP1iOZhNBKjjU6dOg1p9Zm99NJLtmDhQqusrLIuXbq4qKGwYaRNQUESWYTaWQcZ8g+vXZhpD5sAXt8iIAIiIAIiIAIiIAIiIAIiIAIiIAIiIAJXHYE4h9c4d8f5O3wkbK66n1IdFoHvHoFkzj7pWBLR4jP58Qln3Hn4tH6YgccjShe+hIKcwQuJcgNHoxTh7DzzJPk44c868rD8GA9O4PMIsiAjbXxiH6nhMZfhYoRMLV56uS5pvBC+Yrt8GcYDXgDFQmQHr/nIP0kGSo7wQSvIwP6zX/n5BZaH5cuYxsOlAgpyn5gGHxdrxZghLPw586I/dVhCrL6OAgTROxQeXj+EC+tIiSwvg3GTRS3y+z0L4QgRS+bLp1FChYMDSX4LtEOR4pyYZqEPHCm5nDhxHALlC3v99T/anNnzXNgMHDjI7pk6NVkSbbyVlpZ4H4MoQ9SQ74VjLsLYfy4Hx/Hn5EDssF/gQE6xL+REGccl6k4iomftZ5/ZH1580RYuWGhV1TXWtWtXm3zH7fYXf/lXdttttzkL/i5EwHO8doiolGPhtw4REAEREAEREAEREAEREAEREAEREAEREAERuNoISNhcbb+Y+isCVxGBoChih8NEfZys50Q7j/AS8it8+Sw8fYHLhhgdQ9HiE/0848OiIZoiiSbB/H/iQ/CUAiRE4rBWHum24iQ/q6IkoTAIaUFosG7ex5cjhQPrC1Er7BefBeGBO6uD9GGEDAUHD+4tw4/X4fcQNnlh2TBUk+oL2k5EFNvkNevlnjcNLmDCUm+sk4LGBQWukRUHWYazf3MsKFsHQcKIIYoi/qmtqeEwrbioyAoRocN2WNY/PtAgbFhriNzhswCTkS8nTpxwYfPaq6/ZbAgb7jHDJdGa7mHDMjUQLnXY96YeH/aN8ozRMAVgkZdXgCYoldhvjjFoFfbXu4FWa9DXk9gz57NE2CxatMiqa2p9SbRbIWp++ld/7XvYePQT8rMc2wgCjPWy7xxJWGbPb/QlAiIgAiIgAiIgAiIgAiIgAiIgAiIgAiIgAlcRgTgnyXOcx+SZc5i7du3CHN1s69Gjh82YMaPZqLD9QjJDhkfrsc8BC4wbN84LsBIePmkZJ9JSaf5QXyIgAtc0gTCF7nrBhQBfGUGQxHdDmOivrKh0GVBZWemiobCgEJEbpVZW1s6Kiot9Ut4VBAQLz3yVcKKeYuLChUospXXG91fhM0bHlJSUWMcOHa24pNhVBJdGcyXBdxHK8pr9qKio8H1Tqi5c8N+BcqasrMw6dOjg0oXZXdigTC1ESHV1LZbpuoC+nkO0yVm0eQ51nLcL6DelDd93xehvu3btrLy8PT44t++AsZRZPuQFn4c3I3saIkt4zwgTjj3UV2k1FBn5RRhHqbUrK8W52PsDFcGee9+jaKL04DjYn1Nnztipk6esorLC8zBqhXKkvLzcOnXsCCbtnSs7wfY4HizKZrkQSqWlxdahvAx70kAYQapQZp2EsPn8c+4pA2Ezay76V+lLot1119324EMP2ahRI5EPe92cOY12T1rF+bOQLNUuT4oKC609GJBle/wWZe3K/bfMZ8QNPozy8aXuALm6usrl0OHDh23D+g32zjvv2PIVKyCB6rzvo0aPxp45T9r4CROssLDAx9AOv1MpfueCLBHF31bC5pp+qWhwIiACIiACIiACIiACIiACIiACIiACInANE+D8afz4XCLnE30eVMLmGv7ZNTQR+PMQaBQ2eNEwgsP/0IK4peCSZLWQH2dt//79tmnTZtsJ6cu0TthofvCQIXb9dddhH5deWFILUS4uO4JZZu8Z0VJZVWn79u6zVas+tZ07d7kIoKzp168fZMIo69e3HwREAaI8GF0SVAnLUnZUVVXZrt27PKJj/779iJSpdREwbNhwGz9+nC/FFZdGoxSphBg6cfK07UPerVu22uavNtn27duMkuEcZAkjZNgEhU9v7DszdOhQu+mmm+2666/3/nSAtGDECSNhmI/CybsEHJUQLIcOHbLt27ba1q1b7RjaKW3XwYYNG+YM+vTu40KFIyDHGkSe1LgYMRdVe/bssa++/srF+caNX9qBgwdC9A4aoNTo0b2bXQeWI0eOtOHDh4FTrm3bvh1tHrW6+lwrRZ8HDRpgt4y4EePuFIQN+DLixYXNy6/D3s+FGArCZvLtt9vdd0+xfv37gf9uW7d+nX25cYMdPXoEAq0CLqbBigqLXd5fj/HfPGKEXX/9DTZw0CDr1KmT5aN9/oeHvzWF2769e/A7rMXYt9mePbvR5jrbsXMHfuN6K4a44+854paR1r//AGvfvr0NGTLYuZILJRTri78vqPIn1iECIiACIiACIiACIiACIiACIiACIiACIiACVx2BKGt49vlQn0NUhM1V90OqwyLwXSSQFjaM2gixNpQ3uEoiLM4gKmQrRMWiRR/aihUr7TyW3erYqaPdcsstNmXKFBs1cpQVFhWGZbTwguJ+LnxZMSrj6LGjWLLrc5s1a45t3LDRxU4Zolluuvkme+jBB1HHSJcwhYj2oHyJc/k12BuFMmLNmjUoO8u2bNni9fbp08f3SZmKPVp4zTKM4jmDCBIKoS8Q/bEBQmTP7t0uGShr+Izyh1EyHF9hYZFHszA0ccDAgS4WKEpGjLgFdfZ2geR70gQI6FKDC5/tO3bYSkSVLFmyxPYdOGzdevZCXybbbbfeasOHDYWoKPfusxhl1dmz5+zAgYP29ddf24YNGyCQNtsO1LF37147dfq0CxFk9aXQGOnSq1dPG4p6huFTUlyEfPvs0OFjiBzKAe9OiI4cYw9Mm4o+9/PomjpEDJ06ddLWrdtgryLC5r335th5CJvevXpDRN1kgwYPwVgLIYf22c4d273ds+fO+DJs5JCfm4/oovbI3wt5B9uNkFeMkBlx883WtUsXj5ThMmgUdhs2rLdZ781CW+sQaXMS/TrskU9A75FF7RAh1KVrN49a6oR/G+PHT7A7f/hDu/GGG6x7t24QckHosV0JG/7qOkRABERABERABERABERABERABERABERABK5GAhI2V+Ovpj6LwFVCIC1s6huw/BZm4OFa/KCw4Z4m57G82KZNm+y11ykFZtmxY8c9SmXsmDH27LPPYYP7e/yeS2nlwLn4viUQKYx4YYTLgg8W2qvYY4XihpE4XEqNSzP+zV//td1xxx0QHR18mbKwtBn0CNrkEmL7D+y3BQsW2AsvvOCypxxLdnGJrwceeMAehOxhVAfFEJcO2w1Bs3TpMps9d559uuYzX36MQoNRMvlo0+vGqFg3l/mKHwqNvqjnBz/4gd3/wP3o11jrBvHAdI6/AcuO0cKcPXPWGcyfN8/e+Y//sJ2791r/gYN92bFp9/3/7L2HfxVHsjZcIIQIImdMkAgm52Byzhmcvd712nvTe+//8r2/3xfeu3f37l5nYxsMxmQw2SbZ5JxzDhIoIITge57qqXNGQsLCC1iCGpjTMz0dqp8+p7vVz1TVZJA93ZQEUui0jnvQ7LkCgmuzrIavl23btqtmyp07BVoe/cZUo6kw/KNpM/qyoVZRDRA1zZs3g6myOpEJtVz0QVVpDNJj7NjR8v4f/wASpBNIHPjkQftugfjZCyLs88+/RN8sRp5cadCwgTRt2iwismiK7pZq+7BTg6+doEulvn2gCURMaqJPMkDajIUptYmTJkp3ED4NQbzcvXsXZNUtaEht1X7Yvn27kkJ3C+6qCTr7/lSBrxq2h/59GoJcGj1mjMycOVP69esHbaYWqmGjGksE0w9HwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBBwBR6CSIuCETSXtOBfbEagMCNiGOzUfaAqMJAUPbqurFxeQCQUgGWj+av68+bIQZAVNjpGZ6da1q7z11lsyafIkaQSNDPpEIdlTDWQEiQH6TeFG/6JFi+T71WtgTuykFkzyhBogv3v3d0JNmbYwo0USx1QIWT+1Oo4dOy4rV66Ur77+Sg4ePCjNmzaXyZMng1iZLH1695EmIDFI1hyH6TASI2vXrpOdu/dA++Oq1l8PprhaQmul1UutQGA0kTRoAdEXD7VuLl26qCH9ytB/TnNopfSGls3wEcNl0KBByPOS+tehpg1Pahnt379fli9bqqTViVNnpXVGBrSEZsiUKSRsuqsvGDgOU3No56BFs2XrFsi0Vk26USPlDnzrpMPnD03IkSRq1qyp+tLJz8+VU9AOOg1zc9dvXAcRU6Ty0xzZPRBoqWm1IV8LGQfC5o/vvStdu3bWfioCaZOdlSV7qGEz9ytZApNot0AspaamwRdNTSViiGl6ek1p2KghiKjGUr9+XTW3xnZfu3IVOFyGJlMW/NoUSnqdejBp1kaGDR8qr86eDfNsPbVPCmDW7vChw7J69SpgcACE3TX9DlxB/gIQOlXRn9QAatW6DcypNYQJtHrSB2Te0KFDpSPMuzVq2FAxBMw4XL+GKPjhCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgCjoAjUDkRcMKmcvabS+0IVAoEkoQNtU9A2IgRNqRr6EuG2h/31OfKMmiXfPvtIjmATfscmEVr176dzJo5S0mUthnwXQKtEBI1JGw4cNFfyips8pPo+Wn7DpADV1BiKDezXabMnDFdpkJTpg8c1teH/5jwJGzp37x5QzVyli1fJitWrACZcVrawcTXO++8LRMnTpTWIGHqwAQZCYf169ZBE2elbIUWywX4fKmWWl1eatVSzZR17dIZ+dpJc5g/q5GWppo7JGuokXPk8BH1K0NNnnsgSajZMmjQYJk0aRIIhz7SonlzNQtGbRgSHPv37wNhs1wJqOOnzkjrtpkyffr0QNjAjBjNmtHPDtP+/HMw5fbDDz/AD89pJT4aNWwkHTu8rP5i6KeGhA3bkJebC98wR+QgtJiOHDmsRNXVq1dV46ZqSip85dSXliCQxo0ZLe+99zsQZZ0Vx/v34cMGJtH27NoD7ScSNsuUsHnwoIqSKPQV1Bxm2zp17qg+ZWj6rGFD+ulJgWbOLfVLdAhEzH6YkDsP021QPJJq8CfUu3dP+acPPpBRMGlWp066knDXYdruEEy7nTl9Rk6BWNq6dZv6NMqFJlT1GjVQfnsZNHio+gWivx1qP7XLzARR1lS1r1LwvVC+BpIzJJHkhyPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgCjoAjUNkQcMKmsvWYy+sIVCIE4oQN6JkEaUJiBayLat3QfNiN6zdg1mubLF26VDVZzp0/rz5kxo4dK5MmTpKe0MYgwZGqZrGqqgmus+fOyHeLF8u8r+eBjDgkWTfhtyXChv5jhg0bIjNgNmvC+AkgL5opycMBj8elS5eU7FmyeIn8BPIjG6a/evToIe+//76MGTUaWiN1lEw5dOiQfPzRxzA7tlrJm4LCImkFjZ1RI0fJyJHDpRc0X5o0aQyypoaSSSSl7ty5I9fRnt27d6kGz+Ytm+X0mTNqNq1jx44yBua8SNpQ46ZWrZpoUyBsDqOuFcuXyzffLJDjJ09Dw6YdCJugYdMd2kbp6bWVfLl48SLKXSVff/WVkhz05dMAWiZ9evdVnz9DBg+BH5q2IFWgvYOTmjRZIKjOnjmtRM+K5SvUdw/90UiValK7TiBsxowZJe/94V1oJ3UG6QF6C2RaVjZ82ICw+XLul7JkyTL4tLmtptBI1rRpmwESZRDqHAOtnC5KqNWqVUPbU1h4V65dvS47YaZu4bffyrat2+V2Tp7kF9yRDiBf3oX20/hxY9WnTz365gHhQg0hEmn0x/PVV/NkDbSHckE21YVGzcBXBsvbb78jr6C+6iDMSNrx5PeB/mvoa4jYV0H38toJG37L/XAEHAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBGobAg4YVPZeszldQQqEQJGoASRI8IGkUHbhRckbeDHJjdHTY+t+X6NzIPGDM2D0QxWL5Aa48eNh3+VsaplwQ165uZG/rHjR6GR862e586ch/myO0rKkABKT0+XLiAR6Ofk9ddfV1NcJAXox4UHNWC++OIL1WY5d+4cNv1TZeDAgSAS3pUhQ4bAJ0oKtHxyZOsW+lb5ULZs2aJERf1GTaQ/HN7PnDlDBg4YIM1gCq1mTZI1IAlQLhU7iopI2uTLhQsXhD5ZlixZIhs3bVJSiCa9Xhn0irw651X41xmufmmqQ+uEhNGhQwehYbMsEDbQ+KEPm6BhM0V6qIZNuty4cUP27NkDbZfFWu551FG/fn3FaczoMUokte/QASRHXUgT0L8PPCjPrews9fnzPcin71d/L8eOHpfsnPxIw6YliKRR8oc//A7+ZbrECJssEE+BsFlMk2jZt0ECpcLsWkvIP1L9C/Xu3QMm1UCmKYECXzMgidinJGCoZbQYsq5Gvx49dkLNo9Es2uxZM6HJNEE6d+4kjRs3BAlTTXG7DX82rO/TTz+TFTBXd/t2jhI2Q4cNkz++/wFIuOFK1CjQaB/91qTgTBA0aDL7IHEPFPxwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBBwBR8ARcAQqCwJO2FSWnnI5HYFKiECgDExw+rAhVcNYO8Mlnc+TjPhx84/y979/CKJkizqZb92qtYyC6azXXn9NNWBI2JB0yYKpLpoQW7BgAbRSVkREQoqWnwczWiRt6Mtl+owZ2Oj/o7z88suqjXEfZMq9onty6OAh+a+//Jd899136vi+ObR3RgwfIa++9qr069tPTY+dPHkSflVWy1dffqkEUj2YVevSrYdMmDxF5syeBZNoHZWkCX55AnlAsoAHZaRmy9mz50DAzJf58+eDLDmhGkXdu/eQt95+G6beJklT+MmpCZNfWfAVc/DAftUw+gZtOgGTaG0zk4RNz549pAE0Tc5D82jN99/LIsi9CSQQ83UAQTMNpt+mTpkiffv0ha+YOooDMaYc4SxSjRlqsFCTaTk0mdatWy9nzl2UGrXrSYuWpRA2D6Bhk5UNAmW3fAmTaEbY1KqVDo2arvLGG2/K1KlTob3URKqngUijhgvaD15MCRMaKaMfmvXrNyDvElm/YZNchLm4l1q+JOPHQ3Nq0kQZMKA/zKo1VbKHfXYL5t72gpAiYbMMpupu3QJhU7eukLD54E//BB9AI0AIVQvlk6gB1krQKOq4jkIPHAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBByByoiAEzaVsddcZkegEiCglAwIGh5B4wFUDTb1SSDQdpWRGxyESG5QI2MXyIFPPv5E1sJvDH21NICWzXBoVbz51pvSv18/9VfCbflLly/JFpA7CxculE3w45JSNQUm01rAJ0pdOKw/pxo0NCFGfzTvw19K79691Z8L66H5NWq+fAjNmbVr1igxRAJi4oSJMgkkSpfOndUHC7VqSOisW7sOZZ6VZih/xMjRMnnqdBkM01wvgeSg7xQSI2qOCyHbyfMB2sh2Xbt2HUQFCYvvVEvn6tVr8M3THgTU6yBZpkrb1m0gcx0QE9nw3bNPlixdAnLnGzl5+qxktu8gs2YFHz7dIF89aM2cOHFCvkWbqWGzb98+KSgokN7w0fO7d96R0TC11hbm2mqAAGLdRTBpRp83xLtKhHc+NJOOHz+u7abptf0Hj0i1GulKbo0ZE9OwQXq2IQtaObt3JQmb2yBQGjVsDC2jAaolxDobNKinpteIQdWq8FMDDZugcVRVNYcOHDiIdi2Vr+d9A/lPqm+d4cOHyxRgPXz4MJhFawGtmZSEf549MIn2ySefyDIQcTm3YRINGkTDlLD5kxI2KVUDYQPgBdVFB9uIeu1LZdEeOgKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgClQgB7uvZaXuNDLnHR9/PtGZD9w9vvvnmQ62qgoxhNxaPaKaHGfr3768ZWAgP3byMksXj9KF/OAKOwHOLwC8RNhHXEfzYYMApulckR48ehdbMQlkB7YoT2Nynn5JBgwbJ7373DkiSwTAh1oCDipw8eUJWrVqpAxTNpzVo0Ei6dukCfzJNVRvm559/Vp8xw4YPRd53dcOfWiTU5CFh8f33a1TzZceOHer4ftjQYTILWjODBw1W8uLy5cswT7YChNAC2bVrF0ie66qFMm78JBk9dpxkZmZClnogJzDO6fgWDYUgLXjYWHfr1i3ZD80Zapls2vQDiJ/z0qp1K/WtM2P6dOncqZM0hP8ZklMkbJZGhM1xmETLBLEze/YcmQLNGZJI1DShT525X3yuptNOIU1VEFVDhgyWP/7xfTXl1rBRQ2BGbZdAIlFrpej+PfBjIDQA+L3CQrl+9ar88MMm1WLZ/vNueVA1DWRUs8gk2rvBJJoSNtSwiRE2S5YqgUJzaDRN9uqrr2qdNWumadkc8olHVe1Yar9UkdwcEkQnZClMvX362Rfo32PStGlTGTp0iGoYjRo5Qlq2JGEDM2r4DhAHavSQsFmO70AOTLbVg2aRath8QMKGJtFSFV/OMKyD/9leYu6EDb99fjgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgCjoAj4AhUVgSMrGHI/S47nbCprD3qcjsCFQgB5TIgDzfzTRPlPskDjYMmCq55f/9+MN9Fvy/r121Q02CbN29WXzV9oVnzh9//XkaOHInN/maah9ol3yyYD+JltVy6dBmaJZkgiwdAW+Ml1WRZA82Z+9Aw6dGzl8yEv5TJkyZL5y6d1c/N9p+2q9+bdXBqf+rUSWnSuInMmDFT3nrrLekIM2c1aqTJaZAhNLfG8xhIpBwQCU2bNZdh8NvSp19/qVWrlqRVr6YaNoEoYPtgCg0ECeulxggHVfrauXjpklDLZMfOnXLp4iX4e2mhmj8zYK6tL8yvNQcjTmLnAIid5cuWol2oEyRH67YZMhukyNQpU5WMoibOXmiffPTh/8iqlSvkKsyN1YavnpHQ+nkfZt/6gSyvCblS4H+H+JI3oQz3QdjcK7yLGJhtwxPKtH3bVmgYfSSbftwmBYVVpDF88YwdO1regw+bbt27IlXIlw3TcwkNmyXLQKDkQSuorYyBTyESNgMHDogIGpA1qcF/DSpVs3MoQgkbkjQkbL74cp6SZWzvcBBp1GYaDqKsefMmmpfkC02i7QROn376CUi7VZA1D4RNfRkydChM272PfMOlevXqwB0tIUHDloYvF+HnnR+OgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgCjkClRcAJm0rbdS64I1A5ECBpk9xTD6bCSB7wJDNMwoYH09GPzZ7de9UUGdX7SMbQXBnJlAkTxktmZqbm2QrC4ZNPPpYfYQ7tHvzS9O7VW0aNGiOtWrWCb5Z1SsjcggP7jIwMJUdmz5kjfWA6LBvECP3SzJ07F0TELhA4+UrSvP7a6zJnzmxpDJ8y96HpQ/81X3/9tZpcOw8Ta0xXs2YtkC0tpSEIHhIyNIem2iQUHAfbyAE1+HIh+y3qSycvL19u3Lwply9fEfrXIfEzZsxomT59hgyB1lBLaP5kZ2erTx4SNgu//VaOnzilhA21fugnplvXbqphs2vnDvk//+f/g3bRKpAnOVKvbj1t95/+9Cfp26+vpMEcGuWiRFCBVAIDKCuJBDpJHgCrgjt35Oeft8NX0P/I+o2bJb9AQNg0DYTNe0HDhibr7sPXTxZ83pgPmyWLl4JAyZe2bTNl3Lhxqv0zcGB/+AaKfMqQIQJZRQyICwkVynj82HGYRFsun34+F0TUcWkG0m3YsKBhMwIm0ahhwzJI3mXDBNvOnbvki88/l5VoI33Y1ItMov3hD++ppg0Jm2ogparCBJoeqI/tpXYN6/TDEXAEHAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBGorAg4YVNZe87ldgQqJQLBBiPJGhII9LPCQcg22klunDp5WhYtWqQmu06cOClt2rQBuTEVxMU06dmzhxI2JGX++t9/EZo0a9CgoYwdMw7EzCQ1x0jtGpIt58+fl/rwgTMSZrfefvsdeWXQK3IF5sBIiHz26adKytSF1srAgQPlDfiUGT9+vGrO5MOXzgkQC1988YUSNlevXJHCu4Uw+wV9DpggAxuh2iuBGyDpRJKGbQBJwj7hB6J4r8QF0uOxys2QpBB9skyB5swwaI5QKygbpsf2wyTaMmiiLPx2oRw/aYTNbGjYTFHChibYaOrtf//v/0sJm7sFd1X7ZPToMfIv//IvSthUrw5zYaiPFZIAAe8S5KKJM/yjT5vCgjvA7Wf529/+JmvX/yh5dx6AsGkGEoYaNu9Cw6YLmgCSB4TNzVIIm8zMdjJ+3ARoLs2SAfBlU6NmdSVLSAwVFd5DvQ8SJE7O7Rw5eeKELF66TD7+9HM5CvKmGcghElWTJ0+U0aNHov0tJRWm76iZlJV9U03QEfuVKwNhUx9m8IaB2CFhQ02b1NTUGGETfZ+0nUlzbOwGPxwBR8ARcAQcAUfAEXAEHAFHwBFwBBwBR8ARcAQcgcqGgBM2la3HXF5HoBIiQKIiHCQ2eBMIhft8gFMJGxAcJEYuw9TX4u+WQAPk73L48GHVLKFvLJoQGzFiBLRW7qmWzN/+9nclOdq1a69+aiZNmqSb+Zs2bVLCh75teLwCHzjvvfeeDBo8SE6dPg3fNd/IvHnz5BL81GTA7NhkmOaaMnmK9O3TG+bQakoBCI1jMOP18ccfy/x586H1cx1KMw8krWYNqQsCqA60WkgwqEIJSAaoCGl7jCRh2wJpwqv7IG2QFiSK+pMpeiANGjYA0TFQRo2CebXefWASrLlqohw+dFBNwVG2oyCM2mRmRj5sqGHTBcRUA9kJgsoImzvQlKlXtz60dcbKv/7rvyphkwrtk0B+QQbgShNtHOSrVYs0fkDY3LmTJztB/Hz44UeydsNmycm7F2nYjAFO0LAphbD56suvZDFMolHDph0Jm/ETZJYRNjWqK85srbadgTJXAlNyOeqLaAkIm48++SzyYdMEhM0gEHCTYVptjLRE+4kPCabb0IravWe3fEofNjD7lnM7T9s9DKbQ3oefHvrOoV8jtpEEFA9Wxb6wSkP7IwEY7Ycj4Ag4Ao6AI+AIOAKOgCPgCDgCjoAj4Ag4Ao6AI1BJEHDCppJ0lIvpCFR2BMjN6Ia+7qXTDFrQjuAgxF13UAqqrZKdfUs1SEjY7NixS+6BZOjQoYPMnDkDZrvGqL+XVatWw+zZQjlz5qz06NFD/vmf/wUm0yaoFssumDpbunSprF+/HqbIbkiv3r3h4+V9GQTi5vCRI7IIGjZLly1XcqBXr17yxhtvyOhRo2Dqq636f7lbUAANmxPyEXy8kDy5eeOmQk/NmD7wp9OlW1epXbu2pMIsl5IUlF8bF0gou2amKjDbVZVaOSAY2I5CnGkghTIzMqRTp07SulVraMnUA2FzWw4ePCBLlyxFnV+DsDkhGSCi5tCHDUyide3SVerDNNiuXTvl//1//m9on6xUMqR27XRoqYyRf/tf/wYfPv3Vv4tqArFu4BsIG5JG1D6hxbIimETLl5/hx4eE1zoQNrkFD6SJmkQrQdjA900WTLmZD5slkI2ETWZme8V6dkTYVE9LZVP1IBmn3UtI8C83JxcaU6dUw+bDjz4B/kehBdVUBoOwmYZ2jRk7WlrgnvIxx22YUNu9e5cSNivQxtvIz3YPB1HzwQd/gqbN8MhHTyBs+J2hJhNzM9SvEi+CFCqTfzgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgCjoAj4Ag4ApUFASdsKktPuZyOwHOCgBI00DyhvS5urWOfXjf3Q/OqwF/MHfkBvmk+gZbFhg0boXFzRVo0byHTpk0DOTFKzp49I2vWrJWfQDrchgZHv3795X+BsBg7dpxq3xyD2S0SGtSkOXr0iHQEMfLHP/5RTZ/tP3BAVixfoeXTHJs6s4f2zWCQOSROSKrQ1NjZM2e0/nkwrUa/OtSwadcBmjy//71MnT5dGjVqJDXS0uAvpqqkKGNAsgbaNhF5UwRfMYxOSYE2CNLQbNrdu3flDsougt8e5q1Zs6ZUh3kvprl1Cz5s9u2TJUsWy7z589UkWrv2HeRV+NaZNo0aNl2haVRP/cn85b/+U1auWCHXIVf11DSYfBsl//7v/y4DXxkIDaEaQbsHYKonG5WH5FiQh9jTJNp2+AD6z//8sxI2BUUp0hRaLuNAhgUNm86aN/iwAWGze7fMhYYNySQSNhkZ7dSfEH3+0CQafcpoJ6LBoc7Qk/zMy82T09BqWgz/N3//8GMlzEjYDB06BETUFBkFoqx50ybAgIRSVSXjdu7aIZ/Dh80q9OEt9C992JCwef+DD1TDJiUlBdiyLpI0MFPHC7ZT7zVWr/3DEXAEHAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBGobAg4YVPZeszldQQqFQLUs+DBT2ymhxu9D6bReMk0USqYxcq/UwBtk4OyDGa0vlv8nezbt1/JCppD69u3r5CQ2bZ1u1y8eBH+U2qqH5j3338P4RAlbM6eO6+EzmeffSa7QDbQ5Nhk+IHp0rmLHIGps+3bt6kGTZ26dWFObDTMqf1OekJLpwZMnnG7n1owZ0EyUNPlm/kLlHDIy8uDBk4beeOtt2QGNH1at24jdeD/Jji615YhJ5sC3y8gZDiwBtNcAqKmECRUPkyR3VHShqmpoZOenq5kB32y0BTYPiVslsiCBQvk5Kkz0rZde5k1e7YSVT26d5M66XXkwIGD8vlnn8oKEDYkle6jPpJN/wwfNkPh36U+SKeqIDQoh6KqRAawpTyQ7969QrmVnS2bN/+gGjabt+6APbEa0qJFS9V2eff370j3bp2DuTeYnsvKCoTNl3PnyhKQLnkkbDIzZTy0mWbPniOvgCRKA/nEah6ApArEUBUQMIFIybl9G76C4MMG5tTow+YINGyaQFNpEPwJTYIpOmo2tWzZAsRTNaIHHG6rD5tP4WOIhM3t27lSt15dGTpkKAgbmkQbhrJTJKUaia5A3Cj6aBzbF05++uEIOAKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCFQ+BJywqXx95hI7ApUGgUDEhE/ViUgQNthcV182aEpEKjAVyY6CuwVKxmzevFnmgijYuHET/JakSteu3aRdu3Zy6tRZ+LY5ouQHRobOpQAAQABJREFUiZPx48fJG6/Phj+YXkrYXLt2XTYgD02qbd6yRWrBbFjfPn2kRcuXQL6cUbImLy9XTaBNnDhJXnvtNenQsT20ZaDlARKAZsPOnzurpMiiRYtgEmyPXL16FflbyJQpk9WXDk2pkXigVkj8iJM1JGwKYF7t3LlzIFfOSlZ2lspcq1Yteemll0BUtFT/PNSKIVFBnzs05bZgwUI5CTlJ2EyfMVNNovXs3h2ETbqSVYtBYq1Yvkz27d0LIihPevXqKe+++y7IpzHSuk0bkB/VQeTAd439I+bAFbcgjfKhoXQW5uLWAtsvZdfeg5Kalh4ImzGj5ffvviPdunUBYcO+CITNHpBeX375pSwlYZMXCJux48bJnDmvyisDB6m2ELvwPjSWSBTRulm1ainAJhAwx48dE/qw+eSzuerDpkmTxupXaOLECdAOGimtXmopaaqlI4Gw2blTSNisXLkKGj25SowNHjIYJtE+kOHDh6GPqkm1SDMpkGKBoOH3i9o2xXsk3jt+7Qg4Ao6AI+AIOAKOgCPgCDgCjoAj4Ag4Ao6AI+AIVGwEnLCp2P3j0jkClRiBBGWANuh2upIGoUFkEXiGQ01pkWDAzn8htUCgcbJ3777gfH7FSpgpK1RfJjSPdetWrlwHKUNypW+fvjJz1kwZP260tG/XVgoL70kONvm3bt0qf/7zfylxwzLpo6VWzVpCMicrK1u1W6itQ/8wk6HpQQKF2iokbUi6XAdB89PPP8ny5ctl9erv5eSJk1K/QX0lfqgZMnnyZPWrU60aTJ6BJWCeIhA9rItmuqhhwoNEz5o1a9Qnz6lTpyKSqbVqigwePFjatGkL0qaOmgI7dOiQ1jdv3nw5YYTN9BlaV3eYRKOGzYUL5+VHmItbunSJbICPnuvXr0HjJUOmQJ5p06ariTJq/kSqJkqgULbgy+aBZEO7hv5rli9bJt9DrhOnzkn1mnWkadPmMgbm5qip1B2EjUiREjDZ2Tdlz5498vWXX6vGUx7M1WVkZIAcGgvtH2rYvAKyJU27km2H7Tgl4th++qXJQT8eP34MJtGWQMPmCzkK7ajmMIk2ZNhQmThxomoFtWzZXGqgDJJfJK527tgRETYrI8IGGjbQnvqnf/4nGTFipPY7MSY7EwgbVIuqeU2NJyds+M3zwxFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBBwBR6AyIsA9Nju532Un9/i4v7h48WL4iG4mb7755kPNq4KMiR1XbuoxAx1fM4NtpDG0ZPG4h0rzCEfAEXjOEEgSNqRrlEVIjBhsavyGJrtCevp4KYD5sOMnjqm2CZ3dnzl9Vk2VVU9LU/Lm7t170rBhQxkL4mDOq7NBpPQC6dBIfc0UwATZrl275aOPP5Lvv/9eSRr6kUmpmhKZJSuEmTRqy0xRbZl+IG4aNUZemhHDkEZNkdyc26qNs379OvkK/lvox6V69VRpCWJnyJChMgskEX3nNEY++nBhPhI2PJRMQFtu3boNE2b7ZeHChdDWWSmXL19COpGOHTsg/ywQFpNgWq2VapBkZ9+CGbgDIGKWyTcLvgmETWY7aNgEwqZHt+5Sr05duXnzhuyHZs0SECDUtDkLTaA6ddKlZ8+e0DQaL+PGjYe2UAfV3CHiJKGKYNqMRBb95Bw5ckTWgqhZCx9Ahw8flhtZOZJWq640g9k4EjYfgLDp0aObascQh2xoBSlh89VXsgxmzXJycqVN27YwnzYWRNlsaNi8IjXpN0f7F/2HvmO/YtjXkzgqYYO8n37+hRw7flIJG5JVE0F8jYSZuxYtmkt1Jb4E5tpuyc8//6yEzapVq1Sjp379eupr6IMPYBJt+HCpBg0bOOqRB5ywCDhr1+sQUsMnxOtD/3AEHAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBCoNAtxntFP3vLjvhdMJm0rThS6oI1BREQiDSyBmuIUeBheTlgNP8uB1uCdxQpLhwsULsm7dOiFhs3nzNrly+Qq0K6pK0b0imNyqLpkwjzZz5iyY5pot7TJbQwOltmq23IMvFZpM++abb2Q5fL0cPHhINVi0eGpmQI4O7dvLH99/X4kTatekIy/luXfvrtxD3UXwY5OXmyc/QRvlb3/7b9m0cSPIngJJBTnTpUsXJXvoU6djx46q+UM/NIEmIHEDPzwwVXYafnA2It+3334rW2CajflrwudOt25d1YTZlClTYVatscbdukWNIhAxaOu3i76Vk/BP0yYjUwmbKZOnqI+dhtAuupOfD2L8NDRklsPHzjy0bb/kw78OyasBAwbIJJhsGzpkiJp7o5YLCZvCwuBDh/L8+MOPshpEyM4dO+V2To7cu19VatSuK81btAD5NVoJm169ugPHMAkEDZvdSlothWwkVFpBK2gsCZuZIGzgw6Y22kTtGJ1IQNiw/0JfPlDi6/jx47IE8n72+Zcg4U6CHGoGomegmpcbCzNuzXGfwvz4lw3tp59/CoTN6tWBsKkHHzaD4Kfnnd+9I4NA9BDrajD7RjN5VUHCVYOmFbWt9PuFT+1hZ2yAhB+OgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgClQ0BI2sYOmFT2XrP5XUEKjQCv4awYR6Y8MKAlHXzpuzavUu1U1auWAVNjRNSRD8pIFxq1qgJfzYdlLCZPXuWZGa0klq1oOmhbPMD+Gk5p9o1S2H6a8uWrXLlyhUQMUXIWRUb/tWlO3zC/Md//IfMhAZLOjRUqoMECOTGXWWrqe1D1vro0SOyAMQPtT1IPOSBMGnQoKGaQ2MZvXv3hm+drtCUaS21a9cG4XNP6zoGvy3UTNkJfyw07XYR5BPJhYyMtjDrNUI1e/r3HyD0X0NihYQNiSWaKpsPDRsSNm0z2yPddJkyabLK27BBAyWSblwHLruAC8y1rVnzPfzaHAV5UU3NvnV8uaN06dxZ2oGQIhmUmloNpEmuynTixHE5eOCgnDx5Um5cv6HE0v0qKZJaMx3aSc1k3NjR8qc/vQ+TaF3RB8Efza1b1LCBDxv4u1ny3RLJpQ+bjIygYQOybMCAgVILbeDBfjOeRLVccHM7J5hEWwINm09gEu0YTKJRm4cmziaDXBo5aqS0BFmUCsKFhE3u7Vytjz5sloHkuYnvAE2rZWRmqhm5rpCtXt166o+odds20gym3NKBOzre+D4B16Tfgwr903DhHAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBByBUhBwwqYUUDzKEXAEngQCxQkbkinUgrCDg0/ySGrYwJBX2LyHhsvJEyfUB8y8ed/I3n37pODuXZRQBSa/6knXLt3ULNf06dPkpZbNJC0NprKi8q+DkCBZsnjJEnV4fxYECMkUaubUB/ExENoo//Zv/wYzYuNAmEBDA7KxVmrv0BSYSooP+qDZvn2brFu7VrVkTsM0252CoClD048dOnSQzp07gbBpo4QNzaIxD02PUcvn7NkzalaMZE2TJk2EfnMmTBgPjZHBmodkBA8SNurDBoTNPBBEJ6ENkwHSZeaMmTKZhA2IivrQsAEroubiLly4CE2Un2TlqpWyfds2uQZfNtSkqQXygvVQa6UhiKVqIGzyoIFDnz80yUaihmVUQb358EdTcBfaSmm1pAnaMm7sGPmnP4Gw6R4IGzWJlpUFwmmPzJ07V5YuXir50BLKyMiQ0aPHyAwQNgOhKVMTZup43L//ADga88+eeBARNtCwgXYOfdiQsGkOwmb48GEyZeqUhEk0asmQ8KEWEnH4/LPPkWcJiK6LkDNfTby1BdnFvA0bNZROnbrA/OYA6dmrJ0xwNtf6+XUimk7YKBz+4Qg4Ao6AI+AIOAKOgCPgCDgCjoAj4Ag4Ao6AI1AJEXDCphJ2movsCFQOBIywwUY6CJFwWEjeoCRhw3RMFcibAhAjly9flg0bNsgnn3wm20FQ5OXlQhNGoBHSXAa9Mgjm0ObAb8s4aVC/rmpisB6ed+AD5+zZszBHtkg+hC+bY0ePQTvnPrQx0qGZ0w6Ew2h1zNWvfz+Vw/KxahIVNIlGAoEkyM0bN1Rbhv5wfty8BRoqp+QuiCPKT82WNBAW9GPDMnjP/Ldv35ZcEE5sSw1oA7WAFkmPHj1k1KhR0LAZLq1atVbtGpoS45GdnZ3wYUNTZyfPnJZ2HdqryTaaROsGLZ569eqrTPdBCt0lNpcuq7+XTZs2yY4dO+T8+fNyF/KyTraVMjIt5STJxbpqwXxZy5daShr88Zw5c1au3sgCqVNTmoLgGTduLDRsPlDzayReiooKJSvrhuzds0++/upL9a9Dkoc+bEaiHTNAJg2khg3K5BHqYR8GLSnil3P7lpwA6UaTaJ9+FnzY0GfNCBA2U2ESbhQ0bEgu8aCG0z342jl16qR8M/8b+OhZov5vbt7M0m8EsaXGUL169aR7j54gvibIJJBZNEtH7PGhoZpE0xL9wxFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBBwBR8ARqFwIcI/NTtuzZOg+bCpXP7q0jkAFROBxCZtA4HDvnaRDIUiTXPhZ2QYNkg8//Eg2bvpBboA8oWmzdjAXNnnKFJkyear069dX6taprYSNgoBi6IfmBsxp0YfN//z9f2T37t0gUHKlTp06qhUyc9Ys+GwZC9Nh7TQLCQ1Vz0BeEg0PQLpwEOTgSNLm6pWrcvDQQZgi2w3yZi9MpR2VCxcuaJnU3FGyAoIbAcM4mjtr2bKlEgo0n0bChubT2rRpA5856dByqapVUgASNvugQbQMGjbqw+bUKfiwyYAWywy0cYr0QH6aAiMu+h8yUnOGpA3Nrx08eEC1V06D6Dl77hzkvSI5wI5tYF1NmzaVNtACokm2Vq1eAhGTJWvXrpPDR49L1dQaqrkyFoTN+++9p9o8gXQpUu2gvWjvV19+qdpKJKEyMzNgEm2cTJ8+A9j3V5NoSphAOu06CGgm1fJycwJhs3S5fKKEzQlpoRo2QxOEDTWVqGlE3CnvjRs3QRLtkfXr1+PcoBo3xIfm8FgP+7ALcJw4YaLMnj1HunbrRkhQNwmb4MMGt344Ao6AI+AIOAKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAKVDgEja2y/0UgbJ2wqXVe6wI5ARUOA2+jcvKfyQ9jKty19SspBJ3mEtOHe8lFTpEh9u1DjQn3RXLumWhgdOrwMp/VT1LRYBnyZ0H+NmhdDVmpqsGxq2WwF2TN//gIQNruUwCF5MWzoUPUh0w0kCM2GUbQgH0mDSI6IPKA8JGE4IN6G2bIzZ88pYbN161YlEmj+jMQICRqWYQOp+pSBaTISNANgfq1Pnz6q2dMA5thogo1HvP3UyKEZtY0bN8o6EBVnz59VXy+joAk0cuRI6dSxk9SB7MTTaB7mp1wkoi5duiTHT5yQ/fv3yf4DB6A9c0blQiXSoGFDEFyZ0NLpBvNtnaUxTIrR7NjHH38sW3/aKSnV06B18xJMoo2Vd9/9HXzgdEI199W8WS4IlyOHD6tJs7UwC0cNp1YgfoYOG65m0di+NORXsgTShW4OhA3beOdOnmo6kXhZuGiJnD59Rho3bgxzZn21XTSp1gT3ZhqO8pIgY5tIYC1dulR+gmbVxYuXYL6tQL8+7MN27drDp81wJbM6vgx5o4P1B50li/HQEXAEHAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBGoPAhwz89OI2sYOmFTefrQJXUEKjACRr6YiEbcGEnC+DAIMdRrhJbqPgYoEiKXL12Rm9AKUTNfsIlWq1ZtbPw3kQYgXGrXqgkSBDQGTHHZoWRG0X3Jys7CZv9ldV5PE2v06dIIBAFNlNWtUxcmtlJZeyAcAtug2jXU9qA49PViA2Lh3UIQFvlyC+TK9ev0CXNZrkCThddG2rDemjARRmKGWi2shxokjRo1Uh83JHJ4MB0PG3RJ+LAMahCxvHyQTdWhodOocSMtq3atWmpu7QHaVAhTZ/SVQ/mCRg/82hTcRZ58JTpyQarQ7wv9/TwAoVMNbayJsmgOjlo/NPdG/zd/+/vfZev2nyS1Ri2YX+sgEydOkDdef1064pqEDSBFQLly5crVK+r/hhNDGky80Q9Qo0aN4VumDtKBgEJ76MOGfmsII9tFDIvuFUK2AiXLLl6+qvIR8wbwxxP6rz4In+rIoHDoB2UmHtSqoZk3kmK8Jj4k49iORuh7ai8R33Ro3CiO7EXUGysqWahfOQKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgClQABI2sY2t4hQydsKkHnuYiOQMVHgMQECRmTNLmdzoEmHGUQNnycyBc8k5BcCbv7JARCbiYLZI2Vw4378OzBg6D1wgGNm/3MWxVMRArNkWmi8Byf+oy5wqB4X8swGZMDJX3cPFBCIS8/T25l35Jb0LyheTKSKExPUoR+VurWravmyOjfhlo1JctiXSRcGM/y4wclheh66DM67oGMd+BD5gqIIppjy4LJNxJADRs2UjNjJFFq1EiTFMQRJ5qOY7stP6+pdXTl8hXZBE2ejz/9RHbs3C11QJ707T8AGkuTZeL48WqyDVMAMCKOAVPSIFWrRLJCkqKEvIyHoBBP60E8b3kykqSNPmcboftC7PiPbdbymIzpEWeHYVEEcooEXWHh3UBIAW/mrwmCrgZIsbTqNdD+FGBIbENBIXDSxrD00BFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBBwBR6ByIcC9MTu5h2anEzaVqx9dWkeggiIQiIawv6/b6Qk5OdiEIwxCuuuvEWHTP5GQO/o4i7jZj+JoPqsKyAMeWi4+QlHBB4qSASBCyAFQQ0c1P5gAJ0kblsZNfl7wWj+RzogCjQnJE4NjiAt1GmFj6Y0UMfKFYZygYd74Ye1mfrtmGB+ISdiAaoF8EfkQyUftmz3wx7Np0yY1GVYNRBBNnQ0fPly6dOsK8qZh0BpC+qKIrNFySVjhJLlE/zvfr14tCxd+KwcOHZZm0FSZPGWqzJwxXXr36ilNYMpNMSQ4wIuYkXQhwULAKBWKV2x5oTLqc82gTdW24QrROEJLoK7EGEaEA2UAdZYWpYuiWbjWETRtWBYPI8RSU6mRE2SgBlSCEGI6/Oe94aoZ/cMRcAQcAUfAEXAEHAFHwBFwBBwBR8ARcAQcAUfAEagkCNgeoe6vRftc3OtywqaSdOA/KqZ9AViObnIm91x90/MfBdfzAwFqeOj+Or9h0RmAsY143mHMiR3coOdpR3iosVA70VKQQVMkP5A40iZBimR50cZ+FEGCgHKE22KVokotDM9DaGUEOZN5VEskkTbIGP8dMYa/pThpUNZ1yF38k2lBk+BfQEzFQX0kTq7BPNjPP++QFcuXy8YNG9VMWLt27WTUqFEyAr5u6FOmTmQijOlNLg7oNE124fwF+fGHH2XVypVCPzwXoK3Tqm2GvPrqayBsZkjHlzvAr0+DgA/biDKqgIAKJIjJGWGPWybR/uBn+K+JrG+ZUjEnpEr4xDBPYFg6YcN+oNx2hGsQNJCHGGnd7CQWyZAB5YmuNcI/HAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBByBSoSA7ecx5D6XnU7YVKJO/DWi2oYq8+qXAGFsKzXxRfg1ZXseRyAgwF16bNcX20RPfsvi38Him+wh38Mo6s58Ijqx368xHMBYF/PGDyNOQr0ln8ZTqqCJiJDy4b3/QBRYMpOb9XLQ1N9SJIMNppamZB67j4eWFiUFoiPCj8TJA2gYUcNm186dsujbRbJ4yRK5eOGi+oIZOGCAjB4zWgYPGSIvvfSSmmJLgbkwgAKTYoWSeztHrl27JkeOHJF1a9fJD9DQOXP2rBTgWYdOneWtt96WadOmSivkrVMnndlQH/zk4KC2EBRZYofdWJh8VDwGGAIL/uP/sgkbihnwTpYUrkr2p93T1J1hpSl18uIYVlyCkuX5vSPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgCjoAjUJERsP1FhvH9RSdsKnKv/UrZbLOTHR3eVudGaXKD054ztLfYf2VVns0RiBDgRrzu1kf3D3/f+CD+PQzpS9vAZ954/qhIDZ4sYVMWgcD6Iz5Ga43Lzd+NnXzIZ/HnmiGKt2v7zdk9Q+YxwoY+YLRMkDVseU5OjhyDSbMli5fI/Pnz5fjxE5q1UaNG0rHjy9Kvfz/p27cvrjtKg0YNhf5zsrKy5MjhwzCltkd2w5zawYMH5RKInsJ796QetGn6wH/NG2+8KWPHjIFJtfqah30QkA5haItWFX0U7wt7kuwdxqAPFazQ/w90rLEUfGa52ObYTTJa2x671cuAP7HlbVReuLG7kln83hFwBBwBR8ARcAQcAUfAEXAEHAFHwBFwBBwBR8ARqBQI2P4iQ9tf1P1CvCx+6tQpWbx4sTRr1kzefPPNh9pTBZkSu2x79uzRDP3799cMLIQHQ0sWj3uoNI946ghYP1hFdm/9wniLszR8Fn9u8R46AuVDgEMET9uot7D4d634d8zylKyBeeP54885gIXhKDkq8fnD39+QKp43dh1ljoav6EGUA0G4SsrABCa7/XYstHiGFhdPz+t4PO/toFYKSRutkfWSuCkKZs2uXLkiWzZv1sF5+/afVMuG+erVqy+tWreWjMwMadGyhZpGq5aaKrkgec6fOy+nT5+W8+fPq5bOPZI19epJpy5dZeSo0TJm7Fjp0b271K5dS6pRMwc4UDZIjvYFE2TEhHIBUVZX6vHQk0Rn4InmRzYFMSBphRTH22IpRvF0ySdWUxQWC+xZMrVfOQKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgClQEB7ofZyX1FO13DpjL03q+QMd7ZzG4bouz4+GHpXNMmjopfPz4C3HC3TfdHaKcUKzieJ/kgEAXJ72nxvfwkYUNmIP6s5Hc7WWJ5ryAPB8pE8qQMFsU67LdkcQx/qe7S8jAf/c+QsGF++o95AAadps3u4WRI8mXDhg2ycsUK2QkTabdu3Za7hfdUhqKioiAL7ZhFAzyx42+Z5s2qVaumJtPaZmTI8BEjZMKkydKpcydo1zSUNGjkpFRNIYLQwoNJNOSvinvKAGFCuSoXpQyHIUNUipM5rJ+x0aGEWphwLCoZcvJJ3hFs4l0sLva47EsrxMKyU/oTR8ARcAQcAUfAEXAEHAFHwBFwBBwBR8ARcAQcAUegoiFg+/IMuTdopxM2Fa2nnoA8urGqe8/JzrYvAIu3zWWL4wZvPF5v/MMReCwEuO3Okwc3/MOV3sV245PRdmVhLD3ysww74mWxDtOwiddj32nLUzLkdz1+lJ4+mCVL1p2UIZ635LUNqowvvVxIXaJ+KyPQNZCNgzJFRDoOyiRuSMjcvnVLzpw5K/v375O9e/fK4SNH5ShMpV28eEnu3Lmj5fL3bq2rCi2ZmjVqSJMmTSQzM1M6d+4sXbp0gYZNF2nbrp00gGm0Gmk1JBVkjpIzqI86PmxpGAeSbQ7dZiUT+XDNFJDWmqB35SdsYtkeuuTE9FBkGRGW0MIyknm0I+AIOAKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCNQARGwvXnbW+S+Ik8nbCpgZ/2jIllnWzns6PiGsd3bl8EIG0vvoSPw+Ago2xBl+2XChpv/SFVGNUYHhM34ONcRyBrmSz7j9/nRB2orpaqQLZmX5shIS4Syk/GPLjtJxlCOsmSJ//7i5RGHB+bXheSJycmQ2jcwj5afny/Xrl9TbZvDhw7JAfimOXnqtJI5hdC2oZYOy68KTZvq1dOkbp060rJlS+nUqZN07dpN2oGoadSksaSmpUkKiJpq0KRJiUhak4tYKIET4ZrERgVRkSkrDyJjPaQRjCm3hk3IUfon8Sv9ycOxltDCh1N4jCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDgCjoAjUFER4L6cnbavyNAJm4raY79SLutkZrfNY31jH18A3pOcsU1ahry3dL+ySs/2wiOQ3NQPUJSPsAnkSCAB4hBW0c3/5EZ8+b+fJeVIlmrf+WSMlW9h8gnTss54vfH8pcXb8/izZIllX6nECcLGyB/EAgOSFyRs7t0r1IGaYV5enuTizM7OlqtXr0oO/NYoXwKNHJo0q5NeR+o3qK+aNOm1a0vNmjUlDRo1YGPk3v0HMJOWqto1lEgJKsQVHwNQN9uuIvMzienjEjahDuYv7/HLhI3hrJQRAYokLW8Nns4RcAQcAUfAEXAEHAFHwBFwBBwBR8ARcAQcAUfAEagICHCfy07bi2TohE1F6J0nLINtarKDrdMZWsez03nw3g57ZvceOgLlR4Cb8uE7ZRvo+LrhiLb9Y9+z5PY9t/95l4xhDh7PhrDRmrQ+kzO6KRbEfyP2u9Kc8TaFxupvLZ6+WEFl3CgKCcIm+Xs0WGgajcRK+H0CZdzfR330b3P79m0pKCiQKurDBq2ATGnQoiFJQ7NnKSnB1CHFYx72UBWYTCNBE7Rp8AD/tWw80/ZBhNAGk4X9w5OfIeQTo3T0Afu5FA2b8Cwq125+MXw0aZPsA6YLkvxikZ7AEXAEHAFHwBFwBBwBR8ARcAQcAUfAEXAEHAFHwBGoYAhwn8tO3Z/DXhdDJ2wqWEc9aXHinc6yeW+EDTduLa74W/Ya7R+OQDkR4EZ+aYQNs9vGeigqbPnbNfPEY0J8RSJsKFEgBnjFQZRhOJLxSVKCvy874s8tLh6SbwhECnDANZDCP2i8IRGLCWSNacZprP5+tQxktrrot0YzWeHMi38sjVf0haN30MAx0oWEjZlAC/WFCYJFqNwUTg+2x9oUcvNJKDuk0MofQdgwlclqOZIhSkKBxZ+HuGSa5FUyHdMESZJP/coRcAQcAUfAEXAEHAFHwBFwBBwBR8ARcAQcAUfAEagcCHCfy07uc9nphE3l6L/HklK3VW2PFTnZ8bppW2Iz2UiasPEZbdQ+Vk2e2BEgAvyyFSdsdBNfwSluHi1OLITt/9gXNSIGngZho6JEH/FNf5PTfgNMEp6HATMeHy+D18WfMb3l5TNNUTJLsXvmT1JWxAEDs5WBwvi8eB2WncgF3PRT6wrPTFnH8vER2wPLZ6EuhJSNZA3/se/0CvdMZ6eNDaGeUBfThtPyhjopd3iEMCGA5bE0SEKAHjpQFsUox1E8P/MxYzkzl6N8T+IIOAKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOgCPgCDwrBGwfjqHtAzJ0wuZZ9cAzrMc0aFglO5x+L27evAkTSSmSnp6uZpOqwfm4adgkNnfLu3P6DNviVVUGBLgR/2sJG2ufkQHc8y++EW/fT0tZdpgso+w0ceIg1PNw+SQuaIoMspTJBzxMNIRB9lF54lKF/DB2ppQJn/BKP1E3L5VUiQRQ4oSPcbAeGEbjld4HISN5EKWtYr7Y4zhhw0yqW4ckyJVIT22fkCdkVFyMgNGaGM/TcjEyqidkYUQkQDwiRPOTshc/IrmLR5Z6Vzwv82lLS03rkY6AI+AIOAKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAKOQEVGIOwlhheouc9lpxM2FbnXfoVsupkb+ahhdnbwhQsX5PDhw1KjRg1p166dOiTnNQkb+2LYG/W/okrP8sIjgE14kgw8IoIh2rXXqGJ79Nxjj46gvZG8i9iCQCHETGyFjXlL96iQZEBJQuDh9MmNfw6EwSzgw6lKJ2yK5304V3j+MAmRzMc8yeckbAJWoawYPGQ3NJLtZ36evC66XwSNmaLwnHjDhw3NoilOyKIl8gPXLELzs4xQRRQfarZ8TGcyajlIHepDpoRQLIEn5A+1hIcanexNipTQtMFl/LA6knFJLJJx5bnSSsqT0NM4Ao6AI+AIOAKOgCPgCDgCjoAj4Ag4Ao6AI+AIOAIVDgHuk9nJ/Tg7nbCpcF1VXCDb4AybqOFZaXGWi89Mw4Zm0HJzc2XPnj2yYcMGSU1NlX79+kn37t2lWbNmlqVYaF+MYpGxm0fVHUvmly8cAty1jx+JXf54ZLHr4jmK3wVCoFjyctwUL+OXM/ySjI9b3i/X+HCKGJGCh4+SyH57pEbsWsvjgM5/pWVGEyw+tCYkYn5ePWpcsTosf5CdpVhFFiIqDlUxrZyQ68l/xup+8oV7iY6AI+AIOAKOgCPgCDgCjkCFR8DW6xTU1vUWx3te8yz5N348jTXS0tm9h46AI+AIOAKOgCPw9BGwudrmYZuznbB5+tj/QzWUXExxs9Y2R0suyqwiI2zu3Lmj2jXr16+XZcuW6WJt+PDhMmHCBOnUqZMu3Eyzxr4gVqaFVl/iHgs+OyzO7j10BByB5wuBkuPP89U6b40j4Ag4Ao6AI+AIOAKOgCNQeRGwtXq8BRZnf6vrPd51KuuFuHg6XpfMHy/brx0BR8ARcAQcAUfgySLAeddOzsN2OmHzZHF+qqUZecKOtA60TmXFFs9O5fMbN27Itm3bZOXKlbJp0yYhgTNw4EB58803ZciQIVKrVi31axMXmvlYDg+7tnJ5z4PlM87IHo30D0fAEXjuEIiPBSUbZ88s3sYHu/fQEXAEHAFHwBFwBBwBR8ARcASeDgK2Frc1uN0zZBxP2z/Qa8THn1Eqy8NrK8f/1icafjgCjoAj4Ag4As8GAc7Fdtr8zdAJm2eD/xOpJd6BXHw9oCdxO6K3ZpiGptDu3r0rJ06ckO8WfycrV6yU48ePS2FhoWrWTJs2TcaOHSsdOnSQ9PR0KyGxSGOELdh4zTLtsHjG2RfJnnnoCDgCLw4C8XHBWm3jg9176Ag4Ao6AI+AIOAKOgCPgCDgCTx4BW4vH198WZ7XZPdPEr/ncXvKM/03PNJaOaeLPeO+HI+AIOAKOgCPgCDxZBGzuZWjzLkMnbJ4szk+1NHYWD3YcO5LEjHVoSkqKPmMakjXUrtm+fbt89NFHsm7dOiVrmK9p06bSu3dvmTRpkppGa9myZaKM+Bcjfs06WC5D1mP1myxasX84Ao7AC4UAx4OSB8cGPxwBR8ARcAQcAUfAEXAEHAFH4OkiYGvx0tbffBY/LQ1Dnvzb/t69e2oxo1q1ahpHaa3M+LXlebqt8dIdAUfAEXAEHIEXE4GS87XNu5yrT506JYsXL1Y/9LSWVfKogsyJnTk6sGeG/v37awYWxIOhJYvHlSzM7389AoavhVYS73kSd5I4JGvYTzSFtnr1ajly5Eiib2rUqCHNmzcX+rKZM2eO9OrVSxo0aKBETGkmzlhmvG9Zp/Wv1e+hI+AIvHgIxMchHxNevP73FjsCjoAj4Ag4Ao6AI+AI/PYI2D4AdgSKWeCwv+Ftr4D7BLS4UVBQoEKnpaUJyRq+kGkvZVprLC/vfZ1vqHjoCDgCjoAj4Ag8eQRsnrb5nPMuTydsnjzWT7xE6zQuwvR/jKBhZda5vM7Pz1dCjUQNWbh9+/YpgWOLLoYkbXr06CGzZs1S02gdO3bUOBI2PO1guTzsy2LxHjoCjoAjYOMDkeAY4Ycj4Ag4Ao6AI+AIOAKOgCPgCDw7BGw9bn/r276A/f3Oe2748KQ/2+vXr8uZM2d0/6Bdu3b64mb16tWLvbzp6/pn139ekyPgCDgCjoAjYHM3Q5u/GTphU4m+G9aJ1nHsPB68J9HC+ytXrshPP/0kS5YsUe2aS5cu6Zs0Zj6N6fkmDU2hDRkyRGbOnKnaNvXr1xcu1uJaNvH67EvD/H44Ao6AI8DxgQfHBj8cAUfAEXAEHAFHwBFwBBwBR+DZIGAvcrI2W4tzL0DX51ia491cFYRxPLkXcPPmTdm7d68sXbpUzaHx5c2ePXtKvXr1Eho2JTVtWJ6V/2xa5rU4Ao6AI+AIOAIvFgK2925zru2/c/6mhTM3iVZBvw+66CohW2mdySS0Q3v48GFZsGCBmkM7cOCA5OTkaG4jbJiXC7HatWvLyy+/LNOnT5fJkydLZmZmscUaM5Ws2xZrFmrB/uEIOAIvHALxscHHgxeu+73BjoAj4Ag4Ao6AI+AIOAK/IQK2H2CbOhRFCRuY41CyBnzNg/tJP7Q0g0Yz6d9//32CsOE+wKjRo+VlWNvg3oCRNfEyf8MmetWOgCPgCDgCjsALgYDN6QxtDmbohE0F7n7rNIrIzrIjHm8mzEjW5OXlyebNm+Uvf/mLbNy4UbKzs5V0sY5mZ/Owsho2bChjxoxR02ivvPKKvPTSS6p9Y88tZH3xg/H2LB7v146AI/B8IVDyt19W63w8KAsZj3cEHAFHwBFwBBwBR8ARcASeLAK2H6BrcG4T0HJ69Dc743itBE4Ucl+A+wPz5s2TH3/8UZ8NHTpUpk6dKiNGjFDfxPRpY4ev7Q0JDx0BR8ARcAQcgaeLgM3pDDn/2umEzdPF/R8u3RZeLKgk4WKdSrLmxo0bcujQIX1rhupS1LRhfDy/pbcFGH3ZtG3bVhdpc+bMkf79+0udOnX07Rqm5Vs2PKwMy8eQcXavifzDEXAEnjsE7Lcfb1hpcXxu44GF8Tx+7Qg4Ao6AI+AIOAKOgCPgCDgCTw4BW5OXDG0tbnsHubm5ujfw3XffycKFC+XkyZO6bm/fvr0SNtS0oeUNmkZjXstPSePXT05yL8kRcAQcAUfAEXAEDAHO43baPMzQCRtDqIKH7ChbdMUXTjR1lp+fr3btqOK8fPly2bVrl1y7di3RIna85bEFHR/Slw1Jm+7du8sbb7whY8eOVQKHKtGW3gqJ+7ZhnJVTMp2l99ARcAQqPwL2O7eW2L2FFs/QxgKGdh1/7teOgCPgCDgCjoAj4Ag4Ao6AI/DkELD9gfja3Nbh3Cfg83Pnzsm6detk0aJFqmWTlZWlfmtpbWPYsGEyY8YMDdu0aZMwi0YJfU3/5PrJS3IEHAFHwBFwBMpCgHO4nTb3MuQc7j5sykKtAsTHF1/sLFuUxRdiXHSRpJk7d676rrl+/brQTq0dlpb3Vl78S0BTaKNhv5a+bAYPHizNmzdXMsfqY1oSNnHSJl6O1eOhI+AIPF8I2O/cWsV7O+Nx8fHEru25h46AI+AIOAKOgCPgCDgCjoAj8GQRiK/Juf7mEV+7m8n0HTt2yOeff66kzfnz5xP7BGZtY9SoUfL222+rtY3U1NSHSJsnK7WX5gg4Ao6AI+AIOAJxBGw+Z2j7aQyfe8ImvmixhUwcmIp6HZebMlpnKYnyAL5oYKeWb83Qb82xY8d0AbZkyRLhgowaN9bR8fZZHBdiPMxkWt26dVUNety4caoW3RGOB2kazfBiSG0c85djZZaU0eIfFVqZj0rjzxwBR6DiIGC/83jIcYjjz927d/WPPj6rVauW1KxZU8cJ3vM0khd36vjUWsVxwMcCQ8NDR8ARcAQcAUfAEXAEHIEXHQFba5cHB1tH25rb1ta2V4CVtu4T8M3ctWvXyjfffCO7d++WnJwcXcNzjc49Aa7fe/bsKe+8845a22jWrJmu5608k6U02eIyMF3Je8v7a0Ir69fk9TyOgCPgCDgCzycCpc1FZbW0Ms0jNpczpNx2PteETWmdWRk67VFys8NItNhmKU2fbdmyRb799lsNL1y4oBuldBpYvXr1RDpTiWa82aelZg7Loq8akjaDBg2SV199VV555RVp0aKFkjR8xgVdaYQNfxilyVrWD8biK0MfmKweOgKOQPid22/9PgjjontFcvv2bbly5Yr6z+IYQU29pk2b6ljB3zjTc/ywa+LIOPv9M7Rrx9gRcAQcAUfAEXAEHAFHwBF4URGwdfbjtD++xrY1Ncuxv/tpdePHH3+UpUuXyvr16+XMmTP6jHVw7c6T6Vu1aiUzZs6Q6dOmS69evaRRo0bF1u8mU3wdb3EME/FU8sFLpU/qsDY9qfK8HEfAEXAEHIHKiwDnmsc9Kss8wrbZSZntfKEIm4rQWaV9yX5JrngepuUijCbPsrOz1WngqlWrZMGCBXL06FG5c+eOEjVceJF0YZqLFy/KrVu3pLCwUBo3biw9evTQjdQDBw4IF3IkbfiGTUZGhowZM0ZmzZolffv21TRcyPEZT9ZNWRj+ksyP+0Py9I6AI1BxEeDvnpOFhRx/jhw5om/sMSThO3z4cB03SAjTxIKRNZoPf71VrRL+MLRW2hhiY4rFe+gIOAKOgCPgCDgCjoAj4Ai8SAhwPfy4h/1tHl9L89r2CmiFg5o1tMJx4sQJ3Q9gHUxjhA3v69evr1o2U6ZMkWnTpklmZqau4/mMh9Vj1xpZxoe1g3lsrV9G0oeimZfn4+Z73PQPVewRjoAj4Ag4Ao7Ab4iAzX82B9oc6oTNM+wUgl/WUdpCo9T0eHOFb7eTmOFbMps2bZIVK1bIDz/8IFevXtXFVYMGDaRfv37SoUMHNZe2d+9eJWZovqh169YyCnZquam6Z88e3XTlW/Ksi3FUiX7ttddUJZpv11BNmto1PClj/AtUVls83hFwBCo/AjpaxcYs/vZ52h+BJHy/++471ezj2NK/f38lbbp166baNjSPxoN5eNikozclPkob/0ok8VtHwBFwBBwBR8ARcAQcAUfguUTA1suP0zj72zyel9d8serSpUuybds2+frrr2XDhg2Jlzfj5fPlKh60wNGyZUtdx8+ePVvX9HzJ0zRw4ut0W8/H62QZxe6xXwG65rGIF/yVodo5LCdeH8su7bD6TJ7S0nicI+AIOAKOgCNQGRDgnGanzWsMnzvCxibv0jqlPJN/afmeVNyjZCutjpIdhnWPLmSoKZObmys7d+6UTz79RNauWaumiUji1K5dWzKgKTNhwgThxunGjRv1LXhq0nDxxjdm+OYMyZzjx48r4UPihhuuXJRxsTZp0iSZPn269O7dW00cETd7C8faYFjafWnylxVnect67vGOgCPw2yPA3zYnCPu9WkjChlp5p0+f1rFl2bJlQlKYBM2QIUNkzpw5qmnTsGFD1cyzlth4xnsbNziuWLmWzkNHwBFwBBwBR8ARcAQcAUfAEUgiYGvnZEx4GYr3XJtzzc51NcObN2/qi5lco69cuVIOHTqk8Uxr63Fbg/OexE16erp07dpV9xAmT56sL3HSvLql5z6EkTAl1+5Mw3p5xMvViMf4YDk84uXHr60oS1cyrT330BFwBBwBR+D5QyA+9pe3daXNIeXN+yzTsW12UmY7Obeegi+6xYsXC33Mvfnmmw+JVQUZw+yJR9zcZwa+Tc0MBgBDSxaPe6i0pxxhMpS3GpO1vOmfRrryyGz4Mi2JmXPnzunbMnPnzpWffvpJnQpyscY+GTx4sHCh1bx5c30DnmrQXLixs0nUzJgxQzVwaC5tzZo1uulKXzgkbaidQ6KGhM/EiROlY8eO+taNLb4ML5PZ7p8GLl6mI+AI/LYI2O+cUvC3zjGGJ+Ppw4bmFTj+0D72/v37dawYOnSojB49Wvr06aM2sPnWXvyIl2nl+jgSR8ivHQFHwBFwBBwBR8ARcAReJAS4Pv6l9XDJNXQ8D//OZ37uE3Cvhn/j08ft7t271d+kESqGqdVlf+PTmkaTJk10j2DmzJlqJp33XMeXrLdkGfbcyrTQ0pU3tHLi6X9tWfEy/NoRcAQcAUfAEajICHD+s5Pznp0vPGHDTqsIC4HSFij2hTL5rAPpj2bfvn2yfPlyXYjxrRmm4dsxL7/8snCRNX78eH3D5fPPP9c0JGyYn8/po2bkyJGanirSZOvohyIrK0vfkqfz8BEjRih7R9Nq9E9hKtOsx1SWKwp2hpOHjoAj8PQQ4PhhhA3/uONBTb+zZ8/K5s2bhW/xcSxq2rSpkLQZO3asdOnSRf1m2R+DNpaVlJLxZT0rmdbvHQFHwBFwBBwBR8ARcAQcgecJAdsL+KX1sKVj23lt6S2eVjV27NihL20uXbpUzp8/n3jZKo6Xpeff+Fync1OIPmszYY1j6tSpam2D63hqzNtheaxe+3vA7imL7ROYXJa3PKGVXzLtrymrZBl+7wg4Ao6AI+AIVFQEOP/ZyTnPzueasHnU5B5fEDwq3W/doSanycgOu3Hjhr7RTv8RfHuGb9Hw7RcuqGiS6J133lENKPqm+eijj2TRokXqw4am1KhhQx81JHToYPDgwYPqA2f9+vVq4ohv19SpU0cGDBig5QwbNkw3YLmAoywmh13b/W+Nk9fvCDgCTw8BG4cs5O+eYxFNo/FNPpI2JH/XrVsnR48e1TGEmn7U0uvRo4feG+lrYXwM8XHk6fWdl+wIOAKOgCPgCDgCjoAjULERsDV2XMpfWh/bWpoh1+V8sYra77SuwRepaD6dL22ynPj6m2ktL+NtXc+Q+wm0tkET6bTY0b59ey2bcvG5nbxXwiYy2W7PWa6VzbSPczBfeY7HLbc8ZXoaR8ARcAQcAUfgt0LA5s74/Mm5jvM19/ufG5NoBNgm+0dN5uVJ81t1Vrxe7bjICZ+1h5owXIBxIUYtm5MnT+riihujfKudZAzfjmH83//+d9Ww4ds2ccKGizD6q2FZW7Zs0YUdy8zLy1Mbtq+88opq2AwaNEjNGhlhE5eN1yZTyXi/dwQcgecHARsv2SL7zevYFP1RRvNoJGqoabNp0yb9Y5HELwlkkr70p9WoUSOhLWz7w9DKZHlW5vODmLfEEXAEHAFHwBFwBBwBR8ARKB8Cti6Opy7v+ph56auWL3XSVPHXX3+t6/FLly5pPP1M8kVNvmhFk+g0g86yuS7nep1/59OCB1/CYlqaWOf6/e2339aXOBnHOpiHaUnUmLyMs2d2TS0brO4fe31vZcYxKOuadfnhCDgCjoAj4Ag8Dwhw/rOT85udzyVhU1E77FGLkEctOuJvwXCBRFKFLBu1Ymiblhulbdq00bfZx40bJyRraMaM8X/96181DRdwXKTRJw01bEjYMF2NGjXk2LFjStisXbtW1abpx4aLtOnTp0v3Ht0lvXa6brLG5Td543Hlxd3ylje9p3MEHIHfHoGSE0hcIr7Rl5+fr0QNtWz4BsCBAwdUO4/+bDiW0JFpvfr1JLXaw3/oxcvya0fAEXAEHAFHwBFwBBwBR+BFQqCsv6lL+7vZ0toz3vPlKbOcMX/+fF2HW7rGjRurpgzT8O9+rtnNqkarVq2kdu3acubMGaF1DpbJl6u6d+8uv//979WvLc2lm4YOrXrYNcu3OpjP5GG/Wfzj9GE8v+V7VDmlpbd8HjoCjoAj4Ag8Xwg8aj4oq6WVZZ5g2+ykzHa+EIRNyY79rTqtpBxlfani8ZTVOs7iuTnKt2BOnz6tPiNIxvCtGRIwGRkZQsKFb8zQL83f/vY3tWHLNNSwMcKGporatm0rtWrVUi0bauPwpNo04/iMaelskGWZHIYdZeK13ZtsHjoCjsDzgYCNO/Hfeck4vkH34H7SDAPf2jt8+LCQtNm4caMSwPwjkT6zRo0apebR6tUDaYO380oePpaURMTvHQFHwBFwBBwBR8ARcAReBAS4xn6cw/4WZx6+lHnhwgVZvXq1mkKnlg21a0jK8CVO/k1PfzRMQ214rtf5jH4nqQ1P0mbbtm3qI5daNtwgoiUOvsBJs2gMmZZ5TFPe1u1xOUx+e2b3TzqMY/W063rSsnt5joAj4Ag4Ao5ASQQ4r9nJec3O556wiU/ocVB+i8m9LFnicvE6ns46ivHsLDvsjXYSN1Rr5gYoiRaqLBvBQsKGJtHo68YIG/qweeONN2TCxAnStk0gbFgWy8jJyVHtHeZPT09PlEWtHsrBzVl8dVQEu/4tcDQMPHQEHIGnh0DJCYM1cQxifMlxifGM41hy7fo12b9vv5pi+PHHH/VtvdatW8vw4cNlxIgR6keL9rHV7nUkfry8p9ciL9kRcAQcAUfAEXAEHAFHwBGoeAjE//7/JelKpuV+wL59+2TBggVqMv3cuXOSm5urewP0QdOvXz/VsKEGDs2pX7t2TckXkjIzZ86UXr16Jcwanz9/PmEiPQMvgtLcOtNQU54viNq+ANfuJgeveZT8G+GX2hF/bmXF46zceByvS6YtK13JfH7vCDgCjoAj4AhURAQ4r9nJOc3O54ywYSOLwx/uS0QiSXxij18Xz/207kxOC0M98cUHr62TLGQqS8OwCJukhSBa+FYNF0984yX+5gtNov33f/+3mkTjwozpXn75ZSVs+LaMadjwS8CTGjhMw3LMPq0uyqpWkapVItImApgymSyPi9Kzx/txJfT0joAjYAjY75y/Wxsr+Mz+YONzxuuYBMKG4wj/SKTT0w0bNihxc/bsWWnevLkSNvSx1blzZ6FJhaopVXVssbKsTg8dAUfAEXAEHAFHwBFwBByBFxkBW4MTg/h1/J5rcFrd+P7779XEOTVlqEHD9bhpuVNDhpo2fJFq4cKFcvXqVV3HU7PmzTfflDFjxmgZa9asUQ15mkbjfgD92/Tu3VumTZsmo8eMlpc7vhztEXAfQKXQ0P62x86Gvtxp90xR3oPtK9lGlmNlWRhPw+t4mvLW5ekcAUfAEXAEKhcCnF/4/3EPmzseN9+zTm9zYHxeo+yc4+kOhS4H6F+Oc3bJowoyJaDZs2ePnDp9Svr3768ZVOMCL1UwDCCGiVNjwssWWp5N6rxhxXYfrhPFx+I1pebVD07YrIP9hA8WjShWgyOZ/z6ekXRgVJWq1cIkzpxIp0mZvIqlt9AehhSsRw97rHksCpGMtyy4jCcLSaP8IUuiTeE24PPgATY4EWEpKTffTjeoq4Agqco2kyiJNFySFSVrDBuoRVo0sUxiK+rDhoQNF2ZxHzbUsCFhk5mZqTZrWQY3TFm3XVtZLJjX8VBvog+V18TRZHbDBHZtIdqiLbZWx0sqcZ0oi3l5U448JYrwW0fAEXiyCPD3nhyjkr9Ji2cYxqT7amqR5tE2bdqkJ//4ow3sUTCNNmjQIMnIyFBNPpLD1OrjH4YlD5bFIz4elUzj9xUVAY7dNvaXJmP8+5OYVnWojz0pLWMsLqQMtYTZhQ91LZKom+uFZIkhM9cSYW7ROZNpI1FtvtPvNCI1Z/KjlLJi4vhlmQiEPin+uGSvFH/6OHcoSfsPH4n1Xcn8JWsreV8yvd1HXwy7TYTlzZ/I8A9clCVDssiQIilT8iqZ5pevfrmeh8v4dTU9XM6TjuE3ztoTZCxLUh0Kourt9/940lg9j5OrLGkepwxP6wg8LwjwNxR+R/HfIxcEyd8kftH2MPq7NLQev6USP8H445An/O1taFk5ybL5xPYxcBUvwDI9dlhCqMfOHzJYk62R8Xu24z7MEhfg5c0dO36WuXPn6otS9EVTeLdQ/dTSFBq1Y2j2jP5rqF3zzTffqPY7//anhg03f5iG+yc//PBDwj8uTaNxfd62bRsZMXK4zJo1SwYOHCi1atZKrNlZPzomYKa4sc+CtGYXhCspohFHhCNg/AwJolEb87it25jGjof7JZT4cHzIYfUly0heWZkeOgKOgCPgCPyjCJQywmsUxtzYsPsg+hvNxub4pMAxX6cQzCeY2SCQzducP+zv9iJc8RkL5Vk1CjkNBRlKTt9hLqEcMUGQK3kwX+yZCsf0yRTP6optsJPzmp3cD3sswmY3CJvTIGz6gbBpDoaHjU+0R4EyPALoAbzQTF5bxUGYkNYEIzLEUvNEOGmRiKxaJUXzBn8J7DakRh9VYcdr54dJvqjoPsx73dMFTNUUbAKCtEnBgqQqJ38TlNfobP0yPMAXQB+wHYkEiX4JMVE8ApM1yqLpouUFJYrKsHIQpZ1uX6JgNogy3wdhg8K0TayWcvNNmCIsfIgRNy9TqrLN1GwJ2i0sn4eVbu1R8ifCnhWyLC6gaBLtr3/9q9qyJWFDQsh82EyaNElNE9H0GdvE+rhwe9xDq+UHhNK+0AZHjUZh2pfRDwu9iIjoZEXWEKbjPY9EHMu0pR5xfXzZQoH+6Qg4As8SgfCbD/a0+cfh3r179Q9EattwHOrUqZOaRxs6dKjazObbezTnSO1Ajn12sBxOUAw5Nv2a8cnK8jC7MBgAAEAASURBVPC3QCCaYzm6czhPiMArnFFfc/rAf06HekChMyRGRHJOCc+KTRSaPySOslpGhFzwRIs9xsa+V5jwk8VADj5TGbQ+lmfyRPOPrhdsbufTkIZXZR3F6isr0QsUzzVSGBeAncLHNVg4LDQ4bPxgQus2xoV0+CyWIbpBn0a9FkL0WfGDz+MZS94XT528029m4tZK5bchHBYmkjyFi0gGYGD1P1yJrWPjUgXZHpKwzELw4CHcHq4pGWMlW5h8UhGuwktRobFlS2hPOAaEtXd5f7sGYyiBdxZTntYn6y1Pak/jCDzfCHBs4yYMfkX4GfEMR5if+TcwjzCPhHnZ5m1Lp+N7lI/zRnzusN+mxWlZWklyjrFyQv3lGwuSc1WUO16BjgeJhoQE/+gnsdFyWVAYQ2hp4x7+ts/Ly5M10K7585//LDt27pS83DxJxZq6RYsW+oIUX9Ts2bOnHDt2TObNm1eMsKH2O5+TtKHfWq7Zv/76a9m6dauSOmxWgwb1pU+/XvL2O2/L2DFjJb12uq7ZdT0F0Igb+yTsV4SG8lVSnpy9sYrXa/urninYq1zvVUXeqigAbwczOvSd9iE+NAoffIb/WgfKU6gfMV9pNi0t+YFc0Y2FyWd+5Qg4Ao6AI/BrEeCIy7/5Odrz4NiN8z7Hfhuvw3BOwiY5j4WhHRGaj4oK4QozxwMoXyAt5xTsUmPfnGVBwUBgVQpnSIcXfcEPcDbhFKF768gTyuGcgkd6sFzOOAyTAS+LycwIlV0vcBmlj26fRcB1hZ1hTiV+v0LD5tcRNqGzbHHDinnoZhyIBWqXEDKb6PWtau0kpCNZAZBpkiuAGNAPZeFa0yXj7nHxUngPb4lwo49mwlKlmr65zU2/KlhgkPzQ2rVOlknChv/00D/8wyU/E/HhRkHUp9G3IP6lU/kSOaw8ysZrhmyj5kY5lA/ECtrOL9h9kCwkVHjPOlOw0KpWLVXbHeAJm5a2cI1K0U7Ur5smwhXCQNjcf+qETZALkkRYcHONP6aHjkRCYE/CBu0LUew//MeppFnoRo3TNkVlsf+Lt/uhGjzCEXAEKggCNs5zHC8oKJDr16+Ladps2bJFzTBQndN82tAJKoljatqUPFgW/5lJRnvOeJtHLM7DioYA54Iw7xWXjJNgNBHqgzAfJIb/6BGXZpo/MX9EpegkamVEeePFWT6tJZRh87jWkZjjLZOFnIuwKOJ8jHVJ+I5hiYd1g37XTMBisgeZ/LsY9U0ZQeiFBIBIFWLYL0n0y8hs0fo94A1yJDJFF9qnoUz+0RDKt4yWh2ktY/w6nq7ktZUZ4q0FSamtvJL5nuR9UgYbW4uXHr6fyVTWyiDbQxJaI4oXgjs8iNZyDz0qNcJKtrDURL9Z5KMJG2IWRAtfq+JtKM/v2WAMOePol6fJVp+F5cnjaRyB5xUB/n4wbuPHqP9w+wBjeph7Y5ssfBqlsVGOIfNU4RyAH7X9ovj71r0EPmO8PYggDL/78CsO18zOvzW5pgh5dI8C13xZqLQxoeR4XDzN444Jyb6lPKHsSPaojckU4crScd8jPy9fNv2wST788EPZsnmz3MzKknp16sqAAQPUlBnNoTVo0EAOHDigZAz93JhJNGrYvPrqq0rYUAueL1ZtRhlLly4Vmlajj9vGTRrKwFf6yeuvv461+wi1zpECvLhWCutz9FOEHWWvihdOHwC3QlxjO0bPQN5AdqRTogaXJGs47bCXddsNEewrnjwYaK+ysdHBe33+iPkqmdpyReXobYkvQzKJXzkCjoAj4Ag8NgIcce0lTYyvmI/DuB1GcC2O0bgwwkavo3GdaTHLYlzHiQe6l0zCBlfcHyZhg5kB/zg3gLB5AItUOmtw3uHMEe0FaCrOISxEa9AyKQ/ThDMZ4Ao5WDLT2nOEvOU9o57xwfnTTsWDmODkeuQZaNgEkAlK8iAKwQwYVXALCu7ivKNaJgSIm3dpNWpIjbQaSlzohr0CDvw0KxuEVQA6Re9ZGu7vYIMwO/sW3gq5jkVHFhYN1aRRo4Zqvq1e3TpK2AQZqO0CeZA5fAlYKM64iBrDeL0IIT51IYW8xduTTBRaG93rF0YFZk5GohbmJGHzQOWlf5krV69ITk4u3ohJlYaNGulbLnWw2KKWjebCh75lrsQVSwm1q3T6EcpmmUVYHbFjn7aGjTaNouCwN6ENFUpDscKPABdIHNITHfzArFG8QB/oQtcyaa6w6RKinLABgn44ApUCAR0fISlDju3UHKQt7f3798u6detk/fr1SuLQn9a4cePUrw39aVHTRscB5LWQDWY5vC8tjs/9qKgI2BjOGTZ+hJkhGYNZQueDEJOYBjA36NyeeIgn+jDKz7k7yhcFYdLRRCGG80+8bo2NFm4sm/l5xr9fSthEDxgf3tRh1SyrpOzFv6vJNvlVSQRsbZDsZeubgGrJ9A/dW2frA/RDomPZkZaaF+F7ZzEhjPpNv1CMie6LJyrljuUlCk9cJSVOCFFK3icVVUKGEjiEJoWVFlMGieKfJeRINufhB1wTl/uwtltY7ozPJGHZhA3nkrgIyXGkWGzxRPFHem1IhaJ4ZzEPJS0lwgSwsJQkHuUIvDAI8LcTrRcwvoUhjr9TI1AMiPAbK/5LC2Mff0nxXxPn7kDYID7xIJkzXIWR3Nasus/AxNEYm4ynLIlCVBh7ZpIxLJ6GNSTri6cr65rVstxQNvOGeq3qIBbXRdF4H73wStLkDvZPDh44KKtWr5LtIFm4r0BtmcGDBus6u3379roep/YMNWxoKv3y5cu6r0CShoQNtWy4FudeAzeG1q5dq8TNrVvZ0qhxQ5jC7yMjR42S7t26w9RaGqVTwobIBFIL+xUQEuJAdJA1RQ8kJ/+u3M6/I7kFeJEWD3QVBfm5riJJw5OaNamos2ZamtSulYqy4Y83JWo7n+OfHgEAvVdMHjFflYZ8ohwrL5Tqn46AI+AIOAL/EAIc2zmH8+QcytE2hHofDeGs4r6O20yPa+zba0rMFyRs9OA4z7NKoPiZggoYNndg1kEM6mEy0vw6H0brABShpZAfAKnDWM6XKos+ieqIAtYXZiVKw0g7+SSWiLfP6LA1AEOVnfLjfEaETfFWxoWgNkh+fh7e9Lgm586dhd+DbNhLTZW69etL02ZNpUnjJpjA01XYB1CtUig5w7On9ASkiFQCBCRFFjYGDx85hgXLz1Dr3a9vfvTq3QsLljEwAdZesyjRww7XwvBBMKxjIlIoLjGBKn6we8O/4vHhLpIyWSaiQwdEdeL+Pr5IFCYb7f3555/VCeApOAwkSdOvXz+8FTNQ2mZkQEOoWoKo0cUr5bQve/T9UumihQvrua+EzTMyiRaarG1he6xXGIaDPzHc8cejB9HRDgxxeFTF3mBm2xKHlcQ4nlGexHO/cAQcgYqIAMcgO6g1yJPETW5uruzbt0/mz5+vDk3z8/OlR48eMmHCBBk1apT+oUiinuYZbcyNl8UyLd7K97AiI2Bv2/D7kJhhI4ExpuvXJCyywhgfHsX7nP0dcmJu0fScT3jgM5qXGZ+YjaO5kWn4OKTVjJorFKFPwz3nS2wicLOBB+tmGq2XdaM8LT+avxKavpo6pIsuPXgkAoke0lQB5ah/tFOI9SML0L4pnqK0POyvsJAvXl7U54nI5HegeJkl7yicCqgP7Cp8Jxn1C0Jrrn/0o7gMpZdW/Hf0SKmsEQ8VhAfROvKhR6VGWC0WlproN4vUvwn448WRlLDkd8aeWBjGABP6UfONwRhy8s5iLPejQqvPwkel9WeOwPOOAH87/BuRIX8Tyd9FmH/D75Lzhv4mE+M4U9pvGnk4f9tvHmkSv1/MCaY5y783WQ6TJTRDUKMerBbxLMMksLVAlEIDqyMex+tEfSEVPtme8h9q8h2ycs3Msvj3v5XBe9abWK9AwKL7eMsY6xeaRaO/mhyssS9evCiHDh5UwqZ5s+ZqgjgzM1Nq1qypL05Rw4ZrcPqwYVqufYywoUm0du3aqfYM1+f0g0MTatevX5N69etKly4va9q6dfEyqa7TsW6CTEX3uJ8BDRnIWwVvOz/A5hu2duRWTr6cv3xDzl+6LteybktBIZGneWPua4SdAJI13IqrmVZdGsPsWotm2PtpWgs+cmhhhL3LfsJnBGXom6jPHzFflYZ8KI2SWu/y2g9HwBFwBByBfwwBju2mQ8nx+v9n772/rDiyfN8oiqKgcIX3UIXwIAESRl6ADAIhgZCXWt093X177lrv/iH3/fLeer/MWnd6Zrp7Wg55A0ggQN6CEAghvC2c8LagLO/72TvjnDxlADkaSZlVeTIzMuyOyL13xDYBtvWr4dsUyi2wsEkQO9QgCmzOi56B8N2oQjRPeTXo5Gp/IjtkJ9JBLMuBew6m8lhvsuZ8XvTRnKklih9JUUqcVCa5UHPPhYAksNmV3C/PAY2LZ+Q/uF5mgY1ALVhYRUzT4rxMbetMy+Mr7Y3DHgfbtm0zxmJw5ZBwrQQXUydPDYMGDhKUBM6EnyvChlYANsKt/MizUR1zVpvjbVX6pUvf1oLgR+HY0RNiPq4Kt98+I9x26y0SgAwmmf4lOCEz+oX6pDrGB1hhpwCowoPu9b/CcH/K52FDywIttsokK6uvDabGcPLkqbBhwzfh3ffeVfs/sA0Bx2kR8+67Z4UpEtp07VoeSqV14oADfhrWNhrJFjiQIf8+WoHJZbOwoQYGT6+L1UB14Wr39i756KzzqGcCE72zPoBJt/rrx96hUWW3SS7xwRfUFJgdGQQyCFzhEIjEhmpCZDiZhB48eDCsWrXKBDZr1qyRVeHpUFlZGWbMmBGmTp2amywyiXT8Dr7zb785Hr7CgfArr54tmCaLMAnWz0HE6QOh0Ap+OfQs5J8eOxaqMI+npzzBsfhOLBKaw+SdjHWFHkJHPGcrTSH2UnEUqluxIHYgTMT/+4EDB+xkL6XeffuE3tJO7dixo8WxxQUbh4UtudCYpB0cF4pjEX4VP5FjSoCetLlwvYX+bx0YEZ75GC3FV/4Jr5GPxx2DIc1DUNAFCsslpr75Osc7H5FEupQ8cpl9z5vCOrScSeQAvT4XrFVsRLOM9KKwQ5rFKAyIpcRr4dt/9pPjH28sNcz3Wbpmse7xmn534fsIRk/JUwy5cDp/G8uL10tJk8XJIPBLhQDfTjLBN5yq70JB7i6dNvtihdEAvSoSLXbL1/z3w12af8jT3fhdigaJJvufIluwpfI5NcRHYU5n4CGgL86PcM3nF+NQr8IjHccLiGUXxmvtCT45Tec8P8pOyqR+SozQiQOeOsIIssczCrBHjhwN1bp2Fv9SXt7NXA7DR588eVLrDRvMJRpCG3geBC8IbO6/f55coj0asHxHIENe8OdYx8MflZaWhG7dy6VA28FctlsNgAsVSZpZlOy7yy4DCGyOnawOO6sOh+279of9B49JYEMvI6xxgY0tyLGopnZ37FAa+vfrEyoG9woD+5VpjxwX2LjQJimHotSHPAGTAnplsIkVoVIOI+7ikc+l+bsYJ7tmEMggkEEgg8B3hQCUtRWBjSHrfH55gU2Crx2j26/TYJQ9mcNHei36r7eNZmUDDUhIDsmFyrkk0215pdKaEehdAXLYqdMFOAQ51vff5EGhpCf3JDN7kcQpuLeol+XH+BTV32gdNDY5L5vABmB6ocBR4EkW8dDi2LF9R1imzfKee+45LeZ9ETp17hTGXT0uzJo9W0zE/WHs6LECEgybMrF8gBmNyQtAautqbTHw008/C88+u0Cb5X0ujZBuWgi8PcydOzdMGH+NrHV6JD0GwadzGAyWJRnq8C61ruJFchQyYQT6IIp3SbTcxee83uGRQciX5IwGdUfjB63z/fv32QLmC8+/EL7ZsFGbBPYP99xzT5g5c2YYKjPmcgltEu5Eg09/MhW2qiYlekleYWDEfjjA96d2iWbstYqlZD4vDn6BLod3l8Oa9psmDQOP92o7H1PsB0/Bh+Ym8EkWyoU3nOnFFj1mRwaBDAJXLATixDPiTnA+B+7R2NNm9erVEqwvDR9++KHhQKwKcY920003Bfxpt5c7TA4mmZwxHwtMfloKS7/P7v+5ELAFU+F5owgJwwYmd1rIjdMCHxlOFyNhYxHCmBVrAu+UUv9ok+oFmeg5H85LgmycWVTRGwqD7liJvOUgE4Xo0fgJhZyTogeapmvXrg1farNexttVssYdNmxYGDBwgOhv11BWViZ3qtIeJT2/sT3JlZzTRxzvMSzGj8+/tquNBeuhwpZbFxEUu8dgWxgnPjWFqfdDfBuvyshwTcwwKcH6KVeaIns/xlStX8kn5pW/83FAqnSerefyw94U1qHlvHz8G0xajpAPzTcnH8adMa6tvSyM6k+x7fHaUpx/Xhh4xnENOCDfY4U1utRxUJiKJ4eU/0boN4/VWkiEWby2Fi8LzyDwa4AA35HzChE3g+/9VIjwt81txU+A3l2BotiENqQkHii+8Cu3N/YOCJIHaaOAg/gN8BM6cD0O3Scfd9Hi8cnP89YdCZKDsKZH+r2/8/KbxmvtmSxZFyBv6sIRy6ZxaB17udTF8Q91JR31ZM8YC1cAYR7DMjHYkfbUqVPaT3JjeF5rDc8//7ytmbjApn+YN+/+8Oijj4ZRo0ZJUbSrpfd8gEWsG/sGeL3MklVPcc5OHVCKsGUy8VjyhhaOnjgbtu08HDZtrQp79h8JNfVgyhJFc88hplzRqH0KJLTpXFYq5dz+YfhVvUPFwI6hS6fEwkaFW/tULofBhBC6I6Vg4LCxKP5jEVLPRKeOdsRr8phdMghkEMggkEHgB0AAbhsazmnIWejZr/6cz/p8kehW8udYOdrWKL7RL38LfodeW64S1tSLqNRpfRtXmw3yuMUR0Ty0inttU28Cm2LR0NKSNnK16VgfypNbRfakSYWMoumeKwcvm5724rL9QMviCV8Rz8smsKGlTlATQps8V2ujvB07toflEtg888yzYeXKVdLu6BzGXXO1CWzuu29uGDN6jMDnjBOdQod6J8HYEN5oWiBrv1ob3npzSVj69jJpjhwMY+Rn9b6588LdM+8OA/r3leVOe3UkvarTM9LaDd3oHeqdRE2Tu9h/HpQrk8cmrywGQyzFPygs5muvkzSkTIaqGDAETWfOVIevVPcXX3gxvP+BFjBPV4fhI4aHu+68K9w9a1aoHFqpxaJ2xlTScTnGJCnMhpaFO0MHY8vC1k8psKEVCa8bFZbNP219vT4kvbRW8qPKUT+HDf0mmNB/+gRL5HqwpJjFWAQ3HMBLf/6Qb6d8FJpvYYuT/WQQyCBwpUMAwpIW2sQJKFp7NdpnjM1OsapcsmSJcP5Ka87YsWNlDXm7/GRPCoMGDTLLQiaTHBHvkSf35Mc1O65cCDiVi7jd6+k95m9iSMT9sSWQDQiI05CEOtPX/q8XFiOJzj05sFDgOVlKi2s5eGaWIe89DjQSd6yMI7ROcdWHv3asfBEowoP0k+Dw+qnXh9vlTnVIxRDbT89clCaWGnH8xasyzx05Gp2EtBQnF/lXcGMLUfRB7CK1GVpfeOg9XZY78rxALqiVmzS8W0YLTcviuWlYS5nHMdT0XUwbr03f/5jPXod0Gz331uAT6xSvl1qX1traWvqYf7y2Fu+fFw6m4Wi9hi29uTAcIvqxuWiSd0u5WMHf+SfmFK/fOYMsQQaBnyEE9M3pw8p9r0Li4Ds/883hmXknFiYICrC+4GvNCVl0b7TW8orplEZ8o4Kcj0zSMEfmj/jwAW6xE3GFvj8+QdLEPJPsmuNhauB5e5T47RKu0/49jr9v/Rf+mHoWa16siln7qZ+3iazIDBiw4MXCVCLYsSx5Z/8W3+ir4vlBXnj0OBlwibZgwQtmZXPw4LdaWyg2JakHH3wwPP7442H48OGyyGE/ScHV4OZCGXI2OKtM3iFs8fqgfEptEKTJhZtcoqGOKc/s4cjxc2HbriMS2OwL+w4cC7UNJdq/oER1od5wbORNpetCx/YlYaDWaUYM7RmGDu4Quv4ggY0qGLvBauY/BhO7beFlKl52m0Egg0AGgQwC3wUCUAMTrSgRmBYcqzMntEnyEr4nntF2YkBMFAcKYqns2WIojPVk7YWsReWa2oZw6szZcOzEmXDmXIOENloXRgRjNNDxObSkbRvZcLYtCp3K5GKzuwT/HUtNUMNKklneUA0rzGmWP3Afn4mQ1L3gSvjlOZzvUevV9kj/uUKP2Vtu4cKFoU+fPrKIfaRZhbS+nlBjvVqrRbZdu3aG67So1lcJoNwOKr1MogFvA72uqaQEpp6JdN6EFdu3b5PAZoUsY7CwWRU6a7Hk6muuMWHFvffeawIbpGVkz0IL+UfmCiakpuZsqNqzJ6xYsSK8/sYb4au160JZx87hTu2NcM89c8LkSZPFgHQMJWJM2kr8Zma4VlcBRJlFuZtV22rp3WW3qT60dMn7VHASQmtUtxZeeL7+6xox3gb2samvq5eksFH79+wJ72rBaNHiN8PqL1YH9nK4XpsFPvjQQ2HS5Mm2eWBpO7lGE7RN00ZlceQ7kycNcsHHmCxdL4fAhlrAttVr8euMNhc8IRPos+fqTGjTaBJQxTCGTB+yYM4Hg8kaks/OnTqELp07SGgD8+3MJ30LpIAl/35kApsIieyaQeDnAAHwfpzsUd8oYCGMSSmWNmx4irCGRXLco/EOoQ1WNljcDB48WHt6MXEUjQEx6CAORwyL4RaY/VxhEACL55B4rm5QXA/Pv/PeNRJhKUQhFT/PxNmjhTh9sMw0xjwf4rIAEHPhlrzJxa/8Om32OAhr6kR7GT9Hjx61PeTefPNNs3Tdt2+fCQtxE3LnXXeKKXrYFjGw+iqR4gRjmSOOvXi1wOSnKd/TUpx0/F/6vS8wea/GbkpxjvnmG1+Wf6RPk08/HdjsPg/vS4tPvn7Ea7MskwAfP83fxnTx2jzGjxeiOuTGejrX1toa6xSv6TQXum+tra2lifnHa2vx/nnhtIjj0mt4cRjEsZYX2LQ4kr3g7/wbaxqv3zmDLEEGgZ8hBETJwf2G55y2gvcJcz4SHpJvQosWxNNLaKp/Jc5RoAgIzU8fxIl5eLieFabpsY58XMtL4VodyNEb3lqJVo6XZKmsHO7i4fnkaXyMS3jSrhj1Ildvq+bJxmN4uyK+8fZSd/fOQVbRuoV754V0Vf1iXMINpqoK4adOucAG6xqsbHBRHC1sWhPYkIW1LcnDnwn09iG4oU4enhfY1CnoyLEaCWyOho1b94d9357QIls7Oc2R0MYW6NzDRrHyKZLApqx9cRg4oI8ENr3C8MHtJbCRFQ6wjyfFcdA++9M9dUjAHeFkAbELLEH+J46YXKL8q+wug0AGgQwCGQS+NwRA0IkQ3/JIsHSLAhsUExyhEwuM7o+sBfMsNJ/kRo41tVI2OC1PGN9qr/u9B8Px0/US/vt+aeyZ5ngduidhTXFjaN+ujYQ1ncLQIX1Dn55dg+Q3dhoLQd3IPNKvXGleH177ARGJ6wqtEJQY9Ue+Apt45uif4AJ/cBkENgIn7RU8osACgomwBR+p27dvN2HLggXPhy++kEu0TrhEuzrM1D4uc+6dE8aOGatOkNaGuCw6FeGNMzQsuriLHbS1X3/99bBYiy6HDh0J4ydMCE/+9rfh5ptvCf369pcApNgENvhhNcbPGAwxg6qHV847hN9451cv03s4iWpNsR4v6CYYJhsDvM8NxhhFuek/3wn4m9WgU/2BDVrnO3bsCC+/9Ipcwy0ILBoNGzY8zJ03L9wl12hXj7ta/vQ7Ka4zrMCO+rWRJo4zNV4O+V+OPWxoPSwaJwKnaglrDh05IcHZwXDk6KlQKytnrYmpvxUB8zeZPAeZPgN7PqauEtQM6NdLPmt7iVFra5Y2MI7xg6JtfnBnLY0B2TWDQAaBKxgCOdxnSD+P82KVEdiAx7ii8YfQZsGCBbZozn4ikyWgxh3kLbfcYkIbhNdR4BPxJ3lFQhbzza5XGgSgiBCApkekHs3fQTbNTQlanLZo4QxTOqZTBIUQWXSQd9qe134Zcrznx8vmrZ9Om+1tqNcmudBcskBwiHs+BDa4ROMZF2i4RJstt6wPPDQ/VFZUmhJFSUne0pViOBiHTY/4DcTwluLEd7+G6yULbACGOsXhl/CN6swWQJwHm/owP84uEjeXKk4QPKD1/onjJ5cwuYl9Hq9N3/9YzxEW1KO1o2mbY53itbV0TcNba2vTePE55h+vMfzKuUaotVTD9Dea7/+LwyCmywQ2V04/ZzX5uUPA8RzzWmgy80S+Sb41FincGpYwLGkTmpsQBb5tvdEvcX0hiLSwD/Y1Q080F7X8FK9OPCZzVvgLwuxUatYVcoqhJNDB3DotFLFA/UQc4M8RZ0QsE6+UGfF3THnxK+scXnPmz3jhqDMY+PxYeVvbXGmEenhbWQfIh3k49ddeMpqIs9YAD40LdrewWWACG3gdrIYHDhwQHnhgfnjsscdsDxsUpZodqhZ9430k+KltWDq5Iit1Zj0AxdpiKWy2kXAGCxsENsdMYLP3wIlQ29hObdKpfrA6SnzTRusDbUKt9rCRhY0ENiNlYTN8cGko7yzXabSV3qWfvAirhJ4URolJoO69T2J47g2Rcgfp/EjK90xy77ObDAIZBDIIZBD4PhAAFzv9zqcG3xq2zgcJZ58vaiqwAX8ncXUReXbMLvpCcHVNfTh4+ETYsnVX2Lh5Zzh0vDbUiZa4G9DE0saoSr0JbDp1EE3r3zOMHzssVAzqHdrJKEA2AqI1STWoqhEUbtJnqppGK6CpXq/0m5/6HloWT6N90D+dl0lgk29erASEH0DBQGzbhsBmuRbtntfeBl9IMNE5jBk3zixs5klg4S7REHAwGNyNibvKOW/+5zdt2hQWL14U3n777bBly5bQoaxTuPXW28IjYj7GXzPeBEDFaixmxpj/aghY2QCAwWBVsU5xBs37lEHmb2KdqbGS2CFw8uQPycVfeN5JUmWexFPC6NaLNgB4mEuuHA26PypXLO+++2544cWXwqqVX1gH3Sht8zmyMpo2bZo2Qe4T2mnBiLJh5DhhwpoKbNjD5nK4RNMnp09E+1LIXO30mZqwd5981W7ZFQ5Ik6ZWvmobGpPBbkxrFNicD6XtikLPbp3lq3ZQGFoxwEzXzMoG5k9tgrsGzjZQgbExouSVHRkEMghc6RAAX3IYfk3ueY54NL4nrLa2NlRVVYVPPvnEhPYI3kl3DRaWd99tljb9+vUT3m5rmoDgfcMLCSLmPjuuXAik+xoGyekm9YVy+Djxi9NJhotN5oX3uUJjtSwQYzo5VZz4hjzguxgGLE6w0GOUQs9Wlo0/IpCLBVK4CWxQ9kBws23bNhMYLl68OOzevds22O3du3e44YYbwqxZd4fbpt1m5seMvSiwsUz0k42/CIkLX41TsL5QH+i/9cNfMm6sv4iYi69+TB+WXzoAHiv9nLpX0vzYi5Hi9WL9mC433se08Zoq60e9Va3jGG4x39ba3LReTZ9bygz+OraP963lHdPGPOM1hl8519iadA0L2+h1LRg3gkF6rBS8S5oW8+UxN06Tdxe+pFO2FjNd26ZxLvSuadzsOYPAzwkCCFx8jh9rfV5eKHAZfvz4cbk9PynvDefEB5Zof9quoXu37qZYUdKOvU6g+j53NNqRfCb1ElQwH2buzfrBt7IoWbvua3kJ2W08JVbcuADr3buX1h062nxaH78OvEGA/9I40DP13/x3XIhPCr/PlnCH5Z7Ds+n8WagS5hEMqHd1dXXYp7311q9fb3vsdenSJVylPW1xl96lcxerv7mGszUAtT8pmvoAR/CSKToBmeT59JnTYcOGDcbvsGcwAhsWvRDYYGHz2GO4RBsm7xedExzo9bM2arGEP8eNCf8G82UhPOuQtrNEXy6wUdDhY+fC1h1Hw4Yt+7SHDQIbeQlpU6opPW7RlJMsa3CHVnS+Vn3ZNgySwGaULGxGDulQKLAha/LnUJHUw3m62A88++v4G9/EZ78mkZKXGf9WCJ3sKYNABoEMAt8XAvl5XszB8a3h6xyCFt5Hid/WABJcLtrCjA/cziQOeoXegq3Pi3afkzu0gzIE2Lxtt/Z536b7WqMl7IVWbPuhsa4vmndeAps2DRL+t5HApnuYMG5YqJTAprQkEdjEakF/rWoUmD5jBK5EgK/gapF1vTyH00boo6CS40Muo8DG4KM25ytCw6PABpdoy20TvC/kDgz/qaMR2Ei7db42whszerSlg/FCyAFvxuIJnYqG9qeffhqeeuqp8MnHn4jRadDeNWOloX23Fvxmm+/5EmmQiKdJMWAMDQDh2joJq2E94Ro1sYMYMU0nsbF7yUPx7N+v1u82CElnzYs/ioc2D4uNaKC4xguwIKJprOjKfj4sWC5atCgsXvSWFo+qjDm7Z86cgGu4YUOHhU7mIkgMmNJyui9fZU+VyY2iL7fARvvWnJTAZufug2Hd11vDnn1HpRWk/pGmjXUWdTNBDNo09Saw6d2jSxg7ujKMGiG3R9pssK0koJhGq/JqHH0s6MaBykdDp2dHBoEMAlc8BByvgZOEZYWQ4jMVj8/x2+aZiemuXbvMwmbZ8mVh/dfrA+6nbr75Zgnebw1Xy9qyR48ePjmXtU0U2lzxgMgqWNj3yeTewZIjkAwKC+LCMgOMWq1oCpsL4gfdYzKWREj04K5PdNUDdI9NBtvpB5enZn1rdIOoigzd4VA86LWTIlmhisbU1NSZwgg0969//atZ2bApL2Ny6NChYY7o7t0S2IwbN9bctJI6E9g4OL/7b2Q+PSV92vygdzjilVjxnnCek5TJmCHUD/iFeN/kqiQ2FgqCiZxPAD66+JEqP5f2UtJdPOfWY0T8mbS7IOIF2pyLF+sXr7kXLdw053Xtq2k1aXwRry1k+U8OSkMt1jJNj2L1CtwdX+LYink7ViGnWELMtek1PX6avruU55h/vF5KmixOBoGfDwSY09t82L5B8QD1tXLb9W3YunVb2LJ5i3nPaN+hTIKLYWG01gX69u3jCplM8MVfkFY//inqilIG1jQIa44dPR7WrVsX3lr6dli/YWMo79o13HDjDWHabbeFiooK27cu8qVxHQCyAO3If+PNv/KW8IlD/ML42aqZ+pR5Zn0Dy3MUSU6dOhk2Shl1ydIlYc2Xa+TmvSxce+214TbVF/6kaxe5ekkUmYx+xbzIx+CgNY9EoROySTtQkN24caO5n3/22WdMYAM/jfvXh+R+nT1sEArh5STdBmuj8mAeTjH2rEU3coVrM7jrDoENLuY10zfe7bD2sNm8/UjYsHm/BDbH5camnRZvtJdwsa6K2agFtvONtbqtkeJmW+013CeMvqpXGD2kowQ27YymA3vKtJPiuDd6Tb8kAbwlQuqIb1JBfgsfmRyXRvdj7OyaQSCDQAaBDAKtQeC7CGycdkS0nczvEyTOFhusb9uWJaJhZyWwOXz0jAQ2VeFrE9jUSWDTXnRAc34pcDDvJ7+iUC9LmnrthyaBTb/yMOHqqySw6RXaJwIbpyGRiEQKAWXknvTpgyfytVTpFz/5PfQ1npEn4Qp/9CO4RKNB3mAuRkvVSGN24EJS90boia1wmKjTp08ZM/bOOyvMRHf16i9t34IxV48zl2j3zZ0rl2hjTNhiAhsxM+SLKzBcmrD3y4rl2v/muWelifJN6NO3ry20zJw5K4wYOTJ0K+8mpkZ71yQdimsu08KVQICqGe1OhAHGGJC3CjBCngOaT2RzdbfWqnOTeKbdAwh0GANhbfY28uyP3CG0YQDwLjI7MFfOpNVIe6hqT1X48IMPw3OyNlq16gtbqLxZroHuu3euXAVNCf369bf2m0AjEWrk6qs6UMdGrXKhpfNT72EjVgtWTUKyoM2gGsLOqkPSXtoiH4PHtNhWIka5rbXZNGGsvW7+XFpyXv4Fy8LYUUPC6JGDQyeZQstjnQltgBJWNg5bxg5HHm7+nP1mEMggcKVCIOJJvmHDR8JFEBqbPBoedjzFO+JE12j7pU24evXqsGTJEtsEHrdU48ePDzNmzDBLGyaVTFBNazDi6CsVCFm9EgiAwxPmQ/TJp9688nCLpHHAE37NxRKEc/KlcfrMOfNZe7a2zlxrFBXR7+72zNxrKg3qFtC+jjJ/7tGtg1xrtvMFDBsbVqp+oLPwJAmB1r1psIpoISjcv/+AWXc9/fTTJjBkLOKCj72UWLy4a+ZdYYC0T0tKRMtUZmm79qJT8tOeHd8RAqn+Vkqemh+xj+I1HSOmSPKJj0mUXP8q3EdTOm1L95SRLyeXvqWoubCkbHsuTJ+L8qPf+LfTMsScx75wkbGN8dp67DRPmo91sTIuFxzyNfoud7lhom+3pcP7PdWvum15/DSHQ8wxj9MuBuNUOS1V5qJhMf94vWiCLEIGgZ8dBJx/BO+JD5DL9E2bNoZ35Hni7aXL5UFja+gu5R08aGCBjTJP7949nbdMBDYm9NFXzFfCok+DaP2x4yfC2jVrw9K3l4X3P/xIbruP2brCfXPvC3feeUfo1bOnKWMALF8fwHOFUwjwQf4bd3Cmv0Cvr4fHX9ANuCXFdsRXrV7JxwRMtsZRJOvzurBj53bta7s4LFO92W8GXmTG9Blh+ozpYaTWN9jXNvLEZBxxF/VN19ELLRLPkxfYPPPMM+HAgQOWftCgQbKweSA8+uhjOYFNs/obwnOenvysbYnQxhbqVCJKNXK4oZNrUTgsl2hbth+WhY0ENgeOhxrtO9BYJIENFjbUNxHYtCmqC506lshFes8wemjvMKaic+iWFthY2d4PlOut49leUBv7p17xiG/is10tMA+ZS6P7BTlkDxkEMghkEMgg0AIEbEXYeG0h2gTXGi0ylMsPp6hUTtgfcXmkKx4HS1Nm7o1C9myFXl3TqC03qt3CZuN2uUSTq9DGMiXGq4a20tBZJH6BvdBK2jaEThLYDOhbLpdolRLY9Awd5NXJXKIl9MLoBkUlh9NNKmyVtlCnMb8ogU1fXwixPigknmlGxfoPEOgGcPAO5gSBzRmZ6OKSxC1sXtAeNqtN0+Xq8RPMwgZ3YKNGjWLJ3tI3YuKsRRiYqhNiwtjzBt/zby9bJnPnQ2GM9rvBD+vtd4gJ69UrdJA2DuXWy/XOyRPHxbgdlfZKvUnkEOR06VoeuvfsFdq1kzaHdZPaoQrSoTB+SiyDD99rAbNshERo56JpWy5hUFdp6ZQkaa1eyoRFpMjEoTFz9uxZWQKd1nnGGLJ2pe2Urou0WDpaPsUIj/RXL22gEyfFWK79Kjzz9DNh+bIVZjE0dNhV4a477wr3zrnPrIeiGTQ+eIGDM1Zee9oKg3o5BDa4RJOsxgU21Y1h154j4at120KVBDY19WwsiNmzWxQRUywcxtJy64bApoMENoPCmJGDQmcxarhEQ1PaPw//cOyjok/Q6kkEXTxmRwaBDAI/HwjYBFp4CTyFsIUD/MhJGFcmqiygI4DHYvKDDz7QRH2T8GOJaRXeIXw+QfuSYWlDGOmipQ332XElQiAyQPS16lfQT97/1NppJZY10o8RycVac9/+w9qk9kg4VS1ftXKviQsNtGmMyxBNRRADfRDZCL26dwzDKnpIaNNZdLxEAhXohdPUuBFxeoywMF1fVy83KyfN3cgy8Q4LFy60exY/8N8+adKk8OSTT4Zp06eFbt3KczS9pKTUxt2VCO0ru07xe8/X0n31Myz4fr2/eBufI97wsDh8iKvBxL8NKgaWH/AI6X6O4ZdyvXC6NH6J5aXDLqWE7x+naTut8YJXwefUJHsDjcI8Tr6usfZEh8vl8Lj0D+o3HiPCgzhNyyGG1YmETXgzckyXQf7xiLXw9nhoLCfG+bGvubrYTe4pV4yXT7hO+28eJxc5d6OWxMZYmD9EeLYOASJfSv65gprcxELjtcnr7DGDwC8EAoaLTGBzSl4b1oU3F78VXn3tddHoDXIN3luKFHdrz5UHRKevMysbV1pEK9e9V/CdoUgJDcFzBS7Qli1bLuHHm2GzhD7duveQoOZOc3k6ceJEuUMrM96Ub9gVgvjE/VttDUfFr9BxoX/ZhbgyjzsjjjWsGxMmfeX4kEDWRFx5k1cIm44cORxWrlplSkwfSdB0RgIX+OB5c+dpj8ebQ08JmsrKOhrdi3iV+housiwL8Q0CsI0bN8nC5lk7cYmGAgqCoIcewiUaAhtcosnChnwK6uoP+baokonVdKTeWEfDx8GducDmXNiyE1fpCGxOhnP18rxRpLWWNnLtrrgI5SSa0ry/Tn2AwKZXGFUpC5uKTqG7BDbAKy6ZARMOamHtE7xiH1loQV29PyxBwY8ipUDSWt8WJMkeMghkEMggkEHgohDw+YMh9lRcx9YRc4OA8wIbx+aGkyEIRnBYCxAdsXjYzEjR4FxDOHT0rFnYfLNRe9ic0P71jR1UhgtrbC3I1gTqtF+NLGw6FIWBfbtKYDPEBDalbdMCm0g3VHZCM+K8x4mDCreDl0gekjomoZfjAi2Pp9Fzo8WCi/iZ729hI2sWZw6SJjglb0Lk/V3Twu1ZvVJ9ttosbFg0ee65BbaHTWf5ap04cVKYLWHNPXNmhxHyMWvUXR1IhRvq5fdU9/v37Q9vaKHl9dffkF/WjWIC2oTrr7/BFlrYsLpMPmlLtADDYuDxY0clGNoSNolZwdQYBq9MwpwRI0eFq6XF3UWCF7qFepl7BBZ1JNiBIWCfhW3bt8tiZVM4dkzWI1rs6Sxz5BHDR5g2CvfkV696IQRirxwYTmCD1cy3B74N27fvlLZMVTgplyvl3bpKCDUioNUCY9S+A1q7pGGfnnrF3R6ee3aBtWvv3r0m1LnhhhvDH/74R2PS2uIWSO3yjmSRgnFO7WmA8lC9L4/Aho/K97A5VX0+7JagZt36nSawOVsjxiyw345ENFpgM61ofXrFOktLGhOBzcAwZtRAmT6XyqUNn15eYGPyVfqCXlHbMoENnZsdGQR+eRCAPIKvOBBuM4n86KOPwiuvvGIuLBCKM7lGaIMbDHx5M6lmcZ3T8WCC/yyXwh9wOkcORxa+zp5+MgioT0UHjSuK9ElP1t/qE+MBeNIE31io3Ea11eGbTTtsP7QTp2vkRkM0ThN8KARdCW0VI2B5i2yGwQPKw+Txg8Og/j2koNHeXKOh1EBJ/OYn9YSQgWiWeIKD3x4yweBrr71m1jXQ2g4dOth+NTdp7zgENjfccL3lqWysbDbojUJHwrKjEALxW4uh8ZvzBTh4K2i5v3WBjPrHhLjqpSbfqfF68A4K9/2JEoEMi0Tqfxs/yTgiX9OySoQ2+XIVV0d85t7LoS7ep/6e38IjqY4CqXNS6cIozZ6atqFZhO8RUJgn7fE2tZqVXjPOqT/1tnamIlt4wleRFc/Exz0NDzGNXVkua9J06sPCKPskGswTIbxH81/7zlJlpt9RBnm0BNOWwlLZfOdbNS1/qMz0kS/Lx1L6HfcePYGHYZEI0yRmHBOWbR5IafdqTfP8Yc+xjHj9YbllqTMIXJkQ4Dvzb/JM9WmztmZ/uVdfeV3360NPCWzunjk7PCgBw+TJk0LfPn2ES/gIJexorLd5O/PxNtLmwLXY4cNHwxpZ17z6ymva53a53KvUhqnXXx9+88QTWi+YKsXOnjnlH6chfPvKT3XgS0t70ACvWVE5wDl+hVZxuNJA7qXdeF7KTn+RluRQB8XAz/DGaBekzdceFGDKlwcPHwzvvfde+Pvf/i7hzcrQs0evcP/994f75royaw8Jn0w4Bcz0Z7RSeVH5WDblcX/qFAKbDbbWEvewYb2if//+tofNE4IJe/qgtHKxw9tDLNrFL3/OywE5lG+OnqgJW3cdCZu27g97D5yQwKYtKpuqW0mSxpynKbb2sJGldP/+2r9GAptRgzuFbp3Ym0jwtT9rTlIUpTgOdN7O4ZoEWb5eH7/1mDEYPiPeA/PCt/k32V0GgQwCGQQyCHw/CIBkU4gW5GyP4FvRiZSFDfN/DvCy4WPhZJveKwwLG3lHD2cQ2Gg/tM3b9gYT2MjCpu58B+XKenuyDqQVZwT/JcUS/rcPYVDfLnKJNjhUDOwpIY5WD1QMXjmsIjki4PTbqkYl4nvqqCfUBZqrDFjEn/QHWh1PYBJPeIMfLLCxmgOABAieeWF7mhbOM4WzQLdt67bA3gUIbLCYQWAzYeJ1Yface+Xe7B6Z/o7wyis+Fjb18mt7ToIQfNo+t2CB9nt5Mxw9etQYuekzZpiWyJQpk20xD4JfV1crTZUjcn3yseIuCtu2bAvnas6Zdcy06dPDvPsf0F43FTIvbicGgXozIRXzJwEMC4mkfeutt8KbOg9I+IKpck9Z7+BLFvPkYVddZcIhBDy4TsGXvjOcjWZy/OGHH2ph6CMxSltkQXMqDJPFzNx5c8OUKZPkR7dcmiUdTLBkMBFc9u/fZ9pAb7yxMHz+2UrbbHGc9vR5/InHw4zbb5cJeG/byyZqmtsgN3D7okfjZRfYyCWaWdjIR/A3CGyOh5zARq5s0BrC9Fmfnn1QpW0bpRWNhc1AuUQbELpKk6adJKAIbPyj8j7g1z9lbnJ3BGdHBoEMAr8QCERaQHO4hyZgafP5558LDy6zCTs+ta+55powbdo00zDs16+fWdqgWREtbUgbT/KKRI57jjye9Ofs96eGgAtVvBTh7wSHwwqpq7yvEgyPKw2m4bVaMz58TAKbjbvCpm17wvHTsrCRwIaNat0/uuLhN00L+bjOhAEbMlC+6K8dJMFNN+19hAVMsrCvd9BycrZbKpKQEWj43qq9gYWgF154wcYYFrS44WPBAuHgww8/rLE23qrNuCJxJrABiN53fue/F/+24qK4M8gxDwcrfAtdpL6CObVOggfz75m48VvOvVP/kyhfLvn6SV7xPpYT87Dn3HigT30senze5g/HJTx7vnYXx3BB2YVp0vHyb77/XcRp3lav86XlRr31qQAQO5Jv0B4dtqwV8ujtJ8z7yca7ohebuwGUiZxfVwxbELQusp5K8iSPWIauxGt6pN9bh6cieNtSAS3cWp1aCL9gUNJfXp3mdfK0hBe+c5gDF2I4HD2u/8ax6XAjkuJECBSu6KaT/cB7h2CE9A/MLEueQeCKhYALMbTnShTYLFosBZ7XRKe/kYBFAhvtT+sCm+ukYNHb2oFdBxYbKD26QFmLOpoL40ZtkdYIli55O+zYscvcpk/XPPr+++fJI8dozb87uhIGKECfmNEP4QO+/fjFcRO/bwrLhVvJ/tMcP+XxBmsd5Bs9YsS4fiVePi5hNmdO8sarx9qv1oYFzz0nTyQrzLPIqNGjzF3rzJkztf4wLJSWlkrxIV8r6hpRH6gtlnf6NC7RENg8Z/vYoBz13QQ2jidpS3MgQLPBpPREIrA5WRO2mcBmnyxsXGDTGEq1Rpe4lRW9YV1AqzQmsOknl2gjK3uGUUOwsJFVtf7SQhtH05TgbTXezgKtg1K9kcfoeah436Yrfil0pyDT7CGDQAaBDAIZBC4BAhAJpxceOWJisLYmHvFdEsXorhHdlgU2ByWw2bJtnwlsDp+QR6rz8p6lFYA2bdr6nF/5RYFNJwQ2/bqE8Saw6WECm7biy01gY8SLGqXqk+PZoS0cXqm8sCbGtZc/+Q/0Op5xjsH1Bwls+snCJscVAARmf5BSa1u8etsiw8ATBfOcE9jIJVq0sCkQ2Nwzx/ajGSlrFLgdmDAENrUSthw5fNiEO08//Wx45513yTWMlOu02ffcI3Nh7XsjH/SowlAVLF+OyWft+9JSeUYb7X3+2efSujlsmtoz754Vfvcvf7BFwO6J6xM6C1dmdXXS7q2tsQ2xn37qaRMOHThw0AQ2CE3wI3vffXPDTTfeZFq5pIH5wVqGPGC6tm3dYgtCCxe+GbZu2ylBUa32Y7g2/I8//8k2O+zStbMYFfeLb5NlpcOKB42gxWJSX399oSRqu0JlZWW49777wkz51B8lLXOY1sik0UZvqTpZHBMu3C6vhQ172Mgl2t4T4atEYHOuVr5qpUnDkppZ2CQCm2IxZlFgg3XN6BFNBDZqC0IbH0M0TOMq96Dn7MggkEHgFwUBaAG4EroQrRcQ2uzatUu4/R1zB4F7NCbWWE5ibYNbCCxtwIHRygagRCIXAUSenBzxGt9l158aAqwGczoOh12CILOYS5D+kx/6x11o1NSdlybNWTFlu02b5vhpNhfU5F4Cm/NyiUY8F9hooo/QRhRm6KAu4eZJA01gUyp3o7Yo4qUQW6eXqZFg5esxnJWbFBRFXnnlVe2d97xorGizlEDQLGVszZ49O9xr7lhHGi/g/AsCG/nKRXv1V344PAqBcOHvKwpsPE26LxgHNh5smNBbrmWcDA4TOMDXxGfKhnekb8EX9AnXWD5R0TKOz+SXrm+8b24JAa4gth/EQ5DIEfOKV97Fe4/tv/m8UxmlI3yPe69HujxgEY/0fQxL2qFXfGtRYGP1pYEKB364pOPrINxGtE1aHLbw2rSgDbDVe/qHMA4T4iQa3PY9Wmj8tLzd5Nz08Df8WsVyr1uCY+5l6ibCNhV08dvYobnq5G6UtqV7rx/woc35BVbGAMXpR//+3qBnL+yVwdKrFIv1px/r10uxCvxYWWb5ZBC4IiHg9II9V75e/7XNhaHVCGywsJllApuHZGHjAhvHx85vILBhzsnc/ZQ8Wqz8/AsJJxbIivZT+1anTJ0aZom+sw/OwIG+P53hIH3vhrf48HVE+UeCJZPvn68vfoeKFNOkvn1P7nlYbEVP43DKirjMr46DI84gzHC2siAu585dO+X6fXFAiZO1AVy433zzTaacet11k+TirZuUPhFwJG2w+pA2qWPSpkKBzbNmzd5cYDOiFQsb8L+3yxRslHceFgBC9VYIMbhiYXMMgc3uo7Kw2Req9sulvFnYoHzje9jAH+IqHQubDh2KQ9/ePUxgM7rSXaJFYY1fY/EJ3tVjjoZTKG1NDq+lP6SCDaenQ63fY6LsmkEgg0AGgQwCPxIEwMItYWJoRJN3oh2QKMPHIlpivy1Gg5B3veZgWNgcPOoCmw2bdoUjJwsFNjb/M1oiC5s2taFTWVEY3J89bAbKwqZ7IrCBppNxrBOUwU/qw1QvPsV6/+wFNl+t+8oW0q6Tj3dMkZ0joF8EBJvQJUyCmm7MQqrrI5NCpxhTol4xC5tkDxu0PhDYdOosyRgWNlFgM2J4kheTTG0WfOaMFlu2ao+X5eHll18Jq79cY2kQnMyfPz/ccustYbDcjVE+J9YyZ6RZsmbNGglAXg/L335b7s22hHZa7Jt6/Y3hoYcfCTNmzAiVlUMSms+ktUHuzKrl5/5YWP/NN6aR8oaEJ2xeyES3c5fOYfSo0dJymRnm3z9fQpRRlpZ0dLqZY8uyZ/26r8Pf/vY3WecslXnwKS06dhKjdXP4wx9+H26QOXaJ+dxnEQgQ2rBR+6rD3r17ZNWzNPzXf/5VGjEbc25aZs6aZYuWgwcN1mIlpsWU5sPLukB9wILWZRfYyMJm554T7hJtnyxsCgQ2slRCi0b9UKzNBdvjEk2bRI8ZKZdoOrt2koWNpDQ5Cxs1KTLL1rjsJ4NABoFfLAQiLaCBUWCDteLJkyfD7t27zV0VeBs82FdKAghs7rrrrjBixIjQvXv3AoFNGkgsqqXz5D6boBlILtMPzJEzSEyxc7QKWqcaON8E/YKBE5OmwHO158WYSWCzCYHN/nDstBQ0EoFNkFam5aNdbRHWnJf7EwQ2Vw3qHG6dPCAMkWs09ofzyT0MoJXitITi9RcPrGnYhPill142noDNdxkv3bp1kxu0G0xYM13Wt5WVFbHaVl93W5LPJ+b3a7s6bAtb3dK3lY9HX7AIx3hgKCC8Q6hCHt7/AJj49RLeomTDPalwX8eJ0Mb+lAdKKbxsJ1+qWFUC3j/GAABAAElEQVSVmmWV3CNK9MC+edQlp3Gc5ENJ5Gk8mt4XS8jDkGixN5U3ZfvhMVpqX4xxqdc8PPIpLpZv0zT5+L6omc+JO7U7aZDBr1nbJRzXOGe/RL6lEm1abUJIpbQNoAVb8m8UD+mKNg4FwtpizSgXlPZ90X92UFiLEEzeA8c8JAm02KmgfHtySXI36bZfKF4uQZObWEzTGkYN/ibR9UhMxgiwdXrkIeTEHfBlvBJNLSOSnmLdIuw9nr3SW4uth3zf6OF7HLEV8fo9ssiSZBC4kiEQPzPRCujFGc311yOw0d4zbmEjl2g9+4S7Z83WnisPmfJjXyxsTNjMvBe3aA2G40+fPhV2bN8hZdAVUpp8WfzjpjBo8BDN9x+Sq/V75OliuCn98M3y/drXbfn4veGBhH/hvahK8i3zHTv+izjW0qdwoofThiTv1CdrKMNwAoERh3g84vM+j0cIKZIHkSNaG1ktGLwsa6HFUjg9EsaN8/16Z9w+IwwdepXooNSKkzpafamz6sS9lxMCe9hs2PCNWdcUukQbcAku0bxPLC8rxyBGBR38akuhwEbKp6dqw47dx8JGBDb7tLdtHS7RJLBhDxvlYfxhYmHToX1x6NOrWxih/QjHDJXApkvCyxnsaZkfXqo/5QQ2epWmFRY7SRDTkdphnw9x2Hi+2W8GgQwCGQQyCPxYEHDeOJ+b413D+0b/Il+sGELMKAFEusdaAG8bFBAFNt8ecYHNxs27w1EENsEtbIq0LlAMnWPuIsF/W7lE69JRAhutB1wjgc2QARLYtDkfzMImJbBhnuiH6DoF2z+1i/WGD6DO8UyiX4ZL5Cu4Om8BbBBkfUeXaBcW2HgHONDzwI/tc4Lq4bFCZySc2L59m0x9l5tQJApscIk26x7tYSOLmZEjh9uiC8QZN2XsR7Nq5Srb42WFrGuwQOkhv66zJMxg0zxc5/To0T0pVpN7CQrQttmtjQexsnlZizQff/KJRkRRGDVmrARD98pF2X3hmquvFlAYKUzYtFCkNFVVSvP+++HVV1/Tvgofi+GptnzLyjpo8bBfuG3a9PC73/7ONsZuVyKtERg8dTisC/VcKbc+Tz/1VHj3vfe1EBHCUDGJd8lKZu7c+8IYWcqg3cK6AZNgGxf6YTJ9WppBb8sd0P/7//x/4csvv9QeO11kNTROFjYzwxxp/mIGzUJFboQ70G1R43Ja2KhJ1i72sNm153j4ij1sENjUae+a8zBl2vhR1jWcmD0XF9UnApv2EtYMloVNocBGntEKLWySXswuGQQyCPwyIRBpgdMHUJpIpnAwFg9oSG6TQB+3kh9//LHwcZWsC3vJneQUaUjeKpcWY4TrZfIqjUOOmLY1SEXi19r7LPzHhIBoofqR05g0+B4daLIYS2TsAoGwSInARhY2B4+cDes3VYXNO/bLJVpjqGmQoIY9bHCjAZNFntKi5UQJAIHNbVMHSKOmq42D4py035kvyz2hj1YB/fg+SR/Lr/2r8mv/trk9ZWzgag/rGjY0ho9gU1+SclIscahvdgAP60ADhcOlECrp9yy+KYXSMCbgAdWPCUzJBkaU7/2IXNvuP7A/HNh/QO5jTwTcwaDUg8CGgzyBPnsTorCC1R3axX379g79xI+Vl4ML2ptliPWZjTRLmvSa4xbeRZ6rWW+qPozYfD87s+y5/LDfQpgk9bIxdeF8Y7pCOFNHYMqVI89ze1B8VxTOnjsbDh48GPZonyb2VKwWTLFQrNRCX//+A3TfyXhs+gf+86isvPft2yf3vPu1aXd1wCXlEC12Dhw00GBuVo2UJ/7OPg6vQKu/Ds/ktaqbhnlhm/JZ5NvlYa3Fy6dofpeHTPpdhEt8m3qnoFhXYBi7hrqgqNUgQRZjlX2XiuX62IVXwCHmle+DVK6p24u9T0VtdhuhFq/NImQBGQR+vhDIfULcOL1oWWDTVy7RZsklGhY217pLNH2fuEPjxBIQHLZHSo/vykL7zTeXSNixRsKfs2H8hInhyd/+Rq7FZ5iyj3mp0EfON1/kSNNwE1/Y+WQtgK8b3GNh9p3zjGKA4qgscIO9h6aRTuHgCNMHUFgeh6TjEzfiIfCMf9PxahnZD2UFw8Hbt++Q4GqRlFRd+MTejtOmTzMvHyiBlpd3M4Una4vV2fOl5l6/IuOnN2zYIIHNs7be4i7Rii9xDxv6hBm/l2AW0wYVSrBggw6xOLGwOY7ApkoCmy17TWBzrq4kJ7CBCjiOd1fp7UvbaG/b8jCisnu4eljn0L1rXmBD/oWnlah2xUEDbPP3FtujeN2UnsOjJC/03BzeFi37ySCQQSCDQAaBHwQB8HHEyY5z/cnpbcza3hTgbtEPPSN+YQ+bBs0XT8vC5sDhs3KJtj/kBTbaw0ZeNxDY2L5tKguBDRY2XToVy1W6BDZjtC4gV+klUWADZcqVBQ1P6qVynASTR6y3UU5VkzgeL9b5p75Cy+IZaTfX7yWw2SlXNdddd51pO+cIHkBIAOENjwBovWkUDkO2bdv2sGJFIrBZJQsb28NmUiKwmZ0T2LQRca6TEOXbb/fLugatmRfD6tVfhuMnTmqhBQ2Rh2yj4KFDh5qbMdQ+0KZE+ML11KmTYe3ar8I//vsftu/NWS0Q9Os/UEzP7eGxxx+TZu1UaV3SMQKW4rP3zfr160zAs/TtZWHTps0S4iB4kLahhDNYy9xww43hz3/+c7jxxhv1XJZM4FRPpa3avUubBb4bXnrxxbBSAqb2ZZ3Djbfcqs0C7ws3Sot3wID+PhRUT8rNufUwUDaGd999N/zv//1/a++dT/WuWJPlQeFOaZY/+uijtlDJBskR/nYF8Erb8E/ZwyYR2HyzwwQ256RJU38et2gIbJyRLjpfJ/i4hU0vLGxGDQqjhw/KW9io+lFgw6i5vJ9I6+M0e5NBIIPATwsBiJPvdeVujsBnWNkQhtAGSxuENm++KdeSsq7sI+tOBPRYQYwcOdL2I0NTPH1E3BgncjzHMx0vu/+pIJBnjow1SBB6ocCGsvEsmxLYyMJm/aY9YfP2A+HYGVm6yo2G72GD33Mm+cpX46IIoY0UAYYO7Bymm8BGFjaiy8Wy1nQexGKToqCBLDfs3l0V3l76dlgoFyMIAnFDyp5wV2k/OpQ+2L9m4MCBprWaXxSIFCleC7LNHlIQiN8cQX7PolaeRzS2OPnmz52rER9YHQ4dOiS3sVvFc30jXmuT7eWHRjACA3AB/BGMPIvkHeRGtpP4r/LyrtqToI/6bai5wa2suCp0l/IOvBn7Cfr3rnFAl+mHtDD3zlzkxwWvbYwWjJXYz443aEs80u2LYRHfxOd4bSlufNdamvj+wlfqn29DLq6CzKWOAqzdaix7PKJshaISPPCJEyfCYAlg2PNxkqzlbayXusIRMAfHohS16otVJuhBGIaAHOuzXr17+XehfFm4jNrmufJbueG7y1VXwI5tj9d0spZg1lK8dJqm9wYd/Vgv6if2JpWIY7JZGtWLcYArNI5Ik+rq6qWsdcYEiIxDtNk7aL8rxpjx7bmGAfN8SU3z9+c8fmr5fWuhMd94bS1eFp5B4OcHAce/fD98uRKE6MTbRKGFjVyiycIGV+ZY2EyadK0J6/mecZcOb0D66rNnwrp168JT/3jKBDanTlVL0adPmDZtenjksUfCZO0f21YuxOAZSYNCAEIe1gnAD3zTuDaHJ8i53I0VNEwCH6JaKn69NDHBEwhzqQcnOACFynbyoJE+sB4FVYBfiBd5XNZCeKZcyqRezPnJB3RCGUeOHA4fvP9BePGll2w/XnD68OEjzGJo/vwHw4CBA2wfXksQC6UsZWA4Sld4aQQ2vodNdIl2qQIbYCPeSw2Afitnu6qGXprKMrqexKqXgNsENlLkRGCza8/R4AKb9hJ4uzKnAUO60uw9UNquKHQv7yKBTbcwfmSX0KNc+/JQd50MCUrJn15mnjdTToJf/tB7jxIvySvvtxjv4rg6xsyuGQQyCGQQyCBw6RAAH0ecHGfhjpQL3whV53C3v8HCBkrTqPlFFNjsP1Rte9hs2lKVuERj/Rv+OxHYiEgUnUdgU6d90duGikEIbPqHQXKNhnWN72GTX5OAfkEkrGjRGCcUkYJRDygb9Y2nbi/TAS2Lp/MAqonq+J0FNmu/wiXazhBdohnBo7G0OgE6j9ZYu7beQpgcBDbbtyOwWWFMxCoJbDpLYOMu0dzCZpRZ2ABAtDCrJQipkruwt8ILz78gZm6DCVEqK4eKgXtYgpfHwyBpAaJ9CTFvaHAmDKIPU7ZFE9EFzz0f3njt9YAblPZlncLk628ITzzxhPaUuVULAaU2CcOl2Nmzp7WY81H4+9//O3z66ada1DkuwQqT2vMqU8KH4rbydz8x/P73vxMjOC307t1HacVk6X1NzVnb3O/tt5dKOLQobNi4OfTqO8DcvM2bNzeMlgu18vJytYkFCM48nIzxUB4rV60K/+f//Ht4Z8W7VnYPafvers0Sqev48ePlZ7aT2gjTxCDzDjUoaXGDjt28eXP4y1/+Yi5fYO6ANxsqP/jgg9JQulsm4cNMc5LynEFkAF/a4b0h+Cp6vRwNnjorgc1eWdh8vT3slq9aE9g0lqjHosAGbWgENm5hEwU2Y0bkBTaUnhbYWE1Ut4ypurQ+yWJlEPg5QiASJsN7SQPS9+AtFhF37twpK8ePzCJix44dtsjIAiK4d7SsFXGPFtNFIhfhEZ8zXBIhcjmukSeIzJqonWid/1E+VATegX8ENkWiG8Fcoq3fvCds2o6FjQQ2DaK5ZmHje9gYn4GwRicWNsNkYTNtyiDzWdvOFlChhd4+L5kSKckKUjmNYbuURHAtgtBm7dq1JrDBimDixImmEIFlL/TZ6aLnJ1KU0KIUsfZiftW/8ZtLAyH9nbEghgIM7k8IrxdvwgI46XBVu0Pf9VdfrTOXtVu2bDG+7ITcIWLhDMx9nPglLnDh7gwXXVjWYe1sghvtKzj0quGyjJoQrhk/IVRUVIQyKdEQj6ONFuBMIYfBobpwwDfxbwNEP/m2OD9lkSyC3zX9JX5sa7wSJ59P0xRxDDUPbymkaT7pMqi0vzcgKWO+IsoWp+zNs7qR5tuD34aPJJh86823DIdiQTNixMhw//wHDH9eddXQ0LEMlzrnzRUli53sK7li+QoTlg+pGGLuhvkusEIr61Dm7Ub4lToo3wGaCkxu7StMqkolY1viNZ3iwu1Ox2z9nqJiPnS59XUuzBdILbW9I7ZP3vJpXGnAFQaqbKETweIgKU7hTqlf//62mNtGINDQsmbn2s+NtdVDrJzcj+OT3GOzm5bSEKm18GYZZAEZBH52EHBcr1EOjyC6znwfYb3tYRNdoq1jDxtZ2Mx0C5tJ2g+2T+ISzWm9FC3ra8MhWRN++tln4ZmnnwnvvvuerOFKw5TJU8O8efeHW2+7JQypGJzgHwl3tAaBpcnBg4fCSSl11sqqE4WAvlIEANd1lwUnfADfswmowROqHfWt0X66uCfbK8vFQ4cOmwCmpKSd8aKVlRVaE+hpwnPwDwIYEsJ/cMLXnpSyKesZhw8fMlwFD4IyEuV2lmt40nEQFzy0SW6B8bzxyiuvhK+Fo/v06Rvu0lx+fmIRjPW58SxKaHRA6cCvUWADH41rYSxsOL+bhQ19wknrQXisPUhgn8JLLLTxFvJjAhvtQbhTApsN4ud2VR2V5w0sbKLABrpMbBfYtJPAprxLWRgugc21Y7qFXt0SgQ0lKBqgyJ8JYCw9eRTiVELiEWP6c3yK1xgru2YQyCCQQSCDwI8HAbBwxMROM8HT6VDKctwe4/lbt7ARHckJbBrD/oNnwma51kRgc/ikrGjlEg1hDWvybWRpY+voEvy3lYVNeZeSUDm4PIwb3S8M6tu1UGBDDcyyJmHarcGiB1TEagf14h7KFuNcXnrBHCSe0O94fg+BzVpbOENgg9adTbZoC9wBJ820tuUJKMEeZq9zPzAhUWCzXBPDuIcNjMr4ideahc0c+ZodNXKE0guIYnKqq0+HHTu02LJwkTYLfkFCia3qqLbaQ2Z0ePSRR6Vt8rDq1dfcFWgJSP5sXWBjDJc6Yd/efeYP9/VXX5cGzvpwtqY2jBsvM+knfxNm3nWnJv7ayFqLAHV1NbZ/zYoVy8J//Md/mlYiiwMDBwwyBqyqaq8YvbM2cXvgwQeSPRWG+0RW5Zw5cyp89uknEpbgSu2DcODgkVA5fGR4QJow9913r1nXtBdTCKhMT8Vg5jA0KApo677+WhpCT4elWlTaJbdv7WVRM23atPC73/1OpuCT5QqkuzFiNhEG7pYb61iXT2CjaW+ok8DmzLlgAps1CGzkEs0FNvJXqw9D7Kn6ToxjFNi09T1sxmJhI4FNOXvYSFIjxWjbxwbhVfy87buhbS0NoNxIym4yCGQQ+LlCIBIm8J6hQV1t0QxCRZiuEComm2jes5CIm0pc/DC5xR0E56hRo7R4i9azCLhW0eIZ4RKJXnzOrpcBAglPYBhdnYnWLPQq0iyrgWhEFNjUSA7zLS7RNleFTdvcwqbWXKLh/lPasIwILf4jqAmynG2j67BBcg2CwKZfV1O2MI3MOJCUglL9VBa6ZbFk75494RNtQoxfeIQEWHegNT9hwgSz3Lr++uu1aNPBqoc2LOTH+RgyzmVu73/tP/atNgFCml7b962+QmiDW6l6CdpOa+H7sGCOdTUWH1+sXi3lm/UmrGGRjjT0RycppeC6q6ysowleCMdFGvsTYPFwTq6+yBfXsu3bd5C1Tb8wdtw1ZvkMj4S1NYI3+o9+dKENTDhjgp7UC7ozGaf5ttDnvOC8+JFuL7Hz+RSmbRqv8G3zp6b5FKbXV2T19rYwuL0pKCol7dOEB+EWykksXL722mthhdwEHT58WFbaY8Ojjz2e4l3hRxvliu6kCc+wZlzy1hLx2zslCBsanpAy1P3z73eBjfCs748gWBqc8nV3iDWHW+6bp2qqd2xLvOZzKIRfS+/TcVu792L0q6pQG6cmMe+0wMbhRj6MT04OFLyOHDki/LBV8PjS5gto218vJYGpU6aGispKjTu0+6S9V8J4ScrINV03NjG07JIfj5cOKbyPieO18G32lEHglwqBBAXrOwKvRYHNGQls1ptHDN/DxgU2M2dqDxspHl6HwEbuMPkG/fuTomWNNHFF03F1imDjy9VrRQN6SsgzOzzy6CMS6I+TFWY3iw9OOqH97IiPBeJm8ZfH9NxVNGfMmNFhoviBEbLgxnsGeAicRz39PC/LRe1xK4vQVVKuJA8UEeAbKioqwrTpt6msq40PJS14mH3YzCpHax/nzp4Tbt1hvOxmee8AtwyUlQyuWFEE7devr3U1ZZGmRh5B4FM+kZXkX//6V3nveD90lIBnkugcbtKxNsdVerT6s3Qqh4P1D+rwwwU2YtDsgIbGM4+rwJygzyiwOYHAZu+J8I0ENjt3I7DB64YUA4pkYYOLW8FfYnGddbKO1t7AndrLwqY8TB7XM/Tq3kHwdpyq7lWc9MkTpXFyNMer+Tcew39jXeM1/S67zyCQQSCDQAaBHwcChfjZn3Iru7kiHLdHbO2xENhwh8CGPWxOn20Ie789I4MLCWy27kkENqIPCGyKEdiwjw3qA/WaJ9aGbl0R2HQL40b1DQP7dDGBTbFkCDIfEKVQzrZ/DfQroSpGDrxsp17c87Y5jbMXP/EP87p4Gu8A/6DzewhsUhY2EoxYZlQeGGghhEP58ptcueddDPdnftMCGyxsnn32OTE+srCRwOYaMSyz59wb7p0zRwtxeYHNGU3Ut27dYlYjLzz/oqxzdtpEfbw0Kh+VKxN8z7OngVcBLRZ3hxZdjh2TpcnKz1eGhQtdu7Zq34EwfNTo8Nvf/jbcd+8cWwAslRnz6TMn5Ud9n4QlS8Lf/va3sHHDZtPoniamCKbrgw8/Cnv27Au9ZWZ9+x23214710+9XpsWl2sQhXBMmwS+uWhhePa5Z5T2GzEnxWHchEkSKD2ifWjuMsEQiwy2WZIqm2c4HT50FqbLz6uNb0or8ptvNlgH3nTTTeFPf/qTLVD21mKlC6IAv086gfPlEtiw+EaP1zW2JLApkSCHfWxYAmBw5AU2pQhsytnDZlDAwqabNhcsTQQ2UXjFR+VmcvSkD1aHTPabQSCDwC8RAmkCBQIHz3IQHq9MOHHX844WHRcuXGgTXjSecY82Y8aMUFFRYQu84MXoWiKm5RqJH/fZcRkg4F1nBbF2CZ2CbnDnJ7+G7TXRb2MWNiaw2bTbBDbHcYkmS01cop03DVVlgsCGUzQFBmz4oC4S2AwOgySwadtWgjoxIM6DUGwsi3uFaywhsMHVyuFDR0zox14duNyDH2GBf+zYsTaOGD+MPfLy/HRjR7wmj7/iS/w2Iwj4vpof6msJXNl/EKuZWi2EH9ZC+CeffmaLavB8uKhjvxr6AEEryjHdunWTm5eB2ih6kCkHdenc2Ra1cG27S1be7F148OABswghbaOEFGhAl3fvoQ2Zr5Y18oxw5513yaqYRSxc04ih1wpQifJmjPjIi/XN4xivf+Q54tVDm7aX0JbbrNwTvNXae8+x5d902vR9YWxn6mmHHSxa6gaBA3Ck+Ci0PiAN8vc/eF8Cm9e1X6QENocOiq8erQXMx8zN7ogRw0P7lEu0dbJ4Wirr8OXLl9u3UVlZGR4R73qvFgZ7y5KJRUlwrAks+C6Tg6+bw6EaYZu8TF8S2FjcFsZM621OZ3Lhe2qSro960kOsiv7Wc9A9s0b943qIE7idkPb7Oi3iLl26VMKud+Wi71uzSp8/f77tXTlkSIWNJ/j9du1wYUQJKiPXbG50UpQdjKV439o1RojX1uJl4RkEflkQiCjhUgQ2CF/wFBEFNsXyUe/fsNysS6nzc1nXvCTXYeC63bv3hgH9B0lRcl54+JGHwrDhV4lH7Czc5dYhx44fNfdpixYvMkXQqqo9tgfu9ddP0bx+tglC+sh7BntW8S2DX6kriycIXN56a6msdRfZXrM1UgAFN44bNy788U9/EL6cY8/gE/iOWr0HNzeIfzkqGvjxx5+Ev/3t71Lu/FRufaWoKs8ZeL/gxD0rB+VYGqWrOVdjOOnf/u3fwiJZHYFvBg8ZbK4tgce1clXPvjzFCU52Hhq6h+tKzdNlTRT3sPl+FjYIbMBN6YWsPK4C1XGiL4CFzYkz9VLkdIHNjt1HwtlaKXFiYSOBje1JaLER2MiVjdYFOpW1Mwubqdf0Dr17ILAhb7VS+VGiP1lIUhKlcTTHrfk3HsN/Y13jNf0uu88gkEEgg0AGgR8HAmBgx8I+x+epOd4Ft3tojO/zGpu5JwKbUxLY7DlwWhY2e+UWba8JbOqwsJHClFnYILhRJrjWLJHAhv3Phg7prjXmPmFAn04usNFagNuEUi1K9HkLdJGS/Ze7lDKXUjCnjDXUzWU5mP/Ek/rF8zsLbNBC+UF72AgePokSeMTAuIXNDtvDxgU2q6RV6RY2s++ZE+bIBcNICWzaJBY2pzVZR8v6lZdfCQskzNizZ2/ops32Jk2abNY1TCgxYfYeAPDqdp1G9wXqUydPhY0SfixZslSb970i36rbwoBBg7WvzFyzfBk7dnQoF+N0/PixsHXLZjFji22vHBYUcLszd+48MVDBrHQ2btwcOkirc4q07e6RYIkFgt4ySUY7aL8WgV5+8QVZDT1t5tLdevQK199yW5h3//xwi7TBO3UqE8PoGjtKoJppMKmSNumjsgrbvHmLaQgtXvyWXIZ8ZQsa7Jnzxz/+0fyJYzaNVg6pOWIHN4qxo2N/SpdoABhxDWXXNbZxCxtZ1qz5eodZ2uCrFoEN0lHrbxPYYPpcL+FMg3zVlpqwBqFNdwls2ktDkQ+O2PHk46LjXMrJB5YdGQQyCPySIQAO4zBcBubQpI8jEqza2lrbf4FJJ0J+tA2xtBkwYIDtrzBt2jS5+hlhQhsWaWM6rjEfu8l+Lg8EYn9C0lRi0rvJXQxxjN8QBTbaXPDrlMCmVgKb8xLYBJk+G8NnAhu3rikWDRo+2C1sBvaVJs2lCGxEG9GeZ/HjrLRc4UEQBDLmOksogGs9rDqiMgQLSFSc2vo4ymgRgyd+q9zHI35nPDvdT+KJ/jfK8heBzYH9B8LqL7+Upce79v3uEm91Uu5e+F6B/eDBg+0bZsEKYSyuYHH/WqI9BziwsDl27Ki0jQ8ab4UCzxa5f2Wh7YT4u5KSUkuDEs/MmTPD1KlTwxAtalkeKgMrG0cHSacqT+tRe6SXOZxJzl899GJt9ljNYVMAl+SbiHEvdCVdLDOdR1Kb5F1SZ41T2sGCIq7jCGUMoxW+XxY2UWCzYsU7pqk9OhHY3CUYIbDBHTC8MhZOaHzjBtj2sPn2oG1Kfcedd4Rbb7lF2uk9TKAGZ1asPomLg9TJeXvuWj4inNNvY7tiO9PvuI/vm4Zf7Jn2x/rky439G68U4DEBWNyPggVOXCB/8cUX4dVXXg1LJLQ5Ls37q6++Rou+j9oi7pAhQxy+ms+1K5EGu67AxDqBbOONVZQaWC3sqfCnaXjT58LY2VMGgV8kBOInqe+xdQubDdqLpq/tYfPgAxJQXOt72LQpllBCnw3pEMAsX74sPC13aJ99tjKcPnVW+G2Uebe4//77tRdsfyl5ttO3y0KM05Oqqt1hoRQsn3rqH1pf2KJ5vRQKtRYwT+sC8+bNM5qEEgE4yrCF1gDgRdeuXRde1B61CHWZb2Nhw944w4VP//infzE3kj169pAwXAonSluv9whrzslahvjLtDcuVkC4oOwpvHrHHXdYGix8md9zMJe3U/N68DoKS08//bQpLOHGrUz7td0oRc7fPPGbgEJnWceOhpcMn6pM8GdcV4DXwSXaM89czCVaJyu78AdrHbfY8cUuwS/2WYyoslhoQ2DTgMCmul4eN05pT8KqgMCmOhHYYGHDhtGse+ASDYFN22I5uWnf1gQ214/vF/r27Gg4lPqDouktMKOf+rX/fAUgq4qaO+KbVFCSWhdbsMs/5hJlNxkEMghkEMgg8P0hkEO84HZ/yK/qOjbORVEMcHvE0TbXVhqzsFF4oxA6Sv+nREeq9p8yl2hbtktgcxqqoT1scIdmp5THlIlWnCWwqQs95U7zqoru8uDUK/TrLXqoPH1HdaxwKI/fSFF0awe14lS9ISZ2rzhGp2INLeJP/mN8hurA1egfNFDnDxPYyMKDTAzcmugZg6BGIpHyQpq3yyuiFEqHMIXJIVoqbmHzbFi5cqWEGb6HzSwJbNBwYQ+bImnQFCn+Kblr2PDNem2893J4fsGL0no7IF+zfcWo3CzrmgcDk8q4NwydHwcAHUDZZ1UePmPfWf5OeEoM3Zdyx9GlW49w0803yax4Tpg+7bbQX6bIaCCuWbNagpnFEtoskenzkXDdtdeZCwlNU8VkvSproNWhRkwbgpw5iTXQgAH9zK/tls2btMfOAgmWXgrH5S+8UpsD3nXPveHuWbPDBJk8d5C/8LaaFCKkacRdiNrGxBdLIAOpQLdt2zZZEi20OqyWy5Dq6nO2+IDAhoXJ/okPbYYWB6ykM3bqi8sgsHGRjQtsqmuCuUL7Mgpsaku0WTQWNhK78EWqfaYRrU2h2hVHgc1As7JBGtrBBDbeX758h+maj5P8x2XNzH4yCGQQ+IVCwGkI9NLxdXxmAY0D/EYYe3IxycXS5sMPPzSXPyygIbBn0lpZWWnu0RBoO41ygDldurzE10v+Nf6C9506MT8WdtfJwW88jXuwd9CKs7UhHDgsv/Wa4G/apj1sqrVfXGJhg8CG2Oe1cIGFTZvzWhyRIsfwIS6wGdDHtWbTNNQXBLxUSqY+Zn2AIFD/LG4zJuI4YXzxHDVSocU+BsnDmSavseX2q/2J32VTAMRvjffxtDGghTQsbM6cOi3XM+ts/6D33nvP9hQ8J63jthKk9OrZyxa6JmoRbsqUKXJJM8Z4OzSWyZe+YcGKkw2mz56t1ne/31zZoKG8evWXYbvcd508Wa3Foka5rx2g/Qpuk6vbu+TGyhfA2Aha3aiD/vQxYG3g0cLsKfmhv52XtbfJWE7HiPfUL8IkXuM7rhEu6TDuLxY3nS/x0/mYMlJBndUIxqvaXidYc7gFjAQ2sg7BjeSruESTwOaw9luAb8XCBoHNyJEj5M6XPQO0EKn+2LNH358WFHft2mU8Nzw1worRchOEUJPvA2G6b9yNaxugZ0C0+1Z/vIoFr2MbLwSLdLsLErf64LXJ1cj6DgzUwgF/Su0VJwpsitW+Y+LbP/tslRS2XrA9M3HBZ3tcyY3ctGnTJQQcIuEgLhnAH86369f6wK7NiqL0lmoQw+K1WcIsIIPALx8C8WPV93hBgU1vCWywsDGBzUTRCLlEk8AG3NXYUKf9ug6YBw4sV9jjtm1xqb7b6+RC7SFTrOzdu4cEzcl+ZvruGxoaRTNOyKJwidyf/4csZdaYMGawLDvnzr0vPCYcyd6vuOhkIcnwldZ0EH5gIfPf//0P40FZiyAv6jFQlqGPPvFoeODB+aGiosJcrIETwM3EOSaFUJSNsBL/6MOPpGxQpXb0kxXjw9pD7xGz5APPOtpiXi9hjfge0rNfDusSb7zxRvj66/UB+nnddZPD7//l97Iyv93WP8BL4GdoBDTMrIlUsSiwQTn2mWeeaWEPm8etrSg3ND/cTZ2Hix8Xv+YEJ+m4BH3J4UYisAnhpDSjWWhbv2lP2L7rcKiuaYvKptbKsLCRkNsWxxDY1Gjto06CrTZheEW3cOPEARLYdDJlA7IFdklpVjx94OFx0Kg6BiyF60U+tCnGjS9JrSO5+EP2m0Egg0AGgQwCPwgCOeQrIhl5a2FvuG8/HT+Deu1U/IiG43o9uJwputOSNi6w2SdjDVnYbN2xPxyRwKauSG5KEdaIjnDCsxcXNWh9WQKb7qVhWEWPMGp4j9C3l/YxVckod1ILpyP8pikKLabinL9EgY0m1Wx4l4O0AAwzQYPzix25bjAiqpdGVKGrPgHThn+JwIaFNxiIKLC5Wi7RZs3OC2zUFyrKGatv5NMWc+cXXnjZtDUHDx4iQc1dYq7m2mId2rEIE2D64hCxya3KZUJ2XJpzH33wYfiP//ovueVYqVlt21BRWSm3OjPD4/LTfdXQSvm4rxIT9r42aX1TmoafyY1HbbjxxhvDE9JiaV9aFl5/Y6EWC98V87TfmLN7JbB55JGHQ4U2Zz0u7c+PP/4wLHz9dU2S35N7tVNhohiqx578Xbjx5ltsIaGsrNQENsACgQ1wY3NcBDjUWT9hpxYfFi1609xYfC43brj9YCHj97//vWn4wRTmtYCVQGlpZ4PayOLGT25hI/jyAdaLcavWQtvu/cfD6q+wsDkRzkWBjb44V2ZRTPawwWStjQQ2XUrD6JEDwlhZ2PRAYKMNB/FBaG1XrlhUMXqcMYsfFyMoOzIIZBD4JUIAIu10AZwYFQAcB9BeURjhBF+4RbsRoQ3WlghsEPjv37/fNPNnzJiRc4/GpJc8WaQnTxbmI878JcLwymqT6BgELum589Anq6CHEc4BnqdncYmG4H//Yfmt36gF4+0HpKEpxuy8JvfRwkZJ2LOkTU5gc94ENrdNGRgG9kVgo4k9NJRM7dA4ohzqoZMxgHVNoxZN3DLAIyIwgOHjMJpjeUBP46IH4f7ea2xRs58WIQCcHW4OdtFy8SXVsmLauXOnfasovGyQpi8WTqWyUq6srAxTpk6xvfnYjxC3WwgJSiVgaSv+jD7xviBfxo20jvVNn5WmMm5ut2/fFlauWil+6yPth7PGrPC6dOlqQol7ZKU9RxbQFdpoukOZu1kxPGOTCWVlYwrelSFC3j4G6GeLZyM0/84ipH48Tv59zCMVJcknH8L4TIrKBcZ8CIj3XOHpYtwY7mOamNTXTwvTky3uSZObartApViCLQQ2H4RXX30tLF+xPBHYjAkPP/qouY1zl2honbuw0vcJOm3fCvwk+y907tTZ9hQCh3JQJ/oGoZD1jQGS2jgMLVL8sSD9qD3+Z7d6m/tQPQ9lGuFHW+MZw2J2F7ta/HzWXtlcvcg3nQM19nLrJehCIMgcBgsbNPSx5mc/jJOy3jLXy1rARRCIwKa0tJ1g3MYENuCdfHNoZdODkOah+bCW3jXNI3vOIPALhUBEGyawAR83aH1AvID2cl0s91+vvPK67mVhI/dkd8+8J3GJNlHrEL31CTFnDJojV8sDyA7zivH3v/932LVzd+jRvafm7rdIofMh4bo7ZCFYLpwFDPneKNTXIT6TG7XnnnvW9vjat2+/CYJmz56tNYHH5CZ1nPax6WjuxttIsVJshFl5Llu2PPzXf/1VygKrzUoXuke+vXv3CrffeXu4d+69Nmfvi3Kr/rB0wcoGAQ0u2LDe2yTlTvDtsGHDwh9+/y/hYQltsDSFT7UaKktwOjwseR8+fMiE7wh7PtA6BlY2o7Qf2W9/+ztzDYy1OXs5ggOhkeBQ3LmRlwtsNlk7sbL5Vq4yEeb07z/A4PnEExcR2IgWWUYsdqUFNgktpfX0nPUemtES2OxGYLMRgc0h8XclLrAxl2jKwzyn1CnLGq2H1Gn/4CIX2FwrN6g94d2l1Kq31D2eunUez8IoUYcu4PDcQWRLEX/jG72waBah6csYKbtmEMggkEEgg8D3gYDhV350Gl0AMzP3j0KbfKaG04mWBBHLMbnz4w2iXdHCZrfWlTdpDxsENkerRUeLZIFZhBcV0TbRCQwgiovqtR96vbnTNIHNsO6hT0+5b1a+BQIb0Sbq4yXH+UBSZ6Ne8R5GIcZLKnkZLtDueMY5EFfWL5hDQ/vZv/mRRx5pVhu5fWdq5sc6MU+7q3abphl+XYE0ghp3w+Xmskx2OL0grjE1N8qK3CzQN5BFiw+BDWa+n332uSxsOtseNghsZpuFzQhjsBDYsEEgfqVfloXNS5pIHZDm4JAhlfL5OttcmmFKjH9ahggCG9fU8eEC2afBZ6vPhlUrV4W/SgPn3fc/NDcanbR534zbbw//81//Vf5nx1obFy9eqP1j3pS7jS0yMy6TkGSatG0et/q99dZbYaGENuul4dJdPtPvkTXQb37zG9v4eu++veE1mTkvWbpE/mK1f43Ad8tt08If//xnCW6uM22b9qX43NMAFgOEb1uAwoKRgUrxgR2u52BUX3v1dcHlMzGjtWHylMnht0/+NkxTXRBUlTB5ViJjaZSGsi7HHjbUN24Mhduzail0Vu07JoHNtrBzz3Ezfa5rKNHHxqdisfXtanNgM1mrD907t5PAZqAENgMTgY3kZjDdSWwbO7TKxgmtyw0i5ZYdGQQyCPySIQAe43Qa4ngttje+Y+GdTaHZrByBDTjypCwwKysrTWCDgH3o0KHmvoc04H7cWkRriphfvMby4nN2/aEQgNY79odCoS3jh1Fn3fKOA+YJxkwWNjXnw75DWqSRhc0WCWxkLGECm6I27SX4d1qijlTSxCWaaMaIivIwfcqgMACBjegFJMNO5WxjhXJEY6GLuDXavn27Fj2O2AJM167loVz7znXt2tXcm7JIYkI90WJPm9TO8qUBuUZQ8Z/8AFL8Ox28tOJaSkNbOMjnQu9pHi202Kk0pPXHVD5N3hOHg2Dja2zxSqUpoF7fKprBH330oVkso1XMYlFHuXOpqKw0y+Hp06eHq2V93LdvH/fDz4KVFrjoy3jEW786D8ICGBrLuIpZvnyF+fbftnWbrEwazAp5tva3ekiLYKNGjRRvKK1d8rUMqCiuW2S5U+cCXZVmC1hEMMGHWfTAR4o/UzrctiGkAEq0MfaLwVd5Gnyt/d5uhB0IAcyqS3nAF7fV4lkUGsPztZU2tENdUwN4QOMLHffRGaYIpbKoE/G8DOtFAzb8I7gQd3NRUMZYpxwEVCUl7cJBuTVzl2hv2IIk43+kXKIBl7tkgYQbybL2CC1lLaL2AZOGetfOtjok7ad+PKvxNkaoPwuYesyNq6RmfHIJDIGt79vAPIEDYSnfWTvtF2kLk2qb5cGPcopwJW4cu7xB0MoRN/Dm3vpTV/A7cYE1+yRhhUWaKMAlT/A/8IgHYcCPdPyZwEZpaSN7TLDf5WuvCmbLV4RTp87Iymh8ePChh8XP3xoqKitCu1IJFGVd065Ek0VWjHOH91MCKoVSk9h3ROKeazwKHhRI/JaOpvFaipOFZRC4XBBobZzyPfIuz8Pla5Qfw/Hbtu89ycrxB9/5ee1HI15gvQQ2UlxEuPH119/IJVofuURjD5sH5PVigi1cuMsvCVHkDm3tmi/lgeMl84KBoHrgwMES1MyUQuf94WZZX5eXYyHotQGHUw4KQN988014Y+EbUpB8w+b13bp1NRdl7I076bpJ5mYTPIaLSebiVVV7bH/Zf/zjH+ZmDBrHZ0ubupR3DROv1T6898ySEGW2+NBKQ5u08/QZ3JJtCK+8+mp4XXuKsYceApbJk6dobv+kCV14VkZWScNNwm2UTXrctX/xxSrbN+dNuUvfKY8hFZVXyZrnAVuHGCMryHLxNRzQDtI4XQkSKp0xRU72r0FB9oBcZYJ/8dbBHjgXFtjQM45/DUEq/yLXxqQkirNfeDmVKkVOuaC3vQdOSmBTFbbJwuaMBDYN59srIUo4dILykyJnkTxvtCmSMmdbFHC6hZsnDwn9eqmfqLtO/yNnTv78SN/7uIkvvD4xXhKqCyE67bWuzSPko2Z3GQQyCGQQyCDw3SBguFU/CONzdMExNxk5Fs9hYoUIiydpQMcRj6NqickFApuTZ+rkwUkWNltcYHNM6wINCGxkYYPAhpVz6R3olP1mSb2sajqFEZXy2DC0W+jdvb3WnfVO+bG+7GvMzDcozU+RmOSgzrHeVOpnLrBZL6Zmz56qMGHCBNOCVPttosSEsb6+1piVyCDACMTJGNCwyR43wAEiLCCdk4blrt0usHnqqae0+d5noaMENuMnXGsWNrMksBktl2jKSvHPm1bL2jVrtP/My2LgXpNW9UEJLiqkcTNL+9DcJ02a60NnWdiQty8a4G5MggJNpigfJrKuts60dhYseF6ac8vCju07VfeGcLOsX/6v//W/wiQJRfZoceGZ554x0+Pj0uAcMHCAhEIz5Qd3fiiX5uY77ywPCxYsCJ98/HHoWNZJ5d8dnpSGywRZBm3dviP8+7//xSxwDh85HDprMWimNhH813/9H2GshEHtTJsXRiQCA4Do5B8mTScT0p3aVHfRosUy734jfP755+Z+DcbxiSeeMAubiopKaaRgWmyNJTM7zGWIGLWf1sLGhzVDmzn0OXFnVfuOhlVrNoUdVWwuqD1sGmVCLgkogh0+Pn16cmEjkzVZ2HTrUhJGD8fCRgIb7WHTQc1o28ZN1uwjSqSmNCjNlFkDs58MAhkEfvUQsImscAuua9DCXLZsmZ2HDh0yd0r4K7+FPRekrcjiIAduLbiHRnHGI+bVUjhx0nFjmux6MQgkdK0gmk/6+Y1vIxsFY1Ytgc1+CWzWb9wtgc23mvBrEfW8XDXJwsbjearzWKXiEq24KIys7BZunzo4DOzdKddP9CxMIDwAJBV2Eb/xW7ZsNotV9uLrJSsOtFpHjRoVKiorQ0/tlYL7LfzNs2jtizm+uPzP6v/0uASMl1IP4yFScdN5kD79rAxTz+RPKToENOIRP5bJM4viQJMFMp45EDCQzHvUguzH45LVeS1SVZv7WfYV4Ds9KGEN/dGvf78w7dZpAU3mW265WcKavlq0MWYvydH726uVlJcvgkL5p7qmpfzxxx/ZQtRHH31s7nIRxN1xx53SRHo0TJg4Xn0sdzjGTNI+FsEabfHtrEy72H8Aa5EOHbQpstp3TlrPp0+flFbyaQkBGoU7ymxvI66MD/hRz0uLUxIOcBiP2YDSjLuwwW0bFta12i+JBTfiM75wzQYM28lCA83tNto0EzgjTCgyDe5koU0NNwts1dXgrDTAHz6WPQLqVa86WX+jOX3ixAnjbQ0Y4nnL5Ha3W7duthh4+PBhs0R8Y+EiKSl9IIHl0TB8pPZ10CId+/yMHj0qdOzQ1iYzlIMAq6YGnh5hlb4B4Uw2s0bgodc6DOqqiepjg0ZXBTEGCDM+W/A8d7YmVJ+rVl5ndZ5T/ZgjSHAuHhhhHa53WJyMOJmcaZ+PO59bIIChTow8xgxCG+DMXhAsnoLTecYNHIum1fSb9kSqUVzKAuYmiNK46izevbP2yATGCJrIFzjCo1L7hoZaCXvOhgblc1h7o32x8ouw6I3F4b33PjSXxGPHXi2t+XnhxltvCUMqXWBT2k592l7CPOVPfey7URu418WeLcy+JYUBJ5WX/rZUeOrwuqQCklv6niNe/Sn7zSDwU0KAscoR6UC+rBbGKWObuDr5PsGvfH98n/lxmx+/BXl7MXwc7pVB+BW85gKbxSawWScFyV7aJ/buu2fJYuYBWbxdE/pJwI8LMNBQ1Z7dEtovN5doH0gp4NjRY+GqYcPD3HnzRWPmhIlat+jSWfgW5kAH9WN2iqAWV+nsq4a7sI8+/NjoAK7WH374kXDbbbfZPjbxmz1x7ITqtT4sEj595dVXwm4pV1I+OB2hNHhyiJSFmPf/RvN15v1eXkNgTeCL1V9YuiVyt35EwnOULxHssF/OFO3HC06LuMPgITxm/Iiup04eN4HPm1LmfEGKq1u27Qh9ZSFzuxRO2fP3xhtvCn1k4WMVUqHgY7f+E38leLIugLAGBdnvJrChb/kjT351gsdomP0SyhO4GoGN6H5NvTaLPimL6V1h205copWG+sYO6l+nc0WJhQ1Cm9CIB456s7C57frKRGAD/yWlAGWLlS4wpgSGJO2y/rAQ6qDD+jWpW6yjv7BfT00OOlmw4zY7MghkEMggkEHgR4QAOBiOnasfdpcISeyiYHC4H+BjxfBIepEI/fUSwb8JbPa6wGabLGyOnRF/0aZM9NZdopFLsbZNaVtUGzqU1mtbky5h5NC+YaiE/z27usDGhDWKaFSKciAmSflWrJ68AtTbFcskBtJ9mnchzk9/wBfFM/IcXOFXdn4XCxsWxnZXVdlmf3204BEZLp80nrZJI1rOTKxKS9uLueppri2MAVE7c4UDKP1jCgwD8bY27XtBm/dhWsxE7upr5BLN9rC5J4weNUITZIcvm82uXbvG3BTg3mHv3gPanG+ABBgzAgt0t2kPmnJN0s0Xnog3zBgDgU6irsagqXfQanlXzNkbspL56AMxdtK8vVbCkH/5wx/NhHn/twfCP57+hxYX3jZm89qJE8O9990bbpcWaEdNMD/RwgACJjTvSopLtOH1TeH++fcrj+skgKoKf/mP/zSNbxaZRmpBaJY0bebPn2ca3zCvPhx5C9NB23zI2JgVY8KBwGaxNGhel2s1LI9wiTZpMhY2T4bpM2a4D21tcGjDzgZffoJLO4HrX/7yF0uP+yAmvvjiRZPmbjGSLFZhWQRc8ky1FX1JP9TShrZ+TGCz/2j4Ys3mZHNBmT43lkoKKm1N5U8ZtNl9DEpg07ltSmBTqo8sCI6+JOcfcVy08SfamB0ZBDIIZBAAAhGXc492JEIaNCQ/lgAdSxueKysrzY0le9rgvgZcBx2Ki4PkYXgpwZ3p50inyD87vi8EoGl5VshzcZrMb3zrWF8Lp+LizpyTwObgaU3wd0vx4UA4dQ4NGy2gyw+9xyOVCI4WpLHYLJFm+ygJbKZPHqgJfkcrwtkr6AYLsC6wIQ17ya1atcoULT79/9l7z3e7iivPv5RzzllXOQMSIIKAi8jZ5GiMwXZ3P8/8nvFM/x3zZl51Pz3GRBNMzhIZRAYJBVDOVznnLP0+n1Wn7j0SeMbGQE9Pny2de/bZoar2qtqrVq3vCoyRVlj3dyZk3nDmwYvq62Pe19LUY86HNrDMrY6Hf4/NMelWXX85Vt2ev3S+HC/3VP+O8R694HPyLwSR/E5EnVUVlPvjyqBF1XUn0cZymt5PFfcakGzBW+HjTz4j3v8jadaHH4Z8aPFjyVNzN2FoL7/s8mQ4WRX4ykfVo8O25XEELSr0iEo8XnVIGVSP5hkz3kBue4ccgCuQPbtFfsNrCIl2+umnhTzqTcpb5i1YilfOkiVLI7Ttvn0HUP71Q7Y9IxRvKgpXLF+WtmzdHGBOmzbtIrxOr169CZs7HFDY0G29ItyMPEjl4laecwUKNIVqQ2rt2rWT43uCBqXtLTEFU6EnMNQVS2zH3PDhI5JW0j1IPN2W49n6K4+/WDz43IWu7B8ACFm1ek1Ydjc0NADAbMUDBGAIGTGHJj6eOnZonwYMGID3zMjwCtcDSYXku++9Tzt3pNHmsAHIugzAxjyRJnz2nToMULMRj5wVy1fxHGuiXD3RRo8eHTmGunTtnIGbytjJw4S3jXYJ8Gh9bj6Hhd8uSmvWrIF+W6DDDvp8fwAjKt/0VFKOFlAytK+gqTzacMYCQ/nJ8xiLsQf4ISC1ZOnSyO24BTBFYEbaTSVMsCCMnv/KvNLed918lQHYMJ58nw2vJ32HDq2jvrGhJNW4SzAKjC3k4717d6WGtavSgnnz0mJC9q2Fxt+SB0NaHGXl2Kdv/wg/NKSuLnXp3i36auTIOizwT4sQSm1aA8TR1jxmeQpfBjafwWMCRI2/K+fiwEl/aMxJtC0nc1llVJSjte8aBX5KChTeX8ZyU12njFN+Zr5ddYXjnnHuh914D/wux8qVUbbFseX6vIgQXsXDBnDiBTxSjGhhrjOBkJtuuhkA5rQIiZbXryktW74U3cBz6TUiY8ydNz/CGI4bNyHdedc9eBJeEZ6EHTugwKm8ShHdgvW24JIyY+Skedh1/9vRRvmExgQ3YahpOMR4DsQP9Qeu/c0l8/kXnwGi7InIHl3gJ/v2741wvV2790j16CV+yXrdOcXwic5tG9EtzCJkukan7+IZvg9jhjPOmBzh2C/h+rq6uuCPFWJUaJYBG2UajQjMkTtjxoz09NN/TouYv7p260n0jank6LkmcrbJ98MzJWgv4GEfJEDnAxitLA3ARs+gvxuwoT2UTFObPs5dBbDZd/hYWrdpT1qwcFVaunILHjatMTI4GbA5AVBTvGyaYYRjDpv6c4al/n3gzUwWYWybBwbPYH15jFiv/DTXX2mC1wXvVDPhgKoMKg9XtTH21RpaWG2rUaBGgRoFahT4ESlQxX+DBSMZOAEFUsMsVuG7jbzbmgurrnwf53rN4AKw2XsYRwDkf0KirVyzKe0EsDmaAGzQvSPdBxtvjsF/y+aHWcccSwP7dUmjR/RNdYO64RDQpgK7oEEIls8cEnX4IzekVJ0bYdv/HwFsDIm2loXRZHPYEBKtCFfmotmwAXelpctI/LoiBCUXRxMnTojFaHcWNyGUQYoALCCUgltY0FDm66+TRA9PEgGh9gHYnJ6uvJrY49ddG4CNApaTte7AhkRTeDMOekPDhtSzR++wKhGIMLxDV1yZ7YcAQeyZLCmGUOZiVss6LfDmz18QodWee/b5tN78B1gc3oTVjqCIVtt/euqJ9OGsD2NReTmxb2+++aY0GQ8arSO/njM7PUxItdfxgNEacezYcWHhciZC0yYWkw89/DDA0rzUDaHtQpRBClLTpp0XFqQOEb1+FL4UpGIQ5XET40WaKkiuNIcNgqoeNoZmELCZOnVquu/Xv04X11+cBg4amJWPPGKhrUCNwMxPDdg4wP3E0OaPgE3DBly15y3NgM2hFnjYANgkXaAN+6a6rQLY4LbWpUOLAGwmjBmUenZtexJgowhmQkKRTeuQNIU87Na2GgVqFPhPTgF5pHzOTd5XwqMJ2phv4H0SmsvDBajNXyHfHDx4cCgEVRa6WYafwjvdL5vHyqccq33/rRSAnoWmjQxclY7/tGx1i1kwFvgCNnsPHk/rN+0ND5tlKzfx+1TAhluYOxXXnE/atG4ZHjb1usyd3wAAQABJREFUU0hSS3JBtwLYWLdjpCTfXU+oUseF1qUCe1rk2wbD5um1eu1112WlMYCNMopbzD2Ohf+LZqAybm1SaVcZw7a5ehzHM9D+U4+VMuI455VDGPDenT9+RfmW0FRmKEi8jE0Zw62pbt+ZOBT1ed7PQbxLVmPk886774dVr3kKVbAJoJ5LCFvD0F54wYUVTwsNNXIZ0ZZQ0HigtMtvNiti1+FVnk2wYBPKMA16Pv7kE3IAribUXdfIP3AWSn0NVAxz4+jz5o3Iqx8AHpnbZdHCxSGzjhg+Ml12+aVxjfkHF377Tdq8BcDm8FEAvjbImj0ZI0MjV+LVV12JbDss6t8Or9GQyfyKs2fPAUhZHCCKoNDBgwfCKySaHO9DzlfYnnBlAhaDBg1Kk047LULiTJgwARmxX2oeXoCMQa7XM9m3BOwL0OAIIXx3YxS0BvBxdvoUQ54VKwzxtzUAimN4tZjjSaBCQEjF3cRJE9NQwBBl9Nkk1P7o40/SNizEx5H34E7AssuuuBw+OSK1bwPtuTfC5ixeSlLsz8gp+VXkv+mPJ9T06ZeQu+VCDKT6EmoNC/B4ltz/7sqDBanWrG1I81GWfvH5lyhQl6MIZYGF3H7o8MF4r3yz7H/Bcz2ghg4dStLsyXymBL82f5Ggje9g7lv6GuKZo8z+EvBbBJiyGxl+LEDPdddfHyDrHDzvv/ryK0C4pQHuHILuJSSQ9bWjvm6sQ0aOHJXOOefcoPcI5ge9bppRl7Tdtm0L42c2XjXI3IC6rhN2bt+JQhbwh2Vf23YdwmO+XccOqSWgW3u8o6ZNOyfddusvALQAvdp3QBnsAtIx61C27Y62vF+Ox8m/+CePz++ezmVGgd89WTtSo8BPQoHCX5v4fKmmapyym/lqOed3ng8ym/Ba54z4qswTni9jOr8fcVXcwB6nvhewATC/gogWN93IejxCovWGV+S6Fi76lvn98fTGjJmhixCEn3TaGayZ7yfk+aVpKDJge6wDwcxjk08qi2hYsGfP7jQH/vjQHx+KyBaC4nrd6qX9a+7XyyZ4Es+wbOlywq49Ezlu16xdHfxq0MBBAPg9k2Dyt7SjOV6Y5xO141e/ui9NPUdQuWOEMHeemPHmDHQYz0d+XB+0/qL6dP/99wf43L1bzl8TTEOqQo881wqDkG9n3150A8sxcp1BLpqnkJUWp7btO6dx48cTEu0a6HJj8Hv1HNJQGhfwxnDwfz9g45xkT/ld+s/v/PHocWQ5PWz2Hz5OTsK9af63K9OSFZvT3gMANkTe0MMGAnEPxASwMb+toI0eNiOwitbDZmDfLswThAyl/U31WTqXUrZH/3rApql9uZ0UEiX5XdtqFKhRoEaBGgV+PAo40TNHlLncX871FaQmuLgsOTaPV3FjpxZ48zEOx0qf3zv3HCIH2nZk+43kSN+SdprD5gT5SAFsmN1izV8N2Awe2C2NGdE/DRnQBecAwuBTotrk0LVHu6LiqMf5MaqMQ+7Z9opuqQL1VLUurvqp/zi/lY/tKx/lAA3S/uocNirEGhrWhrWJCysFnmOEg9A6Re8YrT4++uijWByOw3ryuuuuZ0GTLZxNgJo3iQKRaJTeOApJWqq88cbrJJFeQmLTTiwyTyMk2jXpegCbMXjYhGCFosYF8KKFCwlr8mKALWvWrGPR1y0WXybrE+DpziI4C4950ER3OCYUzhDMXJwZFm3tmob07DPPBvCiN0sfFqLnkvdgIovnw1hrvk6emjnEw9WK71bAoDtI8DNiODkRUPgtXrwoPYq16Avk0dnJ4rcHC/nJk6ek83jWfQhFT//5zyTVXZIGISAaJ1zAxkWxFpWO02rAxvZ5zD/RSbTRb60/XwSs0e36a2h0BEvV8847Pz3wwAMhRLpw1lrcG+N+9uzQnwuwkboxtPlzGBPFho3msFmWVq7ehiUNgM0xAZvsYSMAVTxsWpFcsHM7LKN5oSaOHZx6dW+HsoBwILi0ZWDOp1GizlJ1eTYO1LYaBWoUqFEgKCCPlN/57YSmJb+ehHPnzo2wS4bG0Cpb4wI9Cl149+vXLxTDZQK0IO93czHu5u9SZvV1cbL252+gAHSt0LZxgoqZWXo7Q/vP2U9RSgEMwIaY5+s2AtiQw2b5qs3MIyzhIyRa67jO+SYAG3PYEF6zfZtWADbd0wWT+5GkFosbhRvLpF7HRuS8YF9lrB4GyifPPfdshCRRSaOiR0XHfyEU6g033HASoOe8E6VRZlRLOT/XeDh1DEYD+OPxstmW8tt9t/Lb/VOPeU6alOON1zSWk5/P425Z2d1UTj7q3yxvRP/RHMuzeptWaUa0o8gh+wAKvvl2YSjRXsQrOoxyCD03ZOjQdOlllxEy5q4whAllmMVL54oCjh8eYfM77+cnzc9uHcWLyuc7SNgt8xxuJxSOnh5t27YLD++uXbuGUq1FJdeI15o78Q28cWa8MTOssffs3hug7uQpZ0Q4rS+/+pKEzptybhhdMHi4zp26IAOOjPC4t99+a4B95kUyVPAHH3yA98p7AaDs3bsXQ56jYSTk8yh3ugV9+K0Bi8+h/Na+Q7sI83MmoXBuvfVWlHvn4GXeidw2LEa4TnDSMpT/LDdC8WAo9NEnnyK8r45jAjVuAhO+A+awUd4VFOnZszsybL8Y25s2b0kLkbH37N3PuJ+Q7oL2l9EHI0YMIyxtBmwEQubNXQDw/Ta5wd7FS6Yh1dXVhWf2dYCaltWmbet4NhdAKgbtg507d2EENZ/cjq+mWawBNuGls2cPdCBU8nFk93wtPJZnyfK89GgOP87PrxX6VeQbOvPMM4NPG57Q8/a71FqzZm30l3kl58yZE54/I/BMmnbBtPCyFAg0mfc+aJTnBbhFkN36aCd00cq9GwrRoUPr0kV4y1977XVpGP1prh9z3mzeujF9/tkn6SmSj3/04ax0iJDNhw8epi9V7EJblLDNWcc0bwkogzKxA6DjZZdPR9l6b5pI2KOOGJuZn8c3JLfad8N9nrXyz35y/FW/hx5r2rw6Gt50KPbyyLek2lajwM9FAceqWx6v3zM2OeRYj62yX8Z33BPHyv35Mv96jXOwW3il8Z7Eu8LxXB4K/wP7Y75QP6CR5oIIiSZgc0UANmcQ+aJv396h2NdAU/3E4wA2MwRslq1IBwBtJ082csZvw9tlIGv59vAuRT2qY7Nx8mMMC5g71D88/tjj1PVS8C+vMPz7P/zDPwUYIk9yzvlmPvU8/liaOXMmRqd7U11dHcDQJHQPXal3afpg1ofhOTMVYPg3v/lt6EC64pmoN99ivGOeBezxmfQY7NK5a/Dge++9N02aOCk8D4Pv0S7bWHiZPFMeegCaNAASGf3j8cf/lBYwt7Zs3Q4+NiLaeCdek8NHDM/yrOTlHnmfm4CNcpBGK3+7h03uF//m0nLhed+/+ePZAGysD76ZAZtVJIzemPbuN1R6OxpkKHd5uzwawCbmOIGbY2n44K7porOHhpV0q1bkOnNcxPjK9QZnlTb+owy/Y6t8RZ+GZiKPraaTpY0eabw4n679rVGgRoEaBWoU+LspkOUFeW82MCgFZm4M92adH7gNLDh4d4VNN/Jxjsbd/BG00cNm+y4MDVZuJfznxrQOx4BdB9DhE3nDUNLNMZByJhGwadHsEKGdj2GY0TONGzUgDerXKXXpSN5RzmseYMl5TWUNZQ7wu7IfBoJqGfx4jXdZermW3Z9hk4blo0xUPsoCfxNgY4JnF0UKMX369KbpOezIxo0bwmr1+eefJ3fLu+E5M5aQC8Zk1aVYD5RWrYwL79NmYtkgF3mffvppADDeZ2MMizB5ylnpGhaHV2HFOBLhQyWNiz4t3hSIXsEC7hnitxqqoG3b9gBIp0WsWcOSGeM2yycq8zLhFXTy5rBAoODBTbxqMsNHH3k06TmkMDUKK7mhdXWxAJ0NWNOwrgEryuGxsL0N0KZvnz4syJqn1bTzWUK4+byr8IQR5avjutOgi4rD9z+YRSiIrYT8GJ/uwSX6sssvS70JD+fi1O5HRIy/7vurNE/BMVsEn0iLEB6f5Rlff/2NEERd5F+Axc4Dv3kgnQ9wI2DmAl2iluEnTYui5KcMiWZ9ZViryzgMILkewGbOfDys1mxD8daCYyzqT1QAG19ens2QaC1JLtipbTMAm34ANkNSHxRtHdri2EY8WwGbiP1d9ZIU2gSpan9qFKhRoEYBKCAPL5sTmpteiFpiq9AzV8bXWF3LDw3nU19fT+jKc8OLQkWpH++rnhgtw99lU4lcyi7Hat9/LQWgY6FlIxPPi37nvPxPEaoJsNmzX8BmD4BNAxY1htAwVJrWmK2ZQbS4p27kABf5ziVay44BsDn/jL4REk3FhFXFHBrh0HJfGnp1NnHjDUPy5pszw8q0OQpXQ2ZNPWdq+off/Q4P2UtDRilK4lAGRGH5ecs4KN9/LRV+6HWOw+q6qsdlOV6O+e0499sxW8at1xWZorSj3OO5xnJCgvCK/D7YPz56vqbc+X3flpGPl3LtI/eVvXxHNcr55NPP0wskWH4PQGPlypXhWaEns95vV6J8G4myyc0x4RYKJoTn/LvIR9H5XGQbSx3WkxV/+TvXeRiDHJWBGgmZF6alCnaELdsqSGfbQvB99RUMYl6DT8wLWTQ8XgYPjFwryn5ep0e18p2J7DuSX3EKCsCrAIDr6y9E+d+VsbQsvfX22+k9vLf0/BAwUrHXG1mxN7JoF8AigRNHunlVDF22DdnQhNiCPRrv6L2jN6AW0obuGQuI2BnPDymiEosBHSDUnLlfRz2zZn2EcnFpeMMIIihbDhkyOPXEo1s525DEW/EM0utdL/bWhP8TXDAkjqDNIbzC9bC5/Y7bIyTa6JEj8EBhWYOHza5du6HHXOTOGYBZbwZQEjLwXXciz98YgE3rtnhOM96CltBZgGweYcTefvsdZN8PwgLdnEDSvkeP7gBhA1P3Ht1CXj3BfRpeyac30xZBKLcBAwaGYvOSS6aHR2Qf6Bd0o9McAfbXy6/YX6+GIdVuwPgIT8f6wJB0DaxLHAPmLPPeDoSEE7jai0JVg7JNJNg2L5F8v3PX7qxhJrM2wfP9wgtTXV0dY4P1yK5teFUtINzR22nOV1+Rx2Zr2rRhE8+3k1A+zcIbpy+gf1fqaAvQJtBz5pmT6bfLCZNXBz8SNFYmj0ei1Qw4/tsud+J3PhXvVmX3lK/KOD/laBQUxyov3HfO1w7UKPDjUyCPXUZfMPpTxiY/C892zDfu04wY8hyJOaTqPXBOKGXGmKZcbWQLaFN4vt+CE4amfPXVnMNGsNp3PjxsCFNmSLS+5LCJdTASwoJv5gfgMmPmm2kl/EIedNZZZ7Nm/gcMdi4kKkhvojkAuGIc6OYjlXnKdfvahgbqejWiWhiBQ2PM0aNH4WHzQLr++hsipKLgtAaUjz/2WAD0zlVnTz0Lr8vzAmyZO3dOeg3j0w0bN6cJEyame8ltK08bMLB/6A7MX/PwIw/HvXoeDho4OPLX3HHHHWkU3n+tmbOaNqgA78+yLnM7PEqPzfXr1sFr30yPPfpYmrsAb56WbcLzU12LBhB6EUoTaSj9Yz6lDwwTqYfNE0888SMBNkHFCl+TL2VenT1scuSNTdv2pfmERFu0dEPavb9l5LBJzQhxGxOyeoGj3KVRgoDN0VQ3sHO64MzBaVD/buFB3YL5OpfqLJA/fEVdIaNxNLaT2GLWrzgim07m9jX9rpyqfdUoUKNAjQI1CvwoFMhze+a/8ms3uXbZKhw8T/0eL7ycCzKnZxnimo7fx5i7BGy27difFi7ZGGHSN27dQ6h0HAIaARtlB9eM5qo/DGBzPA0f2pv86IPSgD54xLcH9KesgF5CKLFCP6VNuZVZTvG4NZf5I+6qupbdn2GThuUT7YIOfisHuA76qz1sIofNmtXEXD0jABsLcTMWqmFGtIIReHGRbqgHE77eiIvu2WefFXGvy7PaMXbKtm3b4/pnAD8EblzEdcVD5rzzp5Eo8Bfkark41RHb3AWrn/3Eh3XBr3XKM3jHLF60lCLx1iA0wq0kB9QTph9WNFpmuABT2MneLHQPAkKZ4CWGC2eTC/4Zb5hZs2albSw6e7Lw1RrzGITZAAh1lAXspEmTImTK9eSw6UKsbNthOA1D77yMEuKrr2Yj2O1M3Xv2wKNmCM+VCAu3CqGwBYLcOemee+4JK8AOLPAMx+PgiiHi4PHjcK6MHes1aaGHv/nm2/TkU09ixfNmxGMXUNIt+4EHHkjnUK4L0xDKfEYqLR2r4GnH/tSAjS13WPtCCdhs2CxgsxxFm4BN83ToqIAN1pc8DC2MZ27ZjMS+CGaEEU5jhgvYDEbR1pGXjMSwvnDQoQbYQNTaVqNAjQJ/kQJlMvOCwvc8phLR0DybCUshaKOXzYcffhgen3ra6IV5DhbsJVeC/NP7S3l+lzJhWPzPE6XX1La/lQLQskJPaZk3VQjSOH/nWQFZgDn82PFmac9+lCYb9qRvl6wjZq1JavWwwbumWSvuaAJsTgjYYFHTEcBm7LDu6bzTAWwA/mMsRLWUzxwYv6ltJ8pdlbCPPfZo+vzzz2N8mNPCkEiX42FwO96zeshGqypt1jAjZupKOYIg/15bGZfWXz0Wy3HHvUprf1eDkV7rMeWBcp/Xuh/PQ78oc0grDsZ493qHe8vKu2FfuWWy5P14M6JPfT/idNTjveWT78nKfEPUPPn00+FNrfJcoxqtpE0cPYX4/SrY87sWzbB4yslCs8fR5NiCaERuR2mPtfiMPEfjcyHkEx7NT5b3EOgB55QJ/VZqsezVq1ell15+BeFXD+a5aTuyqLRTCR+hWNg3B2MPPFRaAooIdrRv3xGg5uJ0cX09uVf6B19R7n3yqaciP5IybZvWrQlXOwhgZ0qEIxuCl7Ve39JpJ7lcNpD/YPHixdQ5J5RnGi0JMHVDnpuEZ7le4tfimd4fACO/GXhNI9MZXuyJJ59Mr6BQ3ACIYO4D80T2IGyP91104TRAg2FR//59e5D/FhPW7FOMfb6JkGkqGq3nIB/RqzF4wN+MvGw/jB07BhkMReHxowHYzJk9Fxn7DbzeZwZgM3LUqHTXnXcik98QIdH0VDnGtY4hQ60ZAu5FZGHDGjesWx+ebR07do5+HT9+HJ7h50T4OOl77NiRoIH8WY8c5XmNpxyjhnC78sorYs0wHtCqOwCUfWEfr+C6lzDUUvafR7jhneQHilw4hCWLfgMkMx+OYeU04OoFXQyLtmnzRmTpBREuTVBnP14zglx9+vRjLjgv3YZF+rQLL8BzvgXK0H2AOxvTqhUr0nIUm0swmpr9pSHulhDS7VgaNKQuAF7z/3RnbLSGDv37901jRg0jv1F3ysB6X0Ukm7wkxq77MWgd1HGKseB7w+/v3Spj/TvnyvXl+zsX1A7UKPCjUyCPXd7AGK8nj81yzkqr973WvDCuS5Wx5BOel0f78V33nW7Be+i5eB+Y3+M7XhPPm3NlXwA2vvOGQF8wH8CGvGFXXpnzypzOe66HjYC8hgnz581NjzDHv8maee3adVHnOeeeF4CN0Sl8R9sSRrUANlEJ80Fp0zY8tJUPrE8PwzXksBrMmv7WW28HsLk+jH302v2SUJFP4IX3MblsOwGEX3XVFXjJXBr5zD759JMwDNHDZ9CgITHHXXOtRqtjo7r3P3gv/cu//gu5bz6n3a0C1NHT78YbfpGcK6RBsAuuZo/9TC/37YJDeAKpf3j77bfSIw8/kr4GWErNW0VdV18jYHN3Gj1qdPBEeZBgTdCY0uTVPwVgY0szr7Ptzlr0P0PlIH+2bN8fgM1CAJtd+wyVruekYdHyXMyVXJ1DohkebciATun8yQP5JkdYG0JjohTIJj1hssO1btRGPdLK/dgqX/lHGaflu1xXLirf+era3xoFahSoUaBGgb+XAs5XllHWbvHDmYtjmecGR66w3zJn5LOV8xSgsR0q5XQcZflR9AJbthFZYOE6vGw2At4cSHsPtWTG0MMGvTHyQ8xx3CFg07H9iTRiWN80AcCmb692qRPRnAKwoWLn0CyE+11phEfZzW3gcLQ9t/8/vIfNwoU5JJoLRHPU+PBOmoae0Zr52WefxTLv9QBijKGtFZ+5XwxJ06cPsa85VjYXwCsBNrxeTxUXsQI9xo294qqrI5SYsa21iol8NAhkCisNDWtDoHriiSdJMrsIa8hjFaHqtnT3PXcjuAxm0aoVjR2RF+d2VAwO2hrf9JdWiLpYa7GnBc/iJUsiManJiBUo93NeK8lLpk+PBeR5WNC0J8QGUmjazWJRi8IZb8ygLa/HojMnksWqjwFnOI7eLAgvwHrP3DpnAVi1Q7mUlSCSjQHhyOajzBFyB/uR7wWBUHlWIOjf/te/Rdx9wSTjsV9MWyKJIYBZ1y6duS8vDiko+kKhWIHSAf9TAzYOaT8KZhhsZsBmwQoAm61p9z4sUo9ihcmr4otnuxTjApQ5AQrKMBhZ1zsAm/69O4OCtqwBNtCpttUoUKPA/5kC8hM/bnmBm3+XfT0qVAwbHu0dEruabF6Ftp6hF8KTDbujclBLeJV95b7q8sq+52rbD6FAnpOa7qzMUTFXeVR1ArNCLPABZhDMdu87mtas35UWLl2fVjVsSwcOtwCwYT4GsNEww815xES1LQVs2gvY9EjnntYn9enB3Bzn+UNVBbA5SqijjRiCvIgxyR//+MeQMxwLnTp3wqBiaoRCu/SSSwgjMiIEvxDdnK/o92gxk7H7Kj3+vbYy1q2/jMdyrHxnJRhUqijAvNZzfso9XqNySqOO+EAb5SC9MARu9BDoitdIR5TgedT7t6kfSzm+eu43vhrS23+V+prahMIGT4/nULY9SqgZZTzr6t9vQMhUhgATLNBI5jtlUl7IfT5IbE3l2yRlnAzq0E/cLCBhm5T7jiGU5PPemI+1QlGnYi8ejOvXYHgk+PEyoM3sr+ZE/hV5QceO7VMd3hJTAHgnnTYRPtE/aGoeGxVsA7GIHoBRkMDD13hha+307nvvp1WEWDMvj3lVzsfg6Nzzzk3D6vC6aN8e75Wck+XwkUPh5WGeAz1zPiHXjqHU1mEx3Qqgp1//ASj+Lk933HlXGj1mdCisjhNSbDO5eVQkPoec/AHGRXv37CefSjs8wkeniwCQppKjZzAgUXfoKMBx4MC+tAeZ3HyThgr7jHwsy5avCLn8MFbnR6HV6DFj0+14zVxG3sdRAJcd2yMzI5eaI2f2V18jE7/eCNgIbN4JYPOLG3/Bs/ej7jYVw6JK6F7AGsHxxXgYmfuhKx4syv7nn3cevHYKz9UXkLxTKG+PHDkceRgMX6lHkm0TvFlDXh7pb66h6ci5Wotroa487tiQvjMIU+x64SvCL7tWMDyxIZSHkqPnDPrrbOigN2XXLl3Cq8gwZ/uhhfT9mtxC7+MFZTJywxa3btU2jcHz/5e/ujdAMtuncvQARmEHuWfTho142eCp+eZb6dNPvqCcQ4Bc49PV11ybzsYrr9/AAdG29vCgLp3b01etWTBiAACfijVGHrTxXuRRyLvkyrCylXep/G56T3lYB/h3tnJv+f7OBbUDNQr8yBTIPNdC83g9eVwWPh/c35c0xi3j0yHKbw+pVPFANp6Ub2fQxmPyqrjNfa4rcxfcO8oyJJpRPVxj6x2r10sv9AFXoR+45aabA5gtIdGMwDEPwObhhx8KsKWhYV3MAQI2v/3dPwEaTwtvPwGbFhUPGyoJoCeHkMwePfIZvbP/9PgTUV/v3ugA8L7VG8+wZ8cwBBCkf/rpP0d9A+EDN9xwfboCPtqC+WXWrA/xYFE3sTh1Zs2uvPkLgO5zz51K2LVD4eH7IHLIAsDqnj17YYRZD/+5jjw2F0UEj5hUJYof6GhOMudtprU4pYeNEU3eeetNnvXh9PW8DNgMQO8hz/zlPb8MryCNDOiBoGnwJebG4mHz94REy31s85r6O/M7O92P2QWFYTDkpBu3YB29YNFqZLoN5CJAL3BMwCZ72Dgm4koMcBJ6AQGbwf06pvPOGJCGDOyOPNIWmspRkQWoTw+jXEv5m2U0h13IItUCiQcbP6VtfruV7/yr9rdGgRoFahSoUeDvpUCRF+S98uT8rdxbOZI5crBfTTCb+HLhyE4rsU7lfj1sjnDvZrxq5i5Yk5av3Jx27D6c9qkXSK6pnMuRG7iuOc4ALZsB2HRIadRwAJsxg4ne1DZ1aEO6Dcpy9d4UDs3WlBppG7ulJdTOuSx//IcHbL799lsWP2vDoq8nHiVFSaCCTMsNPWxeeuklFooNoQjojTWMiy+t+Mxpo2WlQpmuyirUtLB76623YyG1adOmUCL0R5F2Cwv521kgjmTx1plkfUFaOv8oiz3jihuL/k9/egJLu/ks3PYQHqwvwNAt6V4WX8OHj8A12YRE3qXI4iRvB5VOQrygh8xTs3rV6vTuO++G8PU5C9uwCuJKK7SdxrcXcLkcYcyFoKAMD82Cbn9ah0D43nvvYrWLWzKKQRUGKgns6nYoPgyH5mL4SoTLCRPG5wWdbSqCjt98xFyipexnxYNW4sciDvj//J//k3Ain1hlWFxegsfS7ViJ53jZKFVCQPG53LKSwj7RsunnAmyoKnvYbNnJS4Vl4qqtWNJwTA+bAthU+gB/GzxsDpN7ABR0aK80fszANKhv19S5Q6sqwEZq+HplBZm/aluNAjUK1ChQTQF5pR95YF485rPltwrp9VizqxR1AS7A7jktty+99NJIci1oY9gdeb2b5/2UrXq/HKt9/y0UcG5qmp/c91/Z3BOw0cZSwGbX3qMkFSRPHYDNmvXbM2CTDIemjUzpI+5iYd8Kj8zOndqlMXjYTJ3YO/Xqlo1BcvgPamI+tj5DLi3FGENjEhUVKm/tb71pzaGi19WZU86MEKNxb1X/R9up7tRxUdr/c31Xj++/VGcZq0Ff2lzej3KvQI0hA7WyNRSXiep3EZpK7yPDeOnBq4eCctrAAQNDCC5lZgE206Gpft8V6/FI/Gl8D0vdyiJ6TesZ8hDWwMpbvpdaLd+OZ4OyjOBIBwAiy5H+ykNNZVpufs/1PNZ7zk8Ba0LWyVVXjvlDflApw2+OCAR06JSBk+he2i1AoHfNKwI2KPN34CVtsnsTR5977jkRrm3KmZMrSrRsIa6BTJvWKJFQNG7Gc0MAQUXiPORYrbN79+qTLr/i8rDG1vpbGdn+aMF90jIDTFhJA5IZrsycN0+RONqQvIcx8ulEOOCLp1+SfvXrX4cysgNgzyG8PhYhd8+YMTO9Ts6d+RgZ2Y6hQ+tS/cXTI1SPHu+CQq14TmXdowBDEkFwbMH8eRGy7R3qEiQSUDmM0nFMBbC5FIBo1Mjh4WGjqk15evZsAJtXXo86V69eGyDKHcjjNwrYAGAZJu4IQNKhg4fSZwBJD/7hwfQxfHYP9+r1Y9mGdlPROW7cWI5VLOnROh45fCjAHrqlkT+/AR0FlhyX8uMpU6aku+66K5SZAfTwXBmwgQaEGzJfpkZiAu6CZ+eQK+IKPHOmng2QAqAkWHscD6CsJM4Jxb9F8fsSwJJ0XE9+nYMHD0cYoV/dd1+6+VbDHfeGflAA3tKS/t2Bx9UXn30RXkPvvvN+2g1INnHS6ekWxuyFhMMbPHQoeYagN8pfwzFId1WLEdqJPbfy7vmilH/lWNO71XRdPuaI9XPqlst0fNe2GgV+Hgpk3mtdDOHvbGUsO179J8+J77hSJYo3+U7p3Xc4PPw0utRIwLEumC3PjbCVLQCMee+8I5dhRI2cw+ZVQNoXAP3lk+oQrrrqmnQzgI0h0QzNnoGf4wGgPBSAzVsYdq4HmD5GeMVz0+8AbM6fdkHqiZFp2zZa5TY9im1zbvRZ5Bm79+xOsz6clf71X/8tPBQNuX766ebXynkQD8LzBKdfxttv1aqVYXCgjkAdhwYBH374AeHGHoswm05mevzddPON6eKL6wHMt6U3eBbnDEO2DRs2jHM3k+P22jQefUE3APd4fiewoKfqBtt2CmDDnGpINAGbufO+Qa/UCr48CDD56vSre+9No0ZVQqJRjrJOUWpJT/UCykHqLYyMogeUeXp9hrsjnNrI1Amdy/dvuV3SKveRv2kxMpz/Yp9nFoZR+hKw2baLPly8Ji0ipM323c2IvEGI0eZtOe9Er0Et0I6AzXEAG0KiDerXPp1zWv9UN6gHyje8XQFsgrf6LOiAAriivswHrds2+JPrGgdphX7SMD5V1+eL42/tT40CNQrUKFCjwI9FAfl5hR9TZCNgIwdv5Nlw5AqQk+cMObn8OW8xD1OEjgDmsDmCXmDjFnLdz18FYLMl7UZHsP8IoTWJvNGcuavMbZFug/zoTl2jBWyI3tS7e5vUvnWzRu3BXwJsrD63wbY7c2XdwX94wGbBgvlhtWd4mT5YnghwuABXGHNhLpCSkwMuiDAHLr5c4GnZPG3atIjTbRgDw5GZ/O4LXIsFbVSiCPp4zlje9/zy3nQbMV1dRIVVZIWACi87d+wIq8SnCbHx6SefUe/GiEV/7bXXp18SL1YBydjizt0xfzM4nPAbJ/YYHC4cCZ3Dwu1TwkY89vjjsXhWkFQQMeFrG6w7zHlwHwu6c7EUNB55K6xWbMNhhE4VQVra/Ou//EsIdiYtVJHgYroLsfHP4V5DW5zPc9fVDWUxmJFA2+FAdog6OP12gIQQRIMVHvft3ZdmYtn3P/7H/whlQkdCsY3BpdrYveYFGjVyRFhfNi3wLMtFqmBNLvfnBGzCw2ZrBmyWCdjsxcsIwOY42KYvni+oVjIEvmkEbBTIxo8emAYTq7ZLR+LMNzckmgKZ0nT5+CLVthoFahSoUeBkCsSCMdi6k6y8PnOK4KMVocF5yblF63RBG6255a+GRTNfhN9a9wvEe7+L278E3pxce+3XX0cB+yb3T/4uC/18t2cKYKNgtnPPUTxrdpGgdj2h0XaStNaQms4jWumq9KafXeAL2LQ4kbpi2T4GD5uzJvRMPbsK2Dh/ZNErPE3pa61RDbeqIYljQEMRvRmUS66/4YYI9TR69JjwjnDsRDsrY8mSorzKb07+bFtuS66ujG1/lfFdjpXv6nPl3pK/xvdAZc0OZCcNYzSuMbytSiN/79m7JzymlXfMK6N3ieGdVObkspRPKvIUNGkkB8eiPdHHTcc9VhRhDbx/KoceeujheBelad2wEUF3DXMMnauBi2XGG5xf47B8zgAHZeEpYcJkFfp6VoScQ91F0WbbPOYnb00LBOtr36F95F7p1r1JLtTDxpBoelfPJYfNNsJydcJbYywAgyFuriM0mQo1vUncGAnBG/zWa2v9+oZQvulhI5iwF5nNcLx33313hMHp3r1bAEU+jh/lxqNHidPvg9Lg3cjAnzEuH0d59umnn4Uyrz1J6y8BTP71/Q+EzKzh0W7CqH2EAtHQvR99/Elaj7xrXpwLLrwo6lE2HQjwrMQU9EIGNeSP9Sib6w3++eefUc8TWH9/nLZu2xEy2djx40JZqEePeRratub94t0S5Ckh0QwLtBpga1jdsHQnAIqATd9+fcJ7xfG0fdvW9AFtMzSPXjLSv2fP3gHUmLvBcMIqZVsChOTHPg6YdCTa6diyfa4DVGAK2qyGjvbvsGHkjcT7R6v2IYRE1nNpJWCfIdcEbAyJtn37tggjdxa5kK7HQl2FqfTXsjxojQeU1ul6tVvPGkLgaalvny8khJt5aQZilX4nSspbAGz00unQHkUiSlsVg7vwav+KPDYvv/gKytF3ADb3pvETJnHt7emCiwRshpDsW09+eY5KR2V5PAWa5TCbvhK+f272hesCz8f74rVxLA/2/I7FpZXrufk7W7628pZ852ztQI0CPz4FmK9lrmy+v6du5RyDP/7FN9c79zZDs64yxfFqxAlDP2okoHfhduYh82ppMNMfr0JDRrZsiQIG+SvkOtaJlrUPHuO7riek4RC/JUy4XilXXnFVGGhOnpxz6Wag5wRz2vzID/Mma+fVa9YGCD7lrKnpN7/9XbrwgvrQJ7QVPOZZyvPIs2IeoT71BP7Wg0/A5h3ee/USAwcOCgBazx7n0FfJffb++x/Aq3YRSnUyxge3cf6y0F8YJu0xvEk/QTexk3xgzm833QRgM70+eJ1ROeTHehhOAtR/4De/iQgkRhYxBFjMDxIasseaPsAkJCAabZsPYCy6YT05bN4CsHn0EUCsRakZtBtAG69Fpv0VfFf9QDNoaRneI10t1zn0xwVsbCgVVAM2/D5OXXK+I/zZDmDzzZK1eNhsArwhB89hjF7xsEnwyRg1BbCRXwPYDOzTLk2d1DsNG9ITg5wOADbyTfumEhLN56nUae2ZxUKboA8nY3PMVn88Xj5eUK5zv7bVKFCjQI0CNQr8/RSoXieWtZilFt6bOXme47O8nM9mfpznA+VmpxTWL3w7h2zYvCt9NXcVYZHJa3vgeNqHXuAIkTda4GHj3ObcKGDTqsURdMnN0mjyo48fOyj17tYmtWvlig3ohbKsvWleqNRMPR5VNs/nbHdpe9FD/7zzRVkj+F3WCX4rm6xize6a0zDihnI/dQOUqkhsnDEe9rqGtek0ABgtXTylsKNCYBsLqHl4mqgUMQyN1nxOpm0BPrSCGcKCyHjbLtQUgrZu2ZrWYQFtHhtddZUsBiC8hQsxCQUvIK60C+gQriqEdiG2D+WCyQffQPB5mcWXcW1NBmtSQWNy+z148KDwsAnhJ5pvB5RuycS33bsQIucgnD2KQmEmOWm0NtX7R6ufriy4jVf+61/fF0JZbgudy30uvF14zmaR+of/9YcAkBTADkEHXcB74fGjxZ/WM4aG6IGlZetWCoqxZIuhkdtjsxymGbCR+AeIsa079+uvv5H+8OCD6duFC1Gm9IzwLXrrGL7FBLOlPO8pZUWyJoTln8PDJovU+eXypdq4dVea983KJGCzczfh644ZEq0C2PCEBbBpjjVN+9YncHnumsaNGJCGDuyRunUhXBzh6zJgQ/9UQJuf9zXJlKz9rVGgRoH/+yng5FW24PP8KJNambL8VrEoaKMFtwYFhtjQor+ER1OpWO1pYxnVn1JH7ftvpcCpwlGIYycVEstwFvvOJYcLYLM2AzZrNuwgObohNbOHTRao6GMW9YbPILIJIaBQsA/vkSaP7QFgUxTrXEN5WtkfYT4WnFDAmTlzZnhZaZmvEnkUIaduIr+eeVQijCpzvjfG2GHcqPDI1vJNTS7jrOnIT7dXxrA1lHo9Vo57zH3fAz9lzLpfPGlCqY5c0tDQgKHN2ngPBGgErZRXVIipFFIhbkg4Aczzzz8/PCpyPo48A+eym56VqmOrbo/EK9eVdukNoyzzyKOPAtg8FEY9KvGGDyfEFgCAVr0DBgwMTwnlvIpgVKkoP6vt08Bn1coVAQosRB7SWMbQNMo7mQ7IITy3m/JXAFVaiqAUMoyZ8pIgzOlnnJaroC7BgRdefDG8bPTU1sOmF8pAE0hfe+3VeEdflhNal3EQgmhWgik77tmzixwni5GJF4bsexgDIPMvTsbbRaMhvT9iPFXaFSDCkZxnSFFUA5/PP/s8kj9/AmCjIlBgSS/q+x/4TXjYmA9n29ZN6U341uOP/6kxnJfAxLWE5rrtjjvxGBxHSK7O0Y+O+QgFqLxFeyWp4JLhdwSG3sabfC39sX//wTSacGC3IfDrZTYCD5t2rTEo4m3bA/BkiDhz2Lz11rvQScCmLt0FEKXiUVne8HIqK108qNR86qmnI9ydHkLjx09IV115NSDK9QBzdbGgMlyRMqrtMfyQtNDryb5zPJpnTDBFAyi9IgVTNU4SPDR8ZTdy2axctRJF6asBks0l/LKh26TDhcj75pe8qL6eHBV4/TNeBGiOHoXWUY/P1Zx6NsIDZhB++cU0ByXw5s1bQ1msd408QA/6Ll0Mi2YS7ONpJ++H+WteeuHFMJ7avmNPeNjoaWRdg4YOJtm34IweVL5/Pp81ZeUojxrjkwLjOX1WFZtuvh/5XWn6HScqf5osE086Ws5WH6zt1yjwE1Kger75/mocy77bfvwXUSbgjwWw8fQW1voaZn711Wz4xKIcFpN31/lGbzo9Lp1vBDVzOfLy48HnBU1fwNji9ddnBNggj74S/nIL72zJpdvCyBX8/6YANjMFbNaER8/kKWcBgP8WvcJFqV/ffngiGkaFy7netgUoIt+Eb1i37TZ31ZNPPh3hF50nOnTohC6gnjnk6pjD9K4xx6zhGi+Ydn7MY55vSR4so20Y4n3GjDcBaFZE3pwr8c45D2BdQOk9PB317hRkmkpI1l/ff3+U3QGwvnhI5sZV2gb/lsYqpWyfeX0a0MG8BWCjoemCbxYCHLdLdXXDwgjpbvjT8OHDQ37hhuAzyjHqHtS7KA/90JBo1u+/2GiTe3mW4e8JV/hucE/qkprqBXbsPpAWLmsAsNlIeDRCTh4ixGUFsInL1X0c18PGsGjH0oDebdKU8b3ScAAb+XHLVnJUeat+O4Lj1EBFpd7CU4O/2qmx2ZLqj8fLxwvKde7XthoFahSoUaBGgb+fAs6jlVlAJh08uPDd/C0nzzOHe5kPl2/vjQ/HnUMEbA6zjlu/aXf66uscvWn/IUKXHmlFSDSNpQBrnNuYzwOwaXk0devUIo0Z2TeNwxmgFzqBAGwoJ+aOygO6H1tUn3/lNmS5I7fbKyg4Prmdcc/P8KeRDtCjrBP8do3rmuuvBmwWLVqY1q9rSBMnToqQIiU0heHF9uBKbLgyhTITkH75+ZcRcsOFbO5EqZY7zQVvscLRrUmApEvnrqFEu/76GxCuLkhDWewZo1rihazCtws9LUx05TWh6iOPPJY+wNKldes2AYxcedVVEY5i4sSJAQy5aLXuWDDyna1lXVDlBb5AkQvup0gaa2xsE6buQyBqh0LH0GyXsZi99957I4xOW+pwoBmaIxbFlGEM7idiIfxOhPvQyrIFbR46tC7diECpQqJfv/4s3tsSSqMSFsIOpy0xKGLcMMj9xzHdxly4z2VB+drrr4VV0coVK2NheRkWhAqMLmCN26tw5/PkLQ9+y8ih2Y7/5CHRTgJseI5NADbzv10NYLMFIQ2lUclhwzvg8+UXBisZAJt2rY/j+twljRneLw0bRGLfrlhhtiDwDQvf0kf5Rak8Xu2rRoEaBWoUqFAg88pgno0TWiGOk5rn3cp1frtIVmmvQYFWhnp/ykvNr6aVpBaOeng6MRaLRPdr2w+lgMKP/VDpi6r9snh2VtCaxoBChs/YuftoWglgs0gPm/UANscEbJABmuUotLo0n2D+aEb4jLatm6VePZxDeqbTR3VlDsE6taqphkQ7iIJCy389PN59992QGwQx9KqacuaUdCOGIcZ9N0G5IcEEE0LZTz3hao0kGGOpMp4cDz/XmChjt9TpHKohhsfLsWIso8GMx/3t8+mJYvgVP457QQ7zxwhcel6jmS7k+ejctUt4Dg+tqwsZR6BBT4NuAA8+f3lW3wc/ZSsK5dJGyVPa5Cvjcd9DAZu1gEWGiHmoAtgIqBi29q677k633nIrclZ/YtUTwjY0MbkGy3NTZrAs5crPPvsU7+0X8XbIXlIq5b3C8tx8dp/Ld1gZ6CiADnZX5CVsy/t9Rvrnf/5veDxf0/hMCr4mstboZ968+WEBPoiQcFpCe920aeeFp7bD1nLj+ZHPHK95TBOiDbofwCtbgMw2q8BrhbWzbfCeTIPDITMewcjHcECHuVZjH417HJtPA3bMmfM1QMleAJsOAaA88JvfhoxtQs2NePK88vJLoZzTCKqZ9ANcuwXaGVZuMGCUsqCycQZsbOCxkKNa2m8oBpetWJGee/Y58ie8iQHQ4rQV7xHz39xNvoNLkHGHDRua2hEqSMBm9+694dX92mszyN+CXFsBbO6+5x6AjZvIzdgrjIU00FIJO2PGzPB8UUHar1+/NO38Cwg5eXkoIwV3VLy6PrBt0tB+y0AK7xs0EoxbsmRpY5ghFaQag5111tmhgFQG7923T1IOngHgopyukZVgYx3j1vCWvsNnc73vtXQ4ImBjnSgADZckOLSTZ/b5n3vu+fQFa5QNGzcjR/dL15Nf4ib4gGGGu/foRtcC2NCZ23l3Zn+JspgxMuONmWn7zj0Yqk0GuLqHMXJxGgTdeUni+VqGR5PcDHqzgMyLv/we+N46ToM3VQAbDjS+L47vU7fyfp18vHC38n3y2dqvGgV+fApk/mu5f0kUyuOXMayChv9G3ZAXOx9oAev5Fby7vrfqBTScOQYIMWHCxMjtovxlODDX8AWwkWdYmOHJDGn7LADI2wDDK1asgjf0Tlezzr+FEOga3Rh2XcDG6UljnIcfeQQ5783wejx86Ai5raYQKv3+dNGF9eGJ2K6tip78/uW257rM4SaPkic5Twq46JWb23s8+NmlgOnOpxpTqoPoT/hFPWuMenH22WeHkeiKlSuCl/35qWfwnPw81v7mNNMwSH755VdfxjzQF5520UX1hH6/I3id4HxZz8c3r7m8TFo5ucT8CE32MU+sXrMK3vxm+hO5cr75ZhFGsR3ISTaGkGjXBJBVV1cX87dMR84T4Bl86e8DbLKeIjq5caDJi/LHMPNyQMdAAWyOQtodew+kRcvXE+Z2Y9q07Qi5gZkXk2HRsLgJYYGL4LknIiTa8dSvZ+t0xpjuaTgh07t368KcCk+NkjNgk/UI+bmKMi1z13jYSstoBPfk/vWQbbS63Nb4UftTo0CNAjUK1CjwI1KgSfci/81ybOG5+ds5IsvE7lX4cuVbfh0f+LTcXsD/EKjNuk142MzJ+dEPHG6WDoZuOQM2zosuHQVsWgPY9OjaEg+bvmn8qIGpJ44AONRWIBfm0MqTRr0VQCnmLI7nljhv5GfIvxUUvCv/qtz+k3810gF6NK2rfwBgo4dNQ8PaEJRcVLkgVVmgUKEln2CKFpVffTk7AJWvsYRbB8Czb99+roH6PLcN0DLSRqkU0KpwEO68kxGszj2XJKVY3AjW5CSlLH64ToFF0MWPlpWG8NBVWsBGq2m9Unr36Zvq6+tZUN0VljuxaEYy8/YITcBOBgPoLtph/S6etTw1lrghGT4HZNJt25AThma7HC8ZBcPhw4eFIOfzBmCDNY6L6Ya1DWG1NwNFoN42W1nkdencJU0mJv4NN/wiFt/dunWPBXUBbGyQC0nrd3OwuflbQVfQSKWibuCff/5FWMOOHDU6/fKX95IP56qwBjevj6HiXACHEBJl5oEWIBpahZ86JFojYEPbRUI3b9sNYLMqLVu5lVi1xMsHBTWHjbkJFPVCnOO5i4fNwL6d0uhhfdOwwb1Sr+4dUmt81lqQSDpeqlj0xl7QpvanRoEaBWoUqKZA4Z95EZb5ZznvueDV8GuFA5VoKqpXrlwZCgCV9yqxBW0MAzV9+vQAb5zTVO6Vsgt/LXWU8mvffw0FpD1zPn+btrLvpJeX3l6hM8QRAtZu330krRCwIYfNWnLYHDrG4r6ZITQQzvCWiLLCGvMQedCap769uxMSrWcaP6JT6t6Z69hyyZSJ0nYnXhNa7T/44IPpo48+CotdFcYqli+6uJ45WuOQC0lSnpO1h3eGc7zzO4ob5/jYKs3+ucdBGYe2wf0yrm2H+8pehrDSyEO5RVlm2bJlyGAN8dvjehQ59t0EafQmGzx4cIRq6Y1bdffu3eNjGFnPd+jYsWLtrCCsrJTr8jt/clv42dimKBzKe8yNpvHJoInKryeefIJ4+4+i7FqNMv1YGkwoqrtRfJvIXu8mc7UEqaPMKCEKKc+/desWlF+fpGeeeTaSNgtIZWMhBdrcR15r++w36/e8onpbcqqYkP73/+2/AgBczflssCNgowJRwOZrQqIZando3VC8Oq4BALgKefR05MAuMZ5CCe/DNQrv9sXRGGPKbI46ZVoVbM0JyWX9GgNJdy2iBSV24LEh8GSIPsOv7du3F1lvXfrm2285vhkg43gGbFAKPvCb3+HNMQnZ93hau3plADZPYlS0YuWquEZF5a233sb4vTFyL0kxw8bJ80LOCp6nbEkfIidmL5ZZ4U2krCtYoYeNXjMZsKkDsFHWzoDN11/PRSk5I82cgaU6AMqwYcORP3/ZCNi0wltcjxVDDb6B8vId+Kk5Hbsj607AmOuMM6YQlmdU5OQJuf149qqRVscFbCCQ8qsyuu+jXjVapn/xxZehCBV4MW+EQIwKXT1pbMfMN2eGh03ksMHDZjRysSCScvFYQgbrOachmGPPNYlytiHRrEv6v/XW2+k5AJgvAGI2bNhEiCQAG8K83Yjn0Bg87roTMs+E5Mrk23mfzGHzIiDhW4SG27lrX5pEm+64865UPz0DNhqbuShtiexaxr7qRd4EuyS2MoYdE7LCpvcoX1POVy6PrxpgU02N2v6/HwXynGP9ZXyf1BbGs/JVHth+Z74rH3STb8qGly9bgQcdYc0CAPky3nnzrJgX5vrrr8crcTKyGKC9/DVeC3UFJ8Lw4P0PPkh/fuaZ9OEHs0K30Kdv33TN1deGV9zEiROysQU83flD750nyZf2Omv5RXjJqHeYMOE05pp7CdN4WazjO7Rvwzvu9OKzKX3kZ8vy4tGYP7YD7ho+V3DXSB67CG1mbje9ZDZu3IRM8QmyxJ7Iz6V3nqDyqNGjgkYaSeS8Xn+Ef86MNboGChqArl27Bp3B2tSRUF/ycPlWeDiOGBm8OtrCnwLY2D75ut/F+HQPfM/cOW++NZN59Snk2CVEIukac5whJKXpYMKwaYAir8llCmg1DwOWH+xhQxtCnquwtszj/FH9ka6u99W5oBfge9e+g2npig3hYbNxy+G09wBjoiLTBcEsl/khvGzg1317tEqnje6azHHbAwC9tZ6fjAWNCRxnzvbWKJdt2ivjhkONWxm7jst8baFH4yW1nRoFahSoUaBGgR+JAsreWQ6QV38vYMNpU2RkrlxmEfmzc0fm2YZDCyNOWL6ATcMGABs8bFau2ZIOHmmOXoD86IREC7AmABt0xwA2bVsfC8BmDCHRxo4cEIAN00dlzshzh/VYa86jU9qaZwjPMRH5J3bjzsY1X+XQz/DVSAfoUb1eUEZx3fpXe9hU57Dp3bt3LBDNBXOMiVbZQMFiz569eOGsR5BYHBYlq1evCpdoY1Pvx+JVSxbjTLcjPrcKAi2bhw4ZGhaFhg8bMHBAgDUuskJwgXhavmi5E4IgD6EF3Yrly0OgeuXV14hPvSasSM7CyuW+X92X6i++OMLeGA7DLeSWGEh0GoKL5UoUrTB37Nge4Mbs2XPC0lIlR0cAEZUakwGPzsZtuQ/PqmLA59M60Pb4e+fOHbhhL4iYtwvxtjFOr2EhxrEY9r5x4yZEO1T8mMNGS5cshGXAxnY4qG2Lz3eQhLCLlyxBMfFMWPG4oLfxWoLf9+v7U319fi7DoVmmygkHLQVAJUOjqDSQXj8HYGOdCGYMfhVum7cD2CzEw2bFlrRt51Fi1ZJ/IEKimYTQF4LBxxGiD6a2rY6n/r07ppF1fRDM+qQ+PTqChPrS+VJZahbLagIWpKhtNQrUKPB/pIA8tHrLC3A5D9wEXul5ldsqtGfNmpXeQ3G5cuXKUFbLX6dPnx6hjHr06AHLzQKE92UeKz/KWzlXfte+/xIFKgv84OflmtJH0tcZIc8Nzh+HEc627Tyclq3egYfNhrRu487wsDnR3Nj2zuPMnwpTADbNTwDYtCVZbt+e4WEzpq5j6tbRa5xjKJv/h/F8MK/ee+++h2HHI2EpWzxRRuChcANWsVehMJl02iQSFXaKuTkEuRhHRU5ALVAZC+UJ/C5j7fvOVV8X19Im5+OylXvKt+P01K2cK/V4XmW37TfPnh/lKRX9PqMflfIqvgVrlGFUhDt2VWIrYymvFbBG2aYviq+ugDSe18NZWUn5xBQB3aUAAEAASURBVDFv/UzF+Zv90o44Dm3Lb9uV95ueLx/Lx5XXbNszzz4bcf2XIbP5DvbvNyDdSTgvw6KZJybon4eEt7NRnjJN5Z32eebO/RrjnDfi3c15bEqd3ljqywZEymF79+5nGADYoAjUw+b3v/99uppQZ1pj26UKvi+8kAEbPZqtw1Ayhv0yfNqECYQaQz7NOkTvsZ4yHpSzBCEAbQAIlCP3IvduQcmvsm8HgJJg4d59ewBuAGzwWlHOFHgyHN0uzh86fIjzHt8Z/arFsaF9L7nkUgCb34ZFtsZBK1csxYPlxfT0038Ob5fu8CfzQd6E9/bll11OH3aHTrxrXCu9NHBqZoguWmub9bARtJs3b1487wuAVBoFjcIiW/pfShnDhw8jVJDvFzlvaOucOXMBRmbg0QJgs2pN0OUePGykjZbhehKZR0Hg2zBleruryHQs9QUMHTRgMB74vfAqbwONsqGV34Iofvs+CJwJ2Piyqvx0nKwh74Sh1jpj9DRp0mnxfpprbOCggWHZrnGW1u2z58wOGo4VdLrrTsIPX5lGovQUgPd9UjINOb0C2Pge6DHz9tvvpOdfeAFl7BdpHfX160seK/jADYA2Y/E46t69C+M/sT4BPAZU+xwLeUOiGUrOHDaTTjsDmt2dLqwnJNoQwi7HGgXAJqYH63WEVN6f6AEOsJVxXB2+J4+nqnP50hibHK38qv7K4zyYW/Xh2n6NAj8ZBRyH3zcWHbdWWs7x7YThEb7iHXSu9h3gt0achkl/lnnA8IcHDx4mL8wAwpfDx/Bum3b+NOaAzmEs4y2uiY+S00QvFo0h//znZwDV5wavHIAX5A3X/yLdfJNhDEeFF6S8xPdWXcNrr72WXiYE6mcYO+7csStp7HjTTbdFGDUBni6dO8S1wTNtbPBLv+DptNn3Ul6uQY+hzcyvtQ5+2ZucvcoNgjcrVqyIZ5069ewAsqfxHM6ntns/RquGS/tf//YgYO9LAdqrT+jYoWNEHDnE3K1X6cXoKAKswtPXubnwCFmJbQj+QHtinglaZuI69+ktO3PmG+gJnkN3sQyvoz7Bk/SwuYC5wdBv8n31FBQUc7qhY/Sw8V5Doj1OODXpK2/sDyBuNJC7MXbVULUT7f3uBn+TXtHPmRfRSltb9bG7NV5QL4DMwkfAZvmqTQHYrN90IO3ZjzzRrC0fTJ+5NvqhMpeCTqU+3VukiaM6p1HoBnr17pEjk1BnVuFl/UDQqLFe+C08N35SYtNW5Acaw8nMb3O7m66p7dUoUKNAjQI1Cvw4FKjMEVGY65DMezNzlmvHdBtgSfWZOO+BOA/fZs4SsDnEBHLwCIDN+l3pyznL0qo1W8OI87CATYREg/PD0gX0zYEuYNOzWytComXApnunNtnDxjmesgv3d946FbCx7jgfc22lMTGv5NnG8z/X5jxbPkUW8Fu5ynXrXw3YzJs3N4QiFVwmvbHQrHCwc3gcPsfRvhgGrUmpsAWBbW1UtIGEqQcOomwBrNFqblhdXVh6urhzsdquTbtYDBYBLAgE7ZzUXey5OHYQWJUL3w8i9vVrWM9+nDZv2RoJY+/95a9iARcLRqwrVUBoZec9LhZz2XRCpVzDWhheQStIF/lF0WFYtBC0sDjVXTmS+NkghRZuVhjyOVVAuOD0W+tRLbSNR9upU2csRzvwPCTvpQ3Sx4/0MgeOZSgsURDPhUcK3j7m0DFMxsMPP8Ii8R0vYWE4JJSJt9xyWzodS9FIPsjddmBTWDTbFK3C6lMh72cIiUa7Fcri5cKqZsv2PembRWuwptmM6/OhtP+QljatsKbJgpmCni8WKWABbBDMerQLK5pRw/qnvr06JfQFhESzb30WCEWZWcjKP/lb22oUqFGgRoG/SIFYUFbOVu+XGzwmn12LlaPKRkOkGdayMyC7nhYmrjacpgpseXPMHfDz4NMUIj/yWNm+r45yrpF3lQP/6b4V3pwh8rwXsxOsnZkT7p4nQ/dc1CucHWFny7aDafHybWnJik1pw9bdWNfoz8z8SZipZIx05m89NJunw6lj+5ZpYL+eeGn2SqOGdEhdqwEbyjTX3fJlyyM8iQYQKqwFPOyX0884I3LTadgxEAVQ+0h6r6JBD8+mPrZ/vd5P2fe77Ff3scfcqo/52+POx35KWY6hUmY5V31/GW/ZeyPLDCpbVNaYe0ZwRqXRAoxFDAPjeDavi5tjtyMyiwqkYcOGBQjpmDb5sePc3CplfNsXUW/lGe2X0kZ6qHE/CuZPdRvzfuX+mLPLVSc/8xZACgXMJ7B8/oZ4+wIjfcjxdxMKtzvuuCM8I/SosIgMjlhmLotmBd30RjEUjaDNt3ik+KyZntIxv4/KbYIg5kZcsOAb5M01FNIs+tY8Cf/9n/9beNhYJofJibMKb4vnG0OiCZyorLqVnCYCNmPGjorcMKWvcr/mmx3XKtIcjwfhJ9vJw7hq1eow+BFMWIx1t4m1Dcl3hNBcGjUZHs6waMqNgkat6SdpqJzs8wpedOzUJUJ8mcNGT5UTyLwrli1B8fd8eurpp5G914altgDF9XiHnXfe+QEqSS1bFgX5zgFUSNDoT1Y0AkOGHTOk3FMCP2vX8ayjyGFzG7l6rsBTZSR04l3jvt279xAS7evIYfMmgM0a6jQEm8DIL268kTyT/Qgz1zosxfVKN5+DSbq3bN6aFYMYY7UiH0XrVoa9yYuM44CsxcvdfssGT/k9C8Ms3gdBL0PG+e14cMwaAvi6666LsG/2v8rY1wyJhowsIDZ23FiUjHeTMwfAhucxN5k0zYZLytgah2XDLy3f9QR6Hqv5WR9/ghIWwKb/wHQdFv6CtxMmjEX52yVo0JpcFDu3YWWPB9HLL71Mnp730o5deyqAzV3hnTdw8EBSJFl2NjTyudxaAryVd1v6547xTPzKXUQby3vmmZM3B3/lBTjpRPQwR8r3SSdrP2oU+IkocPJ4LHOAlcm3gp8G8878xuO+63leg5fzWjhvGW3D0KQv8T65XnZ9LIAiUCDwbG4aAdd4L/ByMzS5USIEWM01tnZNQ9RXVzcs3XLrrenmG28KvtCRd75Y2m7btiW8W/SMkU9s3rSF+X0wPM5Q6b+AX57LO945PGxspwUGv6CRvlX5vT0ewIZzqvPWk3ix2A5z29o+5xl5uTltp0+fnn7z2wcwqJwShpqWZ6QR55ZHH3k8PUs7Nqzf0Ojhal3qFfQukq/dcMP18NbhqX1H+FboNhB3KuJltAe6SkeNOQvP3LlrZ5pLCE1zfr38yqsYHa1KQ4bWATrfGDm/JsE39Zr1etvpMzm3OI/tI2S78q6AzRNPPBFyhLKGhhz2gzz+fwvYBF9yPGQelDlabmk+llfvUBXOi2zHpbv3H0rL12xOC5cAyK/bE4BN85Y8L3oBgR3n0AD1NThgnujZpRmADbkJUbr1J/R7a0LJW6raBI05s7eNfccnNnU7EC03KcZIPl4u4DvOxWxYuaf2VaNAjQI1CtQo8ONSQJ6b51Rl7wpLlmtzvDJPMB85P/ivnC9MWw6tPlkvTaM2mb/mwOGU1m7Ymb6YvZRQoNvSETxrDkd+9JxiJAM2rKmaMT+3OUa0ptZp3Kh+kcOma4c2iYiaed7wQStTQq4nt9XDjZsCTbSrcmHMK6XtjVf95DvKCeVT1gl+Kwus+uGAjRYlPF4U7mIlL0J8GhdMgisCGi7CBEO08tt7EqiBkoWFWVEitGCh14KwJ81ZzGaX4Ey0Un4JdZC9UlLE/NZa2rjUhsrQM0Xly3UkO70K1+CxuDB37UpMagZBkJy2KrQo3MWQoY2WrXBpmDUFLReLtt2FpsCLVqd+a60iwWJTAo2OLc+pAOe9OSSFEleruC/HMY86uTcT3lszQOO39Us3yzMJrZY8s2Z9hED1BOEmPovQbOciZPpMF6BQNDmjm6EevMeyBYMKTQrd7difPiSar1y0PhRuW3fuTYuWrEtLl2P1u3l/2gdgo9vaCRaw+dXNAle8WHjY9OrWJg0f3AM0dADeNl0CCSWYSOUlzrS2p2KrfOUftb81CtQoUKPA/54Cmbc2XePvCMkDbzRUkQpv46O/h6eN4XZc5KrYNUSa4XUMj+ZCtwA31RNnmQuyEpIZppG/1xhVE8WZ36qENmeKPGN4hXQSIMkeNobOELDZtHU/4dC2pKUrtwD67yOvjf7MeNi0wGDCe1jcN6sANp07tEqDB/YOwGbEwPapc0cVzhQUc2qKUFQrAGzsX0OxCNgIejifO6f+4z/+E319TiiO7OfqPnQ+dQ51K8fjR+VPGVtlHHj4+46V456rPu+4OTUfTTnvPSpaVOQrN6lo1nNGBZJeCP72I3CjTCUIJQhTvGgct9UfDWu04FXB1Lq14Ff2zrXt9lB1vUFjyOx39BDXVG/lWu/N+0UCqL4q00L6+RGgMazNs889mz5BUd7QsD6Sw0+/5JIAbXzfVBjlmiwvb5m2+begR6bF1ihP+li2XS1g47W+277XyoSv4nWt18chLLlbY7RjmN3f//ff55BoSvj8N4n9CySgf/nlVyMcl4CNYbxuu/1WrLEvJydAAWygBeUXYMibBWsOHcoAmrlVDA02f/6CtJ7+sa+2oewXPDIsHx0dDyQfad2mNWBah9SzR4+walZmXMYY3YzxkbJy5LC59PL029/9Q+QUMg/LyuVLGb962ADYrFkbhk6G4LmekLvTpl3QCNgo48YWcmWWD0v3CXAtXboU5SeAzVMCNmtRGo5JtwOYCdiMErDBYgZ1I95AAjaERCOHzUwBG+ocgefRnQI2ABvmbTgVsJk9e06EedM4SWMlP22guzQLjxrl1TDGkXZZ+ajsKl2Vx30nHU+C6dLXHEpas9fXXxwgej/qXLlyZXoVwOZ1FJVzvgawwUtpHHz67rvvibBpowDbpJ/lKAfbRy4aW2p9Tzu2bdsaIL3K3A8/yoBN/wFNgM1EPKoMiaZ3kh42O3jHPv/ks/BKevvtdzCmIofNaVPSnVihX1B/EYZUAyNfpWuJzMWUXQnJxxrG55LXxVvEGG3ct0MqQ9xrvn/zgspFJ11Qri/fJ52s/ahR4CeiQNV4ZNex/H1btqbNZ+Id5D2MOQLli++1fNlwZc8++xw8cl3oB/RamTLljDT17KkR3rAvc5W84MAhQquva0hfz51b4a3fwP/3BiA7AUDilptvjXCJ5shq1zYb11i/Ho1LlywNoMQQkmvhXT179k7nT6tPNwLwTJ8+nd/d4v3OXq/5aQQ3fLL8SqIsOnokwAx1C0+RY0z+5jwmz/KZlBecUw3X+ACAzcQJE2JutQzndsNDvvD8S3jovEjOsG8D0M/r/RyW1Hw2NxFK7drrrg1+Kt+k4Kjf6QnGEb+9x3ktgzUc5qQGCR/NMrzlaxHKfSPhLfWWNLzl1QDcQ4cOCU/HrC9RR2CYUEEPjFjQvwjYGKL0CXQMGn78/YBNUWhlvpRHi5JfNsTZA2Czcu2WyGGzumE3HjdwxRbteZisFxCwyYCUz0lIm04pjRvWLgM2/fughDMcrry0CbCJMPlBJutkHvHpovMqbYghWsYpfezpfFXcxZ/aVqNAjQI1CtQo8KNSIHN/50Hns/iOv1VzBHw6n5Gn+8lbyAr+4vzxAtiA2uw7RP7hdTvSl18vS2vWbgfIaY1eAEcATP+zbl3ejqNEM9ZPbU+kPr3aprEANmNG9E9dBGyYO5xTy/rIXGvRLo7n2d/6G1vBMiXLAvkYd4VBYDmf2/pT/5UW5ZPXnsxe0gU55W8CbLRwXL16VYToMg58LEyi8MpDUmgsRCBKdYX54SUPD85/+9LzMYlCTf95zE3ieo1E9ZrY+LaxbnGaPyYy1dpEF+s//OHB9MXnX4T1iqHIrrrqmlR/8SUoAgZSHmVzL7JWBl+sgE7JC37aEHXFnyi/UkOlDT6PlVadt018PBTPymlbmZ/XIuKEOxz0vnjquNZ78jPpZZNBHttn2/bvPxBu2G+8MTMsigSghg41pvp1IdyNHj0mrHhyPRUhjtIdtOFBRNk+k0CZC9afGrCxN0Lhw/O6v303ypIVG9OSZZtBRHcTq1aBrXU6HoCNF0szFsMioS2Ppx5dWqe6wd3T+NED04A+XUkYRf9AK1+sTFeJxQ83dmtbjQI1CtQo8LdQoHH+4KbMG1EcVjaVCCox33777fQBCmUV4ioKzzrrLMISXRIxy/1drBPlSYXfy3PdYg6RZ1WUxuV8pYr/5F9ldihM3Pnf/QozZ250CW6P+DEk2sZNe9O3SzYTFm1b2rpjP0kHs4dNs1A4QH9lgABsDqUueNQMG9qfHDa909B+bVKn9uYuoYaYZ4h9iyev4cG++vLL6F89M1T623cROvXX9yWtUUus9zwPZ2Wr83sZO7nfq7vy+yej6uu9uvwud/q7jA+VMEcripgyloo8oFJFQGbr1q3hRaxnwfLly0NhbQg0QQmvdVwasktPGuUELWOHoKzpSy4/ARsNYbQIPhVwtD2lHe6HiOJO1eYTKpOdupVn8v6ynyfp3K8cjs1zRb7Sutr8fobOMh+Acokh0E477XQMa64mH8E1gCOjw/Ck1KeS6eQ6pGdF/qtqVm5Dvla5R0+cuXPnp4ceeigUWir5VABmwCZ72BT5cjUeMYYHe+VlctigGDSUWSNgo4dNADadYrzQmCp6qNBD9ty5nT5ZgfLsI57t9fB2EnQxl419Y3gwP+3oA4GytigWDTXTvUf3AKjss92E13mXkH3msdmxcxdeLu3SpZcD2PwWwAYl4DGAqZNCojEWzDt0cf30sKg2abVGT/RGkE7SuK/iy37J/Ywik3G/GEWmIX4EbNbwXpik+vY77iJp9hWEDRoRIdHMy2P4NkOiCdiEh83ahgiJdifJsQVs+vXvCxjTKt4tPRRfwcp79lezY7waymzo0DoMpwYFEKqiUIBGSdF1j+2JnAwhj2fAxmOOUcgWQKV9r7d93779GCOnEVr47ODLy1eQB4MxpOX811iYC1YWDxsVp6NGNAE2jgXzIsAwQjb+awCbCRPGA6RVABvasB2A5zMBG3LeRA6b3fvIOwFgQxi5C+ovTAMJidaSflZR3KIxBF0Gh6rfHceoH58zQL/cVfE7Ou07f7ygctFJ58rAL98nnaz9qFHgJ6JAZTzylUfl941N322rd6znZmTe7H5WNGhs4BxgiLH58+dHmHQNInvADwWEx48bH3zDcIt79u5CMbEyvEdX4eGn15+bobv0KjQ/7AUXXEDuuc7wWj2glccIlY5OYCMghHmnHiJKhZ6O7dp1SFPOnBrGAYZA7dsXQ5yWWVbITyS/LO+o6+k8twnQmP9OgOn9999H77Em2uAfjR9GAnJfe+21ACV30v4RcZ8P77/Nm7cQivV9gPZXIoSnedwKXTSsmHrO1PBokW9169Et+IIQh0+hHCItM99AT4AhadZfyC9TGASYN+xlyp4NH9yD18wk8uHcd9994anUs2ePiLzhPBS8FXknjBooW+De+VfvGj8lJNoP87Cxw8u8mPc5wNP7aQJs9hLmfXXD1rR42aa0CsBmJ11pSLREmNsTTArhXcMcEfMEc3z3jifSqEGtA7AZOLAf/YfRFDw2S4vQyP3KGMt81r8xuUR7bINb0/jjYpvHlq/Pe3Gg9qdGgRoFahSoUeBHokDm/mUWyJOeKxIZsLMbG5NYuSqfycw8ZGSu08MmABuWewcBbPbsP5ZWA9jMmbs8rV23MwCbIwI2GEY5V2ePSyJnAdh0bJdSvz7tIyTaqGH9Uuf2rVMrvHVbUr2wfmxRHYXHJOJM5fEyM3Ay1pleVGnzf2TApjokWsRs5aF8ZDsoT5A8uFJFhQCey4IHRzge5zidhRfPSigOVI7lrvQ6i4y72SmlecjytKw0HjYhFLBiNLbtE0/8iYXv+6Hg6IFAdBku1nffcy9Kt/EslgnNQE0KQjkcmjXmNkfhlRqa9t2zAT5VNIS/tqkcO/XefGXWfFSeIC6tXM/l7lVub3wGlS6GecuDrhlC6S4Eww/DoufjTz6OUBtTppzJIvlGrAyvTCZbdIFYrIFsnfQwRFooZajA3yavVVnyswA2PJtqFD+7iVW7CgR08bKNxBrcjiu09poioVrS5L7TEjFiDeJh07VTizR0QNc0YcyQNKhft+xhw0vka13GUHQTv4OAfte2GgVqFKhR4AdSQL5YPhah8lvvGnPavPXWWwHguBg3PJoxxlUYqhQ/Vemtkl0elecj54PMs8qxMlfEif+0f8o8Wflmboo5K8+GUozZQNuYJsBmHUD//IUb0wrmj+278VBg/nBh36w5QIrIAnN/M8JmtEiHUjcA/5HDB5PDpk8a0IswpG1yf1hHzPD0tR415q5QMeFHIETQpg95OKaec04ozltQdjSJ5jlH23f+yxtlucu56Os4mGWB6j5WUmmcq7jGc44z73GLMi2XTxkznlcG0JPGcSiwYZjXFStWhDeQseYFaGyvso7AgwCACp8hhEk138rQoUMjNK1jVIDG84IDAgZeX8BG6yxtKO3xO7fOvZO38vQnH8008Fh5jurzPFply6Vap8/os2ld/c47b2P5/FyaQ7gt3x+tq6dNuzCZG+Vscg/qJeR7ZtnKRE3lWaxlOVKsuyL4x34c8U/QyLoMh/bgg3+MUGd79+wLQG7y5Mnpnysh0WydTV0N+GG4q5cBbFR8ZcBmZIQJi5BogEhdupjbyOu9o3xYSBzYj0JxBYDGjGQOxYULF4UFtjQ3h2G/fv0jjNhoQt+YL6gz4X8MQ9ehQzv6qF1YircjDN+iRQsj5I7eQFsYm4b0mj79knTfr+9Pp6GEU2ZdvRIPG0ICaem9Sk9AvHPOO/f8dCMh5S4D3InwN8rEjBHp6j0CL/yJfem4B0WdORmk/9NPP4Mn0EYAm7ERWuhyZMuxgFNt27RgofRXAjYoVVVC2qeG5vniy6/i/RoEUHP99Tek+vqLA3DxGUNeB9CwD+N9YEwI2giqGA7NMapnu+cEeLxOWgs2xrgO/ts83oviYWN4pV3Iy+aKzB42gE545BQPm2rApsjG8gHDYD733Asnedhca2gigKiJE8ejPC6ATXNy2Gwlh82n6QWuf+uttzEO20efTM4eNhddlAYMHlDJYUM4OzxycqjiqrEZo4a+iH95DDm2fbxT30ceuGrz+f2cujn+3Mp3/lX7W6PAT0mBeGcbK6gem6fy6HyR18v33YIX8e0x57e5AOMzZsyMXGR6ehzBrdb3U95vaLO2hjAHfDkG2HrocJ4XDxw4CK84wbvdPgwsjDZx4YX1YaDQAV7qXCGP8dVSPjAUpZ4xjz3+BMYas5lfD6Ux4yakm268mbX0DWkQoQz1yinvmB4xtlevjSLXuaJ1Llm8hFwxM96M3DuGYeQStmbw934hH16Hh8y0Cyv5ayxRPsy/3XjjfbPg2wgr+fzz5plZ6o1R/qBBg9Mll14a4Pe0aecRBrMTfFeemI1Kg480vuLQMugJoCEIzf4qQqI8iXfMiy++HLnI1AlopPrLe+9NF8GXOlOeOYJDJrHSCiuxL3ymAtgYFu1vBWzySj83LutG3Hd2zN+lOut2BCjf7cPLtWEDeYIJlb589c60fRe6mxPQH89pJ9f87BmwsQ+7djiWhg9okcZiIT1kcP/UoR1ewSGiWSNzXJSaa22qu/Dd3LZoR+4sd7ms8NNyPrc4n6z9rVGgRoEaBWoU+PspkLl+UzllXmjiu0WC8DufzUecM+TK/j1+nBDpHD5w5DgA/2H0ytvSvAVryDu5C8CG9UIiN6ae7LB94wJo9tmi2eHUqUMzwmh2DLDf3OidAGxaU4mATbTAqrg+ZBrXJDEveKbAOVzAXJwnzXw8Ksl3e/PPstm+8inrdr+VU5z/f2AOmxwSrTxBECF+QPagDo9dIVDjNWWH71i8qJ6vXBunvJ7fHgrhpzLpaiHn0XwMgsYDZeuTNWvXxELMhePneNlo5Th16jnpnnvvS+eglOlJbNzWLKQVhBQG+bJ0iovG5briUHVD3K/+za/yUHa497MFUdkvAku+Ji/SvCSOl2IqzxI3InSohPHj6SNHyKuA1aPPYHJZFQC6XF+DBY/eQpMnT0mdu5oQlbKxBrL93pcFVUJKhKKD35w7imBnx/7UgI2tOMYzqUZxf9+BoySK3hWuz0vJQbBzL1bEuK9FrFpaq2CmsNWi+XHAGeKU83INGdAtTRwLYNO/O3ltWPjyUNHT0jo/Zu4GH7a21ShQo0CNAj+QAvJq+aIf9+XVfkqYCJV5JsZVEWm+D+eO+vp6EtuajLp742Lee5oW9vLgPLlm3p8bV73/A5v7/8htFeEHGhWGnmeuLJi53wTY4PrcsDPNWbAOS8ydGAAQYtRFvYBNCFTMDMwhzY4fRkg7mHp0bUOi8Lo0ekSf1Kdrs9TOySPmxdyv8Yu+LqCIwIdKI5UWKohM/GscffvSLeZ0x4X/nH9iK+3mPLt6rrp5vlzjt2OgHC/fZayVa/32mCHMtHQ1pJmAjCCN3jT+dl9LZD2D9CCwbeajEaQpH0OcOT4N/+q+QI1K72pwpoxvAQTLKO07+VmjyT5MZefkr+87Wsopz15+l2eWikFJvjzn5zBGNZs2byIc2keRw2DWrI8JkXMkAIyJ5Gm55ZZbASmmBwgl4BT0UkiL+x0/TU2MplJ2th5u6gfJL20F6Mzp8+CDDwUQo7eIVtyTCYn2X3///xFG50rkv9zfq9esDu8QPUS++mrOSR421YCNMlfe/M77u/DsXjDv67AWN/yaodBst14h48aPD6B3hB5PgDX2W+QypI+Mx5/7BHAEAOrLL75Mf3zoj5GDcSt5cARs6qHFfffdH232+rWrlkdItCefeDKtWLkSxWXHdNqkM3JIHfKv9EFOjD7m+W2qLdSgyTcs504kjA5gxZdffBH5IF4hrNgWwuqMGTs+3Xrb7RFyzOTdbduoPDsZsPm+kGiGJ9PDZuPGDenjjz5G+Tojwvg6Zg3Za39ee811WKCPirGp3Kf6rsjzKkb1AHJshOU3DRawyQZMKl5ziDQBLsc0B5CTyeWzYkVjSLQAbOgDDbJ+OGCzEYv9Aena6wVsboh8GsXDxrXC9q3b0heffRY5b958820Aon1p4qTTAfTuIIcNIdGGDs6ADcOpNYCNfeC/2Bz//IuxHD1SxZMq70aciwGdb2n6ywXc+90tj70yBr97vnakRoEfnwLVPD6PS8em4/n76yp8X55crvGYhgkCBHMAx/Vq/vzzz8OreRdehr77wQ9y0XFfM94pvWfa4yHTA4/RuqFD05l4P18EWGPI2h49epInKwO8eR1sfc0wcNgdoTf/TJj09957n/l0E56nw9IN5LC5mfxkI0cOByQh7w3XWmfwJ9fhPE/2lJZ/4q2DF+VW5mPB9D/84cHQMQgE++zmhruZkGbXXGvo9TGpW4Re943nAbjXaBl6cMo3Hn/88cht5vwkIG2o9ptuuolwalcQ9nJcaossckweyXnbwGNThYQoG6VS5lHCrh/Yf5D5bUF68A9/TK/jaWj+tN7MORfV16fb4OVnn3UmfLwNeg4QDpviHzZ6K55XWVdDEL1rBGz+1pBoQjCW5mbJWVpS2RWNrjreBNgcQB+zccvuCHO7eDlew9sP87zKdeoGsmelehDnQ3PUdGl/NNX1a4bSbUAaStjJTh1OBmyQooLLWmdTvRW+G+2IZgTN8h5/G+mZ254p0ni2tlOjQI0CNQrUKPB3UcAZobLebyynzAtNfDfPG3n+yGfzEf/Kz5nqmB+IYkBR+wi7sXXHgbRi9Zb07eJ1pNrYG3PH0Wbk/9TDhgIEa/wYEk3AZtCAToRD65eGDSEMOGC/gI1zqnXl6TDPp3rfNwE2eSaJZjMXN7WOeaWCPcS5n+mP8335lHWC38oQfxNgM3/+vAiJpsViH0JvKGAEmfm2glM3D5UFr5Ud46MQYeXNsWxtLkqWC4lbmwQMy0O5RgFW4XWNDxD3Z5JqibN//17CRqxMz5GY9Zk/PxNWJ8OGjSRB6s0kcL0c4W4MC+b2gBrme6G0+J8XkFGpFdDOaH1jfBA6kI6qbltc66U+FAJAaU9jm70+WutS2dL4RXn58fL1lld+FyHFcnbu2E5c7q+xpHw54q/r1n366WeQ0PA3uH5flPpi0dMurIkQaVAahXhCu0NQpSYXiz6GwqdC5c8REs0nzICNT9ssHeTlyjkINvByrceS5kigoSdQuGlpowinQNackGitWxzDXQ3Apj/JZccNTUMG9gCwQUDnweIar7eCTMYKzfhd22oUqFGgRgEoEHz4L1Di+/h2zD8sDN087++y70JWC3STWgvcqDhXKX4plpBXXGF4pDFhCRp8Nni4fDx/LKPMBaW876vfc//5NmnsXOsMUWHmHDGpoEdCNIKOx/gcQkJbSVLBr75eiwv0rohde6KFlpgobZkVVLBEfNnjh1KLE4dSz25t08TxwwBs+qYeHU5gBGCJWXEaU3rUmSmuQsi8J2EgQTmGQVMZXDw6vCr6MFqVlRscCSVKVgY5K2UvDuel6r5v7GvbxzXlt+PLT7nWugVrzL2hctvxZl6dlSjhS24ax6HX61kgIGNyYseeCiLBGUP0qdQv3ii22+tDWawMwL51Br3ZrwZxSju8p3G85gZ76KTNw/HAfHlf2bzP7fuO5eOcj/9Z3vF6n3v/gX2EsZqT/vDgH1BivZV27dxNfxyNkLX19ReTWwZL5WnTAtxQXvTjc3ivm79zXzGaKnQtz1iey9/Zw+bb9MgjjwYYY14aPazPOvOs9I//9A/pCpRkbfAOUWJZRWjf1whlJuAy+6uv07YIiaaHza0YyfjO62HTMWgbjci9G7TbDAD16ScfEXrnhfQ2IO82wBaBszPOOIN8gzekiy+ZnvqhSBOcOUKf20bbnxc0yjYZXDL3zR8feih9BjCwE5rokWIo3/t/YyLrM7EEb53WNwAsvfxyehyr6iWEcFQuHT5sRHjY3HrbbViMD+aZyPMUhJfu9L/KL2imB4sdYrjHN2e+CYj1cvqIOrdvB+wYP4FwPvdESLThI4YFYFMdEu21197IIdHWNETIojvuJIfNjb9IAyKHTasAGQVOLHcmfepCwnB8l1xyGcDYNdF+larH9YZHTs/9xdihn2ha0Mb3UpDNPt23bz9eSjuDTo5z3wHHuWPb8IErViz/i4CNPLrawybeAep0USbd5R3f8bBp2JD6kTvpOkAvAZtJJEDv2Qtgnntsox42AXKR80bgavee/WncBHJP4NlUP/3iVDeiLrVkLMWzQObIJUkvRDfwp4zPrASOF40RdPK8U/0e5THmX9+x/J41HXOvvIfl++SztV81CvwUFHAcN215v2pKaDpVtRcjOOahzLstQ34veK+RgiHRPvjgQ8KZzwoPZ+c9832FMWJ+VXhvMcJgzduPMGiTAErPJxSawLvepd27dY+5Lc+4URt8kfmfew/g/Wg+MYH4l/CeXPjt/8/em7/nVR17vkuzrMGjbMuz5HnAEBzmJIQAmUdyEjKd06ef7l9v39v/TJ9+ns7pc5+bM4V0gEBOICHMBgIEbGxjG+N5nkdJ1izd76dql96t15KxjS0SvJe03733GmvVGnfVqqodUiM5R3P6N4ypccutSNKhhszHEfMl8wXf1Mw1jD3mT76lmZ+26QDA//gf/2AHAHp7fS5fuRLbWT/TevIVk9JlTfb8HBakeo6JUYSq3X/5l38Vk2qjJIb6RIOoMZW7/+nv/97WOyR9azWHGOCMeZVpuyKBBs748y97qYwV4wMG0htv/Cn9/H/9o+zXvGRSSWuk1vWrUvWG6nQkDqtE58BFuwEXcxBzIDaEt2/bZgwb7AldH4YNeATK7JudstUWvvPTvk5SVKfPd0u65lR6f/sRMW+6RReA4Ea9ker0OXFIczwq0yfX96dFs5PsEMxLS9oXGsPG+HKGn0zCRkV6iSCK8nVZe/LsLupvbwXDJtBS3AsMFBgoMHADMMDaZ1/0ubyzuZn52ZyvEUzZxPZQXzP93cODYdPZO5COitm/SzZtP9x9XFoIuo1hMwjDRvwD1vxKfV/oC0Gam/qlJr0yLZw3WTQBMWwWzjSGDec4WRENAgphVWVPM4phk4VbsMPjACql1qhcKDFuuLO1n/VfF+t3XOxTrophs9kYNvsl8bFOH2daVYUF36iMVQcVKOaCF+7xbE1VIu6eDjQaKkcyUAr3kbch1uKrIdUXvAJEZZPlH4IDIsScFsMDfbO//Y//0EfWuzodM6QPxrtsE/NFnciZ2TLDVIVQEnsyNgb2YUvjyY/NBn8lWICx9O5wKNgcaazlwyO7kyZi6MGicPcOYPXSq5VDPPkDexei1xIPf/b3vzdiIUZeZ0jtxf33f9FUNaDLHJ3vfPwbiMpupBwaVFkZPg0nIpAI5xPDsFFZgkWY1B8bs+F0+lxv2r7zaNq89YCe+0YYNugldBzyMQzDRtxQGDZSibZ2VVtqm9+SGqTSpgbCDIgDZeaoXam+9lL8FBgoMFBg4CoxwPwT8zhzO88sgFx8mKOzHNVNEFFRkQYRHdUXEJLvu+8+U0GFvQgIgEYEzAjkkRfgkKevG8xZPnddJZifsui+ztpGTuiwtY81gynemDYiM2vOhywPw2bXvlPpzxv3yQZah8ShtVBXw7DRujfsBIdK4TeJYVNd0ZtmTs8YNrJhM7VBdkOkqxaUQ5joF3EDIhDGgyHC1NZwStPbjXYZeWYzEG0mf1/JgNMd+UWf4Y5qlniP9iX/eCaMvgRjBhtJnCoGDpg09Cc2W9ikgSEY6tmISx4QfWDKoEILZg0qZ7FFw14g7NGg7owL+KNMIOWZfhywxT0kbCJOpIlwW2ajsl5l+8XL2irnN9Zj5HNpWODJx1i/2mH/vr0mjcE+B7VoMDmwVbJE+v+xS/CQmBxI3CBZzMlqJ55RN28DpEZw1MHqCyNAkhlAChzgGgLUu7Kn8hsxUtavf12STBeVV60MW382/bf/+/8SI+HrGRNDKtH27zfbMy5hs0HwnDU1O48++gMR976WY9gIAMrMeoWKklHpg6YODPsm619br3K6bH74mghn2Em4XfvjJknCADswAh/MJ/asENToI8CK/UUkqjl1jtRVQ2OznZb+u7//z0bYmzK5SYyDE+mF55+ToejHpP73PTEOuuyw1Jdle+ZvfvCD9BmpbaSPKHMwIXzA6JKqXT3BREB6e5uYg0+K8fDH559PO8T06RBeVoq49+OfyIaNmB3LZY9hUj2HogYzGzbvyYbN741RcWAchs1ZMbhQa4QqyaeeflrGvndZH161crXl+W2pRkNtH20J+Y66Wx8VbCrI2he8MO9+8MEO1W2j2ZzgdDp9AGYlKimRcGxqbjLGZkkl2sbMho1Uov30b83492iGDf1P5ahcVB7CvLqUYXNEDJv5Yti4SjQYNjMzhg0fG6c1RrGL+eTjzrA539Gp/vmZ9MMf/cgkbBa1t4mXTN6SsKnO5ifrJ0G8ZDZhZ+wEUwEjx8hyF2Mx3kt32tBGZsnLniJt3MuCi9cCAzcAA/Rhd/S7eB6roFK/ZY4MF8+MddR6dfd025hHknnPnj1iJh8yJs45MWu7NAdis6VOjOpp06ba/gv7KtjEmqdrltbDRkmc1okJz9zm4ynK5T6kdbdHByIOm22xX2rOfOONN5Vfg9aAOzXWv5sefPhLqX1xu41Eny+pldIKPt4ZlzBh8WPOYp3a9N5mgxWmE+HMtxwEbWtrM3js2xxoRECinsSDAX340BGTaGG9Z/7nkCpMGhgrc1WvRql5q1A9KIt8jVljCMPHkWhYlx/rw7at202l3FNSkbn1/fcFR0t68KGH03clsbNu3e22Z6gRnYA5j/yAJfKGaQN+seWHhM1jjz02wrDBNtAPtJb87Gc/tTUQW2uXOmfBxBzmWAdmIGQfRQqe9Sg8ROw+7ZkuXBywwzgbN+9PR47plLRs2LiqdK0NxrARlMI17ddU15sWzBw2hs3SxYvSlGZJWiJIpQKgHYAj9njBsHF0qVzVz+tMXJzjzx4FjzuHj+exnrJIxa3AQIGBAgMFBq4KA8yxQQ2OhMyyceHHmsT64LOzh5DO52dbgRUIXVna0GQHvV+0gDNpp2yj79l/Np0RTXkwoQ5NttGNkQKzRt9USNiIpgzDZtH8KZKwaU3tCyRhI7VNLB22rtudErXWAAQ8AFsXfAUjxJwvKBmAWtf+2hk26NH+rEnYzNYCqfrYT9Q2dxdOOKXCJo1WMokabU6IXnK8jPIwZFqcEW89KC9raPCMP/nqBN0gH/F67tEmbdfu3enll1+WqPDv04c7dmljNFcfct/Qicfv6PTdEhE9mq1Y48plGyu2FcpSEOjPMo6Nh/uWw2YZ0OCWClhGgLQg9/cOYGkhSglw87fOQZf0NPxC1EFMnFMzjz/+a/sgRy3I3XffKzs8XzbjyGzsaqTWY1RZlgUbGGWi/KmTHoSTiWHYUFkwZ/p1rWQRyQYr0tmO/rT9w6Np45a9En3uzYxG60SrNpDAR1slbBBU9krCJokbKpVoK9s1uKQ2pF6ENZ2qgihXwmqGq5IHRReuwECBgQIDV40Bm4djatc9mDXMrczTnNBH8uH3IipzOhJVVRAdP/e5zxnjBkkHbNxANM8T6gFk1Pw8xvtVA/tXn8Dne/9017PmcJvG+UC3ddG3Uc6wkYSmTmLu3HsivbVhbzqkj/regepUUVPGsNGqUyGGTU1lX5o1Y1K6VRI2y9pmpsmTxLCp8vWQgxicREVyBUYJhAoYHqi149Q+jBHsYyB9MtJmrOMBn+GdNYg9gV6oBn/WX7j7RTTSc9lpXBFpkBZA7RrMGFScQYzmgmCDlAMEKk4YQ7wBBuCBCQgBCKIJ/Wvp0qXGsEHKALVQeaYLZUeZPOPiPhbDhrjlzB1LlPtRU1zi8Ip9yiWBmUeUO1647RAMRvYJQ8LDGdmXed/sgbBH27Vzt50abhSTYtmypenee+8zNYRIFHlbTRaOZH8Q/IqYAxMCB/Hdce4SS5x6RvUZDJDd2gNu2LBRamz+JCbAh2Li9BmzjpPZ//2//z+S5PmGqcUlPYwzjGBjw+add8oZNiFhU7Jhg2QLjirtF+Ptj8/9Pj0phg32BoFhjSRWHnnkkfR1MYXWSC1ajeYI3/I4UxgJr34ZxVbRUtvTKVxssf0qEjo7xUTplnFmVPR9QQd1/vbv/pPt/aZPm5K6Os6nt99608p6RafSUZ2LnZx16+4wSRZsbS1dstgImEZoFK4HB8WoFKzg7ewZqfaSOrRfS0XQG1Lvc0J9sUcMzZVSK/Toj34i5srXxBRZlhrq+bQRw0awYdPn2Wf+IPsNzwtPB0dJ2MzNVKKhYvDI4UOqwyvp36Su7X2dmgc306bPSJ//3OfTjySRc+edd1jfhrZKn4dRNTjQD2DWhryb9I9sTmB3Ytu2D8xmEwxLjIp/RTZ6sG/U0tKS9u7bm5OwyRg2Iny6SrSvjZKwoW9Sd/qgSb4IgDOnT9mBqBEbNiKmOsNGEjaPSMJGB6NaWqZpvmBciWEjw+F/futtMbp+Y1Jh59THbhFD8Yc/fDR98aEvpbb2tlQlhg1S+7UwbtSujJkRyXxlw7zBP3MQDvzQ97jGd4pEoktcpIn7JREKjwID1x0D1oct18v1O2Jl/VY3nhkLjAi+EQmyPi8vz0Xfi5oPOzo7bG08cfyEqaTs0NoJw6ZeDBvWRJg1jH32XHWyb4NKMc8bgLwEK40yKETjljm240KHqSH713/9t/QHSQB2dfWa+sMHH3oo/eDRH0pSZ50xzhm7DEVgQiUZc4Yd5tB8wb6FvQTqGnu1jiDhwhpLfNZk1mbbB+rwhDvKVz5KY9/FonsMaE/DgQzywIEDVHSaukek81QOca1OCrO5J6uL+1kqkz5iL7r+1dfSE08+aSrfTkutZVv74vQ9Sfw98jffl0T4fKmPaxCtQAyb3IEOx5fjypg+YtggXXPtDBurimDOk8E0v1n1vXVLDBsxr2SPoLtPEq2Hzuowzu506EiHSdg4w4ZDAsI1OBDuuTfU9KR5LQNp9fL5mtPbZOtWeJYwjhUg3HK8R+wo4UplZZsXmD4x7xr6DERvD3u0Od2eRn6i90SPHAkoHgoMFBgoMFBg4CoxwEoDm97nXV8J+I2L7KCz+LRNLA/hydOwpnCYE4XOOr8ptei9YvSfSDt2H0sHtW6c79ChLzFsBsWwQfZU1O6MYSNr6WLYTJ1crcP/U03rRtu8ltRkDBvlqkKZ71kyKApIh0cYNhw3ICBbCWwBIZL82Lfb3t3DLdIE/Nh6KDi4s2eIi/0Hhz6v2IbNJqnw2C9VEp+947Om/sAQkEOCV9oL4JkCKQQkxYd2AAN6HBA/GRF4INwSGI4cUeHFPSPzKEtjGdgdcWqMkG6WKPTjjz+RXnl5vTZug4LzLjtV8/nP3ZsW6COQ3IK5QRn84bibUeOs4RxGL5vwQBjPRLH49lL6KdUri++llSKQio2DKmGi3wrv6uq0U4p8rEI8gHmDft7vfe8RGRB8QJtVThQhXRMnapUWHAgGg4kX86AewohuE6cSjY0pW2RK1gZcG7MLnf2yYXM0bdBJmuOnelLfkDZknJA2hg0R1WZmNLonNYnIttAkbBanxQsQX6vSSUVtxMCY8vW24qVwBQYKDBQY+PgYiDmauZN1ydYmZct7OAjsqOxgToZpA+G9vb3dVKPBuIGgDtGfj+LIg/S8h1RA5HVz31mrWB18hWBCt+VPD8NaK9gmIXk5pGtA8/3F3kEZpj0uhs0e6art0tohhsoIw8Y/0M3grBg2tVUDqbWlIa29pS0tbZuVJtehEs2x3StJC1SQPCv1dthx48AI0irLZFOEtRWmCJIsqFuC8DLS9gJJYNgaZWuqZydoAZyXbLOp9ZZ2j74E4QnpDvoNUjQwZdATD/OA9RzCNoxA4tFHYEa0tbUZPMACowZJGhg0EKaAC2ZSwAZ8ASNl4uJe7m+BWThxIi33gNerk+WJv9c6ktqd6lq9R/mWyi3zHvUaMDk26QPsSyB6dZtB6HfffVc2WX5rJ58PHTxs+xWYV7TR6tVrJBn9WVMHCxMHop1JaCgP+pLVgX1f9kxbHxdh/cMdH1qbM2737NFhETElkK5BIgo1Y3fecYckbP5b+pokZ6qqfL+J7cP/+K0zbJDKyUvYYF+gpBKthCtwIlDSQR1aev75P0qS50mzP9MtOBYtajMViqgNu1tGoCkXPAIzDoZTj3AAUw9mIrZfYHbs3r3HykZdMPaKvnD//envpDLn7rvvSjOmT0s9F7vS7l27JNHzkkmyvLd5k3BSJ3zNMZsO35FKr3tls3HqlMmqm8aTyqFMPYnZ06G++EF6+ZWXTXXZNtlG7JQEC3YOsaXw4x//NCGpA66RsOFTqUMSPO+9t8kMZo/NsGk1tXIcNuK0/Ftiavzv//1PYiq9LemfDs2BVZojl0ld0Nd0YOrrUme0VoTNakl+IxFVYtiAE+w0ITX0xBNPmHqksGcBkxwbEUgrMd82NDW6hI32yM/KziPSOKRdVcawYV4G2z4vXw3D5nuaS6QqafpUw4G6iKlEe/vNt9NTYso9//yL+r7oTMtXrJLNm+/olP5DadnK5alWtnywH1kHw0a492+IbOzYnEEt/d26gYCLMekhY/1SA+8zo0Mjw7iPDi3eCgzcCAyM1RPLy/E1RDHVyW2OZv7J1i3rzfYTfV+ps/WI+QCmLRIpMDVQi2V9XxsF1r/aulq7M6e41Ih9HXrxylPZqDy+KZ2RwjuHA/o0N7H+PqGx+8zvfp9279lv9AA0g/wXqRl/6OEHbQ/HvGSMViWkbObOKn1rUxb7FvLmEGR8r0edAMDmWBFzsIubTfQW34HzcCWXK+0beGOPSH0qVLDjBuxl+Iq9EpOF6hJrKevc/v2sV/8hRsuvtN7ttH0C68z3fyDV71/+ihjl02x/4ZJHKkhZ2H6U6cILMikdDiTBsMmrRLs6CRtqkWVpX+usisIBZRgi9Kj2i50feztpS08HDp9Lb72zW3epQx2q1Z6vWnHAczBswLcOslR3p7nTnGGzfFl7mjalXusHc6syUjjrmqZaZ9hQnjFtqCSOezwbQOZrG09/GvkF6ixwxK94KDBQYKDAQIGBa8EAq1jM+qRnho35uDTX2jSuV2ZnD+XJ52r7atEjKtGQsDnf2aNDnEckBHDYVGl2dmv2lzpNbNuagQ19U4h6II0bzrCZPqUmLWlrScsXS1PFnOmpUUQBjnhoYcnKcjhsxc0xbBwOYJZTdFuZP10MmzvsA5sNkjkhJD5MbVMDenJh3jp4ZPFAColHMiAXNW+WxjZ9+Ix4EG5492y1IfDNlAwTaxEnPif4jopAgpHAt978s55PmO7ae++5Vyo37pUB2AV2Gs4Xb9tJkaPnq9/RDBu8owI5OMzbmhtoectyADbvGBZFcJc6K3GysJEs1YHUIThhhFoJ1LlBZILZcuutt+p04f32McpJnCqdyIHZZSdKlJ5iQIufMIl8dVcZE8ewcRxQng0y1ZaNWYcMRe/YdSxt2LQ/HTvVrVPSIrFVZAwb6s4OdpjTn72poXYwzZ8zOd22ZklaYgaitEHPGDZsy0dQBUILV2CgwECBgY+JgVijyMbWmLJ1Cz9ORHKa0fWsv2qn9ZHU4OQ/6tHuvVdrifSoI7XBB7Gtd5qQ86caPyaYn5LkrElO9GCVYEIvZ9iwkEkwUx/0UvnRM2B6av/8nhg2Jy9qPakVw6ZeC50YN/bHmsnpyv40qWYwtc5qkv2zRWL2z0hNtUO2dkCoPa+2evWVV6T2Q6f+t74v4k2fjAJPNamH+3Vq/x61H4wSmATRZr6aAa+cgVp69j5DDMGq9Zn+AdEdVWcw82DSoN6Fi2ckaPDnNCvEKJgvMGk4LYyqJy4kCIJRk5f6CSYNZUa/Mphs7bSny/5EOiIZ3OBcf+Hy4eRPLW1vEhGyOyny6fC2/LLw8W4j+7WRCF4CxPoBSVZgz+XgoYN2Qvill1819bXgDIZXff0kMWhaDDft7e2prW2R1MMtMLU44IhDK+yD+nSCulP7JqSYjmu/d/ToMVOBs09Gng9J4gPVOq5mrkoMkMlpcfsSqZe9XyeRv2eqfJ2xWmGSHahDw67LWBI2K7BhI5VkftAIopxjhP3XsWNH0+vr15tUNIwUGBWTJ08xJshXxAB54IEHTLVdkxgN4A2CIrYVmFcOHNhvp79ff/11U09z4UKHJMT7rN81iOFwn5jC2Ja55957pAKoJQ2qvtg43CpbCk8J1pdfedXqDqFrwYKFYtrcke6+8y7NT8ul+tcNcUN4vHD+XNq3d48kjt41CZsdYmqdEs76BQv9ePmKlZKw+bHZeYRh09iIPZZhk7DZuHEchs33vqd+2yoCWo0IoDo5rnJ279qd/uN3z5gh8W3bthtzctr0lnSLJFbuFiMJVT1tbQvTlKlTbM86KMYlzDvmVIiqMHyw4cMzuKWtIayiogcGOWOnWmMIJtfvnskYNpIA4oAWTCdXieY2bJxho32wNqZjSdi8+NJLJjGz/vU/SV1RzoaNmGzGsMG2heYY8HBOY3gD6vUkYfPcc89J7fL5NHNWq07n35Hu0QGwlTI2jnqm6WLytM6SFICeGb8+f6ivqL/g4o4/9eN9xG9knOQfFMlGZt6P5xjHcS8PL94LDFx/DNAbL+dK6wcdnH+/j6TJxgHe0e99LfE1zvq1wnA+Ntg18J3o4TZ/Emb9v9T3/Z1UvsdgNaMo4kMLgGn/7rsbTcLmj8+9oHn3UFosRvCjkvxDCpL1BdVfxm9RQpOw0XwGA8aYHvpW9bqRqY/ZkLK0WjKYcRQKWHr3umf+hGXpPAKRcFl8S2Q5yS9Lq7r4quwpPHqF9jTnbT/6pKRb2I+vAABAAElEQVRrnvrNb02aFFWi3/3ud40pfouY4nXaz8B8MmaTwRNMDUBT/rrYs0Bv+Pdf/nv6d9lEQyqVOevKGTYw1NwJ24Lav9RHMWxUNrWBwsIFXUDa0tPho5IUfWdf2nvwnElPDwzrwJOR0/IMm6FUX3kxzZ7SZxI2K1a0y17RpFRfyxloZaI2MYYNJYN3K183iGsKcRf3XDuw8TQXd17oQfl0HqP4LTBQYKDAQIGBq8VAftaPtMyweWquZnGmcU27zMQ++/Lk8zK/qEMzho2WmnMd3emDXYdkE/2QNDb1p+5eMflNHRoMG+Wl70qzXyOGDTZsWqbXpZXLdDhThzjnzGwWjTkYNipLBduSQTrKyxg2lBkrRiwTdjgA30+FhI0+ODGIioHR0gYsCDOqo7AS/sKFuUBIbBzycdwv0nl8ExMGp1levrBmmYFhIVyfnSIC9OnDzPXKstHq6xvQRu2UbBHs0QegTtXow3Th/AXSPb1WxlLnanNCTmw6tJUwoGwbYBmXGDY5+G0jYMFZfJWcbeQMJuUBOAYSv/rPslUH4cneLA0nf6iPEfmyLNlA8SHKSWB05UIE5HQieuynTtXHowiCdHAytk2k0oMv3Rw3Fki5bI2EEz1OhA0bL4sNLU927tk2Zp3dIrrtOi4JmwM6JX1Ram4gyInHyYkiQ4Vgh2Ez3KNTif1prohut69dplPSrbJpA8NGG07lyRAHd449eyl+CgwUGCgwcFUYYK4sd/m1CQI/H/exzvAcaxSSERiFf/zxx9MrYgBwohybCqGmp62tLaG+EoJ8Ps/8c3nZN9977At8vWVDZB/6mYQNi4Iwnnq0ThuzX6LP727ea9KZ/UkMG9mwqZChXiOUsH5qg1Zd0S/1mSnNm92cblm1QHbQpqVJ1YOmThMpFoj4SC/88z//sxnYvShd8qguue3W29J3dTqeU/urVq/SQQiXrmFdYm3GRdux7tB12BBi8JyTwKhEuShGDSqjYMjAmNmzZ48R0jm1imonmDTkwTqOdAjMGewgQbxmzwTThjD6DTDhor9F2dFneYe5EC7vP1Zc+jKOsAjnnXT5d/zC+frtY8TX2vyvx4pyeYvn8fIb29+J5+zT2ChjkPmQcLV+/euSYnvBJCwgHMFEoP0Yg6iN4cTwEjHW1q69RQzSha4eTpIMSKkQf9eunWbz5JT2e+Rp6raUByeXsVmEerGFCxeJ6P95Y0rcIgmKWbNmiviPjv/KhITN78RoePppVKK9a1IuyyWF9eiPfigiGBI2y8U8aM4YNuAlrmTSVO9tfDf9n1/9yvJgz9kn2GE23CnmyZcefEhSQreZJBc7TOoFg2L//r1p06ZNprZtv6R0sH3DHo86U3+YVndKBRh2UmBWzFXf8d3VsKkye/6FF9Mzzz6bNkoq5eSJU5p7ZOtBeMImwud0KGmFGC+TxWSCmXJIKt82SJppw4Z3TIwe+HD0kj6dZl+6dEX6gdR7YQMBlcHNzfWSFkFdW5fZ1HkGGzZ/KKlE+8lPfyLJ7zzDRm2l8YGky9at2ySR86wRR2Gg0aOmTpkqQuA89f01UnV3l5ik7TZXDmksnT93VjDtF0EV+DYaI4v2g+GC7RrUvNk4lUQcaKff7ZMqgBEbNjBszl9Iq6V6zox/fxWGzRLZAAoGGXthVOj5vpk9NyrRYK6hEu01GDaHj0kl2lxJ4KMSzRk2LS3TVRyn9sS4Unuh5u3Jx39jbXz85Gm1VbVJw61csyotXbnMpOKWLl2cbr+N7wvZpGhoVJlW7Mg4HDUmsu+JUX6eIvfrozLnkT362PR+eGlo4VNg4EZgwFeHsXM2wkdZkPdSTsJqplFi+nq+vzOWnZmq7zx9F7LOEk9D1OKV8vSS/Z1JwP5tLoB8H8yTEY0ZWv/4NicW9INerQnnNDe9/Mr69POf/5PZNpuigxtfklq0b3zzmybBiB1eowkIRgABLhgBvJIPtyjH1W7hb1ENVktm71klLBtBLFiM0c9MktEByAc8WEzqosvzpyT9CXZBQAqLl/lavGOaU9evf1XM5qcklfmq7UXvu+9z6b/81/8qBv99abr2Gg5zMpVozsywSig9+KAtxLDRegOtARs2XEgAs8dgX/LRNmygmwTDxqAUpBnDRmrNqM0IPqwmwbCRXT/h4MjxjvTOxgNScXNO0tTYKMgxbES4QMKUOtdVdqVZzT1p1dJ5WoOXSOqxQeuipJIoTfWAYQNDnT5gZWYwKJjk5md9xquPh6JFoEVwP+CNtJlPcSswUGCgwECBgWvBALMuczhXOOZYJmKn6DJfs+6xJDIT+wwec7P89AjDxpj8WmrOXugRs2Zfev+DQ+nseWlpGMB2TWbDhrz0TWkMm0qkbPrS7JmNsok+36RsWqZOEl2gqmTDxgr2QimRNcFt2NhbtnZo7RVwvnTo1xg2DiVJJsqxZscV+yfufOPzHXTFKtE2b0ElmmzYlDNsaCQQIpffnJmH+QsJqrcD4XFig0Ec/OU7sumweKBNkRxd/HoR2ZPCXKWNG6P1roJYNac1UYfR0dmtTlGhjyidgtNGraFhkp1AqVA60pac52jlW2mEyA+QdHmt3IsQD8oC9erh0chEyHyyjRo++PFnsZR1lMhHKicsL+i0IKdCq0ScQi0GFyeAbbMIKMrS0hjS9MJ7lglbJ2fY+FDhw58ThmzMfv7zn4sg8bQRmCAKoBaGjdnXpK4CVROUA6791GkMKof48r9eH5VstTLBNKH0Iqek95xIGzcfNIbNRR0clZC5BoaLgPs2j0EmOwQV3eKCNqZ1ty5PyxbPkb5BVKIFwwZMMbh98xp1vTxMRWiBgQIDNxMGfN24uhrbR7OtN54un0eeYcPcDFEeAisMm7fffttOj0NQfEgf/ffcc49J2jCHmuoJZRd5xSJ7dZB9GmMHEYU1V/+2dLGg+Qc+K+KA2qJb6/b5zj6TsNmwZX86caZH6wYSNhnDhsRa0yq0QavR5qx5UoUIpJPTmpVi2MydqlOZg9JhKyOFYph8KHVkf5ANIgzPc2p/QOpWaCNsYvzo0R+le9Vuc0WoDYYNK1j82VJK39BFW0L0wfYKhI39IoDv3bvPpDk4vQvTBiI86s6ICyMG5gynVbmj4gv9+0gIxD3UsAUjhnSxX4q+R1+yQx3Zopd/Lu8hkTb6HeHlfvFOWL48nvljI+FbCf+1PPgZw0U55XmOEXUEDvALIcwJ6HpTO2KQGcL+li1bpVLsdSPaQ8AHn+yD2JiiCg0C/lRJZsDgqqp2ghd5oSKmQ20N8wwJKjsdrfrAPJ0hKYn29napV7tFUlW3mpq19vbFaYaI8ZMmcQpZ9dQ/NmxKKtE2qj3PpmUa2z8Sw+brYtjkJWwcQeDHcYS0zP59e9PzUpuIlA6qgk+dOm1Mpdk6yISR7BapuZsiu4mcYO7v7zUGw2kxDU6dOqk+0606NRpTr14q8JAywlYMBMxbxVj8jhgIzDFLlsg2jQYNo+ViN3Yad4nZ8IbZAcIe0DkxLThRPU04QhpnsphUqPlBwqaj44LUep3WmLhgTKppOgDU09cryfPjqVP4X7FyVfrxT36mch7WyfP21NRYq7zcvs6GDdiw+b0Yn2LYSBXPEu0Vf/rTnybUvc3LbNggXQMZrX+gX/PkSWMivSQbkn9686108MAh6xK0B5I1c+e02vgA9TBBezRmzkoSCjWCMF4YQ4xRGJtf/OIX7WKeRU0ge0zGxp69e6Te6HfGGEJl24Ucw+ZrYtgsgWGjfmL9Wv0HyXv6Hsw5xtAIw0Zqkl57/U0xbI7aWEXFGfVau/YWSShNN2Ig+9RuqaLb9eHO9PRvnkq/eeq3Ysge1gk/7dHVnk1i5DWLMYYE1WdlH+nb3/6mMehmSYUxdjZGRpRVGCiY6VR5G2ulfa0h6ZIfasxV7rzvRR8sDy3eCwzcCAyM1RMpx9aOjypQiS0ed9Y6MdNt7chlyrcqa4KvKQQQTxMR3/cw4DVmXGqEcZPlw2hibJljbJHO9xqsBcSDAdCtOXPzZqlJf+JJY9ae1Hy4WHMZ6hq/851v2/xaZ7ZkmJgEBwwBPRrTJsvbPBi7WdmMvyh6ZA2VX6wrlg91sPT8xqUnS5jVI/K3e/wAux8ayjLQYZFBs6v461//Wvua5zQPHdLBg9m2PvyNvuXXaM6EUW37BGUDHL5Og9OMSQRCBFOn1kuTsMkYNhx8uBaGjWOccpjVYJ34fi7wAnWFdufOLIyEzdETnaILHEq7951NHRd1OAN16Rzm1F7QD5l6qtqKrtTS2GUMm1UwbFqaTF2nrYMZw4b2thKpH0/WNhQqXOvf+pxeR/oUG08iZY7wPNzhX9wLDBQYKDBQYOBaMKD53vbczONMw8zFTMa+RvCKr63hNkfzZj769bnZ1m09Yr8GNZpnL1xM72/fK4bNQUnb4FenmKESjbz0DaKrOmPYzNEhzttvbUtLFs5IU5tqU510Z7oAgOZ75Wt7kKzsPMOGcllT+HeY/Pevn2GjD9N9IxI2s1U5VSwQoNpSTXfxJAzgQETmZXsHUkYUBRrCRjz8PSJYGZ6LxxvJkAdf5GkIM/SXbdYcEjYrvmGxDViWPyc0rPUAahTEADQCVPQhKzO8YwMQGwIgyGpoaUups7xLgSNZl+IoLXAbzNSDDuh1Z/MVRB3DQz4fCjXnnt7R9KyMwcHEMGwcdoNA5RrDRi8XewfSLhmO3rjlUDosw9FdPZyuwY4NDBtdgruiQidoU58GmkSfp09Kn/3MCmPYYMOmrlof2IrENswRRkvm+4rXvPgtMFBgoMAA8+XVOubwSJd/Jh+fi5mDPV+kalC9hJ2F559/Xifx37EDAaitRM0SOsQXLFhghOXIi7R5BritFcrO146rhfavO759shsuc+1kj7Zqae10hs1FKaw9c75XxgWPp81bD6aTZ/vMsGCFJC0qkbDRigAhpkJEYuzXTG6qSPPnNqc1krBZ0Dol1WjTViNiEEyUt99+S4Tm36eXXnxZzJWDSpvS7NbZ6atSVfWTH/9YBPzVabLsfZj6IoX5Ws6v1k4RR7pFTIdxcFKE9RMiaBw9IpVbUrUFgR+iLWXQL3AwYJCkwSYOUrHt7e3WH4JBg8RFuRRW9K1g0EDEUtFGlCaMtR9pYQyV806/iYsyIz3P0afCL94jXv4970d8/lhblQlBo9ylPqXggCl8oux4j3upbPZmTkzzfP0XuwUQjDZt2mLj64MPdgi/B4xJCiEfIj5q1BiTXN5SDjN2BmyPJIkHGDscbmlsbFI7zDLC/WpJZqxevUYEuWXyazXmSI0YGZx6DoeEDfZQnvkdNlE2GQNh2XJJ2PzwhyZFt3TZEiPIWxLDUSkt9mhg5G0V0+TV9etNxdsOEfcvyGg2sNK3qkW4rxExsEaw0tYwopBIAda5c+aaytu2tjYjLKIOFykfcAJz5KGHHzYJk1vFRCA9MECARPXazp070/rXXk9/fvvPdigHRhFSR2b/QXjGASkgI2mE1AgqHGEkohJtkw5dwSxZkalEQxposfptfT32HCTpplPYm97brDH0nNScvWSMJMIf/dGjInJ+R8xIbNjopJv6EPaAaAcOSR0R02nz5i3pDakk3rxps6kIhAnWJyaR22D0Pq4ObBJrMMTBE+0GYw5coHKSC8YNY4d5lP4F/vaKQfZ7MWKBa4u+Q8DFSjGdHn300fRl2ZRZsqTEsAFXRnxVHwc++iLt9arUyT2F/aQ/vZWOaFzPFfOJ0/bf/va3xdxbrbEsho3of5zg7pdE3bEjRy3Ns8/+Pm15f6sxu7DbM5jhmbZEeuhHP3aJKOxkgnPGVWl8qeeKIE2rlMYE7UMrjedy8+UlUS6X7pLIhUeBgY+NgUt746U+lxbi/ZTxa9IjSkKf9zm41Idtz5WND1sIlZHF0RhkHDOWXLsEJHpf/0gd44f8fbSpnGzcWzz9DIryA4PjlVdfFWP9mfTOu++mmrpa7d2+mH4q1WhIQU4x219O9Kf8yJvhWRqiDi9lyZfszVndSKOIAU/UIeIQ34hFmQfxLJvs3XPMfjWvQPTiUALxYFid15ry5ptvpl/84hfpTc2t1Zpf7pGqSQ5dIoU5X/tP1hn2CwGDrZfgJZv7KBBYOdCCNDDSNXkbNpdK2DSXwB95gvXC5VgyrBsxjhYyrKl8CxZGHAusRqRAwubY6U7t7Y6kXfvOiBDXLylPJGxqFKr1Ue3G2kg71qSLaXpDT1q5hDVysZjojalxkughQqJL2VjJXqLqbNQClWttY4jVSwaHAt3RAOb87tgGatKXR86iFrcCAwUGCgzc5BjwNU+zZEzu4+DD10KXavf1gIjMrdk8a9Osr322PmTTLvR45n0c0zdbAdYLkZLT6XPdaYsYNlvFsDnfhVSmmDU6yIkNG1sFdICzCoaNtG5UVfTKxMaUdNe6Zal9wXTZSK80mgAMG4dCGasAoDHJGlsTWBetaPtxkIiD41frg9Xbfcx7An4MlwKMO3iPi3X96iRsYNhIJAf1HhApIiPqkG9Yb9uJqmQJ4/ZUes1QS4XHwjIRxwwYK/IN9Qvc5Qv5qAGSjxvPbIyRMqJhb6yEjdrbaQN2qIVHhupFqaTbc+Bkeu/9A+nAkfPpQqdUyog9U6EBVlEZtghE0JDOwcqh7tQyrV4SNsvSiiU6gdlYkSZJSwwMG9rF//5SWgiYCldgoMDAXzsGYhGMesTcG/Mt7/kLYiFEetSj/U4nvJG24cMXNZuoCMKmDTZJIC7ah7IyRt3VCEOgbOHNlxtlht+n6c7qKkyOrLBCAx5650eECd2d0V9hKjJOnkVf7fH0/rbDEn+WbTrpqq0U0dtOrOvDfIhjN9qg1VeLYdM8nBbMb0prVsvGyUzZxhBRuEbECSQVnhOz5rk/PmeSUSdOHJdu9zpTs/Ttb30rff/738/Ua0maQO0FQLQZ7UCbIzmBNM0Wncp97bX11ubHjh+TFECHEX/q6upNUgDpGYjgK1asEFFhVabCdKpJCYSKPCPaiyAdBOOx2jbf9+I54n36+kZpc8x+jJPVMDFQC4taLWwAYTcK5ihMCdqBcQYzAOI+45C2Ap/Y9quXtMwkqRCD2A+DrL29XQS4z5j9E5ioSGzQBrQzabjncUp5L8mmyYsvvphMWkV2ApYtW5oeeeQRk/BgTMOQw5Eun5a2AqbOrs60TxLnf5b03Z/+9CeD/8TJE1YviIfAyxiAeUNeSFstlSTIHVKbxryByjBU6T0rNWcvvvBC6hKTaqakNNatu0N2Fr6R7r7rTqWTHUP6p/4gbGE/CbV/70ny73Uxbja+t1ESLQcNj729PYZX+iASPKj8WS1GxNq1t0oCZ6ZJ6Lz88ivGFIPBAaPiC2I8ty1qEyEQ4hWSJd2m5g/cYFvx8OEjmt/mKe530le/+hWpBJsl/Ls6v8At9YTZBMPq0KFDkpjaYIysD7bvEMP7lMFMWyONoyKsPbCD09TUbIwkxtGdd96Z7r7nbtmaXGS4ou1oN/a0MPzI98UXXpRKoJftlHiXcE8dvvnNb5n0XFsbKionKX7GGKIgc04kpo+9JcLnH2SPBsb78eMnjPn0sJhjX/7yV2TTZ7mkuab6vjP7LOhWeyBZh52dF1543hhS1LFfbU/2pmpRajLB43333WdzAn70FdrLiaiKKqCAC1fel9y3+C0w8OnGAN0/G1aXVDTGhgd4TA0hGzf45efeSxKbRxx89EHm+Xk+xqSQ3Zann3rKpCGRmkZN449//BMxbr5gByxizbZxm33Y+nrBnPhRbryawTR2eEbnwFoy2oe5xA9bat/DmqF0zB0wh1GZjg2tX/7yMa2Lu7SXWZb+9m9/Zgx9DiM0Nzdne6RSplZqVrbXyfGDml8YNjBrHnvsMTswQT1LDJufmRYO8rzU8ZXPxdRnuzjAtidKLpVODN4kOaM76m1Qc3PybFfauvOYGDanJEHdLWlF7Zdlp3C4osbWLA5BkH+1tG9MaxiQ4eg5adXKRWnOrElSl14hmoEOCIAX5en8F5VhiOQOJJJM0p2SFcN+HQ495hwx3XmsseJEjOJeYKDAQIGBmxUDvm/1GbN838r+lv9Ym+1QnrHnUVlps7TC+a4Ge6x5rBn+Lm9bO/iFCY/WKy9FPgocGNIhMGlnOiGawJbt+9P2nbJNehGGja8XHPm3fMXkl/4CMfkl+Z66pR59arr3rlVp0fwZqb6mwrRucNiN1QBnq5YVBOAwitzXNX7Yi34Eiy8wCg+GfuQQcW7sPfDOPfDOnb0B/JcrV4k2BsMG0K0xvGXsOfxubLWK3MsxQANDSKBhbyTDhqa25uauvoxJAoxHd/cPpn0HT6VNOiW979A5ibFJz7kkbBIMG23MGDIM0CqxcbBjM2NKbfrM2iVpxdLWNG2yiBoZw4atlP+RonAFBgoMFBi4MRhgzsSxhoXDLz6c8YPYiDQANhcgGP75z3+2eRbj2Kjw4Y69Ek5cQ2SEAFBOICaffBm8UwYO//IwC/gr/gGrbOocq/p1D/N1ho0OF8gPxUpdPUNSmdGVdohhs10f9ecu8PGtNSNj2Ogcv59SH+rXRmwgTZ2cjGGzWvpqW1v0ItVlIsebCrR/+7d/lW2U53SC/rDUQnWImD8j3Sv7Ht8UAfxLX3ogtUpCoNqI9wJFYMU6A8EEY/Z79uwx4vsLIqDzXFdbl6ZKNdOsmbNNFQkEa9qaAyswbniGwEGbR7vHRou+8Glr14/bJQM39H2YMMGMgXFz5MgRY2BAVINJCpEdBgUSTXmGDcwPJJdg1vAMoZ22gMnCnfaA4B+OMvNtQZuQ965du+yiPBhDMFSQ7lgs2zlIeNCeOO/HPkbJiysIbKhmg/mzT4Q11AWfPHUqXRQzoa9fEjWqH+p8asXggIE0Q/m3ts5JCxYuEJOvzeA8q3puF0Fxp1TYYotnkuqFPZSVK/XhsWhhqq/T6Wl9dVBm4Ir5CJjpn6jpOy6mIowEpJLAa41UCYID+ibMKwhy4AlVftSZcUFdkTZrb2+3uubLOCmmk+e91/CEajIYk8uWLVc9Gg234DCPU8oFLgikhw4dNuIiDFTwTNvSjoQTj7ah7VAjOFOMpLmCb5Ek1Gg/2hVHHObQYRFQqfv58+eMkbdnz15TK0de06fPUB1WWR2AEcaYfUhaDv5DWkY5zCyYYzukMhHmDzigfFQDUy+YfuCoNF5dsof60L6oEkLKDnV+2LPCwThEcgk80mfAaR4n5MUFDA6HIMn8LIPip8BAgYFrxkB+TMUYC7/IlHkC5j8qbbFth7QKDkk+7GSh2pa5g7k+xirh+XHM+9U7n7cuTceckPPV9MScBdyxJvLM2gHcr0o6CLg3SWqR/SVwf+tb37Q9J3Moh0gqtcYAe96RR/hF3sx5H49hw1wKtJC+uOKJGZYwHHBwlWzbImVz6vxFHcg5mXbuO5mOnUT7hsKHpeJGatE4vIFUEflVD/emKZP605JFs9OqFQskSd2UpjRVyn9IezxyFf5EdOAPRJLGNpfS3OEMmyzMYBiNE0W02NwtPQ8Wzx6KnwIDBQYKDBQYEAZsXmVtyq0jsZ6AINYqHH5ctk8XswObcjG3Kml2sF+ztqsqsAnYGDbZ1OwSNjBsPN2QVGQOiKCMnbNjp6US7YP96YOdR9KFblRrSrpGLBp9VWntUNmiB1SLglAnlWi1lb2SrJkmhs3KtGDuNJnXkGSuyrBiLW/P34vN1gz52/yfqWg34AqGjbVr8TMBGGBwTRjDRuPVur1GABslmDY9YtjsP3w2bd5+OO09KNHn8zpVabpq2QwjYaPhwcZLEjYVQzBs6tLa1YvSyqVzUsv0utRQJyKADXjn0FZSAGl8lPlL8VtgoMBAgYHrgAHmy3D5zQj+XGxKIg6GwyF4cnr8mWeesTsERWwYPPDAAyZxg+0SPqrx58LFhibyj/wIi+eIg9+nxbEBi02Yb4qymoFzk7DJGDbaLHVcHEgHj14whs3OvSfT+Q7hPZOwQcqGEzvs0Cok/lxfO5imTalIixY0i6g9L7VOb0pD0lU/LMnS97e8n/7hf/6DnYZHTRR2NtrbFqVvfftbphLt1lvXGlEVVU7gnnXF8hZotDUMm127dqc333rTmDbnpJoLSRpsmyxZslSEb9kmmdFixHeIu0HkYc3lCiJztCdEn8KVMBD9HR+eg0AFnrggrnGBS5gPEMu5B8OGNMRjjEHYpw1M2kbvEK/AP3G4aIPAP+VEm3DHUUYwg2KcR97kmR/DhOMiLe9+okyeKgvJF+DuErzYjgFuGEDEqRUhEKJaQ2ODMVGAHcZCpeYHVPaSl9Vb9lHor/jVikk4SWlqxaypQg+z6kydmIOIH3MLjAMkTSDEUSb1gdBXI7U5qBqDIQHjBqkPHOkpizt5gD/CqDcfXKqM1TFgAveRn+Pa4wIjDB5XOWxZWzxw6nAOWJv1qk7ggXzKGTYwsOLKw0F5lM/Ycrhsl2lq1LqVFyrWCIfIh6o1+kG0F2XjAi7ew4/4/WKiAQ84QP0h+depPeolhedtUmKwRlraPPpj1AG8U1T0PeDnov+Fy/e3gCEfFs/FvcBAgYFrw0B+jDLeeGfOhWAU7zF2YbrCrOF0KocZYQ5/Q4c4UC0WDG3ShIvxG+9Xfy/tHUenBbbMR3MIe6RwzGsx/zFPYYPvKUkGIRU5IOlEDhMgDbhu3e1mJ485vBq1sRPEsBF2M1BhlDizxJ/wxgfHL5czbEKR2mkZkN6x51Rif3fkhNbIi5LAEQFOp3I0l7P/KjFsJtf3pbZ5M9KK5fNT28IpafpkrYNqV+gBHMyxtslKc+wpA+0pMy9i5ODQY84Ftj0OAQ51LkrxWGCgwECBgZsaA/k9a35dBCm2zjJpy7GH9nBWMvzcn1kVaRn7rGBGVjzmWtuiKxCmDa7EsOFZbwro1+n/zp7hdOyUpDI/OJA+2H0kdYhhI9aMcq/RuoHdM63zMGxES55UPZQm1QymxQunpXvuXJbmterwlopzhg0zvs/6TlWm1PDjDiCKbABldSgkbEBS4W40BhhIbFDZ+N1ICRvqwcCDy0kXp68PqZP36Mj04aPn09Ydx9Lu/TKwK5G2Pom3DWNAWvrmicgArZIYGwybac01aZU2ZSslYTNnFrpq0dfOxi9j2KicbFzfaNQV+RcYKDBwk2Egvymh6rExwT82Jcyl4SB0crJ9k9QRoTLorbfeMsLkmjVr7LQman1g2kC8YyMTacmXdxx+8R7lR7lRzqfhbuuC5nr/MPZZnJORhludhkRCISRsznf0ab04KRs2J9L+Q+dTpzZnQxV1Roio1LpRBcNGa43O3kjCpj/NmFYlNU5T9EE/N82a1pgGdIK/U0bIkXz6x5//3FQ5sXBgC4YT+NiueVBG3BcumG/EfsO/wgzvggPosP0GYwDi9+kzp812EWspBO+pU6ZmxPYGI3BDKImNKnlEf8Ev/GnDT2O7fpy+GXjK4ybGAH5BrOKZcQL+Yz+DXzhw7MQqV5sVeOeOy5eTT5NvK+JQRlyERdvl75Ge+Pn2tP4d+x/yUv9B5Vc/DKdBEfRhMCpP4OR0GZI22N7hTv5KKqfRQf9RWnT4MyYIgOFgzD/iyaaBoigNDGCHOXDGHfwEI4e64GzMqFzKhvHBHUd4Hsf4G3wUYBCNnveinKhH4Dfgxt+SKnXkSxqqgfOT096OEY4/+VA/7vmL/MJFWUbIswyjvbLMFZHoxKMOTrT0tiePgKEcRrfxCDE1cAUM4IH5gCsg8LyoT9Qj7tQFF/iLOkRK3vN1wT/e8ziN+MW9wECBgWvDAOMpxhbzW358xZyDPwwQmDbYC9uzZ4/NP0jXwQRBSg7msc8VPicDTeRb/nxlkJZg8fmylCpgLPn4E/ByEQ6TG4lA4EWyjwMJSPEh6Rh7TOY4VMaW5vBSjnm88AwOPq6EjXJRAcy/WrMgcmVrBnd8rJ62m2ISHc2wOdvRm3YdOJt27jmhwznndEhHa4ZU3AwNu3obbPcwJ1dJwqa5tjfNnT05LddBzmVLZqaZ0xs4U60wco0dJXfK8Z2mM2yAwaHg1y/zGvnxGBEH79yEPxKreCgwUGCgwECBATDA+pFfT1hLYq0trT2aWU3NWDYfM68aE4RZWmuFNtasV3ZlUzPrq38JldJAU4Zh03FxKB053pHe33FATP5jqatXX/+SyBxChaYYNlb+IKrQ+qWZaSg11ydJZc5Id65bnObMnmoMGwnZGD2ZNSJWC5/tWQXiyoCxdQA4tG5bfGrOasM656nwmQgX+A6cx76EOhcq0SaiBSaoDBo4CBwTwrBRvSgT6RoYNn2iwCHuvH3nCduYHZehwb5BqbVAHVqlJGxg2GhTVjHcZwybKU3VaVn77LRy2Zy0cN4U6arlNDXDw/8mfqhMUEMVxRQYKDDwiWOAuavcsTiGf2xMiBNEOIj6qEdDxQZGsFGTRhgGYB944IF0++23m1omTl+TPp828raNitJEOcTB79PkShsf6uV1G2HYSKhZWz4Tfe7XhujMeezXHE0f7j6ejp7UiXypyxgWw6bCiNsw8bVtUlOhFb2+pk92PmpTe9u0tEyMfhg2vVIHdVC6XV9+6eX02K9+ZQy1GhGEG5sb0x2fXSed738nOxefN6PiEM3Zq7HRdOebOYix6FGPNo+2I54Tl10iItow2o62DzdeG47nH+lulnvgjDs4CVxSf57BOSMyRgLx8nHzeAz/wB3vuMgz3sMP/5EycnEJj7i0ZTzjHy7y5J1nACzFc2jtk4MPGfY3WTyLm2ViEjlZvZHq8noCkz4+lAabN3pRxpa5zSlWjhg5uJh/6LzRN/H3Z5fCGUGcHqyKysp2Uso34I07aXkGRocTiByHkb+XWcKJgZfhznK2tJ4s6uf5e70oA5fHQ7wTLy6LlP2U4HH4PL8SbBFOvQznGTx5ZgvMMmN+KU+XBLIWUXkQUQMex2Pk5/VxIDxLxw0+WRF6Ih93QFR6c1gJibpyd9jdL/zzfp5T8VtgoMDAx8VAzCX58cU8FnMZ4w8JOdSMISmNlB0SiKiNRKUmh2xwMc/m88E/xi/PV+aYa3w+ZaZg7gxX8g8fAkvf7pSFdOK5c2cNXlRKwlBCbeOMGdLPL9ttzGvUrUISKiWiWeTn81HATHnEvfEMG6upgGBP5AwbLNOwgl3o6kv7Dl9IH+49LrXpp/Wu+sKwkQYOiHTM18Na6yrFsGkSw2aW7NsubpMNtlVzdZhzcqoRfQH5RUhoTMhgzOvHOsA+m/XT8c3aACR+6ZZzuRbJfIlXuAIDBQYKDBQYCAywt+afOZb1gyueWUs4WMB7rD0cLNOMrou52f+Yf51ZwwEm/05WNmRrUzPpmb3xsfL0NqjvHxg2Fzr700HZQd/64UE7zHmxr0rrSJ2uGn0iKS/t5YcGemULvS81iGHD4f+l7S0ysbEotc6USmytj0jZOD3Z1wuHx8tzKHgevU6wljgshBQMG5qqcDcIAwyqiWLYUAXbJqlM7qhF65d1wROnu9OHEn2GAHfk+AVJ2HA2BgkbVKJpcHBiVCdQkbDBmGCbDETBsFmiwTalSSo3FMWH0CfB27xBDVNkW2CgwMBfJAaYM/MuNiX45cPy/syx2GfAKPdvfvMbU4/Gxz+Gp1GzsXbtWvuwJg/SBREg7viN5cbzHyvuX7rfqI1PdtIGYjQ4ZWPHtmhA7wM6LXMCg7QSff5w98l0+tyAVGvq1GWlGDYiRrDRM4aNKhwSNq0z62V7oiUtbtfpy6mNqfvsufSOpJ1+K3UnL730olSJ7JFkQbUYZ7PT58Wo+clPfpLukgH35uYmk3Aoxx35G+7VFWzryJqWEXuAlzDg4DQrz+X9orzdrI5Zv/K0Y7d3ORw3w3vgJvACnnF5HEacuBMeHwY84wjDRbqIG/laoH7wD79IE+nyZUdYPj/C8UdSSx3DnIWrD5QYMwqjebO+Yy1t7+pDxiDwjwWSw0ggLgwblwhxZoidBFdcVJ8JWL5FbM4wXqDKD0aDMR+UgX1QZOXl6xcSIKCGuYYLF3WLe/gzjwGPlWsx/YfycCG1Ah4ChxaQlR1+ni9jRKEKg/iGJEuUB8Mzj1fyICzCY17EPx+PcH+PuMo8i4M/9bQC+dVzxI12i/Q2h2QETvyi/qVwiJ9k5vlHfaPPkTfhxMfl05uHfgLuuBN3VPwMN8SPOJG2uBcYKDBw7RiIcUYO+XEX44zxjOOdZ+Zb5j7Gt0kyZoc3CMvPRZHGEl/1D/MfZV6aMA+vYlgc/Pzb3VU9AofDSj4+/+LHRb6lNSHWktEFkV/Un2fyup4MG82CqhjzYVxeT4eCMA4iaH+nJxg2Hd1Se3usU6elj6c9+0+k851IpNaYgWlTnUM8tQsMm8aanjRVhzkXLWhJt4kAt2CebBIoS9mRtoM7yphlyyRXwR+MGq048gEWw6j9Wpj5lH48RsTB3yEuxSieCgwUGCgwcHNjgDUj1hDWkXiPtSTWStZQ3NCQtCEM9Sqe2+2skm0y5lZmamN86PAl+VheCinNw75+xMwtErI0FKR09kJf2iOzGts/PCR76GelvUlrhalE07eE8oK+MDTQI1pyjxg2w2n29EkmjXnLinlp5oymVJNj2FCaz/L88uxlOhT4ldYA8mU1wRUMG0ND8XOjMMBg8k3fjVeJRh3YBkfXp5v3abSdPifdu/vPSC3aoXToyNnUOwhH1Bk2bMxg2CRxReGMNtVXSPS5Oa1Y1iqmTWuaPkUnhxSF4cOWL+56LFyBgQIDBQauOwaYM/Mu/5GLf2xUIk5sWDitifFqJG1ee+01M+bKx/RnPvOZ9JDUb61bt85ObmIjgjxDBZBtWLLNEPGjvMj/03P3jQ/YRbLGnO6GTzZbWi84TYPKzOOnO9KmrXsllSn7NZ1J6wgSmUjYQOi1M5VaC4ZN/Lm+ZsAM0a5YOlsqzqbpw742nZHR9T9K2umJJx5Pm2WclxO0nJhtb29PDz38UHr0h4+mW29ba0wch0PZK19bYPCwPhCEE16dqEMQbeQuq0OWqLzd4j3uWSK7jeWXD/+0P9sGOBtm4CI/5tj4845/OZ4invWZbJyOFS+fLp4jLe/59DHmIpw7MJS78nIiX4sXdQAmPbuznm791MtTH1c/gkljZSkefY7onjfP9EHBp/L9hLHgyOJF32Q/5MS5Eo4cFSpP/5QRzIYMRZZv1DvqgVo2/qLeUd+ol5cBHrwdPF3EAgbqYeCVPEfienguQDDxEedtm/ePZ/KjjGiPgMPLdZx6mRn+VFmHPcOzRQlYI1c8PV7k7yHE86fIE9wppjyzAD3ny/bYPj+TlrpTn/I6kSYYO56mVI49KS14xwVMUVfzLH4KDBQYuGYMxHiOsct7jFHmFq54j0KIW56OMOLhYk6KOPFugVf8w9xXPl96YvItudLcVIJz9LwW8SMc+IGJ6QvbALaOlDK0J9LEPMMzaa8Hw8ZnMuHP5jTqwdPo+giD8udQAwdynGHT1TuYjp7SYU4xbHbtOZrOXRiwfd4ANm51aIccgmEzqbpHtIGkfd60tO62JWLcSKpI+m1qlG0luKNulCA8eB2ZyQuGjVBSuAIDBQYKDHxsDNjeODetQ1dGTeeogw7SVuHfNKwAfLvoIL6Y55UctGQNsO9+NsCxRvnqYZM3P8zfNvNTmj/ZmqFvJsxp+MH/I+mQ7NtCRx6sqFcpYhDZ2ieJHknYVAzr4L/oyAvmTEvQBJCymTFV9mWVPd9OXL5GceclSqJyXPhZJN3xCX98fR3Lh1ukG/zDeh1XrHHcWcMLlWg3GPkTmT2N/EkybPrhjMoewd6DZ9OWbftlj+C0bcqGxBmF+GZ6bzEuKGYNugcb61JqkejziiWz0i2rF8g2QaNOe2qQCWkxjOJ5IvFYlFVgoMDAzYEB5sy88w9A94lFM+74xjPxUKtx5MgRM2b7zDPPpM2bNxuj4Itf/GJ6+OGHTd84Kixg1oQtCdKz8OL46LYPb3v7tP2IYGGbH9XLNm7gzvGnX7Z3xrDplfjzkRPn03tb9qRd+07Kfo1WBqnKgGFTKYkWVgLWAj7pK01f7UBqWzA1rVoxJ82d0yxjgykd3rsnPfn4r9Ovf/3rtH/ffqk+uWjqTm6RpBNGhb/73e+mFSuWq3wI6JRcIhTTHgDGTU2qu7cPd9oYKQMc7xYXeOQf/cT9SFvyjzBLWPyM4C5wFCgBd3kilJBobR3hcY84vOfzIH3+PcLzbYJflEPm2JDJO/KO8UhekecIkY7Ol00RhOMi//xzPiyYH8SLsgkPwr7Vkrrah4znCQwQrHDR5/yZ33zf8/KVtfkH/AETeQI74fhRbswzPEd8Ugc8DicExpIkTdSHeDiPkyFiZAwQQj3GDg+JH+oVsES+vBtc2I6hYcjJcEJ+JTyXYKPswINFVzyk3pypQjxgKcVxuMjLgrJG5NnbJ+risJfK9vy8TpTjsPBEGdQl2ok01IH5vdwRN/L0Onh5+OHiXp6ueC8wUGDgyjEQYyvGGiljjmNsxjgjnktF+vyCf6QlXqTDL59XvEcci3hFP+MwbJh/srnIs/F5KrI0OG2P4vtD/AWSYHKGBnUDlpB+ZGcUdYw8PI3XI55Jd+MYNlaKfpjbuDJ85hg23f1DkqTuExHuWPpw12Gpwe2XCvVqncrmtDQX9dP6Iwmb+qoeMyI9t3VK+uxtS9PiRbOkCrci1cqIjavGLc2lXncwyr4u89dTCRZ7GfnxGIT6POzxRoKLhwIDBQYKDNzUGIh1MY+Ezs5OO6B6QerH+W7AtitqOjmQivr32trqVF2jmdw+rzT/a+7ngL7Ns9qnx5oQMzTrWX7dwl8rltZo2bDRz/FTXWnbh1orpCb9mJ77BqU+E4aNmEFIVPrBfylUl6amyY1VaanUZ64y0xrNpqnJ7dd4DXw1itpQipfm60WsWaVwX58z2G2diLUi4tzYe+A/9h6xH2ENLxg2Nxb3E5o7DfzJMGx8uwTD5kLXQNov3YOb39+b9hyQrmCJssGwGRbDxgexNlZD/am6AgPSw9I9KDs2YtjcdsuiNMtE2RjIvo1imDDYJna4TGiTFYUVGCgw8AligDkz70ZtInJhxAvCYxACeIdpg6TNm2++mZ5//nlj2tTX10sF113pW9/6lhm1RUc6fuTNoosLYul4Zef9/zqfnWETH8ag0i/fLMHDGZB0TXd/SoeOnRPDZlfafeBU6urhI16c/Cox+VFBxp8SohG9SkLRjfVDUp85I92yan6aOXNSGu7vSh9u2ZJ+/X9+lZ5++ul0/NhxGWHvS9OnzzB1aDBrHnjggbRw4QKhESAybKp8a3u9WxnxTjBxskWH9iKeE9cvJY7k+wtJeY8+VR5G+M3qHIclIhm4yeMq8BK4M7yrIVx9mG+52UjTVpdzgfMoj7iRp6WjnUekXgj0cNIxriNupM/DmM87H06+EW+IAynB+ABW/ceYt36GpIt9wFCupyM9cZwRkGPskFwXEeMjAoAp29OS0uHPw5NNMVYfj+G/AX+pT5O5p6ccKwsM65n8IoxKeFovG38fDx434nmaiGuxRoinvDHnjeWirJhXR8cZzdTLw0Y8h6OUIvLi7jgyBOZwVoI5Uo3Gpfac1kb0VSuhDBeRirIdfyWf0hNhXIKCJrM87N0ztbDLpS/lVDwVGCgw8FEYiHHPmIr5LZ8mxlrEY66JeDzn537S2fyhgatRfI1jlcnDJpASGDYVlPll821ECph4L5/b8Av4/Tnmqku/komXrzP5flyGjdcH+JnXfF7Fz2dB9ycsiHNiv+hQjhgx8umWqM1pMWl2imGz/cMD6bSYN739OjU9HAwbSeNoP10p+7Z1Vb1izgyY7Zrbb10iY9KzUoNOUddq+YAQV8XcChS641hvjYhn+LYWM/+xfgL7ngMxPI+x4hZ+BQYKDBQYuNkwwFoBHRkXe3ZoHOvXrzdNIoS1tramRYsWpTlz5tjztOnYgRPTxs4uaU6V1KS+qLI1jOOWNmNbniNzsM3ftkO2cOIEw+bI8fNp07bDaffeU+nUuV5p4qhVHNGRM4YNEpWVw2LYiME/TQybVcsX6JqbWlvqzRY660S2PGTHB6xo+1EpugMFVwkuC5SfQxTrSHm4x7qRv7b30PodazjrHBftUjBsbiTmJzhvGngiGTbR5Y00p7L7hypSZ8+QibC9t2V32r3vhIuyDTPQJGED1xVqwrD0B1dgQHrIRJ+XyhbButsWm7EoMWpNjC2GyaVb0QlGalFcgYECA59qDDBv4uIDMCob/vEeH9P5ePhx+oSFFKbNq6++mj744IPU0NCQPve5z6X7778/3XrrrWnaNOnh1mkUCAPhYiHO3yPsr/8emyKfyUEx+HSciggr734xbC72Dsm44Nn03vu7xeA/nS72iWGj9aLCJGxk94xdlz7IsYRWLZHrpknDEn2WROYaJDLrU39PR9q1bWt65re/Tc899wcZ6T1uottINn3zW99M3//+99OaNWvEwJmeodTb2l6yR9uaqZzy9iZOCWbefOOUb/94jrS88xz+pCqcY8A2wjn0j4WjwHcen/kxMxZuI998GmsHNuRl5TFew0X5kQ7/cj/LJ2vPePb4pTamb5ec50+fsm+BLMDT+FigS0dgvmxXbRZEOPqa9z+vhL+T0tJQpuWDj1zu3eHxvmpBGYDldbN09kNdSm/xlIeNwkrw6M0SRKJRCFBy/CMs6uBpxoIh2oR2jvDw83fy9woGDF6l8CvDTVYB4FeIgcKz18fxMqq+lg3wEsfhzLIYeR+Ni7Hrko8T9SAf/HnnytcrHyfKK+4FBgoMXD0G8mMvUpf7Md7yfjxz5eediBPxeL+2ccqkwpU5Pdo6Fe92J+9RHgZPwEXI5cqPuSoOAORzIo+Am2fmnevJsGFS9fowx0Y9qQyX73GDYcOKKI1o0r4hhs2+Y2mb7BWePN2TurXXGzQbt2LaCBHi1xgRrrayR+rP+tOslqZ025p2HdCZrVPU1SZlg6qbKl1RNyHMq82p6wwOYdX9xviNNvA4xBs/7hjJC68CAwUGCgx8qjHAWhEMm5Ae379/f/rDH/5g9I2DBw/aAVSYNjNnzjSGzdy5rWne/Nlp1qwZqblpcmpsbFYcqSarCZoH86wzcHzKZb7O1lbWZb2xXsCwwbQGNIENm/envQfOpHOdg6IV6OB/hegC2MaxuX5AufWnKjFspk+uSWtXtUnCZp40NtWkhjpJvas41lZK5cq7KM3Xi/IYrBAZbCOpy3PI53b9n2P9jzU89gC0S8Gwuf74/sRypIE/GYaNi38PiArX3Zek4qZDg22XnabpGajVKRqJsyVn2BhRQgyb6oohnZgZlOjzQFrc1pLu+MyyNLdVg7xWfFmNj2DUxP0TQ2pRcIGBAgM3JQZi4YzKx0diEN0ID6kbJG3YyLz++uvpt2IevPfee2bD5qtf/Wp65JFHUnt7e5oyZYqJD0c68uUEC5uiyDvK+uu/5xk22hqZiLS2QvaB7QybPqlDu3BxIB04fDptlg2b/TIu2K1TlzBsUIlWVS19Z9p1VWDIUCdqqvURP7khpdUrW9OtaxemKc11aaC7Kx3Zvze98dr69Pprr5u005mzZ9K0qdPSDx/9YXrke99LM1paUl2t8hzT+WbMP+BHRwDWaBeeHXZAyjaaWfTLhY3O8eZ+CzyV4+9KsTJe+vDnjiN/iHDhH37c83GibfFnTBMWxLuIR1jEG52fdU1L42mj3Ev7hkJyeTgMkSdpeY6L8vJlY8QT5/GJZ6/j/jgKsj6tyDFXRR7knS+rlBG4iz4+upyAh7y9/FJ4CSfhx12fRN4U9lwqw58ChlJa9w+48A+4vT0I9ww9X4fT8uGjSuV5Ws9nrF/ShSq00W0ccJMKvDnuAraAKfLEH4f/eC7S5sMjnwiL93yc4rnAQIGBq8dAjEnuMS7jTm4RXu4XY5E4ERZ3/HDl7+57Jb/MEz5XEDtgKKXMzzsl33hi/uPKz1VWP9aSTALQ45bmrEjL3eJmcxTP5PVxGTbKxYoQBLpzRR01H2ch+HOyGpdn2GDL5vzFfh3iPJ7e334wHT95UYd1RHKTbYIhncYeEqwCUXu9gVRb2av9Xp8dyLll1SLZJZgtuwT1UqFe5XZsVDSq0UY5I+K5D5CN5xxSmEzEKqgL4+Gp8C8wUGDg5sRArBfcQxPI4cOH0wsvvJCeeuqp9Morr6TTp09bGOrQoGssWrQwrfvsrWnt2jVp/vwFae5cMU9mtKSGxsZUU+1Mm2rdndZRwmuFVFRjh82YNfJGIrNPi8W+g6fSOxv2pAPS1tTZ43Zth2HYoKlJc32FNDRhVgOGzYwptekztyyRhM08qUOrTJNEQ4apb7O77uXrQcGw0WJL4+JigxN38yx+JgQDtMEnwbBh4zaosjEi3SMVN8dOdqYNm3aavtqe/trUL4bN4PClDJuaykFtzvpS26IZ6Y7bl6b5c6aIO1olPfNotfW+VGypJqTrFIUUGLgpMMAceaVrU6xpgZhIh39c8WHNvItRvgMHDqQ33njDNjU7d+40yRokbR544IG05pY1xkhA0gYXsMRHeZTz6bgHw4ba+P5AaFOdWS349K+wU5fnO/vFqDmVtmzflw4ePZ96B8SE0XqRKrTJE8PGPqyNwa+1Qgz+yY0VUoc2O916y4LU3Kh4g/2pp+NCOi5bQkeOHknnz51P58+fN0IH+Ea6ZtKkSfYeePWP9XjT3bYuvq3LtzExyt/DL9qfcC7eI8weip8xMRB9PvAXOAs8lyeKeHk8jxU38B/38ji8l+dFWfhF3PHShn/ciY9xzUiPP83v/oRZ0Ei4v3k/ibFQXiZxxvLzzklZpXwdDn/PP0c5cQ94/X30b74s4gFzwM17qT6Oo9Gp8XN4SXOlMESZ+bzyMMZzxCu9U3sRAClULuC0l1E/JRyZt6JjtyIPq0e/PO6yYqxewBLwkLYEkzdyvHu+pd/wz6cthTru8u/Fc4GBAgPXhoEYa/nU+XE3Vng+7njP+TzGizO+P3OVz1fEGQ1D2TxVnomSsUfyNOPFDX/mIZ+L8tmQNuDn+foxbKiTys7KBM5SPQUHum4J07wpCIzFw05wQD+dPQOSoj4hhs2hdOxEV+rqlh3IQSm6hWGjdJyurtB8XVPRp3tPmjq5Nq1ePj8tX9KaZrc0pmYZLKwRbQAFO0KOQRCqOR0OFWSlch/bBbwOf0FdGBtLhW+BgQIDNysGWC/ssrmUdabCGDQbNmxITz75pKkeP3r0qKEHhg70jKbmpjRj+rQ0Y8bUNE33lhaXvJk3b36aP88ZOEjkoB6+rg4GDrZolDen86Ui2uZ/5cg60aufvftPprc37DZtTd19YuwjYZOUDjXpSlIphg0SNqwVLVNrE6ozkbCZ3CDVmRLDNPqx8iOu/nOOdSPWLO6E5mOwQri/rxHl4bmsbtDjCP6zNRz8c7GGFxI2Nwjpn0S2NPTEMmzo2L5dY7PFKZpeqT48caozbdy8O+3YfdRU3GBEGoYNg9I6ozpelSRsqqUWDYbNwvlT0+1SibZo3jTXP6h9lEnZqJMWW6pPoicVZRYY+HRigPknPmQ/qoaXi0sYjjtSNiym5AvTBkYNqtFefPFFY+DMnj3bGDZf+MIX0rJly0w9V4gakwfpxmLaRBlXCi95/eU4FGVq92Uf8A4VKOPyEH2sa604e6E37dVpmq079qfDxzukq1YMG60XMGxMwkZJh2XzrKZyyG2eTa5Kt6yQSrTVc1NTQ51OWuoU6tBAGhocSP19MmTb15sudnfbOjh58mQ7/QNu2ZTZtkw/2VMJVdaUpU0b+C7HPe/5dohwMsn7lzItni6HAcYLLnAX9/I014Lny6WJsCiP93guLzv/Hunw8/jemX1z7/1a3YZQhXN3R39355vueCu/jwWH+41kMJLEYfFySnBFuVH4tGtS+QAAQABJREFUpX2YDMrL8fReRsBtfnjJAz+PE/ln77nwEcBGHi6FgaDL4ZkyvByPR9zwc7jy4ahWGCnMHsphxBO/6GfYQor6RMrRaTxD2ivg9PDSe+TJvTwOccMv8s/H43m8OPn4xXOBgQIDV4eB/DjNP+dzCf+8H8/5MZuPk/cvT3Nl7z6natRbdOYVf/Z5pnz+GslT8SIlMeOZcNJ4Ph7bYfT83Kf0S12iDjwzD14fCRsgokytB4Iu1j8v2f1tzydgBcEIw4aT0xdFHNinwzlbxbA5fLRT9m6HRIhDBW6l9nzOsIEZg+pbbBNMbpJ928Wz0/LFrXaYc4oO6MCwgakTEFBHrhKeKLX0VsKIPzm8xCGHgrpQjp/ivcBAgYECA6wZ+TWkq6sr7dq1Kz3xxBPpX/7lX4yuwZoScWJtqpJoy6RJdampqdk0W8wXw2bRoja7Fi5cKJs3rWLsTE+Tp6A2rSnVybZvNariq6RNQ0ycfn0W9vQNmzmNP2/cmY6e6BSdoNoYNsOiCVRUoiad+d/toNdW9aVZ0+vFsFmcVi6ZKxu3YthIHxozu/GCdGfNZLb3n1gfWCO4sjWLcHOxprFCxBoRYRNzD9yXcOtrHPguGDYT0wYTUgoNPHEMG+/YvjXyzRYSNr3i2pw43ZW2bNsjhs0xbcrQX1ujoZGJPjMRqOMhBOdEuKE0r7UprVk5N7UtnJGmT5HeQw16Y9gIa8WWakK6TlFIgYECA9eIAebdcP39/fZhDNPmtddeM0mbvXv3SrfrrHTfffclVKQtX748NTc3m0hxpIWBE+LH5IV/hMVHaZQR4dzjozwf9pfxzGbON3SoQ3ND3pykkVCM5n+ZZpc6jJROy6Dg7v0wbA6kY2L0Dw7LLo1O07A5q66W7TOphBoawAjtkJj5lWnmjElp9bKZsmMzMzXqxGWFjLwnMWuGM6bZoN7BSbVO4lRWuTFhiLVxElMPvm8zJNk2Lvf+l4G5mwUK+vdfbv+9XCvE2CyN+9Gx2WDj4/2rdB8d66PeHD/jlTFe6mstk3KutqzrDYPPe5f2iRJsTLWO2/HKvtT/8mnAV+Ds0rSFT4GBAgMFBj4uBmIvl89nrHmOuWpsx5rjIZEu7mPFz6+tPH98hk3ZmmcSphlAzMkAYROzf7EHWYz9HrFQ7AnD5uDRs2n7jsNpj/Z8Z8/3m10CDnMOaI8IGY5fvUn9raSpm2rSwrlTxbSZldoXzJC6mzpj2FSpbFTewBgKHGSQyBNYeIuLR0XOpnjVwsJ8F1hQF4SMwhUYKDCQw8BYc3Uu+JoeY566psTjJALOmOcj/7H8xkk+kpbwyCcfF79w0JWRqnn88cfTP/7jPxrzBr8oj3hMsSwBHJDkqhJNo1aqyOvqpM5SqtE4QImtGw6trlyxIrUvkd3y1jlp6vQZqaF5SqpRvAHZte3qTqIJHE/vyazGibMXRROo06pQazQBKZEX/VgnPYf7RCce0CHOgdTaMil9Zu3itKy9NdVJukZKmrROOA0ZuKiGTf/2EytTrA8GNdFGXNTa1whLNBI2EQ+B02gT2parYNhMBPYnsAwa+JNi2ECU4xQNDJvTGmRbP9hvEjZnLgzKLgEmoxFpc4Idez02VdU6NV2nQdc6syGtXDZbxgVbRJBrEnFOg13jxLikwt/ED5kJbLSiqAIDBQY+NRhgUcWezZkzZ9KOHTvSW2+9ld5++22zrTJjxoyElA2MmxXasLCBYWPDvJ1n2MTmK4+Ucj/S4Mr982k+2Weka6SHHTjtw1qzuTYdQI0f8hU9Ok5z/PTFtGvfqfTBzkPpxBltzlK91hFn8HPqZlgMmKHBXtk2G05T9QE/u0VrxdKWtEhSmUmSNxelDm2g92Kql5h1k07sVFW7TaCamhqp1kQwOnAUH/daTfyfEA+33+KnwMCVYqCMeHVJMvoant6/SvdLIo7rURrfPs7HjXhJwLWWSTlXW9YlhWce1wrDePldT9jyZVxvOPN5F88FBgoMFBhwDMR8Hvi43L6tPG6kCf9IG/cIz9+JG+E8Xw+GDetDwIAdgZH1Qo8+k+oXSUa9GVlM/uzz4JdAG+juG5S69PPpw53H7DDnydPdxrAZEG1gUFI2pCFP7BXWSP1tU0NVmjOzSQybmSLGzUrTJuswpyRsMCgdp6eVQKl8X8kzzt9y8Fm+HlYwbBwPxW+BgQID42NgZJ4bP8pVhcRcfFWJriByfp4nesBNeagD1nRq6wDv+bCxnkkfcEY4fjjWD+zWPP300+nnP/952rp1a+qWJotR8fjWl7Oy7Al4nIEDfYNv8uamxjR79iyTtGmdMyfNnDVbV2ua2To3zdC9cXJLqqhuTCdFF9i550g629GnNQSbtvWpokpXhWjJUoOehmDY9MuExqDWiEnptlva05K2OcasqRXxuMbJDQ6Fg8XCIFcwbLyBco0FWqLheS7cxGCAwfNJMGxsw6RdGfoH+7QzO3u+O+3YdVDX0XT8bF/q6hEzp8IlbBgu/LGtq5JKtJrKgTRzep1vypbMSnNnyTC3RhsMGzZerovQRtrEILEopcBAgYECA9eIAeZgVKQhaXPhwoW0e/fu9Prrr6c//vGP6eDBg6m9vT09+OCD6Utf+lJCRLihocE2PWxouFg389d4YMRG6S93nbVPdasbzHz+fEbXJ7PqKDkYrQs6tXOiQxuzE2nn3mPp5LluY9ggXYNEpnZokpzx0zTSfpamybjg3Fn6gG+fkWZJ0ubc6ZPp8IH96cK5U2nqlGad3plrkkuc5mkUXmvr6nQoACOFvnF0nAV+wayvK8XqMl4vK/zHxkAQruJLYKxY3s885KN6GPmNlYd6qBHGxg4b2zfKivvYsS71BYBxgLg08kf4RNlx/4joHxl8PWHLFxbwxT0fVjwXGCgwUGDg+mAg9mvkdiV7tnz88SC4XD6kj3CePz7DBijy87A/GwNEjz6D8sXOUzBtlELrFztBtG/0iUBw6kxX2r33hCSqpRbt2AU/oANtQF/6HOjEwbCpltqbhvrktIH2mTIoPVe2CrSnq3Lpmgrqp7i2s4QYaSn9x98C1tFzu8FrsUnpcOaSFo8FBgoMFBi4LhjIz+ExF+czzofn/XkeKz7+kWa8cOKEY84nPodCic8zF8+RPvKLNNwJI224iItt2GeffdYkbDZu3Gi2YiMOd8ubtYCyrLwsVM+V9i54lK/AMe0Xdfo+b26eLEbN7DRvQXta2L4szV24JE2Z1iotG7XSvtGtw/5EnpSqahp1GFMMGx3CHBJtJQ31prrqQa0RQ2nu7IZ025q2tHjRbK0PzqyBqe/rg2DIlgBfI1g3Ym3gPhIrA9ZXOV5YIUYSj4Te+Idop3xbRZsUKtFuPP4nrAQaeKIZNl45uLluMGpAss8XOnsk8nw87dhzNB065rpqZa3GNmSYDNSIFkMGw1CIP/fr5EyVBtv0tGLZnLRg7rQ0qbZKp6N9qITxqAlDYlFQgYECAwUGrhEDMQeHXRuYNh9++GFav369SducOHFCYsCt6d577033339/Wrp0qTFtMNzHxipOxfDMhYvNE4t2XJSD4/0v05U2RTBscNzNV698wHdc7E8HDksSaffxtO/gaZ2mkYqMSqlBE8OGlcHiS/wZ44KTJGEzU7pqUY2xpG2GGP296f1NG9Ib619NO7a/L7Vp/Wne/HlpxfIV6bbbbktr1qyR4cMWO9XDegO6YgPE5jG/Ebs8Bh3PVgH7uXzsUrzi6dOLAf/wKW3tx6qp9xMfn2P3mRjDnrq8n7nv1Q/vKCvuY8E2lp+NzLECrsEvyo77NWQxKsn1gc3xHTD5/aPmz9Ft5EB9VJpRoBcvBQYKDNz0GMjPI1c6f+TTXA6BY+VH2vDn+YYwbGwPml+32N8xr7Jvze/1nGkzICLBhc7+tP/gmbTp/X225+uXuvRB7fcGK2Tj1tKZ0hvTvoFU9bTm6rS0rSXdsnK+SVfX6fwNhzmRx6GMbBYfhZ6sZPl5nFIg7Br/g1lTMGxKmCmeCgwUGHAMXOm8m8dXzLV5v3w+5eH5+Zk0+bjlYfk87ZlJLzftRt42s0GIlcOPK/IdiaM5O/IfK9wSZz+RNvygZWCX9xe/+IVpDYGWkadN2PPIukPZAOI0DJPIzFSk8/1dVV2pA6o1onFII4ZoH5MaJkst2vTUMHlGqmuYluqbWtLUlnlp9tz2NGV6q6rLIU6pOUdrhmgCFdKuUVs9ILWZSTTj5nTLqoWpbf4ss18jrWjZQX+H3NcIocweyhk2xIkY/hyoLRg2jr/i9wZhgAH2STJsBiVdg5RNZ09/OnT0jE5OHzf7BGcvDPimTMMoNnRVGsi8YWBwcmOFVNxMllo02bFZ0CKuqQaxxjlD3bd+NwhhRbYFBgoMFBi4jhhg04J0DXMxEjPcESXetm2bSdqgIu3kyZNp3rx5JmVzzz33pLa2NlOPhg2b2CTFqRhAI8/YgMU94sVG7DpW4TplxbantPUhUw7tsIlj44TO8nMdsl+z70T6YNeRdPh4R+rs1gkcnaYZqhTDRioyiC8RG32g96d6bc5mtzT6GjF/Wuq6cDw99+zv0rO/ezp9sHVL6r7YZThd99l16aEHH0r3f/H+hMHDmlptCrXWWLkqW3tFw2V+k5bfrgHnaBf1iFhxHx2reLuZMOAfPaX+PV7d/aMp39fyMWMMez4xVvIxSJ9/v5LnSBD3K0lDnOjnVxr/cvGi7LhfLu6VhF0f2BzfwFSC63LzZ6l9LoXxcukujV34FBgoMHAzYyA/l1zp3JFPczncjZUfacOfZ/aQHR0dafv27emXv/xleuyxx9Lx48fNbiJ70R/84AfpZz/7mdkXwL7iWA6CoDNDCNWczKZqZI+HH6HMraHeTKF6HaR8hXAAp7t3SLSB82nD5j1ppw7q9Mqg9IBU4A5KEhopG/KAIVMlCZtaqUWbItrAkkUtae1qGaye1ZgaagnzUoygpjxtNrefmNkDLu5ZgJ6ANf4Kho0hpPgpMFBgoAwDVzrvliWz15hzecnnczX+pOOP/3w+5JHPx0P9F/+xyhvLL9Ll04yVL2mhJ/f29pr6s7Nnz6Y//elP6cknn0zvvPOOrR/GpMkytLKUhhkZCRube41hQ11cYgeNAcawEa2jQqfyB3TCf1BMpgp986eqSam6rjk1TpGatDltaeHiVWlB+8o0vWWOUIGGpuz4PnZqJWEDw2b61OrUvnBaWr1iXprfOsMlbFS0sYlUrEGRLQGsRSDV1zCQmyEYb3MWe8TXV5YscUSZgDt4jIt2iQtcFxI2E9AAE1VEDDAallPd6BpE5yD2FDjxjbEnNmZf+9rX7GR3U1OTdQwIhXGa+8phtSnFOjddmvFoxqQ1Ki72DqaTZ7tkmwDR58N67pGQM4ajRIRjuIgYBxENsbVqqUVrbkhpXmtjWi2x58WLMCYtfYcamwXD5spbo4hZYKDAwCePAeZeGDbcmVe5eO7s7DSbNpxQ4Tp27JjZsXnooYfSAw88YNIhdTLQF5I13G0DpCrFZor3PCPnk6/t5SDwTQcxAn4O/3CxTeofqkhnzvek7bJds03qMVgjegc5PimGDSoyZHxwiI2cGDbVUptZK4maubOb0q2r23SfLFVoO9O///P/m5595rfp9MnjWt/6TXJp3e3r0pe/8uX08EMPpwULFqTaGm0Ebc/l27TYjdl+0gOy3/HqArRcsXGL+3jxC/9PPwaib9MvLufiA+uj+oyrLrg0J9Jf6nt5n0gQ98vHLoVGPy/5XPtTlB33a8/JU15v2K4Mrph/x4I+5rSxwgq/AgMFBgoM5DGQn0uudO7Ip4m8Ii1hXLyHX8ThHmHxzB70ejBs2Auhjsz3ROXrH3ssrtIxS1SiAQvkOr79+6Th9vjJzrThvd2yW3jM1KX3DVUbw2ZYUjacyFaNlIN0csi+bXP9sAhyLdr3LRKNoEm0AW0RjZLAjszLURG2TpbWyjx8+blesGR/BcOGnlG4AgMFBq4HBvLzceQ33rwc4XGPeDY36Zs37oRHvjGfQ0/I+9uLfiIP7vEcaZn7caTNh4e/Beon4nPHway5ePFiOnXqlNEroFls2rTJDp7C+IeuHGVZXlk6SwyjxuZy7spPYczP2B6rEke+Ug+sFQOiS2O/rLK2IVVWN5gNm7kLlqUFi1en1vmLU/OUWTp0iW3aejGBxNSXUMDwQJ/OcXabDZtW2a9ZsXRWWrakNc2eMVk2bCqMrswqogIcFD3q3w4P4OmUgFgjPE4WUzcP5Z11yFPyNnEu3w7RXtzBccGwmbh2uOEl0dATJWHj3Z2BmA0G1Y55gdM0vTImfb5rIO05cDJt2LJXdgq6Ur9O0QxhMEqbLGPY6A7DpqYSXbXDMhxVn1aJS7qsfbYYOGLYaO/Gdizjqd5w3BUFFBgoMFBg4ONigDmYhZV7bGbIk3mZjc+WLVvSK6+8kt59993U1dVldmw+//nPp7vuuistXrw4TZokw6oyzhcLNWnz+fCOy4e7z1/ar288HSo2PloqtFYY00bPsj8rHbU96f3t+3UdSOc6B5LYMjI42JAxbGTnZlCf+DoJwId7bVVfmj+nOa1dtShNlZqMD7e9l/71//un9OILf0hdnR3ayA2kBfMXpAcfejB9/etfT3ffc3eaNWuWVGuW7AIZDPpxaPwXv9ITbzhWt3DxHLHiHuHF/ebDAH2bjyD1Dese4/WJ+HgaLzwwRybRz8LvWu9RVtyvNJ+/BBjGg/XjwpbHRf55vPLcnzl8PDfWnDxe3MK/wECBgZsbA/m55ErnjnwasJdPRxjXePvACCMdz+xJPx7DJtvPZXNiiehFCeGcCGdf7iIw+azt6Ug2pMOcAzqoc1J2bDZt2a/DOkfT+c7B1D0Ae6ZG0tV89IuoaFIz2LHpT011Q1J1M8MYNqi+aW4UTUDSN67QrGDYBOaLe4GBAgOfHAaYY8tdfr4uDxvrnTzKr3y8mOuJwwF8HH5xiLM8Le8Rhzvv0CH6+vpMYgbJGZ65oEWg8gw7NefOnbNn/P5/9t78u67juvfcmAfOAMEJnGeRlGRNtiXZsSz7OYrlJG+lnafkeWW91eut7l7pf+n9lNcrq52hncRRnDxPsi15ki1bE8V5JkiCBAhOmIGL/n523Y1buAIoyhIgyqoC6ladms8+p+rsqm/tXQA2fDewxCGVeenSJQdrxsbGvP6oNwH5XqXGcY3QAmyQqsQFpGGOM6MNmHwZAGwalEZTfJ1P0y71Zxt0hs0+l6hZ1bXJ2pZ3SdpmlZ9d09Cow8y0iVPb+5Vf3sqEFppHtZFTawK9K6UOrVegfrd1rexwCZtmaEK6upm+muJtiC8TV8nEsyNBfMNq+SPVUrlBT9x45rgFsFmqJ7BE9USH5MEutoSNL1d4pwCx1MvNC6Vr9NROSi/aqM6FOqNzCV79zUk7f/mGFuiEjgohneEg6Sq7BRjTop7V0Tqt8wlaXaxt766Ntnp5i8TdEmADSwYqW0yhQKFAocDHiQKMxwHe0G4kb9D7+uabb9pPfvIT+8UvfuFM0P79++3ZZ5+1p556ytV6IfkYkjaMq/Hhzu89PuR52P3lr07wZxuVvg/aIONL3ZPaaTlwfcTeOHzGAZvbo6KVGLOG5iRhM62J/bTEpZVLhwtWrL1l0rZsFHMmXeYtDeP221/91L75t39jP/vpT2xyYtwPM9yl84Be+C9/bl/96vO2d99eqZlb5cyin1kjOqbPyLs/JnNDYN6w9SZShVsfX64/ORTg3WbCxHui9yHNBOpun/eEyQrB7/XOLPTO1RV5T5dRV7j3lEmJ7oc2LNTWD9q2oEW4C9UzN5xxdyHD+FtMoUChQKHAvVAgH0vmGzsiPo+LMMrPw7kmDkt4fVzERzjpFg2wiSHSh0N+tEjnYyMLXzUekKGUcwtRhTt0c8QOH+mzI8ev2MDQuA1PaCFRgI01YjXx53s6M6XNmhNSgVaxbb1rBNhsta2bV9mq5Y3i/wTYqA7ty1Z9un+VTZVeLTc/+y2b0zgPjzaRG0v+YgoFCgUKBT4IBfKxOsqJ8Teu7+ZGftywefooCxewBks61glC6mZKmxanNLEGgGGtITbvUw7pAVgCgAGcQesH1wAzqDtjbQIpGkAZ/AA3o6OjXg/fD8rNgaJoE+XTFkZSQBt8gDFI2DAC08ZmSfcA1nh+bcIEsGnRrvyW1g5b3bXOdu07aI888bTtfuBhad9olbr0SRse15qJ1o1r6tCShFDjjBaXp0dU5qSfff6pB7dar9YGVnZo3dglbPgmyNIe/5VHhs8K34YE2OAPQyuxKUe6k7l5I+VSuPH8naZ63tAZyzMoEjZL8QSWqA4ecHTSjwKw4ZVHH+EknVvrGef6huwXr52ysxeHbHRCAw3yMjBkQksbEINTJ2mSSrQ26SLsli7Cg/u22P69vUJKmxWW2KkC2CzRy1OqKRQoFPjQKJB/dPNxmY8uzNLhw4ftBz/4gXGmDYzRrl277JlnnrEvfOELtn37dmvVQXwhwpw3Kj7eObOUx98//urUWN8kZ4TEcOANwGZck/T+gTv2pgCbd45d0LlnYpmaOquAjfSa6/sxLeCfvZcANp1tUzrnbJXUZm60sTsD9spL37d//tY/2Bu//bUzgqjYfOjhh+yv//qv7Stf+YqtWrXa2tpQMSd1n6qbNvAcEsOWGKCY4TsfN0u4lGb2ctYTqcKdjSieTxwFeEeShA2+hQCbWh99r3dmoXfudyFs1BXuvZZxP7RhobZ+0LZBi/dLDz1WHy/mb1Pt2c4fX0ILBQoFCgXulQIx1tzruBLpKX++PMRHOP4lA2x8kS6NtwGO0EaGUjZ0AtpwduHxkzq78IR2a1/V7u3RaZtiBzXnGFQBmxkOlp6ZsA6tDfhGnQe2aHFujXWtavFNnk3i6Wb/+DxUh3fAG2WkxszG+B8tYkmuADYiUDGFAoUCdRTIx9a6qAUvY6ydL0Fe3nzp8vg8P+FhGb8xgB+Esc4b+eIacIW1BcAXLBIxXLO+gOU6wgBpQsIGIAd/2AB8AvShXtpNG6I9hGGiDfh9CKbNXACGM0ZX7ex4rHgF6XzfRlu5apVt2brdPvXIp+2zn3vGdu0/pE2bHXbyTL9dvHJTm/9bbaIirUzSztQgML9R1uuYFpIzPWztrVNShbbOHnlou61b22GdLQ0CbDgbPaD49IVI7ZltoGJnQ7ylqW14vXTFJpfcH4UJGuMG/YL+BbD5KJ7IItXJA14KwIbXPSwvf9rhkl5uVKJNqWNP6gyCi1du2K9eP2unzg06U0YYu2g4YMoZJqVt0oJci0Tb1qxstAf2btaC3GZbu6bVOtpgqJJKNECbYgoFCgUKBT4uFIiPLu2NcRkXEAaXHSwc3PfSSy/Zq6++6jtdDhw44Oq8nnjiCdu4caOrR4NB42ONwc2vPfA+/vHpse41DIyQS2Fq0j4sEcxLl4fs7aPn7fjpyzr3TIyoAJvGlg4H9pHAQcLGAZuWitRkztjOLatt78611n/xlP2Hzq75wf/6jp04flQ7bRoNqaTPSg3aX//1/22f/4M/EEOo3TtSLdeEio0q/aIpXDtFg67RQHfjyxaB0X5yhI244n4yKcA7kQE2vBfxmjhBmKjg8Z/M9ch5furfuXmS3HPQvdZZX+D90Ib6NsX179K2oANl5P4o895cxup6E+NJfXi5LhQoFCgU+F0oEOPMvY4t75We+CgL/5IANvroxfk18Hp8FFVzIoeG0bRZp8FuD0/a6XNDAmyualPndVeHOy3AZqZJoA3AjfjDGQ6WnhnTmQQ6m3Btp+3fvcH27uqxDeuWKcyXA1VDgl1yejtWw8dYUji1j3Jt/HeeVPEpJysLtbi8nOIvFCgU+ORSIMbXe6VAjLX16RcqJ9L7eKRJcaSbDdeYjTQK67kAJwGecA2gAsgCQIOUTA7EANBwDUBDHKrMCMOGijPyUQZlRR0BCNF+2pC3gzUL5tOsPcxK81TbRRm0nfRpnVYjPiyzl1Fbu5iRVI2CNB9vtPb2Vuvu7tI56nvtU48+Zo898aTtP/gpW9G9wa661o3Tdl7fBenW0Fpyi74bUmvOujFnnKkumx6zpplRW97ZYPv2rrdPPbjFgfy2phl9G7RBky+DkrkEJpV6e2gTd8fYT0Bu4jp9C9K3i+TpOk+5FH7oGTaeBS7PqAA2S/EElqgOHjIdiAe7mBI2vN7JJuVm8f7TNyq0QT0W0efLOlzw9cMX7cTZAbuu8wrGtGsaPbVNTW0qQd1KPBWATbNU3Kxabjo8aqMDNuvFoHV20PEKYLNEr06pplCgUGARKBAfXlwMYzN+GLDBwUEHbV588UU/yI84zrP5wz/8Qz/Tpqenx0EHGKUAasJdhKZ+6EUGG+RMqd+/mA4xQVOaud+4NWpnzl+1Yyf67KyYs9GJKmAjEenpmSap1RTjJwkbVGO0C7BZvaLRdmuH5XZJ2Rx9+zX71j980179+St25dJFMYDtrkrumWeesb/6q7+yxx573JnDZgE2MJrO5KneUI2Wvl41ZqzmgwTxdQtyxDWpygQ/qPLJdnknYkGI92LuG1SjTYSHW4uZ64t3bG7o73YVdYV7r6XcD21YqK2/S9u4//dLg4XqL+GFAoUChQKLR4HgD6khFss+SG2UF+Xgh7dclDNsGJoxGmpdFRpgi/gkX/Ry9ATVoTIK54vJOTbDkqi5cOmOHZWUzYkzV+z6rQntrG7V2kCSsOGM21nARps5u3Rm4Y4ta+zgA5ts+5YuSVyzi5oa0p+XX/2ZA9g4z0kDa9+CGSXgrwA2OdWKv1CgUOCDUiAfw+vLqo+LsZlxGQtwgoshLdesEaCyDAvIAuAC8AIQg8uZuBcuXLBTp065y3oCQE4AMVFmfd1RB+ERR3vms6w3oO0DbRVYztgljPoBhpDKoR7m1s5tM75qjGcCTnmznwcFNSsfZaxZs8oOHTpoz3zxWW2ufMZ27nlAke2uZeOCJGveOX7O+nSMBhI20zrXFpVoM1oT4NtRoa7KuL4Bk9a1ulUamdbpDJsNWj9ukqrMAGy0tqyK0bpBO2aNexn7o1URE9cprX+7FFW9o0i0ZG48F9z8mfB+nD171lgvWr9+vb3wwgvvapNU0vlXzyPQvU+Gxx9/3DNQGMYfTDVZHuaR5WfJKMCjWjrABrZHz57OqXrdr2suk5RNg/ULLX3neL+dFGCD+ps74xqUdIZNo0vYILiWzrABsFnRUbHdO9bb/j2bbEMPhwtqd7QSSDOaL5MtGRFLRYUChQKFAh+QAtlnc7akYMyIww9Dhr7Y119/3X70ox/52TYk3rNnj/2BpESQtNm6daszObGzJT7gpItvLf770QQbRNvSFyLtspyYnLGBwVt24vQln7BfvjqsM8+0bbKpwxoE2MxoJ432HbiEzUxl0gEbzjjbta3L1ne12S9f+aGfX3PsyJt259YNSdcss0MPHrLn/vA5e/5rX7N9+/Y7bdjNEwAXOnWdqXSehS9Pzcy9otVzW56uSVUAmxrVPsk+3o97BWzmvl3zU63+nZs/1b2FRn3h3luu9I7n7/295psvXdQd7nxp3k/Y70If6v6w6n8/bS1pCwUKBQoF3h8Fcn7xw+DrKC/KCX5zMQEb5v2qUJ+RBNb4oherZlpySyYBNizmjYxV7FL/qACba3bkxEUbvDkmwKbNARt4P9IA2DRKJVpr47gta6tY77pO7aTeZnv3SPK8RYCNqvK1B6pVNX6vHkZtBOj7LBqk75o3jgjxloRhWI4r/FyiRfktFCgUCArkY3GE3c2N8TXSMBbF2Bsu833SRdmAHIAdADGMyyElAxgTNg/HD3AT6styCZsAcSgrQJpoC/VFGyIMN8ICkOns7NSZrzoHRnaVVJXhrlixwpYtWzbrZ2Mk4M3169ft5Zdfttdee83blAM2gCEA9z7K6gc/59W0Kd+qVSttx45t9uijj9hjkqw5ePCQbdm+05at7NIRGo02dHvCN2++ffScXe6/pfWATq0lt2itRNI9TQLzBdNXkLycGtWawJSt75HkpQCb/Xu7bXmHzjZTrW3NScKG6Rn3BmgTI36aDnDlXyfIUDWRgu9ELbYANlXyFGdxKEDnXArAhtZr+NGvrDNXDERpEIAhIm5KjNvg0JgdOzMgnYTXrK9f+hRHJYInpNRVogHcqDM3N85Ysw4XXNauwwU3rbE9OzfocMHVUpHWLlVp6qLqQ7BVC5noanl86nYR8t4pImVxCwUKBQoFPgwKBGNGWcEcBWCDSxgMBX4O+uMsG860gQki7JFHHvGzWJ749BPWs7bHd7ZQVpQbQERefh4fdXqYj9O1dhC2FCYfefFjOeNsfJIJ+5Ad0dk1Zy5c08GzEzY+LVUYAmwaW9r0/YAxY5eRxK2nxq2jdUaHCq6w7b2rNYGfsJe+96L9w//7/1jf+dP63o1bd9caB7j+5E/+1J566imXtkn3lxhnZ9zENHKeTU4X0sz9VhASLcWPiWtSlgm+k+QT/xPvBO6736BEHsIXipuPgFHmfHHvJ+z91htlf1j1U97v2oZoy3zu+2nfYtQ/X5tKWKFAoUChwAenQPB1lFTPo+Slk+5u8ZE2T4cfnvLDB2xUm4Zl34wjfpZxPy2FVV0HR5CpDi4qLeSNjnN+oc6xOX3V3jpyzq5dHxVg0+6ATQXVaBWl0wHVHC7dIsCmrWnCeqQm/dFP7bSD+zcLwNFuarFidwNsZjLAJhbe+IKkluDCG2qTkLeO62IKBQoFggIxHt3LWBN5Pg7ufPcVYbQ/9+f3Q3jE5TQhjLE1bKQLl3DWZNmciRqyOCcGaRlAGsZk1JchHYOtP2smB3JCmiWvizbm7Ylr5rxol2CjZ5r/JnVmhIXEDAAMEjOoE1+9erV1dXW5Xbt2rWHXrFkzC+IA3iAdQ12nT5+2//k//6f927/9m7efe/N6aQtWYzNjrW5dQItUoClfT89a27Vzhz3+2GOSqvm8Nlg+qLB11tImUEbz6pHxijb4j+n4jAGB+Bfs6uCwNWg9YFoq0SpaM24CsAHM15qATd+25Vov3tK7xvbtWWc7t6/RGbdaS9bKQZxhw7o0QJFL2dA4tUxBMu+eR6SvUkqTUnhCvhBcLrmJdwcXeofluSMwUyRslvyRLE6FPOClAWzSS099vP90zBSiF1yMUkWM2ox67Y1bky5dc/xUv3TVDtmtEQ1eMzq/RirRGl0notI7YzZly9Qfu1a16pyCNXZo3wbb2LNcYE4CbPJuQ5XRj/CGScE15LTWCUkRKfOSwh9ulFTcQoFCgUKBD0YBHxvriuDDS7hbxTHy8BFmZwxn2vz2t7+1//iP/7A33njDmazHxNx86Utfkoqvx2zDhg3+4SY9JnTKxsccFxPxEU5Y1JmHEf5+DGVEHfjDRFhc5+kijBZPKQ8uh86OS93ZxcvX7Y23z9q5CzoccVgi4RJ/bmoGsGn1b9i0gJimRgE20yMO2ADkr1+7zPr7ztqPvv8f9oPv/rtdvXxJNzdlmzdvsj8VWPP1r3/d9u/f7zuEqBta0J5gWOfeP/dQuw/S393EdyLcu6cusZ8ECsT7U96JT8LTLvdYKFAoUCjwcaBAzofhhxf6wICN831886pLWfLGF5DFsORn5l2NV1i6ShQD2OEcmzFJWLOb+sRZSZe/dUqq00cUjmR1uzJIBY7WDlB9Yw7YTFlr85R1S/3Nww9utQP7em1VpxYdWRtQsXK8Nr7AqX5+WX+gtrQeMPt1rrZfamM8FwuAvrqoq2IKBQoFEgUYL7Bh5s6bIrTm5mNNLfS9fVHHvcwh6+vI8+Kvj48yI120pv46DycunzNSBmFYpEiw+F3NtjJGWLgBzACsxDkzADOMuwAvQ0NDdu2aNq/39dmlS5dcpRnxAB5hKYPycHN/1E17aUOYeDa4Mc8NUAapGSzADJIyADNIznR3d9umTZs0b97s6woAMwA3gDsB5gDO4M8BH+qgTRy38Td/8zf27W9/20EmpH2SUbuURv8+GJO+rb3NNmrt4tFHP2Vf/MIz9plPf9p6Ve+q1WusWXP9maZmATZSkyntSxcuoybzup27OOQq0zlWw9VrCrBxV0C+SeNGU2XQuldII9OuLbZ3d69t2rBKEjfajKm15GaN+6wbQws3KiNolBpG0xApgIaJjjQ3fbGqeXSV8npE8i/hb7xzuNF2XN7NAtgs4YNY7Kp4wHQoHuxinmGTXvSE0vhrLy8HTXknUIdJgE2TDhecsrNakDt28rKdPj9ot4Z1dsNMEn12wEbiMzOVKe2kmbYO8U4rJNa2vXelPXKo17ZsXDUL2AgodeN16ccv9ePXc4haY9Do6lmuqp+c1cLe5VaTFKdQoFCgUOADUiBnqiiKD26Y+CBzjR9mDAbv4sWLLmnzyiuv2LFjx5xZOnjwoH3xi1+0hx56yNatW+dMVJQDYwZDFWXj5vXm14zNMCWRNsq4VzfKzcuMvHmZpJtzrUQKMkEvNi0STInpGpuYEYA/aL9987Rd6Lthw2OM1u0CbNq1I6dZ59dMasI+bi1N7KYZto62GanK1M7K9gZ78ze/sJd/9H37zauv2s2hQe3ekR7bfXvsz//86/af//N/dia0TQzqjL6B0BWTg1v+jeIbou/U+zPx/MJ9f7lL6kKBQoFCgUKBQoFCgUKBxaZAzofh/9ABG2ef0iw7gTUEBG8kt+qNJTD4Lm+HUmltzm5L28bpc1ftN2+ctD6pxJ2cbhdoo7UBFNugxoZDqrUZp6lhyvnArlXNdmB/rz2wd5N1r2q3zpYGB2xcRkZ8b7BziavTgpwC+HOe14mtGHhThTlDSgMLYLPYr2Ep/2NEAfoL//OZfE5XH5+PNfVxcU0aTF5OhEWaejfS1peft5M085UTeSmTsW8hk+enHNLmLn5szNFDFRmbLON8GVyAl7ARnl+HHwkaVIoB3gDkMO+POmhj3m7Cw0Q4bovOZgWIAYTBArZgCQt/gDMANBEXeZCWQZompGiIZx0BuiKRUr+uAE2w1E17WVv+27/9WwdsAJ64Nwz5lURWZbQ0W5eAoF07d+q8mkNSgfaIS9dw3dm5THP9NM5XlGFSY7F/D85zfs2AXb4qtXAjE1oroExUmqGJRM+Zx1iRmsyZQalGb7AH9u0SaLPJerqXCdTXl0Bt5Ex0CfXI6hwbvkEANj7We8NopreTtnp79UsyT+PwPykUEqQncolNvA+40Dwsz6AANkv8MBazOh7wUgM29Aq9R7J6w51JCgmbZh0uWBFqelMHS1+RWrSr2kk9YxPaSV1pbPGFuUYHbBB9npbYc0VSNjO2VapvHnt4q23dtFqMmgYndb73C9gkJjF6HBTHj01dk5Dkz10PLD+FAoUChQKLRoFgwuKjTEURBjPEmTaoR0PS5lWBEuwUAbAJSRuYLMLYkQPjhiV/fNRxw0QdERf15Gki7Xu5ed7wR568POLmXCuRg/lqluAX6ao112F+5sKAAJtTOlzwpgAcMYuNMI3t2vDYKOZY8I70lzc3StR6etQ622fsgd2bJUxzx77/3Rft5y//yM6cOGZjI8OuDu2JJx516ZrnnnvOmVCYTxjsAGxipxDtpX2pjfoC1EhF1HuYSBzueyQv0YUChQKFAoUChQKFAoUCS0yBnA/Dz2LPhyph49NrLaJV2SEHbVSPz6szxmoWsNFqW+xrRl36yLjZmYvX7Ne/PS516cOSum7TRh3ttvYzcNImpAaBNahFaxIfuHpls9Sl99j+3Rtd+8aKDp1xq9pyKRtI7M2iJkdwUnvSkp383j5cVv7U8ALYQLJiCgWcArW5UVokno8spKk3c+Z71fj5wmJMIi78lBX14s/zcY2pD4v0hNfHpRy1X9Iy9mEifeSJNkR54QY4w3w85pHMtwFpAFpQYY6kTAAvqDIbGBjw61Bzxjk0pAfgiXko5WOiXr/IruvbF9fM90PSBTAFkIZ1gI0bN/pZ8vhDnRnnz6DeDBsqzQBkWCeIzZ1RLi5tgT5Bo6iLtJigSbQdOpw8edL+8R//0VVznTp1yqWHKMuHVDnNqmulQKF9+/bZl7/0rH3u6c/Jv1eq3bt9wynjsUs2Nmq8V74JPZ6bw5N2/NSAHT7WL3VoozaqTZ2VKlgDpMIjRFVmg74H7Q3XbePaZjt0YK/t2rHB1qxqsxbEapSIteQmjf2oQwM48nY52alTlntym75GogCtkeU3vlbyx2uesihu6UxO8/xZ8YwKYLN0z2HRa+JBLylg43dU7Uyq2ztCg8QG+RMzNCqm7PK1YZewQcrm+i3tJNdZBeipbRDCyuKcVuSsSZ2spRHQRupt1i+3xx/ebts2d0nMrUE6CZNaNKpKHY0OlnpRuvZxwlvCT3TAWo8jdL7eFz0xXNIVUyhQKFAosDgUCKaH0uOjHH4+xjB2MHgwhJxl873vfc8OHz7sh/89+uij9uyzz9qD0v0Kg4YJJoy8UXYwdnHtCflhmKsOg87EzEbcmyfKI2/4I2deHnFx7eNz9buANDOiz6hDuzU8bafOXpVKNKnD0G6aiSmdXyMd5o2oytQ5MxwsWKmMizkbkx211ctbpApjq43evmb/39//rb3605/YwJXLNj05YZt6N+r8ms9JuuZP/RwbGFWYTkwwyiFhQ1it7bSTkHs1kTjce81X0hUKFAoUChQKFAoUChQKLA0F5vBh4sHgET8cwEasJIyT85LMtqvGg9IaQMzIg1PC9TUB/bJ06lLW2ouDlPVrkrC5cPm21gp0kq0DNtq8o7NlqIPFN9SiNencwlXLpX1jy2rbu0tn3Oqs2zUrtAipgllW9HqiIbpIrYgFOWJZMVAC0YHY5CqcM2y0qFdMoUChAN0iOpG6RTY5WigcmuVxd6Nhni7KDpexKQxhWB8v2OknE+kiTb0bZUfeiI/wcCMclzAs66UAM+MTUmM2PuEgC+NkSMIAxgDOBEBz69YtlyhBiibOo2HOHmrQKAtQg3ID7KEev6cqfevbyXyVuXycKYOLRSImVJihxox5P5Yw4pCiwUV9WeQJVWZRXmxWnKWhhj2kaGavq7QI2kQ4Lu3m2URYrC0jVfOd73zHJWxef/11B6t4XbiPzmWdtkUqz1Dj/qTOkz0kDSHbtm5xAKmjQ1o04hxZjb1IUgLYjGsn5/VbEzq7pl+AzRW7fkPPYlpSNYpPEjZqi3RpImWDxGVn003bsqHNHjq0z7ZvXWfLO5usWZv7GwTEK5e+G7IM79TF1wG600Bs+hKk90u/vGkpFArEt4A8VYoQucQm3s14b6A/lmdRAJslfhiLWR0PODrVkqhE85uhY1cHQF1r2VGWDtIkBqxRBwqOS8Lmsr1z7KIN3Ji0sSktyDVyjg02ATZ0shbla7Zx7Z7psEcf2mE7t3bb8o4ma2vWYVLVTkMfYiBPl3S1mqn1q4jPY2vpar7IEW4tpvgKBQoFCgU+bAowPucmrmPcJo4PM2P4hQsX7Mc//rG99NJLfsgfO2c+Lf2vTz/9tB04cKDKAFVFmVVuML31gE0wW5RNPfHx5/r9mGgr+cMf+eergzjSxZiN6DOAzai201y/ydlm/fbWO6ft6sCITUrq0hp0oKEAGxg6RnkAG86vadZkvWdNuwM248PX7Tvf/pa98etf2uDVKzYqCZve3k325S8/a88994f2yCOP+O4jGEfaBMNMG7jG5kbRXk8ednd/fCfCvXvqElsoUChQKFAoUChQKFAosNQUgO8Jvgw//OFiADZxX7OSNuLdYgbuQi5K4ByTa99QO3QNYINatAuXh+z1t8/ozIIb2sRjVQkbNnQ2q+3KB2AjhTlNDZNSlz5jvRuWS8pmne3ett7Wrlm2IGCTVgmoCe6zxq+ls2sUJnr44hxgDbaYQoFCAadAjBs+c6Ob1JkYUyKY9JEnwnAJy01+TRlhI3/E14fXX1MmYbkbc9+Y9xEX5TKXBkwBVAFMyS1hSMGEujL8SMYAygRgw9myly9f9jNmCSMN9UV7o55wqTvmmoTRVubkACk5mAKgApjCZkLCkZpBVRkgDH6AGK6RkkEVek9Pz6w0DWlCbVncM/Xgr6876Bdtpk15Hq4xQdP8uRNHPtLjB4jCIFn08ssv27/+67/6GgWq3LmHrq41tlXgDOrbP//5z9vjjz+uM3LWW2dHp2iQVJSl9qRxF0BGcJCvCfRfH3Ow5oi0Md26ozN8ZpoF6CQ4paKPC0KR2GZt7F/Retu2b+qwhx7cZ1t7e6QWHRqL7tybvhmUimHzp78p3CLvTHWs59Lv078PCp79YsW3QGnj9U2vGsUtmYHWYeP54fIszp4965JN69evtxdeeOFdbdI3Trmr5s033/QMPAgyxEPGjWR5WOQr7tJQgGfw0QE2vOMayBoQfabDNEkvbZMN3Zp0CZu337lo/YPjNjopwMbBmqqEjTqEcFaBNdJXWxmznq5We+iBLbZ7R491SdStQzrRWMILA2OY+hDdLJn6PpViIjZyLuRG7nAXSlfCCwUKBQoFPlwKMGZj+RhjguFiBw+MImrRUI/29ttv+y6ap7Rr5dlnn3WmiF03wbiRl3L4/sY3mLDwzxdH/L0a8mMoL/yRN+rgOuqJuAr3p4uQsBkem9Z3YNROnu63I8cv2MD1UQE26LPle9DqTJYzdzZhlakRnV9TsU3rVmpn5Sbr1Fk2p0+8Y6eOHrYzEsu+3HdRByh2SV3cM/bkk0/a9u3bxTgK/NEXQs0UYJMY62BQaRNtTe2lVff6jfCc5MZTTKFAoUChQKFAoUChQKHAfUmBnA/DD3+5mIANrFFwVLNclTxwTM411QE2qMa9MnBbGzn7tHlHO9gHx7ReoEU6SVoD2LiGda0JsDYAYNPROm3r1rbb7u09tm9Xr63vXmGtWl9jM6fXkVUai3GhFs3r5ylV25PYPjKmhcP78gGWRhUKfEQUiPlduDSjNm96d6MiXcwDuQ4bqfNr0uVzMuIYn3DzMvJ68zT4yV9fBumJi7SshTLmIR0D8IK0DGADwAt+LH4AGoAagBvAnZCMAaAIoIey8g2AgDBYDG0nDpc2Acbgcs39AL4AuAC8oLoMLRChugxpGYCZCAOMQVomAB1AnbCExYZMyg9DHfU0jbhwgy5cR3pcwsPEdYRxnZu4f+iF5o9///d/t29961uGWjTuD1wAbRe4O3fu9HsCyGnW+TJxrAXloaoM6RpRyzdx3tF5Zheu3NZ6QL++BQN+pq1/AzSys34gMiYtllr8bWmettUdI7Zzc6c9dHC3bdq4VufXCLAROZI8TgJs/JtQldSsjfe1++EbwZ+3R26KCZrqKshSy+Jpl+InnhVu/qx4nwpgsxRPYInq4AEvDWDDDaUFxrg1+r26llvXVqsOOSXABvU3LM699U6fdNXqMCkdMG2cVaDd1N7LdOldV6LPjZVRiTo3277d62zPrh5fqFve1iwwJ3Wo2MWj4UnVJlimvj/VrqPHRQvDrQ+v5aDMYgoFCgUKBZaaAv6R9nEu7aSAWYR5PH/+vP30pz+1n/zkJy5pA4OHejTOtXnggQd8B06IPedtzplZwvOPf57uXv3kxwRTl+fLGbuoJ+LJBdOF6DP7Je+MTGpn5S07oW/CKUnZXL857hP1ikb5BoE2zghLH61O6XHAZuWyZqnC6NY3YYOtXd1uI7eH7ObggCRsrtmQ3Pb2Vtu9e6fv7IEZDiaa+mFwaE89LfL2ku7eTPk23BudSqpCgUKBQoFCgUKBQoGPigI5H4YfXuiDATbM7ZmHwwfJOjsYPJEuPGjuVkmPTVGKV27aodzTmshPaAfP0K0xO31uwI5KA8e5vus2rs2cADZSkC6eUYu68jU3iX8TYNPaNGGrVzTZDmneOLhvm/WuW+Uq0wFsWGYLaR4almqh4liS8+Z5m6NNHlIAm4/q9Sz13scUoJ9iw8SicVx7r6pFe3A+p8rzxhyMRORL40eUpF5YBwp4umr9kdfHDY1fXAMasMbpasyqkjNsbsQCuAAm5OfIRBhjH37SISUTefDnoAx1RBvCz3W0E5c5JhslsYAogBJYzopBOgbQBRCGuTrADFIyuCFBQ7qwocqM6wBqomzmrWG9UfqBFrQh2hPhCvCRmfigf7j1aSP/fOGUF/lmy66GBWDDugSbSX/0ox/ZP//zPzvNH3zwkAM1hw4dcq0X3G8AV+k8GUbc9NLwFqD2sqKt+GjdGLqlDZxnB+346atJPSbfAa0FkA7gnvPRK1KJpoZJ49KM9ayetD1bl9kD+3bahnVrBOLomfAd0HifYCDkdjDZGTb+nqVQYvwdrraHqxTDbzVNbbGZ5Etq4hni5s+K97EANkv6KBa3Mh7w0gE2dL7UAb0r4nXpGlSiYbVAV2m24ZGKnTk3aIeP9tlZF31WwuYOgTUANs3eP9A9yGFSADYrO03n16yyvbvXizlba6s7dbB2jRujEv/zjHXkrHXHuog5l3m784iss+bBxV8oUChQKLAEFICJmBFzErt5YAxhjs6cOeMiyD/4wQ8MVZdI1nzlK19x9Wi7d+92ZhDQJhgwXPLGdd70+cLy+IX8wcSRP/yRNi8zmAyPq34ekK6Rmlp9FUzizuO+iwbm7OKlIbtxe1ISNjBvADbatSQx5iYxYIL7lWFMIE2HvgUbHMTf2KNdleLM9NVAfMamJgTqVCSZqfQtLYmBjrbQjmC4A7BJ7c6Z3Xv7YlBdMYUChQKFAoUChQKFAoUC9zsFcj4seKEPDtgwq4dnEkTivF3wT7qQN82sIyxRiKsUIn5M83wW4Fikm9KC2PBoxc+4ffvIBYE2fTrHRmkaO1xFrgM2jTPWgqobATZNDWO2rH3atvWusYe1s3rrpi7rbBXfx85qb4sWt6ou6xBJuia1gWBvgzyzrZtRRgdsZkNS4vJbKPAJpwDjRZorqYsABPhidyJKhIcb8XEdpCOcsNzm5TA3Iy7yR/oAZJgDs2kxVJjhJy6kXkJt2dDQkEvKhAuQwBkr/f39Lj0DGBPzQNpGPQGC4IbJw6NNzKFDugUQhfSEAURwDUgDOAMwg4QM8/KQoEGaZrPOcsEC2NTAiwQicO/RLupjYGJsDZpEW4kLGpI+pU2jsPvjBuRGmblLmtxG8oXCIh436iUtdUN7/BjAL86vYU0CsAlV7Q/s3+dq0aKd0MuHZP06aOM54zshsEaSlHwHrl5Pkpanzl23a0PjOtMmadxwKRwVUEFTBlZt6JACjc3rzPbtWGm7d26xnu5Vehbp2AyXsNFHQBSj8aoNOkFY2pzaXW2CUhCPxVTzeJpqWs8+N09Ku/i/QXfc/DlB1wLYLD79l6wGHvBHAtiARqqjzIghUwvELMGYiSnTQtyIJGr6tKP62MkrduLMNZ1jMyGgptNmAGwaWaSjUwjVrogpszHrbK/YxnVapNu1TqLPG617eZu1wonN9p00WEVAXN07ked21Fo+KqgN4LXw4isUKBQoFPjwKMA4vZCJOD7O+GFSEdc+d+6cq0eLM21gDDmz5XOf+5w9+OCDLm4NExkfeMoP5ioYU8rL4xdqw3zh0S7yhz/SRT1cRx2expkehSkPgA32+s0R6aq9bMdPXZUajBFDHBrVmeyoQedsk+8qmvG9N63NfAsE3uug2W06cHbt6lZrk+wzk/Qm3QvgljNdPlNP98a9vruN+krE90NZvFkeEIEqpphCgUKBQoFCgUKBQoFCgY85BYIP4zbww09+eIANfFOVd4KZwvj8P3nh94gP7iq5sT4AYCP56ekGG5vUeYa3xrWZ86I0cJy328PavDPTJtsuHo0NR/oVGpMAm3GpR5+0LRtX2aMP77Xtm7sTYKOUzNpTjdSvtYQqP8iyBMb5vaon2iRRHqXDzoZ42vJTKPBJpoDP20SAfPyAHhEetIk5X7ieRj2NOVmkJS6fi0X4fC5gACBArq4s1JeFyjLmwWEZy5CSYaJXoKIAAEAASURBVENjSMgA7oTEDGGUGfNo2hftiTbHXDGucQFWkJIBhAFoYZ6NCyCDRUoGiwQJlrRIxgDsAOCQP/whRUNYXkf4oYPTQkMQo2WEB40jnvCF4vL7yvPV+yN/fZl52d6WakbCIy1uXIcL3QcGBlzShvvbsGGDrVqpDZWtLT6mpnI1GrMZX2UmWvMeMR4DqbAls9FQjXmp/6a98c45O3vhht28U5H0ZauvBzA+M3YD2FSm9Cy1FrKso9F2b2m3g3t0Xs7m9bZ6pc7zEajvEjZKm0omV5W2+giggo0Yb4h8GFKk35SWfClBfA/qXc+wJD/1dE+0TKBZAWyW5BEsTSU86CUHbNKb7+/6jA580hKj/NMu9jwtwGZ0otEGpJ/2lHQTvnO8z3fUsItmprFNll3hdCYVMjNB93WmrHtNs3ZVr9NB01ts3cp2a4MBy/obg1vqXImuqWtFB3svWtPgaHT4o7xw36uMEl8oUChQKPD+KcAYfS+GdDCbGFyY2ZM6twUx5FdeecUuXLjgzORnPvMZV4924MABZyRhGoOpIi/+kLahzPj4E/d+TLQ7LzvyExYm6sD1PIqryE7ptidlr12/o4Nmz9nxk/2uLnNkAlBKHJcAmwYHbMTcMVHXN6StZcY2b1hjO7f12PL2KWuXHvM1Kzq107JN6jCaddigdP2KW2M3pW8WUJ3OGOr7oJBo0jxuMMC1ds+TqAQVChQKFAoUChQKFAoUCnysKBB8GI3G/2EBNsy70/ybgik98VCJ3yJAvJeC5qbRtSfTwptSVASWANhMSM3N7ZEJO3rysr15+LxdH5Jqo8lmKcPVhk7tvgawQZFuo3jBZleLNm6961fYow/tsZ3SvrFCi3chYZOW5KhflvUCOEB3a0txHqUYWpdsAWycHOWnUEAU8D6sruPztipFcn8QKeaQeVzMCwkL1Vn4GXdCagYABVAlXECVUGUGAIOUDCBNftYMYA1xzH+xodIMYCbWOqmDumIeWt8uwpkXhnRMDqoAtnDGDMALbi4xE4ANQE2oNwu1ZrgB1tRvlMzpFH7alLcxwutdpyMDVfYc4r7yOPJx3xjCPQ3DGkOfTOTBn9Ojvh2zeevSkQ+T5+U6yoX2YYOuabxWGjZNemI9f56LGpXWeaN5SXEZawKj2r9/Xpo2fvv2abtw6aYNjzdpo38CbChH2fX+TFUBmylDRfrB3SvtwX1rbf26LluxrN3PMeOMHOoE4kkjfrXtyr8wYEMjIVgtDyHJUFrYCFsaN54RbjwfXJ53AWyW5hksSS084BjEUJ3zP/7H/7Bvf/vbPggyaO7Zs8e+/vWv23PPPSe9/7t9cCIPHY4B7f2ZNEimPAwYDEgMIAJtXMJGbRFgMyEG7PZwxc733dBC3RmpRRtSCg4W1C6aRu2qbgKRVaeRhI1YOOkonBBS2yBxt3V2aP9W27BGA2rSnObdh/rmMoN0uOqARQouMfS1eU3qoCkq9yeW7y4Z5y2tBBYKFAoUCuQUCMY3mJs5cXAgMvExzuPy9MRjIy1+GFfGdc6zQRT59OnTtnHjRnv++edd0mbbtm2+CwiGNGdiYSgZ4zF5HR5wjz/RFvKHP7LmZRKX0qTvQzpcsMHBmnFxaJev3bbXXj8pfbX9rrN8YlqayvWdaGCCLq4LxovdMmLbrLlxynoF2Gxct9wGrpy2seFBnWezUQfOateTdjetW7tWjPRycWlQHHqxkycN/N5GyDfnO5C+E7X2zomM2yluoUChQKFAoUChQKFAocDHkgLBh9F4/PCD8I9Hjhyxb37zm/Z3f/d3rjYIvrC3t9fXBb7xjW/4GgGLke824ueAW6piK4lz0q+u4b3kSa74w1mei93NXhB8Fx7K0AqBdOROCLCZ1N7OUam1PX3umr195KJ2Wg/ZrRGziQbVz2ZOSpSqdE46aGmqCJwZF++3zB48uNP2bF9rXSullkgS164KRwWnhTr36KLaHm9VtR0eRUNoY6x3pBbSumIKBT7JFGCcyMcNxoyYR0KXJCWR5lAxpkQe4vEHOIMLMIMkRg7KICHDGTMAM9euXXMJjb6+vlkVZqQF1CE/lvrDjbrCpc6Yj+LW5nUpnHjazPwXqRfAmVx6BkBmreaQzKG3bNni6st6enpmNz6GpEzcN2MllvrDRFy40R7i8zZFnmgj1+GPsnAjXcTnaerj8nykIz7ScJ3nzdPOVw9hefooK8rhmufAddxr/buR1n+rIINXCJ0SrfhF2sZJJ81KOsTcpWvujM3YmQuD9vpbp3TG+W2pQ2vVZn9UorEeAK15/pxnK/0cEghYs6LFPnVgrT20f511rV5uHW0691ZDeG31VvVX65zTBLVbjfcgfmpPUMF+FSG4WNLmVpdLZIL2uEF/XOhdAJsleghLUQ0P+KMFbDilQEydAzYwZuykaZZu2gYxY7fstTdO+/kFk5U2LcehEk1MGYANuSoT4rHGrbV50lZoDW6XAJsHH9Dhgl0aZFvphPxFF6p2PO/95E4xKQXXMtUk6SL/jQ5JGABTdM5al89TF3+hQKFAocD7oUB8cONjW5+X+DCkCRMfaK4jTZTFxxpGFmYXSZuf//znLmmDvl4m3I8//rg9+eSTtn//ftelC1OFofwAbPLyPfJ9/ER7KC/8kX3+e2B3TVKNiVA0h8yOScTmYv8Ne/U3JyRxeVWMmb4DM80a+zXl1gS6UWVr/u0SNq2NFUnUzNiqZS02PXHT3vyNpIpOH7HurhUCcHps/9699rTud+/ePdbchn7hGh1Tu2BYo4XJjfuP3T53+UjMzViuCgUKBQoFCgUKBQoFCgU+BhQIXoem4od//PAAm8RY8TvjgE3Mo6uEgfHyf/1UWV2COIOASyRspsQPssN6TKjNBe2wPiLtG2fPX7GBm9qF37DKNXDA0qG1gxMOW5rkNozbmpWtOs+w1/buWGebepbbsjYtogqcidl7qpA1iFQxsE0yc1cQaryfKimmUOATTgHGCCwm5ngxbuASFvM8Fu5ZZ0RCBgAmVJYxvmBDEgY3wkgHGJNL1XAd+UkLuBPrl7QjbxPXYaJ9uAAoACtIxoTaspB8CckZ4rCEh+WaeNSWERZqzvCjpSLmzPl9hz/qr6cL7SMu3PBHurifKMcTVn+Ii3y5P0+DP4+br7yI93w0pTr8RZsJz9vl6fIwMlTz1Kf1slVmvs6a15eq09jr70u1HidHKpPR378XAmuY77OB88adKTt1bkASlmfsyjWBdZKuqZgOqnHARpI4+mZUBNiwob9R68rdq9rs0YNaG97fYyulbaNNmjbYiurVyE0SNrScm1AoTtWrGyfCDUGYFBKJCAk/MbklbmlMPFfceFdwC2CzNPRfslp4wDHgLZWEDeOMv1R60YXB6l5hlhKSOg1jJsBmSgty/QN37Fe/PWPHTl0Tk9akHdcSe5NaNAAbyqioQ1plXLuqJzS4mu3crk55YJtt6emwlW28uHSf+EtdyTM6dVPHygeSak+ch/bRIYkKRpP8wfLhL6ZQoFCgUOB3o0DOxFBCMEh5afExvlsYabB8qGOXEelhcgFtkLJBRVqANl/+8pft85//vECMvc6MwnRSN0xt3obcn9d/N3/cE3nDT/r6smbjNIx62zXcSvOFAza3R6dd/PnXkrBhV82MpCw1FfcJPJA8zFIzOsv1HenQgbIrl4lxmxq2c6cP2ysvfceOvPVra21psI3r19nTn/2s/eVf/IU99dST1qaTCJv8Xmt3ACjDNyMMbcHS3gLYBFWKWyhQKFAoUChQKFAo8PtEgeB1uCfnwz50wKbKXIm3c8kb/c4aGC//r83Ig/8SC6b2SNGZ1ga0X1uLdtNaG5DkuNSiHT913q4MjNmYAzbtaSe3Nv2gIhdp64aZMVve2WBbN3XrfNv1tktq0VaJRxRL6At11J92S9cAG1pXaxmqetJGJm8gjSymUOA+pwB9B1M/94ow3EiD/3cx5I85Jn7mnKwlhiUOG+fEoJIMMIZzTJCUQY1ZqDILNWbEhx9ABombKAcXE/cU7Q/pDcLDH5ItzGfRHoEbfqRmAF9CUmbdOkle6NwZABjUmAHicNYM16QDoImzZig/7hU3zQ2ZH9bmzNEuwry9ysM5qxiPq45zcZ2X4Ymq6QiHpphI4xf6iToiLsLfy32vfPXxcR33Qvn1YVxj52tj5Is80b64dhIhEVPJJXFmU2kcFg18hEbRpTZwSh36taFxbeK/pnNtL9jVQam5Y124QevC0r7kY7TK8rVhHZnR0jxj67qW2WOH1usMm7U6z0ZnBkkleqzcUlPyV0d8vjUE6n50Q35PXGKqKapfAK4iJPw879ySa2lMPIP8OUD7AtgsDf2XrBYe8NIBNunFjs6aXniFSd8s3ZLYtJOm2SVtBoZG7Y3DF+yoAJubwxr4pwXUCLCZnhHaqm5Gnsr0mIoZtc6OGdu2NUnY7NjYaWs66W+wWrLuVklKR6QmHBbo9Ifhco5JwdWgah5PVQNsNEQrPjronNzlolCgUKBQYEkoUBtPU3VxzccaP5YxHkmbM2fP2i9/8Qt76aWXDLFyxLq/+MUv2h/8wR/Y9u3bnWGllGC0ws3DUi2136hvoTTRhojPy4xSPE11FGa0deZMu2muSUf5qXODduTEJRd/RvpyRmfXsJtmmoMFK0zKtZNGFrCmR8zZ4NUL9utf/NgOv/FL6+87I321Fdu4occ+99TT9l//8i8dsGltl1g0Jw/OGpiz2YvME+ERGW6WpHgLBQoFCgUKBQoFCgUKBT6mFIAHC94MP/wjC6gfTCUaS26YKuhRnUo7JFKVaPFoTxSzcQ9xvhUfh5KzgYe1AUExDtgM3hiRWrSrdvT4Weu7OmxjtkY8Y2ua66PyTGU3oDJdgE1by7R2Wbfb/t0bpDJ9s/wd1qrmxM7qtPageb0WDt2IEYQXpKliNvWTlvRqm3Y8Vfn5mFCAdxkT73Z9s+vf+4i/W/o8z3xlv586o77fxZ2vHsLyhX6uI10AGtRFmkiXAxGEce9hI22UEeUxpwSMQcol1JixMTAkYEKNWX9/v6svA5yJeMAbwJg4myZAHlyAGdy8fXk7czrRRsAZQBlAFcAVABks4AsqzABgAGQAZrDd3d22ctVKW75suacjHxI3lBHgDi7lAvLgBt3inQga4EZYuBHm1xo+GNXyONof13Ev9dcRjkt5mLul8QQf4U/c8702YfaeWPfVuJvyc48aa/1bwNb49N6y1ot0DYD97dFJ67ty206cuSZ7xQZvjGvcF1Df3KFnpPVhvbvkNGlfmqmMWUd7o23Z2GWPHFgnCcs10sCh5ynwLFZ+qapaHYSWP9E63QcPbza27tZIF2lzf/pWZKXW5VucS+gXlvckLP3mrNZ8XnzxRVu/fr298MIL72pAgzLGndibb77pGVC/QoZ46XAjWR72rtJKwKJSgGcQg+NSSNjwktfeDt2aFtPoYOCnvDQwZZxPACgzdGtchwv2u4RN/+CIdNcKqBFgM1UFbBB/np4e947Z3jqlRbk1tn/PZtu3baWtX62zbvSOkSYBNgyb1Y7lDfBhlAbI1rqeX/Azp59GhwyXNpO/ADaz9CqeQoFCgY+MAvEtzRsQYbhYmGN2LTHOI2nzs5/9zA9t3LVrlz399NP2mc98xnbu3Om7imBSMTDPMKvY3MQ3m7Copz6e8EgXbaj/7s+Jpyws47p+Riam7OLlmxr/++2MzjEbGBqziSkpuxBg0yARadqG6HODdtLAoK3oaLIuTcwvnj1mP/3xd+2s1KHduXHVAZsd27bYs1981v5c57E9/vhj1twqJhxdaozjc8b6OXeRxUWicPN0xV8oUChQKFAoUChQKFAo8PGkQD2/xmLPhwHYpLly8E1yff6d5tL8unHwhjTMq5MhGX4OoZ4WaBMLdxNTFbtxe9zOXdAu66Mn7dylWzYys0Z8IYt2sZtdawquEmfSJW1WdJjt2bHeHpYGjvVrl1mHRGzY9plqZGlQc/o6wIZWINmTABst8S3MKJK0mPuUAjE/Wej5Rfy7ms+jn31B9S5Wn/+C6ZW8Pk1cUzZvcF4eYfUmT18fx3XUHeni2l/krK05wEGaSEe+yEuaSBdlcx1pcVkbRA0Z4wBzR4AWQBrUlOEPACZc0uAP98aNGz7H5PwZygigpl5aJu412hbXuITFeTKoJAubgzOEIQ0DSIOKMq6RkMmlZgBtsKQhHjAm6sMNUCavO+qvD+N6lk56qDFqRRjPI8IoO+qZjVf+CJuv7E9GGP2B9431VF7eKp3odzIJsMEFUpfmJa0N37wzLqB+0NcEzkst5q0RSV01Mu63K4dGdKnDTKknfV141Yo2aV7aYA/u7bbtm5bp+AzN+3nWPBP/o9aq0fs+u0ZMqNIkE24kxKW90eFyfwFscioV/4dMgRiUGagXF7DhpY6PQdYBnFGrStgoSUUMkos+C5S5NTxlZy/esBOnr9m5vuvqnGKrpBJHxwmqM8FA0d+lFk3n2LQ0Tdnqle06YLrHHtrXY9s2Lk9IqtI4aKPa046aaidzh3aktqTQbIFxDp2JTSkYDPBzVQCbOUQqF4UChQL3IQWCSQwXJvqtt96y7373u/bDH/7QGWpUon31q1+1z33uc7Z9x3Zb1rnMmXWY8xAlj/w5A5rfLvHYAHfw1zOlhMUkAVCI+EgXIysugM0d7aY5pV2Ubx+9ZJcHRvQ90Jk8k+yyTAcQziAiPyMJGwOw0U5K6SzvaK3YmROH7Rev/NCu9J22sdEb1t7cqENnD9jzX33evvb883bgwANSqyl2TR+GuJdZ3qx6Q7SJEb4Wnr4T8b3I77v4CwUKBQoFCgUKBQoFCgU+rhQIPoz2B5+2OICN1yBWKhaS5fo6gHNcimT/c9WID4MVE16jObd4PzFkU7q4PTJpFy7qHIO3j9qpC0N2Z3q11uuWW7N2xrP+xyJgozR3NOtcw+aGKWuXlM2Ord0643ar9a5fbivadY6Nl6g6q5tGvSJm9qoj+L4C2MSD+P12E7/Po59982b7AHdemyfU4oMikTf6T17Ge6UJgCTKj7oiX7hRB9fz1UPYfOHz5Y8ycKk/rG+A03XEM/dDSgYJmQsXLrgabaRkUFnGHBIQZmhoyG0AOkjasDEw5nhemH6ifVF2hAetcAMwyWlBOuafqChDhVlPT49t2LDBNm7c6NIyADDE5RZABjAn5pe4lEH5MTeljqA9bcNE/X5xDz+Rj6RxH/gjPMLCJS4MaeYLj/hPhsugngE2eiY5TXgqDtb4mnCDTeoMs8FbIzq77LI28l+ROjSdYTSp59rSqVFbatJZDtB6ACrSmzTmI125du0Ke2DPFtu3Q5v4u6Qer1Hjviz18IcBpMGXrqk1vQ/EEZpMuNVLTxPp8jwFsAkKFXcRKMDAsTQSNrzUSNHI9V0r1Q7gjFrCUBk3YcxcykaAzZ2xil2+esdOSF/hyTP9kriRTkyJv7HDWl1S5fGRUi+1SXXQSeuU+Num9Svt0QMbbM+2NRJvVGfmQ6B0gDbqpuqodE7aUCWm4lN7dJ31yYjGraGu+NOyYkr80XTOasuLUyhQKPAJp0Awh0GGnOGJsHBJC5MKU40OYQD6X0g92ssvv+zXDzzwgD3zzDN+ps3mzZtdxJz0wejih6kPxj6YTuLnY4qJj/Y5g8RYWzWRl/BIM4Nf8YywOlfWbgxPSEf5FXv72EUxZ9rRNc6hsy0O6POtUEYxZTBm41KJJtC+YULumJ069qa9+vMf29XLZ21y/I6tXN5pX9A5Pf/lv/y5Pf3UU9a7eROZNYTzPQiroMxEmxjnlUQm2h5ulrh4CwUKBQoFCgUKBQoFCgU+phQInozm44ff+yCAjUoQl8V8O/0lsoh/gndz/ss98qV0EU/qBNrQBqWBj1RYWGb8o+MVu3Rl0N5466idOKfNnFOrVEqn+FA28zAvl2G3dZU/bGuesi2bVtvBfb22bfMa69LO62atBbCKwMHUznUmprLKF1KAjNYqAIq8PYkR9ODy8/GkQI2vT+2P+VJ9eH538Q4TlqeLvHnaSEO6heJjzhH55itzNozuQh+QWai8iJvN46lTW+nDYYhn7sZ6H5Z5IJIySNAgFRPgSy4VQxigTUjNkJY82JC0wQ9IE+uIUefd7pP5ImrHkJwJiZmQkgF8QY0ZAE0AMkjKEI/0TEjQoPYsVJnhotosLGUHOBPz15wOQctZmonOPkrV9fHZ+Grmd+WLQu/iRh6S5OXl4XfJ/nscxXsdm/jx6wlU6c9Vslpx1Rgs7eg6EsPs6vVhe+Ptc1KRfkWgvc5M0vk1DU2SrpHGDR++hdpw0llT46Qx5vdu6rKD+7fajt5l1rVcW+wF1rhCNK8nfWmoKK08yOXT4zXjhvFAXdAZq2Gz6VIraxEfzZow71VYaBiWvlhUosVz/D1wecgx0C6+hA2ADR8QXnzeeCy9BVaMjqsrWeAbVKKN6ICpQR0wdercNXXQPj9samqGQ6d1yJR2WfMh9e00AmyEv1pzU0VnGHTYYwc32AO7e6S/sFVoa9qt06SXWJuqnfXyHTVURP30UPfSlpohCItJnZk2Ymh/NW8qTddz85KqmEKBQoFCgcWmQM4ARl3B9MR1uKQNhj0m4++88469+G8v2s9/9nMHXZ588kn74z/+Y0PiBgY5mHpE2YNRz5lz6mL3EsxzMNS47HCCaQ7GIXejPdHOuAcAm+rUWXrKzQZvjkv0+YrUXlx0dWjjUodmOlxwSjttUJPhU3gkLCVd0yywptnGbWL0up048ob95lc/swGdZTMzNWbrerrtj7/2vP3VN75hhw4ddP3FFRfDZvgP5iZaJVfDu39bZoNqjGQZ62eJUjyFAoUChQKFAoUChQK/BxSAD8t5suARf9czbAKIYVmMv2Tk+sRaP75Zk8sMsKmuC6T04vPYNi1Du1i4IyvrA754d02S4oeP2bEzgzY0utwmZzh4OklfJz5N3KRU5loFDRwTtqFnme3bvVGq0db6QdTSiuZrAk0CbJzz9DUB2qXao7leZwFs/CF8TH5iPhHvct7siMvDIh1xeTzh9XEer3fD389wozBdA66QhvkFaTx/NTzKCzeviyLq64pi0zpXugJ8IB15Izwvj/kd8zMsczcswEqANMzhmMsBvITaMsAYQJqBgQG3bOZDqgYXaRryY/y+qpv2ou2MEeEnDW2hjcwJc/CE67A5UBNqzEJ1GWAN585w1gySNFhAm5CYCamZoEHUHzQIcCauoVGja+PRQ5Bxunk/T20lnYczsjC4yEQY/vze7hYXeerT30se0nwyTRrNnfBOez2L6vNgfk9QWhlu1NhuNjYhkP7qbfvN66fs+OmrfqZ5paFVi7rpTFtl1idFY77Os2U9YLnONd++da20a2yzzes6bHkbkjds268C+p4+UT7eAq7Y1D+vqX6bPC5lkDfWgyNPAWzmpV0J/HAowACzlIANL7hDH/nL7yxY6qDclbNP0lc4NtVgN+5M2unzA1q0O2/9A6M2Pt0qfbYJsKFzp4+WVvfElDU2TlvX6jZ7eH+PHdi11latlM5CkHZ1LrqRi0AzJngL6GB+kVz9+rV+iQlLv0w2QsJVRAFsIEIxhQKFAh8hBd6LSYymkS7AFhhf/DDpR48etV/96ld24sQJ27p1q33pS18yzrUhPbszsIjEw8Cz45JJAHlhlmFUYcxhuDnMEVH1HTt22Pbt2/3MumC0SRfMdrSHsJzRdbBGYWwoGxVYf21IEjandYbZyUt27fqIQByN4jrDrFKRagy1zafvOsPMpketrVW7KadHbOjaBTt+5Ld25K3f2NDAFReP3rljm86u+TP7y//6F7Zt6zbtypKEJqLYaki0QdUmo8Dal2i2pbPtjG9ExBS3UKBQoFCgUKBQoFCgUODjTAH4vTn8mPi7DyJhI2VL4qXSwjW7mpMRowXjFUZgyRzApjrb9kU1eDEBNolPQxWuODPCtAA7Nd1g14du2bETp+3o6QHru96s9YJWrQE0uzpdpGKaXOWtuEqdb9hkY9a9utV2b++Ripz1tmndKoE42myElA0bRinY2yqX62AIC2ATT+pj4+bzodnnWG09cREfcbgRRrLwzxefx9Xni7ykwUb+uCY+5kBRTrjEkT7KzMPxM9fCBhgSYbhRD+t4gDFIxAT4gptLyQDAcA1Agz824pE3NvOF5AzzPMKibdEm2oA/rj1B9Yf7y+eDAC9IygDCIDWDujJAGuaLgDNx5gwb/gB0yI+LxExIz3AdQAxu0CmvN8LCnS8ub3N9uoXi5rtH8t7NzJenPj1p6ttQn+b3/zqNubUPAnRl/q93y78KsVLbKHBmRirRJ+1835BL2Jy9cF1yNG02LcAGOUmAet4NnXCrb8a4AzarVzTo/JoeOyg1mBu7O2xZSxW8y79FEJlmyPh6gl9UAwhz8Ca1ixRuWLuuelPmAG2ILYBNIlL5XRQKMHAsDWDDS83gj8uAH52A26KDRCdJLB6HTE1ON9rtsWkdLjhob75z1vr672ghD4asTamFrFbPsbHKlD5m4+KzpmzVimbbv3ulPSDApmftGlve2Z4AGxXvqtGUk/E2DQWq1jtetOXdXXb+GNpai6kWorBiCgUKBQoFlp4C8zGJ9YxljPW4wXSz4wqm/ty5c25hkmGyYdbPnz9vx44dmwVrYPLZmQUTjw3ABoYaYAZGnN1Q7IrasWOHS+ns3r3bNm3a5GLswZTTLtoQ7aAtMGluFceJZsOow+wftRNnrkrCUuowJW0zLgCf82sqkr5Mnwx9TwDqZ8asU7tnJkeHdH7Nm3ZMgM35M8dt+OaQrV61wh595FP2v/3Zn9rXJGWzfv06a2xKE470lJgkaQRnONewHl+h/AnW0zGPe28/BRdTKFAoUChQKFAoUChQKHB/UgB+LHgd/PB3HwVgA1gjrsyJlPjExJ/5spjzaJxj02C37ozoHJtLdvTUgJ24qPM2hpVFO6jNF+/SgeK+4KYd100653Dlsgbb2rvSDuyVWrTebp1t2CDQJgE2jbrftDYBDdIagQoTQ8iaAItwaaHYG1V+7msK8M6EifeZ6zw84nFjPpKH5WnzMkgTcfPli/hIk18TlpeFP2ykxw3LHAs//ZANcgApuMzNAFpCQoY+CviCxY/ETFzjxzJvIw82/JQTa3/UE22L+pmXhQ3QJAdTAGZCmwLzvxyECTAGkAY/c8NIyzwwLPkoh3KpP29H3h7oGLSKcMLqTcRRTpjIF/cV1+GSLo/jer5yCA8T8XFd7+b118fF9XuVEel+f93amJvukTE/bdhMkA3zfa0aawwek4jNtcE7dkrrAcdO9dvla8MKb9XxGFoHrqrB5PgLABvWBFqbJwXQN9suAPq9m239Gr1z0sCUpK3mAVXUlPTF8a+MN2eupE1q26xmqJRY6ervYZ6yvbTF/Yn3Fzfea1zGjrPacPviiy/6BtoXXnjhXQ3RZoVab3nzzTc9w+OPP+4Z4iXFjWR52LtKKwGLSgGeQQzai6sSjY4gSRh/wauAjTNDaWBN77+6KYgmL5oYpCntpB6TRPP5S4P2+ttnhK7qgzRGh25XOukuRLxNoA1q0aYnx+RMCqBptJ1bWm3fri7b0rvBVq/UQYRKQrEANon94qVOZHU3wCPeyWoLU6tSJ05dNY/Bn0Jrbiqv/BYKFAoUCnyUFIjvKm2Ibyt+wiOOcGxMBmDeYew5SPLUqVP285//3M+34ZvAjqwcpKGs3OR1wOAD+qxfv94eeughe+aZZ+yJJ54wzsQBzAnQhnopkx1VWEZUZ87UJmD9W3em7NzFOzq77JrG/wHtrpmySQE2fBNmJH3p466Aeld30SjxZ6mxvTV0yX7z6o8F2CBd029T42O2dUuv/acvP+sq0Z56+rPa5bWqClbld7CwP7+39O2KL8PCeWox1Y9MlRWshRdfoUChQKFAoUChQKFAocD9QQF4w+B38MOjLTVgk4CaKn8H36R25Mb5RM3XteHaRsbGbWDwhqtEe+O4DkeXBg4W91pbdeB4ixbymO1zT+IomxunrKN10tavbfMzDXZtXW/L2iQdLg1qSNjE2oCrS2edgAUDr58Y2lMAm/w53M/+mOPQxvx9vlubIx1pIn+4hBGPJYx+EWFcYyOe8AiLdJ64Gp5vdGPew3wJwxpc2JgbAc6EZX4GQMOmOeZog4ODvtHuypUrvqGOzXWoMaO/kifakN8X9eRtjblXgDKRlmvUlgGk4Iafa+ZvAC+xOa+np8ewSNGgXaG3t9f9gDdNzemA97hH705Sn8D9BW2CbqQJf7QTlzBfvK+qmpsvTdxruOQLQ/r5TF7fQmnIR5kLmbvlWyhPCc8p4KM5RK4G6ln5807rsGjaYJyfUncbkY70i32DOhrjkl3ou2nXb0v1X0Wq0KRxo6FZrhtkbVg5HtdYP2XrulsF2Kyzvbs22dqVOuOoEcAmrRmnZ/fud6Nas79zyU/BpJONdWL8c7IyHmCr6WZdBS2RiXcfl3sLSz87WwCbJXoKS1AND3hpAZv0YrPolhbe0k2m95+OC2hD34ApU9eT6PPF/hv25uFzdvbiDbt5W+cwSBQOCZuKdtkA2NCxKtJVC2DT3lqRuHNFemrX2G6pwlnbtUpoa2IDEXtGx6EzZ6qDsRwbHZE6MUolmwwhYYlJHRM3QsNN6ctvoUChQKHAR0GB+ZjLnKnM43M/4z+TgJMnT9ovf/lL+9nPfub+mAAwWSB95IHZxV/PdHPPhDMRgLEHoEG65tChQ/bZz37WHnnkEdu2bZsz++yowkT7cAFqpuTyhbg+NGpHTwzayXMDmohrh5gOmZ2SZA3SNYz7FTg5jfdS7GYtOmCwvXnCrl4+bT9/5bt2+vibNj5628f5Bw8ctBde+HP7T1/5ku3Zs0uTjY7ZOtMYntpMW+pNtK0WzriPvVfDtwETbroqv4UChQKFAoUChQKFAoUC9wsF4N2C5wn+7oMANuIQxS0lfuluKtHgqVI60sIrpUVs5uVpSq75uNqG8XMO5YX9m5icsjvDo3b87HV79a2r4hO1aZO8nGkgzhA/99PIGTWSwm5pHNV6QKs9dGC77d250VZ0SH2T2FAtD7gWjsR5ukJ25Yu2sBbAGkMCbbwR5ef3kgL+DqbXbMH7432KdzES1fcbrvMw5kkhLRPgBdfMuwBh2BCHumk0HSAZw8Y5wul7XOPHhmQMWhGYk2Hx53Gxlhdtw40+jRv+iGeuBhADAAPogkUqBsBl2bJlbnPJGdIRjmRMSMmQH0sYYA5+QJ6oy126tWhbT7toR7Rtofj68Pw66omyiMNGmRGeu/PlIb4+T15Pnj/S4t5Lmvq85doppx9m+zyvmCOntVe6ITEA8JNaGLg9PGFnzl+xd45e0Dk2I3ZntNHXga1RG/dZS1D2tL475Ro3VnTOSIpyle3Zuc62bem21Z0CHlXirISNf1iiTlVUNan29N1KfiJIJ5u1MVVYzVS9h3RVTet5In7x3Xjn8/eed7kANotP+yWtgQccg/zSSNgg5slLLYYqdkrr5U78UbXz6oIkADYT8vRLFO7I8UtSjaNdBdcnhbjqHJuZFn0E+QDBTMnMqFztuG7Wbuu1q8Zs1zapRdu/RwcNdllbC6Kd1Q4tnbik51wb+qz3Wz4khFY7cQFsIGgxhQKFAh8nCszHOOaMacQThh/LxIFJAWDNyy+/bD/84Q/tN7+RhEpVqibS5XRYCLCJtFEnLow70jUANl/4whdc2mbLli2uu5jJQqSlfACbSeVhZ02/Dhd8/e0+O3NB+pZvT0o/OfFSd9EglZioRtO2mxmB9M0NADY6x2bqjvWde8d+8cr/svNnj2h6PW0dbe32mU9/xv77f//f7fOff9rW9qyxFu36ghlM3w2+BMnQ9tzk7aqFk2ZuulrcfL4oP9z50pSwQoFCgUKBQoFCgUKBQoGPjgLwQMH34Gex58MAbDRL9790Z+KFchZKYEoKED9a9TlgU02Dmhtxa86rkp9gcgCrTKt9YxNTdur8DfvF61fs8tVhnW1DHGrT4fMAa6hbvplR8YQjOuO22R7UmQb7tOt69YpW62zVAemqwtcDOLBafCPAzVzAhjaIV9UfbSnm95MCMX/h7ugH2NiUloflc4XIgxvSM+GSl7W1AFZCpRkgC+ALfSvAmqtXrzpgw/kzocaMeCyADKrMyB9rddGG6K+0j7BoNy7zNOZYbI7DAqTE2TA5wAIIwxkzqLJGMwIb7QKwCWmaOHsmwBrKo/yoL+r37pH3b0WQJuiEHxNhflH9mS8s4uN+8+soizD8cR11RXjkqXfz+oJ29Xnq683LyPPn4fX+aFd9eLmO0TxRQq9vGt/lYexmDdjlZaQO7cYtqUc/fUmAjSTJBidsfBJ1aFKrATivd5z3jlPOGnUsBirS16xssn271zlgs6FHZya1aZxPXwRVxmIwdaZ3MdWefhn7aUW1BdUopfM160iPKxuXSp1ajluNq0VWy1hcJ975eI9557AFsFlcui956Tzg+AgsPmAjKRjvEOowDtZUd614Z0gvPSLJFQAbUQJ0dULprt8aF7o6qA57zfqu3JaatCYxawmwoSPBlDUqT2VanbVhXDtnbtmOLcvt0MH91rtxncTj0gGELlmjj+i7JGxUlw8Qs50tXcfDcIZPF6l/VhnM6MCqO2IifXELBQoFCgWWggJ3YygXqj8YTfIyITh9+rQDNf/0T/9kb731lk8Ootx8wkIYNhjQ8Ed5eX0RB1MPaMO5OE8++aQ999xz9pnPfMYAbVrbWq1Z+medsVC5IWEzqSH24qUh+9VrUoN56ZbGew4d1ARd4syNDRr3JWEzPSGA3gGbCWusjNjI7X47d+oNe+PXP5GkzRlra23W+TWrHST6v/7P/8Mef/xRa2sXc8duSzU0RKOjzbQ3TNxfXNfc9I2qXb+XL30xyvfhvehU4gsFCgUKBQoFCgUKBT4qCgTPRv344f3gD48cOWLf/OY37e/+7u9c7RKLwKg++vrXv27f+MY3JLm8xxd469vNXD8tfAF1ZLxQjdVyfozZNjYFp9k27Bh5EmCT2uPtqqZLcjDa4CNRm3N9t+3VN6/KvWHDo1pjEI/Ixh6XxFZBLMKhEq2pYdRWLZ/RIt56l7DZ0K2FPO28ng+wYQdpanN19i8AyFvk8/36Oy3X9xMFgpev5+P9Dcvevfp43nfyBhARfSDKIxyTh4ekDGtoAcjgAtIAsrAZDhAGVWaoMQOYQY3Z5cuXXaqGeNKG6jPKocyogzaFzWkcbeIecksb6Z9YwJmQegkpGcAYVJgBzADQYLnmnBnSkof5Wti4Z+qgzKBN0CHC4jp3o10RRpsjLO4vwvJySY+Je8RPvrjGn5sIj7CIJ7w+zq+Vnb4dbYl887n1+UkT5eOfL57wezF5OfeS/vcvDZ2R9dRk2KjJObYzqL9TUIVnrg354xMVu3b9jh0/1WdHjl3Upn31tUq7hvhOLSO3VLGUdBZZs+b3TVKJ1tPVZg8f3Gy7t6/VmN9iHfokANikJ88KMGbue6SnqZB5AJtY6/X05Kna2ey0lxJr34x3l019i2d4D8PGe43L2HG2qERbPMIvdck85KUFbFiWg/nBAtxUX361I43DADaJ0QNamRRgc3u0Yn39t6Ui54qdOntNTJleREnYzMxIFM7zp+4BYGMzI7a87YZtF2Dz4IMHbPOmddpF06zDBcX80aHmU4mmlkR38zZVr6PTVVk2hRJSABsnRPkpFCgUuC8o8H6YxkiLy06vM2fO2EsvvWTf/e537bXXXvMJOXHBTEb6+W40T5fHR54oAxfx+a1btzpo85WvfMUlbZgsMEkgnjx8GcY1EI+KQTsrgP7Xvz2ddk3qcMFJgTRTDvKL7ZKbJGwmxIRNCoO5Y4NXz9qpY6/Z8cO/sqHBPlu+rF1j/2b70rNfsv/23/7KHjx0QOfR8q1JgI2ffeaMV97y5I92vzsmfSXeHb5QSHB04S6UroQXChQKFAoUChQKFAoUCnw0FMj5Ofws9iwlYMNds+SFZaZd+03zc369XbPpEmDTd23M3jh6w05LA8f1G3fSOYcCbBoAbZxlE2DTJMBGEjadbeO2af1yLeStlxaODda9apkktNnAoxozCRsAm9SGansKYCN63P/GF055dFUTvDzhLATnJuJi/sG7Vd8HIox8ACk5CBPSL0jAIBWD1Ez4uY74UGUWUjWANFjSAO7Qz6gHE65f1P1EewFJmE8h9QLQEtIwoaKMORVSM2GJxx/gDVIz5MVG3lBjRtnUEzZvU4RFG3EDsOEeMNFGv8iuIw/x8YwirD4t11HO3dJEPtw83UJ583SkiXR5OfX+vFzi6vPUx9fnX+i6vpyF0v1+h/POzwVsoCfdtKIfn6kLgBzV+TWXrgzZsZMX7fjpfhu6pTXritYNmjs0cAPYsH4wrTF8xlVctjVPSbNShz18aLM27nfbsnadVabza6oyl6qz9o3J6euhWntOXxvalvpkcvgaAdimr5K7eN346CIfX64svhq7FA50CxvvNm4BbJaC+ktYBw95aQEbOgTixbz8WQeY7RuKb9AOA8UinCyZGanDabCBoXE7LHT1nWOX9SGcdpVo6C+cqQj40T1gpwXYzGi3dWfLDdu2eZk9JMBma+96W94BYCO1aOrQ3mnFmHnnVPUxcFJfakJqU/IrUMa7oAekVC4uHUCTOkVK4UnLT6FAoUChwH1HAcZ5THzUmXygN5kza/7+7//eQRt2gbHTK0zk8WsNczAjs+Olyot4RkBFRDaNo1XGhTBPp5FeM2J2be3cudO+9vzz9kd/9Ef20MMPuyh+yq6xXulHJV5z486EnT5zTSrRztq1wTGrNLb7+TVTAm1mZP27wSRnasLPr0HVxXUBNudOvWXnzxy2W0NXbFlnq+3YvtO+9MVn7c/+7E/9/JrULtGBIjR+Q5JgbmYbT1SVVnkY/nSLiY71cfNfUxG2mEKBQoFCgUKBQoFCgUKB+5MC8D05f/fBARs4Luba6W+WF8pZKAdGCEiBcI6x7DWbXrHOsqksWLPakpp4RknYXB2asqNnRuzEmX670j+oDT+qt6FVh55LbY7P07VY14CqnGHxiyO2clmD7dzWox3Yu2xjz2qTAg7FU4dKdtBGbWbhztWtp/YkjSDV+ygs3Uf2Aue8Oe9qfk2juI6weJcjXYSHm98EYcyJeOdxscyFQuKFawAXzvpEQgbL/AmVZkjQYAOICT+ADdIzlEn50Z5oJ26E4WKRNMEGEBJ+rrGoIgN44ZwZpGSQdEN7ASrNAoABhOE6wBzAGsAcyqIdtCevO+gRddKOSBftJm+0lbz14VFG3IdXsMAPaSN/lBlhZKkvIy97gSLnBEf6KGtOpC6i7jx8oTwRHu3M84R/vvIirrh3owBjfozmSeoFWgLYTOtnWn5rbNL5NeN27sJVATaXpGlpwG6PoPC8U4O2pGxQiabxm7VfnW4rNeiNtlwATe/G5fbQwV7bsmmVgzjSy1EFbBi8feu+3NpAji+pwWTsp12YcPGmfLN51Edqhu8caePLxRry0hroFjb6Dy59vUjYLO2zWNTaeMhLA9johaYDZp1gptoxUrcIpo4UpIV9kuobeSe0SHdTnfaYROLeOdYnkbhRqcnRTuuGNiGtSfSZc3H8Q1QZt47mYetd32EHH9hlO7b2WNfKVnXaBhd9hjFrhClTW8SP0UurfTC6HE2sMmlE+19qN6HeOu+rxHgBnko/xRQKFAoUCtzXFIiPOtI1b7zxhv3Lv/yLff/737djx4755IKJSaSZZVI13jE6MuIl4wEEKsR/FJzGT5IwSsbI6YOrxlOGWQ78W7u2yz71qYftq199zp7/2vO2Zes28WSC0ZVgShlvj0v8WYfHnjhz1d453qfdNJKgaVomJk4sl8rxb4j01DbMjEsVmsb6lmnpIm+yNSt0yKXA+qtXLtjlvrM2cueWda1ZbZ8SKPT0009pYrPR21jTTc7kRa2jYe/LpLu7tyzvt+x7K7WkKhQoFCgUKBQoFCgUKBT4sCgA3xf8EH7m0x9EwoZ2Jb4R3914obk8Vc5pppj8l7KqbKCCK5rL37gzbWf6hu3IyT4t7PXbyITWDUyLeU0661ZTfdLA9zXOTIpnHLU2Sdts27zGnnhkj23b1CWV6dpMpKk8y3gATIA2QQs/6xAe0ZvAPcje7Va8dR/dT7R7KVtAnfUm3qP68Hu9pswoN8rimncSS1geTlwAEsxhSENYno668zKmlG5aKshY/4pzZpCQuXVbasxu3HQVZoAyWIAawJmQmmH+lKsyC3CHsrD5eTP5PeftwR+ADJvZUEcWZ8sAumABXQBfsGvXrrXu7m73A8LEeTKAMfn5NLlaszi/BjAmDDSYz+RtIz5PR9xChrhIe7d0eX7S16d9v2Xk5RX//UGB+Z5r3rJ4ximMdwDAhneN96sK2sgHWDPlttEGbozY8ZOXdSRGvzRu3LLhMb1vAuQbNb43CNBxI8BGitMlQWm2dk2npCe77YG9Uvm3bqXWfbVRX2Wl0mvjxrsGcn/F1TeUNkzttc/f/9wfKSPPfHGRZvFc6BqWfhWWcbAANotH9yUvmYe8NICNbi3eab9LLQy66rMI1kvmnBAvvCz9Ri6ADQt5w2PadS2U9ciJS3bx8k2hrKhO69QOmyZZcVs638B1H06Pa8fMiK3rarU9u3r90KnNEoHuaNHuBJXsgE21Ib54V62O2tIf7aR+wJhkSBdgDSEJaKK1DBazBaTE5bdQoFCgUOA+pgCTCXQp//CHP3S95K+//rpPTvgO5B/9uAUfFwHX9e9jtDNX8unbEYANaTA+dLqPcVG7shgfFQjjw+RkWWeHbdq0wb705S/aX/zlX9ihQw9apyYf7KTR2YI2NDypSfcNO3FKOskv3bA7Iwps7HTAJu2WlOiz1KBxsCCT7+XtM1JxscJ2bZcu5jXLbGzkpg1cvWK3bgxpV1mr6tpoO3fs0MRnhdqQ2ki7yrjtD6n8FAoUChQKFAoUChQKfMIpAO8Xi6j4Wez5oIDNByXp7Jx8voLEznHmwfBYxS70D0sDx3k7eeaS3RlnB3a71gcE2MChOp+qAiqSHp8a0zrAhG3esNIee3in7dzKGQfN1ioNHE3BH1Y3dMK0poUv5XWeV25tBW++Fn3kYfkzjMY4b15lfeP5Rtzv4lJHmCgvDyMuwiPdvbj1ZdRfUwZhvJduNKdgfhHX1Ik/ABtcgBTUmIW6MoAWwpgDEY4NNWa44UdaBoAGoGZgYGAWrAGkiXkSbcjvk7bl1xEPaAIgA7CCOjIkZHKQJcIBYABiAqzBH2rPAGwAaziDhjAAGsqlvrAhBRPX9W1xmpWfQoFFpMB8fSCvbm6fZhwBfCUF4zSQSlppRccS6wE6qlYgzW17653zduqc+uHNcRtDglKSNUhQumpz9XkgetSjL5eWtM0b1ti+XZr3b+uy7tUdDtQ0qixKru8T9de0JG/jfPGkud8MbQ5Lm8MyHhbA5n57Wh+gPTzkJQNsaKd/61NHrQE2fHYxuPJVmSPvhkrK2QZjE9JjKHT1mFDWE2eu2PWbE0rWaRNTjQJs2KnQoo+1uj8MWcMdW7VsRmcYdNvBfZt0wOB6W6HFO9Sh0XFd/Jlq2E3jndir9qYl5obWaFHP28pHOYaRCEhRBbARmYopFCgU+FhRAJH9w4cP27/+67/at771Lbt48aJPcmIiwsce5h/jkx/GSYZEDX8NPjZzoZGaCYoHEuUjp6dLoyQDbEqHS5mtLZoY6zyxjs52+/QTjwuwecGe/tznbWPvFmtsbjLxYXbt5qi9/Y501QqwuTVcsfHJFgH2kq6RJCW7cBiLBe2o7gTYrF7epPPKumz/7k0+3re3qE7ttmHnHGlbqpOlJk3IE2BD6wpgIyIUUyhQKFAoUChQKFAoUCjgCz7waRjWBVjsud8Am9S6rI1qJyrQLkt17ttHz9qxE312e0xrAtYmrhVFOOL1lJzbqkzqvJAJqUVrmLQNa5fbwf29WhtYZ+u7l1trs1RR6b7FJiaeFr7WDTwnHv3Az7o/xdyPvzy3eIa0L61npOfJNXF5PGELGcqqN+TlvaBcJObvtay8nLzcyE9YfXjURX2YkBIhnbchu1fCmKuEJR5wBTXPqC87f/689fX1+ca0HJQBkMEyJwLUYQ5EGdGeqMvnQdV2EIahfWG5Jpx5UwAnhNFmABokZXp6emzjxo2uygxpGUAYwBmAmZCkQdUZ10jJkDfKxw/oA0gTczPK532M5xBtJZj0c9IRWEyhwCJTgHeQd/ZuJvoPo1MNsNG5tBpfmeMzzNLjdZStNCnNaIP+kL32+ik7d3HIRnTIrQM2Op+suVXiNIzWFYE1WthtbZzWum+T7d6xUdI1vRrjO2xFh/qB1gF8xj9P2+Zra619qY/f7V7ulzjaHDbGDFzGwQLY3C9P6UNoBw956QAbddDgAVy6xtmJ6l3o4+c+fmXpuPIB1tB5x6emJRo3Zid1sOA7J85b/7VhLeK1C4Ftlkt3RDWaslV0EHXDsM6xmbDuNR12YP8We2j/VqnMaROjpg8ZjAZMmfx6nZVPhiplqM9bJKQ3taYaodDUGqVQ568Z6vWhoBZUfIUChQKFAvchBRi5KpqMMHn53ve+Z9/5znf8DBt2kvEdYFKC4UMfH393GRkZCn3ow6PR0f2Mi5jqOK6L2SQekwJQHcnkocmtxmBJO+7avcuee+4P7Wt/8if2yKOPOfM1OmV2ZXDE3jp81sWfR7RTcnJa55Rp8j09I8CGcjRmz1TGXP1ZW9OkDo1ts+1bu3WI7Abr1c6azjbpelbjGK+9pTo7x13CZsfuGLOJKaZQoFCgUKBQoFCgUKBQ4JNLAXi9WMDCz2LP/QTYJG6txrN5G9VOnUltA7fG7cjxi7KXpEZ3Sufeal2ggQOppQVDfCfz/enJcZseH9E6wJStFd+4a3u3dmJv0Dm33eIbtRAu/lDsYprRKwN5qCNxtfCRCtB/9ee+fFFSe+c2jbB4rt76jL+P6zyeMPKE5RpDmvp0cR31zl4zE/DJQMp7t9/ZPE7rlDLK4yrawfvIWhUWCRkAGSxACxIxgDMhFcN1nCUTZ8vk6swif6g2C8kZ6oj6aFe0LdoRYTmAApCCdAySLwAugC8hGcM1wAzXAc4AyCBpg2RNqC/DH1I3lJeDPrSH67DRvkSp9Ju3k5BoZ56m+AsF7i8KMEBU12A16vpIW1331ZD+/7N35s95HVXeP9olb/Kqxfsax4lJQphMJu8wBIZAAmSZLIRQLMO888tbNX8RBcXOzxQpCqiiKAoIJGGS2MSxnXiVYluWZFu2ZO3So+f9fk4/R7p+rCQmOPZj3C310327+/Y999zbfU/3t89pgTVzNjw6pX1rztlf3+q1voHLNqOFmzNaoD+nfr2hQfvX4KQRyf41LTKPvnZls4M1e3Zt0FYYLW7ukj6doX+6Qjrl/X6r29L7la2VvOgTCKPtE2bAplae0HWigwd8YwEbtRwXepjkSxN9fuiJFaHI741Jw9ScmUac0SZUl8ZmrefMJXvr8Ek71XdR2jXN2sOGvQ0AazSVx4bU2t+gsU5CmcyiLRO6eqca7n17t1vn6qXeeH3aT8X0/ZMD4VGQCPAoqQ7xutjmR8qm0ALd6STyMmCTOJR/MwcyB2qdA/T1mANAu+anP/2pgza9vb0+4IH2+OgX46RVuuaFbo8lMEpM/aKX5sedZym2IPRQToONGPxUgJM1WmH24IMP2le//nX73OcftZa2JTYm2+OnB0al/nzSTvYM2IT2KSuVtYpGe5XNuUHLBNjMzY5r9D1urRLQ2rWBbMfqNtvYvdK6tIHsimWttkwaPG2tLa7Ro8tWXAJs0nEGbIIrOcwcyBzIHMgcyBzIHLi9OYCsF3Ib8ZpRitgXAAApEUlEQVQBbBh+a/yNJCkC5x+S06jJOs3r2aXxWTuufQ/fPtpvZ8+N2+UJ0S8Nm3KDDKGnwb5r2JSmMYk2YyuW1st0znJpZnfbDq3IXrG0WfMG2PSQBrdm+IpaC0wIpkv7pIGuv0DDPDE1EnF5vUJLMR7PNcgkrzq/ukyAF3EO+UUf6cV6SKNM1L9YXvV5USdlWTTGGCU8xwHSAK6EB6QZHR11QBGgBrNlmHkeGBhwz74zpKNNQ1046ueeitoyXBsXYcQBZAJMIQzNlgBqYq8Z8gBrMFWGBk1HR4ebLiPe2dnpxwA2lAkTZkUwxi9euX7QELwr8iXoqi5TPD/KBM+jbJTJYeZAbXGAjh3AhjaYPEf4Wf2MTc5Y/7kRWVQakObkWTt/cVJAjRZvAs8w0YCGn//NaYZhRvO7Zc0BLLG9d250ixsrljaZ1oaqdKV2tf9rcbdiu4k+gzD6DcIM2FzLE7+FyvCAbwxgQ2ORJ/BvZMA1wazU9CqZKjcPk2ilTFKPG5Wt2tP9w3bg8AnrOTVk45MN0q6RnVpM5gjVmZNmTL2ELgCbBvk2qdRs3bRODXizbVq/0lYu0ySeVOeSSTQIUa8wTw90cM3UcXh6hTQn17PpSjgBj8uATeJD/s0cyByodQ7w8WbA88Ybb9j3v/9938OGgQ2rzeLjThlcCC0cO6xOJxjd3jUANl4HwpSEKvpV71X1A3BDt9mmvWz23HWXffM/v2VPPf2MrWhfaZelUdPTd9kOvt2jfWzOC5AHsGnW6Xh2INO5AuStlMyhNZhsU18esOELZwTezDlgs3lTt/Yu26ZB+BatbFvpJtjSvYh4DbzrNBhPNCW6oDO7zIHMgcyBzIHMgcyBzIHblQMx2cP9E0f2u5kaNtCAyJl+XfqryKVIkxq9I5syf6FCl2U/5+zgqECbc9rr4IIm96RNI2iGvQ7mJHNSTurlVleSOV1Z4UBe7FzT6hN7u3dutJVajd3UwMSfVmsL4GnQnoouN3LeLQTYOGMKP85D+Mh9yBGGbB/HnlH4Ib9YJspFXaHpUSwT9Uc1UTaeUWjvA3hwHunMOwU9pHEMCBP7xhCiMcM7iCevaM6MNIAbtGMY10Q8wJ4AerhW0BH3wvXCB82E0IemC1oygC9dXV2G+TI0ZNCMAXhBOyb2mSEsassA4ADmRAhIc5UpM72+jIi4Pi4AnDgOXsaxF9JP8TjKvF9esXyUy2HmwI3iQPEdfe93MfVL0MTcK7MPWpvve9dcGpmUdo22wDhxVvO+IzYyKrBVcwEYRQdaV7NW26lzc2h15Wlb2mqysrHc7tqtvcu3ahuMJQmwQbOG0b6Ca3LvTes1nX5TCsHr8NAfnn4vm0S7KY/ko7koD/nGATZqYTSa9J26qv3wEfNMb1gFwEapbs9QktnZ8yNagd1jJ7QB1eUxad6U0LCRps0s5VEbLQuwmVBsXHZpy9a9boXdoQ2odm7psA7Zqm3VvB/2Dr12hBfYuvDjB55aaNypAZOAp0uJON1A4LeKZpc5kDmQOVCjHKCfZ8Dzl7/8xb73ve/Z73//e1+FRjouBjbEGUTg5gc6lT7bEz8EYONgiQSJAGyampts+46d9rVvfNOefe55W6vB0fB4nR3pHbYjx05b/+DFSt8u7UlpUNbJvIULIQA2MolmpTGbnbpkfb1v27HDb9j0xLAGVW22YX2H9se53z798Kds5/ZtGlgtqwgvCDQS8xgk+cqcDNj4s8w/mQOZA5kDmQOZA5kDtzUHmAtIY12NcBVH9rv5gE2Ca0L8TPQxetefZvZ8/kJPbVJi4aXRGXtX+xwcfKfPzlTM58xpoQ8wTKkk2U/3BCAju2iaI5DJ9PYm7WHTbXfesUnm01ustblOljkk+0pGbJCMmLTCEy98vgDZMeYobpE3Bf7gCCPuCYjBaeLj6jwViPfAy1bOJx51RH6EUX8xBKQJbRbGGAGkAK4wDgkQJsAW0jBjhmYMHlNmlCF9YmJi3gRalCekzhijRAidRVAJGjkGjMEHmALQEgAM+8zEMSBM7CmD5gxmzShHGcAYAJ0oTxxAhnoDeIkw+AVPgk+kES8eV6cFj0nHFcsX86KOCFPpyvvq72qk5DBz4MZzYLF39Woq1D/RRel9BahhdhXP3jXnhsakWXPKARsA+Ilp9eMmwEZ9OnvdzGk8jzZkUz09/LS1L2+0bdrPdre0JrdsWC0AR2Cp+nP6bv9+pK7wahKqUqrbU1V2TR7C6/DRXxDSJ2bApiYf2Ycjiod8QwGb1DpFrItABaJpVBWByBtWyieaTKIB2qgRXxy1t955145pJc3FYdmqnWYPG2nZyCQa9mrVfiVsCbApT2jFTMlWaeXM1k1rbc+uTba5e5WEMqs0cDDa9DGEluSDHF270rgXGi8JeFacJ9BGYp2OM2ATXMth5kDmQO1ygMENA6FXXnnFfvCDH9hLL73kg6Lo/4sCVvEuPJ0uMtzfANho6IQs5sKEQyQcKNLY1GhbtmyzL7/wVXvmuS9b94aNMnlZZ28dHbKTvQPq20e9X59Vv16Sqct6ATY+CJPqM4DN7PSIjQ/329FDr9v+V39vFy+cFQ4z56DNZz/7afvmN75uD/7zP9k6rY6LPpw9bNI9MmDK/XY8zhxmDmQOZA5kDmQOZA7cvhxANgpZiXhNADYxEOexSHZEfGSsLklufoKqpGH5jIRKQJuzAmr2Hei1k6cuam5AWhvSspFejZW02LNOcisWOMqzUyo9JXO69bZjS6ebTe+UKZ0V2rC6uZE5BARW5hECtKlMIyL3VjTGoaJWXZJxF6iLZ0oKeZFPeuRFWuRHuFAL7GdeJJ3PuxGONI4DoCFkTIGPOFowAC9ozITpsv7+fjdfFiFmzQBsAGKoMxxxrg0oUqSBfI4jD20WHLQUNVsAWcKEGUAL4AtmytCg6e7udi0azJeFNg0ADUAM442ii3sPesiPNGiob0iaWUXAhvMpA01xXtAceVEHYbhIo2yYUos6KFOsoxiP83OYOXCzOVB8n3lHF3WVd57ZX+9l1QSYYR2fmrO+QVnbOMx+tmdtZEymL7U4v1yPeXTtRO5NBfOVWqCvcX+T9rNdJ9Po7Em2c9s6WdtgcX6DgHm1FdXnXwyft1iUiisS35PWK0rV1kH0F4TRHxDSZ2TAprae1d9FDQ84JuyOHDli3/nOd+zFF190tVQ+trt27bLnnntOG0Q/Zjt37nS1UM6JFQXXfnFaWAI6aELRjOJ8GlRyyvNoSomzUHtmHfjFyxNu0/DoiUE706/VFxMSxgTYMLGnpquPLKtjpnSFCYWztlz72Gxev8r27tlqWzaukZ1Ds5YGzKJptY03Zq6a6EnXT8epQ1DOFR1NEtwyYLPAqRzLHMgcuDU4gOkzTKD96U9/sh//+Mf26quvupkB0unTYwDCRz4c6e7oInGVvpk+MyWRUClDrFLOhYboV5WdutFUrkGCVINsi3cJpHnq6WftmWe/bFukDXN+pGz7Dg1Yn8xejk9owFeqd++akyrvpiyl+oxJtMmxCzbYd1zaNfvs6MHXbWR4UCvn6mQ3eo09+ugj9rWvvmB79+619hXLvQ/nu5AEGW4CIh0+4iC7zIHMgcyBzIHMgcyBzIHblgPIejHeJY4ceDM1bHgQ8/IncfkkR1YkT9FIPgs6ZyXTzUj4PHdh1N482GfHZBZtZFSb00uGZINqFv2k6QdspwvekRy5pEUWODqW265tnTKhu9bWrlmiyT+WYGryXXWxetuvh0k0XT3xprblRnHENY9EsPMmnmeEpIcjLfhbjEca4Xt55oZYAAYIwyIwwBZMmIVWTHGPmTBlhpYMYAyeeHjKRh51UjfXraaZ4/CMVcKjMYO5sjVr1rj5MrRiMGEWWjFozpBPGtozkR5aMuQTxxc1ZuJawa8iX4K2SIuykR7nEJJWzcdIL4bEoz7i7rjniBfCxa5D9nulF07N0cyBj5wD1e9x9XsZ+bzb9Otq7uqe1U6UMKuF+ZcnSj6/C2Bz7ORZuzyufn6uxeoaWt3EJTeQ5m9LCmespbGkPcna7Z67N2uB/mpbtbzZmjVhwKxwnfffXGmxlkRNV7pqWq/Mrc2jYv8C/eEzYFObz+tDU8WDvrGADcIPDcebktOdPknebFMeURqvfl0AUVjSS4hgNjI+7fvYHDk+YMd7ztnwiASHEoCNUFepytWj9ip1Zytrn4PKRlTrO9rtY3dtte2bO2xZm8l+rVYtiARAm3QVVew0KVEX9cuTpLQrGy85EiacEs5lRUdeqQ2nssscyByobQ6wyo3VbWjW/OhHP3LTaAyUArABhOd7wEc+BAC/o6Kc44gMPXYkpsFs3DnZLix4fhqoUJJVi/TlykwrxnStdR1d9viTT9vTzz5nW7fvtIFLJXvtgDYXHBq3GSH0s6XKQFurGrEpjjmL8pxWR0rDZmxY/f+RNx2wOdt7xMbHhjQQa7Yd2zfbF7/4mBYZPKMFBjuszVfKpX48rXyDUqeoEnKcXeZA5kDmQOZA5kDmQObA7ckBZL4Y74YceLMBG54EtMw7Fzv9Z15GRQKdlVzJXjVDlybs8NFzWtR53k3qTEwxEYhZNMm2KjjHPjZzsz430Fw/q71rmjTBpwWdd22SOd2VmhdgMScje8mcCl1q1YloZyfe1C5gEzL7FfyqMC6ea4QkR/linDTk/9CMAUCJPWIIGSvgQ2MGsIYxBVoyLAaLPWcAcgBvAsDhOMYZRRri2oThyMcjrzMmQcMEX9wbBnAlzJMBuKAxg4bM+vXrtWir048BZgKMIQ5YA3ADKBPaOlxzMXqch3rNeAOq84PO9+NzdR51kFb0pBU99cZ5Xp7xEv+FdkkZ8oouzom06vxIz2HmwI3gQPX7yDWL7+RV+XrH0ZJk7oBZ1SkdDF+Wecszl+zQkVOyuDGohfnMHrRYfVOrl1GxBNiUBb4LsAF837Z5jd1/z3bb2L1Cx/Xel9f7twMoiDYT/v25UKT1/UvWTi48DQ/94TNgUzvP6LpQwkO+cYANkAviFUJPADapKfmXia8TjYpAIQHNdI6XUS8hoM3ETMkujEzZMWnYvHX4tCb3pmxqpkGgjewaymwOK7frpVmjZq9LTQtlLWnvmiV2151bbac2ololO4dLWpKt2gXAhm6i0pjpOHzWMdFyZeMlDcAGXR/i2lvBQRvOzS5zIHMgc6B2OcCA6cKFCw7YsIfNn//8Z9+ok8EZLgYxfOQDtPH+r9Ila8xacfR3MnXmxwmw8SjlKoVSkE6ktHevnAWgjukCgTDrOrvtyf94zrVsujdssb4Ls65hc+HSmK+IZN8atGvoap0ODbTLJZm6lIB26cIZ2//aS3bs0D67PHRWq/0u2+pVy+3ee/fal770BXtcfsvmTdbc0qxzIUCO0AnFTFvtDryd1vyTOZA5kDmQOZA5kDmQOXADOMBcQIx3iSMD3gzAhmvjgpa4dU+topH5ASb7mBsAsBkZnbaeU8MCbM5pwu+C9rnVPirWpMrY61blShq7C4Cp116IjZIj25pKtl4bVX/83m2a8Fsn0zqaFpRpqybkVARHXa8sPkALWtrJRRiU3fwwnhf7OjCXEbwLXhYpJG2+fEXWB5hhHogQjxYMmjGYKEN7ZmBgYF6LhjFEUZsGMAYAh/MZS1T7GE8UxxTQF75IW+KzJls1RgBYAWjBY6Zsrcwbr1u37ooQoIa80JwJDRn2laEOxjThAwBa7LpBA3lFV+Qf8Tg3eEjZSCueF/Hi+VF3MS3OJ4z0YjnSquuPcsVziWeXOVALHCi+n0FPvNMcR36823rFNd7XYky1PXajGZ+Zk6bkhO9Tjjm0vv5LNjmFpmSz5g4Yz2vsrmbqS/ilKckcL8A7ptDu+9hm36u82fcy13wDKL1q9Tld5iyq2nf1cZE+4ovlk15rDl6Gh+bw9LnZJFqtPa2/gx4e8o0HbPgoasULDWiedsQxvFIIFKaUtNobYQyhbEZS1+hUSY35gv31wLuuNjc+KVu1c5oErFdjrmeDahoppnOmdZVp38fmzl0b7Y4d3da5ttWWL2nQpB8raZh21FUq10PQwcGTFF0QfDzDC2bAJvEi/2YOZA7cShxgIMaKOEyiYfoSTRvMETDA4gPPwAYXAI73g5UbdAybeKWvpINMoAwJEhYqHXkANl5OWQyS3KlP5Rr1aMroOmUJXR1dG+y5r3zNHn/iaWtZ0m6nz03ZO71alafNYxNojsYP1etcBWw5WCcNm/LsqMyhnbTXX/2DnXjnTZueGLI5ATnd3evsU//2fxysefjhT2nFXYcEPAxcyFXoS/RnwAaWZJc5kDmQOZA5kDmQOZA5gLyHjIYjXmuATdDlBBZ+SqI1AJvxyZKdPTfuCzqPnOizoeFpzQ0IsNHeByUJqQuATUnj/xlN9s1ax9o2WeDYZDu2dmhuoFlm09msGtPqzAhIAK3In86bCn8Kl6+JKM8rnhl0xnGEPEvmedCoB4jBA8YBtoRJMvI4LvoAbRg3oC1DHueRDqgDUMN4gfpx8f4sxpTIA0xB8wWQpWi6rHqvmcgnLPowbYa2DGAO2jbUWQRmuBZjD3xcN3jBcaQFncXjYpx8ziu6yK9Op0zkFctHuWLetaRFmep6I71YX/F6OZ45cDM5EO9nkYZ4V9VLqUFd2aZoXbKC5oA7ffSY5ndP9w3b4SNnrPf0kPazlWbfLPPFTZrfxUe/nDQll7bVWXfncs3vdtjunR2a721NmpLebzN3wBXoE9+/3Qe9RfqD7sir1RCaw0Nz+AzY1OoT+5B08ZBvFmDDFNyCo1GlhuVBZYZNYqMO9Ss50jcPVJEpmct5t++i7T9w2k6+e9GGZau2VNbmNBLKfDNpNhbUCpqybNWy38Gytnqhr122Z1e3bd7Qbu3LtPoCBFZXBLAJ2IhNCXEpSLRc2WBJy4CNMyn/ZA5kDtxSHKCfZ7D18ssv27e//W37wx/+4AOwAGwY3PA94CNPGI7+d74f9GT6SQkFXsZFMA7dFQEbxKoAbBZWKQoUEmgjY7S2fuNW+9o3/6997rHHZaNWKtD949Y3VNZmsapKnfCcvAsh0qzBDm2jVj82CoCfHB2yd08etn3/+5KdOnFYJskvq/yUbdu22Z584ouuYXP//R+39vYVuk7qxzHJNn8P/t1J95Cozr+ZA5kDmQOZA5kDmQOZA7cnB5C1QkYKObCWNGx4KkW5NJ4SFjgCsJmeKdvQyIz2PRiQBY5eG7gwrvkCNGxafJGQr+SWfFtXLmkOoOSAzer2Zl+dvWNbhzarbrclrZofkHjYJJmRRZ24NOlHpJLgqR/dT9xnPI+4UqRzXIyHzE6IPI+sH55jtOsBZtCMQVsGE2aYMgszZgHgoFEDMBOATGjeBChTvCY0BH3FEJkf8CTAkuIxWi8ALGjLYL4sTJix70wANAAy7e3tDuYQB5gBzAkHDXE94txzeNK5BmFcl3iUIwy6Ij3KxnFch+MPctSHD8c5H3Re8ZxrKR91X2tI/R9Ew7XWlctlDvytHCi2hzg33sfiu+/laGP6Z6wP5Ms+ZKPav/Z4z6Dtf7PXzg6OaT5AQLtAd8xaMmObFGzow2cddF+1QqbQNb97x85OmUOT6cMl0qyjDaBJqTMWrr0Qr6YrjgmL9Me5xfxajAdfCaE5PP1i1rCpxSf2IWniAdcqYJM+g2nyUO1YQpm8Eqek1dx/fszePjpgR473e3yWDanqW9UiMbcDM2TmRw3atCK7pXHOutYutV07Om3PHV22bvUSa1ZLRhjzhq3SesXTaf7LleX9g+2VceRpqVsBtFHD8A6ETiSV8SL5J3MgcyBzoAY5wKCLQdtrr71m3/3ud+13v/udD+AYzOHiI0+8KABcmZcGP6lDjF4xhax8cadviqfoh8ERjvroU1Fnrmtosral2ux191778gvfsH9+6GEJZkPWd27aRqaWyMQlKyG1ao96JHSVSzPSoEFbUgKaAJux4UHrPXbQ3nrjZet795j6+EnVO2P33HOX/fd/f8s+/+gjtr67W4O8ZqcDUxZct55rO4kuxome3G+LCdllDmQOZA5kDmQOZA7cxhyIyR5YQJzJnusB2CTZ70rGvt9EWJSvLhPpUVPkJw2bNCKf1bB/eGxWFjhkMv1Qj/a7vWzjkxqjN7Rpwq9e+yJWzP9KBGzG/FlT2ZYvrXfLG9u3rBVw06kFnUt8ZA9og4TIwiQPK5OL0BG0BA2EpMVx0BhhlC8ew1/SkZHxcT5hzMlcKT8vPJcAUCgHqIIMj4YMz4t9ZDBjhukyNGPCF/eToSzaMWjJcC71URdp1Bf1BxACTUEf98B9QltotRAWzZhhpmzNmjVurgzwJUyXAcCED+0YzJjF/jRoyjQ3N7vGTGjNEAYf4tpFPgZtQV+Uhca0Gn9Bzo8ycX6EXpbn+yEcdYZ7r+cf+RHGOddaPs7LYeZALXMg3mtoXOzdJr9YhnldHIYc8RMyhzZ0aVKA+6AdevuM9iHTHruzjZrNlXn0cuoj0+JLevMZAeuztmZVm+3Zvcl2be+wtas0t9BcmZlV3T4loeYZLTRoirBIS6Qlim6t3+ArIfcRPgM2t9Zz/EBqecAhHBw5csRN5bz44os+kcdHe9euXdrA+Tl77LHHtInzTl/twDmxguIDLzBfgCaDsASOSisNoCM+ktGklOXRWNNCeQmP+kXLRso1MoumzQWlJodZtEPvnJa2zZAatSbn6tpUNXvZ8MKm69VpU6rG+hk3g7Zdm1Ldu3ezre/UplRq1AhkbEqVpu90jtMVdCj0D3jQIQKcsNS1pDhnxn2Qn13mQOZA5kBtcoCPN4OzAwcO2E9+8hP7zW9+Y6dOnXIQB4rjYx/Uc1x0SaCpADbKoOdmmBzlioBNyk1np/NSXfTN9Q3NtmZdt338Ew/ZE08/b7v33Gsnes44YDNlK21mrqEyYERLUt8MaUrOSVMSwKZO4MzQ4Ck3hXbkrTfsXH+vNWjz2LbWRvuXf3nA/ud//p89/PAnrW1Jmw/WoACghkFcADYMv0WFcuLbk+jMv5kDmQOZA5kDmQOZA5kDtxsHivIfceTFvxewCdlwMV4mufDqnDinmB9pxdKRL9hDMwtIoqzENjepw/41h95+106euqQNrKWBUdeqxZ6SK3VPjOubGusF1tRZc2PZWptLtmJZvW3btNr23rnZ1q5cJplSO9RKPPRpBPHCZwdcXKzIsaT5/ECiiDg0FtPIIS088yzI32iv4JlDAdgAuACkoFwAJKHZQhgmywBTOJ8wwBYAl6IpM0AZgJrwgDdhziy0ZpjXqaY16CYdF8cBykAfmjGYMgNcCWCFdAAXPPlowwDIANaw3wyaM9wjgA0eLRrKUD6AlbieXxOe81fhZzWdUZYwaCWOi+OgPaUu3EsxPcpWl4njHGYOZA787Rwotqtie6OmyIvQ09TBMidADzsr9AZLG2f6RxywOdFzXubQtLeWtrsoa153ThPALOCk222oK2nudsYX43dqMf7eu7ZJy6ZD/Th9OrOyssahMT99iU7yrwPXC5oiLNISaZS71Rz3EZ77CJ8Bm1vtSX4AvTzkmwPYMGGGx9GoUuzqXxqbBC7/JSahTEjr5fGS9Z4ZtgOHel19bnJGKKzMos2V1bhlcietrFCjlijXqAm9ZmnZbOxut3vv3mJbN660FUsbtcImIbGYRRMFTgNhRSSrkEIKHQqN3WEjxQhJBazJE3/OjPyTOZA5UNMcoK/nA37s2DH7+c9/br/61a/s4MGDPqAjHRcffeLx0S+mebp6PheEvNtOqwRVNR3ivPMSlUEXPSqDM+pj7xoAm83bdtnD//6YPfLo49bZvcXePtJrp/q1mqauXV2q+nHRMzszLXpQe8YY5rSvppmbGrVTPe/Y22+9LnNo79jIUL8G3Kb9atbYZz79Kfuv//pP+8Q/fVzXqGjT6JoJsElCTOrciWegff5h5UjmQOZA5kDmQOZA5sBtywHkPGQ0XMiKtQLYxEMp0uh0VmhlpXaaG8ACx5wNnh+xd46dtaMnztnAeWmRlJocsHEpV/fYpL0NWzWzh2zZWD9tS9vmbMvG1XaPJv661y3XRGBa0OnckGyreUKu5B4WhWzsqS78pjSOwwUPQ3sF4AUTZMePH7eTJ086kHHffffZpk2bHMRwmbcC6jAnAxCDCbPTp0/bmTNnXGsGk2U8E4CZog8wBkAnwB6ui6fe8NCEDxf3QRgyegApkQagBPiyYcMGN2EGGBP7xwRAEybMAGQoDyATWjNFLZyIx/WDFq4VLuKEkV+dF8eE1WWKecSjvur0fJw5kDlwfTlAW1ysvUUbjZCrEtdUrgM2mESbESAzdEn7j/UM2PGT56xvcNxGxwXGlzFpKQhGG93Qj6GLUy/Nmib5JbKUuL6z3T529zbbvnWd+nH17RraM6e7MDNL35L6PWgr0lekp5h+fbny0dfmvKz07XGPhPArm0T76Pl/w67Ag76RgA2Qi1qf/mhOeDmkrfdyILCSlgBsKuISzdXGJst2ZvCyAJseO3Ki38bGafACa6Q6NxdaNt6wtfeBGna9Jvw61iyxu+7caLu2rfV4i1bZpH1sUgOHBF9RozBdLYhiohGaoR1fAXjm7+F96I8qcpg5kDmQOXCTOcAHnEHjvn377Je//KWDNmjZMMjDxYefeHz4i0IN6XTESbiJXpKeuSIIqSuM8vSYKZ6EOM5hpUxL2zK7/4GH7KlnXrC773lAkHqLNGz6ZG98xqZsuXpYSVw+yBRNWkUjQxZaTSOTaDJ7NjU2ZO8c3GcH9r1qF/pP2dTEJa2qabGP7d1jX/jC5+3Jp56wO3bvhIgE2leAItSoAW6So79eEOcqiTnIHMgcyBzIHMgcyBzIHLjtOICsluS6JMMhK/7dgA1cVL0e+C+SIvIjB/5TSU1ByI6emwpdkV88cOlTVVM7fg75UuGsJvaGR6d8w+qDh89Iy2ZIZnYbtT+CNGyYa5Ac2KAFPc0CbQBsGuqmBdDMaH/bVXbf3h1a2Cmtl1bNDaiotrp1F4CNk8Qkgf6r5eNq3iFTA6QgbwO6sFAKoIb42NiY7dixwx566CEHQbhIaMIAxBQ1Y0gPoCZMmVVr2XAc8zjUBR+Lz5M0HDSGdRSAmdCQAWiJPWQAYSKOtkxRQwZgplrLBmAmNGyoDx9mzIJH6erX/hu85IzF7mOxmorvTuQX64m0HGYOZA7cOA4U22XE50PZLKPfZquLae1gMXB+VFaTZEFD1pOGRkrav6Y+LcJnLkHfozL72foMwYw1aSH+qhUttnXTWs3rbrJN6r9bZDmpSUN7umi+BnxhfB7Cb1eJHHsn7lH/CVqq0xdK1H6MewjPfYTPgE3tP7u/iUIecnzob4RJtJjioxEtTJoRfy9XeRHV+GhuACegsZOayxsYGreDb/dqJU2fXVTjnpZQVtbqbK2Z8Q7AG7YkrnppxtTJpE778ibtY9Nhe3Z12eb1K2XrUHZXdekr9GS8TdPaU+NOTT5o48ipUELqCtI9RH4OMwcyBzIHapcDfMBZhYeN61//+tf2wx/+0E2kMTAPwSXC4l2EMEPo+ZVBuJdRV+gCggaAuDkNmH1E60B86kc9Q31mQ1OLrVrXaf/+uS/a8y98y1at3aj9a0akBj1kFy+Xbbq8RGYrVKE2hZUYp3BGXfGU4BUBN9qPbHR4wP762su2/7U/2/jwOV1s0tatXWmf+cy/2VNPPW7/+smHrKu7y4U717Kp0MQCgTCJdm3fnURx/s0cyBzIHMgcyBzIHMgc+EfmAHJdyHnErxtgI6bFuB/+xcg5xRY4Wi13Bi0LJa6MUd69kpkXoD5ETsTPSdlOH9A+t/sP9Giv2z7tYyMd7ZIv/VE5bUovudD3sK1neZAm/+qmbH3XClng2GHbNq/WPjZNWqktqTHEV63T9CsoIa7rMm9FHiYtNGkAT8JM2eDgoJ04ccL2799vr7/+uh09etTBF0CRe+65xx544AEHRAB2zp4963I5sjlxtGuQ1XkOUT/XwAVvyMORjyM9ABlC9pXBF9NC8wVgBfAFEGbtWi1i7ehwTRq0aYh3dnZ6OoAN5s4AYYr3zPXiunFt8kNTJ2gkD1csm1IW7iOOo1z1ucX8HM8cyBy4NThQ3eY5jjR6rHIFsJnVmH9iumx9A8P25sEeO/nukBblywi6NCPRlWE/21JJiI5b3FCfrXmBpoZZ65I25O6d6+2OHV3ah2y5CYN3c5Y+E0GfKF/ssxbrV4KexfJuDS6nvjV4G/dLyPcha9jcKk/xGujkId84wCZppyyQhQiUJvmSOLSQk2JhbkcNTy8fZWjk2Kmd0j42Fy9P2vHefld97hsYs7EJ5da3aD026s/Uiy4O9g4F2GiF9jKpy23a0G577uiyHVvW2fI2CWUq4CtpVC4atyJXCJgc46AgxRMtKSWlelb+yRzIHMgcqHEOMOhjhR8DyJ/97Gf229/+1s00hPmE+PDHbYQAwDF5jmVXur3UM+rX+2dK0Efr2DNUSGG9hDJWNJY1ml6xUnbC7/uEAJsv2YP/+lmtrllivafO2+CFERtFQKtb6uUAa6Qzqf47aUeyCtJmx+3S+T776xuv2Fv7/mJjIxfUt8/alk3r7emnH7enn3nKdu++w1a0L3c63bSD6GJlDgsEEmAjmpxWvgy57+aJZZc5kDmQOZA5kDmQOXD7csBlu4ocR/x6AjZw1eVChUhdSfIK+YtJPEokd+XE2ULe1emikRNBaZDzFFANyyplFU373E5otfYZOyTA5vzQpBZ5auKvXnMDc8kIeqNkUuYGWAzUYJO2Wqu1d23r1ARgl23oapcGjhb5qDZ+g1LCOanbwB8cPAJUwaMFA9CCBg2eeH9/v5szI86+MphFQ84GQAEIARwBOEEbB5k8NGiKZs3iWYR8HvI4YTyzMGNGvQHCUDdADPvIoCmDqTLyCNGg4fqEHIeWDGEAOoSAPdQdIExcx29eP5GeTNAnLsVzCjqjbPCM42JegE7V6RwXz+H4erug9XrXm+vLHMgcuLr9FtszfXdZfQs6M9MC2Nnm4lTfsC/CJ5ycZuG9LCYpv+R72VJSJtI1N9Agi0mtTWXtO9Zhe/ds0byutrmQpQ2f79WPuvbUZxc+LP/IbR2+ho++lTADNv9grZCHfOMAmyROJRYS5wMfntT0wScWold6CZXjgiSTgSbTZ5q700KTsSltUCVE9ujJfm1SdU72D7XnQV2LleqkZSP1Z73CLnA1aIaRib3W5rKtW9Nmd+7stt1CZFevaLJWoTWaT9RKmiSAzVNQ0bAJOlKY9IJcMgy6nS4nOP9kDmQOZA7cEhxggIkptD/+8Y/24osv2iuvvOI2sVkdyEc+Pv7cTBIAEL4qt6YQ62L0hvSx6c9Lelqll1R6KkMX2ailL80tbbZl+077zCNfsE8+/Ih1bdxp5y9O28neQbt0WRupzmowLQDHr6OVNPUygZZAmykB61KpnJ2wi+fP2MH9/2uHDrzugI2UJO2OXdvs+eeftSee/JJ1r+/SasBWnQeNadNBTGoGdR468ehVzvf2np9/MgcyBzIHMgcyBzIHMgduNw4g86VxNrLe9QJsQmhMI2h+k9QVUmIlfV64hOvKq4hmKTnqWEinFJrc87IdexzoHMrPaeyOiR02sD7Re14aNv2aCLyoY5lW1/wAeyWU5VlIhPWNBNhMyQya2YaO5VrQuV77IXTKAofkR69QvFCFs7MzNjk57qALwAvgCiHa6WjIAMj09fU5QANQg6YMps0AcgKoSbJ0urnQgAH0CH4HeBHlAiAhxAOgoOkCmAL4AuBCGHvGRNqqVauMvWYAa9CgQaMnylEWkIY0NGeaVVeD6l7MQU+MB4rvBmU5Droib7E6qvPimDDGGfHukebpjB7isS9W6XVKC1quU3W5msyBzIECB2jXRVc8Jqek9g4MMzFdsnMC1XukWXP0eL9rR06XmMdlmwv6+hn1FZhGZ0/bZA6tfWmT5nHX2913brGOtdo3i8kAFUZzUlh7YXi/8M0p0vKPFH+vfpS+u6enx37xi1+4xuRXvvKVq277/wMAAP//26AQlgAAQABJREFU7L2Jf13HcedbuNgXEtxAEtx3arMlW5RsybGskTRjTeJYcUa2sk2WN/PyZt6/lOTzlkxm4iReniIvceI4Viw71mJLFElxJ7iBJAgSIPbl3ov3+1bfAo4uQUpOJJkAu4G+3ae306fOOXWq+9dV3TAnZzV38OBB6+vrswMHDtiGDRusoaHBcwijWDEt6uXwo6EA96BcLlu1WrXjx4/bn/zJn9iLL75o169ft0qlYnv37rXnn3/enn32WduzZ491dXX5fWtsbLRSqfQLdpLHIh6Naq0uz0PRU6JYrlBF5ebky+prRUVm9DM8OmOnz161tw6ftf7LI1axFitbm1UbmryVkppuapizxtKctTRWrHtFk+3eudHu3bPZete1WGdbg3EVDaIDrevidKBKdMkd/azlkTjXoF/VUOhlas9zrXAOMgUyBTIF7ngKwNunpqbs5MmT9s1vftO+853v+Hd6eHh4/nuw8H1euBxnjzosibH6N1y8kTR49px4It+EUkPjfIX0bRf/bWq0nvUb7cCnHrfnfvMr9uDDn7bpcoud7rtqp88N2vRsg3h3m/h6i747VbU1Y40NZTU8Jd48Za1NFSspbXT4ih0/8qYdP/yWjd0YtPbWJrvv3n32pS990Z5+5ilbvbrbWlpb/HylhoXvk7NzdXSuqs6KfzeU1MdC/nyHcyRTIFMgUyBTIFMgUyBT4C6iAPJezMUQZ05gdHTU3nnnHfvLv/xL++pXv2pXrlwxxv6bN2/2eYHf/d3f9TmCFStWLEop5EKkQ9zCL2PoOocQ+Z4OmTMVovjcXNX7yEC8VGryvktytEq1IlnSbGK6apevjtmJMwN2/GS/XbsxY1XJmLOVkuYQYogvudJmrblhxlrkV3Y12r7dG23vrl7rbG8yncBmpmZsZnrWxkSLwcHLosElu3z5sl29etUGBgY8ZL4E2XlkZMTlauiHjI2HjrgFeTrJzkFr6NnU1OSeOI45GfJbWlo8va2tzaBxZ2enz8GsWrXKent7bceOHT6vtnr1alu3bp2tWbPGVq5caZRfbI6GNmPexu+35PhGl4XpYOpj9DfKRj8J47mgDMf13jtPU7r+8KRRjvMSFl19mcgnHRfHxCONeL0rlivm1de5VblinRzPFMgU+GAoUP/+FVuFK8KnZ8W/xyZn7dyFYTt+asAu9A/b8EjFZqut4tfM4/I3IwYwo5IzVqpOW0eL2ab13XbP3i3i15ts1co2a9Icryr4fG5jY3wr4CN4HetvuTroHB4eFx4+Df7y0ksv+XfihRdeuIkEmvuucVtlZcDmJvrcUQncqo8OsOHS4wXidU0vUnrFeJkcMlEqYhcgSXrNFKhovGya2NOkG4DNrPo+MVO1c3rB3zzYpxd+yCRb2TSTftasiUMJCS4oaMKwQa9+qSwhrNG2bVrjQtmOzV22emWLJgLTuQhTnxQgWPgpUz9Tz5TwLsAmytDB7DIFMgUyBZYGBfiQM5gcGhqyo0eP2o9+9CP7wQ9+YIcPH3awfnZ21i8kfcrhi2J9KUh8eZ4/1jgmvLLGPhsAbJSP0NCoQVprW6ut37DePvnJA/bk0//ePvnoY7aie71duDxqZyWkXb0+JfC9SQPpJpuZLVlVA3HB8uLd0xK+Zqy5qWxd7SVbs6rN1qxotdnJEbt2pd+uX71k5ekJW9+zxj7xyQftvvvvsfZ2DVQ1+PVBqbNn+lH8jqiTcwJyan30C8s/mQKZApkCmQKZApkCmQJ3KQWQ9WJCmzgy4gcL2NQESNH3psmzEC7fk/ZJnqN/4WnNFwlJ0GPeoDKXABthLAJppnxB5+F3ztqlq+M245OAmu3Tgk7kQlPZucqUzc1O2NTYNZscG7Cutqqt7m7WfAFgzaSN3hi18bFxmxgft/HxG/I6VnxiYsImJyfds/hpZkbAjnyALfQPF2HQljTAlObmZgdWWltbHYQBbOnu7vY4C2PxADSALx0dHQ7EAMYQb29v9zyAGsrQBuXwAfLEpF30IfrBMY7j6FOEUSbyIj3KF0PiRVcsSzzaKpaJeLFsfVqxXrFcMT3qRFgsF2mExTq3KlMsn+OZApkCHxwFiu9ffavMrDLLMKM51Ruj03b8xCV75/hlG7w+rXndJgfXKxqri6PrRZ7RuzwlQGZK87VTtlYAzb5dm23v7l7b0rta8wPNmuPlS1CbTdYifXe1kO/NTd+cVGJZ/ELn8PC58BmwWRa3d+EiuMkfLWDDuXlVA7DhmJXQ+IYaVMPLhk+vmP/W3j8AE2pW1G8Am1kdXB4cs4OHLtiZs4N2Y2zWJmYbrTwnwIbV3r7ie06tV3zFdltLg21Y12m7t/fohV9rG9Z2ugYOmjjvBdik3qifTPh592ocQv3JLlMgUyBT4E6nQFGAIh6aNkeOHLG//du/tZdfftlOnDjhqwUZfEZ5QoSh9KMw+LFLSIkzqojKq4g0V0patecDUg0ke9b32AMPPGD/7uln7DNPPGnrN2626yPTdvhYv10amNRKSHFngezTAmumtSrSwXpp1zTYpAbiM9KiMQdrdm4Rz9ZqmnXdnVYRUDM0eMXGRoa1ArHBenrW2tq1a6ykeIOv5ON7kroLaD8/WPN+w7dZyehX5OXyT6ZApkCmQKZApkCmQKbA3UgBl/EcxUCOu7MBG+4Pk1FJPn03YMOCH+YImBsYGa9Y34VBO3j4tC/snJxp1hRCu5UaWyT9aS6hOmuV2UkrT43Y1ctn7eypwzYxeklzARMO5EyNj9mNGyPSrhmTbCpgR+XnAHkQdAvOFwjpmHR8yJvFcpFGWYCV9evX26ZNmxyIAXhBYwYtGeI9PT2GFg2gDWUBaNCw4RhAhjZoL84bXSmeO84XeRHW95H0W5WtrxPtR/ni9UXZYnvFcvXnJS/yqbNYW++VH+cslou0Ypu3yi+WzfFMgUyBD5YCi73TcQZ49LT85OycQJpxO/zOBTt67LKNTkjDcK5d06xY3BCPnhOsMzct3jupid8J14bslXbNg/fvtt07NtiabgHVLSXN4zKLXBjTa5G+zyXoHPB6ccs49TyvWS58IXgrIdcUPgM287d8eUS4wb88wIZXFseLlCbQEIN4yRZeNPJrL6FeXmXwrxdaq19c9blkQyMzUqUbtNNnBu3ywIgNT+ia5pp8FYtP3kl8A7DR2m1rbZ6z7q4m27pptd23D3S229qaG6xZp28ULVLrnFLnwqtOgoiYsORYBb0f5FNOPrtMgUyBTIElQAE+4HgGenjigDaYc8Ak5g9/+EP7h3/4Bzt06JDMPwz6FSUhAHaYOCCJsEp4NPwvhB5nn0pAw6W1td26V3Zr4LnOHnzwYzJX9ow99MmHBdZssWqp1c5fvG6Hjp6zgeszAt7bBbK0WbnSqBWKErJ8IC7zFdUJ17Lp6igZAtruHT22a5vMPqD+LL5cYTXjLIKc9CnFwJtbAOnVx9q1ed9qDNrZNH32TiLQiI9nk2hL4InNXcwUyBTIFMgUyBTIFPgwKRCTPZyDOLLhR69hg2x281UmuY30hXzSkkcGZf6g5lSfmQWJkjY5M2f9V4YE2JyxM+evC8BhsWebhvDNVpXp3Upl1qoCbCozY3ah77i9/fNXbGjgjM2VR9XItGhQ1vyIymE2XnJyQwNGfBJQxLlD9o1TRxqyNXFoWF8GbRjAmccee8yefPJJ27Jli2vWoD0DMMNCJ0AZQtopmkwLuT3a9OuvTYqQFp7+RF+ib8WQvKKL9iLN261dXzFvsXr1adFGhFG/vlykUy7OV18njgmL5YvpxOvbrs9f7Ph27S1WPqdlCmQKvH8KvNc7iZWkSfnhsWm7cHFYYE2/9Z0bEoAjUL3UId8qXi1uW9VcgAPomg+Qdk1Hc9W2b1lnn/z4Ptu+ea2sJpWspWlOgI34X+1PHEF/CbChx5rtUE4CbIr9WowHkL9Y+vu/8o++JH0OT9/D8/3pyybRPvob8mGdkZt8pwA2QCI4XjX+cKTwErqrombLhxtABsBGL6Qm3cYmq3b24ridOHVFWjYDdm20rNU1UmnWxKFXVVn2sMEsWnOpYu0tVdu4rsvu27/ZdmgCsLtTE4yCZ3mdk+kz9QGp0f8XABuAmgZNFS5o2KhC4gGpf/k3UyBTIFPgDqYAH/AQSOKjzjEm0LDD/dprr7m903/8x3+0U6dOFcomVgpbhDNTxx2s2fmkRKLGJl8NuLIb+9qbtVfYLtu9e7c98LH77eMPPWi9mzZLA6bVtSBPnxuwd07IrNkNzFd0qk2ZsaxKM1Lb1jBIdnu1mESTXyHAZtPGVbZza4/MWa52wAaQvZGeANqrvHcjMfB5YYVsviN+nSpBGf+uwMczYJPuX/7NFMgUyBTIFMgUyBS4qykQciFEIH6nAzbRT58T0EA8yaUSRyUHMpdQVsK0ZvwGro3ZsZOX7FTfoOLTNiULHBUt6KwgC0t2nCtPO2Bz7vQRe+Nf/smu9p+UibQR0YD13wA03qBC5gISYBPnRrYsuqAh6cTjuFgGjZmtW7fa008/bV/4whdcRgasAchhziJAGkLaCZAmzlkMiUcfKKcDbh7J7iKPA/qCi7Q49sTaz2J5kVYst9h11efH8WL1yYvzRz7HxTTS45jyUY54vSuWq8+71fHt2rtVnZyeKZAp8P4o8F7vJPx5Uj+Xro3Kqkf/PH+erWgP8lK79iFvcT5elTk0Fm/OVcY1fzuj+dpGLdxcbw89sNu2bFgl7RqBNTJfCWSf9s+NkT6ADdA9c7sJsKnv02I8gDKLpb+/q/7llKLP4el7+AzY/HLux4d2Vm7ynQDYCE7RNSa4xCfV/FWNy65NthUAG5SeywhcElKmZErn8uCsHdVLf0Srtq8OC7CRiZ0mbXSNY3PCZhk4bJQ8w8bVTTKzs7a71e7dt8n27Nxg61Z3WEerXmkBQakH+nU5jAcfIS1NCDpQo342uEk0FaCM2swuUyBTIFNgqVAAns+HHOeDPIUcY4+b/Wy++93v2te+9jV78803lc+AUR4hgAGkyrpgQGU5V1JRXmOj7HG3d8icwxrbtm2HfeKhh+zTn37MHvz4xwXU9FqTNGAaZLd7crpilzWAPi1g/dTZq1rxKO5a7dD+NRpEV6Xxo/FwtcKGq9oItlmaM+xf09Zg62W6cpOEs409K23dqnalCWRXm/Bsjbq9jw3YtazZrNUlalVO2vC1pE4ms2hpALzAuLma7DIFMgUyBTIFMgUyBTIF7l4KINfFRFXIiB+Fhk1RCkNukzh5k6M/yTEmf3d2ymJyv1ZC+Sy4ZHX2jH6GRmfsvPa5PXHmqvWdH7KRMeYOtNxH5Xy8j1m06THrkzm0N/7lB3YFwGZGgI32tkmAjQIal5yZjpPsnM5Gf+kT508diDDyCYtl0J7BFNoTTzxhn//8523Pnj1uFg1NmtCmiT1u0LIhHvWjzThH1AlQh3Q85UO2p06UJ75YW5FPXpTxyHv8RL0oFvXjOML6cqRHWrE/pOGjnQijnX9rGOesb+eDPk99+/k4U+Buo8DCu8Y7ffPVS3HRxrU2E5OVb7190i5cGrWJqSYB7bK2YQJttIiTOQfZU9LcwJgG9OOyhlTR+L9VW1qst/v3bbPenhXWJu2aZEGpBtj4pASzyAHYME3L3IV4fq0jwWMWe+8j7+Ye37kp9Dk81xQ+AzZ37j37V/WMm/xRAja8ty4kOfIZbzFCwgLywav2bpeEiGpVr6CyOEIg43UknBVKOzQyqRXh52RmR5sLXtPuB+UOFRRgQ2GVaSxpk0EaRa1ZG1mv7GiwbVu6Bdisk1rdGlvZJRVkdSFtXJVedh56nwOknpxeA//17iVpz9v2zPyTKZApkCmwBCgQH3a6Gh/26elpu3Lliu9h881vftN+8pOfWH9/vw8Wu1Z0SGNmg+3atcsHlpMTU7LnLVNk+nYw+MS29po162T+bIP2q5HvWa/yG22j/BrtK9OuTVJBduC8wxo8nz533U6eGZCANmxjU+LpWklT1QC6ChCu70CjQJeShLSmBqk/a/VMo03Z9asX7Ma1fuEx09oQtsN279pmH3/gftu5c4f6hF3vZti8XO3boQCBjd/g21xrcjU+7vy8lpSDTIFMgUyBTIFMgUyBTIG7kALIcyEjEWey598K2MzLY6JncVQfktgHRWafMyicoKr+V5gvkCQ5zR4JQxMym37Zjhy/YNeGJL+yQAhrGQ1YzJCG+cyU9V/ssyNvv2FXLp7SPjaD2i5hWNo32lBBsihmd3UgiZFtstGySY7zpmvRry8eIr1evkwdQ/yErgApHZKJkY+3bt1iK1esdM30lpZWybH4Fpdn0bhBGyf2sMFcGvXYx6azs9PjEVKnHrzhOFzI+aRx/jiOOOXoGy7y/EA/kb5YXpSJsFiXeLFu1CdcLJ3y4cjHR3vFvPr6kRdtxnG09W8No13a+aDbfq++BQ0WKxf9+kX6FHWivV+kbtTJ4V1AAWcF+vFXEi5HQuIP6eoTj0tja7IW3l0vJl6YSqf3WC9OoQX2pZnTQvs5uzY+Z6fOXbNDR87YlcFxmylj7LxVXmbRmBNWs8wFNM5Nir1OaqEm5tBW2/5dvW4WbRXm0aVdw/7j9MU1bNQXnvOS0tMCTvLghcwvpH7e7r1K17e0frme8LzT4TNgs7Tu43v2lpv8UQM2dKr4+t6uk8Ei1M00/UbEa+vF4wOv/7JAmLHxCTt37ry9c/ycnb2sicFJbVqlF9RfegkorABHQwa7tdJ/1kbWFVu/rllme9baPr38a7vb3SyaFm2rSZXR6mweelZnwwjSa17oqXes1odCco5mCmQKZArc6RSo/7iPjY1JLfmEvfjii/b1r3/dTp48aRMTE9KaaZON7V77xCcftEcf/ZRt3LjRJicmbXoKNWUBNs0tbn87gTUbbe2atda1QhultrdqLxsxUyHeDq6LX85UGrRnzYQd0caCJ04P+kaDU+wnKDCdgXVN5JJWjTYQ1MAYIQ1zldXpEfvZqy/b22++auOjQ9bZ0Wwfl5m1X/3VZ+3xxx+zTZt7BQppHxwRvfhdKcb9QzHPxYObR3in363cv0yBTIFMgUyBTIFMgUyBD4cCyIQxgUv8gwFsPpy+vrtVJgd9QJ6SpQlD3xnqs1CIJT+jE2UtErrke9n0D4zauKydVRvalN0q+VR7J8ok8LXBATt/7pRdG7hg48NXbOrGgE0JuJmS/FmemdSUgCphJk3AzYLTeZkGmE+Q1OmTl6REKnSlQHFSja4lLRiyGmVOuLW1TVrlADbNvn9Na2uLm0kDsAGoAZwBrMF8Gp6FUgHYhDm12P8GAAfzagA03NMAc8gPrR3SiBfNsBEvlomJvwijvTguavGQhovQD/QTz1XUIZ20cMX0SJvPV5M+/1Jru5jPPY+ZGdqYr6NC9X2Iev+asL7dOK4/R6RzDu8Pz+TCZd62T9Stb492os0PKi/ao23cYu2mnPx7V1PAn1v9uMUKXxovcixw2fTe8b4nEKQhAJv5esoCtKm952mBfXod4I8svB+dqtjFa2U7LlOVJ04KSB/WXmJzmqfVAk7ZTldd7gCmzrR406YF4Uzaqq5Gu2dvr+3f3Ws9a7usow0rG7KCpMKa5RU/QItG1TgmHdDGX0IB87W+KrLsHHQOzzsdPgM2y+xWc5OXAmAD2XkPedHdmg+8wj/SUn1WwtQMK8QH7OTpfjvWN2WXh4Bd+NozaYhasVZg64UFsKmWZ6ylkZXaDQ7Y3CfVOkzusC9CcyN73aian805huLxRy9qzrMoSEeyyxTIFMgUWFoUKH7gr127Zq+++qqDNd/+9rdd24b8Vau67VOfPmDP/sfPy8TZpwXebHUQ24UfcG3MkYnHNmuAyGDTB4MCakoMFGtgDaYpUH+exl7tlVF789A5t1fL3mPlOUxTyJ54TYjDdFmTGHATJtEkpPXIXGV5csi++9LX7dUf/5PNynQFJir37d1lX/nKV7xfu/fscpAITvxuvg2TTjycnOQVeLwYEs8uUyBTIFMgUyBTIFMgU+DuowDyXkzgEl86gA1SXsh6KawHbCY0OXi+/7r2TbxgZ84Nupk0a+ywxuZ2iYOSNGdmtUBp3KYmbtjM5IhNjg3a8GC/DVzqkz9ro8ODSh/1fRTmZELNR/36YfifpgAU0amZnxDplE8Gk5m4BNhgWph+4d3Sj4pQ0ruuMIEhC9ovtBsaMBEG8EJIWniOkb0BbtDEwQP0cEw6AA5xPOlJi6dlvkxo7gQYBDBEWtQHyKn3nDv6E88NV4Tj+QkXk4cRRnoxjPp+Hxeq+vNYbIs6UZZ45L1XGmXrXdQljL5FGmWLbUZd8qNMsU59GuXry0YbhIu1HfnUK+YX244yi4VRjryIRzsRLlavmBb1SHu/dYr1c3yZUMDfQRgZkQR68G7Gq+n8zblcTWslAJv5y0910/OkUbl4Ga0wJ+stKuHG6KwdOzdqR2Wqsr9/UMfaS0yj/ibx5AbN17IgVDO28rNWqk7K9NmsbVjbbvfv32J7d26UVSTxtWa9K5RRo272jPletc1503YWlVqvAa6Vp9LL0XG94Xlvw/Ot6evr832JN2zYYC+88MJNly+wS7Vr7uDBg17hwIEDRoVgAoRRrJgW9XL40VCAe3BnAzbBJHjReGYQilJaeoawVSuVZu1WfWNkxM6ev2pvHx+yvv5JnyRkMrBUwhYiCGvNhqEErkZNBna2z9q2zavs/nu225bebutql9CjN19ylTzng8X41KTOnF70CBPnUpqX82L5J1MgUyBT4I6mAPweH99cPujET58+7do1L730kv3sZz+zEfFSBmibtP/MfxRY85+e/w174P6PycTZ2jRIlPCjZmQqgtU3CAgMNNMA1VlijS9q2zEHY6ZmZV5jfNbOXRzSKsdzdq7/hs3KDBoraqq0BW/3drSBoITE5lLF2hrLrmGDKbTvvvQ1e+PVf/ZVjo1aNXP/ffvtD37/PwuwedY2b9nkZtfg0LSiXvk9KMaToJZ4eIpTJI69eP7JFMgUyBTIFMgUyBTIFLjrKFCUC4kjG/7bTaJ9FGRkPgC5Nk0sckaXc5EGWRCkqbxprRi6en1c+9gM2DGt5r58VeCLNra2xjblS59b8mmZvRPnZqVAM6mFQSM2MXbdpsav29iNq3b5Yp9dOn/ablzvF6ij/W3m5WjJmoiRC2KnjrHKkSYyfa7C5W0lq1yicZK5S1qY5DJzbbLT5Wm1E22n61mgH+m4kN0XclIMAAWZHTAmtGQCZAFYiXho33AcQA9pockDUAPgUwR4AvwJoCdAoGId6uGLgBDlOEcASxHGNRBGnKsoXiPpcVy81mL5YnoxTr33W456xbLz54z7Wmy4Ln6rPtYV4+lMz0h9Ru24eH6Sisfz/blF3WL5xcoW24omiuUiv5gW5SIvjnN4l1AgsRo9iMFTSajxWcXSPCgMDV6H4zfF0oOeypODdk06Spo1LM4saxXnVZmpfOv4oB07I3D8xrj2t2VncvGLptbafK1qsSDU9x2f1lxAybZtWmX37evVvO1qLbAXv9NcgAM2+k2ATQBInDHl+buH9o3PT0Qf6dnycby74Xlnw2fAZvncY78SbvKdDNgg8LjTQwhDSOp0CUjhoaw6msqkoLRstK/C5YFh+9mRATt+Zsg3uK7Oae+axlZVBV2trV7B3NnclCYFp7SJdafdK8R29/a1tmYV9hAR79jLhvPVphEVJY3zLzCqdKw3w3PyT6ZApkCmwJ1OAT7gcNTgWhzD/3/+85/bn/7pn9oPfvADu3Tpks3MzLi5hf3790uT5TftN//Tl2zL5i0aDMqEhPNkWkiCmHNFH6RyXOPXxFQEkaks1GZiumJXro1Js2bAjh7vV1wmJmSOghU1mKxI/FlBQ0W8tyLzlGad2r+GlTX9Z4/bD/7hW7Iv/rr4cFlAe4s9/PAn7L/8l//Nnn7mKTfJhhkJXIiG9fGUk6468fBUgt/sMgUyBTIFMgUyBTIFMgXuVgog18UELfGlBdhUfCEnE3xpSI4kyKhda7Q1qTir1doj2uGaBUNva7+Es9roujwn+VMLhsqaI2AegWtvQkSU2bO5CnsmTGsRZ0lWNyp2QabSjh56w86detsGBwTaTE7KjBpmgdNcBM+MS8RqAzPqJcz86pxY/0AkTnIxsrEWI8nuekcHe9O0OojCRCLFKjLtXtFEZgUrIJqj8MVQtfsQ94Pz4ZMMzlmT4zjSIizeyyhHWqQTRj3igCkREgdoAaABuAkQByAnQBuAGTRxVq9ebWu1kAu/Zs0aaeWvcnNtmGyL8oBDtBmgUQA3AeZwXjyu2Ifod/QnyhXLRplIi+OgQxwT1pcp5hXjUbdIn6i/WBtRLupRJjz14p6R5s8Cz4TuWbQZYbHtiNe37ZVqP3GOKFvMu1U8zkv4fuu933K3OmdOX2IUSI+mXhj4W+Jb6QoSJ/M4A/zE9cQr07urB6pWnnp4Pe/+zAOrpPnasvjilOYDLl4ZsdcP9dvJs0PaZ0y8T+A62o4yz6EQ82Y0BYg+be1NZWnXdNreXRu0hcU6WUTqEGcXhwfQUdioVfbzC0ZVM/UGPqk+8J7Rns8B0+jyc0UeUeQJ8J2sYbOM7jc3+k4GbJxV1D5swCcwgABxeDAdUlEh0tjL5trwhB0+cU1m0a5rlfikTc7Ili12EX2DQUAbuaps0FanNTE4Zd2yibh962qhtpuF2q6xVplFwyQaWja11l3bJjEA0lIfUkPFeErJv5kCmQKZAncyBYof96mpKbt48aK9/PLL9tWvftXeeOMNX1VJmV27dkm75ln7tV991g488gnZze7WoAthqsb3JKQ5T0YeCkGtxqthkwhocxqMzqrcyPiMnZH247GTl2WaYthGxiRMNbS6SbRqVQNF2fA2ATUN1RkJXxXrlG3artaSzFJctMMHX7M3X/uRne87ocFz1datXeV71/zRH/2B/cpnP6P9ctplRq0m4KlHScCEVy/E0xH9JpZCYtllCmQKZApkCmQKZApkCtzNFEDmi4lZ4kz2LB0NGwEc6jN/SSpFtkv7IgLGzPqioapdGhizQ0fP2OmzA1pEpPSKFgxpz4QGmU1HHqxWkwzK/EAANmtklrero1Um0Ybswum3tHDoDTty+LDLzaOjY35OJgWxygEAgXysZnyeUMkp3x+sJI9u3LjeHv3UAduzZ7cWRXWpeMkq0gCanS1rLgbQpmLTWnw6PT1lyOeTAofGx8eNfSa5H8RJZ1EVczd46vj11+5bxAljAm+xZzvKRZm4/5SNtABa0JYJrZwIQ9MGUAcgh/RiWmjpBPATx5QJHwAQx+QDBOGJkxbAUABApAWAE/2I4+hz9D+uuXhdcc2E4Yr5Ubc+LcoW6xXTot3oQ4RRhpAyvFfhimWifrRfzIu6UY+Qa64vs1i5Yh3i1MFxnojHsWcs8lMst0h2TlpuFPBXQz/+qCS+BScrcjMxTh0veLeeVpsH0FOuovI6LgI2FfFhwJmh4XHrO3/NDp64buevjNf2G9PievFhTKjTrls5ksYj1pC62udsu+Zn79272bb2ag+vLgHANbCGuVoW2M93VRG64eN/b4v2BN6Lzy5XV+QdwRMIM2CzzO44N/qXA9hAyHibIlycuP4wemleyTQNR0leSjRuqM26cVa0MDF46vyINGyua78EgTZjM2IGCGX4BNg0OOoqO7TVKQE0Qm7XtdsD926xfdrIaoU2tG5GVVnt4X2Njr/8nJGzpz4QSy7C2mEOMgUyBTIF7lAKxOouPuTw1evXr9trr71m3/3ud+373/++m0Yjj0HUww8/bL/3e79rTzzxK7Z1S6/SpKosoaehUV7XlzZXhUMGD4zVLHD2xEGrYtLT5QYbvDFlx09etKMn+21waNqmZhvFs1sEsisUD2f1i4bPqjgpLceK+HCLdbWV7PyZo753zfEjbwq8uWTtbU22besm+9yTT9jv/M5v2yOPHNCgLq3eSyTna8DZcSnu0Vp/goMv9Dnl5t9MgUyBTIFMgUyBTIFMgbuRAsiDMTFLHDlwqQE2SearSXeSVd0ih2Q/tLynJV4OamPrY6cu2sm+y3YdOXRG8mupTdfd4guMqgI+pF4jsGbGfXvrnPWsW6m9brdaz5p2G7/eZ6eOHZTZ4DfsbZn7P3u2z4aHhx1UQSMG+nFOtGSSfIwcjIO2kr5F1/337LOvvPBl+/SnHrXVq1YL5NEejiqfwBotMJVptunpGQdsAGvwgDVFwIY0ABv87Oyse+4XcznFtChTH8acD/0pujiO+0+bOI7j2eC4Pp6uO11flC2CN8QDmCEMsIb0YrkAbAijPFo6sbcOcepGnSgXadEeYdQH8Ik46QF0xHXQ9/DFtPlyGkykeR9yEy0IgwYRBu0YfBTLF8s6XX3+KdUvnpe88MU6xIvOz0eflRjnLua/V7xYJ8aDv0id9yqb85cJBeANjsLE9aTxtD8znsQTKP7pwE3xmacEfHQBsHHtGpUtVxpsfGLW+i9dsxOnr9iJCxN2daTsvJJtK/DsOsN/I+cWYNPSOGurVjTJClKP3btvizRtZLKxRWd2wCaBNaX5fjL2Vz26xrwC/dS74v1MR6QsOxd8gzB4CiH8uy/vYbN87jc3OD7ex48ftz/5kz/xvQyYyGPVxt69e+3555+3Z599VitC9rgKLHVClfUXpYRepZtcEmkWy0lF/WEUGOMvHiip3r9U2l9HL0R2VS8wmwv2D87Y8T6t5j5x1q4PT+m9bZdvkceerV5wTTg2StWvMjvpJndWr2yye2QXEfR2/ZpOa9fKbkAd+gXKm7RtUl9qn0kdxLkjTPn5N1MgUyBT4E6lgPNSBDE5+D4f829+85vO8+H/Q0NDPiDCzMETTzxhv699Yj75iQdtRac2aG1qTHy/SasRXQiCG8KPAXAYbNSWFSqPVTUAMWX5Cdmm7dfKRjZ8PXHmso1O6JsjEL06B2CDOQh4KIKWVjWaABtp2qzsbJWWTYOdPPqW/eifvmdnTxyxyfEhmVtYYffeu8+eefope+65L9oDD9wvDWrUnYMP1wQ2v0KuM10r/DrB78SirBfKP5kCmQKZApkCmQKZApkCdy0FkA1DjiK+dAAbX66p+yZZz/+RAeWYK5BDDq1gFk2ThcPj03b+0pCdOnPFzl24LllUdTU/EHJoyRd1oq2iBZ1aQNRUKtuqlW22d89WmePZaD0rJEVWxu38+XP205/82F555RU7fPhtbZrdbxPSfJmZKYtuScZ06dPFT2RPOc0pzGli8ROfeMj+23//b/a0ZNiedWu1cbZMtutsNbHcQzR9kom0FHIv4p6QzjEhMjyADYBM0sqZntfIYR9KwCRk+mvXrrlnXof0iYkJr0ubuGiPsP5c9f3gnNGHqB/PjTdW+wHwCIAk2iRtHghRubimYju0VfRF02y0Rx5pgDUAOEWNHIAZjjs7O91UMmba1q1b56bauru7PT0AHNoIX+xXxKOfnK8Yj+NiOeLhuJbwkRb5kX676406EUYdjoMuxKONyCePB5/xjccp9D5csf5i9RZLex/N5iJLmgLOvequIKUlwMafMuXDaWs+sRLxOiLALgns5TcBNo3aa7xBe41P2ymB5u/INLosVNr4LNqNsG7mEtSumzdLbbB/TbtMo29Y2257dq63vTs32pruVjeZzvJ73jp84rCciW+YB94LMVwdp3ZTP5W3DF28w4S8r+Hhu8zxsDfxhg0b7IUXXrjp6htUqXbrzA5qFQIVDhw44BXi5SeMYsW0m1rLCR8qBbgHv3zAhkvkkZl/bN51zfSRB89Xd9dePk/z8nyheEMRzOa0iqZqQ+NVO3Vu0N4+fFp72oxKHVoCkVbRWIPQW52C4ppOlGU0TRDKVm2n9h7cuW2N3bO3V2p3a4XmtokJ1AAbnSMAG2dRfq50PlpJ3k+ffzIFMgUyBe5oCsA3g58ycDt06JD9zd/8jf3d3/2d712DqQXAmn379tlTTz1lX/jCr9n+fXutrbVFg5bawEWDE4bIrCDERJoLRFy1AzYITfq2i0+jXQNgc/2GBLRz1+z4aZlDuzQs7RqZgKi2aNDXKF8EbNCumbA2rZ5ZtUKbnjZV7dDPX7Uf/P237OLZEzY7NabB12r7lV953L74xS/IHNqv2PZt23wwFd+AJLjFdyRCOkdO4tvw8ewyBTIFMgUyBTIFMgUyBTIFJL5JNoy5mJARl4aGDdJomqzz+6jrcDc/L8DiIa3ulp+YqUgenZI5Hm12feKiDVyfdLNoVV9ApIlDyau0JcnVfalh1jo7mqx3Y4/2Tthg+7d12aquko3JNFn/RS1AOnFMgM0hO/T2QTt29JiAnIs2OjLqsjHzEa5lw5wF0qfETrTBDxz4pP3x//HH9u+eelKAzTprrWmuL8wlqA8FOb14X7gujiNkXiSAmwgDxAHAQZ4HnMFjSo2QdEAeyod2ThH0KaZFHEAo2iUeptqibY5pN8rFnFJcRwA89DueseI1RDnScFEmQgAPwJU4JgS8Cc0a8jgmDO2dAG5i/50AakLbJurGMWHUjZA28eQBEBV9tEsabVOH89cDOdFv+hz9X+wa/cLfx0+xDYrH8xBVb5cfeRFG3QgjPdoqhrfLK5bL8eVAAXhMcfzMNdX4Ti1M4+iYA62NqWt8C4hmTovifZ5AnJSdZvBT02ZXr0rL8fg5t7ZxbbzZZuY0Aat3w1tXM8nqungwoLkAm+7ORtuxdY0Amx7b0rvaVsr6RrNYKltXANr4mcUHnW8LEEeTkca8PfH8DNhkwIand1k4GHV8XH95GjaQktfLX7Gb6Eof0weFD15CSzmuoAXjH0Fs1ao21yK5bVyqz+f6r9ubB0/5JoOTM3qtS0nLpqIyaOOgvmyVGfGVaTeLtlFm0fbuWm/7d2+WibSVUsODEUg12tmMqosrJAYFA9CBd5VEZxc39TknZApkCmQK3GkUYICHh+dfunTJfvKTn9g3vvEN++d//me7ceOGd3fLli32uc99zp555hmZHHvENm+SOTQNNGF1zvoUMhiFhaLCzCarnu6GuxNPrgLYiGNiO/zS4KgdPnbJQXTAm1kNjgFsylptI+sP3o6Yt9qZtLbmCdmnbdGqRvHr8qS98S//bN//u5fsSv9ZAeyThv3v5577gv3WC1+Rps29Di756jV1IHFi/xLUyO5MWvHISWGUvNPuTe5PpkCmQKZApkCmQKZApsBHTQHGzzEpSxw5cWkANpofwARPzB+o7+6SuKfUNIp3LRvJo5Mzc3bx8rAWdJ7VPMGwTUxqKmCuTeVkFk0yLdUb2E8RLy2b5uY5617ZaXu2r7WH7+ux3nUd3jyThePjI3bmzGn7mfZ+/PGPX7E3f/6mzKSdt/ExASMzs24ViPPTtwbNMLa3t9mjjz5i//W//hdpsH/WNT9aWQzloA7lap2uhdyHcHFv4jhCyixWrr58sRx53N8AdSIEeCmCLwAw4QO8IR/wh2cDDR48i78w2YaptqhPSB3GGjHHRB8CWCqGxKN/9dfCMX3FFfPi+m+VzjUGDaIex4wXAuwBZAnAJcAYwBfigDSUI44mDx7NHYAaQsy0dXV1ueeY/AB/AvCJNgL4iRAQB4sFTTKH5+OX4sUoHv2O/sY1clz0ddXedRjX/K5EHUTbkc7xrcpGmQjr60Z6DpcbBeA78sF+gi0pgb/kGEkHz1LoxfmRF1ijt7amZ6P5WfE3+O/YeMUuXBy1Y8fO20lZ27gx3WaVUpf4n55BgBb5xkY4ZjJLWZLVjZ41bTKFtsl2ySTaOu0n1q7J2SadDv6L53zp+WXuAU+a/+snAzZ9WcOGp2R5OB70+Jj+sgCbBUom0SYd642rOfqI94+Fv418YNKkIWn+wVFZGIlMwZqsoln/1RF7R0zhjFZ2D92Q2jAraEzezaKl+nNlITva5Lq1SSZ4usy2beq2+/Ztta2b1sgcj1Z0lPTyCxRifUwyjQYnSIBNYhDp3OIytZ7mIFMgUyBT4M6kQJHXM5A6cuSIffvb37bvfe97WiV42AdmDDT2799vX/nKV+zzn/+8bdq0yVauWKGBi1aOcVkSqLQIEXlIPviheKKnMNplYMUAucFmZIJiYqaqQfGQvXXknF24PGLj0wA5rQJ8mjSYE78uqz0h6G6Ccm5c5ignbZNW0azqarerl8/bj//5+zKJ9vc2MnRV2jdzvpfO7/zOb2n/mt9R39hXB80f0PUkOkbP6GqKEyb+HF+XKElOdpkCmQKZApkCmQKZApkCdzMF5sfYIgLxpQXYIHfGnIFCj6ZjN89bk0mZNGQ/m4FrmOi9bKfPylTY0JTNzGJ9Q2bTTQuTkBd9XSimx7SXjUCbTq3q3r1trX3qYxtt60ZNMKJtrmLV8ozAmTHtBXnNzvadsbfeestee/U1e+vnB7Ug6ooAEfbQ1cSlm3Sfs9a2Vnvs8U/Z//nf/7sDNiskW6N1g4zKH/Ma755bQLRN8mv9s5nK1aculL9VPjV8zqR2jwFL8Nzveh/PQaRTDgAntHQYR3AcYE4cBwBEGOUDzAHsAeBhgRgm2jDZBtgTbTIfFQBO9PN2/QsKxPVGHY4jTpk4Jg1XrwkTx0UtGcoW0x1oqWnxMPZgvAQIE0BNgD4B+ADihCYOoA6ee455tlWrVjnoQz71o23OV+xDxBfrs1+IfuKaCKPPhMVrDvpQJ+LFetHWYmGUq8+7VXp9uXy81CgA70z8U09XofOJz0YOgEhyNf6lOumZS7o1MhCpP8b8jb5/zTVpNJ48NSiTlJet/8qojc4A2HQkfiqG2qAxPvwW31ia1QLOim3esMI+ft9227Z5jXW1l3yPcYAaFWU6otZN3nUSAGxS72IPMWfmanE5P6vQPDzXGR6+nQGb2iO6HAJu8p0D2ARrmH8L50lMP3EIX7WIBxyRpsfVjzF5pi0T7PqI1J7PXbVTpwdkhueG9lHQJCIbC8osmjBcKlnaYLAstBY7iWXbtL7TN7XaubXHVq/Ux7gxATao3iXABuaUXobUH+LBsPz0+SdTIFMgU+COpAA8KwZUDJZ+/OMf29e+9jV79dVX7cqVKz4wwc7ppz71Kbd1+thjj7mqf6uvNBNg46xXnFZhAm1IEFjDKhauGGFJohmsuqICkwJkrkmj5rT48KHjmJ/QwLjCXmICbKqyKQ5Y4wsjqVCWADYq4HzW9uzaJK3HBjv4s5/aj374D3bw56/bzKTyZCbt3nv22m//9m/Zl770JR/wwH9jDxvvg38H0rdAvazdh8hJobh2LT0HmQKZApkCmQKZApkCmQJ3NwWQD2NSiziTPUtHw0aDfgRPufkpAj9OMwNpvbcmDrVgE9l0eHRGizmva5X3gF3o1/zAJPvcyHS65ggafAEQDQFgzKjZGU2olwTUrLAD9/bYzs0rNQnfKu0IZF+dU4uUqpVZmUkbsXPnztmhg2/b66+/YUePHLNzMpE2qP1jRsfGJY1WrU0aNo9/5jEHbH7ls5+xLk3gN2qC3h1zCWovXUZNVq1NvpGf5hy4vpo8mwqmqsyBFI49cZEf6hbLES8eUyXar68eZYuASqRFyDMDuEKZW4E4ADehocM4BB+m2gLcAbyJeSnai3ELaeEDxCEslgmQiLbwlI+y9C/6ShiuGCctaFCfHuXJB1ghjDjASoA3RQ0bwBu0dvChnYNmTmjokB7gTwA/hLQV7QSYExo6hHE+ylA+QKIAjaJMPQAU1xZ992OuIy7uFmHUq8++VXp9uXy8lCjAu4GPp4Iw0uCpijMJ4PlRRod+TDmAGlnyEG+saI6gogWawq7tkhZtHj5yQaYjr9nwWMXGKx3Kb9W7pFnZJhbJ0y4guSwfsZB+RTKH9sA9W6x3fbJ81CRApqR3N8Aanlw9vnJ8vxKn50iMvNDH9K56+jL8KfK04nudAZtldrO50fFh/OVr2AR7CMbwbmK7TVhnGpTjJUXQAazRe4ng4X9a2a2EcaE2165PuJ3ad0702w2p4lWk8jxXEmgz1yhBzFvQhCE2eSattXFWptDabf+eTbZbqnfr12gPBZm0bRRXaBQ3wKcPEyHno2+RRjy7TIFMgUyBO5cC8EgGPgzCWXXxne98x/76r//a43wDGEQ89NBD9mu/9mtuDm3Xrl3+bWjSQKS1pVl8ryaowf/EBCUe6WKTUETMOTGDXAly+NGJWTunDV6Pa1DMHjbDoxLgBNbMSdORvWsqQtcb4eFafVidnRZwPmbr15rdf+9Om9Z+NS9982/sxy//wPov9KnMjEyzbbDHtTrxS7/xnFYnfk797dRJ2YAUc218C+a/Bh5XL2shvYuj1NOUUsvOQaZApkCmQKZApkCmQKbAXUoB5MOYfCW+FAEbxuWMz/HJ6Tq4FkAVSaiY6i1r/D8xNWcDg+N2/NQlO6G9FSUSJ8Cmsc3N/KaFoci7mh9gxbdk37XdjXbftg7bt2ONbVi/WhPk2mxRjtXeLFRq0OojgAH2sOk73Wevvfa6vfLKj+0tATjnL1yU7F229k4BNo8/Zn/8x/+7wsc1gd/uq8tpJ5kWVsd1DUm25jrShXA/cHGP6tM55n7dykX5yKed8JEWYTGdeLhiG1GmGJKPByAIFwAJx5SFPoTFtiKNsoxPwuRagDikAcIUj0kLDygTdTDPhsYOGjx4xjpo7xTNs3Eezomv71/xeqLP0VfywhXjpEWZiBePow5p9T7yAF8AXgLQCfNqAfYEoBNgTIA0gD1unk0aO2u09yiaOwBBaPHQFuWiToBARSCHexXAU/Ql+h59jXTCyIu0xcpEXg6XMgV41vHBSCNMfGMhL9LTteoJ91rsX1MRPyzzrmmOYLYqnivTk33SaHzzrTN2WcDNTKVZsEynVRo0tyD+2aSV8cIhxUcBbKaso61qG7WIfvcO7R+2c72tXd3urTfqDA7Y1Ngd59Sr5d5NsYnXJx6TOHO6Bsos8KWlfGcW63uRb8U7SQh/Y67npZdeMhbjvvDCCzdV16KDBc528OBBr3DgwAGvEC88YRQrpt3UWk74UCnAPfioAJvEAuZFkUWvK73+NWahAJGJF44XEkd/558uJaZ0nqVUlncYy7PaW9BujE67ncQjx7S6W2rPk0J4y3OYRkMFWV6TihVNVLJXQosAm1Urm6R+t1J72Wxwe4mrtJdCs955B2tUgy54N/yn1q+FVOVmlymQKZAp8NFRAH4Y38/6syZemfIpwwCFgcvp06ftH//xH90U2huyvc0AB8Ee82dPPfWU/Ya0Vz75iU+4fW0++GgXplWA4nksawG4EQ+U2KRTyj6tH7AHjbLksf4wLe2ZwSEB5qcv2okzV21geEb8t0m8VyrQGuKqK249jRWKbg6tWraOlinb2FOyXTt7bfBKv331f/6/9tq//MhGh6+JB1ftvvv225ee+3V75t8/Y/fcc69WrDHYTRo2nNjZsnqVQqgBj15w5ODJj9ILuTmWKZApkCmQKZApkCmQKXD3UaAoSxJH9lsyGjZM0umWLch+Ka6rUHoCbRBTfW9FATYzsw3S/p4SWHNJizr7ZRZNoIDMoqH9PacJRFZnO8gzp/mBBoE2kj9XaAJx29oGATbrbOeOLba6W6astPt1TQp2ObaqfRzLAhPGRsftyuUBmf85ZYcOHbY3fvamNto+LpBhyh5+5GH7wz/4A+1l86h1dLZrojJNLC70nRhzGwspPI3cE1x9uifWfqJMMS3iUa9YphiPcoS3Si+WKZaL8pwjzhP5kRfH0cZi5XjmmI8KH6AKIWmExTTK40kL7RzGOAA8RR/aNoQBAIUmToA9aP0A7vDMEwcgIm+xvnBOrgsf8Qjj+oph0CCuOY4Jg2aMZQBTGIuFRk1o0ZBHvD4MoAcwBvAGcIZ4ADwB7ARoE2GUB+wB2GGxHiAPIWmh9RP98HFWAdwp9qPY/4hz7XGtQYe45ihDWJ/GcTEt6hbDyCet/hz1aXEu0qNe/Xkjb7G2yCu6aCPS3k+dKPthh9G3Yp8WS6vvx81l4DMxdg4elMI58dmUt/Ce65YVnMrpH77LfuFYPCqL8Y5OaM/cKyN2WnMBZ/oGte+V9rbSXKxsG9ms+DGmNtCuwZeU2tw4Y6tXNQmsWa/52I3W29NpXR2Yq9R8BO+MPMo4nIyewSoJfW6iru9pzI+GTQZsMmDDQ7KEHS8rHyQ+Nh+2hg3vF8ITzl+u2q8n1H5Seq2k+pYczGGhFH12JoNwQBswfrL1g3gGYIOfnC7b2fODslV7yTcXHB7Vqowqk4bNKiWhTKyhIg2bqoSo5kZNGEojevWKJtuzs8fuv2erNG5WyFSaBDK1D3NI031EuArYgJiA/uJq6EJ2mQKZApkCHxUF4IMhoDlP1Inj2AcQHNfS4POYH3j99dftL/7iL+yHP/yhDQwM+MCGlVn333+/ffGLX7TnnnvOtm/f7sI/14FwJHZXa0gC2yKADeeGG8KLsdc9MV2RjdphO/hOn7RrBqXx2KS1itrYtUGAjQS4OWk4iomLYQu5EViDmvPqFdqjZmOzNGnWWv/5Pvurv/wf9uYbP7Xx0RtquWKPHHjIfv/3/7M99tinbf36DdasgUlJJi4aNJDgvGpCzn84TPF06Klp1WRwbC9AoewyBTIFMgUyBTIFMgUyBe5aCtTLksiPSwGw8dE4crDfOX5rQp+HCbBJI/YE2GhaUCu/SzY6LrNomh84pvmBi5dGbUQmetwKh8yioTcDYCMdDIUJsGnVJGJP54zt2bbO7tm/23o3dFtnuzaNV2ktW5L8KdkYG786vU8OKhyRts3pM30yO/ya/VSmh69dH7S9+/bab0hL/MGPf8w3qmdSPFxcQ5Lhl5aMGuMPrqUYj2srpsUYJfIijDK3yq8vF8fFkDbwPL/hAWkY/wRwA1gTwA7gDOMiNHJCQ4dFbDz7AdpEvWiHtgI8Ih6etDh3hFE2wqjn47NaX6PPXEcxHtcFPUj/RRx1AgAKTZsAaggBagBp8Oyns2bNGlstLZ0w1Qa4Q72idwBH+5k2N6W9e2g/wJsAd0KDJ44Ji2lRnjAc8TiO649jynAti9GAtHDUi2MPqbNI3Wgnysdx1I32Fgujb+RRPuqQHmke+ZB/6s8Hf3N2Vztvfb9u1bdo593dhZfWnjWnb6LxwuMHnb1Fr+Yla8VJ8PWbCn3xvNI1DWuDwxN24uQl36Ji8DrvXtJolLFJB3SYB4B/lhoq2qtWME5rRdo1HXafTKHt2dVrqzq1T1QzIA08lrkG0d//CJN3AqSuqvcLHeJqtORUfV543lRlWbl4LgnjuSSEx/RlDZvlc6+5wXxIuLHLArDR9Thgo4d1tlyxqzKLdkomeY6e7LdLAyMCbJrds+l1g0zp8F5jFq1Rq2haBNp0ts3Z9s2r7L79m23bpjXWvaJNDATAhtceJs0LkQQCZxjOBGpcYvk8FvlKMgUyBZYQBeKDHR/r6Drp8HYcg45Lly7Zyy+/bH/+539uP/3pT33QQpne3l773Oc+54DNk08+aWvXrvUPP2A1fwvCkAbAWm2IQJTEJlbHMFyFJyoqPyvAZkTajWcvXrPDx88rHNJ+Ni1SfW7XOFgrGGsbFaLaDFjTIN+iZjasbbE921cIsFljw0OD9vI/af+aN39mVy5dsKmJUQ1wH7Df+93fsQcfelArwVZIwGegq97Bg5EgOTltyieOTByXxE8Am/hL6fk3UyBTIFMgUyBTIFMgU+DupgByYHGiD7lxKQA23DXkzySnxj2U7OfyIPJqkgn5TbsssqeCFnTKdvqg5gdOnx20o8c1P6AV4L6PTUkrN0ta2KkymPXBY1CtWXvdrmqdtO29q2z/vh22XcDN6pUy06OGkYCRZzGhjrTpp1asUqnauDQ1Bgauat+G89Yv+RszaA9Jht28ZZMmvtlTl766JOtiLPJsug9Ld16BZ+n9uHjeKHurOsUytyu32NTLJFoAAEAASURBVPlos+h5pjkmDOCEMACXMLNGGOnEA+AJkCc0dYp5xbRID60dgB/eJUChMO0WGj4x/1bsD/HitQZtIoQm+OK1Ub5IK/KiXAAii4WAKfU+wJYAejDRhg8NHtIBfQKICVAHzZzwaOrEXj0BAkU7xbr0Kdrxi679RF+5hogTFq8xrpm0oEV9WtCgSLsoW8yrbzf6EnSMY8LF0urzOb5dm1H+VmUivz6M61gsPa6nPo/jqBd9rz8v6eHZU0YklYOuRT7Ec5fS+S226XEyuRfyzBKI/WnxZtX6B0a1d81Zn4sd135hZd/LVho20q6BFzO/CtjdINPnzbJ21L2yZNu2aB72nk22Q3uKd7TqGXHtG4E69Ep1olfMx8I/i855qhI9VF8csBGHXq4u7hthPAOE8LkM2Cyju84Njg/GRwXYxIsGM6h3KSXePhhIKqFnb97Fwwlj8NeWvNo7i71abCbCMFDJGx3XPgoXr9vBw3129sKgTZXFJCpiEtK0QShLK2GYeqzqdS5ro6tZ27iuzfbtlirezo3Ssum2liZed5zK6VwO2Ciezr58UVu/5PyTKZApcMdSYJ4X1noYH2vSw/HRxqHu//bbb/veNS+++KIdPXrUP+gI5/fcc489//zz9uyzz8r02H2++oo63p7zadqTr4E1rD6EA1oDasrit8pKtsIFDJWrsg8+KlNoV+y4TE5clq3w6SraNRrcCrBheAsfLaktq8herYS0NiE22zettPv2rdPKmm4rz0zaub5T1n/xnF0fHLCxkWEBOb32mc88blu3bfVVX5heUwckzMPH1aBfc+qnjuS8gJfxI/+I1L4ZJGSXKZApkCmQKZApkCmQKXCXUwCZ0eUo0YE4cuNSAWxuunUu+vFTBGyYvmMiUV6LhmYlfo5NVqRdc8PePnLO+rTP4tQ0utyyvlFq1spvgB3RAdBGtGjUvgpdzZO2QXsp7Nyxye7Zt1laNqvcZDoTjoA12jcbabgmemriUueClpOTk9LiSPuqNDSWbH1PjzQbOiW7JknVK9BdHbqE6qDN0p5bKI5B6u9PPGf16RxHvduVKZarb2OxetFmfVnS6/OKacR5DwBgwjNfRnyxkLQAXgK0AaDh/hMC4BRBnwBtou3Q5ImQstQNH3Vjzo6+FX30lzTiRVosdlxPj+IxdRkbBhhDCLASYYA6hAG6RD7ADsANAE2YXQvAJ8yyBfgDeEM82iWkbphnox7aQKRRjvPRt/CcOxxpHEefSCftVo486FJ0tytfLEfcAQFVr69TbLM+z+vVzrlYXv054pheztXuK2lRN8IoF8fFPkQeYf1zUCzveRQK4NnLe0LtfInupOCccrqWOJf3EZrKAzmWZRPtxtisLB0N2dvMwZ6/Ju3GVvFgeS2al+V0xdl2Qs8qgM3crOZgy7Ze+4nv26M5WM3DbuxZqTTdV5Vhnhb+GLaNUg+8F4ou3Ge/L353lOc8Vc+MavmBfpebg/7huZ/h4QN9WcNm+dxubnIw/w8bsIFq8WITD0ZB/NbOWcBN2Qvt8HAutMskXlUJ2KqFJ86Ka1y6OmI/f+ukby44Pl2Sip6gGTELRDBUnwFtELJKUn1utElbtaLBdm5dLbNo22z7lh5rkxSGIMZ5/JUX44BRwDiWKwO4ieA5IVMgU+COo0C9YB481T/e6i0cKtzVq1ft7//+7+3rX/+6a9egbYNwiyD82GNshPrH9sQTT7hqPEIzbUR7zkxrApCGMBK2AGzkAGzEPxGWsFeLn5oRSH5B2jXaO6xPIPnQqAYxDZ0CawTYlFq8zQTYyHzA7KQDNl0dzQ6SP3T/Zlu3qkMDAw0WXECr2Oz0lGyCD+tkMpsm1X36y7nRoOQKm5tkGs2F9hAc4c0476GXoRxCJC7xbY/mn0yBTIFMgUyBTIFMgUyBu5oCRXmPOLLlUgVs6H9ymuCrjddJwSctmwBtGrTX4rgdfkd7LZ4esKER7X0yy7SgABuZ7WX1d1X1y9J4aKhOWXtpQnvZNFjvxjX2iQd3224t6mzRZtlpJkGG1iUKI4oiaWK9B4c0yoT8jOTiRpmTapI5KSZEPUcF2R/Sy2vC0vutLF+E5Brk3kT++QUoEPc+xi4cRxrNRHo0SR5pkR5lI4z0KE/IuxFaMMV0j+tmMsagHm3gozx14jgAHUAYfIA0gDMsrgPcwUQbZquvXbtm169fd08a4A/lAyAipD3CSOOc0fc4J2nRJ8JwkcZxMR71F0uvz4syhJHHuCzihBzjAWkCiCEMMIYQoAbzbGySjsfaA8eYbQP8AbSJtgBmAIniOAAjxq8B2pBX9NG/SAs6xHGElHsvF7RarE6x3du1E+VuVYa2cZSL5yfSgr5xHG1EvwiLfVvsXMX82okUpOdUT69OrOfYn+favaz1h7KpZ6lvHCfNmjQPwOh8Vio2V69rr7BTV8Vjz7sWY0Njh/irLG44fxWTFC/UcF9zq+xuW5Y5tKpt3bzaPnbfFs2/rpaVoxbxSPIBbLwnPg/L+XTGmiegN/M9UjzAelLh0GpkPl/RZeQWu9/cV973DNgssxsNg+fGLhfABqYxp0lEwBst9rbrN8bt6IkLWu19xZnHBKto5lrETAhhQqz4FiovhiE4R5tfl23T+nYBNtu1n81GW6nNrlqF2MCnnCWIwdRYmJ6EYBDL6KHIl5IpkCmwJChQL4CF4Fb8gMPbWcl1+vRp+9rXvmbf+MY3/CPOoAABeO/evfbMM8/Yl7/8ZXvggQdcmEYQxEV7CEZsOOjnE//jL41HKZfEKExITM/OadA7I+2aS/bOsXN2Rdo1k2XxzqYuNdaiWqnd0GgszU1LQCvZhp5Vtlubue7e2m0rOiVsqxggebPkLGzbzgi0oQ9t7W3Kc2NqOkIQT7w79ZM+JgEuceUQ5pxruzAJv+Yvu0yBTIFMgUyBTIFMgUyBTAFJV7XJPWhBHLlx6QA2yHr024PCTxqrIzviAV9IwbAOY38AmRtj03bm7DWZ6xn0fRdHtCJ8VlY4qlqMRJmyGmVfxobqrKYZJ6ytuWzr1kob/N4d2hC717q7mt1kT7PESmE3SRpWiNke+uOysuqjpYMWOk5iqyauWZSE/Io8iuzKFCcuzUn4QqiUcMf9Jjn75m4tjBduzivWeb/lbm4lpdyqPueI81AmykVatFefHvmRHuW8Pd0b/m/lqBP160PqeBu1+x5x3i3ioY1TfxzgS70pNo7xAfbEMeBN1GE+b2ZWxwIISQtPmWJ7AEOAQqG9Q0jZ6Bd9iv7RZvQ96BDHhHGdQb8ISS/GGVcGyBKACmmRTgjoAqADeBNADiBNeNLIB5Qplou0SCeM9ohTn7oARPii5k+0ybnpV3jq1zuul2sqXleU8TTe/fcYYwbt3lWvdkAbQdPIJyymFeORV0wjzh//XMOt+lusE2WY20SrMCHOXExca+JL8M9w8zFFamcUb4W/prnX8amyne+/YUeOXbI+mZ4c1txAAmyaxQ+pI5PosrSBtY1GnbNV+9SsWdliO7f3aFuKTZoX6NT8gO7H/JxrOjvnnT83F+mPIClxv0gAsEmzFBmwecnBzxdeeEF0ebfTHsW1N1jpBw8e9MmhAwcOeAV/oJVefCiLae9uKh992BTgVn10gA0f04UrEh+QW3jtFnKKMSoUKtWyFh4xmIlK1BomCMCGWqz4Hp2YtvMyi4aJnj6p5N3Q5oLsZTOjicRKlRnBpGIJ02hqmJWt2mlb0y0zQXs22/7dm2z9mo60uaDOk/pcZBrv1f/iteR4pkCmQKbAB0eB4HvRYnxXYwAQ/H1wcNBef/11+6u/+ivXsmGTTfJYwfTZz37Wfv3Xf921a7Zs2RJNeRjfZhfGWDKIZ9DrPBeRKK17AaypasXM2ETZLl4Zs2OnLtrJ0/12YxztGoEszR3i4tLaAUV3kQ5tRvHa0qx4rUxMbO+1rb0rtbkgIE0yL9Gi5YptrRK0m1lBJahHg1tWKPLNgM83uCCKgBbiMd8X2g+RMsW9PN8ZZ96UzTxbRMouUyBTIFMgUyBTIFMgU8Blp3l5T3IUMuTSAGySnJdkv5DtIkQWrOUTFgAbzKKx3+2EJhWvXJ3UXjZXBdpc0qLOcZ8bmGto9TXfs5pEQM5twALH3KTmB2a1MXury6xo2GzasMJWr5CcKtEUwIbzFaVS10bXqRFNE2ijhaGSXdGswaNt7n1HtpZzoAZE5w6WUxOtvbvv+onn512JtYNinfdbbrF2SLtVfc6Bj/wIGb/4Y3CLBm/Vt2ivmB9N0PZ8+9xcuWK5YrxYp77crdopth1tEeJ5NwFXAoQpatuQx3H4+XIzAnumpucBGrR0QpMH4Ca0dgBpmBOMenEO0iK9mEc8gB3C4tgz+hv959oBQ7g2yoWPckXaFOuQ7u+M6gb4EsBMgDoANkUNmwBfAgACsAmtHsAa/IoVK+aBG/JpM3y0V2wnzkle5AfwxDXRxwBI6HPcw1uFUYb8KEMarv76U+rN6UG7KB/txDH9CRfniTzSiUd6qsszVgNsnI3StwTWwJPS6BqmlVpV151V8Y4JInGeSpnpmaoWy0/byTNX7R3tEXbl6oSsb6hwCQsZ7F2jBkpl8Up5nQ/tmq6ORtu8cZWA8A22e0eP9ghr9b1rfPyvWnH2dGY/7Xw/0lHwTbXdIC0h9Si5bBKNuZ4M2MSTs0RDXtaPErApvF23oJhzCOVFuFgxGArpiWPAMIIBkV4EbCgxJbtow2IcCGSHj57TngpjSsOGrdTyNInY0KCJRJUDF27WS16SUNbVVpU63hrbpxU0O7autdWaVGTF97zJWdVIrOF2/aSP2WUKZApkCnw4FIDvISTjELYQzhCC4enk4VG3P3LkiL300kv2ve99z+MI2QioaNd86Utfsi984Qu2Z88e6+7u9raCnyYBLnFa17BxASgJQfBMAPFqNa1WRGPx2tCEazKeOjtgl2WKchIBrVGATVO7Vu1IM0Z9wxatjESIz05rtWLFNm1caffu32qruppscvSazU6OaXBctc6OVlvR1WErZeubjVpbW1r9+lieyEaILoj6wNa7zEdAkeQTV45jUiXqIXSqxAJgk3l3jXI5yBTIFMgUyBTIFMgUuEspgMw3L+8pjhy5NAAbtBWQSZH3koT3rltYkAuZG/AykgVZZISGzZQ2UhgZr9jZc1ft0Dtn7NLAiCYWJWNqoVHFmtys+pwWdaalSVO+0KittcHWaC8bNsTev0crwddqJXhLQ5ojQNpUf5LmTG1iWn1gsZJr2KgLjVIhD7DG+0oflxBg8y763uEHTCT7P/fAn4V3dzie+QgjN8oSFuP1+XEc9QnDR17Uj2PCYnmOKcM7Rxj1I4x8Qly0R1jvo41Ij/Ick3c7z7gxAJrQxiFE+4ZxZJhpGxoa0p5MN9wD+oS2DqAP5epBnABy6oGd6At9XIwepMV1RBjXE8BPHFM2wJKgW5SJ4wgphycfQKYelAltHPbOCW2clSu1oFBm2TDPVjTRRhnKF9uIeLE/cU7COGcxv0gD4os56BWueL3FtLhGwqAZYdGRF65Bk5r+p7SUzjO1AHaQt8BXAzIp1KchHerpSvMBOkDLZnR8RvuH37BjJy7bmXODWjjPfIE0azQPAFhOnVLjrAJp2WjvGnFaLd5s00L5rQ7YbOypLZTn5XUYaGH+lV64UzDfP/Fyd1yba9YAH+Epo0WftRnbVGh5/Rbvc/H+87z05T1sls/N5kYvPcAmPljppeU1FYvxmwJfcsDGxat0n1BpnpyuSDVPG18dOWtntb/CsLRsyjKLNldqV03tZ+N8UGp5DthMWVvjjPVIINu9fb3t37tZG2Gv0MbYCFmJPdAy5wW0yS5TIFMgU+CXQYHih5o4AmDwdBhURfu8YP/4lVdesa9+9av26quvuk1k+tqjjU8feeQR+83f/E176qmnbN26dS54Ur/40Yez1rhrQZBDqBNYw6BXXLBcKYnHml28fMNB8XP91yWgocnYJB4roKVRvBaBSpUaS+rn3Ix47YxW1DS4ndp9ezbazIT4889et5PHjmrPmhvar2aFbIWvt927d9kD999nmzdtdq2a4LxpJSJceL6DHknCXPQ6ek45eRdUg4fX6lI/u0yBTIFMgUyBTIFMgUyBu5ACIfdx6cSZ7Fk6gE053THkO5+4q5PtEFaRYsl2OTAt+gGwmdXYf0o//VeG7Yj2WcAKx/CotAbmtDG2zPhWtCDJ97tV3ZLJpI9k16bGikz1mG3dtMoekGm0rb2rbIVWh7OfTZMvSOJUfibJyDV6uijqP94PAB1fAFpL9/I1+dRnFjyeLiv//mIU4PktuvrjYl5x8rqYTpx6xbrF49vVi7wI69tdLD3OQ+j5et5ipFJfP45v1w5loq1iufrzkBdpvPOAKswJhgd8CR/gDcBMmFIjrQjycBxpRS2fAGtoi/RoM8pESF1AH9onTrnoC2WI1/c3joMuHHNdjIeLLspFSJnwAZ4A4uCLmjZFAIc9dsKcWmjaUBYQJjRwKI+P9ABo6o9Jpw3KAgyFyTaOSad89IcwromQY/pcvEbScRFynXGtkU5e5BfTiONSEwnsqA38PT1xM9pPS9U9UT+pfb1vyoL1luGpWkN69dqEwJpL2hvsqsenKwJrfG8wgBt4n+YCHLCZdsCmrWlOW1F02wP37LRdMom2akWTFnTq+rwTCZAvLphfeDdqJ+bk+ndXA2wET6ZDwezJLFoUqJVbJkHcZ8K4v4QZsFkmNzgugxsMA+TGfvh72PARffeHNPqxEMYLFeFCTsS8BX8wU1uJYaRcJc8DNm59h2bkyzoYGBz1jbCPn7okLZtxMRap5pXYAKtmT7GKgjQaNtNKmbIuCWTbtPnVA/fvdG0b9lZo1sqY6Blsa4GBRO9ymCmQKZAp8NFRALC6KoOw8PAQxOLDjdB75swZ+853vmP/63/9Lzt27JgLywiCaNT8h//wH1y75uGHH3YhlPrUDUGQOFwWENz5nlbeVGUaAkEIAAYhjsHsTFm2wEervpLm7SNnXIsRs5MVWf2ekwZjg2+gSguYnUQhWubQGsu2dnWL7dqxVvy1286eese+8dd/Yz/8/vft0qWLtr5nte3ds9s+8/jj9sUv/ro9+OCDuj7EN9pB6EyCIzw/8WTlpE+CEpKAF9+bqON3xa8lteHH+SdTIFMgUyBTIFMgUyBT4C6lQMiMXD5x5MmlBdggCCLXMUZfkO+4FnfzgmLK870WJMOin455nmvD0g4/NSB/WeDNiGRaLeQ0gTZaFV6d08IjJOEGrQYP0Eam0zf0dGlzbE0wbuuxtas06apFnexlw7wAWjZ+au8TNNWJlE6QZPU09enaQcr0xaBelv5Jtq3V08GycvP34zZXFeOY2xS5ZVa0H2EUjDYjjPT6sL5e8Zh4HMcEerE+edxbQs4T54p41I10jsNHmcij3ShPvJjOMS7yi3mkxfFi+almqhvlimkRjzzaiHbIIz3yOI58Qq6d+USAlQBdCDkuAjYB6pA3oTHqhEyyob0DvxkZGbHh4WFDi4djxrDUpw5liHOuYnvEi2lxDwijjxEW+xtpXugX+KmnAVV5HgBZAnQhLJprK4I7AbaQhoYO2jtYtwjPcYBCAfxQljggD/WL4A/9CcApQtKCDsSpQ91iflwy+ThCopgg0531NKcX3E+HOot+AD8oxL2HlylPfwzHwUzKsrQxMVXVAvlhO3ioTwvkh2ximr3AMDEpPRrdE+ZmS1q42dg47RaNMIvW3dliO7b0CLARAL5pjfYFaxAAnngp8wbMNkSfdJqaI8Y8gBwnd0dHE+CkJ9Jraam96idz6rVCyyooPtPpHqZ3lPvflzVsls+95kYvJcAGxoBzhlGLcw3JJabC5BzmbyJVh2IWc1o1o80Fz123YycvWp+0bCammxy0qUgYg4FgricBNrOyVTsjm7SztmFdu92zb5Pt3rFB8ZVaVZM2xOZ8sIlgEen8+TdTIFMgU+Cjo0DwvhDMOHPx4412zU9/+lP79re/7ebQLl686J1DSHz00Uft+eeftyeffNJ27tzpAl187CkUQlzSWPRqahzBuKJVh6hM8yczZ/IIaP0DY7JXOyB/2QaHpySCAdYI9S61OGDjApeD4mUNasu2sqvRtsns5K6d6ySsleyNn/7I/uf/83/bG6+9asND19wc2j3799kzzzxlv/Hcb9jHP/5xeqX2tOpGQqNLljqG/Sc+LM4fTH9e4CSBflKixq0RSovHtUvLQaZApkCmQKZApkCmQKbA3UYB5MZ5mU9xZMolA9hoARByXpLx0sg8yXjIwzffySQVUiOZ72FVeNrLZswXdbIqfGLSbKaiOQItPEqADeVlwkeyqwyqy09Zt2TY3TvWu+n07ZvXSMuGRZ21uQE/MbJnOg+/9A8pNMU4e60nClJcITOfzC7UJlF1sORcjEv+rR2P53Gxdm51jqhTnx/p0VbkE5KHj7Qo80GHv+g5os/Rx2J/in2NcsX8iEe5+jKRTrliHun4SCO8VVnq+hOu5zfKwDfwAajE/GK0SzppkR8haeWK0mUVAq0aAJoAekIrJzR7QvOGfEAczLIF2APgwzFloxxth0YPYbEP0V9CXFxvXE8xzQvUfurzSY6FjoAjAcpEGHm0XwRNyAd8wRdBnWIcoIX80LwhZAyPtg/gDiH78cSePGjrUIY+QgPOWQSSisCP9wswZ77PcCgWf9aeCw6dafHDXjYJsIFzwb4w8+j3lkWSKjIjZcdrGv+f7hvU3jUX7PLAuE2XW52PsnMtAPmcEG32DG9smHDApqXZrFfaNft29sqq0QZbv7bL2prCxCTNwiA5YyzEhI/WOuZADfFw9AOghr8AbLJJtLyHTTwfSzjkRQuGeqdr2PC68gqm13ThBU2Ms5argA0CnXPUivA+s9fChMyiXb46ZkdP9ktN74oNjVRkv1bCmCYWMe3jG21pQpENsJq0iqbJpqWWV9LmgmvcTu22LRtkwkcbmzmzgeH4WZbw3c9dzxTIFFjKFEi8LwnLxBE6CSOOds2LL77oGjaHDh3yVUsIbps2bbKnn37avvzlL9tDDz1kq1evdjIg2OGLDsDGRVm16xKa+GPVfeLGmIwYHp2xU31anSi+2n9Fq6Qm2HxQYE1jqxbkSMPGubZ4d5UVimWZlzRbv6ZdGjTrbfvW1TY7PWY/+N537H/8X39mRw8fssmJcQmfHfbJTz5kX9T+Os8++6zt3bfXT88KnxIaO95P+HbwYbVPF3G3AGzI9musfUW8bP7JFMgUyBTIFMgUyBTIFLhLKYDMGLJfyI9LBbAxAShIo0kSRH5NHrnTxdbaPSU1RERCZhN8mabG/5hGuzE2bUdPXLF3jl2xweta+T/FYk8tPJLNjSQHY3qNPRcwjTZpHS1V6+3ptH3ax2bf7l5bt6pDsq0ma1XKJxlVk/Onc+rsdIA8T+DstRzvZJSiHIAN8xhL0/H8fBAunsfF2rrVOW5Xp9hOfX3qLZZWrFMfv115fx5FhugPZRc7h7dZfDALJ4m6hSSPFs+7WJlbnbvYzq3aKKZTfrH2SY9zRPkIi+VJCx9tFceokUZY76Ic6QGwANDgASGYtySOlg4ADUAN/AofgE1o50Q9ygaIEyCQA0WARfKcswggUSbyi4BPgD7RxwjjWoMW9df0fo6hX4A7hAH41AM7sbcOYE09YAOYA+BDP7he2onyaO6E+bUiKEQ8AUfs6aN9dpox15aAJNKbmojLzFuT5kDlGwvaOjwLopxzM7afuCjtmuMnr7jFjWsjAHHtbh4d8+lz6kvVWaH2rSlNyk9bV3uT7dy2we7ds9k2b5C2UafOIyYqC5M+m+sctAbYzPNMZ6bikbcFbPwpVRuATLV54fdzE5ZYmeJzx/MTnucya9gssZt5u+5yo4NR3TmADV+vmx0iAK8fuWkCMJVJzHEh18EXXk8VTIzEX3etljGBNNMO1ryjjbAuXZ2wsUk4gtSe9TJX2fTPARuZ7ZEACGDT0VrW/jVtvo/N/t1btSq8QwxLzEycpJEX4+Zu5pRMgUyBTIGPnAJ8nOHl8EM409T0lL355pv2Z3/2Z65dwyaRCG9smHjgwAH71V/9Vfv85z9vW7dudUGNDiPYxcfeuafYKmCNc1dvVzEJTphxgF9iSmJmVuYmZa/27XfOCbC5bDfGq1pRI2FR+4PBWxtKMiehsmkjVvHWUsVXIm7ZtNLu3ddjPetarf/CBfu2gKW//os/t77TJ50P98gk2tNP/Tv78vNftkc/9SljhYjzehdIEL6SAKYuuPPvAgfOlL3H0fP5r0YqyrfBS6eK+TdTIFMgUyBTIFMgUyBT4C6lgMuNLhdJ3pMchTy5lAAbl1dd+AvZLoXvGqXXhMWQAwOs8UWduu9jk2U7f3FUoM2gnT571YZuaGNG0z630hLHakdVe9xWqwKHqlOywjFpLaWZeS2be/dttc29qzXZ2FabaEySc8xYIJj69CF9qMmp3jfmHXTsC0DpMp25wwEbl8N/wfeEcUW429Uvlovy7xUW2yvWj/Ri2nu19UHkx3kJ49yEkR5pnCvSomx9XhxHOepEGvFwkU8e8fr2ivmL1Ym09xvGOYphfb8ijzaLeZHOhveNbi470SHSKQ//wcX1FI89Qz9RnpB8fKQFIANYE2BLmGfjmLFw+AB04phygD0AP/gAggIUCk2faJdxdwA9hNGPxfpJGn2sd0GfxfIoS354RRKnU+hjdtGxJJ5BfgA8xIMWxNGqAagBzAlza8RJCwCHMm3anKuzE+2dpMGzYoUAIWnvdHXJd3Yr7NZiSmn1dHZZe0e7zx00yLwZVjewZDQi0PuEFsUfO9FvV6/LjN0UZtDapGHTZBUttgT4LmvB51x1xlqapqyjrWxrujvsnr1b7B4BNmu71Qe0a3TNADaJaySNGWYjOFYT/Oov5gF0OE9SPTfzGjbB6cMkGvWWn4v7TBjPCCHPYQZsltH95gZ/dIANhJt/q25BxfR63iLTay9eIrXLr5s385dZB3pxmVjkwa0I1sV0z+WBEbdTy+Ti4PC0hLEOMRL2W2BdTI3paVKyNMc+C7MCaRqE/vY4M+ldL9XDTtmQVCcaUemb7xHMQ4mO9gZD4SrokTwMOnEZHXEFN18F4lxyxbBY9uY6tQo5yBTIFLjrKACfqPEK8R1YjPM57WczMc7eNX32w5f/yb729b+xn//8DQErM15+j/aF+eIXn/P9a+6//35p16yRkAeoogb0z4deuI2HkJQzuOeHuNiQfzdUfqY8Z9dvTGqj1iHx1Ct2/tKITc7IsKRWJTY04lO71bIGuZVp8cwZCWgNUn9eYXt2bhRfXWtz5XE3g/btv33RfvD3f2cDVy5rBU+jberd6KDSb/3WC/bAAx9z+74VXVvqX2jY0CHvlv8kDllL8INC5nzRGq9eqJZjmQKZApkCmQKZApkCmQJ3JQVisoeLJ44suXQAmwSOeN/n7x4CIF5j+ndnuMzocqzSkRA1Q+CmeqZkWW1Qk4wnzwza4XfO2pWr45J3tc+t9rHx1eEa+LOW3Bd2+oxBWVo2c1rUudL27lpvu3eskzmfTk1EylCw5hDSDjlpktH7wsnk0xwAMjYdk1NniLu0qh/MoiXTaCk7/y49Cvh4qtZtxiy3c79I2du1U8wrvs/F9A8qXuxzMV7ffuQVaVBMi/RIi/B27USd+jLF4zQWTuCJgykCFDC3FoAK857hi2ALcdIBcQLAWSwsAkHEo62oB+gDsFM02Qb4AxAE0MN1xrnoK8eRFn0MWni+Lo7rjkWVXp56iWsszCjqUaPcnCZC+fN5SdWlXmjRJG2dFj8mjo+9cQhbWmSCzffMaZYpNQEo7tulPdhmq1bJjPmu3XbPvffZbu2Du0rWOViUWZ4ryYpRWTxz1BfFnzt/zUZlaQOzktUGWTGSliIL4+G7vjh+bspaS0O2dqXJykav7dm9xbb2rrXOVp1b8w/MxjLPCpANh65xRw7kSJTnOheuvJYDg9WVa3421YECfAMAd5ani2eHkHsfnucmAzbL6J5zg2E03NgPX8PmoyFcvKbpbFKrkRmeCoxNgA3me26MzVjfuQF76/Bpu3D5hnRppK6nDbHKRRM+INYwCak+twi06V3XJZXn9bZ352rrkSkfhLEm2bLFIiPsQLi2VpDDYogndpL0nmmDHpGYmIugIC9DKQQ050deANZDStGTC6OZb9VL5p9MgUyBu5kC8Igk5AVvcM4jaWhGRmSvXBm0V155xb71rb+1H/34ZTt37qyVJP10dLTZwwc+YX/4h39kn/vck9K2WWvtbR1iTWgYqk0++Gq5BP97l3wTvCuZR2MIigmJcak/n780bEePX9bmgiPSYJQKuQS0OQlwJQl+uDn47+ykVWfGXP15dXeT7ZLd7/v37bCNPavs8sVz9q2//f/s7779LTt08C1NEoz4ip8dO3bYc889Zy+88ILt3r3b0/hOFYVWP0H+yRTIFMgUyBTIFMgUyBTIFPhXUSAme6hMHFlraQA2SRZO4+Z06TFVydg5RuQasi+4iEvYnVNGBRlVuWXND4xOVCXLDtqbb5+yCxdvSJ5ts1n2slFbVU1sNkgwbgRQ0XFJY/smTQx2dzXZts3ddu/+DQpXWLtM/jY3IiVjPo2anJCe+AkVJ0Vx/fuP5gaYyVzolkrqHNllCmQKfLAUgLfhCItx0mKimzj8Dx/limVJZ940AJciYAO4AygD78SqxdDQkF2/ft0GBwc9JJ186lIPD4DDMT60dgijDz5HK17j/asNzMvKpwz7/lRVDxAk+qpIbc4zXWtVCx3JY1yPY+EjZsUBcWg7ro08b8O5EzyIuQABJ5o7QBMKLZ7Vq1bZ/VpA+enHHreHH3nE1m/cpLG+tBAb25x3XpJJdPazHbqh6xLfZIMJFUhe9RNvhgNOaub1sm3qabL77tlvO7b12uruTu1vi0nJBHjTXfw8Z6RD9NF/a3MSKclT+El8nhIxn0oBrqWuIIWXiYv7ThjPMCHPTwZslslN5jK4wcsJsOE1BR4h9JccsAT1O13nnASialUbZGuSsf/KDd8Q64xQ4KGxWRtnVXhDmzW1aPJS9hl51WX5TC+/NhmsTsjGotnWTd12//4ttl0bZbOqprmRNTfvBmwQyBJ7oAOpN5gD0sn1IoEuJyYTpVI/VdZdMBhS8RHQkflWU3r+zRTIFLiLKQA/w2Ysg0J4Q20FiXjc5KQ2/Dt9VmDNt+wb3/y6HT12RPZ9h6TqrL1rNvfaZz/7GfvKV16wRx559P9n773fMzuOO99CzpgBMANgBsAMMJkUKVIUgwKTqERZwZaudmVbu77X+zx7f9g/aWXLki1Z0rUtK1rBClayJFuJEkVycs4BMxjkeL+f6lN4D94BhjMMWg7YDZzTfTp3nT71Vnd1VUn0uUPEmCz+Fc6Xt8I1oJsax1+JqE3YCGILHCaCT23OyDDY2MS8TiOet9/9/oSduzRl07OIPCe7Nah1cCJyYdalaxbnJ62pfs4Z3nt39ksd2nbZCGu2F579nf3d337Kvvedf7VzZ05LJ/GUTvFstHvvvdf+5E/+xD74wQ/awMCAnwYqEyNBpERcjCH7GQIZAhkCGQIZAhkCGQIZArcGgdjsITdhNnvuXIYNo4g1c/jFktwHWKywoXF1IWHDvgEMm0lpQcPW7f6Dp2U4+6IkbqYlnV6nU+INNq/dQ6TGG2TPwa0iCE51UpPWVL8oyZoWu+euAds53CMtHFZI2cQeAWt7nNbyENA8QWSreVXhq31obKe7oeVFO3se3bPLEMgQeGkQAI/dzJXTPczat1SAuLiIZq2JC0ZK+FFPPAcjJxgw7LEGc4ZwMGWIq7arA8OHeHzyRRjGDKOBUTMhyR1wc6hom5gYt+mpySQRpLK04QwgSRSRf3Y22eIhbhEVRKoJHAPDBke/GUOMz/csNdQ6pTuaKpAUo29sbHKtHP1bttrWwUHr7Npkza2d1tiyQXunbdobkJq05m5rbJZd3LoW4VVJ2Ej6RidGhTuL+lRRQ82MdTaN2lC/TE7s2aX91c1SlY46SbHC1V6d9k1rhZtpP120zpXwJWGwqqNTj003ULqPr4JV9cyeRiqbcq2ve8zReIexJ8J7zQybdfSuecHrjWGT0JF/tfpuYZZAiulZSAPEoQPodvnqpB05cckOarPx1Lmrdm1KBJtsLtQ1tAqxyMCgssNVFv9Zu5MTEtGb9Y1G9CzuGumVzsVGZ9pArOFk1UZ0HyRcBeGnDU8Q4cth2BRIygk4byrfMgQyBF73EKhm2AhPQBQJ/1y5ctV+9ctf2z//8z/bv3zj63bu3BmbmxdxtKHT7r//Pnvv0+9x2zV79u6zRhkRjEUkP/LYdkl0TcKihJ0YcHgLf6p+sClYb3J20S7Kds1+6at9bv9JqUYTkShJRezW1EodGgwbZypJV+3SwrRUn01aa0uNbdu6Ueolt0odWq+kF5fsl//xn/bJv/rf9sMffN/Grl11ArO/v9/e/va3S3Xbh+wd73iH9fb2usg23QhihH6FW8/EWIwx+xkCGQIZAhkCGQIZAhkCrzQEYrOHegmz2XPnMGwSvZpgItrYA8XaGSZJ4dJmnh6UIXLBsEHKBroWtT7S6iNJ8VlJ14y6ap9jsmUzwf6A1PrM6fQ3p8XrxbRBysY3FkUNo+p3Y0eSHN85skkqf1u1+aiT4rXadJSEDVSzH24vaHTodOhu6FbIWPYI6E+ibRXvfSZPdhkCGQIvFQLlNeKt1nGzMrHOJE/5ou5II1yuI33TlW85mDr4wbjBDyZLmaETYZg2riVIOGJO4UlJ6ExOSN3a5ISrXJuUP6ODjtNi8ridHpg9BWNoXIydkydO+sb9tWtj3mb0N6T4YiwxBrRtOIaCQQ3SZLxCkOxu1krKsM7VpjVbiw58Nrd1WlPrBl0brWNjn/X0bbPNfSPWubFfDJxW4VbUoqs2GDaOisG0S1Kvtmh9XbO2Y7BDGje2Wd+mjZJMlBSPt4JJCnYb5Kv9CgwTHAO/VzNsAsp0uXx033ueNjZIWHcu3h9+wAo/M2zW2avmBa8nhg2vJ5E+hBKiwXeco88f5jJMm/HpeUnZ6BTNobP2wuHTdnVCp9Vly2ZJm40LYrwILOI+IwaoEzJLMyo/JQKs1oaHNklXbb8NDWy0je2I7nHWHNILcUGpAXJ2r2MLOuDIrsKwgRsMCgLtUYow93COfvRAbJFSPo0T2bKfIZAh8DqHAPgBwgeGsPCIHvH5OyHi7Jvf/LZ96Utfsp/+9CcymjjmSAamx9Pve6995CMftgfe/ID19fYrXgtQxzFkETGWVpWqKxFVEHSJzkm4SiZrJOacGDZjk7N2/OQVOyCGzVGpmByfYtHbrOoaRJs1aVELgQahNyOVaBMqNGVdnU0yLjhge8Ww2dTVYrOT4/azn/y7/c0n/8p++pMfS+fvhBMZw8PDbr/m/e9/vz344IM60dPlp4EgQMoEyet8EuThZwhkCGQIZAhkCGQIZAi8LAjEZg+VEIbWujMYNt5jek1ADtYH4bTGZnUeLm3kFU8aI4yaKCeN6WLYYBBbUuozss14dc4ZNs/Lls2Vq7I7ITs289ofwHItDvVASQn6vNSiSbG6zih1yWD2yPZu27dnq/V2t7qUjWvp0D4BpHUtDJuC3ob2rvFLUTBsJDFPl53m9j5X+u0N5luGQIbALUMAHMYVjnVjtauOK5eJdWZ1mepnxzVFM1FfuR7yl+sq9ynqAtcSX84XacR7eoHVXEpGkjI4hpdUoYmhLBs97OXOTCemTZLOmZFN2Av2/e9/3zVuHD9+XGtsqSdXeyAbJGyir9F+1EvdMGt8xxK85Pm1x6l1PQwdaVVTpPZAdci9rqnDGlo6rX9gh+29+wHbMrDL2jo3q372A1CHpr1UlWO/Yl4aN9hB6JQt8J1DzbZnuNsGtvTZhg5J5igLKBldHjV0QDiR9umnw1ZRCvjlyaSWXutKjKkOkkl5uOLd6GHduep3GPOI95wlbNbR6+ZFrz+GTYVls0yg8e3rvS3q62Zjc0Y7j5euTjtB9vsDJ+ziVYkhLkoVmtT5JONYfODgGZ2esVkhjxkRZEvWo03GkW2bfcOxt7tZotAwdEAHMGxAafGXJgkILzFswDEFovGkhGUCnRAFYkq99Awe41FK8c5EdPYzBDIEXucQEF4RgSaySRe4RJfw2vz8ov32md/a5/7+c/ad73zHDh0+KHHoGWuTjoa77tpnH/7wnzgjZGhom9ShtQutJKlAxy9O+SQCDeCCrrxqeWAmHqQFTarQdAl/Xrgybi8cPGVHjl+wy6NTNjOHBA7qJBvEqBE+hMhSTI3sgOnconDlvPVt7rB79m23HdslXSPcefHcWfuhiMnPffYz9qtf/dJPD/GbhDq0v/iLv7B3vetdtn37dhk9bFZ/EoEZxEgQKfHsXcy3DIEMgQyBDIEMgQyBDIEMgVuGAPSUr1FVgjCbPXcWwyaGyhZqolgTAQshi0v+8p6Ax2lTjzW6/tiEdIaNqNW5hRobly2bU2dGZdv3jPzLdk3qf+dqWqU6XSqElZf1vjNhRIOjaQMV6c0Ni7a1H1s2GNDWiXNtStYrvk52bqSwww900ot0uIpNSG0xqh6n4/2QFAeSoJvZ3Fy5/ejdzbcMgQyBW4YAeKzaBY6rjo/nKFOdL+IjX9mPtHKZiCMf8ZFWjo86iONazgP+urHrHgWDxusUlnC8tWy/BokUsUKEt9nTTYydRbeh84N/+zf7p3/6oj3zzG9kS+eycHuqvNwv6kz9qPSX3UxnJIPvtJ7H7o1rzvC2FZbdmiapQ+vs6rNNfUO2ZWiHM202dPVbY1OnsrIfgEUar13j036pbHvVqyjM7bt3dtkuMbi7NnZYa5PUTIIjlRWGjWO/ZZyY9lVVgVJ16R8c6pAgqnBljBn7qWkKVOAfedeTH/Mn5lC818ywWU9vWWPhBa8vho0IL/2ByPwTd0wB0kkvDjxFivY17drkgh07ddmeO3DSTp8fs+s6IS5ttNr3TEgmTXpxeH3DcVY2beaFVGpscGuPvUGbjkNb2q2zWTirQCKQWeAOP6RehEkCxmo0IeMScqFLXBGVEAz9jBgyFGFvg7TsMgQyBDIEWGDK2CBMG8ctsqYlpDY6es1+9KMf2d9++tP2H//xH3b16qhwzJINDg7Y40885jZhUDXW0dEp4ktqHMWwSYyVwJEJK4Fu/BKgIYqcvlPEgo4hTqsdbNecPHPFnn3hqJ06O2ozs1IeuShdtVINueSnBmFgqw5XBzFrTQ0LkkiUYdbBHi1mt9lgf5fNTUnC8YXn7Dvf+pZ95Stfsuefe05MKLF7GhrskUcesf/1v/6XPfroo7ZhwwY/nQM+5pQOPi6IlCBOPDLfMgQyBDIEMgQyBDIEMgQyBG4ZArHZQwHCdxbDpjxMKN5YXRMPvRhXypeYNhDO5Ev7BcGw0ZaiDiahGm3JLl+R6vSjl+zw4VN2RnZtppdko0H7A1SfGDYwbRJDpsbmVHLWurt0anznVh1K2iw1P22ifWWvQac669VW2iNQX2hWfUo7BqK9xdBh9yFR22xY6lS6M23Il12GQIbAS4GA771VFYz1Y1X08mOUKeeLuPDJXJ1efiadvI6HQDPFmpX41Vx1vV6W8rpWlE1LXyp0jObYiz0AMWkIg5NIoBz4G3/8+rj99N//XQybf7If/vBHdvLkCcVHRalvtMFFGZg5wZwhlzOIxDiBkVyrtb0SVUjr+3qpRGsXs2XzVusbGLYtW0esu3fAWjt6xKzpEGOmkK4RNlODqgdpmXlralqy1tYG69fhzbt3brbhwQ3W0twohjfYEdyXMF9dgZspC8L0EcPgJpf6SnRm2AgcctXzpfw+s4RNgtG6uPOi1yfDBm6z/hwx6QOXz2eP4yOf08bjlGwwcEr8sE6IHzp2QUazJxTfqHQYNtrMdDE8sAISNiLGJPbcIJsLm3s2iCAbEmd4g/V3N8gOhJCXVw5SSZxhEE98NGm30yMc1ygkR4FEVvqjbpRJ9xTysNer54jyPPmWIZAh8PqGAMSciCsRQRBZMF7Grl3XCZpnpA7tmy7+fPjwIZ2ygUBqsgceeJN99L/8F3tK9mBGRnbIHoxWkSKNYNo4kaen0FsLTQa6cYwpogmpxLhQiTYxM2fnLozZIdn/Onj0rF2SdM2i1EUsLIlZo0UmfYGxgno1VwexNG0tTYs2qBOHLGS3S61k94ZWuz562f79xz+yr331K/bjH/3Qjh8/4brBu7u77YknnrC//Mu/VL8fsJYWiUqLieOMpYKwjHfP7xd4NrsMgQyBDIEMgQyBDIEMgQyB24dAmZYifEcxbHydrDE7Kci6OtbX+L4TUCQSTo4xshvgWjC8BLQuUuJi2CjXrPgn4zqYdObsmO0/eExqfy/b2Gyza+Jgt7DOpcipTXUUm5F1OtTZJts1W3o7bO/OLTa8bZO1t9bLVqM5w0aeb0jSCw6Pexfoq+jsGjFtjEuuJjNsHA75liHwh4ZAwgurt1pOq153xnPkwS9f1TWSv1wGnIWaxRVxRR2U9WUuDAuhHG8jobaUX/liL8DbFNNlQYychfkFuz42Zr/97W99XwCtGwcPHvKugHu8Hq87aa9IDBv2Tiv9iP6wX+kaOWS/yyRl2L5hkw0N77Xd+95oA9t3W6NUotXWt8hujU6xS3UkewEc8ITJUy9mzNLivNqblmaNedmk3WA7R/pt17YuqY6U7XCJ1nDwHTVo2AXjsCfSNklDkXeUXuqZ8YNz2ZOgR4UjSg52Tri04wuSJYbKyqmRa3345XkW8wqf95kZNuvjHfsoeNGvF4aND1jfLR/6nL7jWQWuT2HL5prtP3zWjsoew7iMC87OJ4RUI8QEbgDRGAybQvS5va1Fm49SizbSJYNZbdLFiL0GkEFi2IB0QBb+4YBkAqtQmf4LDOJ+JBGrnH4vMhFVZI1yKSrfMwQyBF7vEIC8g2HDiRidpBEuO33qjBNlX/va16Re7FcSfb5oDY0N1tPTbY8//pj9+Z99XPZgHpI9mG4nohzjuNqyRNpA2DiKcpoIzJQuyLcFmDYivmalE230+ozUoJ2zA4fPCHdel0FW9UWnDueVDm6F6KyXUcJ0YmZOC9UZGWSt1eK113aLYdO3CfHnertw9oz9y9e/al/+0hft98/+zi5dvCQ1bR22b98+e8973uPSQHv37vW6QrKmQjym91/9/HqfFXn8GQIZAhkCGQIZAhkCGQK3AwH2AoKeInxHMmwYsBbc/EG/Jj+tqyvrazIpRVlEsWrMnE6PP9lnYKUvepdrWoc6k5TNKTt47KKdG5WqtGnVp8J17A+IfuZ0O/UhacPVWL8oW42NtosNyZ39tlmq01ubRBOrmM57qnb2B4p9AT2HHR3UBSWGDalZwoa3lF2GwP8JCID/Xq6LOlbzwbMcQAwXeXgu4+ByvKfByVBZ4iONA5dgr/m5OZuenraJiQm7NnbNRq+M2pXRK3ZZ6+pDhw7Zr3/9a2fcnD171nFfeYi0yUWd4H0wENiRf2ciKQBEaoXzGptbraWtU8yaPfbGN73NBsW0aW6TxoxFGY9YBLul/dMl7QPAlKYSpGXS/qlMSzQv2rYh7Z/uHrKBXmkpEkObYfnQ6JQuaimG6i372xA+rjBs0j7u8luiu3IB0fSY8LJ3nIFkho197GMfS4Aq3bVfXZkKcPbg8GA4uK+vb3kyxuSgXEzQ8Et15eCrDAFe1Xpk2HDqBURRkbApACmktKAx+6XPe05f/JWxGTFsztn+Q+fsgqRsJmco2qTvG8QDEkuGr5y4E9OmXqzfrs5W2zeyUSJ9PZK4aRcSkt0GbxFkp0tt+LOQxDLy8xg16BychGpAtPQzrpRXj+E8mfTsMgQyBDIEAgLpRI0vM8UlmZmZtReef94+LVVoX//61+3ixYs2PTNl7e1tsgEz7AyQ//pfPmZ79+6TxE1LRVpFuIdf60AzQndCeUEMsoD384N+4pAF7NTMop29OGHP7T9uB8WwGZuQocMlqYjQiZo5ccHnJdEDAdnUIMKuQRI8UhPR0rhgvZtbbOdwn+x/9Qt3Nit+0U6ILvjiP/5/9pUv/7OdOHHc9aX3dG+yxx573D74wQ+6OrStW7fGgJ1OqKYbMs2wDJ4cyBDIEMgQyBDIEMgQyBC4bQiwFxD0FOE7jmETS2lRs8EEcfq4WF+nVXa6Ew/hyzYk9O6iKFyKs+XHdqGv4kXvcgjp+viMnT57SdLkF+3gyUm7fG1ORVHPC8NGDB5Z3wZuaf9V9K/UorU219jQQI8kyrfY9oFOt9fg8udqE20cfpocWptueMv0IzFsUj/YT0AeJ7sMgQyBlwqB0lb0DVUErrshoYhYrSxlyvFRB3ERT1zEV9e9Vr5yfJQp1+fpwhNu30p4AxwB3klaNtjDnbOpyUlXgX7u/HlpqzhuR44csWNHj+kgp9Q5npEmDB3gHJO0zfT0jPdVXV7uZ/SZdlCJBpZkT5L4kPohtr6xyTo39lj/4Ha7/4G32cNve6c1tfXY5asTOuy+4GokZa1LeFEaMVDVgYMBpEPvDZI+bG6ctw2ddbZTeHGfGDbdnU1icKddT5gtYF/n8qgf4GbX0sF4hW+jP/iMn/7j63HZUUd6JEV7wGRyp5TMsMkMm2I23LEeH+h6Y9igCxYSzNGARJzjE+bZP2ONGYbNoj5gWDHXp+bsxJlrduDIeTshmzbXrs+qdJPyyFiWykP8wYBx9CDmDYRZW3O9Dfe3ONNm2+Bm27ixXTZudOpGp3Vg2IB0QByJOx19IEb1QKUlVKN7ItcCZQk9Kg0XeQIbhZ9S8z1DIEPg9QsBsIbbr5GPccEL5y/Yz372M/vCFz5vP/jBD8X8GFf8nA5J9Nrb3vpWe/rp99kTT77DBgcGXR2aE37CUAVGChQpgCaMtCRxavAU+cCR6PXG7tdVSdccO3FFzO3TwpmXbUKnDReXmoTopBJNhB4q2BBlbhKzRjwbWUCclASiCLThzbZrR79s13RbW4sSVP/o5StiMv3efvPrX9nzsmVz5PBR4dZ6e/LJJ53BtGvXLuvs7PSNgzJBGarRePtrEcakZZchkCGQIZAhkCGQIZAhkCFwcwiwFxD0VNq4W/RDNM/rINDnP/950ZZfsPPaDETaeWBgwD760Y/axz/+cdu9e7dsInbcvPJXO3V5uUxAVGyxxo71NQRuonZJrYR8ma01O+qEfM0tepfNQd8uhPYVw2Zqdt6uXZuQBo5R+92By7JlMyk6l5rZXtSmpPJRp//7ul+q06X2Z0NHg20b6NbG5BYb7NtgrY2iidUvcru6n1Qqtaty2LERwZy6ZGLY1EBAZ5chkCHwSkAAnFbtAt9Vx9/suVxPlC/H3aws+ct5ozxlyvHlOoh3LRqKJD/r/llJ08zMzLg0zZUrl+306dNizhy1o8dkU1YMmsuXLtnVa9dkv0YaMMZ1CF3MHPKz17vgjB7hL9W7CkgqTZfAxT5AXX2jbdCByj377rFHn3iX3XXPg1bftNHGJhft0tVJqUqnX+ytCr+Kke1q0YUUtYsqJjUaimZsQ3utVKJ3SfKw122Bt0vTBtI3jIuDnuyz0i8/9K5yxNO2I1eYNvp33OwdT/u5xIUjp+Pxog4PewaleD2Rc3356V0WsHOYATfmSlaJtq7eNC96vTFsEtED8cPXzdesyesfrcJCDthq0N6iM2wWNamnpKz24ui0HZGOWtT8IGUzO1+n0+KgGrjFzvdNCBXCSn9N9UvWt6HGdg122J5dQ9bf12VNOmgO08aRjRNuQlR8PBB10b7KJoYN04iaEkpJfQxpnJQWKakscSXMxGN2GQIZAq9LCIA50FEL/p4VIbZ//3779re/bV/92lftmd/8RtIuc06Wh1YBAABAAElEQVQw7dq90z70oQ/Ze979Htu7Z58Mova4ijF+zB2fgIAKtBLYCN8NDcJ4FpEDs2ZOjGuIMWzXvHDorJg2l9x2zcyc9NTKfo2U1zpeRX0k5xNFh4lpw+J0xjZ3tdjdewdlhLXXNnW3SVWaGlTf59Tv8fHrdv7cWTFrDtuhw0dEVM7Z/fffb29+85uluq3LbdfAkKpm0iRCLuPD1+Xkz4POEMgQyBDIEMgQyBB4xSDgG2VOF2rlyRpZmz3XteG3Phg2kLkFzbscSvSjRqrxwihhbc6GKitv0pI9mzmdVJqeQXW6bETuv2THT1+18clpmxOPZ0l2G5MmDlSpybHuX9KhpdpZt9WwuafV7t4N7bvZT5O36CBTPUwbNeK9cXirpMqlfQH2F+grDJssYQNIs8sQeCUgAE6rdmkdXB178+dyPWuVL+cp11bOH3nKcZGXNOLxWf/Ozs4mdWdivICTUXl27eo1V3l2TirOjh0/Jomawy5Zc+HCBWfQzM/JlINjE9BLGnu0VWHapBZpJ9oEL3luhxdMkSWtwxttY9cm27F7nz381sftiaeett7+ER3aHJXt73FXk84+6tyC8mvPgHbY+2Spr50B4TxUo89ab0+L3bV3q41s32Q9G1uFI8F11RetpytwdjBtvEvRP5X0flKBXPIEt9R5xXBoXg/O6MkMGzScZZVoPlXu3Bsf6Xpl2PjHvvxF88GCeojgzqesU+PaiJzWaRls15w+d82eP3DCpWzGZGxwTkybmrombXxiQEunzEXAphqkpVHiyx0N07a9t8XecPcOcY37ZGywxhEQZB5MI9g8IC2dp1HTtEfb6lWBYPRQ1EdEQk2BdpaRjZeBb5zyUCa7DIEMgdc3BJyQ0yKPRTVMj5/85Cf2j//wD/aTf/+JTticdIKps7NDqkjfbP/t4//dHnv0MdmHabfWljZrlFhzEG6OXgRKTiRSJxjJUY1jJtWvGFgwM2Jej40v6ATPeXv2+WN28YpO7MyJibNQL8kblD0USE2MGFvSYrVuQVKINdYhA6xDW7vsrj1DtrVvo1RFiJhDSlE110lJLSLdEJbptNCkCNM5a21tkyq3dqluS/2kXzBsuFIfV7775bGsjM5PGQIZAhkCGQIZAhkCGQIZAi8CAWiroKUI38kMm1jrp/V6Gnhs/sVaO1bc0KCstyFhoYA50OkqeLTxyB5BkhyXLZtrs3bg+Jhs2Vyws+cv2eQ0+wE6rCT7jRjh9kOg2lyFYq6v04ly2W5EunxkaJPtGu4VHdxtG2XvtkGHOmvVJjYauLx9YM8WhZPR3GDWsO7PLkMgQ+CVhMBqa8i16g98WE6P8uW01eLKZQiX80SY+KinHEcY/AuzZmpqSmv8cTt37pwdKaRoTpw44VI1l6T63CVptAcwNcX6edbmtIamHA6VYm7/VciNOmF8gG/Il6QE1S/2NZW2mvO+CR1t2LDR9t59rz3+5Lvs7Y+927YM7nTJmhcOndFewIQOvUst+7zwpqQC6+qFE3E6vMlav04M7OZ67Ze21EjisMvuuXubbe3fYC1NUrHGAXdvO/YfQl2kI0LvF8n0m74QDvVoseXg/BjvvuryhnlIF7uumWFzzLBrnBk2Pjnu7Bsf6vpj2BQEGK/GP2QC+pSdGgoSDjkYEWP62ud0zYhpc0mI56Bs2RwWQXbh8rhO1fCtY8umie1FZ9hQsyM+IaLW2knr31gvQ9o6QSMDg9hpQFUaatESw4ZSwWwp+oAXmMaxi6OUIiahm9TpQrWa56eOcj0emW8ZAhkCr1MIoGcWKZhxiTsfO3bUvvWtb9qXZAvmwP4DNjE5IQKrw+66a5899dQ77Y/e90d21767nHCrr2sQQaVTgYGDCvyYyBvwlhxoqGBuozpyVkzta7JVc+rsmKRgzkm65rxOGIqZU9Ms/AnDBiJQNYJiRZzZ0owYNvNueHVwy0bbsW2zbRPTpquzxQm0xYVZ5ZEtMDFgakRQQowp5ATkvAhNnl2cWunhINZWI2xJj/jIm/0MgQyBDIEMgQyBDIEMgQyBW4NA2tBLa1DCdy7DRtSs6FfW0fGnCD1XrkT/prH6epudQC+jJ8Ze5IWuTRo50qHOkxenZcuGjdMzrjp9fhH7jY3KUy9aONGrbotmcVoboZOy2bBo/ZvbxbDps10jW6yvW4emGnX6XLS7q0VTm+xEePNO4/Ku6Fes+XnOLkMgQ+CVgADf9ktx5TVm1LFW3Grp4NKyI0/5Ip19WDRjTE9Pu60ZJGXOSnqG65JUnF26fMnOKw51Z6OjV6Wm8ZrUnY3bzKxUnakc+wGMDtxWV1crG7KN1tLaIpsxndYr6YrezZutubnZLly8YL9/9jmvc75Qj+YIqNRBxsbVqEOTGzZskBq0ffbYE++0Bx9+VPZrdopBU2unz1+zU5Kw8cPti3XJfo3wpTOIKK9D6zWLkqyRHdtOHdxkL2CXVKOPbO9xO7Z1teDAJAUTmJguUFLF3aXX5TFKgGGT9ksTjvTMns+z+6vllt4xeDWFC3yqfYX16spzKd4dPvPqmGwFZ4bNOnnzvOj1yrBJ329wXfXRFkiAQPqsHaWAVvwak3HBExJ3PixbNsdPXhIHecHV/SzIqPai7NmIp+MogM1SuMdNNmXdrUs2oJMzO3fIoPbwJtsog9oNdX4uRySXa7gt8AdIKCGMZSQEVgrMVJpPiVwE2SSyMRFvoVatlDEHMwQyBNYNBMDFQdiVf3SJD0c8LhF4C37q5qc//Xf7yle/Yt/73nedCGuQnPG2bUP27ne/2/74jz9k973xftu8aVNRhcqL6AkcWCbUWNy6U/LiMsPGbGq+RmLP0/bCgdN2+Oh5uzw66VI1NXUtzrBZhIhSUbommUVh0xmdqJm3vk1tUoU25KcLu9obrEU6vFGZNjM1IYmaaWlFk+yO9KE3NsjwoKR+6sRM4kRQrfTfhmPsMWb8MiwiT6THc/YzBDIEMgQyBDIEMgQyBDIEbg0CZVqLMDTmHaUSzYcJDau1800YNqIwlaeyeQfdC0UM+Zts3/iT07RJ2iZtrk7rLNIVSZkfPn7RXjh40i5Kdfq0Ni6XdHBpXhuWHFzyw1DahJyflZ2b+XEdSpq1jbJlMzK02aXMh7Z02cb2xqQWTQ3WwrBxpg0dULsFbe7hNehdcuIy3ZvgkO8ZAkBgtbUh8fGdRHrguYgnz1ouypTzluPKYeqIZ8JRhjhwaTmNZ5gzXEi7IEWDnRkkaa5evWpnzpyxF154wZ577jk7ePCgXZQkDWnkp2y5bsJon6ivr3MV4k2yzdDS0ir7rx2yrd1l/Vv67e6777Z9OsDZLi0bzz77O/vHf/qi1z+jU+nguLID+2F7u6Gx0bq6tbe5a6ekap6w9zz9Adu2Y59r2jh05KwdO6U9Uh3knJP2DSk980Pvc2IAgXphGCE9U7sobRu187a5W6rQ9gza3p39UoXWKEa29kN1aBPcBw5MmoiqewEWpDeVa7mrbDZUuRTDWGI8EY7yFZxfVfSOf2RuxcV8iIu5khk2d/zrrQyAl7xeGTalUVaC/vHzScO0gQebmDVsOk5LKe3o2LQdPX7JDshOw0VtTE5JymZ2AXFnkBJSNhJ4hmEjZNNo09ZeP+dMmqHBHtsj44Jb+jqto60hiT177ZQC7fARcRKdD4vuJBE/Rd/gKgwbWEk4NjAzw8ZBkW8ZAusUAhBjqAbD5we3paXFCTBwRvwAx9CTqPS0/e53z9oXv/hP9l0xaw4eOiiiD+maTrv33nvsIx/5sH3wAx8QQ3nQmpqbvCh4KBFAEJcKJmSUSBwlOS+aODFOWDvOiUt9bcqkq3bcnnvhmJ08fVl9rJEqSdSgcapQdr4gEPS3JAZMjdShNTVI/Lm11ga2dNob9g3aiHBjGwZXdaRwbnpKxKekGA8fshPHjltdQ7319wtv9m+1np5NUoXW4arQ4pQOnWbs4YAFrhwXadnPEMgQyBDIEMgQyBDIEMgQuD0IBJ1JKcJs9qx3ho2PuVhlJzIT9o3G7zBIcHCKU/CY09r/ulT/nL045oc62Se4dGVahz0bRQfrUKfW6GlfgV2FeR1ImtIqf1ZaN0y2G9pseHCz7ZYmjq19G0QjF7ZsxNxxo9vQzzCRRHQvU7ur0L28m3CZBg5IZP/1AIEyfrqV8YK/okz1t1L9HPnwIxx5eK52a6VFfOSP+sKnT6zvkaIBt8KIOX3mtJ0+ddqZNOfPn/e4K1euSIpm1C8YNUmF2cIKxg9twahBagbmzCYdyty6dasNDg7awMBW6+3tdYYLtmA93LXR8drPf/5z+9TffMr+8xe/cDs42LNJmi6Egxir/rFZgwqte++/zx5/4kl7+C1vta2DI7akQ5qnzly1/YdOSy3kmPYBhPdqGoT/sHGrA+7aQHCNGWLCiIWtfdB5qUKrtcGtnXbPviHtBWzyg5uNsgHueA/uDlhzGen5o26KUGTChiSmDMuvYkUBJVdyKBTvK9WdylI+M2yyDRtmyh3s+EDXJ8Om+qONZwijhADwIa1g1nAhQQPT5vQ5ENI5t2Vz5eq0M2wgyhaln5ENTJCSMJs1aHOyWYa0WrQX2ru5I0nZyLhg7+Y2Gd2WfkYX90sMG1T+4JYRzk0YNn5CyHuW9FAmRCPrXEUdXlG+ZQhkCKwrCHDKBgLu5MmTLhLd09MjZka/QXC1trY6ceYElUYNAXfxwiX78Y9/bJ/97Gft5//5c4lIjxoqxbZs2WKPPf6o/V8f+Yi986mndMpmY2JwOAoUCZRQkcMuCElwIYQWp21CkpDsMzK4eu4KurtH7ZCItEuXx5QHBrYukWTgz0RYCWvNz0oEGny4aL2bWm2nJA45UbNlc6c1OgEntRLXx+zI4QP2M0kF/ed//kLi3LPWpzFu3z5i97zhXj8J1K1TPQ0NSNsUTO3iLftS2sdAk6VBFOnZyxDIEMgQyBDIEMgQyBDIELg9CEALBl1F+I5i2DDUWFxr3R2bdhx+LEhGxcUGIBt3MEcScyZtCjoZq/hUMujsZV+1+P6Aqr46PqdN1lGXOD9x6qr2DJCugR7WASal0yLHQHXcSa3MaS9A9hy1adm3qd327RqwHdv7ZdumQUwbqQ/y0+WwaqBn6VPqJf3wKA/Q1cooPCnTvwVksvd6gQDfQOCnWxnzWgwb6iivJavrinY8X9V3V84bfVnr2ySePnC4kvU6TBrUmKHqDMYMF+rOkKbB5/ny5cueB2mbkKaJ/rAeTpI09c6kYV0Pk6a3V3uOYs709/eJYTMghs2A7xuwjm7v6PCDnzB1OARJnT//j5/bJz/5SfvRD3/s7RKHS+M1PzDZ29dv99z7RtmsedLe/uijtmPXHluU6sdzF6/bgcOSrpGU4ei1GTFwmmXnu1nwrBPDZl74T/ZphcbcRtfSnDU3aC+gu1XqIHtt7660F4DkjVCfbNiC0wpcvYzPFJf+HVtj2CvhZ/BjwVBazkuvKy7tCFAnFy78VHYFQk0Z1s2dORIX7zEu5l+WsFk3rzkRAuuLYcPLCc56fLz4IAY+YX28Ys647x8+5q+VCtNGaUjPXB2bkbjfFRnXPu4nymelp3appsU5ybMyrAXTBhG+etlraNAJmgbpYuyQyp+hgQ22b89WqSPa5Ma1GyQUw5am8JcQGI3rpv+E6IvNyIRllFhBJ2EE0XXhego1ZIaNgyLfMgTWKQQ4cYMY9L/+679KbPlZ4ZFt9vDDD9t9OuXSLwIKJga4A6JtYmJCeffbt775LYk3/5M9//xzIpakYkySMbv37LYPfOgD9n7ZrnnTm94kMeg2QSwhGg60ONor8I4TlfzYK72mBmmZZGQV0eZ5/dhPzSzYoZPX7PeHLkj92qjanVXeMLRaL+wGEU3tqgPbNAszOlG4ZMPbely6ZtvARtvQ1ijDg8onYu7ypQv261/+p9vc+c53vyfDiWdd9HpwaMg+8P4P2J/96Z/ZyMiIj7UsZcMrryZI1uk0yMPKEMgQyBDIEMgQyBDIEPiDQQD6Kq1NE611ZzFsfIFdwGplmKeKY/vPV+QFPQk9XNDG8liih/OgR6TIxIKpsVltGFwZnZDE+RnZvL0oWzZzopOT1DknzaGJdQxW7SA9I6PbNfPaB5gRk6be9ohhs2/3oJg3ndbe2pBUo/nBTtg1xeYkzRX0efQl+xkCGQKvDATAc7jwCQcOiDUma2xc5CGdi2fwIuHI4xl1I570yAejBmYIBzFZ28OQOXTokD3zzDO+vmcjHUkaGDnkIz91cIWLdmiL9XBTU5O1aT2PJM1dd91lb3zjG91HogbNGhzshDnTKHVmaf2scRS4BKxEG8/+7nf2+S/8f/ad73zHDh44ZJNTk1JVnsZUo3Y2y9bNA29+2N793qftHU89ZUPbh6QJo8muCs/tP3TG90Uvj07ZzBzqIKXKvAGmjQ6zw7BBzTn4THsRYg+5dOHwUI+ka7Zpf7TL9wKcWeN9Eqycsa33sdxHjVyP9JVI/sCMwGE5k0JrO8pxlV1ReTRSTlon4Zi3Mf+AFxdzKTNs1slLZhi84PXHsNHn7ogTxBcfcCKj+PjTh88WYwqDNsgF0waGzaQoMjjJz+8/YYePnZeexkXpZ9TpGRd91iYlzB1VXecMGxFktQvW3LRkPV3NtnvnFnGT+6xb4dZmxJ5h2qgFNSD0VvQnEJESHREpWi7QSmIu0fdA3PQzq0QDRtllCKxXCIyNjTkh98UvflH2aL7nxNmOHTtsz549Njw87AycgYEB4+QMDJuf/fRn9vWvfd1++KMf2slTp5xZ09rWYg89/JD997/4b/boo49JHdpWNz4YMAscE2gHPOkCg44DIe6SpCGM6cnpObt8RUykY5ftwLFRG5+Qzl3p8Xb1kNLXDcZKotTCUyLQanSipqlhybo3NAkHbtbCdIufrmkR57pG7Uipt506cUwMqW/bv3zjXyRh80vZ3LlsrW2ttmPnLvvjD/2x/emf/qkNb98ufbz1TnAkQi31njD9xZXjU2q+ZwhkCGQIZAhkCGQIZAhkCNwuBNKaOVGIThdqkXvHqERbXlsz6lhnBwTS9p/Tq77KZozQkpEeVHE8J5/kAhoewWqcs+jzWv9PTM7K3u01qUa7KP+KNjNntXeQVAS7hLoMgLN+h2GDejRUo7U0mqtD2z2yxXYM99um7japCdbKXo1ATeuslfKmFtNdkdllCGQI3DIEYn1YLhBx5TVjxEU+niOOfJHXMccyntD3GQtnFSRMmfBhhsCcQZUZmjKQpMFHaoaLcEjUkMZ6H2YN5aLtYMzAcEElemdnp1SF97gkDYwUJGpY/xOGaYMGDuJQiYbtmnoxTliTJ9yW+lYrezJgFsZCOydPnLTvfve79qUvf9l+8uOfeD80CG+vd3Ov3SMm0BNPvcseecvbbHhkh7XI7s3kzLydOX9VDOqzMhlxwa5PcqBTh8ilDq22XohNBz1dUkeHMpGcQTV6i/YC+iVVuGtHr+3Bdk2XmEmys8OeqO8HADsddseOTTgPCb9WHOF4Tn75HVTyRYgaKvWl2JXlI+d68nmvcQGfuDLDZj29ZY2Fl7z+GDaMC0QA0cTHG8wawgl5hS+Uobj0QcOIQU8t19XxKTewfVDGtc6KeQPTZl5qgKxWkjay27Agqgykg9qzOi4ZF2xvrbFtAz3OtEHapqujSdI3Isi41A9Qpjqm9mixaLf8A+Ap3BKxl/rOM3kzwwZIZJchsF4hwOIYyZq///u/l12aL7podHt7u4s7Dw8P+0mae+65xwgjYv2Df/uBfftb33bbNRgqbJZuRgwMvuvd77T/+y//H52+uc+aGiUKzUowoR3HJMAvoZ30Iw/eAzsJY7q/oOeZuSWJPE/Y8ZPnnWFz8tyU1K2J8SK1kMgMLkhVGr8dGBcUJ0Yoa1ZGBud1irBR+mo3Sld3rwytdsuWDfa8hDaVd2F+zg7sf96+8PnP2zfEsDl8+JhN6HRPz6Ye7+uHPvgh2dz5oPTvDtxweikIkOp3Tx9uTsBVl8jPGQIZAhkCGQIZAhkCGQIZAgGBMi1FmM2e9cGwiRGyjuZKTkOUWxlXJDm5nHIWhLMSWJUjZbOghBnZsrk6Nm/HTly25/aftPOXxqU+mAOf0qmBlI0q58+pah3srBXTBtVoHW11Nqg9grv3bpeNx26dQJehcNHnvrpXvYlpU+5l9Cj7GQIZAjeDADir2q0WV52HZ3Bd5K1eTxLPVV6D8sy+KRdMFxg1qDGDWYNKc6QauE6fPi3NFOecYcManXyUKbdFvUjDoEEDJk2H1Jih7gzGDAwZbNIMSQMFPurOYdCQB0kaDjbiEAiKfjOWYAI5A0hMEiRn3AlEqGX7zTO/sc9//gv25S9/xfsGg2ir6mZ/4e2PPWGPPfmUDlHulvRMo0xC1Mie97j2Q8/asZMX7eLlSeE/7RWIWSNujaoF57G+R6pwQUxo7QuIYbOhvd4Pr8OsGZSmjbbmesdzjuvUD/Zoa8WsEa9n2aU3KES44lWCibkqLsZaiYkQBVcU1vON5SP3evFjjpbnKTDKDJv18oaLcQTi4cUeOHDAPvGJT9hXvvIVF9Xjo9+9e7d99KMftaefftp27dolo9DtjmxAMNUiga8d0PDBlpk0wQChh2AHPmB8XcvcXEgrEWMqqr1Im5qdF5KatCMnLtgLB0/bhSsTQlycoGlWvgbBIIkvo6vRnCCTPZtGTpc323apA9q3a6tt6RXXWzit0e3ZQLypchC/2qEPCelUEFElRE8YQ1xFX4uSlM4uQyBDYP1AADyc1Jy9YJ/73OfsH/7hH1yEmhFCTCHqDBEHAbddEig8v/D88/b7Z39v18evu+5Y9Ng+8paH7On3vc/e8c53SiJHkiq1wtMFRQQ2SSwZ0FDCLekO7mORCf7TolNIcFoL0jNnL0vK8KgdPTtho5PS0y3CrVYMm9pa8B9ErErpNA0i0Ig/18me12adGrxrzzbp6e6xng2Nwn1gWeVR3rmZGfvNb35tn/jf/9u+/e1/tVERsNjc2T683d77nqft/e9/vz380MNOqIIbvQ2VhbnNc8KXK985eVaLX5krP2UIZAhkCGQIZAhkCGQIZAisBoEyLUWYPYE7g2EDFRtrZlbRlZV0ZZzkWS0eOjNW2pW1OeUSXUkiZQtJdFQGqx7UonOo6dyFMfv9C6dchfo1qVKfg2Ej9eUcfoKeTpWjFm1BtDibmXPWvbFFp85ly2a4T6rROqytSRuZWuIjaROHzYNxQz+yyxDIEHhxCICzXszxTZMvLp4jDnzHhYv4YHxEHM9hjwbciEqz48eP+94pDBokaIhDeiaYODB0YNJQlnrLe6cwaWC8oOYMRgxq0NlzZa81tGmQBiMHdWjsBXDBqIn91zTuNHZV7y76zwOMYzW8vA8wNztnR9XXz3/hC/b5z31eNnROe9tve9vb7amnnrIHHnzIercMWKuYQguLNTYmaZoTp2BMH7Nzl667KrRZMad1XN1q62S7SyBDFZpJugZVaA1189aqvdAtfZ12195tUo++ybo6G6wRaUL1p044FDxHv/xQPX0u9mHptzqri/SSW/HAcKoiSllTMOpIfrrfkGndRMR8xo93j898Zl5+7Wtfs76+PvvYxz52w5gleFD5cn772996gQcffNALpBeSPojIVo67obYc8apCgHcAMuHFrj+GTRBxfNzxgcP84PMtfEcU8XEL5yib66pVYEL2G06eLWzZyMggYoAz8xgXlM5GuMvUw1RfnFNoXghp0cWe+8Wo2bd7qw3rhPlGbVq2NEr/JMjJyTj8yuYj4dQfeZ4n+dwrfY7+pth8zxDIEFh/EBgfH3cbNkjYfEHEFDpvIfLCQdzBqOHkDUTc5YuXZBPmorCKrMoobWTHiH3ggx+w973/j+zuu99gPd09wlPCc476wDlgmsCDieGSMCSLSy1EhQvBf9OzC5KumZZhwXMyrCoibVSi3outYq6ovOqrF5GWiEWVFrOaxWhjPbhvybb2d8p2zXYb7OdEjSxvwdBRDxf1G3Nt9Ir99Kc/tb/SoYAf/OCHInyFN3Xy566777Y/+7M/t/e+9722Y2SHHwqAJuC3iatMHzjxqT5GXDk94JT9DIEMgQyBDIEMgQyBDIEMgVuDQJmWIsyewPph2KwFA+hMSGSnLH01DqHMirviyAAtKo9NMKX6wSY9j45N2xGpDD545Jydlo3HyWlp4tBGJka4oXw5z8mNk+dJPdqMq0of6O+ynVKNNrK9VwebmnXYEws2omtVJ44wTJvsMgQyBG4NAuCsahfrSOLLa8bIS1zEg++4Ag8Shjkzo4OGSM+AC5GguXTp0rKqM8JI0cC0QZKG9GDQBA6N+kKChrV7V1eXM0lQdxYXas6QoEGaJpg1rPdZa0cfoy7Gs7J+xkdsjDPGBaM5wSXGSjlUsn3jG9+wr2oj/+KFi84oekqHPB+RzdzB7cNW19hiSzrsiVr0M+clXXPsoh05fl5ShVLhpgPrC0vgOMdS8uXErEHlOZo2GsWw2dTdYjvEqMFm11YxbqT8w/EfJep06FPsI4UoCYLEpc7HOHmuvE2FKg8pdyBKf1rtlupjtxVXgGa1jOsijncaV7xnfOZwZtisi1ecBsFLXr8Mm0AI8bXz2VZdJYYNEEmizMI9muyIPp+XDYfn9p+2Q0cvSOx5UggMYq1ZkoDNwiEiyvRByNq20I/O1WjjkpPmG9obdHoGvY2bpSItGReEfFtSPlr30zP8UCyjEWILFKWuBuL1yCJl/aOcymhzKEPg9QgBGDbPPfec/e3f/q3ElT/vosqrwSGIGsclylBbXyt1aM1ifNxlf/7xP7f3ve+PJImzRcyddlK9CuibwCvJZ4EqAlXxS8KBLEDBTuC/6xPSz31Si9DDpyUCLSJtSiog6zb4aRvaRE9uvcSsEWdeWpiRP2etTTVuwwsDg7t3SK+u9NWin7tRDJs65ZuW6rMTx4/Z9777PVf59otf/FLtmjXotNCbdZjjf/7P/9eeesdTzoziJBFj5LcJF+MlXCZKqtN4zi5DIEMgQyBDIEMgQyBDIEPg1iEAbRW0FmE2e+48ho3GK1oWyjFW1cvL7JuAgvwFtVlkT08UoR5PF0x8u0AxbASiAGhKkugXL8/IGPdpaeI4aVeuTpufPq/VDiXqg1Vgkc1MMW1Qi4ZtB1QHd2iPYGRbn6TRt9vQ1k4Z425w+nyZpleD5S3LeC/0J7sMgdcjBGI9eLtjj3J8Q1w83yyOtJCkCTVnZ86csaNHj9rBgwf9OnLkiEvTIEkDQ2etOukrbQazBi0ZMGaQonnTm95ke/fudWYJ0g/YqyFfSM5Ef6kj6o+4eMbHqQmVSxgvpaV2SWOlX8mX8oxdH7Nf/vLXun7pDKbdkuh569veZv3qB2rO5nQwc07SNaPXpmz/wXN2SLa6Rq9P2/j0gqt9rKlrVK2oRpe6R+WtowNi1izNT7l0zQhahnYPaP+zy7rEkG6o0+6CDnfSep00dKAEkjB7EMmlfhUP6T3FQzHGeMTPEjZlaFTmB+855gh+ZtishNMd/8QLXn8MG5go4pQ7DnBSq3hPgRTwdaUMRTiyaJIrCLNGZJbEAbV5eeaq7XdjWxclZSMbNwtNIsQahLjSBid4EoaMtjV1zUvEucY297ToBA1Mmz6pCdIJGgRylIfaQTb6lytYNiA7DyuyHE8fw3meeMh+hkCGwHqCAHgY4vB5qTmDYYNaNE7vlB0/wGXCK2ENkU1CQPWNDTYwOGCPPvqYPfmOd9i9b7zfhiRe3dHWUTBXAsdQoyMZR4EwaOA5owZSgjWum/vSlXE7cEhqHiRhA8E2Md9oc7Vi/oiQq9NpH4izhMNYgCJhM28bJfK8U0xqDAz29bRaW0uDGNgwa1zZml0fu2r/8bOfSd3mV93g4eHDR9XveidS3/LWt9j/+B//w97+9kelm7fTRb6DaC2P33uu/paJkur0/JwhkCGQIZAhkCGQIZAhkCFw6xAIuooShO9shk0chyytoR0Uon2Xowo6mPF6GrdIrMSkJOhO8iU6OiRtoJmvT87Z8dOjduDwOakOuqRT6HM6nd6kvGLYaI9gwQ91JikbW5LqYB3q5IBTr9ShoRZtt2w89G/WZq10ounslR/orFNj0ZPoGvR/dhkCr1cIxNr3dsdPucBtsYYu1wWeYw90dm7WpianDFszSMucOHHCmTQway5eFMNC0jOkccHIZr0+Nze3rOqMflE/a1dUlqHGLNSYDw8P28jIiEvOlCVqSA+VZ6g6Q11adR/L3/1aaagnT1iMtXkBIcdXdKp4LnlIDp07d961eIDp6AdMozZJ9CBZMyO8dXV81k6fvSYbuRfspDQMwaxB5aMM2wi3ofpRLUo15CKq3qQKjZjmhiUd1myyPTvAa722uatFewHsGcCwoYTKSDpHWC5hUuG76C/9qHS1EnLES8HCpfzkDhfhUpmiJrBo1FtOjZLrxY85Xp7nzJXMsFkvb7gYBy94fTFs9HnCrHF0okGu+pUWkc6wEYKTH5882INomDVc09JTe0nqgQ7I4Nb+g2ft8uicVAZhz6HOjXFh84ZNTM6x10mCBimbRnGTO1pqbEjcZVSjbRvYYBs6pO8RRg0XTBuoP3cQlmow/SecywOxRf+KxyJ/9jIEMgTWEwTABEjqQQCilhKGDWrRIBLLLog1fL8ch4BT9KzFXodO6AzLHgzMmjdLD+2+fXfZwNYBqUbrdjVjzU2N0jkrgmsZ9ySB5HmJ2cxJmhm7Ndd0iubUmSvCdSfs7LkrflpwaqnF5mvaXF9tndrBJg1ShYlZsyBx5xotONvt7n1bZbtmk7WLQMPwIHZrnKwTU+eiRLC/9c1v2D/+wz/ar3/1jF2QOreGxibr6++zxx5/zD7+8Y/bQ+pzS0ubM2wgXFdzQWgz5uwyBDIEMgQyBDIEMgQyBDIEXh4EYrOHWgjfSQwbnYn3ZbJDwNfzrJ95YmEdtKJT2p4lZa48e9ZKDcqTYlLmdIds5lQ5dbI0L1gw2iNYlL3bKVeN9sLBM7Jrc93mdaBzYTEkbOKEe6KZdXbd1ad3tNXbgFQH79szaNuHNlt7a51s3mqzVy1A/UavU+vpnuneMjRy+PUEgVj73e6YwWPVuA1GS9iiwXYsDJhr1645UwZmDWqkDh065NI02KUhPZgz0b6vu4VbWKuiFQIGTVwdsv+C6nJszg4PD7skDVI1qDojDebMaqrO4qAi/Y0+xzcfPuPBRfspb2WMKV74qZQvcCB5cfizMJuk55w9AW9XSbUyprVUU2fjs4t29uJ1O3L0oh07ccUuXpmWOQjhvdpGCQ4Kt6mORe0bLOlaVD1LC3MmU1zW1SEb3gMbxazZbNsHN1hna62rS/c9T983kC6iReFQbIALdhLO8a6pJu8XN8Uuhz1Q9NnTVCbhZkfGKZwQvcKkRdm0p8tuK66c4hHr7JbmwMo5A3yZA8zlbMNmnbxwXvR6Y9igekwYqcBRibiqvK5ADPqEC4bIMmNEmUhdEgJAwga0KFrMrk3N2wk/QXPejp8Sh31MNn+sUUSZGDfiRIMvatVekrJZtAaJPLc0zlvPxkbZsemxPbu3yLbDBt/ErK9FwgYEEggn9QwklXBRoBbqLcIRlbLme4ZAhsA6gkDgYIwUImoNw+Yzn/mMM2z40S078uKIddWK8QD+EOHV3NxknRt0WmbLVtu5c6fdc8+9dv8b7xPzZq+Ix3SChjq9HnyVRx3arBjT1yfmdKLmsh0+ckZShZfsGvpqtfCcrZH9mtpW4SdO/wivguvkJ9s1OlHT3S5pws22a6RLRlSbtRhV/8SsqZEqCK0/bX5uRjp+j9rXv/pV++d//pI9/9wBnVIas9a2Vtuxa4cMHb7D/uTDH7Z7732jNTe1SCw8nTRiaGW3GiwYB/HVaeVyOZwhkCGQIZAhkCGQIZAhkCGwOgSCliKVMJs9d4pKtBUMG9HCqPllXZ4o5YKG9ohEP/tKX8+VjcKVdDYlb3ROdasM249pf0DnnKRxY0mnzxftzLmkPv34ySs2MSX7F3Pk05ElNbkoephDTk43a7sTpk1T/YLUCMvWw3C/6Oc+2yLbt22tDWkfQWXSGfTUizJ9Ww7f2McckyGwviHg32x8xhqqr2VXGXJ8J6g3izyEsTGDKjM0WCA9g/0ZrlOnTrmaM+LBezByWJPD2AEXUkfUA4ODC0ma9vZ2tz2zTRottm/f7mFs0cCs2bRpk2uRaG5udmZOMGooC6OHizD10t+otzycGEfkUWZft6c89IlQAgh59b/svM96UmxCheQsxpHKpZLEMUbaRxXa5evzdvTkJe0FnLULMgUxMS2FjmgVAp95XanP7B0vyXYN2jTammptoHejtAr1iwHdIUkb7UdIs1B9bTr+zm5DjfZMFaMLpo26RF91Vd5pqfNKwqWYYnz+REFdHlWEl3MWFSofODpcxMbzevMr77SyH8JcyAybdfamedHrjWEDKcW4/EPXpPXvevmzL+L9PSqHf+/yPbMeVY78iDwjPQNDZkr6gq5INdCJUzp5LlVBZ89flXSNJGwWpSpoSbochRk4d16vDU0IslrEnmtnxbRZlNhzm+2VLseR7Ztl26ZeG5KIB9KH1I/kxzP9oCMi1ZxZQyd5Vq6ifykmjYhwcmUEHnGUKhWqRC+XqURFfamt1GYlNYcyBDIEXj0IgHP4YYUw5DTEpz/9afvUpz61LGFDerhyWCSPE3mkkSNy1RXi2N0yaDioEz0jI8O2a9dO27Nnj5/0GRraZu1SPVYvfblgHpjSkzOLdukKpwTP2+Gjp+3S6KTr516SgcE5a9YSs9FbkNZa4TfUoC1owbkolWv1kiDssb27BnVasN0624UDwb8ijEGMDSJIp6cnbf/+/fblL33Zvvzlr7iY+cTEpBtefOSRh+x973+fPfHEEzYyvEMEcKMTwRCOQaiq4RUu4gMW8bwi0+vuId7+WgO/2W9BZe6sXrpSd6pF99KcvOnPzOoVrhmbqo32youPm/d/zQpXJES9KyL1UF13OV85XJ2v+hmwlPPzfZbbWvFQTniFwyv7sFblKVelT4SiZCV2rdI5/tWBQLyBV6f226/1tTUT0vdVgdHt4P5KqVuFQpmCrpSOPhBDjpV9uBV4VepKPbmVMrfa5+q6y+VeyXbK9f6fDN9svKv1az3CYLVx3n4c8zrmMmFo0juFYVP55UrjdoaNB6vfd2m+aB1eWTlX51sNfuTRWlxVUI4NQZgxoU74ijRxHD1+QQa6L+h0+piNTy5oD0HGuWsalE8ny1XQtXGoGmzZoEoYdUGbNrbYdgx079yiw0+toquXTNsEvq2ZcE3gmKKPZaIihkOd3uWICL88jqL8clT183JCDmQIvCQI+PdUTL3AJVQU8S8Wt1aj6TtIqdSFZEfEUWfgq3J58BdSMdiYCVs0p0+fNi60V8CYwb8g7Q+XL1+2K1euuJQNjBpnROh7pV7qZz2KfRkYLq1SG9bV1WW9vb22detWv2DOcGGfBvViXNikIS/lwtGnuCKONsrtwATiCmZO5FMmR3P++evTTWBW2SLkGACRFbn4svGpe9lRuGiPxQnr/wWt07Gz5fUoeWxy0Y6embZDxy7ZydMX/CDn/KL2CoTH3Fq3c0KEk1QOyZqaBe11aojYrR0Z3KSDm33W19siiUG0D4llLbXoHPSkXaeXtENA26nb6q8PqOhhqavLfS4CjGV5YMtjJrJciFye02NJqcSQd326mEMxj5izXMy1LGGzjt45L3h9MWx4OWCU+FQJpQ+YTzc+bWJSDt31n54VgALjSZN9QWI2c3pOp2hMG5pj9vsXDmlT85SNT83Z5HyLdD22C+FhgLtBm5MSF6SNxTnpdZwSQpuRccF6G9EJmt07tmhDs02q0ZQPHCW4o1rIdTuqv446aRZmjS7vg3NpElJF6gdHP1PPPVCEy0g7xSfEmJB35Kz4QrQ++lS8AhU6tpYwdqV0DmUIZAi88hDgx/XkyZP2yU9+0v76r//aCUnwM/H44crhROyAFRJmcNywnFc4Rp90gxDO5t5Ndv/9D9jTT7/PHn/yKRsa2m6t7R0iwJLtmqvjc9JTO24HdaLm+MnzWmyKiEMHtxg289Jbu+BEGja6xIwWXhMbRidqlmyLVKHtHtmixeaQJAqlzgzaFKJUp24gqiE8x8fH7be/e9a++MV/tq9//esimE+JOTWjk0j99sEPfsA+8pE/sTe84Q3W07NJ/eXEUdLjG0QH42bMXMThwveHfBMEmCPApRoYwu6VqXMD3MpzqXwoYGUtVJCIbeaZ/xQpKhH4UTkE4o2lVsbc+FRVZPk9ezueSL2RK/wb63nxmDSGyFcZt+r0+qPuSr70u1gZXwWO5I381FgOp3nqsYpe7rrypJrKecn1Sju14h2t9LvSB95ZuOhPek69Ii6opYiJ/FRbKl2utJLljgq92HjK6QysMg/XHuZLKbOytpW4PqVVfVu84tKbZP69Oq8j5mr4K3v6h3/i2yrDZ61x3xjPzF0Js5v1nvGqDs/CPZX2R30DbNXGe3Z6XZsklblRgVXpcynq8Br85hsYKaT6KmUqORRaWUGpjRW5lvuSYkt9XTEvaGONdlZWdwc9lcd6K92O8QvPVcHWS6/xEUWpW2nhTs4DTGIeE4buvDMYNn9YqAObmD8wYTibhJT61KxU/16+bodl+/GApNQv6tDT7II2OmtbhDGanHlTW1efYCxpm1oxbOprsGkza/2Srrnv3p22bVAbva1SbyymTYM2OhMOAL+wNuf3ufIbDVPK35mGz9RN0ze+ifCBDTM4roTN0jP318vsBg7ZvRoQ8DlY4M7ytwEuKeOTyFeOoz/xTJg8uHIcz7EGjngkZYjjGWYK4VBZRh1lSZqQpkF7xS9+8Qt75pln3HYL8UjbkBcXdUeYemGawHBBQgZbMzBhsPUyPDwsjRDSXnH//a7JAgYOeSgTMKA++hX10Q59pE2YSBzQ5Ir9V/LRHkwh6uIiTJwzcMABWlOrWleB7gsxZzqnbzqpa1Se4pv2L97BmXBB+tI9dpmigU2zIFzEal9UjYfPXZq23zw/YQePXZUN22tSiy74yN5srduu0fh0iF0Ds1oxeeo5nL44bRuEs0aGum2nbHINDnTbhs4mwQ1sBYun+F0p+lKrATCG1E/2G+lzdi8HAjHn8Jl3cTH/MsPm5UD2NVaWFxwIA/sJn/jEJ2QY+ivObQbBoHfxox/9qDb6ntYp7V0u/keZG7i/r6lxlRk2EDhBpOBXSJSEwEgkVs65wCkPyARcy0alM21UyfXJad/MPHzstMSfL9voZK1IrQ4Vxz6NCDGRXs5sEQJcWpxRdTqb3iS9jhuadIKmx+3Z9G1q1wkajAvCrIFR4zxrR2Cwaly6pvAdoald73/R2eQRE85T46HwGWNRoColPaYyvoBVMEGIFIjCzLBZFWQ5MkPgVYYA+BaGDSrR/uZv/sbFs+OkD03zIwzuxeF72D/z4nuPdPCZx8uX3rQm6Sjr2dQtAvM+e/p9f2RPPvUuG9omhk1bh8ScTbppl+zcxXF77gDMmksi0qZlzwYWADq460XEuSUa4TghQ/DawqQWmrM6GdjszBoYNlv7NlhHq6RjhEKw1YUhQsYD0+a8TjD99Kc/s69+9Wv2g3/7gU4zXVK/lmzHjhH7rx/7r/bhD/+xSwK1tbUnOzkFw4ZxVo+Z54j3QL4VEGATEXhVA4Q5U4kL+EVMzCeeb5thE5ybqKzsqx+lZsspy+EbfqVUgN+k1CelajBpPDGo8JeruI0Avan0qDJu1emNRN2VfCsph5SeYEk49Y8OlOHLc7mdlRuzpEU7hF8NBxArY0j9rLRTgUAxniIpPTHilCOey72twAyQlVMq9a+nUHm8jOvFxlyd/1bK3AivCjOgklZ8B7za0hxO6fGNVHK/cqF4x+G/cjW/tJoCN1Rm8YvXo76Xup9Kvlh5CgiuXjl38hclYRgVz+RIudhQJfNKXLui4aI8uSr1qdAK3JNSl+9ViGWt+bdy3lX66mNIg1CVBJYflpu4swPlsRYjUVR6P3ozNww3IpJfhhuwTW/4RohEqRtT1lcM8Ig5RpjNnsywqX7HCQdhvwH48InWiF5ln2BGouoTM/N29sJV238YtcJX7Op12coQ02ahRhsBNdiQbBCMwRcqqJPpSzrYWSdauntDs+3c0S/1aJttqw5BibR2KRuflwUeqNFmMGt05rdjHuasT1rd6IcmaoXeICGumMHFvC/hgoS/qseYnzMEXjoEyniVWso4ZbVaIz3SyuUjLX1rzOeV9QWeggGCZMzVq1eXVZ2dOHHC1ZydP3/e9zNJQ6IGH6ZJmcFDO3HBoEHNGUwYVJohOYMkDfZnYNb0SHMF9mnwucImDXui0V/6FeOg77Q3OjrqKthgHKF+DYke8Ct9j3U+jBkYNdSJtA7tDg4O+oUt2ibZj6mvVzuyJcv3zkFu1m1gBEFGFzjCE+QrNoFMMQVuL54ps6A+Co15eRg307NzNjZ+3c0+PLN/3E6fnxJOk/kHGEVi2PjeYDCJVbZOmjYaxHRulaaNvp422yP8tV1MG/Y7m2XXtlZr/Fpp4kB1e513Vl2SczyVQqnOzLBxaLycW8w3/JjH+My9zLB5OZB9jZXlBa9nhg2IrMBRDnmecdxTSAHBAOfEDgjJn4hGXDDpqBU9ZtPiNF8ZHZcRrrNSjXbSzo3O2/RCq/LDrIGQguUi3ZbKuyTuMwybOtmzaWpYtC39HXbX3iEbEvd5Y7s46A1i7ywzbEBs9Cd6lfzE/FFd0R/Pw0PEVIeLjEVN8VQaaRFFecg++XzgXh9tqucgZ+9HkTV7GQIZAn8QCATD5u/+7u/sU1KJhuh2EHJ0gB/gIALDJ57v2HGFE0V68h9tfc0wa8Qx7u7aYMM7hu2BNz9ojz3+hL3pgTfbps191tjc6gZTx8ZnRaRdlvSg1D3KYOqMdG+7wVQxoBedWVPgRBFoNWLYIF3TovXngJg09+zdppM1EgPvAKcllZDkXpCEzbyIPcZ0XITzd7/7ffvmN7/lJ5zGxq5JVLxF9nXeYB//b39mH/jAH+lEjmx8NTSJGG1Qv2Eap/Hix1jTuMBTlTR/yDdB4NVg2Kz8nYnfp1gE+O+H/3as/gJYFNzMFW8y/drwU7SirrQgSeWjovBvVutaaek3L1JjTvlvHd/N8m9eJV+a9QGDSh7IhbQwS/3heaXT9+h1Vuqi/pTt5YxhZSurP6kV71B0qgxHYBwu+kG/KlRRvIOUmt54lKjALMYfKevTv93xlvMDkTQHbhc2azNsqutnTvk0W6WJmJNrpa9SpBQVc4OocriU5f9YcDX4vFhnNIbSMGKOr12KzDH3CfPVpC9HWxz+faVvhjyJXk5w1rdU+cC8Dt0KV06I+lI7a75EShYV3mwurZwXUTeFy/OjaIvodePKY9Wg9Ljy3ZbHz6CBAS58wBu42oun5Kp7JXdVwjp7XAELwYXNnsywKb/kNN/c2HYBH6e8dcjR9wm0+zkvoufKmFQLH79oh46es9PnrkrdcI20dEj7hiRtatHCwYEklV8qGDayIGltzbWSsum0keFeG5F6tP4NDW4TgtPoIVXI6X3wgL8n4R3HCSAe5r3PY83s5cka3wZ+2ZGB3/wyfiun53CGwO1DgPmHi98pnsEfERfxHrHKrbp8OQtlqSvy4LMuRjoF7Q0wPkK9GQyZc+fOGcwaDj9io4Y48Fh5LR31803BJGlqanL1ZRs2bHAmDerOQs1ZMGvwYeDAzCE/5ehb+bukXuKijzBj6B8MmqNHj9qhQ4f8KjNsGAfrZFxI2CDNA8MGJtHw8LCNjIzYyPB2SeAN2OZNPdYi+691YtyoMb/S1843zR6e4vx16Ln4/BWjNJzwjuJYmwXDBoYMh9LHxqfszNlzdvDoBTt4csZGx5QdJjHMGl+TJ7xBBahGh2HTXLeYVDoO9thu4a4tvYKNtKfX14FhYNYUe5veJ+LoXhn30F9oqOxeDgSYb3E5fIGxrsyweTlQfQ2WDcTCi11XEjZgJTlQBPZowiWUkZCXI49I8IWYHogEoXm8JrwwGzYe+OlhMTAztyAi7JL97rljduT0uI1OsKkJo4aCYtaIeKsVAsIGzaL0OyJp01A3b10bG51Zs3Ok1wa3dus0uvRUwoFWzXWq18UEabNoG8TrYtBer6KL8ZBluXeVYaXoG+6RIfzIwOjSB+51FVg9LT5BntX5o1z2MwQyBF4tCEC0QcjBrEHCBsKTuGpilR9iXPjLuIH44ocbIrJR+sk2bdpsd79hnz3+xOP2jqfeYTt37ba29k6rka0YCTVLP+2snTg9agcPX5CNrst2dWxGhJxOAdY3CW9KwsZpbnCFJGbmJF3D4lJ6tzdL1/bI0Cbbt0vSNVpoNokB3SAGkZ//E7N6DnFviX+Dto4cOWr/8i/fsO9///v23HPPSZ/whE4mddsjjzxiH/vYR+3JJx8XYwlxcnT36hRiFQG3PL7SmB0A+VaCQMLppYgiyOKhEhtzphJTCZXhnH5jUkGI+zKJXSnBdCtVXk5QOPWIX1yvwCP4DQ1HfJrJEVP2lbacGIHwy/lW9mHt8aXelEumvqtObyjqruRjeZJGQSnSI89KvxoElX4X5eURWtkOEa+Go7GiXa++DMfKaMpjKb+T6FEa4cr3U37Xa8M5anhl/Vttu5xvtR68WL9XK/9iZaKdctlbLRNlK35lY6ISl94h9fNq0/xa+V49L6++9H1F3tvry8q5XenDi4fK4yf37bX74vWnHOW5XcT4fL9ZeY0phlWVrQyvymyPzOFX2kz5i2chxlQm3sVKXLuyUcqEi/qoX1cFYUQG96P1FZGrPKyEe9RNxuhXCq/szyoV3XFR5bGC9sowjsFUwyBgEekVf7XSa+eulFsvIeAX3yxh6M7MsCm/XWaIVAcJLtDFDivRqn5KXUmuL0M4ASmbMzr4tP/QaVePdm1cm8uSsrEa0bh1zSonulp0vXYIROvqEJTbs5lzunpgqzY+JbG+Y7DTeqQ+nR1X8oEi/JS6egAOSjZumdtpvU7PHJssIw1iqi/PETnl45YLpMd8zxC4TQiAKwL38k1w8Y2wdo34kD4p45dyM7HGjXykRb1RJuqEwQFegkkDE+TZZ5/1dSWSKzBHkLSZmppaVjcGo4a+UA/rYi7q5hnGC9IsSMrAHLn77rvtvvvus23bti1Lz2CHBtVkXEjfBIOG8jG+CPNtxvqVfnDo8re//a396Ec/sl/96lfOSILJFH2K9T3jLddLfcACaRsYRDCS3nD3Xfaudz5pDz34gA0ODVmz+oW9WiTvUM2of9UhZrD+wBq+K+l7kapcSIp08IkozJSf/isX+5tTs4tS5zhmzz1/yA5IOnB0qlk2uiXmp/3MJV0wVZKEH7gHRswCytKtvbnGD2zu3SVJoC2yYSubtmAkGDVgluSrYRqXdDLt+/6o2lbj5PC6CWX30iEQ30rMa+YPF9/MMdlF/trXvubSYR/72MduaESHB3xqeAKTlQIPPvigF6ASHH5kK8d5Yr79wSDAOwB58GJfjwwbAA3aStgDpFLgEUc3kEpwouUTr6kLV/rK1XE7dOSc7T8+aqcvSh+lkB0zHpVoIqt0wbARIiJOUjY1NiOu87z1dDXbiESed430u4GulmYZFlMmv1Q3UjbUk+pKDJv4NlTp8vfiFeu5+JRIWsWpMnfhl7OoER8zd0bIM/13VOshj8i3DIEMgT8YBMDBnAj69Kc/bX/1V3+1QiVa/Fbil3FCdM6/ciEEfKRU0LULAYoUy0OPPGQPiMjbKZWWG7u6XS0DJwEnZhbswiXZrTl83o4dlyq0qzLOOC3pGEkMWq0IU+nbTvgIxovUN8g2V72MCCJNMzyY8NjQQKd1I1cHyAAAQABJREFUdUhfrRquK4g0CLP5+TmJnMuOjSq4KML62WefsxdeeEEE9jG7IpVoIWHz2GNvF0PpLmtwwlMMm0JlRIwr/LXGHenZr+D0lbBI+J/3iFtt7qSUdI95ln4TikLMqlRNeOUipd+lUrTy+9zRb4vPSsqrOhY04Ygvqo2okq+05cQIhF/JVulvilt7fLRbaTtqSMsKnqLuSr4KXRDpkWelH7CNOiv9Vl1KTGOmjK5KYmR/hf3UZmWsZThWYivjTWMpv5foUOpxSo+4MrzXhnXkfuX8W223nG+t1m/W79XK3yx/tFEudyv5o9yNfjXDpvL+Km1U4srlK+kplnnJdLu9/sT7Dr/cwtrh6rYj5+21HaVu5le+T8/FdF/lu67UAAAqT6uFKC+IlpIiHH5qs9JO9EGlWBS4q/aJjLiUo/L1Vcp7nlVwQnXJqGEtvwL/St0rq6XG2611rdZeK/Ex1tSfCgzK/St/KzH+8Mv5Km9nZez6g1r1+OIZ+MX3Shh6NDNsAjr4zDfwMxcYAwoBLRzEK6wPjr0CVAxfGZvWQahLdvjYeTFvxqROXUyepSa/liS1TlV1WvBjlNsWpX9YdiEb6hds44YWG9KBzn0j2kDu69CGrVQgSf0RG5/yEvng7dEi85j9hsp8rnzz8W2UffKVLz2WyvKUXYbAS4EA+KL6dxT8sTpOXtlC4BxiCXNRDmYGqsRgcGBvBpViMGnK19mzZ12SBh9mTdiGQd0Z7UedIUUD82Pjxo3OjEFahgtJlghv2bJFtlW3uMozpFzChkz0K/oaz9Xj45l2UbuGNM0vf/lL1yoBU4l9cMbBuMKRP+oo1006DByYNtH3PtmhfYMOX77pTffZfbKds2/fXdYrFW31MJHq2C8EI8AASX7S9aOKoFFAA7giCM4CX8FknplflOagKTt6/IIdOnzKTp2/ZuNzzWLHNDkDKBg2HCrXDqhKYHtL0jWNS7ZZuhvZz9wlO909OpTe3Jhadayk7ElrEWOkpbTaSiohi87QIb94zu6lQiDmEX7MTXzmIvMuM2xeKmRfY+V4wa93hk06GyOMJlgkzAam4aw4jOnErAHdOPoX4TQ1PSebD9ft+aNX7NCJUbs2NikYclpGJ2KEMBcX4eAjacMlQm1+StVOWmvzomw1dNkenUofGujRRmdrYcuGzU5KqtnohiPd9OMViFzJBXKnZyBD99a4RWL45WwxTuL4mU3PmWFThlEOZwj8YSEALoZhgw0bGDZI20DcEY8LvxwGB6QfaPnCIPVisnAaZ3h4xO5/05vs7W9/uz38yIM2tH1IUjOcxhETRrhtVjTjFdmqOXbish04dFY2ucbEYKm3uYU6qXWA6EsnaqgfKcGlxSkXc4Yg29TVZnt3Ddlu6azt3lBvLY2odxT+Uj8hytRR/acTVvQfxs3E5KRdvTZmF85fsPPnzip9wbYObFU/t1uvCNF0ukgq1cQoSqcGfcj5dssQKOP0ciHwP9etOt4decv1qXxU4e+Y9EpcJT+/V6QVpZkH+mP5UClPmjfg8VFtKlW+V+oie5Qo51gtXP6tXJleHk85xWesIqIn5XzFfPbspEeeip++yXhO9VZgQMdTfXg+ryMxZX0V7kX/00vxPpebJDW56HP4KTbeTTmXv78iooyDiFob3lHDK+OX271Zm6vlK8e9WJ+r875YftLLZW7WN/K+uCtvcpS+ARWstLMy3uvUi61+d2nO3e47Yj6snBMv3udy327M/fJhUq6TgTLSykwup6bwjfC5We7Vykc9yY/2ohb8Aq8BK48OmIW/OtzTO0zlE5yVv/yBFp2p1FJE3IJXqXu1KqPG8G+hwtd8loBj6mjl+yh3vDwXYuzhl/MVr3FllD+tnnuVjHd4FPCLb5Uwmz2ZYVN+qeCBtNmqWaUENpbZIyBeYX3H7BNwYh0pm0tXJl3V8KGj5+2iwrPzoq9FYy8s6hS8pGzqZfARm48chILGrpUtm/r6JdHXLbZvR4/t3L7Jero7ZQ9CG7LOsOF0etr4BOfEej35pX76hI1vo+ynPlMyXZTxzKXCOZgh8NIhEDg4cAl+XOATLnAM672yI458ab0454wXJFRg0mCDBoYM62E2n1F1RhxMkUmtK8MGTLneYIrAcGlpaTGYL8Go4SDjjh07bGRkxG3DoPqsq7vLWltavW/0ESZJSPsEToz+xnO1TzrtgjM5/P7jH//YtUo888wzstt62ZlP5ToYb9nFc4yjWuIGkLW2NdkO9fttb3ubPfXOd8ou7Rttg1Set2qMtWLuJLwg+LpUjL7txYSjaId6uRQlxjK7nmbzCkxMzdnJM6P2/AuyzX1W+5mSCJyEuYzNLVShUZfjiSQRWFszJ20a87aho96GtnRJIrBPfrf2N+sKLRswo2lPxdQOewEEUAMZffA0rzPjHwHnZbv4xvArMNa71lzODJuXDd7XTgW84PXHsCmIFI0tCKmAOM+4CprwHHpOZZaRpjY/Uwwpmvhi3ECqETevj2B8UqqEzk3YgeNXJPZ4wcauTwkxYZhLqobkLy7QAqQUP1izQlaoRpuVrYY6MWu6tOE5YAP93bL7UKv4YsNT+UFkEIPL/SsCKSY9RB/VgPLToyqnKGL5cKltdVcul35UveWbllm9phybIZAh8PIhwI8rDBts2KASDT28nBTiO+abj+8+fdcQQnzowk6oIhM119zUJNWLG23f3rvtiSckOv3wQ7Z7z27b1LvZmkTQQajp8F8SgZ5ZdP3a+w+esdNnrgl/yebMonDXogykimEDdnDsyQJxaVrn+KZ0AnBJBgXbpEcX6Zo+29q/0fVsN9bX6MRNBY+CMSFc52XHZlEEIXWxWT2vuJnpGRGuU4qXwUKdDOroaJe0jUSvyeUMbvQCw/CmVHIx3njGf7H0ct7XSzhBuhrjr4X/V4OKYM4886QK/NO7UWRE6beQn4l4JFC8ZZ+rkcY74s9/z5jD+lvh9Lh271Ib5E/vujpn9XOqebW5klJou6p9T6BfOOpTyOddkc9/Wwn7CIo88opeV6Zopa+khvOTuNShf2pwoBVlI8+r4acRFWO4SQO8mZUunlOfI21tmEaO175/M3xxs7S1RlYuE3leGTjx3l783UWb7vvrijK8w3iPTLmIX1Gi6qGSv1y2KtNLflwNVlT20uAlLLI8pOVAVd9W/x6rMvnjWjWshEPCY/FeAmOkPBXYJUwXrVTiI2bZLzYwlsuDMKvcjTFVGdZ8ZERrjeql17pmc6sm/KHauXGsK+da9TyIfoVf6byXUzQ1ptRKnkqokn89hoBBfJOEoUczw6b8psEDxS4Ap9aX8WzlFzcYNjOie6eQYL+szdsj58W4uaIDS6gJ5lCnNkNFUUPngp9RMSQ2jyafJG3ku33IzU06tb5Zm7ODkrpp88NSSLDDsHEV6nSL8t4PNr9XmaWO+9NmaWVmR78jf/hUmF2GwMuDQBn/Bg6pxiuBY1gjcrHGRc0ZzBcYNKgCP3LkiB0+fNiZNNeuXXM1ZzBwwEdI3KDyLKRoqI8LBgsXTBokabq7u50hs3v3bhseHtbBwF6Ph3nT1dXlBxth5qAWjbUoZXH0N/ocTBM+L6fhC/CAGyNPjIckmEjPP/+8/fCHP7Qf/OAHtn//fmfWMD72WXFRLhhC0V6k8Vx21J/6IUaIuoiKtm3bt9sjb3nEnnziCXvooYdsy9Z+HzfHvvmiHbcoTK+9Oq9SNyU6o0Z4Y1bdQSX6+YvX7Mixi7ouCP4zNrMopnKdVKLL3hYMG8p7nyVZIwVq2q+cs9Ym2eXu67R9uwdsmyQCuzqbrVFcGpcCVKu0jIuxpD558ylBd+LU0eXnHHjpEIg55O9JMI1vIjNsXjpMX5MlecHrlmEDxIWYoGkCBZY3KRKqICZd5CojmEgBtUCIubQNVeqaXVi0S2NzdvSU7D8cOmXnL4yKGIOUatDVqFM0bDrSArWIIBMxVquNz8aGOdvU3Swpm60yLthn3Z1t4kxjzwbmTkJiqEZLeCwtFPmhqKSqxhsQeowudY5S1JR+SNIoFbGGI28qT7W3VmaNqnJ0hkCGwEuGQDBsPvvZz9onP/lJO378uBOlEGt886Tj0jdKSB+s4iGq2tvbJL03YHftu0s6bh+2t7z1bbZ3317rlq2YWhknZJnJhXFBP/1XiEAfPHxO4tAzwl2SvNFCEqYzeA4csqiNJXBXnU1aU/2UocJxa590bO8alL7aLtvY2WSNOiIonrMINfVDZcCU9Iuy8/Ni2ki6BvKtUUSxj8PzKIfGArHOgrVWCC8RGCxi4zSPihWuMt6IYdgJZxGzWnol5+snFBB5MYy/OkTSXEq/HdU5lMbi3xtQ7Zof/D75o7KW34Uyepo8pqY7xRCt/FGiiF/5mCKX70U9eq6uvyi9nLMcWHsu0NhqDTLXceogIe905JPvmx6MgHQunMpEFp5WWXSkPmvEXp4SUT7q8Ipe8Vuqnc6VOrhGK5Uc0afwKV5KXWV8a1T5mo0uz6Hq93WztLUGVC4Tearrjfjb82/t3S3XqewrvyveId9OvMtbqS+VWa7zFQ6sBiuaqPTx1htcq67qGm617sosr66h/AyEExwTVKNUBW4pJqWW7+VaKmH9Rvr3VZRffleVHKmOyvOth1I/bz3/K50zeh7+K11/ub7Vx5rmCN9AOS/hiAg/pcec8uVaUSTlKN+LhHXsAYf4bghDb2aGTfmFgwVC4p1f9DQ/fP5osrHeX4DuVRFUDnPwiQ3R46ev2mEZ8j6lU+yT0+wjNIn2hVqGYZNqqZGKIQ52Lko9Wp1OsG9sXbDtAxttz55hbYx2W1trgw5MSd2xM2ywPimqWgwbn9M+celL6o8Cyfljoscr9EDki7zhR6HsZwjcOgSq16SBS5nXgUOozee54oJBA8MF3AKDAyYN6syQQrlw4YIfWmTty3Xx4kWXTKEcbbFXSTjqhDkDAwYV4EjQoF0CRk2fVIUhOTM4OGgjI0mShvRQcRZ+Gd/R37hiXM5U0frbvxr1Pxzp5I3yPCPpgxq073znO/a9733PkKxhbLG/GnVG34MZFG1G3eGX609x6feO+DYddNwpKaHHHn/M3v/+99l9b3yjq3lDw0bCAwV+AkeomJd0P7Gc53R6c0qHNi9cnrAjR89KHdolqS6ftMkppUtT0KJs2MpAjsZHeUqLUSy7NXW1s9Kosejqz4aHemzPzq3W19MuVWgcOi9wmfdAsKI9LwuuCjxXxCktYFeNttJY8/12IBBzKOZMfG/MuWNZJdrtgPK1nZcXHAhl/diwAcFw4SqbK9A1bCmGSyFi0pXKgGXIAXGmHI6k01YkBJnQtAiudFJ9Sqdozl66br9/7rjbgJiYWrQ5iT0vScpmaQlJGzZaIZggsDhBM639nylrk2q0AW147sK44PYt4k63upRNreqlL4g/c7FZxB+x4l27r9syEiSMK5+iDARJmYQQK+NNuavv1M+Fi7zhp9h8zxDIEHj1IcC3i6j3Zz7zGfvrv/7rZYYN3zE/vFzlH+RaEIZcU2OzDQ0O2Vt14uY973mvTtw84qeJ2sTEQTZZaCrhLdWDuobL16bsiPTVHtTJv9NnJFaOxkYMotY2CxPUqR1wDDq3ka7RiZqacUnSzEp6p00n/rba3XuGrHtju3AW1Uu6RjRdOv1Hf4oFosYyJ4bNxPikCOxFa9ZJpiRebf8/e+/9Z2dx5Pu3NNJIoxlpNBqFUZ5RRkIIJHI0iGSSwcbZxti797W73r/ku/f1/X5/vjb4er02YHAiJ5NNDiYJJFBCOecZTdb9vKueOqfnaISEbHxZcXrmOf083dWpuru6u6q7WgtP6eVVvkiDsQfbj6BzZ06xCLWS+U9pYpe5lekc9K9Kr0BNJRXP0HUCr75YKceSB1HMJngQnjUGBrYjvSNcaClwrwbGHQ8b/haCoKUcur9Faa8exuCLCT5jUSmekhvAHjlOg9X7YG6eGrFHjtzFfxlx3cfzE1/46r0ot5WhVPLynAKoyjTLbVMlLgpZDh/YI+Tf3xTYUcSDlXVgemWIyFPYBC/7Rqi8nFHG3C3g/lF25IH0jpePT4PFj4c4jhdPlC2PL9xONGzAD26D96NxPzgs1ZTDUn+VZTiR+DzcsdL4e7kPzOvx62ywdImjMp6Ay/Gfv4f/YHaOvcH83a0Sh/m30w/w7o+H8J7kv5XxlikOPnwdDXe0i8cSZT92+fK8Vab8j/iOnIf9eab5WcvqeaLL5PgLnBrfO8uuQzvlzpxP2VfwEHjhnflmVWCTVzczmBDY0HOZq2rGy4SZPqxPTpEz30btcL82H3X3Dkl7D/boLpsdaeWqTVJD3K15sa7rlns/atO12Qr1RcK26Fq3NnrCJ9Cdt0Pb04Qm1B9N19OSWiY22v0Qw3XnjWboxg3wOyVopXqs8XqLJU9m7DP6SNgFfInmVISJsFW7ioHjYAAaEcKTwU6nRPCgr/QT7qRBSMM6d/PmzbrTdJ0JORDO8M39NNzzgvADeB7C8wRtIl7S42QMghqEMzNmzEizZ882VWeoO5upEyjNzc2mDg3hDOtLBCTHMpEGdqRJeqQTGyb5jjwEfLhRLsr04osvprvvvju99tpriZNBwGFC4BTfeTyRJ9xIKwQ54Z7nCQrEN5scWfefvnhR+trNN6Wrr7oyLTztNCuvC1kUWjTI0rEswLuEb6k603ePZF77D/Xonq096f0P1okPIJXofSNEm7S5U3SpX+WGhtnsRCSCkzPDh/Vrg2ZPamwYmmZOG5dm6QTgtJYxabSEyaYSnSSLhzmNb1h3N9uELj/ynuMg8CCvqvkbMBB4xQan8dCWqwKbvwGxX7SgVPCpJ7AxjqNQ7UsjJxnFnKbk5jXhyyWfzAS0KIo8IY0YkSCkzBYOIoT+RxE9Mam69L6/vSdt2rxXEuqd2kGzTztqmLBJYDNUj6ZVfTboMMnTcUIxP3XFlxG90XWoRhuXTps7Q7vWm1JDXa3tVvd9NxLwiBnrghhyAeGsFNh47sin+mfJWNbtKwaXzLMElb8QT8QVsGHncNX3KgaqGPg8McDgikq0u+66K/3yl780lWhMXDH5gMx7jQQetbU1uixRF5TOX5CWLTsnnXfuebqUcLEuTZyiyexIbZBh0qWJmrozkzUuQ92vHX+btu1LH6+RTuCtoleHdCRdQuY0ZIToiJ+uIR/9/dAs31Uzclh7Gt+YUttMnQrU0zKpUZM0TYJFFn0iZtOzkqCZ8F1dnaKLW9Lrr78hncObUqN2N5GvmTOmpym61JGdUMOR+Ij2UB6fpPKNIEdWZiondcCHqfQL9y+jXcbK4KUfnKpHqPJE2kPLvRQgYORQwdWiKmgn2D4OMe6U02cUlUt42kjj3xF9BqxgDu/hwyev73LVu29l/Vd+l3PCW5Qjfy9G/ZIXL/7h5QgP0oscuV2ZF2I1U5ZE6ZMSFTiw8BFHAft3toqcWbqfLepyvqzEUewskhy3USe5Wwb6hXqNvEamKvOMPw/ug/kdK1y4f5odaVfGOzCMp49b3ncGwhz9Va6ict05lH9/elvIw+TvR6fzRXKx3lQ00Cj/wNzn9McozUll35PwNuH0IFLzHh00gp7tBjve9Wrg2be9lqEjvIfJ4LLYitcB1vHbEwmX8zog8D/kI8oS9ueZ6AmWVWC0m8D1YH3R2lVFVr0EJ9+GKqL7wn8GDSSjvDOPqwps8mqjlTDXifkObQOeQJlOcMLGTqibwEZqgDX57ujuT1t3HEir1+5IGzbuSXv2dUmwI6YoKtQTc29nJCukeDG6z6avM42oOSyeQEoTxo/RnLsltc6YKHXEo7TZkxPtmv/rpI1xJRizoDsxL2MAKb1bTVKb2YMbLTue+MaumioGPhsGfK3ogpU8JAIK1q65qjMEMjCPsTlJw6maeBDShKozwhBv0GnWhiNHjrQHFWaclJkyZYoJaThBM2HCBK2Dx5ce1J2NHj3ahBcIaogn4srzGO8xpvLNezwWTgtc618FcB5PhKOsCGteffXV9Pjjj6fnX3ghrZcgCpVtYSgPhvARf/hFnLhbmoPk18NYBESijMIjHJLGS4PG2ecsS7d+4xvpyuXLdbqoWYIcreU5FcQDrAwUIK51YNPmQfEtN2zx+2tdXWOP/EfqkYYg8QGOoHtNQRHzwJOsGdqnDZp9qWHUkDR5Qr00BE1KnLAZI2FNrbQ8Sjxte9sMWwW/1CKw1Is3MqG4VEzFW9BMy57nsQCtWieBgWhTlW2IdlcV2JwEQr+oQajgU0tgY6QJyiuUF8wYKA/vGXEw2lGCYCHlT6meFB7cYJBY44vBhvSya7xHEXbprpr9h3TseePutOqjrTpi2JEOdys2CWzYB9MrwU4fzE896KodqqPPNUN0waB200xoGiUdkNPSrBktaWLzmOLSLtLU0UMxfURvixR5cYENU0bKRtZCoFPQZMEoXc+yvTuxPh4xJEAECtiw5VU1VQxUMfAPwQATXPT23nvvvem3v/3tUQIbMhE0aeTIEZqojk0LF85Pl156Wbr4okv1vlj6rqUCjd0x0AjgRRx4UNOAKrQt2w+mj6WeYe367VKF1inhstSgmRpHXzgSrl8T0P4+LRxFwZCpjBnZmWa0jEgLF8zWSZ6WVC9hc630oCGwMbppREeUEZol0sEEdp92F7399jvp3vvuTW++8ZYENrqccM6cdPbZZ6ezly1NbW1tOso+SjufoGvkFJpjS1B7j3LqozTp5D13r/Tj+8tsgoqDA2rfqXiZlpffAkvWQvxDr95iCj8TOuQx4q4YghmgL6rd2hltrdQGilRpdxZcP3q3sci8+PEl0GDjU+TBoTz9vM49ziIvWDKxKKl8N89Bf4jX4yYv5TgBdndluXgvwwHrxm2LxX5wLfyENw/Lt3t6XfAdD/Cfj/FclDJ1Uok4PsAL8SjXXqCSjVulH25fRBP5DJuyRHkiv+HHd+6Xuwds7h9ux7Irww8ettx3PB7yd6wYy+5eM3wHcNhlGPcdrC0EbNgDw3yRv6w0apdBJyrzaiWK9mqeTmsq4U7km/rzOhuIQ087chA4xI53vSoIO0w9P4SXKbzdKtzMMQvnkAFafJWtaFODtyXgBua1HPIf9RZlCfvzTPfEyho48wo4uv+TQ48p6sTz7CU4+fbzeZb884i73N5pu1WBzdE4Vp9HWKN5MS2FluF/9qEfaAJ/zIp1sl02D2qI2c2+cfP+9JHujNyoDZ6d4hH09jNPRxUxc2AEOJxK595HnbCp6dTcW+qHdJH31CnNpo1juuxxY3Xnhu6SdNVoPt82WmDzMrVYGi3vBQ2M3EQLl29hAMyfcK/aVQycOAagEyGcQUBReR8Np0xQdbZp06aEBp8PPvigdC8Np1KC90iKQX9oz3GChlM03NmCkIYTMwhjJmvDH6dpTtOJEk7S4Icgh5M03EWDgCfi4t37RLlMg42dAR/5wAYO9zARLnfDD/VulO33v/99euihh+zUEAKoMAHvmxLFPxQTPdyIM9IJtwiHnftbQYZKAIVHQYfY2D1t6rT0zVu/kW699VapSZtj+AB/CGwApgQiQaJFLrTp6JaAadehtPJj1cnqrTr115M67TqHWkEg7JEERkdmWM9DwaBkw8S3HClVaOObRuoqh2bxLVvS5Ilj7O5aeJpwRVky2qlD8Uv99KHlVHEWRv5WRjlDIc3X6FQFXMBX7RPGAHiNJ9oMdlVgc8Io/O8BSCUH0Tw1VKIZeTKC5mRB9VCawEApIBVuwnZyIYJjPj7d4hUiB7kzgY3eod10AnOlMyhEj9y6tINm26527VrfJVVDu9KuvR2apOl0DTvWkT1bWBfaaPuMXHpSjQQ27JaZqt3qc2dLaKNdNE1jhmsyZnvMJdFWyIJp5sSvENhYx0RkpEhlLDsUy770o7IWPqW8Zr4BldlAe4gyXCm2DK76WsVAFQOfFwagw+zSefnll9P9999vOnD5hjZDc/DH8D5cW1qmTG5J5567NF122cUSgpyb2tpmp9ENjRKADBeMCz4ge75o9KPQ+w916QLUHenDVdvSDtGrjk4N8jpZg2oGqJ7TNhbqUlNmT59O/R1JU5qHpfmtY9Mc0amJE5okrBFFUj4gT6G+0U7aFPSKSfv27TvSS6+8ku76zV3ppZdeIuNpzOjGdOZZZ6Zbv/71dPHFF2lHlHYDaUIuLyeuooyaLVJKinqUCRxgBx0O+yjgL6FDUHGKzkId45jMf825+AGGiR6feWh9FnWJT8kIBNiiKaqd0FZct3QsQqiPgXWitqLTYDzuV9il9jawrvN8M5JR1/GQD9L2+Mvp5Onl76V8H/VCWXkw9C1/Czdrj+Y0EC6wiZepRy0WXh6e/AhCHYGdbw5LaRRHkUD0S4v6c/pxbOb5PvGEPJvg27Mc/Q2cxmIzYsv9wu3ztCM90jhWHTuuB89FhCfsYOFzf2KI78rYBgtbCXOs8IOHBd95fXk7Om6cVtHRzoD2mre0i7YdbaHsE7GGS9jh/sW3qWPwZW20eLdSZPUa35TGSyicnkTRSMfrLK8ftQ3Dr/+SQsQOwzXUQ5bdi/pUBjxfkSfixOB6dO6OdjHgUrscvC0BMzCvHuof+Rs5D/vzSps2EGUtcHyMpBzOaRo0eDDceUxRJ1EjXobPuyTHyPY/3Lnc3p3+MaZXT9iUq4Eef0Trd+9jmqdac4FGAKMP+y/akNpZvx6u6UYFUVdvf9q7v9Mu916tDVM7dh60+Tcn2xN3RmgOrn1OUkukNPq4x+awbezk3poxo2vTjKkwSWfo7shx2tgpdUg1CG3gq5J+kaZllcx4i3XXoE58xWOABdyx59sBVbWrGMgxUKanLqxhvdfR0WH3tWzbts1O0CCg2bhxowlnWMdyl0ucokFQg5CHeOIhfugyggZO04RghpM0U3U3KyrPOE0zceJEE0ggxOEUDTZCGsJWPqHODPegbfk7aUZZIn3sMJV5i7ARhm9oJAKpN9980zRjPPnkk1ZWcFKZZnzn8fNOPJjcP0+jnK7gpD7R+zc8QOdZjhvXlC48//x04403pOVXLE+tbW2Kk7U08MqjLE7XSMGGVDQeSTv3HEwfrdkiXuVOCW7adZeN+Jh9okHauAktsntk7QQfG8xdy8YIqUNrHD3cTtXMmzVZKhrrpWVjmE77waf0fHgpoCc5TXFXp5WeJehQqXzQqqL8Clg1J4kB8BkP7SWeqsDmJBH6RQ1GJZ+aAhufrvgvRAMiIqPG7G7+Ge+4ug924Vd0AieYcjNgJzDmJieIoeZiaZ/01H6yaX9atXpLWi+9kO3Sl9bNDhp2zihtF9qIsHGU0aTWvRLO9Kem0SM0GRuvHTQtafpkXZwmosgkzI48F0wzdT+LR5kv8qDOSWaUP6N1xYQNOHO2rPKlv+MSwyIuhbHwxGGPrKqpYuBLjgHvZ5AN+sXxTcADmYfJ3cMvJi1M7tDr++6776ZHH300vaAj1R9//LEdEWdia7RGfZ2JKRPZ1tbWtHTpWen885els848I02fMVO6fBsFx4490TlIg/Lbrwfa1KVF4IH2rrR1+15N1HboNKBUoXVI/UI/k7NanQCEvmhiprngEE2+EsIaqW+E74zqszkzGtOi2c1pcouOmjeM4locpxDQH4W0nTiCDQx1Sh3a2rVr01NPPW0nbF5/7Q2bTDC5Xrp0abr9Rz9Ky3V8mx1TI0bUWhyWaWg0E81STGCqbAJf4ZLjN9y+zHaMZdbWMkR4vcRvQAWAoItxDryryfi32oPXScCppgXX3d2TDuvSo0OH2k1H8549u23BwmKMdswEkXqp0YWV7HpDxzSXgI7T0X3U4I3SBaG1OrJPW2ZBhYmc8V7qJxoniauvr9f6Af0DVQm+oKuzxVpDQ4MJKAlXaY7VNqINUc7cIHzCD3fC4s0dThj7jgBCH2pOwQO7B2Fo9fSILSNd9KNGSV1D09jUwN1RAW+d0XEusUfmblH7D/gnHb5K4fQu95LBPb4DpvKb4BYgC2cxl2IZ8OK4yOpc8TEXPHy4M7W3S6CrRbjdP6UFNHXH5a6omMCQNNk4Fp4HJFTxUapjuR8rPHWPCdxHvYVbxIF/vAfMseK0CIs44z3syrDxHf5hW3qBdzkeK61jhSeegWGUe5XV6l9liaqN9HKbOHkQDPAQT43ancdX1LzczVA39qLvoq7CpRynQ5S/B76R1sC8Hu2PSyVMlB33qJvB4PLYAq4yrhzG6IEExO1ql/Q9drbiNlxtMi4dHgbzhpJTNEOFysDfMRCb5zVPi/dy+YnIIjMQy6uXTN9KSF6oMjpw4FA6YPSAeyiSGE91Rb9pUD0VeSJf8jTLYuMtvkgj3vM3AzzqZ/C8E4e3E2I4RrEtLm8qnuanwR2dMLTR03EBSA4R+Q9buYk2KbC8HqLOCR3uAVv5PQCG8rGgAskywEb+I6n4NgD9EC9+Hu/R7cH8LTLBWKBy/stvEdupaYODHO/0rarAplzXtFcX2OCmVmHNj77sbdE/rZHJGyVBCG20R13MUoQ2HZ29trFzre6zWbNuS9qtjZ19moP3a1f7Eaki6u1zFWpslqoZKi0cQ7SxU/PxEbVH0oRx9WnurKnSxiH1T+PqtNmzRsxSKVTT2t/m7NEXrPV6jrwl03rV3os8quPozXJa1DXzbZ6q+b+JAasTazp/H2qTx5e/U8bKfu60UVBGI92ODToBC13gnfkhmiB4UGGGsAIhDQ+CGR7Une3cudPuo8GftUGcpgkcEz9rAB427TGXZz3IfTQtLS1m8466M9zDZh7qm/ychhNPlCnyyjf5jTIETQt34I5ngKmEy7/jHXwgnHr++edNM8Zf/vIXww20M2Dy9D8tXeADdrCwUBpO4mG8P7ugZLTWG/PmztP9tVelW7/5Td1rs9jWI4rM+ADwAqA/ndKFdkAn/TZt2ZNWrt6UNm/T+qVdfADRnj5UoUk9o532UzJDdIUDejkQOQ8b2pvG1A+TsFibNttQhTZe32wuL/MpPT9Oa0oCG0PzwPbswuUct9Rjlf5Ypf4NP7SXeGhD8dAO10sNISe/6E/f/va3j0pliAKWegRMKAKgCoUA0SCxAyx3Oyq2qsPnigHq4NQS2IAuFvu+4Ie0lSckkBX+ysaJoEM50QkXwQjQoK05Kx62rGNkMQXiv18TJqTXXbpEcMeejvSBLhb8UEKbXfu7TDVaP6dsYKQKmGjYmW4XdOljmPRCjpRQG6FN6/Rx6fQFk9NUXeKlqymY6umBWav0xITliHWkabkiMj3mT0btMYhiMUWnhRBiy6ow0fcsLiZ99ufwVrAK+OpnFQNfRgzQT3g+bYwq9yXHUHwTJsJFPEDkbnyzKF65cqXpv/3d736XVq1aVdJ/C2ykz2R1wYIF6corr0xf/eq1OhI+N41pbFB8Rb+FTogUwMuA+vWKLnG54KHO/rR5qyZpOgK9QeoY9h/sk7BGx5+5t6ZGC0YjlTr2XCOao8kZEzWeEdKHNnH82LR43mQJbCakBk3SQljjJAXap8BK1C5ALegME/l33n0nPfTgQzZR+OCDD0vCpvO1E+j223+cvnLZpWK6w3AXsSPTZlQOI1aDEKwComodGwM+CgQuj4aLOsux66hXPXojsEUOaj9Q42n1ImBgaIMIZA4eOKgF2U67l2jt2jVqt6uk5mCFfXd0tJtQhTYLs3L8+Ampra0tzZ8/3x4uA20p7i8aPWZ0Gql7ljDAR58gMVgKnNzpU3owZdmpR/+AScvuO3bZtba2punTZ9gOO8Vg8QzWx8wj+4m+VEpPfrjFjj9Ox/gJGfUhw0ks/DwSx0OvBKz77bLU9es/kWDjsAkyWlompQWnLUjTpk/zMimIimbxE9oENubm+cUNE4u7WGCSRpg8v1FHkfeAG4C/CFiyWTh6PkpOxYuHB/eeB+IH3yy4WYRySg4hWVPTuDRH6gzZ4Qj+oTeEBU+kHfmojP/TvglzrHzjF2UNnFTGBUzEgV/+HfiJMJXf4R52hOU7zxPu4RYw+Ofux4vbItBPlCePHz+P1+eq7uchIg3/UgUV7ahXAkwEhDyxGxVbNeFxKcAQ0X3qVLHbH6/lfJbzD0TZnS/PD255eSthHNJh4z1sYPOy4k5cuOfxhFuEO156+NNHO8Uo2rp1i6kg2SGmEG71o+p12XCb+t5pJsi1ogsfPjZ5Gf2C7kitbBMvJs9b2dffHCbgHKv+KzfhmigQcK5dty6tW7vOhDb09ebm8Wn+vHlGD2BMOV2h3ZbrWymTutWTfpQP0uSHuHnHzUvkX+XfHM9lGMJ5vZOvclBvE7hhcD8ezh3Sf/O4CEfauHGXHmkHHj3vlkIp3w7rbaDEYCvKF+Ei/wO+yaMJZhwHeTrgEFjKAT2Kcg4IHwUAJfxZ2R2XxDUgvsI/3CMoIQVZ/jyF3xyfXlbeqbeqwCav8KJfqU3wb4Y2aB/h4C2GfsCaHYENEDBM2Th1uDvZXPyd9z9KGzftlhCHDZ8wS0ca07RPd9uA+xrdFzFMGzprhrgadbRxTJnYmOa0TkyzWsdLG8dICaqHmMAGoY5GmCIf5Io69PSx/Ytf+kCh6QMIE/azduA5MTNo/zqxoFWoY2AgcIpdSX/CLQ86mFvuzzswOdxg76QVtJlxNB5gmVNAq/GPPMEnZBMP83DunuG+VVScvf3222nFihV2ssY3+XAKzU2EJU4McfIwHrIBiFMyCGSmT58uld4LbUMfqs5Qe8YGP/IRJspAnGEi3vjGHsw/dwOGcBE2/HI3yo3J/cBPuAELDIIrNBM98cQTttZ95513SvzUPP48HuLAL9ws0hP6YX2sR6hk/HPtFklqE0dKyDVJqtEvTbf/5MfGa6+Rpo2h2jR3hDoUPKdrDnb0aNPmobRuw067c3vXvsOpG3ojPsARnayBXkGrEOpyZQMn/YYc0Z3bNb1p0oSGtHD+DNGeCRIe14lvOVSiI0KgwwM8UagoRJmmoImgbCizgKBX1h7wY+w+cfpTjqv6lmMg2m60K9oWD220KrDJMfXf/J0KPvUFNkZSrKagKSW6Yu/l6biTlsLXrPJ7meyU30SyRXyYKknJme6y2Sfp9VYddV7zyY60WndE7Np/WIRymNSjqeMIZqj0Tw6V8MZiEPG33TMS2NTpmdQ8Ks3XRV5tM8al5rEjxSwVPKFMmAKDJIia8kQERvAggP7OVJKS8Om73yCixyaG1LsbhawKbApcVK0qBgZioNxP6E/WcwcCZF85LM45PP3T/kv9zpl47DxaJ0bPY489Zs/7779vu5ZiwsiOdnYfMYFlQovA45xzzklz587RqYWxUk+mCa3yRbQeteiE+j1TS3b2HersSdt2apImVY3rRJfsjq0uKItOtuh0jR2xVvghpodWE7TUJaEMqtBSGtc0RhO0qWmedNZOHT9KOmzFhhIKfDJgySpNJr5Mbj0DMNv3i5n99DPPpHvuuSe9IrVo27ZuSyN12mKO9A4vl7Dpm9K1u/SspTppUTCxyLihlp+gc3qtmhPGgDAozPN7bGMotkZSwMniLW/WUZ8el88NOE2zZctmCRI/Sh/pWbduve2kY9Hmi7d9YlgeKs0jyEGcsGFBFrvfuRgUFQdzxcRENR56qFmUIQQYjiq/yDoZUpvq0kmtXbt2pRdffNGEfxs3bZQ6hKZ0+umLpFLvYqkEPM+ECZ5nxkLacfkhuso+ybf1S0viaHyxoACGEzfRzmmc8Y4fi7T16zfYIu2555434Q39dJ7K9fVv3JIukro/Z6IPxK0zdz1+Fq2Wj6LMeT7DHbdw9/Tp50X+87KBOBWlFI6P0jdl5AEX9qsfysM3DuV3aA6CqHclbEU1I4twdkfOnNmm3XtXG+1BaIzaRWB98U3iA03kY6DrwK8oV9il8hV5BzrKGvHxHW4RW+4Xbnmc4RZ2wMd32BEm4gcu3CJMfIcd7nkcx3KL+CIsYXCLenB36rbMfAbGak9tEQFmx+GOtFP9YdPGTRJabJOwvtHGhEkTJ0n4iWodmHQKUVok8z6Q8UB956Yyv5amN5QSWOTZ81sOH+5h5/648e35KUVlbnkaefqEiXAWNvsmDAJjTrC8/vpr6YEHH7RTqDBRxovh85XLL5dKkBttnCTsUPCoMN7CCV02lWniE25RlhK0InGhAXj0+vKIrWTmRhh2FDPmPf30M2nzps1GPxByXnvtV9MFF5wvGjg6DUO/kRnhpnjz+vCvUtr6DH/Sxr2SXkRw/CKcl8FaTOGt/OrNc+p5h65hgC37CUbx0KdxJy2MnMwYrIDzdOzdkFvgxODNwQMpds+Ph8vDFgAD4gu3sAkbc6CIB7/Io7uRnqcpcDOeTvFhfu7OL+XxcO4PLI+5KQJ/B1L+/m/vERs+p7Ip4UKF5B1cVwU2A2vc+1K4iSbQqGSgdG54U4tBkIurXulyUA+Ypl362bO/Q6fcuUdS6qO27k3tJrQZoU0ytYoOtUQwyxlfJbBhOq/NU8M1Jx89SvfZtIxN82ZPStOmjE1jx9SmWvk7j4AUeMgHfZs+rEef1tOtPSs/CGz0WO5EI13FErADDfX/aSbvk58GV/U7GgOB20ochnseAphwr4QPuPCP79yO8MDwHt95XPjR1xlLg+ZiszblZDtjW5ya4QQNp2mYlzP/x+ZB5RmbfeAlEl+eHnNh5vjMHTkpw8OmK9a0nKbhm1P44Y4WCU7nM6eOPFOmiDMv32DvwIWJ8sY3dpR9MDjc4gnYHA43vgNX4Oj111+30zWcsvnkk0/MrzIOwuWmMs7c79jv1B991Xq0fl1IO1zzCnALb+DH//STdPEll6T6hjFpqIRiQHCHVoc2be7YfUgn+7an9bpvey8byqUKrd94AMCJZ0Ab0CPxsoQxPZKrdBndaRpbm2ZOG5/mzZqSpkwakxpEh9hUjijNaE8MkIZ2Psr0JKsKK1ZJYFNJp8y3+nOyGIj2hh19Bpt+XBXYnCxWv4DhqOAvh8AG5DtlCXJuE6uiTnKaoyNiA2rK/AY6mb+GOdnyEBHt7R+aOnSXDbvZN23bl1as2pA26Ojhoc4+TdK4S0KTsJpaTcI0CDGZ024bJk3DdWkYF3qNqR+aZmgSNqdtgohjU2psqNWOdxHFQpjCzC/UEMA0LYpicWgEUUzEl5dCIAVxLwFnpRowYCgNjE/yssgz+OprFQNfVgzEIHgi5S/1K7oRpKGYcJfci0j4ZrK3devW9NprryVO1nCcGgY4jCnCwThhIsbJhCVLlohBfW5atmyZfdfrGPQwVnMkQu+XhbDEduCo3yOsQV/tbgmN10hn9tr1O9PWHQelGo3TNSwKEdaIgWTpKC3RoHREK8d+XXY6rE/6akemGdMmpnlzZqTpulxwTB27+QxcsEwclayVUWmKKUjalKlXk34m99zD86v//C8x+VemQ9p9PEaM+7PPPifddNON6Zprrklz58y1Mlr+KYLFHZHqu2o+EwZAIWPApxkn8w4JnLfJoi7tGzffuYyNOrK9WrCtWbMmsXPsrTffSu+v+CBt3LDBGPs9Pd0a0zSuqb2xKOO0FGMObZG4WdTQlrFpzyzaWJSxE/58MTHPleBx9uw5Onk9MdVJcGPMauXDd8NrN6oY1Fu0m/5hHef++c/vsJ1shL/gggvVjm4yFQATxayGCcnElP6S78ijjJXGy+yupXfaHn+yKTdxUYZ89zj5j76MyjBO/PzmN3elP/7xT9JHv1N9scb66L//+7+nm26+SYJU9TEZ6yeyiY/qIU2e6N/ADGYircgj38cyAYN/hDNYKxdvJO6GfJTjwt3jjXpn8f3CC39JDz/8sIStL0t4vEen+k5LP/jBD9J1111npwbYHRn4jlM2EX857nA5th24CIgBeQ/Hwo5487LihXvJzxDsOB7gbsj3iEqwmRs+ebyDwYRbCdZwW66TCJ/DBWy4AQPeqEpr62Y7Q8bbHYx5p/sW1iOwMAjOPtm4wdrdhzqxuFVMk7a2Wemaq6+x008I9SOdIhiF0ittrVzPUd/AYPIw7lLGBX7kOR6+Az5syhMm/AMXuOfvg/lH2IAFPuDCLdKCluyWCkZ2s/7sZz+3U5z9oi0wfr7xjW+kf/4f/yO1trZaeKchXj/KtvXvPC3ijHKFe6QT35F+9A3PFzhzCLDKB/FwIu2++36X7v3tfWn1mtWpr7cvLV58RvrhD36Yrrte/Wb8uLIql6I/WniisD7okVqeqDOLWnGLtlXixJPN4K2eyQpuXufADDQO7/VVxjFBHQ9O96GhZYENaYsec5JOf/qy78DDwPj9y+Pz98HwmYfJy8U7JsJ4nsplxy+H8Tx6Wd29jA/HASEwtFl/8yT4Ltc9Yb08lM/ADZgglJnARXBzP5V/AheUkXfaSlVgc3SN0068tYSApIApNSC1GNb4ckZluuFS7/RqmKftYgjs2dehOyS2SUXxprRrT2c63AkPgfscdYJVl34ruGxUDpGWTtlIRdFwnX5vahyhTZ1SoT5romnjqBsh1Wi6zwbmLVwGVgGs473Van2giPxPXmbIFXC0exyM7SrbPnAo9TP7OMZP9NNjeFedPwUDtIfBTOA09w9aBXz452FzWNzjG9iAx40n3OI94PmGB8hpatakCF4QwCCoQUCzefPmtEFzfmzWq7ghyAGOMDEHiLEDG2EL831O0YwdO1an7cebhqW4k4Z7aXjQuoQ/AhrmlYTLxx/yGPmO/GJjonz+5b+UJTfA4JaX2dzonRloDhfwEYbygR8e5iCUmZNEPOCB9fuD2kDCBie+gY84yEvEHe/Y+J+cYa3um1Ho6xjWKZyyWSrewI9/8pN0+RVSN651Uo1UT0sLmk716Z6dfZ1pw6Y96eO1W3XK5oB4A5yPEQ9gqITEEhCLLIln4KesUn9PGj6kS+v97tRQV5OmTx2njZuT0vQpUm2tk33cs81m8xrh1vJSkA4v0sB2B5JxN5ACLlQ4ggHpG9JvWcBDearms2Mg2lu0WdocD22xKrD57Pj8woaggr88AhuqISYmxQS9omZKdLSgp2rzpRADQA1QxIjFsCZFPl0Sk1RzIY4afiwG6SoRx83b9kp3JBM3qUbTaRsIZA1CG4iUOhP3RdSgfohLvRqGSljTnE6bN02S7AYjlhr7jJyZkAdmGiUo8kQJbDFOFnjH14rHFA3jv27Hu3lkP4QkVqAgnMAdCxaoqqlioIqB42EgBtCYrMG0xs123qsDQ3NhjrKTnckepwjYnROTX+DQ1Tt37lxTgXbZZZfZbmoEODG5tYmtdVX1YMXNX+jMNlVoHdKvu2VvWrFyY9qoe7U6tKOmuwdhjk4ziAYh3KHn84sqtJqh7Ko5rMnfEAlrtCicPTm1ym6W8GaE0gmKGZMBoxvQnoJo8n1YE34mCPfde1/69a9/LbUPG41h3yy1SlddhY7dW8WoP9d2VpH/HE8R7/FwW/U/GgPUo9WH7KJJZEDumzkMeI06CEfaoOuoPpg+/HBleuTRR9JLL75kqsn2SaAYizTqjzZaL2bxaD2oJmK3F/GhsungwUO28As1CbizKGuQ8G6ihDTnn3eeTqR8w4SQLNqGST2f5V0/7ALt7OywxeEjjzyafi4G7Ycffmhq1i68EIHNjeoXV1k7osTkOdpPlCe+scOEmiRgMAHDO4xZhFR4sQiifL7pAVgmwN7WEdigBuGee34rgc39phoC2LN0aujffvqv6UYJk0LPNsJMzw8JeHqelqePXykPir80cisx3COfESbg4xubsuPu+fUwxE57sJLrIwQrGSosTKQBPSIOdk3aiSYJyV5UnfO9aNHp6bbbbjOBDTshhw2vtTRrVGbiDUNcn2aiLDlcuGHnT8DgFuWKvEZ5SSvoafgBz3uEr8xPpBHuEVekE2nhHjSbuHAPv4g74opv4szfwX+odBqYjscXsB5PXoeah4EP+9PObNXN6tWrExfaPvvsM+ljvdeNHJUu1ZjwrW99Ky08baHUVTaQeBTL6z0+FZcbxajXSDfsUqCAKuDxJ28YbMqAwT0ec8h+cvg8DPUE/nITOAk34Ik3x3PEBwzMkl3SiY/A5o477kjvvveu5QuBzde//vX0z//8z6mttc2iQ6WjxacvqzuYHFl5eMdEHfNemcfIv5e7oC8I1Ajrwa2HAcepp3slsLnvvvtMLRp1dprq5fvf/74LOieM17iNOkFwR2oY4dbQKzf+5EFcPHjADMlNjq9o9w7rULwTN6exgPW0PLG8vPg5jp3xQztzWMcRcVi8lj3LoPl7KuX8WzatAAPd3J0SWAQWV16nUaaj8+44wB8/HvKal6Ucj0EJxm0qxOm31zuuVu9qc07DPU7iAjZwE+kYvNoo7pau6Lagirrm7dQ3lDvaSeC+KrA5Vr3T8GhL3j9oh95K6DzeWoRNOVovEKTaMu1Zbpy0OaT7bLboTsmPufxbJ20Q2vQdqRP+EdiMSP1SV+ZxwhjVXTW2sbPPNlNxsmbebJ1UntWiU/DDU50m5+x0RxADJ4Je7X/+FvnxrEIXlG/Bey5JZyCdOVaJc/doJ7lb9f3EMRB0BxsTdC1iCHe+eQ96Ff6D2cBB37CD9uXxlGib0UCPgfGPcRWVnggbEMag8QFND9yjyvwPFdc8bBoBDnjGN6elHg9xMybV1tbaqXnm84zLnDJFxRnrWE7VsHZlcwmCHAQ6zJNjLCMO8o2JsYP3vOyV5cEPk7vzHe68D2aArwyTu1E2cEM5We9QfoRY8XDKCFXNPGvXrrW1O36xNoq4I+8Rd+Qr/AfL26e5UVwrcUFjNFoZzkaMHJHOOHNJ+sEPb0tXLL8yTZ46LdVqvsFm8b37uV97T1qt0zVbth/UBkqNkWzYNGEN/ABPEdSz4bK/tyMNT+2pQYe2W1DDOHtqapWQuLFBauwkIOZUn3acG7fQNqDDRyjiIKaiSqgUAIvH6wm6E268VQU2QsLfwUT7wo42h007hh9TvcPm74DkL0IUVHAQX5gQP//5z9MDDzxghBqCBaG9VSpsrr322tJOPsJULm6+CGUp56E8kSrIW9nL3griIXIB0ciN0RgcCo8gPhGiBGuAArIJGZN8Td8EzM72g0zGdhySWrRtIpJb054DnXbHTd8RTY5EKNlBM1QCHItTHWqoFgdc7DVCemsnSDXabEmz26SGaKKpIdLOZQmhh5JAkTkn+sW3TbyKyZcijHyWlxhll1LeB7wwNCouGQ2X9luOxZyrP1UMVDHwGTEQA2hM0GIizTd+TITfe+89Y8LBiOMUA4tj/JnIsvOIuz84VcNRZ9ShsUsJhrcZwdkk3ya43oNtUShPhMaH2nXB6Y792sW3w3TW7t7LgpCLTVmgifZoQej9HNrB5EuXC9Z0S7VOv2hQvXbwTUlt0pfd3KgJtiZpul9Q9McXA+TRy2E5Kf1wnHrX7l06ifFm+sMf/qBd+o/YLi3GCu6/uPGGG0xgs2jhIttVBR4wxAXhskUm71XzmTFg1VOi4wruqC3icd9jRRptleoA/Sw62F2HfmpOfXHUnxMltE9iYsGFwGXyZB2R16KMEzIs0Li3hpMlMMhcYHPQ4skXNyxqYFyMGFFrc4urddqKO5k4QdbYOFbjIu1SqZjApt3ULzzyyCOal9xpwiNO2Fx44QWm/ohw5IH0gvEabdPaVJS+aGdR/mh38R2248EZnb6ADh9v7+AGmA4pnueeKU7Y/OEPf7QTZfTLZcuWpn/9139NN9x4Q6pV+awpC97itZ3yHp/3HTA5cKEeeQY+3qEbvMd37sc7JmhLyY98yp2eRJ8K40Kb+PKyEIZo4hRBLrCh7vfs2SsVdItNYPPVr341NUlgU6ude4RDVhP5Crsc+8C3PK/45PDxTjl4cv8Ih1ss5nnHnXD5g1vAx3v4RxjsMPgF7sING/eIJ77ztPEPA1weB3AGK5A4HZHDxjtReDyEp7a8LnEbEL98ujKNm7QAAEAASURBVLq70puiqXffdXd6Rmq3OF0zQfdDcVLxu9/7fjp90SIJTEcToUevqOy1+MTKy+NA+jWP0pelG+WJeqAs5Af3eLx8Ciw3TOS3Mo2AD3/sePALvIU/brxHmuEecKxT9u3fl1586SU7kQpNAseTpL/9GqnsY53Crl3DptqRpa/8HaEP6c/yzbfSiTTyfATjyMIVZaN8CmFP5CdwW5TevLfv2C6Vpo+nRx99zE4g9uqEzZw5c9MNGvMukW75cc1NJSEu8VgchgPFYuV2POMOfbQ8A2eV5DiOfGGX80Jc/iCMJpznt1wvVLTDhyoX1m/leo3wDkd5gS/oCnRLf2UYy26WRjnfhFTWisfbC24YwgfO3QU4h4m2FmngHmECBr+BBpgyHH4e7Oh0GR+IJ9oR35XxRdrY8MIsXZWdZH0XcWX6A3NzKnxR5sBL4KsqsBmsZqPtuU3/oNGohTuwMVO9vZifXn1urjYoWBik3VKhvvdgpzZS7UqrdLfkJt0t2dmlrZt9XPytDREwUzVHJxY2VCG04STN0CE9mmMdSS0TxqS2meO1+31sauZOCakrrpFQB1i1YOOLkh+1dPuzjJFdREbwDIxxSjsnz0W+Dej4P9FGjg9ZhajEAP0qN/l30KDwD7/B6GP4ARvhcAvYnNYGLDZjKEIXhC9oc4j7aJjvx4ka7qZh8yDz9lBzFidLmGcTN2Mlp2FQaYzaYzYSosqMDT2sU5mno+6MdR8P61nWDYSLh3gw5CvyGGUxj+In3LAxAVt4l77DP2AiXMBX+oMLysXmNMrJ6SJwkz/QvxxXfPOwjgl1cOCQsKybgpdKmpF+5Kcyv/F9ojalj2GQVkTvxpAO641Fp5+evvWd76blV1+VWtvaUq020O0/2K17a8UHWC9Vutq8uf+QBHSiMaZhA6GwwtsmNt4kyLV5hzRt1NUcThObRqbWmZNFZybqDpvRqU4yHjSwD4N+UD7+bL2mnEHzvHpKeVTFKPZ4yKmMwUAVceeT9d5noz8WsPozAAO0t3ii3WFDD6oCmwGo+u/9QSUHkTn1BDYFBRm0itwPshPGiIhNtuQCPZFX2VfvTmMKcH2YgwMeoXPoFYFNlximB7W7fcOWXWnl6k1poy78hnD29g8Ta1QMVwlsathNDLETvE3IYJrqoq9RSLV1wdcc7Z6ZKaENu2lGSaSNwMaIJHlSWtY5Nfkio+TRmSfk13Nc/vW3ItODWGXiCeH08McLM0g0VacqBr4EGKDfVRr6Y6WpHDzxxy0mvuyYfvzxx+3h0kYmgRgYv0x4zzzzzASDlLs6ZsyYYTuTCM8kGWO6ZmEOagJMjuJh997hLvTVtqfVa7dIHdoWqV/QUXedrGEh2CfVjSzrjAkBrRBj3AU2OgYtgc04TdIQ1MyfOzVNntRoumprUc2gtJ0ZROqUt1zmKH9Pb09ao91Gj4tx9cgjD5tuXya7o+pGGXP+lltusZ3QbW1tJpRiMsGkPRYXlC/iIpWqOXEMeP3zq5pxKwvsvplD6VUotwk2dQHjmKbMbjKYoQhKHlMbZQfZQdUjbY/TMbTHM844Iy3Rji52z82cMTM1jtXdJjq1xb1E3F8Dw5JFEIsa2jp6njm5gaBj/wFX+8dCb/6CBabe7Oabb0kzps+0HXremv2EDReMP/zQI+mOO3+h+3NW2ULw/PMvECP0ehP0TJ481dqP59+FGywIYzHIAhM/TGXb4hs/YDAINPwpLyRxB86ZfPRh9S+dsOG0zy9/+SsJbH4vNU17dOKhLi07e5kLbG64zgWrdBEFcEaqAmqsp41HnHnbj7yFPzC8kz/8oo9Y4OIH//wxHIhpO0R9yvPMaO4jesRL/fLoV4/Hj99QmENyYtHOyZqHHnrQhHUIbFDtxEkB7uPggtgREihHDB6XfR7zJ9LGDsFa1E+UjcDhz3v4W5mEA89jmVZE+XDP3/mOOrUwYkzXmHC6YEAX8IFPYDARjrAY/GlHETduwPCU3FR4VHJRpnIbUjj1E2As7hw+iwPcA4Mdafo3KRVtjkmdDKcWEZqymYqTbod0cm2SmCFsovre975nC/Uxo8cUaZJPr12LXlFYKjgWJtKJ8uAcbryzHuDBn/EIXOSw+XeEs7ISWCZwS7lyf2D4HszNQ/pvhAeOMMSD3dvnlx5/oL73wgsvlE6kMl6effbZ6RLpbm8WwyhKSrmhaT1ipBA+xk5SIe5Ih/h5+MbkaTotIc/m5f6wXRUfxunCUAmS9qc3Xn8jvSr1KDC6+vv6xaians7TKcLTFy8u7rBhs4VwAJ1RfERB+3G6W85PpE9BItnIK37hH7gkj5g4YUJeoz+bh34Cl3k7JX3Fpv4BvWBeYA5eViIRAPDghpwMRcijt6gP4oTW56fsLE7KVdRZ5JF8BH55DxPxU6aAJazHXabjwOMOPLBuoFuOt3DL8RSCUMrGqaM+1QnlIG7gIkz0c8rv9RntztOCfgyW9yITp4wVdUCBeAffzEsZ67iP8Le//a3twAdfCEYRkDIusKmTsfzLYWh73gexy5MttSsTftBvwAQ9xSBLdr8mZn1SbcZsQ1s0pZZoiE7WtKePxCNYqztv9+7tSu2HUVksPsFQnbSpGSE6obaoZouApXYY83bm692647ZfGzobpFp4ik7Di1EuVWnce2sq1JnTF1kjPA9/GK9j2rn1drmI5gXPwyCO/qG/VM3fBwNBc3Kc4pa7V/oFzSYH+AGb09egefhHXMAFLGN5nIphbo/KalSccToE4Uy8484pG07EA08aYYiX+BhD2VDISRn6PGMvas6mT59u6wLsENBwdyWw0IugscRHvBF35LEy34yReZuNMublIo7ATbhHfLjzjsnxxTt+4MS1CJRPzlD2EFrt1kleHtYv2AhlgA88Ekc8pEFakZ8oS4wZfGPC3T5O4ofiFEWyPstY5WZoGq6TTfMXniaNBbemq665Os0WTR6udfc23am9YtXWtP6TPTpp0631vzZb6RQfatDIllqSImU+o/lev1QBiS6wcbypvjfNlAq0ubNnpMktY9Lo+mFSk8bpPY3Nepx/QPq0R29rlhfyaC+UmQS87ObEj9GdcKOGiSPKUYKqvnxGDETbws77Am2yKrD5jMj8IoNTwRAvKvbUEticCNadtBjREbhN2Aty45Mu4gjikr1GMPmJXMkDKiUyplcENuiN7NTP3gOHJd3eJT21m3WvzR654acdNMUpG7tDQpM8J1nsovGTNvW6L2KKiOSs1mZNxlxv5HDFz5KKpH0g8nyRvrvp25y8wwpM7qWM8imTf3t4XCk3X8B7mBwOiKqpYqCKATAAvaw0MTH8NP+gs0z8YICjCg2BDXeDsKudiSCTYJjhF1xwQVq+fLkxxVkY486EN48fpqy4COq3ToHIle/c0701e9GPrZM1n+iCyB0HUme36I521fTrhJ8ULAiSXu4hmYChH7tGJ/xG19eI3jTr3pqputS0MY1pEDNGEyxERC4mIgfgoGw7zVQscmSH0TvvvJ3uuvvu9PRTT9lEoUvMxrFi5sPgR23NDddfn6ZMnaJdggiPfOJnE9vSTNTjrv5+NgxYlUTFWNC8nVa8B3kPZ3CvdxYgLErWrVsvhujz1j5fe+11W8TBrGNhds45Z9sdMpz4mjptamoc02i77KhPoom+ALMMoQ07z2D6wMRkfvHKK6+kl3Q3CgtFmJXjx09Ml2oH+tduvtlUo42X6iCYnbS7np5OV4mm+1R+9rM7TNjT3DzeTpyhEu1q7aqfNKnFSuvMOU8/8oBNvzP1SMoP79F/w2bRA9OP8LjlzZA7ediJ7kZ9xuLTiQfhiLL8+te/Sb///R/Sdi10ucPmDJ0S4oQNeRspFQXqZIZT+jZx1ygt4oj8MefCnUVwLGjD39Mq57fym3A8EQd5DDcKIV/rqNAJ+pepcXIgS58yRz6s8qEIio8FKoK1Bx98yJjiLFYXnb44/ehHP0rX6y6OcU3NtjgkekhQOQ5P33E18LcShnQqDeWg/YUf34TLwxIG//ADZ+AuFvPhHzawefgIS3gMfrjZU3xHW8Cfd8uq6tGZ2soPlWr5AsJQXI5D8RI3uM/xTvzcaUI7xBAveSZODOWOPNFoFL2lYXUkmG61t+d1r1AIbOhP7F69VgL97+uEDULPeqlEq1F7JX9gt4jCGHdev6REtPi4ibLHd/iFe9jhj23lK/Aa8OEf9UC4gMMPOJ5wo96A5Zu1B+U3fMg9YPO4I17w162+BGMF5hI7YYkDoRInjlDBwk5T0g91Yq4CkbHO8xD5IBwP35XlzPPgZaFdejo2Dos2Md7JUfnWeKo4GPt279oths9uCXQ7rA5GjWrQyZrxNv65INv7P3FZG7FceRuCFlJvlJW8W7pyiz5h7gXOKnFFXJw0ojzUdQgnlC2Lz5hffFiryOtQ/kVdWJkNQnAG63CkT5yBP31YPgNHjmfLgbnzY6gp8hNwVieUTQ8Gd0yUL4ejHMAHDHC84w4t5R26nJ8SijABm4fFzY3XdYwVuAFHnrBNkGbvzKsoiACKfAwen8d6qvzmOLf2J3xXBTZ57VqDoFEUT+4X/Ut2hQAEaMIc0foeekQv1QzHTtkc7OBerg7jE6yVNo5duw+mLvEJeofU64QNJ1nVRum3FifzIngECFl7pYK2RszU0dLGMSHNmKrNnaNrJbTRWKX4bXMnbVfvIbTRq4znPQQ2VYapY+Uf9Uu/wkBPSjQmSzzoDHDxfKqbwrKRztqV5s0RJmglYyRzOO6fWbdunQlnmIsjmGAcRYADDEIa5uoxV4UmxtyK8ZU1KKdpOEXDmrStrS3NnDnTBDacrudBQMMpGuBQdUa4mNdSxDxvUWTKVimcMTfcwZHwkNP2CBfxVdrAM6ZQDh7GZebrcd8M5eYeHh7muqzH2VQY9/GEcCvC4M578EijDOAnHtzwz2HIF/6YPP/AYqJs9nGCP0KHwgUwL/FoHqyNcgsWnp6++e3vpCuvuSpNleCsXVfSrtuwO32sO2x37tJ9Q3ZvjYQ12jSumYtFhHrEIbq/9kg/dxHpTlLRlkat+2dOrk+zpjenqVMm6D7bEXaCbzjpSxgM9bI6Mo4kmzg8H5Ao3jC0bqM1Vl4vc+FhPvZehHMa5C7V35PDQLRL7Lz/0PaqApuTw+kXMlQQGyr21BHYZASihPXB3PB0EjPQN8iOvAsCC2RObPnGGGESOFMjQGGaspELu0tCm517DqZVEtisEfN0ry4A7+iSn3a6JyZjQ0Q4JbCB4EE+IZZ2uWBNrxhhw+y489zZLWmKjj+jlqhWt34H44Uc2qMfbCeO2UKQAc/c/ddeB/x4ifm1iYNsQnioY4UZEEH1o4qBLx0GggZEwRkcc5P726SM3lUwL5j8cdQcVUN//vOf7dQBjGvCoPuXiTDCGlTdXHTRRcaUY+IbcWJDpzFMhI9oQqio/ZFbT29/OtDekzbq6PMq6cbesk0TcqlG6+mXoEaCYhaJmh1beDtZo8WjLf6kDq1uxFA79szJvtltk7RjT4u/WlE1TdCMNhWUgYUj9MLJor9DOdi92tFx2C4q//kdP0svqoz7dGycyTPH4S+66MJ0s5jyV1x+ue3SH6qduRifXDjDxDNW/T1ZDEQ7sbFAdTSoEbPRxwr9qhL5sjpQLbPAYQGD+qU//elPtqt//fpPzL9lymRT+XWV7o25UHU5XbvHXQ1TkYo3CI+TeAumRSxWWPSwQOKkDXfivPHGG1LttF0q+EZK3d8iE77AgJ6/YL4tQFlI9HZ3ph1aYJXvsFmpBWKz9ZGvfe1rFmbixElq26Tn5Y1FUpQdd2c6atzNGKDhzw5x60uCgwkrXnrRx/x0AUIb2rd3cx/jWdiuWvVRuuuuu3SHzR8ljNpuu8/POAOBzb/YiSHuqwAFsWhkUY1Qh93a3ubpM86QZWHLg8HPygBZKaqQMuAOLnl4x/DOIjH8sQ2GgHrHIDiwdDn1gYP6fKSvzu3v8gGc8CGweeihhyW4+YsW9rvSQqnc+tHtt0vYeoMt2KFVikb5JK+eF4u6SJP33ER+cSMNnjDxHWULmChXLIqJI4cFDoYCeams84g77AiHHemEDQzuIMfagurbvuUG7vzd6S3pWFnKRVZgi0FwXh/UKWECjvkUzBRbzMuPOGhvwwtBU5yKsHQiw7IJT/owM2A6/EUCmzvuuNOEaTAYoKn0l+9JYLNAp9zQCe/MEafvnkXlg7gq4uUzTy9/z0ANJvBk5SnC5WWMsPiH4DHClOJSBkpCiKL9Ak884AWbvAeDyMoufwzvgXfSQsgfQhhQb2Or8Ivwa9gw70OEs/RUOcBSB8ybI6+RP2wM7vHwTXoD0tTJHiqa/PE4Qq3iiZlMGu0gPksDL4Qu8rV49UY/CaFfv8ZU3K2cRMZ6AWGewiEYD1xBO/qEn8hn4CLyQR4j3257eRyv2i1LNnI8IoAW/QGW00rghNNgpGdwoNyLZWUEDuG5Ga+O4rVcJzhY2lZaB4I2eH68jLxjSIN6DhMw4RffgUfKhwn3aC+4Iazx9uJpRhhgSQdBPDAWlvGhKBhjRYwXeT0b07Cod/ARxmInvnA4he3AHUXkHZxWBTZ5hdOO6Q9Fnyh50TriwVHvxfynBEL7Q9Brf4pB6/7efmnj0JF4LgXfsXO/TtroToxNYqQf7Eqd/aN01402fRCPtUe1ZdEINUWN57THXrvPBubq1ClNadbMSbr3dnQa21Brd0wU1EdwzprNWzDCGp5ynsvtvZzf6tvngQH6VZh4z2l07hf+uAETfRI7wtBHGUNZW4bwBQEM/RaBDPM5Njds2rTJBDbYCHAOtR/SRhCY9OU5JfQwhDMIXjhBg/AFYQwqzjhNgx0CG07TxCka5mFOjwdvSzl9riyX02todnmMDjdgofvY8RAXbsyLmIvzsL6Id3ABDuIJ3KDGjDt6WHMjtDI8SGBFOHCIiXlIpEGakRfwgz9lDQEW7+SDdRPpMWYTBgN85Dls89BPXtZwOxGb/m+kpkRjmCNqPjmiLi3S+uM73/tBuuQrX0kNjU1p+872tEZ3aW/ZfiAd6hD+ivux2H5pvV80gBN76Yjy3N+lPPVoo9nwNFkafua3jkszpzalprENEtZobiZYzRwEA+1TaCsitEnjeTlTVgTPIgDAFbb56EeeisWDy46/8K7aJ4eBaF/Y0V6xacdVgc3J4fQLGYoKtsWkKvbUEdgMhuoKwjEICBBODSE5mSkIMLjCRIeA9PhCwOHx7iuYRyy2IW2HDkuHpBioH6/dmtZt3JF279cuhn6pQ6sZJYo+QgsnwrJIg5KxC7dPE7Je7UDvS+PG1mn3TEtq05HnKc0jU8MoMZcYBMgDofSDXTK2UnKGjAt2whebAa8EqTL4B79eBuLyv4pYy4Gqb1UMfMkxEDQANEAHKk344xfvDJpM6pg8v/XWW9qV//v01FNP2YQRdybJ7Fw655xz0re//W07QcBuYYQ1TPqC3hBfxGnxKw0Ew9AZ7EPt3RLW7LcdNZ+I1hw4JIaP7qvp1lyUxSGTKyac9P3+Pu1UFeNo+NB+LfD6U3OT1CvoItNZrRMkuBklFU+apMl9qGDYrQfxgDpgROJk9KWFpL3qq1uJ7Nf9Aqju+V8/+1/p9ddeTZ3aaQxzZNastnT9ddeZPv9ly5bZbnBjTBXM65zeDIZTUqua42Egp+/UCg91lIeD0rsfVcqih3pllzWMRxYuLGSefPLP6Ve/+lV69913JQTssbbJXUo33XSTCUvYWUfbDOZb3iZJjrjdFO2lWGCxmPlEqhiee+5ZEwi98cab1oY4NUPb/+FtP0wXS62RCQHU7ro6O2yx+dhjj6U77/iF3WHDohHhHwKbq666SheZTrL06GPkg/7Cg6F8uIdfCE+Y79DOgINJb+qDCoZqMPkIP1wKm6OMjkvv04cOtacVut/nrt/cbff97VS/Zgf9WWedlf7lX3SHje6t4NtPBHRb32cHPYxAE54I38b4JRGZYMD6l1IqKi3wGu5RlvDnm7LwAItNmV1dojNHKSPxs7A0IUHRHjwuEYYCDzQL+iqL2Jdeekl3cTyqU4CvaGfeLhMI3HbbberD12vx3iS8aMOJ4RocDmhgkdWSHXnFwVqf0oj6MDfFw3cYyhGLceoL4/XkpzJ45yFe6gbamacBPN+RBu8RZ7gRP7gift4xVgfDNb8qGNjEbcI6hcdYuvY+sLwIBIg3HgQIGNyh7b3FAl7FMuGL18Vwu+uJOVqZeUxfJa+E9jjkYHmnX3Lq6Re/+N9WNwhsYJxcpRNm3/3ud+0ElKlEAzcKw9zTjEUmui2buOOpbFeWomBwjwe8UKYwEZZvcAeDBPwBjx/4gSbw8I2hrPjn8eR1G+7AU4884J0n4iSeyBNYIQztyNIooalcNgtH+jxKm75g/UJ5DpPHHW3B4gW+aA9e/9AGL1vYxAFzn2/iobqpcy8nOWNeLvwXfpFf5u3ESbgQ2JA/BCK0gbxdE1/QqhDYRF+AUQSOwW30gyiX4wlmVrRt4tZuY5WdPCMcJyzhKCduw0T/GNYpC3/RnnmnNAialc0BMIQzNYOgmLLrD3jLD3HILfBJ3sgXxsvvbdPxVW5fwOcPcdEeIq/0JetPKks5PmCkqqVoN5QhTNQxsIQDf5z4ZGMJblG/4NHyRZtTuUJ4lcdlfSqLO9I41WzwEuXmnfqoCmzyWqYdcxKu3G7LvrS94pGQxfFIBwFCP9jMu/XH6xHRCLRx9PKovxyQkGbjlt1SKbxRF4Rr5//h4akHgU1xx4N6nNon8zTSYOuVHp2yqR3G5s4RaYaENnN09+3UlkZdDK47LUyoo/qkTpWih7PMeF8nGjNFnuOzan/uGAjaGP2NthL9LuzIBDA8GPx4z8csxjYEEszbUHEWDxsDeThJwnyBOYTTQJ8jRpyRDjQQ9WUIYJhjsyGEU7ysQ9H6MHv2bFN1xh01jCE2n6ygu3k5It7cztPkPfAATB42cIANDOUNWGzmHginEMBwYgaBFDaaKsBDPHwjSAmBDLjiCTxEnHleIh+RB/DC+BLjLThirQ6OUA+MQIt8fPzxx6VTv+SXOCMuysc3z99qYhiiR2umYQ/XK4ysb0hLlp6bvveD29NZ55wnbT4pbRAd2bhlj65n0FyUe2vEbxyiu7E06ntoU69YCGt0HcMI0Y3x43Q31vTxaV5rU5o0vl4n9jS+Gi0RzUNgTIpK1k4YUxgT2JAPJyhlaiI+ZKm8Kjf8xqL40D7NeAitUE6Z7KP6c9IYiPaVt7voP+vXr5eK64esT8NfqjSqp1JN2aKfAOgYhghER8AOsNytMrLq9+eLAeoAIgbx+nIJbArqMQC9Tm4G81FjNXpjZKn4Aa4EqxcWNxjas7Vv2V09fVKN1imhzZ70ke6UsPts2rWI1GTMdUmyK9Cl1EO0yBcLVgRRQhsdU2TX+7im+tQ6ZXSaO22MdNaOkWR/pC/KlQetLwoySaqeG7ImdobSxw3jL+UeSf5KuTYID+mTuir5NJRUf6oYGBQDMWbhGeNWAOZ+JTf1y94ehBn7bSyE+czpGu7ygJEA3WUSvHTpUruT44orrjCd4LFjmjh5SAvmRaThfRZBjXZgS1pzuLNP99Yc0r0129P6DbvSHtGcji4YVjWpT2H7BYeBccSEK0kNGsKYkTrnPHbMyDR98jjdmzXRVDE2SNUCp/lMH7bokU28lCC0gXSZrEFPlCOQYG4d7R1pi+4befrpp9Mvf/m/0zu6lwdmE/lesuSM9MMf/iBduXx5am1r08kdTRqViRrdeWJ6eC1rJYJFNqvmM2LA2wW1E6Z4L6G1wk+fXo8+9VeFpF06TYGqvocffiT9+ak/2648Fiitra3pCtUdJ6QWLz49jW0c6ww7JeWjYrkvOMOatJypTr5iwUXOuOvhr2//Nd13333puWef0y6/nlSv3Xxn6j6c74j5jMCGBaE2R8uvQzvittkJmzvvuNP6DAsl7rDxEzbX2JyStkga0UdIJ9JlMcvCLXYYohqCxR7w9CcENg0NoyX4maidg1NsgdrcPM5UO9TVIZTiJJj3PxNQKi3iRK//f/3Xr3XC5k+2WETodcbixenffvpTU2e4a9dOLZ6lJ3zTJu2e3WEqGYaJ4TJKDFcWeuxS5CFdFoDgmfzzVBrKEotL8s2CFUYaAmDKg5CNMlIuVEYB78xYLTbV15okZJk+Y7pORU1Nk5QeOybBMeyjYEaCQ5i1tAHusOH+IgQ3u3bv0Qmohek2VKJJYENeOcnAXAf8wYDG5PmOd/JRaXALd8rE3BM6SN5ZdMNkgOnA7lDwHPUKfqCTpi9dDIQoBwxs8oHBJu2IH5tv0kAlBjQYBsZ6rUdoD+AMP8KxKB9VP0oqrJrTFDEppqhuWK9wmW7QK9KABlo6ereygEWlg8qz9o52qxPqY7Pi36S6YTFPOcEzi/4RYprTvqapLkiH9jxa988wr4M5YHEqPvJF/lC3h9CMU2+MHR9+uNLwQr4YMy6//HIJxGenZqkEIy4EOfiRFu1WESrHPN62cvxQnjDkkYc0wQ/4Jz/ssqW9MB4d7jyctm7RxqN166zNgVMY4Ag5YVzQlhHmsvOWvFBnGOqR+Njdir58drjyDeMEXBI3zKEIS/+gXmEK4R94hrZ0dncZQ4Z2Tx/AkI71X6lsrBfjwsJQHrVnmFWoIN2xXWoLhRNwE4wW2h11RZsjPmDZdUx4VKtRbhhW06dPU7ubbN/WT23Yc3wKvTKa56tPbt26zdpXh8qLW4PqlTbUrFOBtCHwSZ309HanPWIsbdq02drkiNoRUm3UYm2P+qGdblCe1qxZk7br9F70A/BB/4NutIomc1eA90fqOtkO3x07thmOwfOOHTvV7tud1iltyk4Y8Dx5cktqUn+iPHHqjzjANQyuPXv3pJ1qe9xh1d3TbfSQPDarXqkvV4FZ7mvRvhgPujpVPtqu6hub+EiHfINP2go4RpgGHaHvUw/0FfBIPVEu6hQDQ442R5kCBjyhdhI4+hL4IG7yRnjaFu0zwkZ/7+zsoksY/aNtE2aO9P1DV2BWjlCc5M2M1a1YSxobSe9UN9R9lJN36EFVYJPXOuv7QmADaQ0vaxu0D+sFZnMyptRiANSHwyOw4Q+Ri+blPPI43N2Xdu9rT59s3CZ1xtvSpp096UCHx2mq04eoj+tUDvMqa48KmcRk1exfl4EfEaO1TiqMJqTWaePT+CZdBD9KmwIUBNao038FJV2l5cJmb+OeJ/Mq1T2QgxnaBCbayGAwVbfjYwA8Bi6BBp+V37iHW9gx9+MkB3QV2sZ8hrErTo3gztgaj41pGueII+IhPWgccx7mgdC9GLsZGxgjGL8Z/8Y0jhEPapyNmdBL6HiJPhZ5r2wPld+UJUzkge/IU9iUjwfazcO4wdjH+EB5KBtzXgQyjAOUDT+eOE2DDd3nISzjCXMpaFmYPA+R18AJ5WN8ChvcMMaAD+YN4AQ84I7NWMPYxJ16bHBjHCPvkR7x8kSaYUdePpvtcUFMoCD0bp5h2kA1urE5LVl2Xrr5699OcxYsNk0bO6RecZfUo3d0aj6pjZtJwhpO17COkQxGIVnbQ0O6JJTp17pOgt/pEyX41VpoQl0aU68NNII1PiN9X/MpaIiKoyyQg6BP5IN3jNM9o2xGLwoKM6jAJmA9ZPX35DEQfQg72hw27ZD5fFVgc/K4/UKFpIKDoJ2qApsgkiVBxQBCkldHQYDkVJCZ3FNu7uqkySdbTrQI4IsPFuYwRZ2osrNdqtF07HnXXh1P1ETso3WajOmIYnsX90mISTO0TpRTQhsiZTc7RFSPEVPZTLqmThieFraN1WRMqorGaYerLh+sQeqtMJYXK4+/o/vfyKA8PZ/lIgBGmEpTLmsRthKg+l3FQBUD3v/LncUGxkBLDJjxzWCJYcBkQglT6Mknn7TTNStWrDA3YGDCsHuJ0wtflYobLnBlYghzJwZcJrFMknlK8aozszcc+tKhi0r37JNQeJOEwqu3pK07Dmp33nDRHalk0mpwCIwiZYe8sFhjsub3ZR1JY0ZJFVvLuDRbJ2tmTh8j4Y1274qGSF6jiRqMRgocVI7FJQt5Hjkr/9A76MoBMc9gjP35z0+m++69N32gEwj9YmIOGz40XXThhemnP/23dKFsJr9+ooNFA2UsJnsFvhRr1XxmDLAY0/ijehgUjUUdHh2tKpl2pMpkLr5G9cfFwkzu2DHGgoi2eNZZS9U2r9Ml59ek1lY/XWODi9LzaTtjEXERHYxftRd9e91y+soFk7RpTuywwHngwQfT8889r0VZlzHZFiw4LV0tVYBLly015vUwjXHdOmHDAuhRCQ/uuPOO9JHUkLFwOvfc89RfvmaXrk+aNFlpulCINEmDdp4LNVavXm0n22B6M8dhwcech3CcFkEQMGvW7LRIqr9OO22B+uAc7SqcLub3eBOwgDfaqi1UlQZ5RuD6n//5X9afUduGWTB/vqkO4xQZffytt95M73/wvvX9A/sPaNyuSZyEgKF9mtRYca8TwpC2tjZjtMOkjn5vcw3DJ3Xr5YMOsCAFJwg1KBeCI8rEN4tYmKtBI+hnLDxhSp6usiFsI715ojETJ07QYt3VWEAcvN6kwlUL4r+88KK1gRclsGGRfPrpi11go/unEP6wSLV7hsSo97mOFb+oc7WDwuRlKLmpLGFY2EIbqQ8YqitXrjSVee+9954JbyhrzE1ZRIM38s9JJvDX2tpq7QEmLXgr4W6QNGAes3AhbtTxUT+4kQfqlTLBnOCi+NMWnmZtYYHuhqFuTPgwQoxcpWHzO9m0bdoZeENIRjlgmlCGFe+/n97X86HeSSPod60YJCNVx1NVH4sWLfS6mDfPGOgwTILZDH7aJQD/+OOPJNx8W/W8xuqZU10ItYivTvUKY2XO3HnWdiZPmWrxLZbQEKYLgsGiQ1pX9f7odUN7CoM75eAB17QlNhSAK9oOjGzig9lPW4Ap8frrb5gwgTJzagHcNTaONrgzzzzT7qKifsAbhjwjfAD3b6s80BbwQh8F9/S/OXPmWL2etfQs9aMF1mZhijA+Rn45sXZQaX6gNs+JI4RiMPzHakxZovujUCc6SSfuiJPyQGs2qV/8WScGSZv+RVkYb2G2QN+8n75l49VOCTg6xeShHY1Q2SdOmmg4XbLkDPXVxZZH3GDqI6SC3tF3aAvE9eYbb6bXpPJxu4RDDI/Tp88QrTpXKgUXpgYTJGjzhOqu43CH8LxSqlFfTBs3bNSYOzZRbtoztBJhDSdxX3v1Nevj4JnykH8EcvNFZziRCJ1hvgCeqDsYd++9967RnXfeedfoLIwu+iGn68ZJUEibXrbsbEtvntoe7a5upISelAejMu3TSVlo7TvvvmN11a4ThdCQc849x/AxcYLCSMBIuYlbGLC+T3Bo/8EDBy3cX//6V2tP0CXqmDzzkCaMQnDBA9OR8tL2EBDQD+h3gQ/6FH2WPkUZYcaBf+AQPNHWiHfJkjONboMn6gOasmLFB4YPbMLSr2hPwJAn8IH6WcYd0mS8c2G244Iy0eeDpvJ9qhrwEuXknTZXFdjktU2v5lQyQhu6SjHWqX1AB+JRaynThiC1BgqjlTHDRkYLf0RheZjHH+rs1fi7V5o4dqZV6/enbbt1KgIPbpCU+vT+fvpocepX/Yx7J1J/l1wkUOXe24miwRLYzJgqwX1zfWqwU/IIHBXMU1PazLtpz/Rdc+a3ZKL+Sw7ZC/kO82lwAVO1B8dAjkcgrD3QJvTkeIWmMz/BZlxgPgRdYzxlvseYyjyU8RTaRl8FHhPx0Id5Z0xjnOZhTIX+QT8RQDDGM/4y/4UeMq9gvM9PLhJH5DN/r5xzRboBk+eFd/IDzadMQf+jfAhooO2Uk7JQVjYvMH4wt2INjdCe8YK5CPOHwBk28WHCjffIB3aUn/kBD3mPh7KCE8ZSxgDsENQw9jH3ZO7AnAZYHuKjPIxLD2o9ExsxKQd5yE3ko9I9hzn+O+t/Nn0U5MXeJHiTOrRx4yenM846N11+pe6InT477Ze2jUPiC7RrI2cHZEL31yL4ZeMmebFTM1KFhsCGzeHcXztFp/PmzplsJ2waRw1JuoVB/kbNSgky3/V7b5xuiZopFxAY5a1EA/UFnPkVeGANWrxWT9hYxf1df2hX8VC/8dA+mctXBTZ/V3T/34uMSoZgUrFMmLnY9IEHHjCCCAFkQXDrrbcacwSijm5LwkDoIHpfdON00ylFmYAUlIPM22TGS1Fy1YtNxuRXkKusmEAx4coFNoKyhIorvdVhIE5MtYRWO/bcrh00OyW0+Wjddt0vsV0Xg3enwz1iLg0ZIerGhc0sAFHbgMBGxA7hjYgcApumuu40Y0JNmjtrWpqmXcD19SO1ww3mQWTLp458kzRpY9FpT8RQIsyJQTts9beKgS8LBqB3YXiPfhU2frjHE3QRmtrV1SlBxjpTgcaEDsYEE25oLgwxJoF26bpUPMHcgb4yoY4Bl85M8k5xfCC29PTD9PSwpDJ79unS0o2709r1u9K2nYd0BFq7lHRnTa8maS7+1TJREyi48tCIGt2VJTlKqq8bnlp0am9O28TUOr3J7q0ZWavJrOK1u22M0e/hwIDTQhfasLNaa0YbA9jx166J9rq1a42R9sQTj2sS+146pEk3u2AvueTi9E8/+YndgTJqVD2EyeKCCQ7lonxR3hynSrJqTggDtBFnBDg47SQLmAtsirZkvsU77fTw4U7V2Yr0m9/clZ544gktirYZY7F1ZqtUfN0oYcrVdoIEZhvqvnJDnVnbp0HIlOvQ3WNxhh9we7QAo32gHq1upE41aAdfixZFU6dNTY1ivto4rfGvu+uw7SJEYPMzzUu4N2b8+AnpgvP9hA0CpEmTWtSXXJ81RebkFovWYAIipIEpC/OO3XksBvEnH25Qx1SrXXW+WENIwzwHZuiFF5wvhvgcY45zUsXbqwSkYvpx0uHXv/lN+pNO2CCwgYE4beo0Mf6W2WL3ww8/0KJ6QzqgBSc77u3OCJWMRR79HoYxuGSBDKOZvh+MceiH4VN5DJvFKYtU5mivvvpqeuedd2wijhuLWxaJA8pFHxNCqAvoCQvwSZOUnpi9CE5hcLa2zVQ+fD6HAAYas3fvPmMkP/TgQ3baarfiX7To9NIJG1Si1dSg6qq4kycYveCeBDMTea+0AUFt0759Ut8oZsMrr7xidJEFOfW2Xwt1VIkx/6RtYsBJjjeY7pwwAXcwnmE8MCcFnvTICzhBuIXw+NXXXkvvSdhAGjCxYYKD04gf+JphCLjq0ugxo40xPkdpwNxedvYyaxPNYvCayjTlhZIi6CSObeorCOdeVjlIg5M11AltDUaLMmRlII0oR6OEQ2NVJzACEHKcd955hmeYBMOUD5gV3Pf0+JNPpDdefzOtW7/OGOHBpLB2JAZDg06U1OrUzjQJGC+95FI73bVUzH/al/cj1YtSz+sG/IShymCyg2vaD3Xxs5/9zAQr4Lutrc1ORgBP3awVjWdnbwhrXDiL+jOdHpNQjRMl1A3lOf/88w1Hf9WYR7wwl2C+RB+ELjCycWKLsAgjqMsLL7xIY+IltvZo1Gk+2iWCGfK3V/3pxZdetBN61CunOFtEA666+qr0zW9+M0GvaAf0N+of4dkdd9yRnnn6GRtbEXa0trba7luYQDCAEB5RX5yucYYEjBGpRFS/gXHjQhLydaHKdU5qmzVLbaRBeKU+YaYmawOcTHz44YfTerUxVCPPV9++8cab0lcuuyxNEKMHAWlXD/ru96lvvSTh+D3K3wdpvBhEl132FTtlhDAbAQVj6W7Rq0MSloAn6oy2Q19mjsBpFYRz4BghJm36NbXxN95AyLPGvmF+9fS4Kjjo03DdSA6TjjoizFeU5sUam6dK2Eddg2faCfX78ksv2ynL1159XXRnt9XFLV//errs0kvttN6oulEGC3y0LfLIho59ezwvjz/+uOUJPMP0uvHGG9P1N1yfZrXNsjLQ92h30DQYCtztR12EAB2h8maFXaFxaceO7ca8o07t5K4mMtQzwpUm9X3GDmg2cymYajA0EeohIKQ80DXGONILXMbucnCJMGr58iuNpkAnKQfzGvCG/WUw4GVAXap+6BdsCrjnnntsMwf1A97BGXyB73//+9Y2oDenvmEsoj/5mGRzVyu0CABEwCmCudi3yCw0wug/3gYDxfN+AhUOgY2G39Sln8M6cbxjd3v6YM3etHbjXglPNYboLlxRSdWN75BXNRGJ2iXz+Z5UI6br8KHdaVRtv90/MXfWVAltmlPTmJF2WTgbtGyzKn1VDFgF8nrWO3XOg4m6r7TNs/iphM39qu/HxwD4izkK74HrHK/QKOgccwDmrIy70FAe5rF84w7Np38yFgPPOEHcxBnxYtNfY77JSVYENAhlsBkLoJfMO3gYWxBGMM4QLuLJ50lRyog75qvhHjZligc33pkPMWdlkw55RyDDQ1lwZ20cJ2goV5ySwWY8w6asMdclD5G3HIekh1880HrmwDyM6czxGsc02ryRuSNlZ9zhAYb5CGMi4bD5RqATa/MoDzhnDvHcc8+lP/7pj7bJgjJGXiJv5CNwoVwFKSCaz2CY0UmKQh/WP2aI1tAj6hrSpMkz0sLFS9OZyy5ME1pmarNmTers5Y4sf/p0Oo9TfTzQA03X0pB+CWw4oTf8SJo8qVEbuaakGdPGpmbdX1s3TPN740OSW+ZDxRioOlRAxRN/5IV8FXBFxo4tsBGs4gJeAYtfK0r152/AQLQt7Gjz2LS/9VWBzd+A2S9YUCo4CP2pLLAp6JuwX8x2qIeyo9WKT1vstSBA/j4Q0MPz6zJkj8RIEJ1F7uonjE6WEjys0FXbLon3hq17pRpte/pk8560Z792T/TXSmijiZhJv53ZQB6P2GkbTdNEJ0cN60zN9f2ahE1Mba1TtLBr1IXP2h2riRcMWMilpc+7Ekd4RBb4wfZymYu7F7/mQz6Noeew/uuwRHFscyIwhPbUj47nRMMfHbLqUsXAyWPA+yXhaYFq/mboN2ECgumEeRdA5i6HgMWGfoaJQRPhCE+XGATbt221HckwJF595dW0TQteJnRMBCdp8nzueefa5dGXXHyxMWxJ07Ki7JRzRAqkFSkhBE5Jm2fSHnRf62TNuk+2i7mx1/TVIqxhusUiD9rjTAqd0rEFHuoWUYWWpDqhQXdkTUycrmmZ2CAVjGIoS5gDOWAXHrTFFqdK2PPiv2TDBDYibkzW2YHapTLBGFktBiw7jtatW6vF5l5KYzu3rtPpoTlzZttiAGYI5YEBTlm9XJTbH1Ktms+CAbVMtTfwaG2nqC2PAUfwXRgB2R+2JnMYmK77dQLkDTGdEdg888wzWkDtN/fTT18spswP0uVXXKEF3mSdENGCTkIHT4z0qEDFofhg+kf6eb9g0mgLF/c0JjfMSIR89aN0uakWT/X1MJ5dUMm9C0f6dU9GZ7uphHr0kUclsLkjrRTDd3yzBDZi0t8sAee1115rC06Ykt1axJEGi7lNm7ektyXQgGHHLm8EQyz2uD+GxdrYsY22MKWNswDcI7VfLBgRrqKyil30i1Xuyy67NJ0nhmhb26xCIKByy5BvBDLcYfOn++/XInOH4WFs01gTKIGTHernMI0bWRhqQcjpCu502ieG9V7hFhowXIKvyZOnGKPxMjF1z5cgqq2tzRaI9DRwauiVvVeL2ffUr1BThtq6YH5Th5zaoFyc3mH3P3jokvqqg4d8l2K7Fr7QgNraEabq60ydUEGVFsxN1D35ReQiJkoHPLz4F07YSCWa0kFgs3BhCGy+Kjz4CRvq15iZhfDC2oFhJ368zaHGCloIPPmCbjDfhBmx6qNVqqOXjFG7UkzB/VrAsxBngdw8ztVPIJjollBk3769tqC3epQbqss4NXTllVeaGj1ODdWKppIGD/HA2GBn/suvvJze/uvbtiP14MEDRntRQ8bCHEYGjIkOCVf2H4B54KrYcJugkwQLddqGernwggt1l88CUyNFnigPOIVxgvrHx8ScfuH557V435g61Y7qtcBH2AAzoL5e9xWqIkkDgQWMii61U+IgDzDPqY8r1McQFiJEAz/k/UmdWIQRz27a/Wo3nRIqYBCawUBAnZ8JbCT4QcixfPkVaanqlz5VjF5GDURZo2Ko5sJ4+yoJbNQmn3n2mfQf//N/GuMBQcpkMXWo857uHsMNwpZg6NCfwPVhtWvyizAJhgaqtjjtcIEEHIwLb6sPwvSl7SO04HQQ+KW/ER+nHrqk6my40pvUMtnwfd3116WLGROl/s7ajjLN6bG9e/ekp596Kt35i18I7++oLfUZ4+mWW25O//RPP5GwaJaNSQgE6dsIbP+///f/NzWPdRIyICCbIEEpJ3VgDHW0H7K8cCLJmFRiJpAO7eSQ+k+P4qe8nOQ488yz0vIrl1s9odff7oRSOcjfpk0b0+/u+32693e/s1MxtPNFoiGoefyq6FRLyyS16xEqp9SNSQDy7DNPpzvvvDP99a231T5G6wTPEmuLGz5Zb0IKhMJ1gqcvIDCgraGSDgEOeOYEEEzzM3WyCIEq7Qpat3btGsMpDCbmGIzP9IXDh/0SZsOzyjNNePjKV64wIQon1iZIOEJ7oR/Tb55++qn04AMPStXLX0wgvfC0hemHP7qtOGXZpvZdb7Cl0yfWprzv7d61Oz3/wgvp/j/db3faoRZymujMt779nfStb33TBXFiljFmoUpwxYr3TSDwxBNPWv9BRSaCeITo9C+ENeQLfFBHNGD6WLvqByEM7Zny0j85LdOo9goTjc0x2LQ1wtG+oEWcoqLdgUfaL2GhcddqjnKd7trjxA79FQGu1bHey70n+s6pZ9OOg45bnxNuqgKbvJ6ZL/EEAc0pLHC0Em8ppfYCc1WGOVf42Zc+LRZ5I7TpE+5FMiz2g+09ac3mg2n1Jzs1l9FdlAd0Fxp3UIhPYHN6wdHmEcIM5Z7JpFOiEtro/HIaO7pWl4WP10mbCVKzqPFntE4OSLU66o2dV1C0ZaXLG3MM6jqMzfmtvcvXsu75D3/PNF+Ze+k14rEZZhEk3AbuwC/F9zm/GN4NV6VMDkjxeP4DgAf5iPB4Rd+pBMvxy3s8MVeBtjMfZO5q81HNuaDBCEd52HTCE9/QeoQZIRgg3XiC1jE3YG7DaRE2HLA5COEMT9xNgxvzk3yTIHknX5iIk/zihl1ZRsYXDGXgYW7HeEPeKAv5DKGSj/XtNu4inMGPh7kDMNjMPxHg4A59Ji7SxYRtH/ohbcrLwzvjNA/jXpTfhDPCA/Msyhrf4Ib3mANih18IZogzcJCXO+oNu7e3R2qDd2qe8YZtwHjmmWe1UZFNFuCwWLMX+bcuY0Xhx8sUZclt86G5Boh1RH2YgBVJCzjXY/fHDJMq37FpRtu8NH/R0jRrzsLUOG5y6tMmcNSg9eju2h6pTNe/8EedQgfYsKnN5eIDSMGG+AB12rA5QSrRW9IEqVZExSJaNoZavuEDOB7Ij/4tY2oRRXwqo3UtfqAo3s9sLhHlJpQ5EwHw3mYc3iKs/vyNGKBvxJO3Wdro+qrA5m/E7hcoOJV8KgtsPk9Ui2RBtpSEBjL9IjTh32iTCB4UDZqF0KZfEu5uSbs5qsg9Nh98tDFtEoP1cA9MVQlt0nAjsEza2J3H7k0MtFo+EtpI9YN2zLTOaJbKDg26E7gQzC8X9H3q7KVXHhi8CUdgy49NKeTgE6YgqO6tNLSAQWJusIQyYprFoLwPNO5nY0ikQiErjA9wuB/tF7kr2xWBq59VDHwOGLCeQJ/UH/3A33gvJiS4Zf75JM3C0pFlYkCMATJ3g0kLU6ZbDCgYVG+/9df0lBgRzz33rJ20oW8jqIBBcoYYJdeImXOlLk+HkQRDh/5ikx3ZEA/vr0xwcMVXNEcvXQhrpCr/k23tadXHW9PGzbvSgUPayS+1CVwsOFQXDKI+oV8ztSNigA/Vw9FnTeF0HFoqFHS9ALvwFszTqb3JTSYAZjJnD6nZ6s4nVk45SFxxkSd1/pjA5xN23JioxyTdJ+IHjBk/ezZMb52eKMIqNpto802cYfiumpPBQEygCestyFuLtxpvR0ULEs2nrlBZBwwM211iXr4qFTz33PNbY7a1dxwWI6shnStm9Y9+/GNjhKHWp05ChhraBnUmy/qR3i0+xRl9gm/qkvYRbSRKBQz+1pbUF2LBBbxXP/7SYa37F1ioPvLoY+nnd9ypnec6YdNcvsPm2muvtgUoO64R2ND2tuiOjWeefVaM7qcktJFOaQkRYUogiOH0zDwxxBfMn2eCitrhw8QM3Jbef+/dtFInZhCm9mrRSa9rGjsuLZbKMnZdX3nlVam1rU1MP/qUGO/KFwKbu+++W6eR7zdBJf1+mHADgw+GPos+djByggYVT6iM2rltS/rg/RXGvN4tAQQnb1BxAAP53PPOT7fcfIvs87TI1kkOLUIxw7TDG6HM2nVr0/0SDj3+2ONSXbfGFrgwIRHSwFSfM2dumq2d/4TtEcN5l4RI7NZfu3a1qY5CSIRgC/UJjY1N6RIJiG6//fa0RIziejFGbRfuEenRVztAiPKwhGQvvvRycYeNBDa33WbMzHESSg3XsV/yxGkmV69CKyj3W6tX5j8y1h4ES33Hgh6BF4zUx3WS64EHUL/3kZjRnalOTOCRYp6yG3/J4iVpnk5DjBbzds/uXWmV8P3+ivdthykMfvDNqQXwhTpJBB4suqk72gNqnWDW3ycG+isSPO3TDntosgnk1A5mCVcLFpxmDH6Y4jBIPtZO/1WrVhrDHNpFmerrG0xoc80116Yrll9h9cnl7X3GbO6V0Omj9Nijj1rdoC6Mts7OTU68cFpk/rz5EpJNMaHY9uK0D+XdJkH+wYOHDD9NOmHG6Q3U/F100YVixE82GvqR4Di584FOiL2nNsqprt0SLgqVqgNU3U1NbW1tdkKNtrbkzDOk6mpJmiVaO2qUVFZZDXgf5dcM3dY6mV6KPoxN3SBAeVqChP/nP/5Dqr3esHRG1NZZG6X+UJ0F84f8Tfw/7L33m13Fla9fyjnn3C2JKLLJGWGCBCKKDDZhjMOE73PnzjN/w53g8b3zy1yTbRMMJicBkkgSGCQBRiQJ5Yxyzun7vqtOdR81AmRs2Xd4zu7eZ+fatatWrapanxV6Z4uR3QgsjDM1ayZBd3EFtkOeQhn0QuBeVz80aNw2LM/QEqJ/397EYuwR7UQQznK3za5es5b4NMSzgucMp8wERq6++mrcpB1HneZ+Zxfte/26NWnihFfwBHAv7ftj8r03rPPGjr0q3XnnHZTHkBBI2OcFYDPtvfTvP//f6fkXxuH2ywD1uDNDqOPSkvYvoDoAIFo3dballuRd4G7BAmOmLAhwdTv5yjFtuqVzzxuZfvSjOwFZTwhQjtspp924XluSnnzySSx/HsdiZHbwkaOwgLn+xhvD1anlZUysndDg6tUrAMUnpPvvux+Q5SPABxQooCmrxbYruCtgOWjQgACAPZZ2teRbuGBhuC3chjDLRYFcv779A7gQXJVv2C4ElHr2yuW8ge9ZgsXJAsAghX3b4QMdAPqOxbLrkksuTaMuGQVdH0a5SRuJOEXEoiOG2XNY2QVgg6u4IwBsbgawuejiUam+TsAGCyNu9pk8OjE38u1daTX5fHPSm4DZzwLYTI44XoPIz3U3XJ/GYgV1CDzYcnfuIV19/NHHoSgwfvxE+NV8xjCZt8hB5HFdsUbrDd31B/Tq1q0z+gfG2loZ/G0ZtLVpE7GUOKfLt8GDhsCXWoQVp+BWKwDCftCiQsrOvHNXhTcKrAq0BwDcohVllXnwTTfdgkXwWamFvJy1NaBNi8p3+oXf5SXzbSqUxX3rpgbY/HlqXFqWj33lIjv2j+12hL1rtuxK8whSnBSDAABAAElEQVQaPnPmAvjrJvhoc3hKa+gceQG8PASwphf9rGdU7kQe0Ko58WtapoH9cXNcD6/u1zF1A7RpzXCCSyGEtf3ECJGqdk8ZhooV+ZxtOq+NeZUmWM2+q/uxus8SecgXs/KnXxKpmjJ3+odyFrMP96StsuS+KB991fly77fZljSr31Odjtddy/Wy9Z7qZ8t+edb7yrnyfPWz5T6vlbWcc2vbUu7mmFUeJFAhIKMVjco4WoVrDSy/FsgRvPBe+2nTc2savlOe7+q4U/DZcZBgjGMcrTB1F1pXVxdj0gCu6U/cer/Pl3ybnqtLybPper2M3co7Sx7KtoBNBZQRdPF7tAhyHOq3OA5wXOU9fkv1+8o7yznTLUt1Hqv3zVuxfvG7HXMLugjMCEz5zVr7Dh4ymP4ju5wV0HGMVsrLMiiL51yr5yslH6WMyrbkU8WQLVs2pM9mfpR+++gj6YXnX8LadnXaumU7lK5cwXF8phWKkTO0DMbZKiu4eI5/zuaD+GxORmvyx4txAwdxk8CteXYM45ihNX15j3TIEcelw4/+XurZd3Bq2xGr5BYov9GvOZtRAWU377Sl43yRPDAPaEGMbCxreuM68RCUwIcN6cN+x9QeP2jm0Tm/S/neONjPzzdd388jtVMHoQSk07JaJ2WVTufPr7lEOwhF/tdJ0kquATbfruzLkKTCSYPR0Woyq6sw2RgGwWj1HbkL0Gbrjr1p9XoEFmrFL8LcdflGgoNrhcOkbLcDMac/sFQGUDY2GbpDtDbNAW3wSduzB2j4kK5oxvcAtOmE9kzL0J5R4CKj9bUNFjbxWebSk1nsWslWXMndQgFr4q64jx8fqKy5844HKk9lJu31r15KR5fLJqftYC2SzT88/PVpfHXqtSu1EvjjS8DJQ1giRDtppL1Mlw6qpNZKGwpa5R47wniVv/m46ZurBy362d+8ZWMEGv/s08/S6xMnhub1gvkL0AjdHAPl9gghjzhyBIKSS0I7fCiDSoUJupUxV9EuIzP5nWGOTPv1VPAFtpswr5m3fGuatXAtAhECMa5Fa3sHA26HWs1bwTcc1GWhbzP4SHMGbDhJA7TZQTBBXFpgpVc3qGcaNABT+M7tcJ2g8JVvrwha41tlPuRmX37S+PW28fLtDuhd5FlFw8pBuX2Lg2BdsTi4doDcyBsa03KvDDL2PVs7OrASgM9DF6U+MhVnevbXmnQbZ7hRgfueqDP6f7aCFVqkPPLIo1hwvBPangYyPxM3SzcjrD/xxJOCPtuqddzCnoY/X1j1zjKZjFfxU/LypXqNZlSepb+rDDAbniPRPfhX3rrFYNTLAzwQsJkJYKPVw2lYO1yOhc1FF15AzIpeTCZxf4SmtYG8p0//KD2Fm7LXiY+zAu1sha09mbwee9yx6XvE4jkCAGXQQFyv4dJIYasuyxYtWpDmzpkbk+TpxLpYvGSZVB9Ayqm86wqAlON4VuGolg1aFMyY8Vl6+OGHEdQ/HYI/89CSSbCTRsFXXY45Wa6rr8dlUx+UK1qmTYAICreND/HeHz7gfZ8C9qxkYtUyAl+PHKn2/nnEjDkq2ot1Zqy6TQj21Z5/5JGHEai/FsJ8hefDhuN66uRTwkKjN+/oAZjVrl1b6tXy2BKa/F8AEikQffvtd8LSSKG4fOEkXLDdetvtART0BsAIn9bQxGrAEa14nheweasasPkhgM0owIFvD9hIZ/IEJ+9vYo0yntgiAgNaTQiMDAfgOAbLDAUMCht6QX/GjdkKQLZ86ZIQZOji6KOPP0pLEfRL8JbvRcQ+ugReWs8z8hkBYy06Xp34Ku79Xg4XU9anbi+GH3pIOhHXSZbxkMFDcIGBizd4klZIggefA9gY/0MrQd0qKUBX218rhIsRVl9w4YUIQ3oFCKjg4R3ayhNPPhHxmL6gbo2hcsyxx1AvJ4dLLK0gBJ0UAmg1ocuP2bNnhfa/Ls/m0y/okk9XXRdccCG0djnffwztDysk6EXrghUIlqfgBu/551/Avdtn9C/bEcp0DXdt0oxl1RMQRECyJ2vnLh2jrdo4Mwd3TJk5QGN7tf3ZePPWtqv1xcRXX8XCBsBmyjTaFUJAJv4tmfhrJaaFyamnngIwaKwPLa3sk/YEIPgO9DWNupwzF9dtfGdreL2WDoIkCpBsf8ccNQK3gf3Q6Owa1m5aSFiPH3/8aVhzzZw1i3Jqmfriout0LCVuvPEGLJtOS83hOQoR1WbdCMgjYHMXgM2HTQGbH9+Bu606vlTehIUNgrD3pr3fANgoyHfVIqkb9R6uD3EfdySWGf0AAyKWC09rWbNwwbz03vtaZ70LvUIH9GMG9xUgvO3229NZZ54Vz+geUpBi0cLFADZPRUwrAZudlOdRRx+TbgjA5uIAuKoBm9dfy4DN+wA2W3Qyz/e1gX9Yh8cCFBs3bAign274WsE77FMVfk0DgJqKNu/s2XNCkCcAKXgiDcufdBNo+9EyqWu3LgBluCuFtufNm5vemzolTZk2LVy2WS8DoX+Bzhso51NOOTn6aTXxl+LSb2IBbLC2k0dpYWM/oIJJfT3t7GsBmxUB+gdgM2ly5HuggM31GbCx3AVsFBw7XvhouoDNQ8EPpJ8dOxlLUB7y0hyf5sTg2wMB1joARArGCUjbxrVC+xCer8JBcyxopLVWAOuCv4JWh2Mtcxh8pR+ATVvAm127sNKChuZQflNRULCdC5xKE8ccc2y6+ZYfoEBzIRZqgHch3EMIDp0rDP+uL9XjOfetmxpg86fXOhxWjvT1CcmGvYf/nZT9ZkzoV67bCmgMMDl/JeMgLMJ2KENQwROLO+6LmbttiNUHjZHXiiF/65ZYbmJpM6Bf1zQU6/mBfbEkaI/rJBCbFqSdx/NsIeksNjZrJMgi77Sv9Fqc4f68SP+cj8O8X7ngaRZz70XGoXFTOfaaKTq3IXPcUsb/XxoXemtlqb6nnPtjtyUNn/NdLuVcOa4+V32t6fWmx+Veny/XyrZpmuVe+1jHDIIzqxib6pHAtVjRuF+ueV1wo1gB2hZLur5H4EHLGF14yfd1a6YyhcB0cXNWzgveCGQUMMLny2qa7ptH8+d73C+r153DqfxQwJhqhTz3zWexjBGo8T5BqOr7vMdzxWrG9/hel1I+JU/l2DGTwJJjOr9Vqxj3y+qx4Izf6djOewRuCohjX+B57/G8c0/TLO8p7y/fXACbcr7kIzJZ+am+5nMqa+zavS0t/SK7RRv34svp3d9PA3hbznfxfbRVqT8vtg/L1rpsbHkNl9mJ5hU35zqKJGyzccXxFkAN8ayasabUBktbXLb2HZIOHXF8Gn7Y0alT196pZZsOSAxR1mSso/Rv5+6d5NNxP3NxrGpaoLjZBgXwHl2RJWKNN7y+bxrQB0srXKQL7DpajIXMlO8tp5puv+l60/trxwenBEp7dWudlFUarQE2B6fM/yqpWsE1wObbFb3M0D+XGOhQljLkwp49H6yW027Rcw8TxS0IW1ev357mE3di1hy0CxmYbdupBQ7AjQMbBK5OlhVwuAjYtAawadkcE8a2zdAWaB/aM8OIPdG1Mx0UXFbxbGi72WDzW+PZ6CDYiwZcyVlu3J7L+S6T9ngAwXBe/Ir8Jd5fzuU+tvFa5cKXNvmZXB7lYk6N30pHXdIv12vbWgkczBLIbdUBU263MZCyc6vQuZYGLrmz08w7u1jSJY/PaGpd3IyUfJYBYGkjCvhWr11FwOnP0hto+k/AuuYjBIBqyDrIdvA4pK4unYHAZwwa1UeOOCpcGimUCTdkkbAMo7KatzB91qoF82YmctthFKvgGZ8SmHTmAlwrrtuA8AbTcXjHXu6NbXwVUyS+T9Pm5giwHKi1w891n54d0qHDCCY9sDtgDW5XAGuQScA/MkBsFnTTYPv01yVP1PJ++VbLycWBfjFdry4P73MQ7OLAwWuWQXnO62Utz8XNtZ9vUQJOtDLt+nDphQrN55qSvjNdOckPwIbDCM6NMPANhOgCNlMR1rZCgDUIgZ4C4etw6zMC8EGQp00IBSCWSt35rtxeshZemfxU17H71m813US9mxcBRhNhMf8ehAUPfd7WrQjR0cp7Ee347BJtVmgOnnH6meES7YILvh+a1wpMtzJJnDdvfviS/u2jjwUQoAVIJwTbaror0DeWxFCEzV2ZvNrWFPztwBpuIxroWgcIirz40ksIAY0ztSpyp2Z5tuA4H4uJw8LlmC51Zsz8LD0kYPP000y+iUkFSNKOSaGTZePRaC1xIrFPnETLMyjstHsnLh4ALBYtWgyg9EYIx3XdtpqYD04sj0Hj/WIsObQuUCMwBOKU+ZIliwEE3sDlgt81BcFuG9wnDg5N8DFYlxwHyKG7p4ivEoWZxwBammzYuD69j6XJ4088GRZUy/jOrVt3pKMR6t5yyy3U70isdodETLxm5PGgADbSCt8hWLOJerKcH300W3ItWbwkwI+6+qHhdk9A5MQTT4xJuC6MtAoSgNqK6yrL4R0E6BMQJltughm6NjsT4f5lAHgnApIpnFDwMRGg/Llnnw13XAo/5LtHAtLoBu7cc88Nd2rdABQyf8p8SMG2VhXmT5/kAleCOLoG6w+IoAu+G2gLww85JIToxmIaP358egQ6+AAgybrVKugivkFwx/iPXXC/Z5VEG2HPsbaxjQSsnnzqqXCTaZsRFDod65qbEPBrZRPgvc2Ma1q+TJo0KazM3pr8Ni5DNoZFwIUXXgSocVMAg7r5k6Zl1IhaeWMWCMS76RcaFtpYbpvsBMPIW8eaAjavAthoYTMV8GU3fU0LwJrOnbuENdIloy+JWFb1dUM4p4WFae1NX6AV/CYAqTFc3kRAL42FKzF4SA+AJIGeyy4bk844/bTUD7CrU8cO0S/t4LtWUjfvY4Uqfb72+hvZTSJ1auygWwAIrKu2bbM/fa15Nm4QsJlQsbD5KMbJWmRdHRY2fwNgMyTypEvFLwE29qPNmwNkEGweQX5xIXfUiCMAIgGgBHQoKLWFv1i2hPqfTL4eJ/7Re2HZtIvxwNGAaZb5Bd+/IAAf3XRJn9apQPETfEdTwObiURmwaQ+YuhN+s3r1ClyiNQI2mzajscXYQ8HTcccfm0aPGh3uGLX8USil6znrx8DS0uSzuCqTf6h9bRUK+Ek/gsRjLr0UV3SnheBOt3zWtfSzCOuaSW++wfc8EbGWtLLp2KlLuHy888c/Cld6WhwKjC1duji3HyxsJk+qADa47bM+LjwgwKbKwgZ6EGgKwAaXaFrYDAdoNobT/gCb2bPnYQGkRVMbLJ/6A2JeEHFvjqbt9sFKKVsc7sZibjVWkZ+Fld6LgMuzZs+FdraGgK8T9Cr4KxiltYxxuyxbv203wqstKAII0L8Er58AaDx37jx44vYAWG+86eY0CjrXAloQR3rQakElhe/6Yp+ceYOsoQbY/LnqmyYKT/b3GxbK3LsEY9TP36Jy55otgODLGdssJ37XdqwQEbsGYNMclqGCZ64nUxaw4R86R4MeWUG3Lm3S0MF909AhPVHubJ86tydQOmQsJTdSc2O+5OfmNA/E2HXQH5fLlvlAyAe4Mc57P0s85ylPKhXx3L43MALkNvIMnyt0Fn0iL43nqm/3+cpS6LEcH+i2Os3qNHx3Wcp5z9jPlqXkqxw3fcbjpuca0qqkb59e5kXF+kQww7GJfFxlEPmiq8eCNoIejpEKcFLeX+ZNAhGOJQUfBGoEZ1QOUinEPtDtwIEDYwxqvyHYUfJlWmXfvJu/6nO+03e7FoU7wRXBlwLKmH/zqBW2Ls1cPbYfKvsCvII7plE9F6h+t+/32LGXW79PMMX8+n1l6/cKwjg2ln/7zY7jBGLK1nOOsb1HIKf6m027+j1RZ9BqmdV6XMqi5KN6Hhr3Ryk1/nhfOZ+fl953Ma5dH65QdYn27DPPoST1KWVnzDTLuYA2ztGC2jgXFMqGY/NZeUUjdXKa8YhLscZRDrAXbzy6RhSwadECpZjuWDIPOzINP/yY1H/wcGQJXRAYYgFuW5M/0B53A9YEYLMXt4nwBgHdzhUrvEPq+6VBWON179wuLPBakBHV8SovjvLLB/v/tTxqy1+/BDItZnq2TspqG5xfs7D561fQnysHVnQNsPl2pZmHJo3MTSZcxikOQ+waZMbucQXmmY0S9Sm5afse3LVsSrPnrkjz0aJZvQ43H6Dne9GM341bI1flx5oqtzJIeAvetmsrTHoXAqfWxJzohPl/DyxtekUsilZM1stATKYbmvKVzmFfnpo7nEo/ETnLA7PKd9jYzXfDyu6XlnLdzqRxkF99W3XHVH0+72eGkt/x5au1M7US+POXgAOk/GeDdM//CChrG/Uw2ottR/rGimUTmrYLF4bLFoWf/RkQD2aA7IDZwab3l0FneXYTgkXdEBkL4Lnnng3tUbX4fZkCG7WhRyM8vuCCi1Jd/VA0fXuGcEJBhAKFnBE5SxncMdgLEEarO4RJO4ybsCHNW4i7hEUb0uJVgL0EKpW/NEcbOgZ1vi3G45o2C98AxGCx0Kbl7tSjW3t4Rs90yFB8GvfsSCwbwRqfzvfqHjFatxMz1lwS+ZdbGsqofK9bB/QO3N1XK9byEZhxccBgeXpcBsNlMOF1nylpVZ/3Wm35Y0pAemmkXWvUJeic33LkGW/MgA3P0L8oqF2AQM/JxqOP/Q5XUh/gF7ljaIhfeBF+/Yknccihh1GPzAUgC0FA0zDNUmfWc5mgVmvR2T5crftSzz7oRCbThr2W/YG5lU5aIqBtg+CBOAXEsNFq4YUAbLCwwTKlV6++COjPTJcDiHxfYAMh3l4mm04UDW5urKiXXn4FV1WzUgcEv2q5G5D8QgR/xiNRi17BBtnh/YAoCIF1k2MsDCfOk7EyMpbCO+9MgabXI9ztFjFmrrzq6nCN1gWwx1gSMwFsHg6XaM+Ghc0OLFcUTquBfzEWH+eeex7C4/rQ0vbLBGyaYeWmdqugxdz58wKweebpZwF/Pg+NfF2bmc+rrx4bwn5jOBiLxhgp414aF0L++fPmhfa3wcYFoXSlNWjgYL5HoJY/25vtPV6q0sfOCGD+MmUy7qWX03vEctm8ZRtxL45O11x7TboQcOEwgIXWquZSjt8WsPF1edwgaUAbZSDkp5Mn8+VEXsGEwOBvfvOb9D7AyLZt24NfnE6d/uCHt0b5lQDt0pZ0sgugS3dYuoGaM2c2z79BHb0Swci1cDhyxAi+4yKAruxacvas2WGN9BQuqnQjJf3pGkNBrHVzCMJcA73rEqvk2jzrrsqYJgomtH5R+P7ee9MQRqwNYMj4GGPHjg3rJC0hDGT+4rgX41s+xjJLgPAYrCMupV4Ed3QDlYOzWzr5TY61dbn1yaefRIwQBfDSXndAhOMA7MaMuTTorTVu56RR86UQRRdTd/3yLujzbSxvAGx69g73VDfffEtYC3VGQOP9juXsO3gy2mZ+rxcqSwOPyHzA+23Lu6BPFRMmAHT9y7/+S4CW9iHNsHg57NDDyddlQf9HjTgyrEFbQS+2Wet5E9qzM2fMjLb31FPPoIQ0l7rC7aZAD8DIyPPPT5cQG+QoLGzaUuYtGaDaDrRKk1doZaT10AsvvsR4eG60Bd11/RB6uJC2a0wfBToqPWxcj0s0ALu779IlWgZs+tJuxl6dXaINqwA2tm0ts6otbLTeaU4/NGzY8HQ5lkzS/qGHDs9WLPIEGw3FIZC9jZgvAgLPvfA8oNxE3PEhgEFD+AiAi7HXXAsdjUoj4CeCSQLGAo+NgM2scA1XLGy+FrB5fzr95/agUa2+Lr8c934jzwuQQWBLMMZyth3oinEu5SO4ZRD4xQC/Wv5o/aTV4aWANWcC9tXX5zhYAj0ujl20Zvp4+ofpgV/9Or2AC791xCzTykaLnL/7+78jfsvFCMsUAjXHJZoWNhPT8wA2kwBsVmJh43d/LWBTyo6yWI31i/GcwsJmcnaJ9iXABj4qn5I3aAH48MOPhIWNwIu0Y3w/40ZddsVlAO1np164O9M1XJ7XGKMLreYliyPe2pNPPk38tQ8A/1aHwM93qRBz5VVXhaWbQr828AkXgTz7KS3d3n3nnXAxOQlQadmyL1L90GERZ2c0fKL/wAEAWljZQKuu0edFCt/dn8y3rUhZQhbq1ixs/vT6hqXYMx9YQvIf7o1w4PDfzVtRzlqNRw68ccyZtyStXLMJ/ohCJ3EqGy1t7Hkd/xuDUlrlQYSzrZEZqEU/ZGDWou/bqyNa9FjhxD1RydRzGTfKyytZpO5dMn9nxzzZGcQ55yPlxjgVP5VHKicq31q5z7yRmt1MtHdvKuNGLzkGdZHmXMq1/b0nbjiAH9Mq9FzSKemXd5RkPJ/HouS06tvcL8+UbfX18nzZlnQK+KFFiWCG4Mznn38eFoG6B3MM5JzJ8ZBgjqv9oGMD0yjvdeu4WRBCCxmVgQRoHMtokVtXVxdghdecdxVAx76y6Zi7zL1Mv6zm233fax60gnHsU4AlwaT5CHvd+h2CNwV8KmN8gRlXv9l0XN033bL4HaXcqrfm0zwXS5kCvgg+CTzp5lXLIYEYgRxXv9Nt+Ua3zi0tI/dL+r67et9j26C02PR802sl7/u9r+q7yn0JqxUtN537a7H5zNPPpTdQPpnDOEh3ndKWbcjxIUVhoefy4aAAhVqSuzSWnfn0G3J787XGr0rNiYeInNC5vq7PBg45JB1x1PGp38BhqX2n7nhLw71xK9aWjEt4Rkvf3btRON2DUsje7Yxx8LABWDOof0+UvvtEvKustMk8jffnkaLlxMJL91cGXirLN10v99W2B7cEpMWyWidllfbm1wCbg1v4f8nUrWSZrBVrp3L33XejwfVsIOYyD7UEnaQ6AdWU3cGvz5QO4S+Z1//X3mUH4F9hbpaLi8dxzl/+Petde2hIilLUoNnBz/qNO4hjsyHNmY+vz6VrCBiOpnq4RUPEuqcF9+VBDuOwAGwSSHmzRMDyVntSxw4t0VZEk4xAYcai6NyRDs1JJ+/yfp6MlcNg/DkXHsmHcz4rR9mEmVP7dmYkwpn9L43XTOurmHbWDKh+V0mtdJpflX65r7atlcCfqwRyWy2pNbQBR0WFzgupeop9NbN1waOGuoNsXWwoVMsuaABaGCTKB6V/+ad8dCmauZMnT0J7U5dCb4WGNqMl7m2FBlT/CMR8+eVX4FrmdHzPdmawmQefgjWRFQd3FYFbtOCKNp3727CsWb12a7hAm7NgBWANcXKIY4Msh08A8EFoq1DKJbc9HaRps5e1arp2ahMu0AR5B/Qh6GIH8s+3uhqIcK8CZT48BFeRHwGbUj7yIu8phZQnX05KZuHK5kM03h3oOwAvpvluHYTbZxTNJ5+vnjxEZqt+voqXVN1S291vCZSJt/Ul+Zat9dV4FIRNHQRgg6DY+lAQPpeJpMG8H/vd49D8dNwqdcG1kzGWshsorQp0r1NoxXdEf+EOi2MFhTsODufMmRNtx8mdde1axgu2QtuKMdrc0vlA981jUqZGdV19XUzUOqANum0bLtFody8iyDVmxQx8e/fs1QdNfQAbBchY2OgSTQ33NcT2MP+/xXLj3SlTwmqlb7/+ISy+/Ior0D4/MdziFGrOgmqtjLIFh23HQOROuNTYdl1AvAjj39TXDw0Q5SpiagxEkGe+G1yiPftMCDR3ks6gQYPDpZXu2kYAInQnDgWfG2Wd40jtijau8HUt7q4Eh4wZ9M67U8PSwImrFiDXXnttWJnYfgRsjK0yddpUtrO4b2W0Md0m6TbKMZmxofJ7bJ8ZsJGfBABMu9bF0etMIBWgvv7mZKxuNkWQbcd1F150YTqCyb8CzUYLm9/jEu3FA3SJxoiDQs3Uxi/l5bI/wEb60ArqlfGvhIBWl0byi8FDhgCqXYRrppvCBZLngg+YnLSCQEBBq+W+avUq4rl8hAXMm2HFKF8RRD/+hO+FhY285r333g+rl5eIfSS45qT/BKx2xlJ/I0eeHzE+fEcl0+Y28q1gew8Al1Y9M2fOACh8IU2YMCFix/hNI7CGVCgu3Qn6KOAYRzn9GvBJujF/1sfI884D3Ph+WGSpGSrvb4lwPPoJ6kdhh0DkB4Bn1qkAQQcAUoUxCu0HDx6I8AFeTmOzXAUq3nxzUrrrrgzYrF8nYNMLIftorKR+kLQ+6AioQTNjiRbGc7T56FA8l6fh7kkn+Tw71lXwfcCTBsBmQvpf/5IBGxUF2rfriBXGaemmm26m3Z0emryCSSbtO+yrFJ4vXrQQAf8LgJi/pV4+T1sA4sKlIvR8Be1PKyhdEdrPCNaAssZ3yzcUlr+CpcNzgDZaKm3avBXecyyxs24NgE1rDIUxWq9uELCZML4C2BjDZg8u1BoBGy1sfEcBbLSW+3klhk2Llq0AO7tCKyekm26+KYCRngABWslkXiafzH3cHvty2s2ktyZHuxHMXotSgvV+2eVXAspdyljgWMoHCxvKTrD3KVyiPf74E8yjDhCwuf9+xhbTAWJ2ApB3wM3ambjkujnczvXBUkzARasp60s+bVnpIvLRxx5L9953XwCxfn+//gMAI0dHzJ9jjj4K2sjWJJl6rGbmeICes6Dp++97ID31NHG3GNsYT+84XP393d//PaD8KMpBoVeLiGETFmrPPbcPYPO1LtEkVKkBPnhggA0u0aAdNe4bYthMmBiu3nRjeShxv+Tbl6IscOwxR1fqSNqV5vyenWk1vHAS7eIJrIbewjXhkqVfBPh7GCCh/daVV14ZAKoWm/7JG60r6d64Po5ZXoZHjGOdB2io5bNg3Gjq1n0FoZZ/CMFjUBSf+J39kU4Kz3BfflYDbP706s4U6++BLQpacbYRCp5a1G/ZtgvlzvVY2ixJC4lVuW4j8f1wj6YbdUKHs8rf5RMqfcntHQNk90dt0abvSZyKYUN6hwvk3ihpdSRORUsfoY5tS67RQniW/2gnmRZsb6brOe7zVu/xRJMl3D/FucZrplkWH/V5e6dYTKdy3bQjfW+JPPENdGb53Y1p5AcP7LekWfJa0q1+utxTzpWxatPz5XpJo2m+BCyc+6i05tY2o8VJcXdmXBoV/wQ+nEd6XV5e3lO2jpEdlziGkfe4ak1SgAwBDFfBDFfHh4IZZQ5qvkoe/RYX56QCQ4It1QCR8zZXx+heM0/m3dVxjQCN81/zq0WN95iW6Zt3F/c953EpZ7fmRxDFb6kGkjx2VaHPfPudjnn9zrIt1jR+m2MnzwvQlG90W/ZLHkp9lDxE5pr8mNdy3a28zaU8634pO/ddyv35KP9W39OYJv2Kf/QtxumZyvzj9ddfxxr2nbDkXL9hY6Sd2xj0HDRuAyj1lefEvsE088oB11UotV37CGrbNAzGeYAyHToRp2fAkDRk6GFpSP2hYWnTolUHplPeo3KkYzQVvwXRtjF+3kqbJ6xCm8ScqVMaVtcv+EGv7rgqbw0/cFzmK6tWX7q/MuCWhuWbrjfcWNs5qCXQSDe5zgpdS+fOyVVkFPC97rrrvpQPxuNBkXHBQJw+oJsDHygV7LbcVn3uS6nVThzUErAOZLpWbA2w+WOL2uFHZnSOQ+JIJuefnK+yOETxrG6GBGHsKjy3fefetGbDzrSAeDZz5n+RliwDtNmsr8lKcEGsbHzOCajxJVq1JF005Xfv3ILgbCeWNm1gun3DB6WxKLoA2jgQ89XczsLTNkUnOZWJjmfNn0vOb97mPJcrXi2s2/2my9ddq75XIWJkpPok+7wnCijno8nF2mGtBP7sJZBpvXGQ5gs8l5dMz5ka868CbQfabyLofA7BhcCN/ZcCVd1sKJDNwjisWljkn2pNTf/oQ9z9/BZB33gECMvQWtZNRwsCGndMJxA74XKEPWfjmqmufhgCGZ/NbS7/kqMATTjrII2p127coO1g0ibIu2HjNixr4BXzlqXF8Ir121uknc3acTNTNGXfpKZgUCGDAk7BGt0jtFCrpiOCzYE9GKj1wac15uNtcfMBr1AAL88IwEZhhm911hf/lsW+q+259NdqfjspUaj5+OOPB3CjYK2uri7c9OgaytgPTi7KBMLny2SMxGMxvcInStrlWm17oCXwxwM2uwVsIBwFyHPmzg3t+scBbKZ/9ElYlow46piwWjDguq6eWkAo0ouaWLmepFepDld90ILtRVpQ0C2I5+TPpdS5wnWXHVi1eM51D4TtpETrEtuWGvUnnHAcQoYuTDA3xYTxxXEvp3vuuTd9OgPABkF1aF6PQUMea5S+fXozYUSggeuqJ8j7fQhBZ82ew6SmJcLVw9MVCO2uvPKqVFdfxySxXYg2IhP+KMBDQK+QsTma5Vo6LEZT/hXcXCl8NTbCGtyVGTdHQf31xGDQvZpxYj777NP04EMPpqdxiaa2tovWBHfccUcIqAVr2rRBO54/S2gv79iLFp7ti8ILkEyBvS6KJk58LSwLLJ/vwSOMmTNy5HkBDJtHXbYZUHwjEz8n1goItOYZOGAgwFpnNM9bk6rt1OTlcQhseEcANuzr1ktXWs+giPPqa28gWNgQMbTGAgxZhocfdiix8NpwP8GO0VJ/C6uPLwM2PyCGzejGGDa8J9pxRTAQw4vIQP6pBmyirqEzBRofE0x3HBr+Wq/MBtjThdkpWAspcB458vsBiis0Lt9j6Ql2WX4KBhQwCFoJ3i1HeN2WulCgoVZmL9LSTVm4pUAo/S5xX7RAqKurJ+2R0MEVYcGjEEFheKZh80vZSYu8Zw+0afkux62XVjbPAsgJdm0E5FJYopulq7DmOOWUUwkEvSW98vLL6TcPPkjspOkRK6Q7QgZjFyl8P/HEk9CGHRpCl/xOebPiNOJ80V62IEjZjjWBNNIKem3dum2AO1o5BAumsbkNl2goAQhaWo/WX48ePQlmfwlWKD+EJgFssMYIvk+9UGKkKJ8uFZKFN3EU5OcFdvhmx4XWj4CN75kwAQsbAJupU99DeN869e7TD9Dz++E+79hjjkWAgrUL+cuWDpQZ/dVOnrP9vQBg8+BDj6RPPpuBZvi2AFJsezfedBMawQgVEL4YIyXeznMCZK6CEK8DwD2PS7V3iCuyARo/9rjjA7C5CCCvU2cBf7TKuXf9utW4sRqf7rn7vrCw0U1Zv/79KhY21S7RtLDZkqahbCFg88IL44ht0hYr2YERl+t6XNudcurJtOWsmRuuQzMlZL7Eu4wh9CnfotWhQLZ0Vz90GO3gkrA4OuXkk1CcahdlsHiRFjZPpd9xXwNgc9TREcPm4tGjwhLwSy7R4FUCNjuYBwhuaf112223Qj9HAVAYq8V2DL+t0KrVJa/RuuYuFOvk2dKTeZJnCFAMH1ZPnjpG/TqGkAdE/ULTs2fNDGswraCWAJLtIn0Bm7/9259BSxcHX9Mt2rJlS8OK6bmwsJmMu8hsYXMzLhQFQiJWVHtd4mX6lLoKrX0TYHNNcYlGDBuoB9775Rg2OzElPpGyvRULq/POOwegDwDTMuA9fG7QnrSwbu2a9DYWZ7oW1BXfIgSiHVGCOR4eeimA/iiUDA859BCeCjKP8tQlmrTvWO1TeNHLtF8VFC3LwUPqUU68lmfHRHywDh3JI+zU+F5a2XzXF+mk8ET3HdPa3wisS3O6sVSAKz3qfknA/ybatkqdClZry/5LgGYLxfl7YIt3MuQPwMYntKzfsGkrbXY1Y/8vkBcQ32QzWv0B2KhJ35IVnmzjsN7oK+lCwzNH2rMDq8Y9SaDm0GED0vC6vqlHF1xPtYKeeYdkDZvI/YUvg+Ctd9OxwdmnZJrgmHPZVXJc8m6WPL5R8abxLM/EcdwQ387TLJm+TE/6ci2L5WM/WIAM+Zqr9xaaLPceyNZvKGl6f3mX2/L+uKdyLE2X93mP16IceNZrLh4XsCJOVM4JcMyFf6iopKzTcfCiRYvCRZYCfIGS8l0lH+bB97mUbxTMcCzjOKOuri7WISizqChoe1PxRJCj2luB6Sm/K2BK+b6SZ99tm1WhwLmaQJJgjPv2JR4LyDg2kycK4jiv81ublrtzO1fTLukXS5vyDZ73OwRhnB/7PWVbYsoIxpRYOwI0jo18zjRK+uXdbku9RGFVfqrPl3vL9XJcytrznqs+X+q26flyb/XW/bJUp1nOkTTtz/lXjt9qTMapU6ekF1BA0fWtSkq7aMS+k+qKJfrmSv2bZq4321uF1rjPe2KNWYttBjCmReewrOk9cEg6/Mhj06A6LGs6diNmIAoQuEjDxAZaU4Eu021zrX+QF+7ZvQV36IA1WNnVA94OwxWaHjbatWFOJxmaMVbHC7yVv3xcyqx8a9PtN11ven/t+OCUQKGhTEeZ1q0baU6eVANsDk65/8VTtYJrgM2fUuyZA8vvHJB81eJwxKuKUxz0uI98KIKHr92wHQubtbiSWIr/7/Xhp3YPJs+i5XsZiDnwcHITk3iGcZo7K4yVAXftTKyB/j2wtOmX+vfpnDqGn1qZLsyXdzS+1TfbCeQfN+Yhr3GSe+kwWPPiUWXhJnNgavF83Fl1vdz3pW15Q9ML5dmybXq9dlwrgT93CeRWVJ1qdHLQtcMTZy2ZGis0SYPWfP3pZ54hhsYLMWn1frWgFcidgusjlRDq6upCY8hBsEI73bW88cbrESBb7W5bfO/ePdH0PDqde+65AdYMH+4E15gDDMBoUL4xCzxtaLmdmR93tbjTDdrajduZsK1Lc7GsWbZ8bdqAVu7W3W3TrmYM0iotXf6QV3Xu1HrKvqzxrIQ1Xmd4RP80oG+XCEgaQgjuBwPOk7YQ3Glhs++SB2SZk9j+o8zImOedDCxYsCAEsAWwcWLixEKwRqtM3Qg54XApgzvTcL8cl+2+b64d/XElkCcDpSylapfMt625ysKkwrONFjYCNjug9bnhouuxxx4HqJge7mCGDTcOwMgQBhr/Q03lom1M9ZmM/5G2Y4glCPpsK9LCzBkzAGw25Jdyr/nKkzJoqZIZBZI7CTCtS4y+/fpG2xAUOO20U9EC6x4WNk42X3pJwOZ+NPdnpO4IqgVsdIlmrBBBVGNhLERD+ne/+1361a9+nRYQh6R9h05hIXQFbnF0QThgQL/Q1LM/rC6LPQCbAgLmT9o1poZuqp59lvgNk99iors0hOOCrApFdUHWA638z2Z8GoGyBWyc9CqwMoaEcWG0rlBjPlwSqaDhZE7AZs/OmBApWBFImD17NgLD8aHBP+3996N8dD00atSo0ODX5ZJ5Mz6O5bsTzXNX03NiW1xDaK3kxFkATiuELcRoMB6LrhjXEOvBNjp9+kcAcR+jSb4Qi4gdaQT1eS0AlDEiDkXg1k4LG3jVlwGb1WGN84Mf/LARsEGo6+TUiXQROuSKlhryEvRR2be9O3FYSzlpQfEiVimC4HOZyOo27DIEpJdcOob6AqiDdygk0rLJxQmk+bL8ogwQPAvabNy0EdBrG2XWEuF2Oyb9xvBpQZrzsHoZh4D+xdCgNxXBdQE3v9V9A5JrLeZivUeb4X26xxP8kz4UYKiB/yRu1XSRtxoLLgURWljedvttATD5TVPQpnwS0M2YImrRtiXuRXfoQ6GLgc4FkgQjtR7rS5yRnr16BoCj2z9jDpl/BTAtsGYhI0EzAdhQBmaRU+Gy0Im/LtHeevv31OumAC4vvlgLm1vC8kcgJQPtfrFtPI/3/BhaHseVhcvxvdF4PYBD+O0VwObV115L//7vP09TsPpqgUJBf4TlAmlq5tmHdUJgZPkI2MT7eM8uym3VFyuizH/zm4fSR59+ljYRT6Q/lh/X8NzNN9+cjHsjYBECQt6tVZHlLUimKyvbWoCEfN+atet517GU8+0BJHTpkv3x67pw7ZrVETfoHsArXaIJ2PSljLWe+vGdd+CGsI68QS+ka/+UAZtfBGDTtm37NGhIXTrrnHPDHeBJJ51InWpl6hfhcER65uuivdLeFFbPwLrNmFjGj1q5chXarPVJAFsrutNOPQUBWvugmcUI6HSJ9vjjxrD5PLtE2y9gsx1aKjFsHsAt4IeUfXP400D6y4vSzTfdSLygw8P9l/zRMratW2fWk4I2Qd77738gxhhcAEw/At50RbhTq+P7OtDhy/ej7TCOoDCirOeSL63BMmCzDIWxXQBjx6V/+P/+gfZxCWXRJmK1LK24RLONTpqUY9jIlw4WYPPJx58EIPAy7UyXaGrrn37mGQF+a9XVGytKQULryHmNBKgbTAG1d995N4CyiVhAyU+6ELPszLPPIabR2FCscbyWx1Y8YzkA2EgbmzZuop5mxnjtmWeeRcFgdtDGtddeR2zBy1Id9dxBEIJmhJ5C9H1k4Du9SF+ZN1hWmWfXAJs/vcrlyF+1lPFZ0+tyb2NQ+KxxarYxTiqgzWyUO5d+gSXEJvr7PQDZe9twH30yDcNV3twMWmdEg1u07E6deOKpV/f24R5tKALb3j0AwQFtpG3bFS+TlVT629xWPO+5hv7C+/LdDb82Rt+ZL7H1/QHWVM75SJxmx21lse90KfRW9qvPe636ejxwgD/Sb1N69ri8x3TLPZ4r16rPxc2Va449HHcIaNgmtD4RBHFsJTgj+OEcUPDG61qkeK/jMp918Z2OlxwHq5zj+M1xo0orzpHq6+sbgBnBDVcBmmKl4jNlvKVSicCK6Zd8+W7HLcUyxq3nXEteyv1ufb6k4775tPxLGbhfjs1zAVPcd3U8VPLpdwjOOH4TkBGIEbQx/1rU+A1+r2Mdv8Nny3nPlbou2yiwSpmVsivnzJ/3laXsl23Jt8eWl9tSv9Vp2fagsIa0qu8paR/w1uyQL+cQ27Zvwep+Bcoen8Zc4l2UUOxfVjPO2UG9aRXrkvObv8O+KS8cR5upeMtgXOj4rRlATKu2WCJ1H4RlTT0x4epS734DUueuxFpri7IObtCMbaOCp+Bp7vtJaPcO+sotALTMy7Gm0bvGkEHUEe2/I67RGMpzL4/ycr/f54okxO8pZVrJ3Jc233T9Sw/UThyUErDuymqdlNW2ML8G2ByUMv+rJGolF0Zds7D5dlVgGX7TIjvOgzO5YxZIMEZiwEVQUNRpVuCn9nMAm3kIZNfg9mg7Js/oyPCMvmodtOUU1IR3iCRyHgHE8FPbk7gUQ9GcqSM2RZ9enUDNGRA4MHBSHdzfZ3NHXBoyWYgpfU4tuq1g2jF+q/4Y3p3zXU42dnCZzZfz+9taLq5NF7/NpWzzUe23VgIHrwSk4vzX8I4KbUv9oWZWIVWp0jatZuEDDzwQwWkdlKuB5KBTbSctR85BiCto4yBVc/fXX3897p05U2H1eganLUOr/9DDDkHIc1G4YDGORpcu3RhIKkRzaGR78oUlVx44wFSzTuuaPbhO3J4WBVizMlwnrtuwLe1Eq25napd2N28LLyAdRl0hDINPYCvAPvyh+U4saZrhAoiJGoM0NWu6dyVwJZO35t7Hqxo1i80AawMvc5+cmDmFPqRqvsoA3vNlMq9gR8GmAJcDfwfx5557bgj5BGwUXrpEWmwLv2zkRX5zbfnTSkD+bhmbij2EKycqS2MJW69ZGKMmphMF62wBwubXoN/HEEy+//4fUhuE4AMQdp1zznlYltwYgj2Fh43axtJoead0gTuAAFdeCusUwYhNTFxdci4c/CMQDe38HHx9G1r4m/HxLMjQu3efdM5556HdTBwGhHX9+vZMO4hhs5zA8uOwsLn33gpg070ngM5p6QrcCqqRLmBj3BFdS5n3hx9+GMu2pQjNe6WTceWkW53zAVB0ndZKV04NZWQ5OCEVMEIJglzqnkeafv/9D8I903jcNOmHulv3boAxJyPUHENw7u8jgO8bLtEeeuihsLBxoq4wXprXCkdQx0loWUw33oF1bK4erCx49xdYcQgOPfzIoxGLQZBi+LBhuNy6IKyCjj/+OPJLH065h9UJ+wI2CgPWKxgAtNi8eUvwJfO9lfLchtXHJs4L1giQrFy5AjeNSyPGxipABy0feHUaAegsYHPhBcawIVgpgA0zu28GbLp3CWsQ27CT4SJAyLWca1r6y3SYS8B7XRUgaPVivJLx418Jmhs6dBjuqW4mKL1C0mEVTe1Gyo0JZEUAr1a9wILpx4SU92ceksFtXYd9hGsyLR1efvkV6mgmrKt5uEq7piLAra8fGsICBQ4uphOZJX8CAgI2gjkKQWYAEAo+qp2mFYiu0g7FJdZPfvrTEPhrFTN79hyAzonhGu3DD6dTB1vII45qSN+YG7oW7Ne3EhCYfqNP3z5YrfSGJyPUgK66dcUymrUDIJI0Y5/h+wsg4vftxJ2VcX9++X9/WQFsNgIa9kbAPyoAG92oKTCRTefPyWO9qBM/r8K744OD/qVC6d+DXDcFsJEH/Pw//gNB+BSE5FhlYnVgbCAtI44ccSSgS7soc3sDuzDzqfB8DWCGlnW//vWD6cOPPw3AZiAuAm+48UbyeDPWYP0pDyyMyKB9TwA2CM/tTwTD3v7974kH9BKWEm+lVdCt8V9uu+12QMJRuPzrjMu+1gg7AGzQBh5P/CIt7gKwoX8UsLl6LIDNj6osbKBl6+K9aVjY/Ee2sBGoq4PezoXPKNA/4XvHE4DXPOXvaGk/Sv4ESuSN9vefI2jRwuYRXL2tIJbL4CF1uO+7MACSM04/rQGwcXzw9FcANqP4hl69ewZgpWXV6tVYFL02EfdkWNh8MJ2Xt0zSpd963bXXhFs9wV75q+UlLUW+qC9Bo6cAiR/41a8ixg4XwlpOSyZByUGDBoZ1jnXrmMA/61oXb/NmzQoLmyd5fjFAtLFijsWt2//4x/8RIIVlzKsiNsxELCWffY4YNlh0lRg2fypgI28cq4UNPK4LFjbOQLSw+Rh6+S1Wya+8MgGru7nkuwXWluekH/3oTnj4SbSRLhXw215NQbQ0i/UHCgHTsIJ7Gmu6VybgTk1eTSyo8+Gf1153fSiN9NWrh7RuKVCvAjbyfOt2Lhrxr1IPWlMa88y6ve7a69Ol8CLdNBrDxtJTSUEa+a4v8mjpzMV922YZ49UsbP602s8U+OU0Gnu6fa95v8Cl1yuz95gLrF6/mZi3K8KN+rIVG3GXhvLHbqxskBXsCqAkAwJ5vKErZGJUMhdotmc7LtP3oFXfif5+YAA33bq0QcEAwbBvoa6D1wStyzVsY+SC/yAJB04exG/JtdtKw+Ddmddw7L55KY+wG2nYabAvXblKa44f4j2RcuOP1wotNp498L39pVnOVafruaareVMm5rhY0F9eIejhOE9rFMcxWtE439GSXOBGQKTI0RTam6Z821WgQ4BCN2D20wIZrl2JPebc0fmkrlRdtTwR5PB50xFQcSziWkAW982TbdO1uDQTrDF/WvWoPOLW+/wOF7/bdP0+y928uS3l4b55FVAp4Ew5Nv9lLQBSAZsEaJzz+S2Oxz3227zfMU0ph1LXfpd5KO+vzkOpi5KnUuOe/6rFa95fninpm67vcCnPl3vKtpz/qrQP9HxJT4tjvWT4fevWrU2fYMXpmHfatGlYYc2jfgD1Nm7A+ndb0Eu8nzZRluxa0DrBmqmVZY7LMtY2rQG9OjFm7DM09R88DLCmf2rbHlexWA03bwlgy/1Gq7U9xhiLBI2ZmfZsQxa4M3XHG48eNoYO6ZP6odjdXg8blfGbMsW85IZqO44sVcq15G1/2/Ld+7tWO/eXKwHpqKzWSVmlwxpg85erh4P+Jiu5dDQ1wOZbFDfl58AkWF4V461OyWtliOOIxWOFMw7Eipu0TVt2pWUrNjMQW4EW7DL8ZSOU3e0IB8AGRqzw1ucUsGaXFAx4EADpGq0NfmoVxNbDjIfV90V7pj3BBdXarGjP8yY7kZDUkEZDY/b9JCqTL6aXGeDxTXmJDqUcxNaPzAxhn9P7PTCdxrQabykFVbaNV2p7tRI4WCWQ22Du2GJIAvll+s50aFsIemVrR+cg6z//8z8jwLUDY/mkA0AHopp5qwl75plnhtWNApt3CGI7Zcq7DOzXxSc48B440Lg1p4S7EsEdB+pZmJsHz4i9GGApacuvzvwB3TiYw06Q2o2bdqYlyzekz+cuTfMXrUSrjkDNu7CfwfqO6RiPGwPHtu4XCeIyGOdKcwIMtsYVQi94wSHD+gPY9MAar1XFBNp7XBniKaTi1Zm7VD6fgggeEV9hiYR4Lo4sL8vG604O1DB/RisktOZ1veRi2Sg80l2P1kgO5n2u8B23TZdyren52vGBlkAjr5WfN+W6ucSlkXyfQisF4NalgMlS6m4yAaKNqfLulKkQRHOEyr3TWQjOfoBrmhOxHpH2WyEAqCg7x4TS52NSRP2ukh5wBfA2sVmc0ArISFwNEyieVxhtO3LyuxS3PF+wqu2sEPvc80ZiaXEJgMdpqX+/XoATW8LtlYLce++9DwB1Jlp8PdOp+wA2vRHcrQ/3No8R20ELoeVotxnX4cyzzk6X4BpHV2u9evUIYXh0eBRpaKFSFrkcsos2adDJsIJ342noLmfmzM+ZQHfCVdmJIdTUSsMJ9syZM8IlmuCAwIgaklqUXX312ABxLRNpnn+LgIU+GGFhjLVsUZTDpk2bcbVIIPBf/zpADCfkxsfKgM2V4R6tDRPeaC6R1t5oc47T5s6dG4DHwkULw33XOqwSnLBrXRPAhpNiVgGIsNBh4r9jB1Y6MBaDjR9JPBZj8qjVf+QRhwPiomoHTXyjhc23BGwsAa2l3sASRQBk0qTJCI0Xp6HDhgefuAyLKa1t2iNUt9wEqfKHyyEV0jqGgSlWCLvBAodyFBCzHrW20CpG4eKEiRNjgmw9GJBd1z0FPNa/t1YsXiu8yHfuQrChQFdhuXTtREfLCsHopbq35PowhM13/vjOEDyrIaqbOifl43HTNWnSpBDirGWibh2YVd+hyzqFIAI48v7uADWDhwyOOBkK6ocNHZbqqXeBHV2b2UYcr1kG5lOhzeTJk9Iv77o7TZ70FhZd64PHXnzx6PRDLJ/C8qVTh0p5ZYKTvi0s/5jCs095VsouU2SFSwi48O0CNjugv1dfezX927/9e/CA1q3bQY/Dwg3WNVgeKFDyGwQPIy4IfU28g7Jai/boOKyaDGr/h+kfBzAoOGJMFi1s+gFUxTO8K/Mg6lMeBI2uWrk6vUM/YiyR1998E4ubNWEBduttt6XRWJt1RWAfgA3lr4WNLtHuvvse6vrjioUNMWzGXpXu/ArA5t9/roXNS7gO6Qi9DUsjAXCvARgxDpRjYsfT1rmlVMpmD2UuIKrbvkfhKQI2AqzS6AWAnLofK4CNtLcUAKS4RJNn7MDa7aijdIl2E9ZYF4eViK4UdwIqBmADyJcBmw8RuLTGZdwRYeVyNe72tJKRBmwCUb6UmfmSVnW38iTAkK4fZ+IWUD5yxJEj0lVXXR1gt1ZdlpWL9KPwJmiccp43Z3b6DbzmCXjWQlw/boXXaTH2P//pnwIwFVCT1yxZsihcW+oSbbKADSDRn8PC5qsAm4+w/HvkkUdoQxPDAlD+pEXj3/zoR4AuJzF+6BzlId2EcIk6k7dtRIj63tRp6elnnwtXfnOwsOmB0FBAbSz0euL3TgCo753pjTK0TWtR6VYh5/x5WJViEW0bl6cKGI+95rpKDJt6XPEB2FAeEaQ9V0aU63f1Rz5QzQ8tpxpg8+ep7QbW2yQ5+fNXLoIe/HsP3DLcpG3bCcC9bnOaxXzA2Ler1u7EtaaKny25rsBWkCC7RXbMgR/W1AL+pMV92r0tBLUD+3VP9Sh4Dh5EnBDkBm3VLaHuJXHnETGXYN9FSx1zYD68J+eXA/7LkQf+BVDjQxXAJr7MH1d9tDvNIX3pTNoqtOZ+Ne3J01y8Xu6JEwf4Y1ou1c96rpyvTsZ3V9/rseMwFe4K8KEVjQCIW8e1zntsF97nav9cnU55r1sBjwJs1NXVRR/qtsSh0RJFJcACgjhesE8UaPE9xXLH8bLjO/NVgBmve968OG51fOJarGYca/rNSeX1cAAAQABJREFU5q3kr3yrZVydz5JX57bOUUssGUEY53Mea0GjNZCgjOec35r3Au5khRPHB9ltWtN6LO9rWhflvNtSR2Vb8mv+y7nqvJe03KqgEHNpHir3mmZJ37Sql+pn3S/plvtLGtXPuL+/6/lcobHcZrSi30I/s+yLZdFXf/LJp4wVPwvLeulI4Mb64tUkyn9zxiAt9KojkNYamuhMWQ9A2WUIY6AeADY9U5su/VK7jt2xqmlPY2XMLlCDFc4uZAWOrNq0Jh4fae3ZtZ0mp2xwT+rasVUaQpzrYbT5Pr2IHYRlDV7a6VN9cc4rj5iBkEc2tuPG/sDL+1tKWezvWu3cX64EqmnZOimr7cZ5jHMu220ths1frk4Oypus6BAiULEOWu/GN7I+fXXzYcehf1p91SqMcMJW0H87lsKQD0rG/jskKqfNY4O8aUCqzXywwPgKb3HNTDEfyF4DtLGT4gqyFDTpEc5i6jyXgdiiJavRogWFx13Czj3EqmCVobYIYUOeXBpYsJnu0ZrvYCDWLPzU1qFJX8dArFf3TgzEmqNh4wDMfMKYWR36lUmc782fYM6cIDrsqnxQJc+5N+GgYfnqDrDhlqo0Ggqo8SJ7pWzKdp+LtYNaCRyUErDNSY95aJNpLw/K2OdSEQKqbeaE/i0EzwI2ChnLoNyOUL7nQNVBa319fQxmHTgL2jgQUxO7Lde1LNG90wUEqT4V9ymDBg9Cy9+AgExIbJL+xOIExfzk1qcrBOPWbN6K1QJadLPwWy1Ys2rt5gzWGHwQwGZPMwEbNYTlC0wOCliD7U2blphAA9bID+qH9AjeoGVNS1wkGKtCAaiAjS0+/+WcRJbMFtnJJeRvvtM7qgcGTmIUajsYEKxyImG56IZEgdaNaFcrgHYy4nNlEJHLnPQpy7KUa+W4tv1jS8BKy/QUfL3J45a09ZzvcRKHgNsJHbSusFatwXfQBHvooYcjLoCuytRIP+nkU9Ktt91OgPszw1pMwb605qKViK+037DtbMbSRcH2IkAEacF2oNBQiwPrXKGo4watQmbN+jyCw0+f/lFagTuJ3gwmBVYuwX2ZFjYD+/cJ38tOmgVs7r4nAzY9elQBNsS36NOnV9rAuwz6/hia8MaQUMBoIPLTzzgLkOXydM6556L91z0AmywKsCzIOP2imvThBotPsl1qpWJaL76AWy1ASCdZ7dp1SN878Xsh1LwQYeBA6FvA5je4FzLg9XreP6SurgLYXB1ASxa65zLPdE4ZhLAwa19q+eG7FFbec+99IYTdjjBXn+XGSbkaMOVU3C4aB8e6Wr9ufQgNjLvw4fQPAQbmU27ZJ/kGQAPTUhtU8M3aaYVQ3XbXGY3ONrg60m3Vhg2b0jrupTqSgblto6MJXH/0USOwsGHix3sOCLCBBkIjl2/Ydwwo/UFlmTwkkVhKexeweR0LDgXBAjbyyuGML2+97VbK9vIQYkhzElUuMxOCX8Gr5FeZdvnlNYI3vq05k1X5qDQmj/7ggw/SIw8/kia+9lqUkfR2JrR7yw9+EJZPWmM4hgphNvkvPMg87lbwAj2U/Ks5KwgoTS1cuCi0aOVnPyPmxzXE/+kAYOMYWcHJTMbOusT8+KOPcOs0O2K6qJWrIMXYSN6Xy6xZAP5a1/REE7UPllnG6DgaixJBwWHDhgZAIX3mPNIXUKeCqXcJ2OA6bC2ATffuPaG30cT5ELAxhg1xS6LcM103AjaehuotrFgaK6e0gfh26l7A5rXXs0s03Xi0QsNzSP3QdOFFo9L1WCwceughAE9ZY5aiYzFR6oWy1/JlHG1Gl4Qf0KY3oUFaV18fYNwPKPs+ALKyjRAAUncBwPGc7kFWAfZMwVJiHK4PjWWzHH4wAkDx9jvuAOwYFTTcEpoT7FhPPzsBy7e777k3ffiH7BKtH9qmVwPYaGFTP7Qupw3g9SULG9rx0OHD08gLzg/A7TisS+RdukILnhb5ikJKe+B/mzZvitgmWtg8DKCwYsUq+rbBAD7npyux3DsTS7oOHbFyhWaWAIB8ySUadaqFkS4OdYuqSzitwFavWo5F3cT0QFjYfIjcpQ3xr46EV42JWDxDBg+JsgpCb6y4oFXHGU9hUSJg89mnn1GgzXBZOCK+57LLL8PSr0/UkV+hKoa0HDQO/c2fOweeBWCDRcnCRYvDwkZL4X/8p/+ZxqBgIRBnHS1ZsjgASNvpW2/9PtxEHjBgQ5+yGr40adKb4erxTejW2DIDAboEbK655lrmkVrYqEUu8KKFzScB2GhhM3feAsqjVRK8/xHA6InwXeP/yU+lV+nGrTSrBee0KdPC4mj8hInU1bwAbC7EZZ3Wg9874fgAymL6E+/SOi/3RcY3mj9/Xho/YXz6LXX7+eezoB0Am7HXpVGXXBrWNp20sKE8tCotfV6mju/mr2VazQ8V9tQAmz9PXTew3ybJNc4Bmlyw7cZD+Q57O3Q2A7TZsm0XHjk2MCdYleYtWptWrt6atm7HdeYux1r0h/SJWT7DA/BBNe3Doh4Bbis8cnTs0JL5QOc0aECvNHgAVh7dcVfVCkuQFlkGIGt3ze2NTEAXNoSwpnSfK/67l3PHtfjjhFsvuLCN2yuP+IyrdObYU1or9Oax/bfXzHuxzHB8Ue6JNA/gxzRc3BagoqRRrpXrvlPQRQsZ+2vHwfb7ekxw6/hT0MQ+XpmYgIn3+lzJv1vTNa+OubSk0cJEAWm4RGXMUWK2FOsT+2vv81tNz3ZWgBi3Hhdwxnx57JzUe13dF9RxzGf+YyxfyYPfFuONynHJp/krFjMqm5gH12LtIyjj6rGrinaCTeU+81vu9/kC1phu03I1D+W91deirIIwvOObF58t5Vv2fcpzZSnpl3Pl/nLdbbnmvdXplGer76m+tzqNsl+ue1yeL9nJaVfRH/3ONupIulI5zb5QDwCOhVejfCJNGbNvO5bzjnTbaFEDUNO5MyBZ9964oO1Bf9iOuRBtBgXNZm274v6sA/FqnP/neXm4TuSV4WXD9iI4i0VdS1yidwGsGdi3axqKK7RB/bHm7oBLOmSCKj5ky5o8rrZnJQULio+qfKlplg+rnGq6+abrTe+vHR+cEpDuymqdlLUG2Byc8v6rpWoly+yt2Bpg80dWA2VXmFvwuBipuEcHHv1J/OREQ1sms8XykEIu/dPKJA0auB3Qxng2i5YYz2YZjH0VWnD4xNzVIm3HV61C2tCeQYsmD2R8/3beth0BjUJiTJ57dCC44KAYjHXt2AYgBw1PJxzkIqxsYOby4MxoHWiYx8pwqwlzLrmXRsryzQzaexvvL8/tuy0pl+2+V2tHtRI4GCWQJxf+2g4baS/IG5Itg3v5oQOq1xD63XXXXQFGNG0DtgPbYBmslgGy51uhMeuA/Si12ImhIdjdq1dPBmJqI6HV3SDssk3m3DhYc5qEogwg7V605hCir90OeLsifTZrIRM0tKh2wifQvEmszZqjec8aMa4QtrXkcbVmmqcd7OeBmho1hw0nuGBvzPHbkdf4btsmPKAi+LAU+JJKeXDEZe/wJ7ODfEfwKPJVFsvo008/DUHLSy+9FBr/Th7iu3HR43frV98Ji5MSF8vG7y3lbNmVxWuuteXbloB1k+tH6m6sqZxerkV/830B2FQmyNweE8Jp096LGAdvvP5m2sykkLlGOhItcS1sRp4/Mtw4tYeGFV6V+ip1KuhjG6CCo363qznGmKI5dW+d+2ZjdgjaOCEVHDIo58RXX0VzcWEANlqrCdicffZZmO/3ZVJMwFQmzC+8+FK4QPoUC5sM2JwetHUx4ElvhN4RHPyTAtg8EQBQLwThp55KrJsrrsSN2fkI8rCwqQgkG8hMIV6AKLrBsl02D424TxGEGjfnWVz9KYTX6uMEAlkb20DXhgKSxuj5Fdrqxi/RPZlAi9dUbjmRe01LHqNA2LYThRntLteHTWkLqrG68DKA+ONPPBlCAAXC55xzdroWQOB0LIlaIajWzZmu2d544/X0xutvBKCkq7hdlLdtK8YEaNtZ1rrv8TuNY9EDjch+xAbSamPdug1ory+IdRvSneGHHpauuOpKYuVcgqXB0eES7dsBNtZs9SJtVB3znZnvpoi/8SYC+WdQCHqDrYDNoYcflm677bYAwwYMGBixfxqeDiKWMzYCNkG90pjEaflab7xQnqLg4r333wtriFcnAtigWWb5nAFd3XLLD8LCxngyLZnsCoY08BvzSJp+f45ppFA3IaxZFnGRBG3mIUhWQFJfX5/+/h/+Pl2HQFj3YJa5yy7ASevkIwCbD97/IH3y6Sch9HFCrpal1mZZIJXdrZS60/JGt2nHEb/H2Chnn3NWWPFo7aB1jUWpkH/S5EnpbgCbtwiybl0WwMbYQsccDWCjhU0stLUYh1o+LCbA2LJxaayc4BLey8fu4tsFbF6Hxv7jF7+gz5tCGbWMuB4CNjdcf0NYYhnnpJJow3ts5+srgM39ADYfYmGzAVBWt1LW7e2330r/ly0dot/hfseiucyzS7QpWEpkwOYNLFkAbOhD7sDCYvSoi7NgCyGD+dsAcDlxwqvwg3sqLtH2Yk2HSzQsU358py7R6sgX7Zr0FcrrEu0Xv/g/6Xmsf9q0wWJIwAbA5aqx16QTEOhnAJrygZ6yNnkuKYEE681291vq/yGE+roUHASYct655+KS8XKAQACbDu2o1x0B6D0DkCJo8/nns7KFzX4Am7CwqQA2YWGDhV2Llm0BbI6INqDrvjqsr6yTAJLJjjTOiciYgM3TzzwbgM0n8A7nELqPM/aK7h8DsIF28qJQpvIo9Stg8+CDD2Ix9lRaAGCzfftO+NoJ4RLtkktGZ17CA1rY6HYuAJu33/0yYFM/lO8mlhHtzvSlrtLmdb22auXyNOlNABssXyY1AWyuBbAZVgFsLPMAbADFtYp7+eXx4dasRas2UUd3/vgngC4nILQiPlXwU2laXsA7eaHuNKdOmYrbwifSK1hdzZk7l3bRI108enQAZdav4y7dSkqj9nnyIvMsT50PfxCwEeDVKqp+qBY21wISjml0ica9NQubz6J+Hn300RgXy1O1MrWv03JRpU4Fu7Vl/yWQW+5XXctXpenMVyv3VR6SR7rshU87VbcF6PpMV+rLV2HdOWNxWrB4DYoYO4iHSx/oHAF+3wzeHZaQNswKb3OO0Bwh7l7cJLVp3RzQpms6pB6ls8HdcTvYEfkB8gXSt0uINk1biXyRB4fqkVTkh7NxIXMlc+hYJ/95lPPMTuUGW5273EEi9tXOH1w89hsFIAQoHEMWKw9BAWmtoZ+OJ775p5RZeY/HJY1yzvfbHwt+yFOd7wnS6OJMi9nZKF0YL8x8eW9Js7zd9Ar44dZjAQxBDhX5tLg+krhfR8DX6+rqAggRhDItlThM11UwRkDIsZD8SDdreioo7y33m9cyxzQvZS3fVfJTQBSf81stv5JPy1VASdBFYEbwSCBJax/z61YLGq+bTin7Ms/128t7S3mYtveV43Lde32u5M+8uJRj73O/+vhAzplG9Tuq0/Ba9VLuq77HfEiNQau8vywlH+W4bE3D5auuey3nO98X+6Tuf17yWNJkVFBQcWoLfY8A3CoAm8UoeujdYBN9me26HW7OuvXog+vcIdDRAMZ+KS1GWXPxkuXIBlHIao7yY8z/taxhZBzpUtaMFfWosweLOi1rWrXYhTeNFPLAQ/C6UzewW+qBC8Q2zoEiY5YBD8dYUXoif7lUKlfZeLmqjPL37Pv7Tdf3vbt2dLBKoNB6psXcrqwb6V2+UrOwOVgl/xdO1wouzL0G2PyRhZ+5HA/lAUlmfjkNXZ35l5ey9VY7Ljhh5VcmWaYhzBEJME7g33Vb00JAmwX4ql2+Ym1aixukrbtA1HGPpm9LhgpsBW1kqJo7E8wYK5vmzTR5Tqlvr84w6D74qe2Dq7QOBBTGXyX3Kvjw/T7nGr1KBbDxKI6rGHRVrvPlA/q1g/L7ZBpf9UC5ULZfdV/tfK0E/nwlgJhvn8Si1UqqrNGUg25TaC3NnTs3tEyNifEHAmVXd4T7JFJ14D0OYDujmXk0gm5jDGg1YNwDB8EtnTFB8iEsjPaX+UAWRzr4YiIGYrN5K8K/lZvwVb2KydjK9MXKjWkTAA6wDA+3ZsCmDwPBIicz7CKIaIl5c2u05+QFatFoZTe0rlca0A8NqTCBVrgCt4FfZKGHnKey5Gxw7ABbbSnZRD4ZrhJ5iUe5XeetwtGpaEULaAlsKZjUvNuBv0HXjUlhLI+u3brKCeLZMoAvg/dyXNKtDf5KhXybbdRQPGjN5tprTMe6zv1Rvs86UCBIbVLnzcI9zvQPpxOD4jdB97o324EQetjQ4WkMwskLASOcgHYHiGwAbOLpRpdnCtHDDRdpm6ardbuLyabvs204adUiZNq0qRHzQiGdgnUtbApgI2AxaEAf6IkYNrgherEJYHPa6WeEwPQiABtj02zetDHN/GwGrouIYYPgb9my5Uw8+0B/Z6UrAQ5HBmDTnYml9G3LcaEcyFvENKAcnHQqfN+MUP4PCFGfJx6HLpo+I13b7slYu1xGOWj9ouXc7Fmz00MPPxTusnSJJohjGSnEOgn3ccXNYZRD0L9ab41CCgWt6xA+GxT9/vt/Fe9TiK8Fh3FyxhKT46STToxg4fPmzgsLC11ByYvWIdhw6YyrIAW0vVi7dcsuKgw+3xPrje60O8HTHuw72f/4408Ax14jBso7TBY35UDlWNgIkGmh0Q7LCS2A9rWweTuE1EceeVQSGBiNILS7LtEQCAcYxyBCXrbvUtXvW8RVlKhgYvKkyQBhz6ZXyYuT1UOw2rj1VixsKNviEq0xPdNCWARdOUoqaUlLfpNlqDtIKzULZPek6QAmj+LWb/yECWHpIo85DT500023pLMAbhQ0tmgK2PBCkoJ2EXIwoQ7a4FcBjoLk3/3udwhUvghBS119ffrZz35GjIxrI2hu8LB4fi8C8B2Adzl2kO3HwLNq6BrbaR1CoXW4s1IwZEyhlWhcCtjJZ1sjHBHQOOH4E4JvnnX2GQFEtnPGzVcbt0hrz7sBKd6a/DYCpnVR3xcBpPwQMLUANtJ2XujnKhPxOC6ATdSHxZVvDC5RuU/ay4DNG+nnADYGc9c9x5Ah9elirLBuvOFG3HYdFnGgAoSMerU/hX9QeOvJkwDs/fc9gEu0j9JGvm3wkDoAm1sDtDFwfPQ/1CUfHfVlofu3Zs1aXLBNiXb+GoDkipWrAIqPCroYBWATLrHoO82f7g8FbO4Kl2ha2ADYoBSwL2AjoCdgs7kqhs2LKEwQdHfYMGKcfB/A7ToAmxMQcmR6tSz8jigTy4l62aIl4Ow56RFo4CFcoilgGVwAmysuxyVaBmzkb1rXGktFwMZ4KDt1iVYF2PQBOGjP4FzAZpWAzasTaPc5ho2+6rWSERC++uorU1jYWLJRVrmMcj+cQtNbwPP+B34V1lxkNR19zLERs+VyLHQUuOm2Li8Z3Ija5tvmz52bfv2rB7AKfBLaXhpavccBahjD5lJcUdqubc9LFgvYvAxg8wL84tsBNm8K2AAsTZr8Vrg+LBY2AjbGsOmMhY3jEAEb3cQ8+tijAdjMmjUntQDEPA8eeOePf9wEsLF+/Ca/uhn1sxXAZkp6IgAb4t/wfbrwGQ1Pux5XdFpQ9ezRvVKnlKP0Bl1Is1pfaSEsT32IMd6MGTNTff1QFE3GptFYGw0dOixi2PhMK4VhX+JzuYS/S79+a+EN7stnFSway7EWw+bb17TUKs1Js00XW3fhxvtcyw81noJPy22lXpW6XHWjvhQL/PkLV2JtsyKtWIMFxjYuoNiJGj51yfzAMX385rajizTnALpDat++Rerbo3UIcwcN7If75M4hK1Dpw27BfAVfJI+Sv92GdNGweFy5I9+br+T2WbmvbJQzNHZQQVum5TzCcYHx4nTxKChRV1cXVsqOsxwzFppseO/X7FTnr2GcAB2brgplKgvZB9svC4wIlLgVnBG4EUAROFI5w7z5nOmUdO3vXQU+5DUCHq66C/PYrUoYZTHv5TsFYaqtZqotZKpBnGyVmwGX8l63ZTXtGHeQtum7huIHVjG+33Gf1jACXsUqRoDGawJKjmdd/YYC4hSLG89Z5qbvUtKP90kX1neFBrxWfU85jpNVP+V8ec5L7pe0y63lXLlezjd93uOSVrlW7m36bPX58lx5Tzn2nqb71c99834eB8V9UU+RYKTZcI6dbFEqPem+F+COscAmAMMtKPNkl2a0VcemKHDs3tsaWtwLnW5gvLiafazoGJruMK61MkC1Piur7c3xTrg/pG3rCr1Lp1apf5+uaXD/7mw7pG4ob7dhDKXytlXrWMfW65qbaKUFR9maa87yv7/y9WpZvul6ua+2PbglIE2X1Topq7yrBtgc3LL/i6ZuJdcAmz+hyGV8NpaY+JqO7M/BifsVJsg2Dr1aud+r5aTsPoskNGlG8xZTm1VrtoRbtIULmeAjvF23RWGunag6MFnDPmvv8oCDMKxsELshjNhBoPHmqV/vbml43cA0oG+P1L0LActa6/qBgRf5c7BW6Y9zdktG3EYnXHLLq1g8Crbup7EUJh3fEsdxuvKTGUc+kHFUXyv75WTZlvO1ba0EDqwEGmnv6wdvjalJwRUCrpx0AKVgJs4yoShpOklVMGoQZVeBbK+VTtCtx2Ut7cFkHTj3YuB+xhlnhvsPNUQdyKuxlIWbtD0FxyEAyO8WZnEChnvqcIO2chX+1RevRgt2ZWjR6RptB64R9zBQi8kYEzFbpcILQVhdHrRqvgcXiM3CR3W/3p3TIUP7YwLdJXXtggm0Agf4UwZsCrfJheC3l+/WBicGgZaFt3HRfJrL6m91EODEQzc9//Vf/xXCxDK5UaPshhtuCDcwCvicCFQ/WyYCnivlWfZzjmq/364Eoibj0TyJ3jeVzGn95T77IOjGevTYenAMoNDKWA0vjhuX5i9YEFp+umwSIAntf8DHAQP6o53JpMLJgqmQRmj+VdLZjaBS4EZLGq1MvKag1W1LBIkCNsYemIpG/fMvPE98qPEMKBeEoDFcoiEsOwcLm37EsNmxbRMT6ZXZJdrd96bPyF/Pnr2IYYOFDQL+AtgomJ0zh1gTaAA/+OBDaIgvS9179ARkOQ3A5mpiGlxA+j3RZEcTkPxaClK15bB7N3FL+PZwBwQAKoii8Fi3sK8jPHaw27lz13Qa2vSX806F/rq6UBNTwOapJ59G4Lw6Ju3njRyJJcL1ETMlax1GEcVPtt5AXY7XygcEG3Qfp2swYygYVN5aUdPeuB0CniOOGhHWQ7obfA5rH8GOBZSVbql0r6ibIsFgrXLULG+nywosawKsYWLeiUl5hw7tQxDhe4xdMWHCq7TdjQA2hwPEXZEuGT0qHXvs0WjjMaagbNasWRXt+bnnX0RYC2BDPBHj3dwKMGB8IeOJWFZ+j98oCHYgiySnW4h33vl9aHtZ77oZU9tefnHpmDHsDw9XHBaE5RMFwo+uXyE09iyhDKgbp0jhSqEp+YzCCDVUH3/8cYS/rwS9SNu6GtMVk8Hm6+vrsUDKbim9JiXEwvt27ZIWcHXCCQEhrSukqWeeeRqh0mrIZU+U94/u/DF0dWXwNu81rwKVpiVvt23sJB1jM60BpFm9elW4tBPYW7p0ScQemjt3TgS0X/5FtpTS6kew7gqAgEuxdjA+WteuneOdWtjoCk13xb///TvU53qudYP+AWxuvS1c2gnUyauzdY3lxFrGpBXARj5rHkP4H2WZ7/F8iWHzGi7r/v3nP48YNoJhgwbXEcNmdLoJAfhhhx0aYID0G0UHD4n3UDfrAF1eBLC57/4HwsJmI8J0LWxuvf22dMcdt+VYIr6fvir3u5QXfa+LgrLfY3GnG8LXcD+6grLWwuaHAHmjoc8ulIPueuxjDDQ/ccLE9Mtf3h3v2U0aCswEbH76kzsrFjYKm3cDiG0KPvOLX/zviGHTtl37VE8ZjwzA5vqwsHEMTdIhkJSSLTLLw3GBMaY+/3x2euTR3wZosxLXbSok2M7DJRo8oT0WNgHYIPwvgI3PCHYfefRRuAW9Kdqzlh5ave3aqYDyi/TqxAnpgQceSB/8YTogWLuIQzOGNnDllVcA2AwiP5UyNi/QnXVm36lrnnCJxrO6R3QZcZQWNtcCel4WrueMmVRANflcUDjpzJ0zOz0ASCRgswz3LNsAGI+NGDb/mHy3Vl3WbQA241+BB+ISrQLYHAY9+i0XYS1s3CktbLxXyxc2De8Q9FyxfBmA1GsB2LwN8KcGsWDsDTdcH/kcBhjSubMWGRng//hjLGwoY8H7WbQ5Y9icN/L89OOf/CRcosnHIm5n0Kz820pirET9TH13anzP+Im6RJsbLtFGj74k3XDTjRXApgfv4WbqU6HWnj0ZkNXibSF9nDFsHnroEXjFjFRXVw8djSXu2RjoBMCmY2eLNwAb3/9dX6rHYe7bt9cAmz+91qG+P3ppYN3xZE5BzEOWWUAbvDYCjOs2eV2au0C3ySvSqnVbuKcV8wmUKoiFYQzcEO7yXPBd2oB9hHOHls13p87td6V+vdrD1wbgmrN36tVNWQFWE7wr+CFtx220cSwXpYtKa4+NHjrymdIneLrSL5h3L0aLlQtxbzzPd0BbKngJlhh3zsDszrOcP6nwonKIlluCBzwW6eT+2vTyEklX0vOax/bDjmUdb26zv2CMIEjj2KMANaFEoSIFq+cEjJzP2L+Yr9IOyvvku+arWKcUF2ICIAUE8Zqr8zzfLxhjmsWdme3I/XIseCOA5DjDpbwrvqPyTZ733ZaBYy23roIs1WCM4It5KvkRsPG4gDZed2xUzvu832PaLuV7PS75KOfdes5r5X7PVS9eL8+VrddLXe/vXHm++lo5V55teq3ks/q+/e2X93rNNMpxSa86nab75Zn9pfvV5wq921dzlw0VouXVVUvux/nNZcU1n4qxY7QP4ibS0orC5oqVm9O8BWtQdkBhe+0WxhOMT2nPO1m9r3xL9L9U4549O6B94tYSs6ZLp9apP27Q6of0DjdoXTu2SG1gBc3JHJhNHlvYWsys5WMuzbMZLnmOk1XZ/4rdko+vuFw7/RcqAem4rNZJWeVnzmFrFjZ/oYo42K+xkmuAzbct5QrTi8dpMDHKktMVrucW5hrXK+e8XGGGYebMZMwbHBZpEikTdxK6dTu+vVejZb9oGX5qmeiv3Ir2DBMcBmLNW7SjQaJpz4NqRjoJadbMYLloCjdDKIT/yg7tMFvv2xPQZhCWNr1S104tA7RxIJaZtnn3bZlPSweZacPZ9+lpzKwZzAwh358ZQhncNE7gvXogS6UsKiVzIE/U7qmVQCmB6o7JcwqIociGTqrcV72NZ2Ki0nhW4a28T/p2UhNtgCOtRV5HcGWw6bcRWurLuCylI/S4aT4852BYbWKtS6668uoI6itg4/kIbOyAyYGyMyAWc57BGuBW2/y6rQQTXZHmELdmLTGstuCbetduAlEC1GpLR5aDV0Sbo503w72BMWkEbDq1V6uGgKKDeqchg5hMdGECAVCrqzRErPE2n86CPIeOcWQ2yEX+bRbCPfJXOVZwpfDLGBGyBfPuRMsymYiQ5Ne4hXoPKwHL0bL5Hu6gfoKgZeTIkSHYNmaPddN08d6yVO+Xc7XtH1sC1KY8Oh5jEN5YvE0SyvdZ894S1cAp25BgwPMIXZ8DpDQWyGpcACmYq6sfioua76crABEOx4WVcVGsV5cIrs57BWecPPh+20WpU11FGZvFCbRgjRPOcIn2ewT3vOe1114Pwb0giIGmLwWwMYZN397dcVuD2wCEpMaw+eUv74k4IT179k6nEA/qisuvTKMuviiAGN3bLEL4r+sqY2gsWrwkhG1HHX10uuLKq0hzTBowEG1NBJJBp2TSJqgLpF07CdKKcF03YrYC4+kInjz91DNYAU2L9wsSnQWINGbMZRGTSpcXM2Z+Fu6F1KiXXzgZ1oWZlijSfismxHto2MbosKwFOHYDCLgvyGF7moFG6QsAIy/jfuhj3ALZtk4gpsSV5Pn7CJUHI7hdSjsT2HoMUEe3PVoAGbTeuDxjxlyKdv0xWNngdpCytT+2PmQtfk9bQBjLXC1S2+ozCGAn41JrI0DC0GGHEKvhkhCMnHD8cSh5KORNCN7XhBujZ4lLZfwKy//IEUfj1ur2qJsuXTtRz4xT+C418q1P61o6KvyzbC3PavpS2Pz+++9jSfEi3/QirvAWhFBft2xqxR/Dt8grXfyOHPtIUF2XJEinqDeBhRVYXU2ZOiVcTzVHkNG1C5PT+voAuBSMPItm/7hxL4frOCfEhx8OCEb6uqY8CiAgg+eOy8xf/EbeBUZ2IlDnIASVujezD3gF8Gfd+nUhkDnuuOOJuXNbpNUGnh6CIVycbGf1XQpC5PWWezQGf3iHQND2bdtDs1/hkZZA0r8uzoyBZDvp27cf1onnh/D8DMCAfv36BO9VsGMcNV3n6QLKWESdOnUBRB2VbgOwMd6LLtG04Mz1wJRe+raBx6LQotSP35zvs/fJfRh0QxnL17V8+l//+q8BdLRq3QaAdhCA58VY2NyUjkBobzyk1q207jZxnmfdA0i7RmAV3vGrXz+YPvzo47QR65SBAIm3/83tBI//mwBsHG/u2Y2vdnhBAJhsLf21fP87CPZfeHFcuMpbvnJVGnE0MWxuvyMDNl06MeZFJYnyM27JBGKd/Nf//SUWJp+Q7xTWSddcOzb93d/+FNCrjg91nLwr2soUBPr/+//8Z1jLtWlLTLf6+nTOyPMiBpEuswQfW7Max8ZPiq+KdqSwelPwHAHV3wLcreQbBR7Ohxca/+ks+JSAjYC0tCzdCdrMmjU7wJDDAFQzYDM6+JSuvQR3VqygPeKKS16l+7hWbXCJdviRWLuNDsBG1zTmpRW0rXDVOtI9nRa1gsOPQ5P33HsfbhlnRu0eBn1fc801ASjrAlG3dQr4AvShhqI9UtZzZn2e7rvPeFlPIahcGVaVWuf88z//M+7ULgv6kQctXbI48idgM2lyjmFjrCmtB40Pc+hhR4SGtm4k5QGBZQSByU93pmU8b2w7gaUPsNrUmkyAWWBWYGmYFjbhQiuD/R9N/yhip71CbKI5xOZyLnTueeemn/70Z8RQOyl1AtyxjoLkoDfrdy8VvxHg2fp96umnw3pw7rx5WJ51D4swARvrtzdAGQ9EfCrd0WhRCfmjDECdIciYgCWe1iOffDoDa7K6dB151NJJoLIDgk7psw0CbOviu75IJ6Xfdt/+pAbYfLnWYaV/1CJP+VaLPDsezHy6MPQ8Z2Duz0VjXW4HuVmJy+RPZs4HuFnOnMFzjjHQxlcjnzE9VZn5APVq3brqHq1Nqy2pc4e9WNd0A+zun4bV90vdO7ejvWU36lK9il6ZN5JIJUe2C7mlAJL9nkoKjf2NOcw5z/fnewV3HA/5boGKMh5QGUXARhBDa0nHTwbH1qK7mh6jr+YlPu++W2nUxWP3BV1KjBfT14pO6x37crcCRN7jONC+u6w+a3quJb3yPscLWqiYN13f1tfXh9szn/Fd9uemKwAULlA5Z19a3uF91Wt5T3lneY/bsrhfQBrf7ypIo+tBx5/mRavuurq6UCJw/Oy1MvZw/BF9APy8pFWOy/dVb8t7y/c3vVbyVrbl/trWEoB27JdYmsXEi3psmIBJU9KndOUOx16uPCH4ogtDZX+OZYxPtXwlLpDnrcId+heMO7dxviUrYA5uEN3agdm+BGDte1vSVpvtBfhDHtCm9V7Ami7p0OED0uBBPVKPrsgAwDxbhbJ2bpc8wvP5zz0XWlRs/anVcUNR/LfZKTzFrfVXVnnM/Bpg89+mHr8xo1awHYsVW3OJ9o3F1eSGChOOszLtcizzyyu6CZVnKgyx3MJZ7y+MnhbWANjIPGXem5HWrli1IS1Yui7NW7IeYRIB79C2b6Yfy9CiIU065EgnrGy0tCF2QCDte1O3zu1DgDuof0+2nVPPrpjIAuSEWWR0IJmBx+CKfPleJ0uRd5LOOa7k26+AVkqH3sjU/SCZfGy+/ife4f2ZoZQ3fP1Dtau1EvhyCRQ6dFv2FTZIl9EenEmwlI6rtDavBelzrQyiMz1mawDPqXGlRoICYAf6DsILvfuukmZ5b7yzkg8FGGo21dXVpZPRFDv55JMjoLQDfV27mMccc4KpEG3GYZ6acpsBa1ZgVTcfq5qFS1ZjWbMpbQagFazZCziriwPb5m6EwJEHnmy+dwvr1hiMtW/XEn+1uEIc3DcNGdAz9ehG8EviVxkwtyUfnNun20Y+ZQnlUmKHRV6V+RUZYwAZ1yqAjeWSrYJSaI5pgTQOSwyFr/YbfpfffRpCa4W7urdS48uJQym7/JZ9f7/u2r531o6+vgTyhDOIilpsmC986SFqFVotNS/bdhWwUdP9E1xnvYz1wwvUq368d9ERde7SLYCBCxBUKqQ85JDhMXkMYT1PF7rIdAkFQQt5EbDYzeQ8T469TzpyYqvLnHEIaLVmUdtbVz4CNrroUljdB8BmO4CNk+5xL43HuuDeEJ72ADw5+WQBmysQ5l4cLtF0M+SE+TkEjA899HBoS+/YsSsNqasLKxK1prXk6NSRYJ18bBY+mEPyB4iioF5gcgdCBPM2HsHhk088lT7FFYyakgY1N19jEYoeccThMSmeMXMG73oQAS2ADXnU0kM3iLdB+2MAiIxLYnD7aDe8VEH1zp3bKHr73BxIftq09xEWPhaWLIsBZmwrZ5xxBsLxG8KNl9YsWow8jYBVt1wLifVjeSrQverqKyNuhcLPjh07Ud+5bMPSw7YabR464H3G21GrXoHoDECfrcSwGVJfny688KKwmjkJkLUD7rcsF4O6vwWY9gKAyltvC9isCu3/WwEGLAOtHQRsBLvNbwZsKtVtiTbwY/v4fN5zloMapwpNdDX3BLF/ZkFfWgadeOKJkbbB2RVCUDxxfxamSDtORuGUfNOOHTuJlzUnGS/kbfIpKQuWHXvssRGg3n1BQAE3rysYGkiaZ599TrjHkz+1A1CRB4cAw9ogo5ZTADbQgmDUMuLX6IZMS6u3KQevDRgwEBo9O10z9ppoD7rrmztvbgjrdaliTJsjjxyBldTh4XpN4UmOjZTLS+sDx9rGhzLg+RtvTkYwPiHA0ZWAFL179QnaHzPmEtzijQxrNsFyXaS8OenNdA9uwAQ21q/flAXTWL5o+SQIJWCTFWfsH6x/iqtS/rmF52ZveTUCNpkP2PZ1j7hTwOa119K//uu/pXenTqV+2wTo8v0LLko3XHdD+ONv186yA9yABrIQQsCGQPMAAC/QZwrYTIeHbMKCQfdhP7zttnTbHbcCwBrDRgIBrOFdvk9a9ZxWbe+8i0s0+pM33pwUoOkI29L/z957Puh1VQe/W9JoRs3qfVRmJEuu2LjIDQwuGAyYZmyqaW/yJm9yP+XeN//EfZOQcr8EQgmQQIBQggmm2Bjb2Ma427ip9zpqMyozI43u77f22fOcGY0a4GArz5k5zzln97L22nuvtddaxFXCRmkMJZdkjOwHhmTY/OM/fo65+YUAi1lzZtMnH0z/V8WwYSTgfpR2PoBR+l+nv/mbvwtmkBI2HUs60w033Zg+yFiWoN8Kw3Est2qA+I/yDIALhJtuGGMyX/7tW98MNXuZ8bA4M2ywF6NKtEkwYWTyrYPR8B2k7WTYrFmzNiRszgMOPvyRj4aEzdy52gaYCOOgD4YNEjb3/Sx9BbVmTz3zHLijLZ0PcVLGpRJWCxnftoto1HYWn8qsGY1kbTBsGDuf/4IMm5eiL8+/4MJgICmVZ9zxSPLY99ZHHBf/9NHqV15OXybP74Cztm7bniVssJ30v//y/4HZ854Yy+a3beuW9MAv7g8Jm5/f/wBqmXekjo5OmJ63McbexSGUywK/SaT1EIr4IDKh3XthLirJ8z0Y2TJsVoG/usGh4qyPIH2orZ2sEi1L2DjGZdj8K5Kd4t3Va9ZFxd96w40hYXMVDJvJMOzMI3YmUS9gnDHazTgIho2Sg/fex0GXtUgXzgjVlB9GmmfFiitC4kgkcZT+7AfXa6PKtuljTNlnMrKVohPXL1wMwwY4vw3G/CLeJ0zMDJtxMmxgGJ3tV8zfNg5XwdlNhs3QXhdjnumVW/RMYxE+D11e3CWUnMVu+fag1xG8PODZc7AvbdrWFVI2m7buTrv39TJneUyLg52jkLiB2OsUGhIxxMtMFtQnjTmE/bp+5q5WbNpMgcjLqfx5M9PMaajUYtsR6tFIRcKwUvqWg9mMp1eBlfweX/GDP3OK84qXeMJ5Z0CCM+PdNZ17Bm3FuI/wluGtn4yHm266Kd31ibvCrpsHIMoeoTwLbLqmEE8rtWKa3s7DRXLGd9cvRbJGf5lC5mMaw6+Svu6uDcxbRomSM87l3jJNlFjR3blcxkyRpnF9IwPHMlk28yi3aZp+PQ/dzMN9k8wW0zV98yv56O6apvgbxnclaJSakZE0Rf+qTK7HnC8iH/qCHM0mruF5F3ef+o3UJsWvHrb5PrwFHFh5X+34ktWZ2zr3f27DCt6qMVTGsE+ZNTJe93X3Ii3XnTZu2Zc2b92Xuvb0xCFOD2nL1Inxwxhy+LmP98CZhzXHjFEl+lEkeFtYZzGG22cw36KiD2k56X3u/w1HCgENTteDjKXY8wsjDTiJ2g37HF7j5vdrqwUKnvEp7JVbPLSuybB5bXXW71IaO9iJx45tMmzOtCVFwo0b9Fx9ZwSYJ0u3GV64FW+f8SlCB9lHcBZUuLvEcUnEiGORNRB6arfuOpjWbOwKjnvX7h7cWYB56l6RZwycs5tlvvDUoidRJW5oz+YIKpJGp3MmtqY52LTpRNx5Ufs0kDiLAeQjVYvmwkskXuVI3LwQjK2RRYiC4V1dFtOFHgHDpSCFRsVKyBGeRosG0C8qHEhlhJBNp2YLnFYLlEWxgRuwKHhm+BzqLvQ53irYxdN3T3xl4hOnXDgp7GJb2wUSSN3ISzB1kW/6XsapvxfidEnXMAZ1IT5//ryQOLnhhhsgMq9A1UlHEFM0fqwB9li0UZxDfUfTrj29aS1q0F5ZszntQLKO/ReMHMWfXYCrTk3dtWx8wAkSlBJSdWMG9sOsOYg6ozHYqjiH8T0bA6Lz0hw2XhPaKAMnb4JZU407y1VO7OaxyC9utohj3XGfySJRi/CwXhLWbGsJ7n67Afoput8lVmvHxg2SGxiJ7tdff3266667Qq2BGwo3ECe6SjueyL/pfiYtMDLDJkNt/i19TifGWBD/Z58M1xJs97H5fQQC9ddgfDzKqccdO3YBHBioxSaMBM5bb317uvaaa/mexcnnyZzoyqe5TVEYcXM+OCbMBVjth0gpoV1JFglAa9asSfcjwXY/hHVtxHTBEA3YQeWaqjCuu+7aNG/OTOIcSF1suO/58U9g2ECgfGVlqLxZsSIzbFTnJSFU6YVdO3eFLaVvYc/gMYi0EiSnTZ2OvYpb0gchsF+GakJPn3taWuKfdbcdJBz39SIhgXqgKNvaNemee34MsfIHEBE2GCR1dnZiI+LDnMq/M042Ote6VpJh4+luiQKuoTo6OtNdH78LZsoHGeuL2IBLlMwtHAybPqQwiGtYJTZ+8YsHUav29fTss8+m/Zzmn8TGXftPn/jkJ0PaxLGzauUrIeXx7X//NifXt0bbLl26hDCfCIPPc5DKkLgQOMEBbrViLaD0IAZOD6gW6rH01a9+LRgwqnWSgSZB8ua3vT2kdK6CsDkRIoEEmb17UU+F2q0fwlR5+OGHQxWYBOFPfwobNkgANFSicXYXSQvzLpdtlWGLWke1aWeewoW3eFRbHz/72b0hnfQ8RtNldEnM1d7XJ6m3KhWLVF5pK1WVKWEj7unBXtHTTz+DdMI/B7FVtxkzUUF53ZsC7yxHDaNSPN/85rdCLVoP9Z+CBM6lEKaVQrB9p8FIHj9ufBDqAwfZZmBa4V9JmV7gwf695557Io0XXngRfN4C4/0NMJZuC/WAMzHY+8yzz4TkyyO0k3aYZFCb/tuxraR9lGkQUgbxH+W0rOZk/+/YtRNVMM9G+v+JpM1aiMfasbkGW0m33faudAtwu2DBfNZ6SKhRpodgHn3xC1+MPty7Z3+oBnwnDBslut7wBhg2FTMyg1vOJ3C9AFFtys3fImS8a9/oZ5cpYdMfY/R+GTb/56+CkSojQUmDW255O4Tsj9A3FwazC3oQ47zqY9I4Rn1275Rho4TNV5GwyQybxYyHTwA3n/z0J5ECm81c5OwinoBhA2wKp45EbfI0GDYPIZGxM9TwfeYznw7VmjJsWoC1Pm0EEfZnEPY/97nPI6WUJWzETTJs/vzP/jQkbKxXsWHz+K8fT3/9N6pE+xEENxg2jJ0bb74JnHBHuuyyN0a/BtOBsnG2IZovGDbgKyVsVqFqUVjSjk2oROPgxc033xwqGd8EnpJQJnzafzJFnRPXIqmoDRvVDn4YJoVMp7lzOQGNekJx4a6d2YaNzJOnnnoG6bhxod5QhqgMG1WiBX6irWzrYC6yxpfRJtHRfP7pi18M5qduF150EfE+EBJwixbKsJkQ444GjvrYx9pmWgP+VCJWRsomGNy9vf3pMlSi/d//W5VotzGeWWvQBjt2bE+PPvIwahjvhnn/s5jf5zNGb7jhpmDaXAdTWUareD5LJglLREQGWGmx5555GonAb4WkphKLfeTd3r4g2iJs2MA8n0KfOuZUofk8DL5vaHeKNcWq1WvR4T8WCcIb0//8n3+SVly9IsKaPE0RhGPL6DqoB4bar3/1OPjxu+knqDZbQx9YLqUTtTF11dVXpXm0u0AvvIkPhQvjH4axtI613b3gon+jf7XTsqhjMTj+IxXDprNi2CSkEJoMm6YNG0eR2PvML0fGb3PFdBE5mmvJOadW9g4ybdzBF1VKm7fvTq+s2pQ2bdmT9vUoGY+kb+KGVnC0Ivq6fQjCMkTcsWN6GcO94D5UKrOP0I5NZ8c87NrItEFaFPu3zvBjGTShUpnUYt0YxbBMvuBSFS+c+XFOcf72Ese7zzp0qC+YK9qqefDBBwfVoMnoUIpUfOIc6h7iM5/5TBx2k3FR1hjOX86dMkkM70EGmTVK+7tPU5pGxo+3ezhxZbEHY1zTL5ff4iyfXr5nHJZDOG/LOJFBIqPGcKYlc6ZI6MSeqIpX1jj1tEoegSdJzzTLIRef3jJflCqexXrCp8yZor7Md6VpdNfNtvB2b1kvaym7+ZR38y755xqd/Lee3slDNn2PbwEHlAxKn8LS8Qwbh0ljPeaYVbKGcYFHH1zX/Qf70xbs1axetwtmzd7QrtGPNI0SNY5bR9Exma4MeA8rGHMMWjWgWgDXMFw5nDkPZs2ypag2hGEzdQoSWR7YbHGPRy6ULVP3XLs5Ih2kjtacdnz76XDw9t27eb0uWqCM9zLmCx4QLzUZNq+LLjy9QtrBToJ2rJtU9WR7qtBT5Z5EUI+oovCqkzj33HNjAjOOk2h9Ajy93M62UJnZkTGcvw1sFwgwMF5ZJID9ineFECO8SD6Qo8sgEbmbTbcyvOPQD+e9h5P227rUabk9bcAI+d49h0HyLKPGtLFQAx2zGBsl0yYWSah4UtqGjYkIvQXGzWSMkC+YOz0MkS/ACNmMqZwaaYOwxsZWhk1m2jgJuJ32VE4NW1OWwNv8iATMw9vCFkJ3rtiwvsU/t8cwdz6jymw0m4uE49um6XJ6LSAOEmf5LAtVY/rtPRy24rsiUEUOVXw3EhI9MtEzi7i7ofj6178ehFKZES7Qh6dX8qrnHenGD2mywRkHUXDmzBlBhHQTIqHjAlSfTEVthypO3GQhCJB2Y1Bw5drtMGTRp4y+2m6k6I5wqiaNlhnL7ZbJ8VJxV2I8Iv486ugumLKHOBmHccEFSNYsnMvYPgfJmhaIDDBsoEBJhJK4YZyQhPCbuvMbd8P2FhjLPKoFHJ5xGTbUE1Wn0mxzN0jf/74Glr8bc4YqrtzUKEUkMetjH/tYnHiXidOcI0pLvtrP4xk2AScZew9mPjjnBJZ3nAx6BfHs8OFDSNasCaL6D1DF9fivn2Cj2gsRvS1OP178hotCLZiSY64NPOHXxgYyjwPgB7j3L88RrC2AGxkhSu94kn7t2jWo0HuS+3FgZyU2PvZAkO6rGDbXo87mnXFyfUH7HNYfh2EO7kDa7T/TP37unyK8hPIVK64K9T+3vfNdaa5qo5DAkFAo4+OeH/807OLICBoD4a9zydKwOaGam8svR/UXG96xqo4K+Ff1FoTqSkXGxo2b0v0P/CLU5DyB9IuEZAmAV0NEfx+E1BtvvDE22NZp7ZrV6Stf/QqSOP/OWmlPMKOmM67f9OY3B1FfezyLUAllu8i0kvA+EIyHzBR57jfPQyy8L/0UgqGMIRm4Spdov8b1VkdnZxAY10Iwdqx9G0aUBAnH35Ilnenjd3081C3Nb5/Pxj+rR3VYSyQZ6IcYzhx9iNOeGzduwP7Jg3Hy/xnaZx9ETjeKnUuWwFi4FfU/70lXr7gS2zfjwC8DwSB78JcPRZsrYaMNIW3lSEBRXZ3q2CTskgF1A1tURJCMUoQu8Yq4RLgSElzHZIZNnEbt7oHp8Mv0hS98IaRFDqA6SyKEau48gX/ddW9K1klGoJf1UcLGfjoMI8VT8UpnfQ+7Mk+itm8c6qSWn7ccRsktIdmkJNWaNWvoF4jnqKiScOM6VtsjN910E8yWd1Rq5OYCB9mWjXWxjErRdHfvB+a2BdNPKUvh3/613qrqUxXWCiS8JkFoWb1qVUhiKlWxYcP6WBvLqBGGhZWFEM+FtxgTNguX+LQXgtOWrVtCckdpowepz+bNW0KS62bKKPH8TW+6Noj8qoA7ANNJ5tlXYYao0m53l3A5M+py18c/ESrRNOKujSa7w7ZvSFKaa96UW0/7pFxVF+EswwYJG5gJ2h6RYSPT0zEvY++WW24Nho12dULdFv1e5o94En8f+4UfUxclbJ7gxPS+7gMwe5Sw+XTc82DYBKOUsDIPZJRKOPBW9aKwdjdSfQ8oYQMDVDVvn/mj/8F4uBVi1eSAOdXGDdqw+dzn0tPPPE+Zjwaj6w4ZNn/+p+lcxoZ1PwLMqD7w8cefSH/7t38XKvJk2CxZdm6od/wATNU3XnpJqLpSYsiyjbHhaCP7yFvYfOUVJGw4uKFheiVNFixYGHPc7bffHtKGEys1Z0q/yaxRReJK8KfSejJSPordF1WdzYHBPR5VZb3g10EJm3/+Snr6WVSitaoS7XzG13uwofKBwBviJ9fnuZ3pQdb33raV8PbFL38Jacjno7NViyjOeDfjcw7MqzZUIca4pA72lN1lH6/FPsxXYdjI4FBtpMaOtWHzl3/5l9jMum1QZat2rJ568olgWv/gBz8MWJUpetnlV6R3vPPdMdZsB+f36FMByjYDVsU3P7/v3iijNrCUrvHgmQybjyA5+OFBCZtJUSbjqQ7yX2Fca9dKho14+4Ybb0p/8qd/km3YnANs00cBxbRJ4G5was9+bNjAkPsu9dGGzSpwsrad3gbD9CNK2HBIRkahxCrLJqPQ9rSbD9K3a8CtMolUN/nCSzJsOoJh8973vg/8i+QiY1x0NraCD6t5Nl/CfFnn+u5c49wtM6vJsMk9X6HxMwIDx+CZXjkffumHDLNVzszd1QgI+kD4AtDSCjylv//A4bR1Oxo5NuzABubutHe/B2XYDLSMZxyOAS9KLyANbueKljEc/EKdOrI6PI8EXWAG0jWLYdgs7ZwLAweJj7GjQlK/LmETpYiyWTPn+VzLgjsZcTHO9JW+JC5dv25DzPmPIVVd1JPJAHFdYBhxltIm11xzTfqjP/qjUCutJInuzuEyaWTuuCdzLbSK+VfbhUpXS6+SeeOhECDsIqUAAEAASURBVKVcDGu6xhOO67BtmbyEdd3LVWC/fMtgKbd+ltE0S3olzeHx/Pa23OJI1zdFGsb1sswZVZppe02GjHWUcWM491HGkSnjXZg8vhdmj+mOlGcp9/Dn8LD1Opeww8MU9+HP4XFPN97wdM6u77xecJw6rrxsF98cB7nNMpzp7frbHbnjUAm5fT29aNVgzK7fmTZv24t0MoeHGKsyaiBPxNh2/09SrO2zhPJoaHxK1Yxm7KLNmHluaozXjkVo15g6EbW1wB63B7OJFbdj1styFQmbvOcPl0i/ClIFiuDNn9dBCwhj5Q7Yq3CQuGpdU8LmddCDp1lEO9mJyI5tMmxOs9EGg4kIRYIZEYqcvUR/DYznNsML10bQcInwZcHAAMtBJB5A4CGwizC9+0Ds+w+q2xL1aBtQmbQRI7bdLHLQbdnHKfwjcOFltCQ4+2YikSPEJYOz3s9J+wHUo7UNiku2z5sSui3Hg9CV9Hf6MIU8+cr4adQgykQhLF653PxZsMywqXkMBrAUuS2KU/1pnWKiqCdaD9B8b7bAabRAWTiXRWNeGA2NqFtjgTsUjo0vw8YRYBgvCcue0HaD+kNOPhfGdXgO+xmatjDtQp3lUWwQSJcPiZsu0NXJf9VV14Qh9AsvfAOnwufE5mk/i7XNbLBWrtmGSoO9oX+6X4OhY1iFjeJZEdscL1FCj9RRpxZO1bSN3su4HoDAg25l7tmzpiFZwykuAo4dA0agLDGuo9x5E2Fd89h0fPI2SBTS/+QMG+vnXKFNDHW/q6feOUOJGzcTnkCXIKqURDEYWvpmWNM1P3/vLfDbMGwE0QaejrUARFFh/rnnVI32k7BpsY4Ti/sx9q00hMazJZCrguo8pBna29shHk+PjXarBHAOcjgjOAaEFTfPwocqv1Qz5UnIlStXhqSFjD5PjLVBdO+E0KptCG14XASDYOaMqRBjD4a9Em1bKGHzErAWKtFg2LwXtTXv5BCJdj4kxh2spDcefvjR9O/YeHjkkV/BaO1L4ydOShdcdCHMiVtCcqeTU9SzYKJKPHX+OnwQAurOHdht2BIqOh6CmfDUU0+jFmgbtRhFHc+PU/43Ua4LQ/qjlXQPB6Hg61//1yDQ7oSQ60Z+AsSGjsUdwXx4281vQ1WV43wGzJBJpAUBuGcf0jg749S+9nHU2/4CthOUrpnD5l1Cxa2Mnze/+Xo28tjwIR8NY/8HDBvVM65BXZBtquTere98B/Z5PhB5iF/GjMk2hSS62xb7adtNmzfSj8+mXz36aBCu7YNeOMRgwVD/czNlfC8Mm2thSE2ehBol8Irle+ihX4aNGZkDuzileuGFF6dPf/rTQXiePm1qSMC4BoiTeoEz8nzfwL8ZD9p++ggL5bb8StZIBL/33vuo09ogqixZsgTG2o3BsLn44ouCoCHBVGxle4uXN23aGEw5JVqyKr0tgVuvQc2ZhsaV/NIOjPD285//PP3g7h+kJ2hnCe2Tz5mczsP+0rXXXgcz5M0wBC4KJqEMFQnh5tEFoVpG3G9gpqlqSelBiUwy05ZB6FfSxPZatvy8GAvbIBSpElL7Ji/BIJQJsQimtbaMbrjxhtDBv4DxIUFG/Kikoqd0bVOJoPdjl+ZBGBTaItm/vzt1dHSm22//QDBsLrjgPJihU0H1HCLARtMjqHf72le/xsnkX1K/nTCHJgdB6z3veR+HAC4IhpJSNudMnshYRFoKIjMdSuvZC84c9gVXDPfcL+HCj3lkGza9oU7ur/7qr4Nh0wIjcNHirDpPCZtg2EBMinXfYGIkeOxo2kudlEz7CmV8kvGzr/tgWsyYVlpMySklTFqYkDR8q0o016hVYWASdoVU192oRJNhoySLMKA6tVth2CjV1Ur7aR+lG4KdUhGfg4H7NOrE+iBgOHY+CAPlf/3pH6el5CmjQ4bNQVSiyRz+7GdRifbDHyF5MjEtXbasYtjcHkbptU0i00GmRtA2ol4WDYaADBtUCP4bY08Jm23AlYyKW8AlMu7efN11YcPGAw2bNm4Mxuq/wzyQEd0HHlWt28c/fldI2IhzZHYJZ7t2bgumxj/DsFEl2hgkmZYvX45KtNtCUmjxYhi9lic6KONnD2QV5tb3OVD3pS99GRs+MmxSugTpsQ/CsFJCR5VCMjtzH1VxrQ5jUOavMPQdmEobYGQeBj9ehs2sv/iLv4BZ9O5B20QyLVe+8hLMyB+GdJEn1lUn1w7T85prrks3gTcuuvjizBxifSPcH4Cw7xh94YXfpPtgnshg1KaMjCvtm9lu2q+544470zIO/slgBDNQ/FGMtxeQVvwX4OcnxFnH2GpFwuamsH105ZVXxPhpAZ6dq1zPBNxQHxk2MlRlYMmwWbl6VTBsboFhk1WiXRmMMvsyw5r7Ftd74uIDobpNaS1x64svvxSwficMpdsYUx3gI+1EebUwfIKZF19n74/4u6zXfBdnNxk2Q/s7j6ihbqf6imF8qkDD/MsaPRNZC/ySEnvzUgbButqq88wE4Ditj2qljahFW71uJ8xWDN/3sBY71gqdAM0cEoGVtDERkhvNoc4xo8TFHvLsB8do1+YYh8AmwfyenxbMm4Z6ZWynqEodiX1nEseTpWjcOnJZnvzIL7wfgkGt7S+Jhs88/WzMY87/O5G88yCcl2sCYc09mPOkOMlDXz79Npxw6HrRud39h7d4SYldmTiuv9SOIING2PX2Ks/yPvxbeC8wPzxMca+HsZxellVmSlFfJrNFRku5ZbwonePcX6RmZNrImJFJ45pNyRn9DVuYM+UgdMnTZylz3S0KwY9uXhGO1vdwUPmuP8Ox9lPSLE4lnfJ9oudvG+9E6Z0d7hneBLnh7ejayjaLXrEvue0ili3pcB8HXdj/b9+xH1WGu9n/7+NA1eFg1hzjoKazo9CWIY5IwN6oI0RkvmUnzoFM1KAxLmdOnwANYBY0gNmM24lpEuoNgyKo3RrBI/Z4lqDcONX+gOb4qnkbIN88mtdrvwUCxiq8JwyWW3wl7vXwmWtDbYMNv9j3VdgSH089GkE91UYoAO2zBKu7DU+s+f3qtoB9UCbMJsPmTNsaZFxhOfFbhZZ5Ex16+RtLHDyzSzhXP4Z3pIjUveIkJsHsk6O4hYgl40Rj430wZXr7jsXpmRdf3sCGaxeGXVmgHEOX6tEWjAySU0srxAUWZhChNeyaF2MyY1iIpV5O3o9CbDJz4j1Bo6TNOCySu/BiLx3PKKdlojyx6avKYzndMMVY5cMy+s5/vEcFTvsnI5TcPqcdqRmw2QJDWkAY9PZyYiqLfhfTuvvtsyyCM1Tn8Bn2cvycBIsjwqv3WNsFqkS77777YqMwfBNgfvU5K4+DAtOWhVNdLtQiHBt+Nh0TJp2DPYDFnNa/DrUi7+e06tWEaUFn7b4wFLqV5z6Ysphmh+/aBoUAAhXEbE+nMsRyHXgZgBAE5Q/JGlQBTe5Li+ZPgEi1CLWH0yBCcyIMwobqDZSaU8pOImyczg1mrnOurVDagLdqMWceMmwyvvKruohQ6m+7Ole42ZLgKNFZ4/TOG17a31ASU2K+p8fcbJV2qlJrPl61FsiwnuHajUF+K78l2+j76H433F7i8QwPARvAbT/qgHbs3BUqmyR8S1h+ZeWqUM1kf2p4XKPR8+ZjpBbCVif3HNThjUOFRCvENuexIE5zglt1Ya7/Vq5aGRtsJWpUg2M5hDclJLQRE3ZMOCF+7XXXoN5vOpvhY+lQz/5QHaXtlS996Z85lbkyTWezexUSDu+GQH8Lqm/aOano/CUhtBvGx/PoRf/Xf/l62OGR8dTLeFHH9xIkL65GRc4Nb7k+XXzRBUHIUxWSYTyp/sADD0GkfiytW7+OMnfRLqOCoC/zRFVal19xRRDQrX8PDJGXIfB9F8aQTN2NGzdBMMDgZ1vesMvEUgroKhghl7zhktTR0RFSPRuRwnjqqSc5ZfpoGJ1XskaisJv1Sxgzd97xISQzbojT6KoFUxXWli2bw5j5N77x9ThRKsHfTb7Mh7fecAOMqHcE80GCqnhOCakuxudKxuTjMBweffQR2u1FiBr7OYErs8ZTe8eQTJqP6pG3wBx4T3oz0hxTOcXu5k57PJZPeyIPwrjZjZTRG6iDhOdbb31HtEErRNqMa8V5ElIDkAZxsW1X3ALCAKyCi8XHnoj9xS8eYBPxg2BASHDxZO1CCNWXIC2gWjDbW6mYMeCcHuBgFfDz6KO/Sr9+7Ne0/YtBuLG+qkCTQax0wZIlS4P4obSVOEl7LN9Hysb+Nd9gDHZ2BEPseiSg3oh0gXhKZoCnc033oYceTPffj6q+F14MeJLpp6qy669/cxC1b6DNZ8+eG/C7l3I/9NBDSD99OwjUEpEk3ChpeDnEJlUyqUZwSeeSqJ+SXMKb9o+MJ3PiBfLx9LHSPtqi+RQqxN797neGZKYEfjfbMuEew8bLV5CwcSxu2riFvh6LxPvydDVjwXrPnTc3mJednYuBn/mMRdWm5Bkvk9lyJ9EzMQfEuLdzDMOYl8Fh+X5O+p/9m8+GijLzUK3ZLbe8I4jtGoGW0RnzHVEzMYDpCImZXcyb2jT72tf+JT319HMcPOhNS2FCKGGitIOqqbLheMa9cyPzmoR+55bdSI08hqTEj2D4/OKBB0JF4sUccPj4XZ/gAMDbA+aU4pMotxdpJxk2X/jCF1GN92ysdee3Y9Pp9g+k//GZT2MfpZP5zjkrqzqU+fr//p+/CfVe42GcLj03M2xuh8GhOrCsIpHKUCYbxvbxEn/th0j48ssr07eRovvWt7/Fye4dSA0tijnufe97b+CTidiLMewWmKFK12ibSYaNKtGUfPkY4+Yd77gVIp3zcxu4rw8Y6ALG7ktfhuny+BNPuXCPAw7vBo6VlOns7KiYSJSHRlbNZJ6X84ESJce+9KUvpeeAa/vCfGQgybRsX9BOPuMY5dZDvM5NvWR2rWYMfR1JFuNvAGeJB65gnP35n/85TOl3BEPJNb8qIndiZ+fHqAT8MqoHtXvVy5zQosTVIhnS1wTukOmpGsfDMBQ3IVnz0ksvpSfAOa4LtmzZGkwr1xcSNpW2up0yymReDtNsCnZpor0pvwybr33tazBsfhoMm1bKr0o07WatWHFFMOzE1Q2GDfDDHkcbNk/Sfqqv/AkHSFRfp7pD1WDKHFpxVWbYhIQxzeE8oUo04V3pq7Vr1gQsudZ7GXyxqKMDxteHGH/vSR2M2UkweW1F7XjI1DvbL3Gk8OTluzi7ybAZ2ut5pTTU7VRfZw45ro6kCYiPylVyFn83+ii6i09DRgyCcSYDlUqe2ndvsSPsYXQfQJofOoH2bFSt7Dm1I9Hf7C2A7bCJARFYw+UDA4fzqf3ZkyECzwnNHHMgBJ+DjQz1bzhuCRgqLh1T8S1Oj/RcD7j/shwcMoXB8swzz6SHwOuPompVJovzvfuIAmu+Ow/47R7Dg0DuIzo6OkJixj2ZUv1K1njLuHEuKPEK3Posd0m7nq5+fnsL217m5zrC8MWvlMNnPb0S1vDiNBkyMl6UllGlr9IzMmc8wKS7bt5+y9gxnldJ3/R89x7p0t/bq5SjvJd49adh6vU6UbqRYJVmeS9hTaNcxa18N58jt0Dpm9IXhoq+sD9iXPDtSI45RHreqKDl7d7bm9ZDx1vHIWxt1vaw/1ezBhZpGF2MIcFC2GAOVdJaRs0YnqMHeDJWMTuZ5mLuYCmM1UXS82CsqsJQu7Vh8oC5LnJ2/RB7vNj9hRs/lKL6q1SuZakbfbgi7/za/H3tt0CBQZ8FDn2KD9x/Nxk2r/0+PK0S2sFOfHZsk2FzWk02GKgsqkSK+XLxIqbLyDCQbfUVjwjmj0hYl7IgMIWcRszd+ono4y8j+H5PxaDDsvtAb+inXbOeE/mbd3F6ho0p6tG0a4NG2liQZUkb0xdh55Mzo2DayJFXPZpIXj2XC+bNwEgnelHRXduKlpPA6cCBMGHZQg0Bk0z+xqlCBhGwChP1iHf843IBUt5P9CwByvNE4ZruzRY4vgWERxdCsdwA2PwuMCoeK+8lZln4Zrg0LNDNRwwzxxjfxtG2gyp0fgHRSnVf6llWZL+kXybCkq7Pkpd+ZQGOK+Mww3aUEoJMK+rRZs2eh5qUS9M1174Vw9SXwJOZiMqQ/rQTmzU9qEHrO8oYZ0M1AHNlgPhxWIpyuWYP/dEuwFi0tcKUmT51PHapuBdODpVoGj8e6wk4svV0jds3CWOWQmPFwYzJKCbXZ7ASNobhDckd/7570bb8SvAobWD7euLN07RuxtxA+XQOUaWTajM9mOFJstIexrV9mter2QIF7m1n+i360fyGtjuQKXhy2ef5mR/EDzcXecc41X8omC0vvfRyGG9XRZKSMRKcjwCD9q12QGQeeIeqMZgMLWOcSEwDe1AQ+ZSwkRgucdt33WVgyqSYgbqxZcuWB0NA4rmnthdAcJRQPZpTn4cPHQgD5J7c//w/fQEm4cswbGYGw8YT4be8DYYN0iaePNf+iIRtGU1PYxfiPuzjSBBfzwlMq6vqojmox9HGg+qJzpk0MTbQhyA27oAQK9NlK7As00dYlkivgfqbbropJF8k1I6DIWPdPDWqbZlvYoz8+6jmkjgpjMt0kPBu3c6B2OcJSu1rTKF9xrCJ6umGAYV9CNVfacPE9pg08ZywdXH99deH6raLLrwoxo7qrTy5v2fPbpgUj8WCW7zk4rsFhonEgfYFC1JHRwcqxBbQB1Noh9EhVbAHgrC2f3aSl/l58lQD2hJdZcAcPNQLEWEGzLHrgoh6441vrRg2x6J/H0ay5gecrn8QpoIqmJQUuAviubY4lPxpo58lemZistBTYKw8A6IGfwruoIkCD6i2REkrmRYaGn/uuecCn8jskLgh80FGylRsz9gOfdj+UQpjI7h5G+rKuoEn7VG0U28lWW5BHdoVHAibCWwYPvATcCpB+8fAzkO/fIh2W5/bm35vn98ecSWmTOXUq8yxwzAXd+4CDmCqbdy0kfLsZdPbEif2i+SgDOmOzg4INdjfoDIH6D/VoslE+RGSNqqcE3ZkLshYFDajHkjKyMi0P627DG/nmc3AjZJrwot484Ybbgoppje+8ZIg8LS0SKzJzAfrou0S1Ua9xDjogyqnBIAqsCQSzQKmzztveahtu/baqyFcT2WMir8d56ZD38S4t1vsp2r88+qYLAwb6/LZz/4t0l+PQZRqTZ1LliJR8o5gJCjJM2E8qvMYA8Y3ucx8OZr2ACe29VcgvD/J+DtAey5Zem76yMc+Gvd8Gfj0TT4YBNGcPONAEvNJ167d6RGkzWQS3g8jzzF8AcyAT34KmEO9XIa5imHDvPPze+9LX4TZof2Xfo6qOg4+IMMGWzmdnTI7JFxl2z+qzfvrv/5bGDb/GVIi5y47LyT5PoBEzhsvu5R+KTatbBVqFG1E3SiXjDSZACFhg4rUrVuRsMG+jBI270XF4rUwLqYwzzkW7M/vsWb4DkzclatWByNJCbuPfkxGZ2bYjEN3SX9fb4zL+39+Hwy4f6atnmUcwbAJCZt3w7StGDY0lK0cTc1TeLOEwqVMyC99GQkbYEI8/AaYve9///uDcSlsy7Ap8YgY5ZOpJIPiX5Fk0YaNePEI+li07fVnf/a/YNjcSt+Op8/dZxwJXCXekQGlxF0weEhrAvhKu1kLUffoPD+ujTqBd/dh+0pmr3hHGHf+V1LGMbsHv/b2hQFDH/zgHUgTLWPcTaZSzDnMNR7+UOLuR/f8JNRxjoWJ/5a3vjV96tOfTlfCsNEWlHsRD5/EusaNCv8HQsLmSRhQ308/QbXZalS+TQfn3sS8cOedd8AwVSUaNmxiLWQc57g8T2aGzWoYNkXCJjNs7rjzQzBm38s4Xxrjy3aUYeO66my/hLGyTvPdtV6TYTO01x2FZ3qdOeiYi3v3zFRw3Ocrr8czjtIl4+Dsn8O44lfVcm8/p/fRwrF1ew+MG6RTkbTZs6839cqTYT9QjSRygInAukEJsmOoWTp2DPu30AzGopVj/PhRaepkDtTMmYpNm1nQDaZwep+9DCrSTMFZxTuYOI4r4MXDotoCldm9fsNGmOpPx2GG3zDHb+VAg/DkHFmuAmfCmrDn7dqmo6Mj1pUyZsQndXVnzu+GL1fGjVSjiq97geMSxmfJy6fME3Gn7+U2jm4yYzx8U2zYiMuUjinMGNd1llF3mTblLurMjOtaxvjDpWcsR71sJW/dh18lXL2uuhX38l6+h8f3+2R+5j08THELj1PEL2Gazwxb0ZeOLPrI96PCFo0jLUAmq+YN+hibqiVUkmbTlr1p4+YumDXdrJfYL4XNWlWhMUa5I57jigOYcViBeXn0UexNMU4njpeONyUYNYsWYOZg2iRs1qBdg3WDdmvr6lQtA0WgLL6VK7+7yhhUj5Y5RAQmTEYnJXDz+RpvgYC9CpfV8YK4Y12TYfMa770zKJ4d7QRqxzYZNmfQcATlTD+/Ij6XP9U1yLDxG9fiQTsPXoHAs38gd17zgDO4CB+H2JRkFCtHXikbfUPnJQsxbdmsWoNoMOqUujUueGwsCB/uPM9RY1Sn5GLETTGLI57qp1XSZixMm0mIUc7GMHkHp2cWzAPZTx+fDZSZrwshNjYi8tFskjzh6hV1pQoSx/CilPU61+pmTNM56VUClOdJAzc9my0wpAUcK2UR6wLbSzcviZPe4jT9JIa5APc9wyWQG2EbcOqnRGo3B2sgavwEVVDa8VIljrqQS9o5jQyzxc2n9+Ak6ebevHiqxqSldXxqGz8JgjFqCGfN4xTnstS55Pw0e87CYNj09Y9Oh3pVWSChG7VGo5GsYQy52PNy4SXxOmMANlIQo1Rf1D53WjpvybS0eD66jzlqI6ElJOAsS8T06abKkawLdyTpV8YrESw7hnf+xj8nEHGqYoSXJ9BMQpxgnf22rYu+aCVq3KS4YSmn1nKaJN9ItDg1n7/XFsiEqNyR9DB9mLtxsDMjt+j5gIMGnGQP4zufGZH4hHGMSSD8DQQ1x8ITTzwRiz+ZNhK8JGjG5hk48Bo+Pozvbd8LGzI0YvMLAW4aNkFUH3bZZZcj6XAZBOulQRxvhXieiWNHSVt7Dzsg5N4TRsZ/g9opGQ2qDtOeigyb+ZxuNLx2ToTJIxBw96Fe6sknn0rfYww//fQzqBLdERt+Cf9CvzZsxgZOgPwXMOwJz3wSfBIEfY3UL19+HtIrb4cA/maYSAui3Kr28VKaR4kPpV6++73vBsNSuFcdlxt17SNs27Y9M3vdeXEJ/hIbo00op+0hU6ezc0l68/VvTte/5a1h60omgvgqxgvhlJjZCOFDlVg/1KYQp9dlKvVBJLUvrY+q2CZPnhpdpzFtT7xbUtvbk+zzYBwoxbQLovpvUMEmQXwC6qEuv+LKIBDf+o5bIIpOZqM3mv7uCsmaH/zg7ngGw+aii9Ndn7grVMPNnDmD8c36wjUCmZT1QAAcZTnRZd9EAaswEmFcb8oUlxEl0VZijkysgrttI2FK5nzgdZg0MgRlEE6HcaQ0jlI/2lOyj5TWst0KbpK5o0TOAw8+EO2m+pQe2k5pA1VhytBwfhhDmvZLP9IPSmaI35QWmz1rTqiquuaaa9NbISBLaLafZWRYm5D42Lc31KH96J4fRR/twOaSEhQy9WybwIPOBYQPW0YVznS+MW+ZP/ORVJM5qLRUlk7E3ksQlPLcEsR2Nl1KDMm8LHZ1NBpvmypBKVFb+1Lv/8B7se/0XmBxToWDCWFHWYLBPuAzu8RTqYPjGDYQ68e2tMF0WRYMmw/e/kFUoi3P0hsxDoQ+4I80JdTtQ/JFZtJXYdg8/uTTHCw6SNyl6cMybLBd0g4TzjGX7SC4xgRnWB5upfB+iSrDH8L0euDBhwI+z0ea51Of+mR6Z8UkVApKZofSU7+AGfvlL38lPUE+qh6TefD+D7wvfRoGT2fHosAHEh8dO08ggaFKtLvvvgfiGdKoMEa0d6Skx6Uwxlo5qWR/i0OqVgr4EYYcxzJfvvHNf4PR8a8wdGHYwPC9GfWI73nPe7G1dW2aBuPBsKpEyzZsvpsZNhAtL0Iy7WMfg2EDM0QJGxm+feC0XTu3p/t/fi/2fr4CnnqWdfZYJOYuwAbTbWHDRrWNwj2NQ9/ZRuDiaCslbPaGRIkMm98gTShOUsLG+sjcageWZKTb3fwTNxNShe91a9ZGPb6DrZ3MsDkK00qGzZ/GOFKVngwsM7StVQ94//3Y9Lr33pBm2o0NskPYM3PsqCbQ8ankiww/JfplyInftUG17NxlUbZHkNZbBSNFpvKdSL18CJVoDQmbXDbtjWmjS9h+eeXqNJZ2estbbwhpsxUrsoRN7ENsA3BoEKN4PbAvq7yTYfPjsH+zmnbODJs7PnQnElArQrIrw1mZ30xDlWjdDZVoSNgoRbSooyPd/sE7mVveg9QaatuYBxy5bIMCPoh4Vl/CcVmn+S7MiJPFzU0bNrnrhZ4zvWIcnlEk12EyNar1WIxkj2Hly6dreK8YEvqz4LPvjMHw5IJAzByzr5tDaFuQUF2HlMq2PXz3oq0jRhFBOKcPvQBZGeDbUrq+50CNqpSwj8FKh/ejMFexT9nOYZf2Wdi0mZimTmpFwh/aAKFlwHs7DxxhTeKa0MNca5mvxM8PP/xIrBu3b9+KJKYMIdPOkiO+FziLEkcZGGv4lzVQ8RcWY/+RKxx1LbBq3HLpZnyf5dbPdLx0M23nce+MZ3Oe4jPdXMPJhJEp4wESVZjJpClSM87Xvru+KuUsdYpMaj+W27teruJd6u9T//pVyuvT+F6lPqXM9TjlvTwNP1K6updrJP+SbwlTT6+4NZ/Ht4B7J9d0tLodFaPTPbwHL2W8HOWw9eFe7HkewA5fDwfLYNJs2NTFWqcbyRo15bj/Z13pmKxgRh2tpEqaSqPJGOXQ9bHDaULrKGh2U1LnIu3WApeoRJswjrmYtYJjMQ53RgkcA5bIP68CY1U5dQmP7B6Mm3jlx4jN63XTAo7bcjtmyy3uWNdk2Lxu+vGUBbWTmwybUzbTiAFEphkVMuFGCAZKMGx4+k3bVi9+RIjyI27XxdPvBTkafBC9MgHEgs1wMl+qcC7G1FOrOrQt2LxYCdNmw6bdMRH0HlHkWaQvQQFCRyB7FwMyYGDaoK/WBdgoTtHAjGfxNRmmzVwIZ3OQtEHn6jgMlTFJxCacfB30gwQZixYlFpmXd8OGR+0nTxI1hxFec2tVCY3g33RqtsCJW6BMTIaoT0wu6CUgKxWj2L2LaRfaLqwzA8EYeWIbXFw5vnTlx/gSQVRxo30WVXyYXsTKgzPyM//hVymHfkEgY9MggWjW3Pa0aMky9KOfiw74JTBt2lNr2zkQB0ezgINIiFRN2KLq90QOGxmIZKMggohGZLimYLiq0hDjgvBzpnDirX3udBZrs9OCWePTtEmK56vf3npYmcbi3ko5liVMOuJc9/s2dEznmhxXoxiiuhrDDwmhlsn6hWe8uyBw/vDW3TZ3QzGcYZNzGfpb0hnq2vz67VrAvhK2je0mIaeSH9VHDhEeVQ9Gz+rghsNT744B40uQE46VOvO05A6kAtavWx9SNquQKliH6jANwGvHQ4ZdWUOUsVH6VliQMO3m143uItReaa9EVVZLlywNRoKbYjfJZfMreB1Dn3rvYaURujg9jUq0L/8zY/PFUHmjYfrbsNdwww1vDVVswQipym2+EhS7YCq9+PLL6dePPwmxHjVar7zMCfktYd9lAKJ8lC83WbzLKFKq5nwIp0pUyAxQBZSMAPGHm/loU+IolbNy5cucPv82kih3p+1I6KiG6soVV4bkhrjnSWxneOJ7H0ROCfuOS5oi+mc86ssWLGwPBs0VME1WrLg67AJ5ilNmp+0e7Uhnetrd9pXZoMo2GTdKpGiPphuiYx8qu0p/2Y+ObU/Kz549KxgMl1IPCfnTpk2HOPw0TKwfhISGqhYvuPAiVCndDoHyXWn+nFmhHspT8hJZfoQ6pEeRsti7d38YT//IRz6CaqxbIGBMC6Ku8GI/BS8AeCk4wjKc+BLqMuQJZzJnlEzQZpjMQCUGhC0J+IVIUdKyq5SekaFlv6hy7HKkAy686EJgaO4gsybCmwUReiE6y1xcs2ZNqG58CibeCy+8EOqr7KNsv6zkINFmFASbtoAxmYmXYhtE9XvaU5k5UwINUozAgarC7CMZHb30rVIFqlTyNLHqX5w3PE2sDRUZTdY18CZZCXeOCWHAuixFCkVd/TmfN9C+SOMwXgZxOu3cD6NqH4yhlyEqq0rugQcehNm1hrrtgeEovpdph7RlZ0d63/vfiyTD7Yy1rJIy42KJ8DQK8CR+yGOTHmPxZrnsSxk2Sqk9AAPt7/7u77MNGxhTwbC55e0wgd4fdnxkOoTkNfWw/01WQtweJKDuhbD/jX/7Zvo17SzDRpVtd37kwxDq70hZwsbjA8wf1W2+AzA2duzYmSTsKyXxKAw2mRLnX3Ah6sQ+FmoPlcYwT206acPmIZg6//K1fwFmYNgQv7OzMyRePvrRD6VO8IvSbB5UOgx8ybj9+7///7A19GNUOU5ENeKykGR7DxIyl156CW4w7OgPy8RKPNeJNpJJ2A3j90UkDLWJpeTMdsopw+amm28KezPXXH0VzENssdC/SmXd/YMfhvTLS+AdGUkXD6oSvDXwn9IoqkTbv38Ptot+gVqzL6IK7qlg2CjF9u53vwvbXO9OC8E5MbZo28G5mjLZdzJQ70YN41e/+jXUP6Lqj+7XPp4q0bRh0w5j1lPdcenpQOBSDaUSNl//+jco4/cp7+ZgAilh88d//EcwC29BSnI88C2s2C9ZymYdKoy0A2S/PI/qsvUbN1H+7oAVky7lVCpStZgXoKpRBvwiVKBp80dpnmeffy6Yaqp7ux3G37KlS2g3VaLJRDnGeHk5fftbrrl+gh2aNfTJ+PSWG24M20eXX/7GSkJR5Ok+xvUHMMseaP/e7hhvSmY5R6i2cxoM/ZCwAeauvPLyKFO0gW1hA7pn46m045rVq8LejmVcSdx2JIS1X3Prre/C1tHyOEBg+xcbR7bj2XwFjopBbVNlQnGTYTO0x/NoGup2qi/R5JldjluZ8a7FZNwK9T69vRql8K2hyohw4ZARs+yXPvYVu/f3pY0wbVS/tHlbF+rSDiFpw1jCTgbKleOwmPshD2iqUjkkbchf1cqWY+yYgTQFJs1cDnkuQtJm4fypaSYS/m3g2SAQO47BgYcO9oRtmWeeeTak8n79+BPMvWtjP6YUXhnv1iDPQRnO/BbevHQvc6TfujuHeZV5tO6me4lT1gzl23glrTwPjon1lUxlGS5KCLoudd3lGs/bwyD6uW9UokY312TOyc793uXbNL1L2etlKW7lqR9Yi54cCg2Wb/hlnLp7PY3hYUu48qz71+ON5F8P23z/7Vsg2tn5hSvv2+1pZxrtRXHImg8la9SiIYNmHdpxNm/dw76qHzs2qifk0ANM06wZh/0L/e/aMohrHNaUDnB0gHE4qj9NnzCQ5iFZs3DBXA5nqAZtIgwc6HYcQpNh414oz5GOC0tU4QThruB2nYXFaszFe5RY9xIu4x1dmtdrvwWEwXI71svdZNi89vvujEpoJxdiS1PC5oyaDtTIQiUQXaySQHX8VZSyBiMDv1j4GMb3jEDjxFwgRxcVNeRosEizbDB0kGEjgSCfoJHZo30LDZbJrMl6avem/XDv+49iWBDkL0dfHbVm4XpHvd7ltH462sdCC4NlUIDnzBD5I2mDirTZSNpMbuPkp6uwwNv8VFeF68M9O1kuJ4jq8pOvwXDFffA5GLJyGf49GLD50myBk7ZAmZgMVF+Iat9B9VwS/dauXRsLbk9fq2PYhXYeV8bKm1E3KcZ3mJimGwIlCb74xS/GqW9tEogbI4aBalc9XxPwpKsqiVT3NHHipDQTOwedneemJcsvSAs6zk1TpkNAGzcJ4tsYxJ81JK0qQwlDEkhacJdhQ30kBnJi2sWaxNoEc3UMC7VWVBR40m3h/BkhBj1v9jlp2kQIf2ymyuQcxauPw1wxRmWj7BIgGmUfOgYboXzjjgWgYcRP4B8RD1fJzzbzclEwfLNUwkSAE/w0ynGCAE3nM2iBqs+IIVbOPZOj0+ND0skbRwM24hjL/ozT0jg7J+UNrxsK1eCh9gai8dat28KuihIkjo/tqBGT+C1hR6mJYE6QtHHd2HpS0Y2vG2A3yHM4+e94lBEyCwkGGTWevA9YsJgWKZ7Y1Og/HMTBF154ESbCrzCguw1JNYzhnrssmCrLlzGu2GAHwwZmk/VU2sAkZDTtRd2UBEdPT69dt5ayb057IOB3d++DGXQo4LkVg9/a47GMqq/q6FgCEX1p2E/xZKUb9sJ8zHM9J1f7eyFGbg2D9k9DoJewO3PWzHTe8uXB9NFI/CvYvlizZi3ts73Sud7Lpn9M4AZVwXkKPRhXy88L4uZUmFaqzYqT5IGTyC3GlRs28AXEZ/W4i9tUTbdh44YgdKsqROPhXqpKUzXdTNKSOdbZ2YFNjyWhqlAGgdICMmE2bdoCvjmKlN+cUIn0xkthFCBh0wpD5CB9KIND4qzl90S9p+O1daHReVXLeZI+cJ6LnEHQsvXLR3laqvqlO6Gq+ukjc12mjXl6q+NeHO4pXRlV4hX7VIkXpbKEnSVLOiEMn0/9OoPgMh5C8xhxJlfuo3iNdpNQLXzK8Fq9ejVE2tW046ZgCnV39wRD3jxkFsqQUXJHOJVo3tkpLCwLhpDzh+W2LLFe4902cB0nI2EPzJPNWzZHHZx77KvdXaisZVwUKTT71nSEKRlA1kXCtnAgg0iGoerZTJqg8ZTIlcefKtu6UMW3CgL3K9GHXV17gmEnvDvWZs2eGTY/rrnmKtoKNVIkkolKzjFclJX/KL952FrhxjrWuU81czKeZLysXbs+GAkzZsyCsXhhEOHnMT7a2rLUk/EjCVNhnurZ1x3MMG3RrEUiTNtRqs6SiXn5lVcAX1NCtZR5Gs+1aMA3zFNheA3MXyXohLkD9Pv8+QuwQbICmDsvmGjWxfLJhFn5ivaMHgNWNlLuBBzPRlrm0nTViivSrJnTGa+5TOKiDdiJ+slP70M93gu07bjAOcvOWw5D9lLsqiyM+pi2GNNyRZ1oFgkmjjnxnXZwnnz6qZDcU9WdDDyZuh0wh8bRHiFhBGP22Wefi7DiRlUDzQeGrrrqGlQKautlQmZ0UuCDB7sZxy8H83Xtuo3k3BL2wN4A4yXGIkREYS3KRIHyeMndpc2rp5/JaobEv5a6vb09bD7JnFXSqgW1jLazl2nY50pp7URa8bHHHg/GYheqFvVdDNy96c1vwg7W+SEBJNOS0BHvKH0jblOt2cpVa9Jq+sZ34VA7EsL1WBrbU+nTUPunnaIlnZ3BhNSWjMwlibYbGW9KRspglTk5F8bOeFTrKWXlmmIrqgFl2MqQ30HaLeBkVcQpdbaYNp4woaGGz/qwG6LNsXFGWyh9+Oxzz0fc7RwqmAQuVzrrCmCuA0mlKTCGok+pE5Foi3y7VtTu0sswV5+C0bqd96nTZgAXb4QJfDH4nLmJcer4aapEa0rYOJa88qjK76f7m+HvdEMbDlgdwrAR4vMavBrRg4nltC0Vb0Et1ot3/k1F4q8S/LuxabNt535Uo3WFpM2e/UhHwxPqPYoNXKVsSMK5Tca460CJxErbaNxcWkHrGCX7UcM0E3Xq86elBXOnphlTJgQNIcHsP8iaat3aNXGoxD2Uh0rWrdsQc6/4J7dcbj3TF6fVL93qV+A/wujuGPQqe4wSP+PFnJZh/Hat5lwoTnKe9ZYJ4/pOxox3kZzxoFBh1hhepozrURk6ximHiEraJb/h5fTbMvkXWLuqi+G9RooXHif5OZM4JZ/hyZU2PZH/8PDN79+uBZxTAAAic9PljjkhQa04RxiTh5H634sKNKVqNsGo2bxlN6qJD7MGl6Hj4WrHn7Ainc8yOHJlqOZxGFpy+J48YVRaNKcNaTf2D/NmB7NmIgdOsmRNnhedvsUWliPAj1fLEg6DjuZRlZeyOqfmb/M1ruPNNWMed7w0r9d4CwT+Ee7sTzq+3OLMdU0Jm9d4751B8ezgJsPmDBqsFrTBsNGRQQK2DdQo1hUHStCIFxChTJv4NmxGyeq1zMjUE/LGzfEjgFEjbv41nIcpXVgFIgaX9kHk3Yc6tHUwbV5euZEFmbowdcdwKBtZRaJNJlSbQWQJdg8nDwPB48N0AZFmFAQeDJd1cvIZXZgw7zEumE/tx0QvEiAN82xM/Lp4H39Z/5Gv4lOeI4dqujZb4FQtUJ+cDFsmJwk/z2JH4L777gsCgMaf77rrrjDK7eJbmI2xw1s5reVinDktNgPiQY2s//3fe8L4sazSCPj3qk+EfpexEO6MLAlxLvhVczQPAsoll16R3nT9jemCiy9L50ybnfYfPBqbpbXrt6auPRC24cWMQXVhSNSMgsHKeHVsKz6dVS9BDERfbUjDwUSdNHEMp5SnpguWLUwLOOE2Ebc2pOXQeBtYI7eBYxzEYDqWMX5lpihVZ34Sphx/+nr5Xo3HHIFvX6o7GM18sng7dkzVJzQUV9nAxAc/pT9KP9S/S5iRnqUNR/Jrup1pC5R+s/dyz0cKOvMdvVx1dU4ZjyBeRYAYFzE2+JRArPSB/ewGuMCJY0YiaFE5KNFLRk0XUhkS2bXNIZHT/nc8uOlVikBGiMwPmTflpKIEctM29wwv5EL5hAm/3e4oZdOH3ZXDUBa0v+J8mVXxZDWHLZSvSCJkqRnix3h2zsrEdQ1ry3RQGmXv3t2UcTtMlK0Q2LuC0HoOKqnmzWuP2zK6cR/HCW/Lb/3L5Yl7UUGGcddMWbe6zBmvsF0DoUBJHSulapCdO3ehKikby1UFj4RHpUHmgx9sCwkDkyZhy4ENkoT/aJPIM3eURFaJ1BJRo60IY9vLGNsWtnA2B+PBvKyvTA0lYObR5rNgkMmI8sR8SNiRh0yaw7Sjhz3yOmIUxA3Um3CHYW2QhDmLL2zvzAgCt2FsXLVhlkEc4u3GcFBCI3oxmoH4plDu3LfZx198xW+xIcyu4hRhylsCsHVTDZ6MGyVthEPbRUK5zI2ZM2GiwTSags0eCTPaUzJNGTZZGiHDT2yibRWKInFX3G4eB5B6se937NyBaqpdlT0czhkjIWAe82B0aetoMnh8NLZF6vU2Hy+xoLCQ4TS/S9x2fPSTj0yoXYyFrcCZ0jeOCcvZQhmFL5mX7TDBZNAIf7arsCbDfzTvNo8EM35pMYEuz1cyTbUPoN2WQ4f6Qr2bBHwraT8qHTQVxshU1dtx4rJx5TnB9DIMS9zSNyoRT9vLMh6mD2SUOW5oVWAhjyPHhG0k0zHagagR3zTowyPAyyHs1hwCJ2A+wUVnHF5oJY5lK/ZrjGPJMoZiTFW4xn7uBbYdq8KmMCfzSqmqOGgU7UNW9iPtcAg7TH20wygkz8Uplk1mkuoOgxlUtZ92tPZh66SXCVdVdqOAYw9EZDtazNmMjzC6bXXMuLqiP2FEyVCQ+Wu7aERbvHAOUjVjYa4G/mSsxFigRvaPt3Gcx0eTl/mIG2LcOLYpVz+qGY/C9BVLUQXu3Bq2kepNLZNtnGGYpvSddgqYIx/ti/Uc6AHeaEv8zCPjrXHBpMutW7Wz8E9Ex9kR6mHbOa6V2hJXttLOrfSPdqnidD31kVnlu2WwvTPecOwcYu2yJ1TDbcOezwHKMJ52nwHh0zEzA0aTOK0NN+0/CUPClMQo4dpxbPu1yZjmUJgwPsBJf3GaY/8I74YNyKPtWsWn9KmwbB1sAPcwgc8EO8Jbn4PAnO1u71mXFuBhLLe4WJVtedQa3zZ0XqHlaQ/xnOoRzds5T9hoHTuO9swqpeMQDm2UGTY5lQIfZ+Mz8Bl97lVgxvm9qRKt0dvC2JleZw45IlC4KcAqox+4dg0m1uQWF/Dmbbrlq1rhAdeOKzy4GEpBPI7UcOzhkNjWHTDH128P+7ddew+l7sMoWhpoI2nTFj/lOUjcJA50vLXArNGuzejUlya2HoNpc05agoS/dm1mTJ2QjvQeSFs3rU8PP/QgaiG/G2pohRtxjjhE3B7jN0ptFSwjaZpndWX//FHeDVfC6la+jVfeffrtHFoYNa47XcuVtae2NRctWhQHVzyMIfNGfFnwkXG9yv6mpH0iN8tSymMYw5dL95JOKWfxO9GzpKf/8Lz1K+7lWc8vPIf9DI8zzPt3+ixp1xM5VXnqYc/Od/qozCvAOEOQcadKQmlwo7AzjYQbjNI1a7dyeGE3KtAYW9icPoZ0mwc21ayRZ2tbx9HqHOf4EJZQI8heyPl5Hho1LlwyOS1eoO3pyWl8G/Opw1Z4dO3C/OaeiKmby+8MO4TgK2MLfRr9pX91E7ehhtFEmQMD7/BoXq/5Fig4xGfBIT7FReuaDJvXfP+ddgHt4CbD5rSba0jAkRk2LDIKHhQZxgfIsGLYxMAKd7FqQdS+s/AoqUd8f3R16cTAc/VVrQt81dcT+YdQSKsxwc3bOE28cXfatG0fTBw2ZbjL3ZfbzwiOjVwhsqhHXFLZaCYZxZ7HgfinTZmUFsxGZUT7eAg+UyC2ZWKPZXAD6pokn77TxdwtVaNMuparKmb5rJ7FtTyHeTc/my1wmi3gJCTOciyVxbqbbicm7c/8J7rwfXeRrjFe7Q94at4NvDBccF6JH8QiN/4Q1u699970D//wD6FqQ0KCYbyc/MzL26vElQDh6dHOjg5O3C5OHZ2daVHHEgwhd6RpM+ZglmZi2ofkm6fbtu/aH2PTk20srdhQcQ9IrI0M/MmjijK6URoY6IO4kcLGVPu8KZysmYrqIgxgop5gLJuosZx8U54uxqGLKzdcfDUuEo6FGGM18ED2ySGMVYUVoQxeeWxnfOUiTj83NDJtBgOd1ottVr9KW+o23K8ervn+27RA1W9EFTNHVw32V9XTQ7uDkPTvYKdWgfmOv0GYIC5wZX/lU/jZ3p196TgsxDaZN96G8XITnAmp4yCmjot3N8im4+27YaLUVdZRvPjBITZAEMHJY6BiMFgv1QaUcZihHdfBOBAGyDtXybGqqgGlUTMhXRs22nfRtsUR3iXaSXgdj40pN/C+51OmmbhoPpa1wK1pm5fMG23mSNCIjRLupUyG9/KEfS/E50MQOSWAq999NERK1ZWNh3ls3T0JX/I04cLIyDO+6dDG0Z6Rc+RtWSSCmqb4SkKlTALrqaqu8RAsNQw/DoJlbm/jUj7yo7qhH1vGVz7Rl/tJgihFo3UzPEgods6X4GqmtolrFesm+rOKuZa57RvtYw/p6W8DT0YBqp/cNxkGStsKR8JNgS9xuUwP4UlGlm3bhn0a+ygz/LJRYIn5xrUslrG0fbQb7tbDslkay2ydVP2lDSJh4DD9I8E20iCAcDoBSQjzCkIvcB/wYP2jwrncJhatyo+woFoy+0zAE16DOUQdzEO1bEdjrqKMxLPfTX8CUkEyQYQBGTWWMf4Jw789T60sOymTrmUUjjPxy6yU9JL4D+ffdO1DduoyR9qwyxJTwWDDC8NVFczJDOLKZTaf+Au4dqzkMSzeNx/7PmCEdmisAUt6wkweo5ZN/euuS/UdLbNF5p43eXo7D+XWM2/raB/l+gSBn/jGlVgedxQVAoZxTda8KF/kFcWk3uQT9WfeFD5d5xrONjxGmfo4vQRtn3rbLlUA+5a7BcCPtF2nR7nNxPJYLseBcGOdSM2yE6cwQaPg5GF9bFIZKOKqqI8OlpnwIcHjtxmQZsZtmXGgJPwABBsrZxDLLuwZJ8JH2vZfKR9n3it8ppMRHR/ClXe5IrvIL6dbGGPiq6iLFSOQceJAF2lYQvvDMub4tAHhRAP2TWHcZMaccJ0ZOxKTlJhpq3BOwDMJ2NO2mc9o9ygrDGDzzJ3kAMrtSz+Zh/AmfBYbgBbEb4PLIA7Yib6xULldygEDSy/MFdgpcDckjvWLi/hV3vavrnmesy9yGpE3OWqQnf+z/gpcWVW04Jwmw2Zot8eQG+p0yq8zBx3HYFGJFjt18hAfSC8AmgcT9AtwdZzzG6QGPOtlFK4H8HDM94Kb9vb0w7TpgXC8K+5tu2HE9pq2c5y387xjIKccYy7UMrN+Qdq/dcwRVKezJ5ncmmajjmns6P60Y8v69PyzT6ZnnnoCKdZVHFLoAle4dyJZLp+Bv2ola8xBZpUrlMPkOOU3j0lwIutF52eZLeXwj4fwXBP4lEmjxIxMGv2LhEyRsinPsoYIvCeOJWtaLspgOUoZcpnERxmf+133L+U71TOnk0OZVvmuv+s7NN9GqsW9uJT4pxOnHrbEP9FzeD4nCjeS+5nkM1L817+bawXm85grmc+AFZk1BzhUswu1nVu2QZfbuhsaQA/MGw699WMzMcEkHdXGnOcai/mGNVJuR9cTHkTJTBuf49GEM4uDSosXTk1L21vTzGltcchqLJOiIzevqWK1Vs2TtmgMujz4Ynw5xsQfXsJyvOBPQaVNxgzogsrbVGXYVJLZvDWv13YLOH7LLRyV27WNtLC7Ud0ts/rDH/7wcRUBfkTR+fLEsxGuvPLKiFAGt88SrO5W4jWf/zUtYB80GTa/XVvnZb6gDvZzoxHJVM8YAdUwCITIOwun2Agaw0URMSBH5PgiUeMbpR7Xgai7/gYgDdGrbhJd3Bz3smffhz7M9Zv2pNUYF9y6Q6YNJwLBvW5UzUO7GLH4IXxsfnSNzaMTBIQrNq4zp7QgZTMJDr76XWdy+tPTftZNokmOR7Z8WwLrYJlyrcMhfnJpG9/1N+tYD1/eTxanHt/3Eme4e/P7tdwCw3H98WUtcF7v4QIXQ/s8iFeFCCZgcnka+zHU/Wj4VwkbVerM5TS7zBr17197zdWc/uRkbBD52MBA9LBMbn48xb8XlSZbsMfxs5/9NPTjr1y1MggJwr6XY8fT/J7wl0DSCtFN4+TzUPvSuWRJOv+889Hxvywt7uhIs+ctCPVn+7v70jaYNJuwNbUtxiQnpF2gjYILw82SLBN5mFRFB0HDcFxyskZ7U6oimHJOG8Y+YQgtnpXa50xGLJq8kYoTc7SMQl0BzxhTg/gkShujJLdotamjGhlz2L65PQd/bYTBi4B558fTca6fhKAGMaj05WAUQ9Xmc92Pxws5tHFP5JdDnH2/9fY6Vd3rYUtLDI9Twgx1F069c1+WL51Kf1dekeywUDEWcvySTH19lueoIGwBq+bvGCiX35ZlsDy8ZyKouQCF+Oebr4rYVoiMlrNW7HjVaVTMMYzR8M9EBE+t5ys/4zfmsKikiTMnZpiN+ZUxYXycKVskxDdl5/bT8ZDbJtcl188YlJ/6eVsnyx5Xlb8ns70dr8HMiLQjQVIkLOEL0dK08mUb5HnTb4mMhlNCIBOmcxlKW+X2JCBxhrrl9KoSRfl9N1/rEvFK/cgjt0cuj+0XBHifhC3tkonhuFRtaRvFRfvZjn4NNoEe4ofKoWqSCB7tRPFym5Z6h1cJTrvltOttW+BmsJ3NIsqa4SankInGhjHPIGhHmNymsbYxoJ6Esd4F1vTLRPOSUj1dy0l4+4OSS0gOInNVTuNaVsPki/eoWvwEHERc8/WKdsnp6BJ146XyjSCDwSpX2ytS56ckM4iDSS/+Sp0opmkq/WOqdlnEERjN27wis9wuOcNa7q5TBzMxV+4qnu++egkfkZDhfa+u6CPDhLfu5BNwk9MtMBbJ6M1dBY9UdAqXqn/0zOXBx3JF2ar2MCQFyuPUpPwjQhTSfsrh7DOjBWzgn6uXw5l3RCnr7CqP6F48bTYLllMinYigW34p7VElalK1y1aynXWtEqraN1xxzmXJUWL8k6BE0MgXZ+vgbRsEvAZeKbjHeENzjHxwyrjQPOtXo/8i3yh8VTbe89gRfkttiVu1wlxtAABAAElEQVSSiGz8sU75mctrmFy37JpDxHjDPUcXBnK7m2UZS1GnKF6VZ1UecWOuvyk2LnGTLpEqCfsuvvApjspxqnGNe2g0IK1SvkjJeLlQ5TPKJgxZxkgVf4NEsMiQNLxwqNCyiD36ROfK19ez+srwEa0SsGI/Nhk2x3e5IHO6V27N0w1dwgmrbOoDtzj/CoHejg/uCjdVPZXhWF8Bvwb8+RXIx1l85GrqMNpT97Iv2b5jb1q/GbuEm/ennXt6Q9JPaUp2HYRyTSJzOxNsjyklx+ExGTYtSNmMOXYo7tEDHBrZtzNtWvdKeuH5p1A/uTqk1bSF6JXbyVJ681XGn546xSXe4INb+ItIMdZVHzsxmDBTUbeo6tsp2DKbhepZ7yydPClLKiOl6l7Mg3Oqpp02dVqW8oPmIWMm0jdt0o33kjWZRZ7V91A/HQs+tSa5jLl8pfi5EidPo0rc1ChDyaP+3ghxZm8l35Jmif3bpF3SKmmc7nN43qcbzxatX4PgEA1c9xn+3gg53Of4b6G+5FTilefxoRthi5/jbWj8MuMNtpewy6gRVx5lTXOEeUmp2YOHj6Rde2CMbkHKnnvHrp7UjQ2b/qNK1KgGDWlfDpAMSA8AtmK/QVLBrNF+1GikeRl+EyfAoMF21AJs1ixCs8acaaPQrpHnQnMuWn2cGz10ZR0sdR5r+TtDa4QOX12j2OFN2EGGje3lLa5Rrep/l5mPqr7OL+Gx3I7JcguXTYbN67xz68W3k5sMm3qLnPm7KD1w5UmjFsR/fCBRrAh2pMkvEx1EsPobTOKCCyIHZT5B7IavF+OC6snciH7MlWs3py3b94SRM21mDCjeiPolxTSPcirNeCLjGOASrl2msUEZN/YYJ2dGpwXz4OYj8jx/7oQ0dTJqI5gJOMDJhEAZOElgWfmJdNxIlcvJzbT81dWnV57k8m9uKd+tTHhXP+WDZyPJiN0IVTyGPxsh8ptpeZ8q3PB4ze/fRwsIV14FngPOam66lzA+CwgF3JQuMwEJQVwlnbwZyQECtoBnN/SeMtdg9Xe/+z0kZH6GcWaM/nLC2QX/Regjv/POOzAkeyuSYzPjZEqVKONB/c6qS+rBJsTG9Dj69+//xc/Rj/9IqOURhnJu/FLIllDT0YpKHtUoLUA3/SXp6mvV/34JG4VZafKUc1ILJ5zd1uze34/agT0s1rrSrt09qHrBACg6a2XYDHCiJjZhkbg1QfUAGzTO5TDGNPJ5JI1nvTRpfEtaMGd6WrZkAcyaqTBrkKxhnLowy/jAsVKH86jZkB/brrTvEI8hH7mWDafcf43v4f4Nn9f7W4FD6zEIZ1WlTuSXYXbkNhnJz0VTcc/9ceK4JU+f3oWwXS+b6ZXy6l7Sror9Ozzq/Z7LODxtv4eg1mG5lRSG1pDyGm/wwreCy+LqM8Z0hMnzw9DtQlWeKo2h6edUomiRj9/CfSPO0PCRfU6J8FXsKmUfhi7jZnjMXM7SDiWPiBVBc2p+D71iGzXEyaJGlMbPoH8jV9PLsNAo02CwwZdSCx1svcZlXNxKW0T75FCRu/3QCMxbzm+IU+1D33INj1ncI93Gx7C3oeUZ5jnip20d7a3vsPKGn+6EKXX0s37pPtQv12KwKSKwbobLMet+Q+PWUy5h663S8M9pDW3dhu/pvjXSrpfJ2JaruJVyD021xD2dMpSwJYWTxzHfRp58lOiVY5TLJHjJZcwBSlsarNGvFSThWPxLKXIsf6swjYwiyNByDI1VvvJzaH0Gizs00Gl9HR/3eBcTyqVuJJm/S9jiPgwv5MaqtUM9/NA6lBRO71n64XRCN/IsrV6QxAg+VYINn6oKAR/ZtVZuPIf38clKFPH5acCaoakLvy59wr9KQHgaTFtYqrn7mnFIxKylV4UrgSNcFbF6DKY51Pms+6q3n++uM5oMmz9UN7vGK9BdA86aa5QsgmSYFkfWALuC9zxaDMFWKQ6K9CGVqRRr196etHLdzrRmYxcHPlFnia2bIwOqMoSQzOn/UdIM0ACgJN1RVbOyR/GQ2EBfT9rXtTltXv9y2rphZerasREbZruQHEA9J/SBwGjCTxmnQfjlcAxuEpYdvXGQpNA0KFse385n+Vb6VPtxV19zTbri8suwxae9uins6cbFoTulldVuMI6Dc2Nb2tibqXYSqe1KclsJPS/Hbh6/PsMl3KNt4zsaELfy1Du/ZyzjV04r2reKLdbwr+5WeTUfp2iB4yE7t3duTd/rfVESw3fQ+WTtTiDDxb4YWhXwmvvIOO4ovHNapX9xqJigjZ7OcUyKP2Ay/jwcEH3uWHJPwxdApUSN9qL7udWe0YP68+2YJ1jPuJIGsBvVg7ofYd+vEvM4aEJcx6PpRqqkp70orNlCi1OSrT9N4nDmQg5rnru0nQPU2FuaiGQNpDtpcVEbAZo7YNxKjHAZJF+DL8Vh2NNG8/Jp2FOFN2zzeq20gPN1uQvOC9hsMmxeK130+ymHndxk2Px+2vK3TcU+KJeDrFxlAGYkWjYjeWDmMPn0q4hfLRa9yGHu2X+IE/170obNXSFpsx8RaG3aHEHtwtEgFMO2VxUT34WQGDiaBMeOOYoRQYybTUTcGT21CxdMSe1zp6CnFrHjcejj9uQ/gYtaCyeboYidssVEaX3y7e+QcDqEi66lriV8ePETgQzIVcLU86q75VBD45T0SpwSvoRtPl+NFmjAK70Wiwl6ReCsruGwrXMJl4nQGcYbQ6D0YwUFLuaFh8EkR4Xe/E2bNqeHHvpl+j7SNRqQ3Y2NAtXQqAZImxE33XRTehcMmysvuxR7GrNJDMYJKjh2I4WzZu3a9MrKVaGn+zcvvhAGvbdu3YKe+IOuhdgIqOdd2zRTQmKnvX0BabSnxZ1L0/LzLgjjuBpXbm0djzqOFlQUHk3b9hxKG1FNuEVmTVc3CzjUJwWjBqkaJGsUgY4hH1WRAYo+eU6vjQ5mTT+M0wEYpW3BrFk8H8maudPS1InjU5tqb6pRoxROOWlX2nf4s97e9ffh4f47f58I99omxe9kbVfClDYsYemdGpyOnNbJwo6UruG99fMvMGj1XdIq5XhVn4Pjj1xqqLXuHJUfdCgMG8tfSsZLwH/+Nmi0WXxGzdjIDAYukU75jLap5tNTtUlpy9xRlmBY+aqyjJTpSP2Tw+WaHB9H3OV18jod73ui9IbmkNsuw8TJ8miU27qa2/E5NtpjaB5+5Tpkd3tp5PjZ//f1W/q09Gd5FvdT5WP4EqcettEWDdeRwjV8m29n2gIj9VG9jUsfFLfyPNN8muFP3QKlrU8dshHitd4fpU7laclLmcuz+JVnPczwd7+96mH9Lmn5frZe1rnU0/cmw+YP2dOnOe9Xa508DzMfNxZYg+vEMmcb1AOgWc3s0VDZtLXrYFq/RUPou8K25sHDqLk90gK9QA0A3uxVlLzhkOgAdvv6DmELcOcWbNasTtu4ZdYc6t6Fitke9jBZjZtyOoN50oR5lSBB2ts1hLpJKrK9DuWy7OzxJKpPmDAxXXrpG9M72Le99S1vScuXL8UNu2aqPCQF0/CQaVYp6gHUHM9nWQs1knWlgv/geqdWOqPFVXOrfesae85wK4FzevmruEWA5s9ptECdYWPwxsrYls4tPjQZ27hq59JN5XtowNqXuciw4Y6rMFt4ksZgMvqRdIGZ0psN/yrvQnswcMApMWLscQiabywQoOkmH5revJXxxL19xwGk2bQFSUnY82cTBRU8kUHQPGTUkI6mCbQT1QINYDx2oqZxUHoetLdFC2ek9nnTsamImmP2/8q8WRMvcXW5s8vxvwWfH+/TdDmbWkBYLHeBCZ/CWFPC5izqaTu5ybD5r+3QjOhPnGdBsmUAhmoCgmfVD84uTid5alPvs1IG6sEuujP3H9DA+e60as0W9GbuQ28t+tSZUY5i4Gz0aEQxR7MQYxGmzu1M4vHdBRDGd8ciMz3Qy6QxOrXPn56WdMwOlUzTYdqMx+CsRs7ihD9lyEsjylLNbs5jcQXTpkyCMR9md4NWZS8qFQY9SiLhUCXIe+PNxAsJbzCjHD1CNULmWH7nOLUSVOGbj1ejBexbJwjhV8mA8u0zRH5x8zJMcSuLH9XQ2G95solg8V36MhbcurAwiqBCH1F2de1Kv0Yy5p57fpzuu/deTrRsDrsPbjC09aDO4vOWL0+33Hxzet97b0sXnH+ey7i0f383+pZXp0cefTQ9/Miv0vO/eT6karSbIDhZjrHAu7YGNOS9GIPXF138hnThRW9IS5acGwyb2XPmpXFsIizHYSTcemHW7N53KK3atDet29SFMfa9MH4ys+YYG6AUYw/dtcCxIy+rHNGoYN8gw2Z86wCqCMek+bOnpuVLMJTJYu2c8dgDUbImYhIbsFbCxst0ylXwRvmuP0/mVw/33+1dOCzXmbaRcYfHN43iVp6mPzxt/epu5Xt4HL+LX0mnxCthy3epx6v+bDTZENQ6knMuo+PdNqgHzx/1OL5neM4b4oLtz6Q+pU3OJE4M+BhHboBKzPJSnsU9P0s+x7d9rsXQ0H7l+breAseHGcn3ROkNjZ3bLc/mI6VSQpdyGyaX3foJYyWEz5JnvT2yfz2YvXSyvHKM3/3XMnuXti7P4j48h7q/fn4Xt3rYRls0XEcK1/Btvp2oBYa3pe1Y3OrP0r7lWU9vJLcS91Th6v7N95FbYKS2HDlkw3WkPmn4/mHfSn3Ks16akcpdD1f3r7+XNOphdRspTAl7tjytc6mn767TmxI2f6jeLXPwyfNvwGk1FzcWMIF/9c9/JZ2Ml9179LP/P4TU//bdB9PqNZvYO+3CHi5qztAG0Ns/Gjsc2c6mqptcIPQjlbN/z860Yd1K1KCt5H1bOtSzO/Uf3p8G+g8gXAPtAKaN9jzyGiKvnfLKR8kbbXNU5RkMk0NaOmEv5A04UDcRho1mFN73vvelW255Wzr33KXQJwxUbQBJJ2qs1EMsSvjyvTbvWGa/4y/axTDcVZvkPP0t18htbk6RRQQzvi+qpeK9eZ1xC8hKabSnLVn1ZbiO3AeGyncV+VRNL6FApmCo+zJOLT6fJa1wrpeG7HPZ+K1gJ/brfHr21JJmHOk4ym5K1hxk/99z8Ejaivrzlas3oV59DwxRmJyjMPg0ehy0t9EcnIbWCp1NsDEN7VMOYFNR9YdqsdEmbWuLhzVb0wJob0s756PlZnI6ZyKHpdn8e1g6WJPkF2Olgr9cnqjUcT8n8zsucNPhddsCgefFazXYCBhrMmxet306YsHt4CbDZsSm+S91tB+8hiPYMhCdr1x4eNXDGsuJJG78tWnTx0/3gV4kbPandeu3hx5NbdpIVB6AaRP6KbWhAddfxk3otSfuaBg2Y0YdhpzcH9I0EyEUz5w+ERVp05lAZqY5M5k8JkA81oYG5c3yPTGvDSmbVcmTEllZ5KhbFpDONVA/dKy+8PSyFt5eQ5eXxTXX3Tjlyin5lZdTJaQuJT3DGKceT//m9Wq1wBDYpN8H4ZcMhe0C33V3ezDco0vpO/6Fn7BDAGAXRqVhApRIK1ShcXTl5VdeRhXad9NPfvrTtHbN2thguhk5gsjZwAA2XhCTnzz5nLTiyivSh+/8YDoPhs3u3XvSb154IT3z9LPplVWr09at28LItQaixYXCWtu4ViRp5qQLL7yAk16XpAsuuDAtWLgoTZ8xO03B8OWESZPTWMTx3ST0sxDrPtCHnto9jLWutG5rT6hEO4J7P2NOtQKOuVEwbLQ9kMX+Xbi5iXGDwxEc9EK3IlkzY9r4tHD+jNSxcHaaMwObNUi7tTHe+A8o9nyZqdlUJ5KwKW1MkOZ1ihYo8Gqwk7Wb4Ubyr8cvYvIlrQyvmfBR4hcVZ4YpcUu6ftfddC9pGP5EV4l/Iv/fq7sDUPTqJRA6UKurOJfv4pMPHBRf61RCNJLSxc2QoYr3b8OwaaTcaN+620jvhfmpX27/Ch9F4FKakWKO5GYNSl1LeoYjnaj48ekd75LTtSy5rRrpZZ8T/dZbb+QwBb4sTx1udC9+9f7J4Rpp1UtCCnicqPSNOL/rWylbKW95mm4pc3mO5Kdb3b2Up8Qp3z5HClf3b76P3ALD+6ge6mTtXG/vkcLV0ynv9TjFrfk8dQucbvvWU3ott3WpT3meqtwl3PA6Df82nRK2pDlSmOJ3tjytc6mn702GzR+yZ51puePfd+ewE5eH7ooweTrOAe3DuKv1yGDfEtLwHpLrY/9/0INmqEfbjDr19Rt3IRnQzX7mCLSEsXHI82hI2BAeAnPv4YOpZ//utHf3jrS3azv3trQXxs3B7t2ooe5JR/t7QxInCNEQC+PycBkaBKBMmHPclsv3KGnjB7dwRvXZhHTxxReld7/rXek9HLS76KILw8M2KHeEJYVIKtqn8sTB1OPHub/6M2J2j99IKLLOCVURBj94afgOpldC4WB7WpbmdWYtYFvaZz6j+QrzLVzCd4QES0PniJnRUnqo+BnN+F7lmd/9yr1fdzced92pipmd/JVWBXOFDDOjKccJe0/QJjww3X1oIG1DUk2G55ZtXRXTkwOjYZ8GOlvQ2qDNkYB7RA+TZjtP7P9hcGbJmoE0gUPS0tsWL5yTFi2YmWbPmMDhTUwSsPmPQ5oUR5pbaLipAK+MaQo65DqR+5BAzY+zpgUGcT0DK+MlcVNTwuas6eBSETu6ybAprfGHe9oPIyHZGIgWi3nChYczXXHLA9P5xoHpZBBCoHGYwNMz+yEiuwhbt3FHGD1XXVpI2nByJmHX5hhE5GOhqolpAMLzaAjIo0f1khp6PzkB0wIDZwJKM2dMnYhaJpg2MG7mzJqUJk9qCyJyTCKRey6bU5kXRaQuuvl0mnSq85R1OV3gWYFMds5xnBi9fLrYyt+N30iJ9HLoCFr7OX4izulkErf5eDevV7sFor/s/NolrLrxq18Srb2ypA29w3dhygRsB4xHiAgX0FXrerNQEmbLli3p4YcfTv/+ne+kX/3qUZguB8BlMEBcSHl6hXwDDvnp6FiUbrzhLaj5a0cqZ3d64cWX0iuvrE57UIumDRzLoIFKJXJmzkQMub0dlWfL0iWXXMJm4aK0aHEHjJppwaQZ08LYYbxoYNBTM/thhu5Akm3zlu0h2bZj30A61D+2Ys6oelDSc/6OMR718xQaizVEoMeMxm5Ny1EMa45F/Hla6tB+1JxpqCeEOcqRmhYGmgu1fMLGUZD/cksPbW8bbCQ8UjVk8zGsBYS3cp2o3UqY4h/4qREtojfgtuFheO8M52Bp+xG40a2EL2HqbvoNcSeHY8ByiVP868wfw/+XXLZXqWKg5Ua+xfn4cjD+B9vZNhkaIscrWFzYruaOocHO+Mt2Op3rtciwKWWvl+106nKqMCXd43FqXlcMdx/+XW/RRk+dKtffzd8yexcYL09TLfWp+5fc6m71OHX/8l6eI4Urfs3niVvAtq639/CQ+tWv0s7lOdy/Hrb+XsLX3Zrvp9cCp9vG9dRe6+1d6lSelv1EZTbMSH4juZnO6aRpuLPlqreP765bmhI2f6jeBV+KUwcXW5bj+LVTKV1Gr6ycYm2VF1gR12SqNGK+xktMbHjdPZ6mbds+LKXv2n0gGDabUZG2CwK0EgPatelHpbr7dekLMm2ODfQjbXMoHejZm/YhcbN3z/Zg4hw6uD/1HjrA3RPPvsOHOLCGhgGkCI4NHCLPw5AViK90jQWIYjImo2bWxELl0o4fNy4tW3ZuevstN6c7sEF6OXZs9I99ooSHiFVq5hd7ymphSSo5fcPEv+HrNADz0UN3g+S1Z/4Np3Atbz5z+1YukQHvjSQqj+bjdFrA5mu0pzjZWBkOfOZbt5GvwMu566LvhoYqMGF6BvLWze+R0sa/cH+qhEr5ckrEooA5dj6i7IHLfkhkhw4fAT8eSjt3owJ9+wGYNbvTnr3dSK45pqCxcThTlYKh/YY4UQolf4B/x5CabBBNwxTBMbRotEBnm5DmYz+6c/HcNG/OlDR+3Khg1oRGG0tAwQKKq8Y70bxlNU7mV1Wz+TiLWsAxUW77vtzO4U2VaGdZRzcZNq/dDo3JieI5aVRz1ODAtNQxMPGRJJ5VoxGWBY1zkEwbxTK37dyPrY6taePmnahsOhjGzwcqHbXHUisxNTTo1Mb0AuF4gEnkGNIJLcwULaPRZAvjZtKEFtQ0TUNFWjvSNlPT1HNaIDKTiRkzCUmSHhMEyWqycNbjygQnpyonKhytBERLF4CNGkVQfmKqrD7ydJndcs2PD9+I53SYr3oaxiuKpErY5vPVagFh1QnCq0wYxa1MJj71k9hcwrZg+8VvrxzedPLi3AV6Iw3Thp3Ior+rqyv96rFfpbvvvjs99OCDacOGDRH/qLqWAebIR5cMOmncuDZUm01JbdijOQyDpqf7AEwfVJG5GBMuCSizZtGihWnFihXpuuuuRbLm0jRv/rw0fuIkQHYM6y+YLhi3FNo9caPNqB7H1w5t4WxMW7fvhHlzJB0emITw/wSSFMatF3FHIRvjoAROHWfHVB1wFGk2GDYTxo+mbG2hglDJmtkzJ6WJuKnXVugdSxsYJzNtJPiXsVPKTqDaZZ2GX7mO2XUk/+Hh/zt/19uq3g6l3fQXdn0Wt/Ks+xlXJqB+xV+3Ave+F7/iX49f3Axf5uiSr2Omra0txk0pRwlvuq/a5VgpqFYwq8FacR6at67GKb62xdAQfjkqcsiM5SPp44Od0sW28LItyvupItWZIo04pZBD++5UaeXGKXW12uWd9KLiJd1GSsNdGnGMUuI3wv8ubznt4/ugnmcj/RHCDXrmfqKEgy6v5ku9fHU4r7uX/It/3a+4lTDN5++3BWxr79LO5VlyqfdFcTud5/B0TidOM8zJW+BEfVHauu5f3E6e4h/Wt17eUpKRyj1SOMOPFLakU+KcLEwJ+3p/WtdST99dazQZNn+oXs3ry+NzP35ONgzdxeUao3r64Cq76PyVf8uKwqc7Ko61xX6rt+9Y2teNpgAMpm/cvDtt2LQz7Nr0qbUJKYGwfUt492qxXYPwfOTIYaRuMpNGhk3P/j1p397dMHK6eO5J+/ehFvoAKtOOHEA7FQwcGD1H+/solzlbAm7ScfUXy6NwgirROjYtXLggDtl97GMfTddccw17L3ZTrqdz5uzBcu1UTeaeyKhDr7w2ySsV3/N3jpW/Swjj5fehKfiV27ZyrzKJsup0okhV8Obj+BaILq56KzdfbtTG+/E9WU9lKEyXdaghso+x825CP1PVvewwDMcVWeBXOrLKMve1HxlCg6Zmauzfg24FnA1AVzvUeyzt7OpJG9ZvTes3Y69m9xFUohOLcAMBi4QjnqrQqqQZYzIqOQgNg0bJmqP9BzlhfZi9fktaMHcm9mpmQ1ubgWaNCahiRyMHEdn6Q1fLYyNrtanXN2oy4k/B4yN6Nh3PuhZwvi63fV9u5/Amw+Ys6m47uRCDXnnllfT5z38+/cd//Adqg3bHKfVly5alO+64I92K8bdzzz03TZo0KQBDQlQhsp5FzfEHqYp9UK7hiDYGIZ7Sep16YioyfPw7Cemip1MUk4xJ8ekpAA7/hxSARs+379qXNqGuyXsnJ2nUUztwrJUbZk2oSVPqhhRGcXSAdEzNddFoJpjR6thsOYZkzTgkbKbA/Z+a5s06B/VNE2DkjI0JJZ8CMOscN5fLYok8SBL3uChXhNAxe2T37Fl79zWnZar5HuZd0oxnlf6Qd1umybAZ3mqv1nd9wjAPJ4vhl25Kv3gLGy2oLCu4xG970XQMF7ADDOZ0Gynt27cvvfDSi+lnP/tZ+ulPf5JWrVyZ9uIW4YinhE0pi6Aj9IxBAXJbG4sg8tB2kwsxr5axLWkSDJn58+aj9uyCkKhRFH/J0iVp9uzZadz4iWwSHCN5Y+MCDG1r6QCna3bt7k6bEYHetnMv+fek/d0H06E+0h81JQ2MnkC+bDBYwIWuY+Be1s0oxpOna1SF1gpzVGbNjOkT0oL2qZRhapo2ZWIwRx1vLS7UlEr7/9l7Dy7LjiLfN8t7295KDUjAYCRhB6QB1EJIgATMHcxwh4H77nrrvQ/Fum+tN4Z5wzAGBiQGBDIg5oKwQgxeQmrfXd3lvXv/X8SOs/OcOtVdLbraae+qfTJ3+oyMzIzMyIgU6Wdsn4JRkzNsal3AauM/1DF/gEX+NPrnfq90e8AKM4dTbgdGNfzKABZukUaEI25j/CyapRX9grl4UbrCZ2aET1NTtmGCiSTY7OysMRVv151Mr371q20uhnHD44tosGS7H+GSoRM/RecqsqzHMoIxfoenhzc45AGLADj5S6/xPluLG2lswcxhnwffHP6Rs4cu40fuF2+7PI8iBRl5BSOEj21es3BzM3IK17IMCt3oGYHMbJZPXYBt/IiChbmNWSnpgMnm7bgx/4gTPpcTN+JU5tYg8HJg3RinWU5VmzWDyh/n1gzuOZxz/9z9j8v1+oid1y1KdLPVMep1uSawCVhghyapGDaXC8UrFf7yGDbkWlJb9XNySYe5O98enjj+ZfsFkgxYWFy1ezfHLsxI2uaCXZ5+fnxG+wVizmjto8WQiBKZIkw4TGcpaE2DCrSVpYW0MId0De+cGDm6D0ffszOTaUn33CwvTqX52WkdlhtP02LsLC5IhdoqGj1YE+llTchaRf/Qtbt37kzvfOc702c/+5n0nvf8mbQbdNp6DYYNB1mohdeE8pS0b1BacjT/RorSY+LqPoTb8HjC5my+fEew3B5uGxKoHDaHQGCdh6gHYQZ4edd/5SnmjRDuuHnr4lLigfv7gUn38YSVc2RuGUV8mXI3ZqbW7cF4QTUghzSnpxfTGTE1T546rztrxtL5iRXdYcM9Ne3OTCyYOwruTEUbV+nP2lvT2r9Ve2xtWv93tK6YZM3unUNSg+ZaNUaGerQnoMPQMCetbF4m7GhnMCy3snqdmv3GGN7Mr3K7OSHAfB0v7R8vc3jFsLmJ2pxGrhg2V7dBgTkPnSrseQliwA0/m95s8LZYmpP4oINiQGQzmOunSA9n9qN5obEQeZ4XIXZ2bCq9oDtt/nDsnDab5yUG3ZoWdengqu6yadX9Gqh5WpZkDZt/bXBrmNU4FaBXoUx0s6dQkcZl6IcPjqTdUpGG2jQuTeMsAcSXny3wMsXAUZsZVRYrX1FeZbLhqVVVPuWkm7sShVR444nvMHGHIKsYNgGhq2mCuzBleMCnwGXGGjag5+bmpKe4z94OMURaxVAxvA8cBrmLJve4/sEE9NKxY+nbTzyevv7oo+kH3/++xJDHNYYta4GJdJji8S/T4ikaMU0dVYHSKlHqFPHf29ObhoaG0sGDB41Rc8/dd6e77ror7Zc6tN7eXus/6KiFcItLAxelOmB2YdlOnh2TztrnXzxpjJv1Vul7VsCVNXTWDqov9Vi+9Eu6kp+OERNURNu6FintbauSUuswnbX794lgO7wj7dzRJyYNJ2okWSNmTZsWJcbg0YKoDWJNhCDYDVabai1ysLFAjtkT4wdOAffM2+GcO1R2g0DAKkwcA5aY4R5u+OMW7rk9wkcfiLDgL30AVRGYvEtLS6biD0bN/Py8MWuQIDtz5kwaGxtL585pvJYJw2bv3r12eOKee+5JIyNS0dehk496gulpH9v6AwbybnwaXZm3eDgp5o8Tku7s/dk6iTwJUgsmu/VZj3RZv9EWeaS8vXL30l7mXsYvS3Dp+GVKG2sSfjGTFfUOZ5mNLmUZSvzLghfWsswb/a6GS5Q6zO3NE5hcXjuoJZhDGp7LTaMhevW5CQQaYb1VODfGa0x+q+k0xqu+N4dAM5jncM79c/fNU7yxfG72+r3c1gAu0d7YK4bNy4XklYh3ZRg20GA+Q9fP07hHP2ApDsOGfQLWMMyac/Mr6fQZSRAcP59eOikaVIfTYNqstaDmSZvYROIwGpoHtJtsmhBYf0l6xvcLyFe0rr5h0izOT+mOmynRuRxEGksXxk6m6ckxMW2mFEZ7EWLwLMzNKsyCpHHWtA5q0/0dA+nOO+9Mn/vc59L733809epeG9ZtLeXpUJWVRR11w/THKS1qkdNchOGNx+2spXJX820gG2r0a2PAxu9IujIvAQFwW62xBfg1NEUtXW/d2mdhwTVWHdH6RQrCbW9p4YmtmfPMoy8I+Wlsa3DHcdSZrSj8ou6inV9cSZNTS+oXkkCT9NlJMWsm0Fiz1iUO44iSRQpN6Wqtzn4D6axxQFMv+2Ic0mzRfbVo1ejS+n9I99Pu2z2sfbRdkrDZkUakEk1nSoXJ3mdBc3YfkCRjP84kyawW5Vq0EQIxfje6V983NwQYy+MFB+KtGDY3WbvTyGwa0bCVhM3VadySUConjXCjBPmga53QJiGGbh/IjQnBxFDMRTbHWFJMEEyGbDGTkCYnheNLNJBO/0uV1PisGDbn0wsvnU9nz0+nWempXdWdNtyQoW1hSQe4yHFsCNvJFxg2ygQVTdxvw+n/0cFuiW/qXhuJcO7SRvPIYFfq7WrVRjTzlDKzOBBybMSrqD6XUigrU+gFpZjxWBWKj9we/pubkXhuEpopGkKuJOY2T6PyuVIQMJy1BhcaaFzh5WGcQVKAcYZN6FtvvTW96lWvEmHerxNVMCOE3+CsHr/vSKSLEGFVIi0wYtj8nhdh/5Of/ix98Z++mL4jVWjHjx8T0S/dyMWEFX0CNHN70WeEhDCFwAlObw0MDKQjR46kt77trdKP/BaTrjl44GAaHRmVVE13TapmRR0H6ZoVHZfh/ifugTp1VifPJFlzRgzQqRktSKTneYV7aqSztqWtW/YexZEdGKgv0Hc6jAHDCTLdWaM+NNDXbvfVHJZkzb69A2l4qDt1d0naSGGD8Yn4NJWwkzYqf5wQ8slYBBzcWKuojMt48vHlMqLddEHBmRwWfG/2BH457H10CreI05hW+IP/4D4MGU6swowB/3npD0iMIdHKyzcMTZgzhOclnU6p8hseHrb7lB566CFTEQEO445/Xq4oz/aYMcZuTL0RerF0Yn7yx+HGqFzDW3cy7zzlLEQRd2sGsGh88nZp9PPvMmePrxKqv/kTZvOYG13LtOr98s2DRp/4hugOu5tlOerdfXBrCNwYhG8FIVStOs3CvCy3gEuYLyuRLUcKHN9yhIaAOV5sDtOGSNXnliGQw5dIfyyMG9OLgvyx6UY6r2SzGWwb4RphGt1fyXC72etOm0d7Y4duqSRsrlWrM2uX83v0R+im5nO5z8OsdoLuipK7z8Z52tZb7A6orWHYsLrHJI0VLXpg2lyYWEgnzkylF4+fNS0dHPxc1rpjvXY/B4fZvKjkwFqldkOt4ZDWbiu6I1RSBd3Svt7T02Fm0n02s7Pn08SF0+n82Kl09uTxdPL4S+nUieNpamJSayDd39HRaeuyz332s+nBDz6YRkZHUxeJABcyAxAGjDgiirNDwOqkL7xxKSLIjAc3XPlreGpgd0uNfrWAWejM2pBC9XlRCBQIc9Ew7ulthz0Hdq2BihTwK90Mr+1TuK09K3tsrezr5VhH4y4UdW/F1w6D7Oyy6RHDkP6g85lJ5zNtnc+B55OnJ8SwGddaTVpqpA59hftqpE1jvW1AexXa/CKyFZX0yN81aqzrvpo2SdR0dawZs2Z4oNPuhT58YEfaKwmbgT6t/zs4KM0+G0kQv6h9dHiZrWyIbPLE2J1752N67l7Zby4I0M7xggfxModXEjY3UVvTyBXD5uo2aLNBFLd4GgdeY8Jo4IfTDgPCmBA2iKtj1qL54O6dlolC39r8JlmCID3DBDSrUwKnz82kY7pY8PipC7pccCbNzC7b5ekrazBskLRBKsVJGSuWNp2ZJlqY/AqRzp6uFjFpesSs6ZeatAGdFBhMu0b7JOIpsWWiG8MGaQGVNiYwyk+5KVexLa2Q9jROQ43fEW5z02op7wAIJnV4eZvam+dT+VwKAo6D3g7YYbQwxrBB/eyzz6ZnnnnGNqbf+ta3pj995zsk4YJES4+1HPhhOAvOFZI5S1KivCrHWamJgtnz+BNPpEceeTT96pe/tI1t0jd8CaSxrJm0ipLat3BPiEj6SM8gVcNdNfff//50lxg2+6QSrUdlaJO0Twv4r3BI1yxJr+CSmDYsXs5PzJr6s1O6s+acmJ1TUjW4pAs56TdrklLjksGkUzbrqBoUAwfiD8k0GJ3t6r9tuguqS8zO4YEu6zOH9o9IveCAMTu7OttMugZCzQHgxCN18HpEZdykbzMSGEfrUg3S4N84vjR4vyI+Y7zNYRFuOQACl8OP8Lx8Qwzxhh8muLggBmJIyrDhgXozzImJCcN7GDYwZ/gOfyTOeJG2IX3up0H96ODgoL1Igu2UiggYnEiBoRKNO5dCJVqUKy/79thZAHnfZnzNn3ANt9qCw/A0XAv8zeNmyZAGL06Zc0S+pBltEQGBy8WfyNFDeXxvY3e5VPzG1OvTK32tZ5efhS1PnXGvhK0HUEliAChi5IZFyB3q7UowQlwSDPUxL/KVlzi3XyTKFfCiXS7dlptnlOPFH5PO5jlUPlcSxnlajZCt2q8RIpf33Qy2zWBKuGbul5dbFfpGgUDe3tgrhs21bLl6OqLss9Amzcrljuqx8iznfUK6T7NIhBY9p7VWMGpYm3M/p5rfmDYcUrswtagDahNi3FzQIc+pNDGje2u0LkKigFs2TSrHNpJ1YDMkbvgWbcyL+rJ2rXvYM4Bhgxr1vl4Od0orgu62QfpmekJSPGdOptNi3Jw9fTpN6ODSzOSUNHjsTh/76MfS0fuOpl2oqNZdpAYLJe/rOa2zbD+hhEltDRWUpAGM+gcMgC2h/Dv3qaVihFOEwNUcat5mqaVb71x9XQoCm68hyjbyNBzq0W6XSrfwB3nt0Rq6YNiw32R7QYanWXoKSmhwCgYL7Br7Vtsua20/M7+cxicXXAWaGDVnzgr/J6XqD2kz1vhI1bRp74KXw85Ky/sndWQNzxUDXG6zZCrQB3VQc6euEti7azAd2DOU9uzU+k7MGu6ptV0xJWDnSimH9h9I0A6xyqRatuZXio1Ps3k6xoxmfo3xq+8bGwKGv+CwXto73ophc2O364bS08AVw2YDWK4rB53x1/DN4M/DlFBMPmbXp3VUnyiso/qUI3cXp4QAs81vETZSwZnmdGHauIiwk2cmpKf2jHRxSt3OnE5zr0iz5rqkA2wDWtGVvqUHx0V5wARCAoBTAIh1Jp0aYAN650hfOrR/ZzpyeLcmoAEjyJh87LSA5jQ/MYBoKMQbaXIGB64Om9rlQ82aPZu5NwuLWzHlyqa8AkabBa7crygEgkgg0bCzic2m9c9+9rP0L//yLyYZg9vRo0fThz/8oXTXnW9KO3bsMNwA58ATY06yeBDBv7CwqO816Ys9nb785S/rjq1/F7PmV7b5bf1BfjxIc/FNvryQQKaOTWmQDpgBOqBK6o477kgPPHC/Tm59ML3mttdItZQYlSRCHN3PhQQYUmksWJBCOyedzn946bQYnWftYk5O3axIMg3JG7FjJFnTqX7jKgXajHCjz3DBoKSlxahpFdHWrhM2MGuO3LI3vfrWPVqMdIvB2aaeIKkfBffeAKHmTABjdsrR8Zmy+SLDy0n45vgdcCdGs4c++Ep/AkY5LMItYMM3eMNLG5jUoWCH+6oY0ivLrtoMXMaNeRQ8R30Z78mTJ9OLL76YTpw4kU4Ld3lh1MCUIU3HV42PwjcYNKgIBDf37dtnDMVbbrnFGDMHpKIPPxiNwaQhDurQ8jTCHuW/8iaYp06hutKP/KfEpcDLjfkqzOaeFjzaIYKVqW5M7cq6+FjhBWRuitTDEma4X8qkBlGL+rC5a7NUAWt93OjfWejMWp/6xi/HZ8+1rNdm4fK6bwxDlUjJ2+kyCtEkqe108jp7DoFTfG3mvp1leSWnncN7Mzjk7bNZmGbpbCXeZulV7g6BRrg2gylhmrlfLzBsrMNWynU912cr5d/OMHl7Y4dGqSRsthPiF0ub2dbn7gjl+L7ZPB1zcrH+sdi2gi9omvCP1DCZ1FkXQdPJEJHghzy1ptGeAfvF7G8vSYRmem4tnTw7mX4vFdBI21yY0oXpOtyZWrtcTRoaBrSSadWhNVQ3W9KiizFNZZoOrCWphGLjWteGSqNAV9q7eyjt3ztqhz77e6XlQ6qi52cmTNLml//1X+m3v/q13NbSO3SwjsN1u8S86e7utjQ5BQq9a1oToMmNIPVf9h54yvFB/tTNXAnr/k6/mmPh12gnXISVCYHmRJrBCpVweRoeu/q9NAQKnLOAtErgZm73VAL6eZoRuubWLJDaaR2cM4YNbe57A5YX+2FE1g/NyV5DMGrAeSTGeDmkeXZsNr0k1ecv6RoBJGymZ6WNCEalNGism9SO9gradDeu1v/sazmuK32pNU/cWYMKNLTSSLKmu3NduN6bbn/VwXTroZ1pRCrRkaqxkgnPo/akwTzF+GvFtH0NbN6fzTH7aTanlbivaiqt6rm5IUB7x0t7xwsOVRI2N1Hb08gVw+baN+jmAywdkYkHpg0Pgy9DvE9ATD1MOrw24NvYjAODvcdwMoZJq9CkKa95yXpOzeryNJ0YOH5S6p2k5un8xLykBsS0WS3S12kZI0pqAwBJki6EGJen6V4bbUZ3d7akof6utHvHYNq/Z0SnB4bs8vTebi5P8xMDSOe02uSpUtgEok1uq4OS09M4pTR+W1VyR69a08g5QZZH8cDV73ZDICYOTCYMJApg1jwhyRje3/72tyYZ8MY3vjF96MMfTB95+EPpyJFbjQB3XAbnVUoIFVkWF5ckCXY+/eRnP03//M//nJ54/AnbEIeRA17UtzFf6hWGHz5xkYaVyR1t03vfvr3pz3SR5Sc/9amEpM/g0LCILPBd0fWjA2SFSoB5SdVIFFqny86hAk19Zkn9gz6CODSq0DhllqTb2U56KV6r6ayFAAPnV9QH1lKfGDMwNg/o7qeD+4bTrp1Io7WL4em9QCmwBrH62lhAvayfCIbqczYBWwCrshWTvk9Zmz2WRubhaWUOlbUGgUZYGa4IZ8BdmDG83DmDejIYMqgwQzomVJfhhh11ZqHWDDfUmiE1Q1wWlzBYYLgE8wUTtWaoOoNRgxQNL5I1vPjxYkf1GUyaaMdIL5g0hh+GL7VqbYOFTqXx3+aAwD2ZBQri2/y5ZACPlpW/iNE8uSvq6mND9KqyCFGCMLeaKVBoDolG18aUG/EQwNb18cYIlyiSp+e5lvVqiCTvcr7EjzFzszD44dkYoCH8NfpshF/0FYqT++Xu16ioN322ObwbKxvwz8OEW2PY6nt7IJDDnhyawZ8wzdy3p0SXn2pjHbaawvVcp63WYTvC5e2NHfqnYthsB6S3kibzdiPFcLF4Pier2WqPWeXsPoV/lqYvh3ydwhrJQuqQHARAqEhjA5u1EHd4TM8taY9gJp3WOoj3/Lgkx2eXtB4iPGrVi01sbWTbqoYk2RxXei3GsNH9ndo3aG+TtE13q9ZE7Wm40NKxaweaBrolhaPdhaU5SdhIPfD5C5LAWRGjZqc0EuyWVE6/mD3aHIcG0X9sbFt5qbXyo5ZI9NhnDRiEx8efenoHt/CLnZIiYA1WVpGiOWSvwYp1H3sl1XN5ECj3pkrYW8ttSMZbst45WstcmwUwN7Uy+016PQj7X7RVmQ/uNCWlIRSSYstC9hnh+bjw/My5yXRWa38Oa07qgPOspGqWdeczkmWB67YU0tquRZfP2nykBP2gJQf80K4hVYDC6dHhHjvMvG+PVKBLumZkSLjeqTWhisTa33bnvKCejtwoKocEHXdZp3g488C/eJrNZzE3xpjeLEzEr8wbHwK0c7w+3vq4WzFsbvy2rasBjVwxbOpAsu0fMYjmGeEWT/3gSkeUyicjeKAZnJgygsiIBZ++LLolwY9CyYFJgMHe7ouRlTDYuTBQNJapepqcXhKzZlpSNudM7Pn0mCYmnSzgsjU2oRHzJA6MG07fsHm8tgbhtaKJCVVPqH3SprTs/VKHtktMGy5QO6CTMzu0Qd0rEehOSY1CpBnDRuEpsWQKFA/CrtlDQQljAT2Avs0FR7O7c+03Syi8M6dasMqy/RBgkgCfMdnc/qVUlz322GPp29/+dvrVr35lm9pIFOzduyd9QFIu//N//o/0pje9wTalwWtOvHh/AIOlS3luXmn8Kv3HN74hVWiPpJ///OdpWVIKqyaLL6zIcYIjYXLgz/DF8Ag7uO/EW0g03HHnm9On/vIv0733Hk23HHlV6tTpLVSvLUsMbXZ+Sfqb59Ip6W8+IdWBp8cmJWmDajakabpkwrTR4kJSNUYI0j/E8DHGpE6KtRpDU6wcmDW9bVIX2C8ptF26YHBn2jHanfq6JSEBsabFhQx7rcQgbyCw6kD9gSN1YoHinYIAXsPCQd/1Tz6e4FM/ptSHvd6+ouxRZr5zt7A3ljvC5+7N3PL0wo4JnJkLYbCg1izUlIUJLsOUQbUfTEi+YcywqYGdFyYNDB7yhcmCVEwwZXZLpcOuXbtMmgzGDO6YSJfxYoehA37GE3XFDXtI88CowQ0zwjSra6RzZUzwTrhoqxTQtOhj4KKsdairkPjyWLgCp8PNfRp+zdNDXDRcQ7Q/7rPELdKhHv6EJcxwD7OokGrp8I/v8C/TUrP50ySpRqdoy4hQG8caA5bZbGprLFdZN4/i5YrCyU1WxhueEpfIGDdM2vllFEQxt/uph5vnlpc198/dt7tcr9T0c3jnMGiEfR6u0S+PV9mvPAQC9pvBHf/N/K58aV5eilGHrca+3uuz1XpsR7i8vbFDD1UMm+2A9FbSZM71uXgroT0MtEgZmqUQjxv+G/O7edC/jZ5TpOjrtQ1icpevvLh0PaQOFrQ2mtS9nRxiO6Y9g9NSD809nqiPQupgVRsLrg5aayHbioY+pRB+6BStA3YfrvYD2uTGJevOtEGt+rAOffbpW/d5dKKaWhFXV8xsU7natL7itcvXVR1qFOQIWQS4cLfvcLBwTepf82ed5Y+ladZwwQw7VuzQwOyPwLDxdC3Kln4uN/yWEr3BAgl+BtOARaNZVieDfM0xQptDY4Di2/Cc9b5ef2jj8gV7rCX1A26vCNEXdMfzjBiQqDs/deq81v46xCymJPthK6vCbWNKCmFtX0x7Y2jXUPKt7G21a19MuLomzQus6cW/kQo03dkkxiTMmv17R7T+HxXTRgfytB+An2mrUdg24RB7AND6oBfjLuMvazvmK1CMsuJXV/eoWRMcbJwXq3mvANZNatDe8TrO+FoNXKokbG6iRqeRK4bN9d6gInZMJRrDNkO2iKDC9JL7MM6AbqO6OSqEjfD4MTnhCanhk5bmGm02S3pBJ2cQ/WSiOiWR59/+4ZxUT13Q5rTU9mhiklInU/vEtrLfbcPkobSQ+pGvpixNTi0mScMGtK7iSEPSyblvz2g6dGCnTJ0aH5QkgU4ZQKhBhFGCNhE8TFLxWCn1g2nJk4uyscGncLOZixBUxQLiUTw4Uyx9hlc+jzVOYBEtzGpCC0hszQSem8EMPyYKNq+fe+659A0xWpCs+S+JuSOJwIY4m81sZr/nvfek/+v//j/TO97xNt3VMaQTVBKzVwsSf0UE0JJJ11wwZs8Xv/hFuwNn/MK4qaRag9Iqnlqb882pLsMCXMNHuC+r+RRE0L4D+9N733dvevijH013/9l7JGUzZLqZxyVpduL0+XRcotBntDCZmFoQMUd/EXK3uvgzfUFFFBHnGGeSDiKwkKhpX5tLrVIXCNE2IBVo+8S8PHxwj4i2YS1GOnWKTCdrpCfQmDXAUX3ICDajzFRIq5aX3fpaUQXg7X2vqDS1y5G8cG6G61sJ1ywMSUaem/o34AL5RxmIE/HCrSjmBnfCkRe6ewlrMFVgcIEXNwtTpI8bD26EjfDmWLjjF3FwJw3emPOwhwQNTBo2KGDGnDp1ylSZoc7szJkzxqRBqiYkZsDhPH+YM9w3wz0z+/fvTzBnePfu3WvSM0jTICkDzoe0DH2AO2iQvMENe2Mdoi6YUY+AA9/50/id+105OzDXCxxl4ySmCqYXPImHcoVD6YovPhEwL6/VzVO0UBbOQxPjKjyU03MtM7Malp+ZTdW3hxhAwnFUw4PmQtM5LXdvHh/LaLNW9Xn+CsgRohYmz5mw7heuYcp5i4+n0Ri4rE+9v5cDt+hnxAQXUfMQ7RT1aUy12Xd9+h4i0mkWvnKrIFBBoIJABYFrBwHG7BijYy6oGDbXqj3KuXprJQgaIUxos4JIUQLRrnlaRofo8I21u3kobhmdSEaRmXo0eUD52Z6BpBAmtV9wVhvbx0+ck5roMd3zwf2Logxh2ug+T7/fQ7vSWuf7Gkll0dqfP1Ll7Bl3e7In0K7DnqalY6Db7vbksOc+aeoY1NoJ9WnsL6BWHUYNR5kwic8TxY2qxrf7Zr/Kp0Z41pxF7xiIoH+I6bH9N8KHGZEsgiXFnkjECd9LmZbNpQLV+UeNwqzzvOIf+RiQJx64lPs3w6k8TtgjbnxjGjNOjebQjLqFiX9pJ3yeRumn+EoAmtqeomksPJgGvtUYNjSZ4zOm7YNp7bKiBBalqW9uYUV30yzocOaYcFoq/5Ae0wHNRalEt/tptRcGbq9pD4B2t5d7a/W0tqH6HDXXS9oW4yDzuu2DDfTpUOquYe2D7UgH9w+nHcN9WvvDcAT/dZhTxfYDm7KovlYPlQmoGJyjXsqjBEfgqmVd/VQQMAiAL/HSP+JlPVcxbG4iJKGRY/OKC70///nP646Ir9gpeDaybrvttvTxj388Pfjgg+k1r3mNbToRJ0733kSguI6rAnPENye9kExQvEwcPD4RmVVt44+mpWwywzUPyz4bxAriziv64XJ1Lld76eS4VKTpDobzk5qwVtL84ppEnmHNIG3DpBUlITUmHggqTr9o8hHx1yJGTrd2qocHe3VHx4Dpqd21qzeNDHdrs7JdJwuc+OpQ9NomNQW2cvuU6hOyApAF1cSIjUEccMcs/PgyJyzZU05yJN8sRBm4JAJKt8pWD4FGGDaDGWF4kTZgPPnud7+bvvWtbxnj5uzZszbWRDpsUN8pKZe//PQn09Gj96ZXvZqL1HutMZeWXMrhrKQZnnvuF0rjsfT4tx9Px44fS4tShRb55A1fQwfhSpxZ8Rq4Dz3Iy+xEUZ820G973evTBz/44XT/gx9Mu/ftl9RMi3TV6oSN7nc6LZHoCYlCL+jOp7XEyZp4vd9BWNUedTYOp7WhrzbNpV7prB0e6tem/YjUoO1Ie3R6bHioxxYmMDWNeUl/UcpO9heEGHge+A2C1ypF4cFjcvR8m8HffBtwPQ9nZc6KTfh48nDhhhmwxj8Pk6dV596Qf6QRaeZhwy3MyIu0acUImzNHamGUT24nDd9kdqCFH/MY98bMz88bQwZGIswZvnlzKRmYMqg14wWH2bCASUMapA1zBekwdGmHSjMYNag140VSBika7EjMYBIOhkwwaqJOlBd7/o3by3muRBqXzhfEoRc5AtHMgUq0Fv/+YCk+IkDgLAEKt7zMbqfV3ZO2r6VBnGvyUJZaBTaUAPyyRwaSpzzOrKkBwtwCd0NCLnDbw+u3lgW4oM9It4BBDidLcAs/ZRp5YDLyzOr9PV9CUlYvr+ZpTeqN9bmcstTnAWzq4ZKXrLJXEKggUEGggsC1gwDjdYzR2JkHKobNtWqPcq7eWglibg0zpyM2n3u5L9RoArIro3qW+oau85K4CfOGc3KLuuAzpBE43MZ66cLErGkhWFhCvRSaEtjc1mJHC6N1e4ty2BqHLW9fB6E6ig1spA4G+zrSztG+tFeqo1CTxpqJje8e1KvDuFEk7YcXexvEJ03qKtNLbdUxa+7AZohKU0eT4C9nW2kYbYKDRyp3VEp619Ikgj2EUNj4LFwvbpR01sXD5b5eHxznOwAAQABJREFUnihX7rMd9nwMaEwfv/CPcaIxTLPvOpgrgJOBpJWHjnriFvYIkJv1MPS0ox0inkxzEi1r7e5pkgqtyYtmjEVJi6H+bGJqXqr+5tL5C7M6oDmhA3vT2vfSHaXaC0CTBphqrEIdXuLeGlMXCA4Lt13aa0H+M3p1WFP42d2Fqj/hsKRpkKzZp7uaRoZ6pVWjXf4qhSoOw8r2AByLVF7fBVCiBU4WUCiqbgwu5eBwj3pa6OqngoBQyvtm3j/BFebwimFzEyEIDVwxbK73Bi2mmZi3bLzmx4kGnHnzYdzscqzNV0UYIzJkJzBzDgwYNEu5tE3SyZnldPrMuDbGJWlzVpOXTs7MFRvWSNwsi1pbJaIGgxZj1IhZg11JtjIRKT1OznRKFRTioBBcEF/79w+JgaN7Gfq14amjBVzxbqKgxKUsGlhWpWrNpkY72UshVTClSWHrNo5sssTfn/yzJCRKggA3n9gjhptl2Hr36qseAjns8gkhQjX6sxn+/PPPp29+85vGrOH+GjbAkUpg45vwvGyAH77lULrv/femj0rK5d3verc2t/vTksJNT+uEy9R0+rWYPo8pne9///vpxZde1Cb6hNSh6UgMGO//ZoJ/tUdEVSPDBiziNcQHW8E55T80vCO9+573pPsf+FC65VW3y61TItEzaVxSNTNzqyLqINoQz9elgpKugXhbtU1N1F4J30VnwXTRKGo5dLSupL62hbRLOmoPHT6QDkmyZpfudOrraZX4vy84OFnTooUSqg4h2ugH9hgiY6dwhVv4eQj/Lbxyp63aA/bg/lbwP8KTfh4Hd54wcz/zyH7KTWCNEda3Pe88buSDGeXK7c3Sx5+0gyGDGeFw584ZXpg1MGVQZXb8+PEE45CNCKS9wMuxMTGo9aLuDDfwl3RgzCAVk0vOIEEDE4Y7Z2DM7Nu3z6RoRkdHTZ0ZDJ3GOkaZMGnaAhFrEDL32te1swDPePIy5e6Myb54ICTjKnNIEc8WHV49tTAI48mZv4fJ56M8D7drXCiAY/ENWFGiq2t6nYsyF9XIS+BV9nHM3fP+5BECbpheL8dt6urw8ZjMfTzMcQEy8Bf4ulvz+csiNfzkMA2vKIe3jNepdPNQeTzPO/oqC9Tm/dzq5MlZInkaG/Omfk0AGQErs4JABYEKAhUErhkEmBNijMbOPFAxbK5VczCxZpPrJYsRc2uY9XP2ZtE5imYTfF0ApWHJRP7sM3hphCFQgMSyvYMFHegcG583bQTHTpyxQ26T0wsmmbAspg3qo9qkNYHDnrx2TyjxdQgUWpBDLNz5wWb3ulRJt6wvpS4pWUAt2m7d9cGdn+wf2IZ3oZmATXIyR77B4lLcoqhmFHarAiXHoj0JmFNB9xieE889FYo66quIqxjFg0PmmAXwNWaE2y7Ta+GV2K48ynTzMaB0LW2X8i9DXszm8ISu3PhEffEJ+ppxyd9GOFAe2o31sqUHDc0fTvpDq4ZT13wL75SnqT8Tw3FqRnc463DmydPjuhZgXOrPZnUn03Ja0Zq/pc3X+2tKgDjkG+s6eJy4s/gH/9bXZoS742I4rkv9eZcxaw7u252OHNprEmP9OqiMlFi76iBD631FBR/1R6m8TnK0p960nlcDFPXCP8IUUSrjFQ8B6wf0Bb3gSLzM4RXD5iZCDxq4YtjcAA1aG7R9kLc5RMV2UwN4NobXrPKs2WUjiYhnPvLkG3ebyMSMmef+DrtccNZUpB0/ed6IsNkFqaiSlA2b1zBsNN8pHgMDemVFlNlEKQJMaTIhIebc1rqq+2vWdY9Nm/R49qY9ewdFfGmjU3fbDHZrQ5QL121woTQUzqZeS4Np1k//KE35OoFV1KZWCf/2T35jQqM9Gbww3Q08b3x88itdI0yjexnilWVrBo9wwww7UIGY4WHDmztrnn766fTUU0+ZZA2b4WyaM3mwqY4ZMN6zd3d617velR7+yMPpHW9/h5g6q2IWHk+/+fWvxfR5wSYbJpwzZ05r033a7q4hvpq1fPKmLdrcscZPrYCjzlmhDziZJDlmubWlru6+dOiWV6U33PGW9No/uSON7twnabM2nSBrkQlzRlJlSNcg5g9RqFyhmVp1UsYWGbrLifulEIuG+dKvu2kO7OhKhzhZs3dXGh0dkGRZh0SmfWFiJ8QgPg3fWfo4EWcJW42iYmHi2Nwe1Q5YWvRL/ES7NcbJ2xK/8M/drSTya0wjvjEb49aIXIM79d68P16i6OZtbS8beWHnZf4Cv5CEQUIGfDt9+rS9MGdgxODO5gMvUjWEDwYiCcOcgTED4wWmDJIy2GHMICkD44aXcKHCjPtmUHGGGdIzIXka8MvhYRUoyo49DxN+YVK/y3kircuJk4dtzC/Sq3cvCEKQlU6gh95kfcIw2Rc3MW6Dt1yaaUGL6kQs4kYepZ1ARUDFJf61eMo6R3nA2awkciYMvrgHTvqhAvyysBaGXl44Kjx9IJ4aDGqRPC8+yQP/CFOWK2JvyCg8ZJZ5lI4qheVDPH/5jZCWJ2MrJazl6+O61VcBwG/8oiyYUb4wy/zcFmH52ixMY5zqu4JABYEKAhUErh4E8rEcO/NaxbC5evCvzynm6HrXzb9iFg9z85C5Dwwb3qACPPbmaVAqwtrKRR+oVJ+RSvXZ+QVTi3bqzKQ0dJyTpMJ4mpWuqdpduK3dOgzXrbiwWVjR+9oMMsRyg/jQDgP34drduLoTpKerJQ1JLdqOEWnr2DmgzW/R51Ir1dfdWaiU9lTYc4AOK0gaSw96K+iO2ma/1l0Wxn6czlIsfSmy1nZeEkyeMCmXP+RRo+PC0codH5cyyzwvFbL0j3KEWfrcyLaAI9Atayb4ZJXy9gM3CQWdGZ7+7b94uRRMpMS+FHFpUu5fZr8K+5o2uYSSYsosSp3fpO5fmjS1fuOT0rIwqzWk/FZWOaCEJhlJiFl2lMnbzXFMuZK2tbvKJLM1zYvJOC9GTZc0aYymA3t26s5aXQUgFX8DkhjrkUoNJGuMUVNUwXGJDyDBX/3juRZulhe1UyW0nop61seovl7JEDB8L/CSNVa8zOHsn331q19Ne/bsSZ/61Kc2gEk4XGCYvJ599lmL8La3vc0ixIINM4LlbhtSqxy2FQK0QcWw2VYQ//GJ12YmJQXREYN8dDMmMptNPCs+y6cc+i24fjyad+p8okDU2VSkaWJDp+eYThxw4fpLustjTOLOM/MucbCsSQ09tYiVcoKmvU2nytkU14RipJjKiIgzBFhLWpZ9RdI2YtqIUbNHKqL2StJm93BHGurvlPioVAV1cHk2sSHlMJnA/BxPuJmn/MqZzctvTiq3jSUUhz+ZARpz0SQXYw3h44lxh+9G/9wvwr+STODBCxxyWNAyccdIhAm4oGrqhRdeSN/5znfszpmf//zndv8H40uEhWHDQ5q4sUl++2tfm+6++24zp6dm0h9e+EP6zW9/Y+a5c2e1wb4gRk8hVWOxvWyUhfaNh/QcP0AmMDFwUiYMGr1GzBFITEZeTn/1D4yk/be8Ov3Jm94i5s1tYuIMyq9LTEkIN50ME6OShQaSBPyZVIyYka3C7QR+yw5+9/Z0pp3C8Vv3DqSDOhU2PFQyayia4bIIN19gOLFHgU1igao0fcr6eQogtwKa4X55+zRNosHR4JS55d+RVpi5X0TJ3aIdIQzCHTeLr+J5f/Tv8I90Ii7utTjyjA1wwuEXEjRIyyD9ArOFFzsveMcLsxBmDXfOxMt9NDBoYM5EPmw6h4oyGC7BqIGgCYkZ7p+BaTMwQBv2GqMm36yO8uYMKcqLe+4W9qhLwIBwV/r5Y9KMckWZIq3c3e2OqBSf/rCqo5JrEtEEv/GJpb8zrlAZAMNGgT2a8KGwyIw8yNPt+IU/8LnyMCKviz15fb0sZXmiyQjDG+UPfOXb4KAo4Cz4Cu6ZxJYmuLjDCIZfWV96iNe/MV3KGXlEmcvyNcCqBqqwhBkxMX38qtWLevCn+X5F/WN2dk6vTgsq6e7u3hqDMhhRpEB5okxRlvgOk3CNT2PYRv/qu4JABYEKAhUErh0EGKNjDMfOvFYxbK5Ve+Tz+1bKEPN9mFuJw+wPw4a88nhOkzSm4LSbh3O1UGxwr0vrBnRfi+75XNEG+Ew6xj0gJ3UPyOS0GDmi1XUIdCXBsJG6ax1+czYLazPW5UrVd8b1TSqsD7kbd9neTjFuuKQdpg2SNnt2imkzpPsgJcnQI9VTne3aDNdOOPsGPM6TUX34B59x43CdPGydZaHqa0sZKEvpmsPCI5iLfoCUP9gsk3DYggnttIVgdUEiQph1ntfkw/ClAESMF5dTEIdckUAR0WvXiHeECfyUHcIUmKstsdFe7qK2M6ZN4SZHkbTGpDFMkvKLJSHpvO7DnZ5ZkiTNnFT4SapmTPfT6hqAuQWpP+N+WrRo6OUgJ3tb0MXkhxQYa37w0u5uFsLCsrTDycK9ns61NKItA5iK+/dyb/OONNjfJS0z3GUjyRpj1rDuD9pZtS0QwXPI6lirWQYLBXL4+L5GfFHb6qkgAAQY6+KlT8bLHF4xbG4iHKGRK4bNdd6gPqp7IW3iYMLyDsqE4lRJNsAX1SGau/pvzHe0ufswDfikRzq4rqmzG6tFk9WCLq+Z1ASH2OixE7qEXYQYTBzURKXWHmPaQIC1tXdrgNCEF4SXiR5rQ1tSNuinRRIBQozL2LpEZI0MdqQDu7vTwX0jEnvWpqgmtw6pi+KxbXYjsLxakFGcn7DEKbeVXX61yS8brBSKmtYTEaRLKlt7gE2e9tZi3ZyhHE8c1tTQcE54gXu8TAixOfniiy/W7qz5yU9+YlINbKwTljGGcMCWTWwe4rIZyF0yqJfq6+1LU1KFNjMzqw12SUCIwFpeXhHxhbSW2C9SEAvxxDfpgROWltIELZyRRHvrFLhJ0IjRgno0CHLlk1BpJhMy3vzbYNh0p+7ewbRr3yExjN6UDh+5PQ0O7UrtHb122mZRHEzEn9tUZlQAWm+R2H7S29a6rFfMSJ0EGxnpT/v37UqH9olgG+1OgyYGLZFpyqzsgQFxVVSl4/3RiD+V1twMx/VhUDbD3OWLY/Fgj+/CTuQ/4qFc+ZP3HRsR5E07ES6YDxHH6qT8iYM9cAE7G/a88USYiMt37hbfkQa4EvZccgaGDGrMkJhBcgY1Z6dOnZLe4fO2ucAmOXF5KC/MFiRlwC8kZ9g0R2ImvkOKhntmuFMpys2mOt+Bq5Qv6h92vnmpU7zhR/7Aj5bGrfHBLWARZoRpFj78Lma+3HgXSxO/KB+mV8Xru6ILNhcE7zlt9C9yPK2AEVJIfX0ujQS+2pRVZGL9x+yk5XAJE4gZ1Awlm8OtSGbbjKirZ+DlcTvlKbOtD5e7w6xZM0YNeAkD++TJk2ICd6fXvu616ZbDt6ThkWHDM0vDYMoYUY8PwKSES6QP3Eu7w4tvFaxWtrCEGeExwci8TrIrwSWNtZMTk+kljd+Ud1HHEGFevu51r9P8vFtjGP2gXrIm+oXjhOe1sbx53pW9gkAFgQoCFQSuVwjkYzl26K+KYXOtWiufp7dShpjvw9xKHKgBNovjmI3HDRKDrxrthgVaRX9WMvtUCBguolXQPrAk5s2sDnpOoaFjfCqdPDMmdVPnzD6/pMNyLb1a+6AMXQfhpC5tndcIl+IgndZ1azoAhLqoNm12t9ndNlpj6cAnmjr6pKljeLBHd9tIU4cOfu4aHUyDkmLgInfut7FH5RLa2mPlp7QqO8waDpGaWxHUDQ/lBFTYPb5VtLCGYbwdfXhcg0R4bZPpOUWO25TJRZM1OrUhRLg1p1MbAjf5dMjxWz6+Ts9rin+Bn7anhN1QTu0ZuOO4p20JZ9LYDpLT6kjWsH6fm18z6a9zY+O6n1Z3LY3PSB2aNDKIUYMWjZVV1PX5/bTCOs+gSAe15eyIgTvsZa2vc/BPa3/hU7cuXRrs7xU+9uqeGh1GlgQYqvu4b6m7U/edsv8VTB4VEBra6WjtRYBILCj071R59MPYRHM4ABPHAH4Vj32MwkWW6qkgYBCgP8YbfRKTObxi2NxESEIjVwybG6BBbaZSOSGcfNoq6AkGeLkx+GcPofxhwMdPbxHVN20ydzz4J4i9TII6PaMfLmabnl0So2ZOBNi4Ts6IaaPL2ST0oJM1IrS4jUaTnUshsKFNXk4I+qkWP42wLtVRbFC3SydUT+d6GupbTTuGRXzt0mbqTi7s7tNE11GcmNGWuwgwUx+l1Gxqpt7CVT+O45t95ObZ+ZRn/pSgmAipk01yTHSX+VgalxnnZgxul2kLjsAjJgQmgdzO5ey/+93v0jPPPGOq0JCqRMKBcSXGl9iAJ52ALX4QXp2dOo2iTV7cl7RZCJPGmlA/7WKqdHb6Re89OlXVhUSWvjt4tamOibREe0eXwkpUHrOjW+49Zm8VkwZmIvi8gPj+3LxUZ0liR91mXRI2LCDau3QhYP+IJCr2puHR3cbAaVVanLKhD7AAAKtNDRpEvwg2lVQq/dbtTiY7WYOuZS0iOP01IF3LnKzxuipegaaGH8Jj0orHT30ZosrdTffDnofEle9wkwmQat/4X94D/PMn2gW38AsTt2b+Fpb+LjgRNscNwjdu7DpMvA6EBS94wRUYM+ASDJlgyiA1w6YB7rzcRRMSNYQPhiCMFlSTIREDQ4YXyZm4bwbVZvh1a+M8wsHM4eUbXLK6ZDCJ+lKvgENep7wuEZdwuXukYYkXP7lbY9p5uMux52leTrythvVyFhfSa/HEPU7A//iJE+mnP302vfCHFwwfgfmttx5Jb3jjG9OB/fttgQKzE0kbnhzHQd/6coOPwNpD1vvhdnWeaGvK4i99uT7vWhgFUYnNk3KD00jUnDx5yu7d+t73vmdMxVuPHEkPPvhgetOb32QMxA4xQSy8YOkn7wQfOURaytHg6cDw9OtLwFeRL9aifBbPPhoKTBhLnTiRntvpQ+fFCP3xj36cnnjyyXTi+Elj2LznPe9Jd911l+5o2m/9hvZoxNdoozAtm+qngkAFgQoCFQRuKAgwtsc4jp25rGLYXKsmzOfprZQh5vswtxIHaoP9A98IdyIi4rtp2w2WlJcnfKFdaq/oDW1HKxXJxWgdsKTFFZI1FyZntUE+LtVT5yV5M5emZjmEpXDSzrEitdMchEP9FJoO0IaAxLatN5UpEg2mpcP2DZC2WUkdCop6qYG+TmPc7BjtM+bNDh2WG+jvtsOg3AcqkkpxeZ1u830ISgh+k26QS04tlfWO2qnCVDd76j4tWKzj6nyyGFfKGmUK80qlu7V0anRuQ/BwZ7yIMaMhyKafQeMSoAa9wuItEq6Y4GdJF+NiL+0srIBRyNoe3AMfsa8Lr5Zh1IghMzG9qMN8M2lMKtAuiIk4NTOvvatV+bO/BS6i8pwXpmEwDtl78gNKSfcqt6D2XIwXNMXwtpvUV7s0xvSbxpjdO/q17u80FWjdOpDMvhVSNe3GIEQqR3VQuVzbABo7aEu9qoNZ5UI/jL5I4Ght+p9W0QrPS/jC1Ff1VBAICNAf440+ickcXjFsAko3gUkjVwybG6AhNXDXHkZxDeo2Q2HqhdFibhh6yuB4+GvRzJcfpkaLVARm0tCEZ5MIk6FPlUyA3FeDrtoLE3PpxWNn0kvHx9K5sVlNiM604ZQCE58RXkx0ikt824y2TNngZ9LSJMYpgxZteK9OiDmzqkvfByVGqsu7JUaKlEJ/X5edTujihEIh6qz9PsXxKcsnP049QHg5seD5KUdmbJ6iWg4UkWkCThAVYRIsiA7sjU8ertHvlfTNgA+cgEdMCOHGuMHmJJI1jz/+uL0/+9nPbIOSsGyiExZ1VJg5TAP2SDG0d4jpR/MpDKr12OTnVHe3mDic1B8aYuNdm/CD/WKsSBx+QAS63h5UVWkDvruHTXcx/Hr7xWyB8afT/Xo7OnuVFszEVkkArKbx6XlbQFyQZMaSvhF9TpyskUo/GDcmrg+DR9/EUXEMjx13g2iD6bgq3FzTwqFdTJrhdOuh3enwwR22iOgQodbKJiwdh74kZMRAMsjR0idWcAh44OewAMa4xlPgcnzWzAiEyQsB50/ANL5zM4c97o1hw99GgSLrPIyX1fMOd8ywB07wjT1e4mEPlWSRN7iDNEyoNoMJg8QM0gjHjh1Lx3WPEUw/JGfwA89g7OTlAE9gwuzdu9dUmR04cCAdPHjQ7KHSDMaNMWTE1DMpKYN5WY8oP+XK045yYvJEvSIM37x8G1NKSQbTivC4hxl2c2j4ydOJ8A1BrvknZeShHVdXlq0dOAk5LqmMn6i/f+mfvpSe/t7T1lf27T+Q3v3ud6eHHn44vemNb9Iiu91exn0BpTY0YwFCBZgsfUEQQFs6+F4MbkWEbTO8zl7v+jJ6lgGT3AQ+Kyurppbvpz/9WfrKV76SHnvsMaOv7r7nnvTXn/tsQjUwTMQOMaKJy18wbOLb4FIAKOZNg5Z7bKwzYCuhJrhttqjz/AzOlgr1k0oTSUchufbUU0+mL/7jF9OPxLjp6xtIDzzwgfTBD34oveUtb7W7nMBz2iT6NuO74f7GEtVcqNO1bMdaQSpLBYEKAhUEKghsCoF8rMbOOF8xbDYF1zZ7+Ny89UwgDnjC9K+L/5JHbBQTklV1xHfTKDaCGX1R+LKWNxdoAfBEH6ylbKHOHoKkjOXGATk2yM+cvZBe1J7BSd1vMy8JHISxlxbZO1B8bZK3svYq1mmGdyoTdKDnKakGSXIj6cAeAFI33HHTyfqrv0MHPoe1fzAqSeAhrQW0FuQgn+gf1KS1645cDn4SDxrO9wzcHvIZ1KOkSlnj4lI8uR0npVGjs2Tnywys2/ZEDmFuW0YbEqYtLvbEeHE59F0tTdsX8jpZLpGXb2QpW1yLtwhLGxpjRlAH5VDh76+3G/YVbVYtCMdmZpclVbOs9f6M4d/58xNaRy5KmkbNqMOYphIdVecc5tQampa0Q5xCXv6ga20nSwwbpaoQy5KY0Z3MOmzc39eWdopJs18aYg7s36GDx71JZ419v4oqqS7aPRDOEU8pWzWVanHXjqngkyN//HMIO2fWUEt/8eJYMvR8QdNvStsrSPW8YiFAv4qX/hgvc3jFsLmJ0IJGrhg2N1qDZpMZkxqf/qNf+7Bfn0httnBX8/Jv69BWbX0XkyUTCQwbggUDCDunGJY1Gy4srqRJnVi4MKGN73PTOl3NyZkZ6QZFj61OJ9gGOCbTlCabYuBgUrJyaR7yaUpipStTsi+JadOuU+5duu+jT5PgoPSADkviZiCNDHen/h5t5mue4tSM/mPK0iRmJVR8nwy9xFRG7nhZbljyyc7Lg2/1XB4EgsiKCYHYjBmopUKFzve///30zW9+Uyftf2qb7myuEzY2+CI3Jo88DcOPYhMwwnTqTpkRqbA6dOhQig34W2+9xb53S60Vm+/tOmplm4XSX4yIcZvMFm2AtuqkVqs2EVF3tioG4vyCFgxTqPuZSecnptNZiUJPSN0aKn+03yycQsIGZo3uYDJGTfENUSSE436O5ZVFhdUKw0SjRbQhVdPbqZNduqdGUjUH9g6lnTrpxUWYXbqLCbV/rao7OMofvY16OlqCrz6ZRn3la9Zc0sb9LEYZzGy5G/GCkCtiWL9oiJJ9kjdP3p7hlgUz/0b3/DuPT5vmafIdbzBlWPTDeEFqBqYMeIO0TKg1ww1JmmDMcO8HzBmYebQ3Uhu8MF9g0PAiMYOKs5CcwR+1ZoRHJRdSV2G26siT4WIB67yuuT2vY7hHXazOAl/cyRJ47OrtvE3Nzdq8sY1LmJNu5BMm8bDbGKkmDvcow9UwrX5ZRlGG3B1YwLBZQSJKR9QuTIynH//4J+kLX/hCevLJp9QHW9NB9dv3vPe96eMf/3i64813GDOCvtpOv7Q/6ldmlNtZpHh+BNgIwzLW1bLR3/I+V+abwyVcl7QTMTU1nbi7C2bNk5JWAbdf//rXpwcefCDd/4EH0pFXHZF6D5jD1LAYD2TzryKvAj62TrX8owzyKPwsgezHcMe+Pa3MK7Pm9ZG9GKPoa7MaF3/9q1+lbz72zfToo/9h94cdOXIk3X//B9JDDz2UbrvtdlMnSD8KejFUCGYZ1Kw5fAKXap6VpYJABYEKAhUErisIMGbHWI2d+b5i2FyrJsrn6q2UIQiDMLcShzxic9jjea6RBhRbI8nhdEOkLjQRGaFQxUYyzJpIkQ107g6Z0UY5kg5o6ThzblzSDro7RHsIc7oTd1V3h5h2Dg7Oad3GhjxMICS4Ax+NTirKwVY90jatUk3VpoNzPdJmMDggiRvdHToyJEkHHewbGRzQprqYN7pXtKuzxSRz2AmA1qQ+qLPC5IW+w+TQDC7Up/njHutOlHnczQM3T0Kx6undTYLVOXspo7R1XtvwkdNtkXyMCfEdZrRPfG9mbkxTsCzgGCCkFbwF/NfTKmFueCZH1O/hynlI8EsKYExaBk0ws1LFNymprjExZ8Z0P8341KLcUPnLgUEdplIEu5tG62ZTfWbrfGeEWP6kXSRuDBdwQ9I1MAml2EMHNDt0lxL3MA9K/VmfJOV7Tdqri30IlQkcMq0whl/gKQcMSVDUucoJG4fDVHZ9gHwtL/ytiaPXEF6bE5a3PNS3jGFjiMO+hJdXAaqngkANAvSxeOmv8TKHVwybGphufAuNHAvw3/zmN+nzn/+8bTawucZC/rbbbrPNF9R5vOY1r7FFO3G2crryxofO9VKDcjuGEvmU5ROBlRBr8XhIHDTY2yDvHrgbYeU+hRcTgoXEMH9n2ARhYbFsYmGCRNKGEwrzC2sSL52XejTutRmXjtrZNK3JEh22K6LYbDLkRLVNY5TDpjP7tkFFG+BrK3NKF9VSCq1Jq0t6aIel/xP9tOgC3bUT4ksqjnpFeEnMtL04KWOpkqRSs0lN8UmjVlWbBb3cTgIQwwcwXPmPZzNCJPwrsx4CjAdMALQhqql+//vfG7PmiSeeSD/84Q/TiRMnbMzAP2AbTBtrd7lj8uRhrDHVoH2SmDl48FB63Wtfl97whjekW289Ykyb/fv2S4pij939EBd2k4q/BSbIcPxcN5Hn6RkYi3OSBJtJ4xfEMBDT5sK0LhhEskZEkBcD5o6YNGI0GiElnHUTfBVxJ2JtZWVBDJtFEWOrwsOWNKxTXLt3DqUDe0Z1sms47dRlmH3SWVtiu8iygkgrMlGaXt/AQ+rPQxliYnUXcy2sDqfSxfE8vr3Wjtu13lDAtkhgy0a0SR4h2i93izbEDbtt4LN5X0jLoCKLBX4wX2DA8A1jJmfYwKzBHRyCuYf0DXhC28JwQU0Zd83AjOGOmXiRTMAdhg32HTt2GBMH5gxxmZMod+BoDlvsm9UzxoVmdSYtXp5m/nkepN8sDHEj79w/t+Ofh8n9iL/dT+Qd+UT+uTtwWIFho/YOhs0Pf/ij9Hd/9/fpcY0BMGwO646We48eTZ/8xCfTnXfeYXWCWWP3ATEOKwP6Qzy5/UZn2ID/L754LDEefulLX0rPPfec3ZUE7fShD384vdHUoe0ooOD9m/o7VBwiBpoaruYjnPzrgRUgNNNw2EZEUmD8avZ4egQrw8umyR1pt7Nnz4gB9+P0r//yb+k73/muJYB0zcc+9ufpHkkI3XLLLdZPaX/mAvoc7drsyfEG/8CnZmErtwoCFQQqCFQQuLYQYMyOcRo7833FsLlWbVLM1VvOPoiqMLcSkTygbTH9CRsmdMnG1KAXCBshFabYRGaFgpd5KyJ7BpDOpiZNggrcbTN2YVKS87rw/Zw21LU+m51F4kZb26ij0gE6NrXZiF9VZOKSJjipI1cyC8pGh+cSd4msLWmttazDWSn19nRobdAttVS6s1KaGNhLGBzsNTVVffLr0n0jSN2woS7DXvYNeKijv+FizvqJOvJd2LW2468GmTxIRNvUpC6bem7iERHC3CTYFXLeDrqtMc0kVWFqXS+x4OfQBKLeCt4aZYUAsanLk2l2hYOlsWT30+iQ1MxCmpjSOzmnA1LTesUQnBJuzXOQGHV7SNLApNGrtb+pUbPkaQ8hhIGWglASX7+Tg459SmpmPfVqfT802K29KalAkyTXbu1NcaC4jwPFQiY7QKzqIMnlr9KyPQDq6OladsofDPY8y/VoMHW853gNraJEMoYNBaScvAEjPKungoBDgD4WL+NlvMzhFcPmJsISGrli2FzfDcrk5q/PLRuGbA3qRbe1SccnILkVzj5tMBEUTmb1byaGmCjtdEt0dk9Qg4Bv0EN8MZ0xaZi4s6RqZubW0mkRXr9/8WQ6JbHnSfSDLurUDGKcEjtFJyiTozR6KpqkGESQcSHzijZo19ekl9Z0fDLQuI5aSTOnXjFuBvu70o7RfqlJY1N8h3SF9thJGslPaIIsRE+Zu4S7UIXMYT5AUUjIKWpCZamZT5IxmBXVMiMGtdytspcQAGb5wyYdm+u8qKzifoZHHnkk/ed//qdJTzCORBxgy4MZsMcMmEc4cyvoECRqjt57NB09ep/dnbBf6pV6pe7M1IkJ75CU8HR9QrIzLBRRWXHyCb21MzpRM3ZhIZ0+fd5Oc01M6L4l6VNGPH9OxBtkGFIS6Ln16lFOt5v6Ib6EY+CmUlSY4r4aLQrQlXz4wK505PDetEdSYL063cVll+2qo6TvDdM8UfU4fZOyE270HPuiEwll6ZE8TrxFnXAJuGCPh7qphPrMXz6DTeQhPR235+nk9oA/oXCPl++IHyaTfR43Nmpx5w18YKMaBj84gcQVL6rNULOEO4wZcCbSwyQPNnyRjNm3b59tBqPODPVmvLskTQVTBuYNDBm7p0ibw1E2TJPckBubxrk7deGJuuEXr/uUv1E/zGZplCHdFmlG+Eg30ok0qCNP+HvszX8DNpcTpzG1yLvRfavfUYcoQ8QLd8pIOwYeXBi/YIzav/u7L+juk6eEzi3p8C2H1Yfvs0Meb37zmw1lOyQRxytst16g5qg9uZ0Fi+dFAG+zWsBrYon+5pkHHKxscmKmKd2ScH3c7vH62te+JonDxwz/77jjjvRXf/VX6f33359Gduys3QVDva3uysLTYKFY1FluLFD5YwFoQLtk/aOsRGDOw2x4ijSVoaVtjYNNc+iq1I4s6GK6F/7wh/TvX/n39LWvPSLVhMeNWXr06FG15yfSO9/5Tutr9Ptg2NAH4wn8y2ESfpjhn7tV9goCFQQqCFQQuPYQYNyOMRo7833FsLlW7RLz+Vbzj/k+zObx8jb2EKJTa+sRuRTRoRD8cZqt+DCSgfVL0LcmcV4cdGMzXGjDHrM9IissFUw22FfkuaRTdZM6OId6tBNo6Dg3maamtT6TirQV2y9AvbrWZ7bPoG1zcW9Il01uDnUZjQTNAtNGqbKxzmE602wgVWlt3C0qdWnDg31Skzaqu/hGpQFhUIe8dBCsp00q03z/AK0dtrmuVFxjhxXZfqz4Vv0CCqy/Ckd3wRO4bEJnlUldAVsBzGiYK5DixZLYiB8eGvd4Yozg2+BRekWQOjMPTwztANkvgVoMWYAl9SzX9wX4HZ/kYzikMhgzT7HR+DI3v2b3KR8/NZZOab1//oK0NMyhspmkYP51K3yHxfHDyuThTEVoXsoCXtG24NPaKgzANZPIQnoLtXv9vR2m/uzAvh3SqLEj7dK+VI+ktoxRo7Rg/sEKooDUUyiqL0/b6mp9CzreD4OSP4/Vz2y4KC4uRdjCWd+E8jKTi9kprCNjLVhlqSBA/4wXPIyXcbpi2NxE+EEjVwyb67tBtTWq4bzYAFRRY3rzUmsADwrJZoFiKrBxnqmjnD4Ij3M8dgiAiaKYFJg2jNVBhzd3/PQqffzMKnNV31zGLmlnk6w5e346neEdk9qpMV0OLsYNl7qt6lI3LhVET60xbcSwYeJB5HkdnVSkqgky1E3ZhW7SE9oF40bE1VDBuEEMdXRYp2a4u6RPhJcUhnZIVxqSDEy8lA7iy+axYnPb6slkXEx0bIAxAQbhEQMaZvng71/17mWIV6INmMUmHZvvbMwjUcP9DD/60Y/smxPaAdscRgFH/HjjO8JYHNEiNMOePbvTu/70Xbqc+4H0vvfea2rQOju61MYwT9SSBcPGW5ztXQh66awVss3MSexeUjRjF+ZM4ovTWzOSsuHuGvGRdMqrNc2vd9qiwIlDpWfEGjgq1FAepGjMPi0AdMmSFgHSldyVxEDslPqtgbRv94BJf+2Q+r6BHjERhNZgtg7hBDnlOGjpOCJxesb7kteYvMoH3HNCLIdLGaYIbB0Ve/l6GOLGNjgwzHEZdPf4xiAqkoq8I2y0CyYLIp7YmOd+GdSTheRMLhmTS88gQRNqzkJyBnwgD5gy3bpnCMYLL2rLcimZkJ5BmgbmDX68SNoEk4ZN4dgYjvLm9SCfqA/u+Tfhwy/MCBNpECaePEy4hZmHC7c8fDP/ZuFwu1hY/PN0+b4ensCLGAu4C+qZZ55Jf/M3fysJmydNHeFhSWHcK6brJz4hlWhiVoCyqDDsMLUBRc/L0LQeZYNhQ22jTbPA2wiEaI96uIMXJW5EGNAlLzdwAd8hjL/yla+mr371q+n555836a/3Sj3cxz72sfS2t7899fYP1CRSIn7N9BrXaqjRUhDgITM3cLNHhvvLowCPjTFmB8Y+LhC2Pg5zrVx4LS3S82/qsKpxFCYrd9h84xuPScrmKfXryfRmqbb7xCc+kY4ePWpSbfRp0oBRGv2yHm5WyqY/Ww3XNHLlWEGggkAFgQoC2wIBxvQYn7EzJ1QMm20B9RYS9bl5CwGLIAUhEARBk4i0aTzRzqx5tPgpnEt/HGJboRbHHKEY9NbSgk7zdXasdCKV3CQHbgNZVTzWZNMzS7pjZM7Wa+fGWLPNGyNnbl4SN1p+wbRBmweHQ1lluRYEowoNR1lXuSYDWwXKDVO3kawvpI40b/fg9kk7B4yaIUnaIHEzPNQjRg5qrHoyFdasA8mjRkpZBWPNZoflWH9ZGKu51d8oWZUxi2VpXPwHWF08xEbfiBDmxhBXw4X2Lttctc4qkrtTltxv43cwbNj/AXr6q8HR15/B+LNDSwI5OLCin0WpHJ7RoaLp2Xm7G8nV8+uwoA5lwgRc0GHhFd2LZAeFhTPrLTo0rFdbRHIjs0if73KtYXtQwh1U7SFR09Otu1Gl/mxU2jNQfb57R58OawqPJL3Vq80paT63dT84QquYSfrFd95vnM7Gz0O6TXbBz1uUiLzKv9anLDF3rsUr4gP3DPakVz0VBKJ/YtL/4mUOrxg2NxF+0MAVw+b6blCdZ1EBjXKxghZDt+yFrRjfC0qiqIymCtvoLcb9wpU4xPJHESOuOYjkUWfHyX/dU6yRwkWucqqJpyohCTWkucU16ahdkpTNdDp28mw6fXa8mEA5hQ/hhaSN0hDDBikbRBhc8kbbzVZG1U3qpxBz5nI3Xp88V8W4kTiqJspdO4Z0wfuoTjgMSWetNn51j0iXmDZOuHFZPWkxUDnZiF0fSqcouwoeg5rV38JST+KYi+oYREm4ef0JtdlDiNI3wjeGLkM0ALwx4BX+zvPdLOnNylyGh8GGVNS81FchOfHMM2LWfOsxXVL9lH2viZhiYvDJwuOV9rIMuPkDfCOcTCOIpW5M+off8Cev10XXH0wf/ejH0qtf9Wrb8Ec8ngh2P41MZWXq99CPvKD7aCDWzkvt2ZlzE1KBNqXvebk77hUsFZFi7WlBp21WxLjhAT+4hB4mhTGExEDkdI3IQ2W1rBM0EoWGaSi1fHt2D6eDulxwz65+4aI2n4VsSNTAfoRhY8Uv6mbVsh9INcfFqKuCGozAmBLn6FMFMAiw4QmYYXqaNRyqEbpleqRbwtnzo214cI/XHAo3NuCRnGDTmTeYNDDnWLAjJcNGbkjMwLhB7RkvjBqYN8whbN7CnIHZAtMFCRkkZbhrBsYMTBncMFFnhhvf3T3dagtXZ1bCReNMUe5mbnn5w45J2Dx8uDWGaeaeh2lmD7g2ph9hwz++CRdueZxwi3CYuX/ufj3ZWeSEpBRtg0TJD37wTPp//+ZvXCWaTrUhYQPD5pOf/KQxbMDsdjFr2sRsrbEWbYz2mtX1DeG3AMYvPaSAycX6xh8PHcstuhjJKTvy9sf7XPFRa0vGQ/pgi8YOQgKT8+ojPxbz+gtf+P80Nn7LcPcOSRg9/PDD6b3ve1969atfLeZvl41htH9I8/mpUeaxIkfGEb3gg+MPfdcL6GV1+LiTFdYienyP4+WPOlhyloblS8Ra/p4uCaxp/GMcpO+fOHkqffep76Z//OI/6j6eX4iRvtfGZMbl10pd5ejoiMbNok2BQVFW0gk8bobj+PNEGP+qfisIVBCoIFBB4FpDgDE7xmbszPEVw+ZatQpzczk/X7oUzPflnN8YPp+Po409jOgLtXWzvBqXJWXqpY00QnoBuoM4TqfILj/oEkw/Cofp32zC60pc4deK1m1z6bQkbs5I2obN9xmpTuNOEu3PO8OG+0VFb1BM8mJN34oqKq3DrCTGcFKCoivb1hdT29qcGDcrynvdpCW4y2awv0cb8Dr4OTKgw5+Dxrjp1925PRK56eTuUa332EMgvXgpqe0nGH3jMCqoL3kRyvcWsG3tUXkbApNqPHjFd30wvupdIk5pRszSpd52qfgeujGViAX+5DhUn3b9Vz1+ya+u0kAQCRsYNkW9DGmQU2G9ZFDX2MPdNGjNgFGzor2HJd2FNC8mn1SbT0yZinNUoM3M6lDhktb6rOvFnEGTi1pS37DvdPJXdyPV1YksrS7CSDs07FJadjizo0VrV6nmH+wxSZp9e4bTvr1Syz3cJUaNq9OjxVnzW8srnaIGXhUvvOxy1b/BTBarvtXLstY3TEJiFg/4yyYCEKgrrDkVgUiTOGEWzpVRQUAQiP6JCX7FyxxeMWxuIhShgSuGzfXfoEEoZMN8UeiNLu5RP/LnX5vFIB74YE9DoIiP6XZ+NSnKQNpmSZMmk+e4dIeelnq04yfPGQE2NSMVOppMV0zSRpMy+qY0ka62dGlydVVGNtgwQ2uKJVtIvXVj4OhyQaQctDuOVM2g7g9BLdWuncM69SCiSydmentaTCIHVWncc+Miqj63eVpKUCn6ROgTLC48tTlTdXC716w2w+uzcLEBkHSMTrPY0SKWknwoe4S2AEUo9y9dsBHO46tk5uW/efzN7J5L7ksCkU5pV4obA1leG38aAyqEER+4U7N1qcvRRt6JU+n72qD96le/lv73//6+NvDP2wYfNXG8IXy8FtUBG8mHl53IosYRz4P1SALj0OFD6QO68+EvP/3f7bLuHp2SsrMvimtMPkVDd+3ElE6Dn58xkfqx85LqQmft3IqIO+7ZAC8h8JGe4aUO0kgrZiFpIUZvLq3cnyS7GIVJ+pDbJN3V3rIifFrTqaweqeQbSYcP7pFove5OkUQNxFynSXYFqQ50skcwq3Wf8LBq+iSahcysEVBFUB/g5dlA9Jqrw8tgzaTsIf23mKT5sP5kZfHwpOnto74qxkpsuuPGNwyXc+fOpZdeesneU6dOGZMmpGaQsIGRg8RNxIU5E4wZmC+oNePdvXu3MWSQlkGahjC8MHJCtRkn9PO3TZ2X2gSRYRUq6hF2TPyjHrn75dibw/VyUrhew5Z9aUMJaygWljAbQ0ZHlXtmjVAwGdZ0/9j62orwVKoHz19IP3jmR+nv//4fTMKGueCQJGzuO3pf+ou/+ES66847NX5rgSNmTRurYT3l6FX0iYYVSmRr+BAZb2pGPcLcNKB5RNoeSvAK1w2dtkxnQ8qKtKr6r4jJCZORfrCwMJ9+8YtfpK898mh69NFH0y9/+Wu7b+uBDzyQPvmpT6Y36j6ukZFhhfWFGgce6EdAo0MShHYJqTIiL8NvwRk8BWSu+gNYWXAc+LcPZjMfz4jM3Io7C0otWQlnX+7ocdxtVQtVGDQ0sjGtlfia6sQ3Y8Xs7JxJ2fzd3/59ekp32SyLMX777a9N73///ekDDzygew1vNyarq6pU1opvr9XAMrW03EZL1j95H7Z6FSWtDxVfETvMcK/MCgIVBCoIVBC4UhBgXA76CDtzQcWwuVLQvZ7TcUrhjythOT83Sw06BtkKCBnWY35PTYtp4kAyYlp7Bxd06O7UuXExb3RATOqt5hZ0Vx7UjDbdueOGTXn71na5jmgqPX+hfXigcdqEs22ibfAx6kj7B1qyic5q0YEZ7RVILUKP7iRhL2Gn9hB2ay9hhHtIe+WvMITlMF6uMi0YOdBH1JIXW9Qz3CgD5KzV05hIBHTayGIRkNjAAK/iLVwx5OYplyGgn6ifRbYQHpPQjU+UyN3Lr4hbH95rk7uVdcKVWHUxnYi0CLjbZ10A87IxhLxZD2BakAiHmyIGdYo/6aC1JQ4C23pfa3zuoJmYWhQuSHPGee48mpEatCXhxXJakBjWkph6SNR4XDLgtSOUKgMw8z0ALwUHMcEaaGvtKyHvFet+1vzCjSHhxN7dI+nAvp1Sya/9JR0M7u/rlLSW6HztFQAdf8kJ7FN+0L6Y9sqoVditNXd5mYv5ewxzqv3gcamHfKqngsBGCFifol/pra3HhJvM4RXDZiO8blgXGrhi2NywzXeFC65Jg9mTx+aGcgLH1X3ChukTCFFMbFWT7IJOQ0xMzZqUDXpqz0pN2qRUU83O6wS//F0/bVdaa+0RLYOMgh5N1qThk3gxKZmDiDwRemywM2F2d7ZqE7jTxJtHh1GTJhHngU5NtF3SNapTECLEIMgguor9QSN1yIJUi5TrTKuh8tbYpocBD4MfD29Ejfxqk7IoMgtiIcofiEORKBYr8vEa8VW6eAzS8Dz4jum7hLaHIn7j0yxvwkQapV151kXnQ26NRSFCXcAikhls9LeYSizuM/jJT36aHn/8SbuQ+vfPv2DqyCCyyvhut3rUsvcMa/CzdHGjzfUhq/tJxY428ke0+X/0vvvSZz77uXTHXXemgaEhoUeLXzAoAn5WItFTkqAZuzAvaRqdthmf0bfEoeUHYxBxaCS5jHCDaQMGQAzKRehn+SHVBemP7mOT6BLh1tG6YvqN+3vb06gYgTslCr1PxNt+nbBBwisurARPrPQFjihFmyAxcfIJky9/mDytnnwS0epfxvFQ/svkyhsTLq7E9TQFpSKtcAt/j12mSRpsCOfSMpycD8kZGDT5G1I0p0+fNsYNkjMwZyJfNqVhtsB04YURM6R2CYkZpGj27NljkjRIzAwMDBiTBqZMqE0ySYLiNH6UN9KP+mJWz8uFQMmUq0+hQFhzDPiGWR8yxiTzLfA0D+EMG+GFbfivp/MFw+YL//CPxrBZEd4dPnxLOnr0/enjf/HxdKcYNjApYNawuc/jYxy9EXzlp+iU5lLrHjYmNC9lEdCMCBFm7rfRnlcp6uo55j718fJx1XwUdEX3vfDSL+jP41IN9/TT301f+ud/Tv8pRvbszGx63etfnx566KH04Q8/lA4ePKAFoe5w05hD31xa1KlASSvCgO7skHRZe4de7vmRaVI7Gp+sSpSS+Y+HccAl4Yi/rAmXualVcUmjQ7oblZyC6VyhTJdc9ZgGdY0jS5qbFxZRHbGgYAqni3h7urqsbzN30R6c/EPN5K9/85v0r//65fT1r38j/e73v9cmR29697vvSZ/61KdNvRsMqHbFJw79lngbYKXiRAmoQTz0+3g4wXrxJ1II8+KhK98KAhUEKghUELh8CAQ9RkzszFUVw+by4VjFaAYB1vKi+zSN2wE6TftrJgXBRj2MGx34lFrr81rPndMBvHPaoOfieBg5s1rbcQcpmjpg2PCyj2BMG1vryc4aT2m2yYTS4uCmMtLLwRTRUGLccPizVfsIaE/o1v2jA7pIflQHQEekLm1I+wh9uq+Eg6F2OE/0WmeH31Gq80ZK0WkZkjVKRD9GxSoLvqHXzF12aDCjp5Q/dbV1DYRShJBd0WqvrPZ4fHxgWWDyyFVp1OJi25RmilSJR/qeYh7XfTzvRnfCR66EIzb19qdMzb4V0GnoyEOuZvVv0qE9ak8RzJrE4ip+YfqB33Xdf7wsZow0PKCKW20+PbtqDJsLUm8+juTV7KLWtTrsJPByT7IdxixaxkvnmTgdqrqQv/8IZmp7U5sHs2ZZ+IH6szXhQYv2jtpNqmbnjn5j4KHNpb9PmiJ0qXKHuHfgEpChCc1UsiWkyJM2Dgyp1biyVBC4ahBgvo43309hDq8YNletGbY/Ixq5YthsP5xvjByYQZnl9BiBwaTsk6DNfeYRtiKc3CBKiIZJPC4XnJ1fTed0n8ixkxfSsRNndb/NhBFkXAi33iqGTUuvwmnS1azuF78xATPJe7q2AaTk7GQO3CARXkZwaZMdgqtLF78N9HWlnSK4OCWzZ9eQTkXobpF+EVpMsuzb688nWyu4/Xht8PM3plm+ISjpD7J5WE3CBRjkVMQsjICLBSS8EYYqpyIUQcwrUqt3JQfPw+N7WbGXcUkzfEsPTy88ytDYGvPIYiutLE4ZrRaksTx4tAgetMeZ02dMoubrX/+P9ORT3zFJmwU2HNV2kayr3xNU5ICoOg9tS66UywgawdNwRGEink0u2qS0yUahu3v60rvv+TNj2LzjT/807di1W/giptHCqpgzs+mUTl+dOiP1XCLop0XArSxzwsaZNC06iYW6PZ0bV45spkJWSXxe+a6qLovL2mzWX4dtItOCnLJZErNmTTqNW+3OJKRpDh/YJZwaNEZgn0nVKFlVyWBsBS9avwBpTJBW6eLH8Sh3cTvuzcLjazBo8MeNSZcn7pgJN9xze3yH1Ayqy5Cc4T179qypNGNz+fz58zU37DBvYPBQLjaNUWeGujKYMEjNoNIM5gxuMGeQouGFOdNVbPjCmKEsPKQTZl7XcI9wZhpcwQ9/LWL18zIgAOwLhGyIXe9qWNwQwj+9tzqeN0sqGDYSs7FxYew8KtF+mK4Mw8ZxpiwDo8ZWHp97ImTgWHznZg6HMh9C5D55jHq7xVFQJFRWpWAdJtSS9HYcO35Md758I33pS19Kv/r1b9RPxHQ+el964IEPpLe+9W3WT2w+ktQoTNPJSdQJTmuTQhebdnarD0mnuvocUmk9YojC4GLksg0ObTYACOzLYrRMmYrCcZ00nJd7a+ru7VP6o4o7wgChMXc1dWv+a5fd41EHbWZorJhQvufGzpkqQxg2Q0OD1q8HB/s1p7ZZfRhjgPspjflPf/fp9OWvfCV9+9tP6PTrRHrzHXelv/7rz6b32v1iB1OvdMGbahL13YphU48r1VcFgQoCFQRuJAhAj8X8iZ05o2LY3EgteD2XlTWMr2NYJrCuZFnvBzihVdg3kJq0JS6TX9UdJQvS0DFeaOqYSFOzC1KDrXVcIWVjG/Za67WgNUH7CCsS2RGJJLpHh8vadLcea03+RA+Ztg427C0bNiBY90llmmgr7iPp6emwQ3lDg35X7i5J3sDE6dc9Jj1Sp9ap9SJLWigjljbsGRiRZHSZrCq3uYef0ZNypKK8BFaAWt/St3lFOph6FAof/fprdlvDenjC8JQMGy+Pu/Ibcd3FV6mkGt/483hO7pP7ewoRojTLdCO0pVTUjQOVZdiiBgrgrU15y3Sj3vixf4C68wW1+cysDvpO6iCm1hTnx12t+axU481Jbx4SWEti1KxyP42t77XWNJOWUOLA1kxrGcFW62Ihl0mRy07+tB8aNGh37krubF8Xg6497RztN1X7+/aO2pofiRqTslKENjWsSeQoF7AJeh86l4dalutY2oG9IvezANVPBYGrCAFwMV7wMF7m8IphcxUbYruzopErhs12Q/kGSt9mVM22xeQTk3E5ZftkFVNwTFruysaQS9ssiYBisp2SdM15nZAYOz+tUzO6Y0QTMhI3Sys6EcwtIJaPJjpNeE4KeRrlZC93y5wTEky8MkUKoHNWd50E0Z0AAEAASURBVFlLt2ibnYwZRNJGaqwgtEbExOFywV7dc2PMGyVhA5hyIDum1ca3mOqLSVikDnAgLAMhzZcxbGIwJADBnDRRiTXBG+GAk8Uil4CUWRu+3R/HMl4ZrkjcHYr2IDX+ypiFdxZtM6vXpDFmGbqoaS1tr7ufpEflz09/+rP03C/+S2qzjtnmP5uOU2IKzBQb/o4LBhAjnh02RfoisKilwcwa1ycV+7YgEGOuIui2174+3feBB9OfsTF46xE7bYOqPU5coXYPQm5WYtGo4VtDqqZg1HhakNVOPFESiHjKtSaG2traor5RRQTxLvxBakv4MzrUZ/fU7JZUzU4x/UaHu+2UFaLQIVkD3CHSDHpUzConiMlqbgXBRjtGnwBPeOIbe7iFiVv+5GHDHTfeiMOdM4zZvDDOYMycOXPG7hM6I8bMBTFhUGcWKs1YdIdaMxgzvMRFaiYYMXGfTNwxg8lrm8hSaZZL2MQ9NTBrQnImZyZR7ihrmFGX3Iy65mFyex62sl8KAobtTQO5D7/e/wjkmOm/eSTGAHMleMPjCyEkr+hLqEQrGTZPPPmUpD5WCwkbVKIVEjZaJdOHNpOw8S7i5chrgMvG0hUFUkAfqyJEmO6/GQ4FvtWqVYvWpLIKVAtPuAiifmgMG0104Py0xsBfaEx85NFH0iNfe8T64Gtuu83qf++999rdNah6ZMxZXpyT/4n07LM/lyq5Z2Q/LWm0Id37cyS9Wffd3HnHHWnv3j0F/BkkkaLxcWdRzJpzZ8+knz+nuD/4QTp+4qR8WqV6bTTdcedd6Z573qO4++y+IB0MtYMKSsCKTVtNTEza3WNPPPVkOn78uBiz7en2214jqZl3Wb4wX0y6Rw3CH4yhX/7yV+nL//ZlSQ79S0LCcv/+Q+l+qXn74IMfkpTN29LojhFJRfrdUxXDBmhXTwWBCgIVBG5MCOQ0JvaKYXNjtuP1WWqXsAnaTCSJ1mRQODysrHxrnA187i7hDptpSdywic/l8mPj3FE6rvtLpBZL0hcwbFB7jWQN95UgbYFq2VapWW/jzhvRMU6/QbixduIlLzJVWfTaIVC5t0uEplvSwt1d/nJvbp+YODBshga7pNFDh2K0n2D35krLhx2oKWhHaCWS9L2JwiQbHNWH7CV/wnkB7MuC8FO4mdUcnd4zyCg+X7EvYt5FGLcXhai5Kb/iKWnXCEP+eFpJilDAnTjysLyC3lQYC24RCj/CRfru7lsi2EknfLNvObo7Y4lqpNfaVyYSNJPTczo8NCtaU5JUYtDN6gAma3v8TOWZ+GrgwopeVJqLqlVW4InW/AUjS462dWTlrcFStdJaZG1lUZkuCSdoe9Sfr5vU1EC/q8PbpfX+DjFsuLMGN6SrukQ8o1rftXBk0Fe1bI+oyCP2RqzuqiT4FrhNmaqngsDVhAD9Pd7ARcyKYXM1W+Eq5EUjVwybqwDoGyUL4QOzLBOtTfDFBMW3uRXOMRXbRK84PllxAgZxVXw1WCisM27QRTqv+0YupBOnzkhN2pSkbTidzATO5fAFYRLizcRV5HWYOHJrETHGU5AJljKnY3gRbbUL42DecEm8iCvuuEG8dUiqrfp0QqZLm/JsUHVICgDiTPuHKq8l6YRWkXakzy4/JEiNDKE+ReVtMJQP//Zj7voRMWCvPIqkPQP9FlHL75oD6XjojbGK4MoYwsj8jUISYMS04mnMp4jR1ADCmz1OljT4W4WBQ0ozUvFz/MSJdOrU6YIxcMokNkJaY3J8UqeiFjSOLKs9YSaguken0CGahBArvBJ5x1zFlL/bMUWM8Sp7ygHBvWPXnvTGO96S7n7v0XTba99gxPuk1KDBrEGaBuJNySsvWshxJCdqAanjI4SiqxkzfbVpQQSciDi1byfq9XpEjIsQ37t7NO2X3to9uwZMNF50u4tCG8xKPHCcCLhbgX2StNYJgs1xFYTxMoA7pT13i/YwnFKhY8KlzMGUQTVZvIzTMFtQdQYDhhf1ZWNjY5J6OqFN1WO2YXxBF6Bzkp9wxCFd1DeFNAxmrzaRkZY5fPiwvfv370/cRQPjJu6fgTGDxE0wY/KyR5mpQ16/3B7hI0zU92JmHudi4Sq/RgjQfxv6cBGEvu8+9OZy1HBb+V2m6LhdfkdC6kvcYaNTajWGzTM/TP9gKtFg2KAS7bBJl/zFf/uLQiWaFkA6shZ32CiyEivGL+F8MfxZBvhEOb2k/lXk7oacojZlXerrsBkOgZv54+FK6OR+hOXPxl2SJ6q5Me74uAYoz509l57+3vfS18Ssefq739WCc05SNW9Nn/nMZ9Ld99yjPrZb9Zfkn+q8tDCTXnzxxfTkk0+aNM6zzz6X+sWwef3r35Due/99YoR8UPfDvMZh4pxms7PYhDH+69/8On37W98y5tDvfv+8lW50x05TQfff/+oz6Q1veKPmwG7pYPdx2+CjMi5KhRpMdqRlvvjFf0ovvPB8GpR0zd3velf68//25+ne971XfX7AxqtoD8aZU2Io/bvuK/vbv/1bu5enq6s33fWWt6SPfOSjkh56MO0/sE964X1zpGLY5NhT2SsIVBCoIHBjQSCn3bBDh1YSNjdWG16/pa1n2Bg5pcLmTBvoFQ7tQR1yLwlrQ901LwkLHRqUSqzjJ8/Y/TYc3EPqAmmcRd1lYuphtUfA2rGltVNLQTQtsA5zupBft8miB80PRssVNBYhYbiYRIVopzapyuqQBAbSNdxrMjrSp419rQ2HtKkvZk6nDvJ16N5N1lRIMiPVA41rh/uUjjFvjGBUvkZzUlundbH5E7RvUbIinH2J3ssZAhxApD4Wt/yJhDKzTL2kX1VVI+rIr8irDhqAwuPpKJLScprXy+HhLS2F4c8fuXsAffqqGB97YaToZX2AimTWsUiSo/KOPQHunZnTOzW7LI0Z01InrEOfU2LCiTmHyt5lSdGY1FWNIScGjQ5lojmDO489P8pMSfQDrIrXyi67uetOxvWVeXkvqq2k2k4Hi1CXP6CDSTul8mzf3h2mmWVIavF61c52qEwxUX8HHlArMgE3gFtI1ujTkjfTauwwKmHrPtVvBYGrCYFaHxXOgovxModXEjZXsyW2OS8aumLYbDOQb5jkfZKiuK7ejLnJJyRzK6w2J1qdisnRBgkcmGx9mwsigbiiuYxpw4kZ7rHhVAWqrc6N6Q6Sc9JTKzUt6PPXXp+id4i4kLhrcZoCsVdO0djJCiMEKE8xmYqsC52kqLbibROxhd5ZJGt6dUn8QH+XTshw502PiK0BXTSI1E2XTeCIR8O44XJBTIgsJW0/PklTIK89xFNRdQIINiWcqLNGR/cnUB7QQsdPEYZPoiugZVeEd9/GBOpIpCIhYsWGXOG0qWE5mK/b4nvTQnqZ9MtpeqRSKBdE17x0yy5L/c/q8oqIZGcirGAXU2Bem5QwdWbnZk1Vz+zMnF1cPad2nZtbEKNFJ2imZ+2dnZ3XaRoYDbrbSG0/g6i7JGWWpL8WRg5t39ndl3bvO5jueMvb0+vfeEfqGxgRXnQYk6alrdPCOKEOfGgtTlchWSObPnnVJNZOnKRyqC6JeJtRWy+JcdEh6ZEBMfZG0q4dYu5BjItwQ5ct4u+E55QNem91TMdPY5GTPEIsGjVrNLajAvCiTWlViDtw1p+YOOMb3GESjTEXZgjqxHjCHQYNzBgkZ2DGID0DE2ZyatJO9MMowx11ZtgJywYrabGI6JFEDHfMwHxBgibum8m/kZ5B9RnMm7ibBika0qDMpINJeXEL93DDxM36gsrON2/+hF+YuX9uD3/i5u55WpX9UhCgb0f/rg8LXrqP94Tw9daqb7N6v/hy08aEVe6wCYbNBUmK/OgKMmyKMVHZeUkb6qPPqImXKMoe5sXxpxmeGWxYuPNEMlk+NhrIPeY85jdULaxoTGTh//vnf5++/OWvpG/8xzdkf9760t13350+/elPp7e//e0mQdOifsKYvS7YTU9PSdLlmfS//tf/k5544kn1n5Z0SPf+3Hv0XpPKuUOSNp3cDVMUx1SOqdZnz5xN3/ve0+nRrz+anpJaymPHT1ib9g8Mpve9733pc5/7P9JdYhQNq98Hwyb60oTGEaR6vvxvX0lf/drXEndV3XL4UPrQhz+UHn744fS2t75F6s0kBcSYr7HJ461rbJlMj3/78fT3//AP6Qfff0Zj/LwYSrenhz/ykfSxj/15uvXILaYWjbGXeTTyizbyOgRQAbA/0Q4+XDS0cQSqmRE/zJpHZakgUEGggkAFgSsEAcbl2hhe0KkVw+YKAfcVnww0VkFnycasLxQzes73DLCL/hNRwJoOu0vPaP9AEhZIXHDh/OTUgu4vRVvHpNZEUostmmRe+wrQY4RPbd1avPWIYPPN/SDqoNVE2ljGrM+gyYy2M/JDuclEBbjtJcC0kfosDoJitrevabO/VQycdt9b0P0m/b2SytDmP2Zvr+711L4CGhnQ+gEtZGtFFSdMJVw80Elh9/JYv1ONCwh4UIWxYBbY15oEseJadFFZfBSEaYQ108IFFWaBFc58iFA4uGFpFOFLKaQIYxnYGg/ZGzLExcBocaiLtxV1gA9mjBpJxrA+p71Y/8/OzZs5P6+DhnrH/3/23sNPr+LY826UR6OcSAIkcs45Z5HBJtjGXvv62t67e3f3ff+Yu+++n8+9F4Mx0QZjbEAiZ5PBZIwxIIJAOY40o5FG+/v+6tTznBmNRJIGjE/PnKf7dFdXd9c5p7u6q6t6bW9ZpWe2HmGbN3Yi2CEv80nwMfes1n0U9nvB+o/r7qdWtbt6p7ROIRGf8oZPGO2pETv0yDzwRp9HM1EbkqZoLsycf4rm++PHaY1Iz3OkJDRhIj0o41L0HjDXiWenuW/SjvdLLvpJSMpzaVxDga+fAryTeTGG58V8rhHYfP3PZ5vVgIeci4dv66DZ//iP/yh/0E5MFghZqN1H5j2uuOKKct5555W9997bi3zkYUGPBbvGfZsowCjMIMiwhctB0jdipsLPxau4E7zeByf5h44j8oMJgY2HUcVpk4UH6bUarFet0sBtU2nSEJB9/JXaZbGuW7swNpBXdmkR3siPi0G8YlqiUFWNMik7BufwEdooB7tddHHOTceYjda86dSCFAIb7NSOH9+pQVyM1phRZsS0+ULvcjJZZg+EHgpEmJIZtBnAFeUfvoFw0Tm2aVZF2wuYYDGS0Yj0Vv4W4RyoZ65hADuuqkBVditHK2Cg6ifq344ZwMA5YfOM2SqbPzIDGzAwYhQbT1a/uqF/6NPVKy2OHgl0EOT0SsumWzu6eySAYccM58aYcZMGDszaOjFxaySwWabFwMVLOEtlhRYGpS0iM3nsBO+REIj3pKNTpoJm71X2mL13mThlWhk+qsMaNTDivAvs2LGQj/eCAUrtoW5mOv3cePt4PxA4SGNm+AaZz+u2Bs3UKZO0Y2pimTxJWlhWdRezzQGTOvvBh3Wrnfhuq94vGEHw+w10QdUzFxw00b9/8F2TKp5o3MBJcNIOwQwmzVJLJoUvTJKJ454Lwc2aNWsMi4CMPAhoUjiD4AWtmLzq5szGjx/vPhuYvNCu4UpBUdaxXk/C1LP1nlbtSJgU4CQM8ZmWfuZNP+PTp1xcphMemEZc4z4PBXj7/CZuBhyx/MZX0h+AuP5u85hIH1xgM1DDZg9pfMgk2pfSsPEX58KiDnyDlVP1B+vBok0J1H4HM6bu53tWf8fAqalmC8zluayI8n1WoqpDClb5BhGE/OpX15cHH3rIu5F32XkXtf/McsWVV5ZDDzmkjJaWmvsICXnQTNqwYb20Vd4sN910c7lXQp5F0tAZP2GitXF+8P0flGOPPVbfaae+TU1aeZ7qgynvgw8+KHfd9cdyzz1zyxtvvKEFi6XWTBwj4exxxx1fvv+Dq2UW7ZQyc+YuZbT6MVOLgoXlQ5lAe+D+B8vdyvvss89JmL6mIBj6/g++X8444/Qye/YsC5ooD2FU0gch/PPPv1DuuOOOcp/yf/jBR2W33XaXoOci8YSXl33330/C4AmCj0ltTmzzOVEL94fy0/V/BtSPa2suiZ/+1mCbtIYCDQUaCjQU+DIUoG/Ovp8w404jsPkylGzybE4BeKwY72PEF5egdyw5gOQCYtYFJ+LZVktow8H0XGjWdGEqTaa0Vmr9YMVKXSu0+U/rB12aY/ZsHF42DutUPtYMxH+IN2HxnzNNEdiwvEEZES+eoio4eBUENqRqTYHNepxzswkT2ghvJAfSHHGUJDKjdBj9aEyoaeGgYzSbQ2WxQOa0EOiMkbZGhzYEdmhtoUObRscoPHIkZwpSJhWKb4ywL5XmSvDtkRy1C1jdh8uU6raqM9QzzmyEMWSeVtOUako4wfwY6NIpqUKnGJ4JHrWoYgXrKJ6VAhbKCIT1gD5ZuGB9B+sZbOhcp+cSc32ZNdMaj02bVWsB3VKVYo2gW2sCa0XeHoQz1MGLSgrzvCi3urdGDUKbVpyCytBal9HzYUOln1WJZ8RzwvwZ2jKjRPPxY4eVqRNHy8T5eM31J2r9h7UfzX31vEZqbUhLRH7LeNNyvp+UZq5DmI2Z7hNdDyocjjiuaEM8y0xr/IYCQ00B96X+RrN/ifezEdgM9ZPYzuXxoBuBzXYm8t8Neg0/cDRywT6kHyO8x1KlieeoOeAFrfeIxXIcvzBHGxXHgO5YdrRoAAYv9yy290jVedmKbtnUX1g+/Hihd850aTTfIKZrg01fhcZNMF/KL9VjGC124QRLAXIENvhcwXBRB93oP7QqyqZ1WtzWDhkN1AzcM2ZIHXb6FO1G1iL2WJgvDd4a4Ed5t0UMvm6KF+pjpwWqzxbqCHNQw4U4HAIqmA2GflxVnyrswV2QHvgzXTDJFNGafkgDg34rhsAhMgpOqO3kux71ylRJ/YAyAw2Kf0FlpvCD6SO5XV7gIKaCVV46f67QrEC4hvaJEiq3A0ypnlHgob2YLwtBXZhEwz6xVKIltFmugwUXLlqiZ79Y5ryWS0CxzoKI1dLGWScmD8HMuAk6R0Umf8bpQO0ROpwbE2i8N2Ya/TIiYKFMVYDnr8VG3gcEeXFJsKw0hHeSz5XpkzaVmbtO8aImdok5vwYhDTCgoKVBAVqQ7xLhpFiEKT9et+oZ6p73Uj+mBweSoy2U9MrJLwKu7GsJoxmDpsy7777rRVzMJSGc4ewZBDZdOhsIoQ4O4QwCFrRg0IpBKDNr1qyy7777lj333LPsvPPONmmWmjNozSRTmcJ16kFcCtoJZ92oK2HSUhhDucRR18xLXKaTn3zpuMelT576/cCwE/WTcFtKT7jG3xoF2v2J6bg1UKfFs8o3+7PA/STVv/ZtpmHzXKVh83hlEm1rApvqw6n6JN6T6pXROxC1bo8xRDjys6pWpcdXmu8ekfX3ivt6Gvfhgm78qjb+1omPvO36GZbq86fKIqzh+3zq6afL//k//3957PHHNUkcVQ484MBy3vnnl0tlNmzf/fb1uLNRndbGjdp2KC1QJoKYJ5s7b16ZKwHKq6+9rrQ+a+NcffWPyiknn1SmTpuqvkmmPVQWZiZ7JfT+y1/eLjfecEOZd+88m2Hj3DDMSo4aNbocKFNomCk7V+d+HXjg/j7XjcmrTY2qrq+/EQKiBx98qMxX2fQHp556avnZz35WjpKJswnjx7lfQbBNmfFObLJZxbdV7rz77iu333ZHeeXVVzV27ljm6BybyxFIHXZomaa6DlOfHxsa4p2CRrjqiTjMT/15xLMALmAHppM73o0v9p6Cp3ENBRoKNBRoKPDFKED/nGMkYXi7RmDzxWjYQG+JAswTYrzn3UoeASYBrqtKqfzgHJiVwZWhOeM5HxgESBjehg2eK3XGzcJFS2UqbZHmUtrctq6vdPeN0cY/5i28z5guG6VLawnCw/wRU2vgwdW5C+avnk9q7gefhoUJAKldWlaIXIqWlnkRbwbLNFKCHM4+YT6JdY8J4zlLV+feaEPgBExu2Ywa1g8QAMQ1XEIF1iT43oiDBrHZUPetWmUFSc2w/KruDlQNiPa0EoTQSIOeRBvApbQbDc4qCx5TOQtidFN/Rjy53LDJGTRcmsaLN4UPjvn8Cp0ttExz+uUSnq2WFRUENqRBb5HSz4I1AN+P1txUl+eR1cybZ0P5wFJ3nlsIa/SslABNzA8yv+eywEYCmh0I9+ra6Lk82ukdPn9odNlxameZueNEa9VMGDfWgjNoTVshG881LoVdcIgJDVO9IDwf/gwNXRyiLnEBxkXdXD+lN66hwFBTwN+rXkT8fDfxGcMbDZuhfhrbsTwecC4iNho225HQfxeoGY5gUmJg8iCVg1VtNEqBTbBZNIx8jLQMgOEcYzzBmuXQaAhwaYDeIHul63o22CzWMrRtlqyR0KarLF+pg+ikccHhgr0bxbTZJq2EOKoazBrqzO2SqCVXDKuMmvm3id0xfWvUiensEuVgQYodMphL65Ra83jZpx2Pr6tT5tM6idfOGNLYGTNCuzUYvFsL+grnAO+1LZdLe6saWHIQd0ETFwuAIeIHyuDwB7QCNAHkVO4SmrDTRLekf8Tlbx0yw1EGKE1yR7uQwOWsJCYOlSf8uACt8MiDKclBgXOHeBAMCEac2cmqOFpFTpi6Hpk761oXzxj1dcyjccjgSkykaZfUah0o2dUlk2iyR2zGTgXBEHJ+0XAtgCKoGTZCDLfOgUAYRGVRoacOFhYozNvgHTIg0K4oTOVhtxYNqgkTdKigzOFNnTym7DhlmHbb8Hw7tECpwyn1YHm+XFUDCagJtKD2PqsMU0I+zu2GaBVNYGYtjBE3mmbiurQ7PYUvKYBBCENcPZ4JMYIbLoQz4EbYktoyqSHDPcIYTJy1NGi0u338uNCgwQwaghzOnBl47gw4o86uvsM8S1z6kZJtizvSUiCT+RO+jjPTEkcdhriB6cQlzJbSiW/c56eAnpReR95bOT9anvnW8mdi+luDrVDq+/pCApsjDtc3JqGoJqchDKBuVC7evXiHotx4HdVvVNXJnXSD12pLbSNzuz31dww8g72H1KX6ul1U5o68tXJUZUNW3wRabpgW4zya66Rh89zzz2uiPqEceeRR5cILL7RG8uzZs40zJrsxueT8mwWffCLzZk+VuXPnlicef1LC2TXlEGnjXHnlVeXMM8+QibTd3AfQIaLBuHrNqvLnP/+5XH+9NHkefLCsUZ+BKQkeMGPh7rvPKmeedbaFNkcfLfNmWjhgcYC+nF2NL7z4os6hudFCpeXLV5QpOqvq7LPPLj/58U/KQQcfZOGQhbpufLxDtL9nfU95X0Lkhx96pNxyy63lmWee1Zk748uJJ55cvvOd75STTzm57L7H7nq2/QU2NDrfRdXQNBj8p/0uOE+8BBVo0j7zpz84pia2oUBDgYYCDQW+PAXo83OMJNwIbL48LZucAykQ86mB/FZCBScgrgEewP8509a4r7kgYozkFlgD4EJgsF4WGdZJewOrDWjbLFuls1HW9JUVnHmqOSZpwHmWaK2bOBPFuCo8bLZDeIBDUEMdSIfjCE0cUnCKIQHHHFibcKglWh3D8xq+SfzuJlvu4OxczsFB6waTasw5WVfA50wVNHVGab6Gz/m6o3T5nF2lDa94KvPwFetDGMERfl5UiTBka1dOEdwpjmin+Ue1rerv71th86Yipk39ij4IueAtWRPEWsZ6bUzqxfS5/PUyiY4FjLCcga9zhLR+062rB2sa0n5ar7wIapRdvCvlia4SvnhDo/hR5vV9Ep71eS5f1VGV8nNXY+h/Wlot2Qg2YoreQWuZqauENZiv8/nFnEfbOUra3p0ye6Z5svzx40aUiZ0jy3gJb0aLxmisc46mBXJeQAlCGCflilJxqVDIR52qIBHmiqvJScxbiKvgFIg4IhrXUGDoKeA+q/qO8v3EbwQ2Q/8stmuJPOhGYLNdSfx3hBymCoENg1UO+vL14efBdQxeThFMQNA8hq7aAjdRcjHk4Su/QIAgP/jEeiiOxfdY2F+nwR77tEt1vs0S2aZdsmyNdmtoQX8tZrLEREi7gp0xcdA8gyWMAIxcMANmrCrcMHcuR4v3m/q6dfWorNg1Qy1grjg2ZAw2aRHSyP7sOAlpxmpw537CONmo7RzdUm02k+VDBhnwY4HfJdCUKMlNanWUVZzJEj+KSQetTJEgW4XDqdDZGBM2QLgjPgQ1lQ+afk4R/SQ5UY5BXEn9KCqyEYFzQniEMx2fW5Ijg+gXTxO/T5IYolmoI7PjFIGgBQ0atEu6JXxY261d6Npps6ZLDLWe42qdX8M5CPjBSIsx7NVz7ZVWjhg7BDLDh+vBiA5oZ/GMU88l0iqBjRi4OPibd1ULoYKEcRuh9o8Qszxm9Ag9Q+1y0g6nqVMmmImbMkn3EyTEGR3PkObSPL+KNJMXVO+In0CNoVNCtE9lmgZqI23vk3AGBnad1PBXy1zZKs7jWb1GbdVZPtxLOJMmzRDYIKQhDgFNmjcDH8KVNFGG0AWzZVO1oDpjxgz7CGgQ1uQ5NCms6ezEbBJ2mqu6y+f9czscW/0Qx+Sidav3p4Krw+a7m3AM9G6v6kgaE5q8B4a49DPMfcIMxGfg6qdeLlH1/HW4Jvz5KZACG3K0uwGe9dZwDEwceB95ed9xmAZEYKMfT/SWLOUMm9SweUzC9U0SHlQaNpdfXg4//LMENmBVHcFNkK+vqsKWBTZba1P/+vNJ193g7xnf8wBA3/JTlaWgofARoujb4Ayut956q9x///3ldzIZRnjatGkWZlwkk2FnnCnBy267mU7xLamvQsNGtFsmM4dovcybO09mzu4pCxYskJbcXuWiiy6Wlsy55aCDDnQ/QFmYUPtUAp4nn3yy3HzLzeWZp59RHbTxQMJZvknOf5sgk2rHHHu8zNBdUU477RT1faO9KAAc9fzTU0+V63/16/L0M8+4Tfvvf4AESueXSy69pMyePcuC6+HCxdjG++LvU5nRIvpYdXv88SfKDTfcaH+UNH8OPuiQcv4FF5YLL7pQ2j0Huh5MspkMp4Ne3MXTzdiBvipYo33/fiGfc+JMfyCO5r6hQEOBhgINBb4qBeh/c4wkzLjVaNh8Vao2+YMCsT5QH++TMjnutzgx+CxzD+nDRWh+KBbAZ9wonXUDuAfA4HM2WNOjt6xYs6EsX71eJtMQ4KzRvEtrCJpzMt/kXBU2iWoLpvDE+gMaH2jwsH5gfEIIPsqPOQzrDBXvITjD8AMQc0LPUJk3csV8tGijKOFhzEk1ZcU810idbzNaghg0p0dx1o3CzP0Q2Fh4U4UxtUb8CO24sVYPQgYvdcS81XFo5yhSVa7qWNUPerScqEk79GNaVWHH6bv2JkMlhPZ3aGNv1OZY5vAIwRDShLAmhDYW2EgYg7CGeGAssNE9mt5YQWFNB9oWWUeJtZ2IY42GtQJrzTCnV9v6JMShbvC4+lE+LtZWgu+DnpEmugrGKwGa2CCsGTGcDZnFQrBOaTSx8RYhDWfTILBBw6ljjDSeoDt4A7WeCWsWUUa8czxlP2mnsVEz30VXpk5WT0yIqCLxQA1uhXlD8zVx3uanocAQUiC+pXh/o9+K/osxvNGwGcIHsb2L4kE3ApvtTeW/F/wwVSwsMYjGOJdDWoxKDMAemuSTYq/6icVuw3kgVucBEg/AjGgwSAxwOcIFLu5gKFigx0btei34r+nqleBmbfl00cqyQGazFi1Zobgewck+rXeBsKBPfpiAkRqENXojPAA/taoGV8rq29irMrUThir4EoQGfbQw7MNUCRWMFevfLPgTRttmyqQJWjTnvBOYgE7bqeWQOu/UUJ7YsQFeWoEprmAGzFQp3HYq0+0mpqJbO9EUdbVrcQ5WDCLYqTzFQH3a2XbcZAQAGW5DOETl5KIa3ORFMMKkhYaNnpzhk3kNaJ4TMDA5lIIGTZ8WadF8QqjGDpsumTtbJc2ZZStW6RmuFLMsu8LSsuF5maU1fLC5NpmmvL3KCwNpFWk9R8qAkeRtpG7ecQOnpVIRlPTxPM3ogZEdN3FA5Gjs1krQNm3qhLLLztPKTtMn6owa7Ahr9xJ2bc3oRnPdZPBXzsya8Cde2kdF0OTBnBFCKJs6w6yZmNV169aV5Tp76ZNPPi3z53+gQfE9mXf7WGfyLFb7V9hcEufN0LfyHjKIpo+AZqeddip77bWXFrh3d3jKlCkWyqBRw4XwBoFMXmbidc9O+IzLyXW2AfxcuBy0ua+bNIPGKXzpl09tz51N5IlF5uqZqO5Z/8yzJX9gnRIu65X3+FuCrcM04c9HAb6o+CpF13gFoLBovPX88Vzi+9/S8wCdceub65MAASEp32sKbG666dby8COPqj/os7bHWWeeXS6vCWywKe3Pl6r4/QyM/euXFU0/Yer133J7sh3t9kbfVc9NePM25jfTIpqzUM02rsCS3wTfE6YMn5Ig5K677raWDd/+ztX5NQheTjjxBJspjG8JHoudnt3GiSbdgo8XlPvuu7/ceNMtMnf2F2u9nHrKaeVCCUJOOulEC2uh+tq1XTqz5nWdd3Nvufvuu8s77/xNWnYakyQc4jv+eMEnZb1MSB5w4EHlqiu/V84/f06Zrn5vrLT0NmhBYYnOCXtEzwYtoJdffkVCpenlnLPPEdyFNoc2eepkNW6TNU9ZJEBo3UffKke/t3DhwvLEk3+Sds+v1c7HNMwOL7vuOlOCodPK1T/8oc7POc59Ul1g03oPhcPjlbEN9tP/GffvI/JZ5/uQ/mB4mriGAg0FGgo0FPgqFKD/zfExxq1GYPNV6NnkrVOg/1gfKfBehNJnzNc4X2O8mHMy58MNq0yIkaeOjXtriAjMawiaY9qyg9cRujQnW14WLV5mk12Y49YMX/NbnZOr9YONEuD0YtpLeXZQ3DBvGNQclJmly4lvIjRtom58G2yCMz/oWWqsfSDysXUGVzDWUZj1wvuOwJoDU1juDRe8qJurezbLIIQZqcUHtEFinhZ88yhpkGBCLeduWIawRrTmyta4UWboFrhMTdPV9VRd4NWhD+bLqTOXLUJobhrzWnxtQJRWDIrbARN5nE+1Nr29HhF0gTYIdwI3dEGgFptnWZfZpA22nqI7IysGglEd1Aoly9wvazZ2tJ15JrwiwBKCydwc6zZ9fRJ8aa7BucSjZHKOeUTpW+/1mQkS0kyfNqnsvNM0+5MmyipKh7SYRDvTB3pCV2FUEGJH2DdEVPGCCbPuPDtFQidVw+bdHWHQFjS5oCvOvmDIlv2mE5qfhgJDTIH81vGjL4g+gW+5EdgM8cPYnsXxgBuBzfak8N8TbhijYDSoNeOSB1oPpXQA5jiU4pENkGpxECgGMa5qRFTmGNZ0z6BWDX6BE6h2jsjJ0rtK18167ZZZJ+2MlVrsXyrbqEuXo22zWrtmpJ0hU2lxmB1Mh/DIXJovdmxoSIaRQOhQDc/BqFCoUqMK+MFgYfeUWoQ91BDiWJAjGnC2Cdo242Q+C+2bTgkCOsfocHrtkEHjZrQuwqN1pggMlTbGOA+MlBkGdsbQdJWMczkRVHkqkyrVXQK2ckRiGwwAXYowaCuBQHVjpBFO6gNdkZ6WGmkIZSJsbFWQ+Hw+iR4fpgwVZwQzaND0asHRKtBWg+ZwwV6fS+Pnol1M68QUd62TqTP57GraaLxmU/U8oESEKdu4JQDB90AD0VSoB56KESXs3dtK6sN2sC7xrGLktHtJdEczil3lE3Wg4OQJ0qSZjGp0p+55XmKABQisliNbtGjRjLL0x8CGBgFnRvRo1zqaMitXrigrZEIIDRpMlqFBQ5hdh6u1c32FBFKYGFqxYqW0ZlZb2wYTRt5NNVpla9EUwQsCGjRnuNK0GVo07MhHUEMcMMBzIZyBOcdBkww7oorLQbkeNzCc31y0raKv8GV8wuNDA74ah/29RNnc1+F5Fp/l6vB12HreLcHU4Zvw56eA3uDqGUaefJZbxVC9+3rCAst+It6Bej6euPuTSmCDlgjf69JlaNg8X26S0GFwgc0R7gvrAht2zdEDhMsyucty0ycuSibUrl/c9futtSP7usQ32Ova/91TywyUdeqHud8NcDGZ7SuLFi0qDzzwQPn97+8sz4kGCGpnzpxZ5khz5eKLL5ZptCM0kZxhupFvI2fRVAIbcCyVIOUhmRpDc+VlnQ1Dnfbf74CCds4ll1xSdp25q/u81atXWjD0+9/fUZ544gn1Nytlhmw3n1tFv4AQBqHxnrP3LlfoXJlLLrlI53TNkInPsRKgY9LsA9XzQZk0+42FPXvO3qt897vftYbNnnvOdr8DndnZyUIBLnc9MqlfjGDq6afKr667XvV92OMpQp+TTjq5/PRn/1xOOeWUWDwQb5AaNn5XhKd6q4xz8J/686Xc+jPIdyPfh/QHx9TENhRoKNBQoKHAl6cA/W+OjTnWNRo2X56eTc46BfqP9f1S+iUx7sdY72i9k8EVVHPAinvMOPC02Yb2GgL8KWa2vYFQ5tFWSNNmpTYSrtKaAkKbtbL84Etm17u10RChDRtCN2kdAeEDF1o44WL2Tpn96wbnFDUhZChV3ef5ulLtOKa1TPrhnZFk9ONm/N1JhiEgazp73hslI8wYLo0SzK5RNiZwLaRROLRRcp4Ifv87o2slvCHIqmqpe75rX6KP0zTn3aRFF2RicUYsApmAEYhbRytUeHUhmGH+rjQXolZrTh+F4ysd+GotAUDD2eeH5ztCF8Idqkq7RA9f0IYLDSXMzYU1lNHSSBpnKyiaS/t8mlGa52uuP0lzfGnWdI5l3ow5uVhzgSKurorzXMM+5Ua86wQMxfMTDeFOzhU1YNSPKMUZMBEQFTTyM2kBkr9xDQWGngL5PuLnO4nPXLMR2Az989huJfKAG4HNdiPv3xniFNgwQMU4FSFGNQarZGCcyk81jAUUY50d4DnAEWwlO4GYSJbnjqaKSIGNwQWKBse6HmncrJVd2uWry4KFS7VjWGeArO7WJTVnnY0SJtJgALRrRmWmybQoAObLy/Qec70QRWXEBFETC04s5KBkMQ1sCcHXpdJVuRDisODIDo8OCWjGdmA6DZNpHRLijHW4w2bVlC41XGzQ+pLKM4fdjxCD1WLCXCYsTZQtTy6I0yKZa5axFeEMF7TbwcyRI1p5E5p2ZbidM/KpRGcyBMyU76o03US8mDgzbxKGiVtDeEaHDzPb3Y2JHHYucR7NOmmQdIvRhfHtifu10qTp0aKkdtYgSENIU/RMoL8FNLqHAURgQ114AvBBaK1sRAsKRrZyZqxEJHZ884z6tNMbhheNJrSlYBfHaIERG8HjxMBNkemz6dKqmT51olSj9Vx0yCPaNiNHqGQ9O9ObUilQl/s8VZJ+D7M/LGz29HSLwUcYKBNnMl22eMni8ol2ry/4ZIEWV5fZtNDyFcstwMG02dq165wXwiFg6eyUQEYXJstm7DhNJs2mt0yapRkzBDTTp0+3Jg3nzeBycM0wi7C5KFxPq8c5Y+0HuHr+vB84eHOfrg6Tef0tVjD1sjPP1vzBcG8NfrC0rO9gaU3cZ1FA7zUvo1y8DVuHj8ec70PmiEnNQAxAGbf6w00yica3igbcsmXLy7M6u+XGG28uDz3ymN7bTWW3PfYoZ50hDZsrrihHyCQauyJH6LvVa22nrzm+Q99ledy06+AkYtyf5d3mfv2di9TAl695vOPg5Zvvnz/f/0iLFvaH2Pwuvw98zq+5++57yu9+9zudL/OyaLHM5uAu09kul156WTnggP193lSWy47BDRt71CZ3QTKhuDrMnN18s83KLeNsmclTywXnX1B+/OMfl3333de2zZcvXyZNnHslcLmlvPLKK9auO/TQQ8vBOvOGifVDDz5Y3nzzLQlpdivfvfwKnS2DmbOZEgKP0wLF6vLqK6+VedLOueeeedKWWVwOOfjQcvXVV5dzzjmnzFBfNFIbDhjnsKuO5p67kmoc3KA+0s/4uWfLtddeZ/Nv2DefOHGSNIhOLP/1v/5LOf2M0y2wSZvrUC2eCz28GiuE+WTzedVpD/3TZXrcx7Mc7L1I+MZvKNBQoKFAQ4FtQwH63+ybCcNzNgKbbUPbBgs8WPJZjO3JFcAQVdRJv3UbHC2gCQ8KwCI36RWM87TnldwChyACU+po3CCgQWCzXLzXcjaBymTaKplL65K57nXdOo9F81vtHazmrpgcgx9CR4aFT30PFA4/4woh3AnBhYvWT7Cr1dpCFRl150Z59T3xTRHO1ruSJCszeJnnRuvULibMEmDsMBwLIcpHgzLjwDDZ0gmGcnGwctA9aE+M6q9CEg0xdqztaK6OaTN/+xRNPkGa5lVDoIc3ELqAxFLBUCe5+hoFT8T1dh2iPqRbrON6UoLm/juI8LqgNubNMXnGRtgOzfExWe/zaaZM1FxfZs8mjpPlDJmsl5AG6yjGL8+aOsYpVNSDX7dDN3KmLQlV24iD5kmrxEO8gXig/g+f9a/2exg0jfwgbVxDga+PAvmN4+c7iU9/0whsvr7nss1L5gE3ApttTta/U4QMTCycy/coFgN2Dn80iphqKAw43fcfrqo7j3GE+6fGbgw6FZICm8sTHvMnVQlgZXGfhX8EBggK2C3DeSicdbN46UqpOa+Q6S0d2i5GDEENzEYIGeiowC+V4eGjNc6OdBHeOaIEBDe0gkrAULgW1NedHTt5sk2uBTUJZkAjPqrNXLZBi8kqh3cQY1HEQCDYQQNnVOnQro/xEuiM007nzrEdimNBLDQ9WLyk/QMvGIpkCKhe0CU9Eh3pH9ffP9xWgVp6QiaImmbGCxqH0CRSYOi8u0Y07pXwhEOquxDISEi2trtPgppeC2agPwcKokKN7VrCNhdGfhEdlfI4Yyh261AODA7CGirue1O7Yu4Uyw4hVMs39PZYaEMFoUGanNtBu4rMsKqSqEVbU0ZxaD5Nl6m6GdOn2PzZRGnXjJWQpkM0ZjeOZGXCIzZbFwwgz5WBa6PaiIAIk0ScI7No0eLy4Ycf6PpIC7CfiImXYGa1zqJRGho2aNWgbYNQxyba1AhsDiOA2WmnnWUaaFf5O0pLZqoYyAl61uMluOnUwdwyn6dnjlBmlDWwRnkxlDBxLIyiibU1lwMvzzzZa+Jw/d+RNpYvGt/O2T/0WeX0h27uvikUyN40ugG+negZ6Gdajji9Rwnrt0vpxDEpZTIW71FkIhzfrr9iAYZwFbOAy3Uu0wsvvlSutfbFI+prR5aZMvF3xumnlyuvvEoCmyPI5O8ZDTe/y/h+j+mgXbp/SSOaizKpc7uOBnUdM4/b5PYJyviqPMYRuMNudtXfGCZxU4l0yqCS6B/6ld0PJzCBn1z0BZw784c//NECm9dee80Cmz1mzZKw5SfliiuutLYNfUE6NAP7ZM6BtiHA75GwG0HPbbfdbg0YTJ2N0UaA0049rfzgBz8oxx57nMyeTZXmzsJy551/kFDsBmnIvFN2UZ8z59w55ZBDDnaZd0rD54UXXijTZ+xYLr3kUglsLiv77bdnmTBxvDVv7pXZNcy2vfbq6+qvN+msm2MlsPlROV0mzdD4C6G4hOOa+VKvoCtm0bTrVGd00R9Sz2uvvbbMnXevx4dx4yeU40843gKbM848S0If2VwnLw9Rrv7ciGnF156BAQUJdLrsd+I+6Z3PKv2EbvyGAg0FGgo0FNhWFKD/jbFfvbLCjImNwGZbUbfBE+N78FJtarTH9XYoUtucAff979oQwW041T/iG7RmwJ88zf4sJzFva3Npmrd2ay5rqxBYhtDFOTeLlq6Qxvgq8TuY8MYqBLzQCPthFlvfA/wNi/aaVaKJs4l1BcrhL3kb/KohVS2iqv4NOCdXMDTL1a5BwYiCxt8jJUiAAV82eL4aggonXuI0HaqyW5G1slpwVV7akpx5lEj7lKgKIKzJCxRVkwN7NUkAmn/7Son7LEX3BLURaBhzek0KOE/Ywig0aoZt1PwayyaaY/tMGs6fHa/NR7JyIqENayjeqKnNRaPYEOvNmJQHUtfS5cX8gQapdF4CHCBV0Pe1W/O8GSk/2kWGdBmGBu044LK/3Ax5gjV+Q4EhoAB9RV75/uM3ApshIP5QFsFDbgQ2Q0nxb3JZGoE8WjESZT1bAY95EZsDGGDtcGQSPGgyewtR4DF6pXr3tAU2CQmrEJ0OWVvDrzodBl2EDBsklFmvRXcENMtlKm3J0lVliZgsDhiMXTLrJVzYoJ0yEixoUY2D7DftMEaMQMVYefCmjKgfeF06905Twapmu8MjQeyLMgSk2T+3iNYQxwAumY0YiI1F/IQFOSzqjx4T2jgIa1DZRRsElV2YDGveYDoNAY4WuvSvi3BcaT+1XY+qTlUtXGfTSD/pqKrDTLSCju7AFUu8w4pHqIK9Ws4pwDYwQgzuoRe7p3t0OCMaNGvXybRZDwuLccCgbd0i9BAOeRaMxTOqBDLUTTQ0haiLAoRNUFPMN3EfLwGNEpAWMnW+g02dKUfukgmBC2fTyFyPmLMOmaBDUMPZQpg+myZtmqkcMCjV6I7R7OLX89bOf3D19kr7Z91aXV0SOEnoIu2ZbmkCrV0bFwKZFStW2IzRJ9KgWbhwke+7u9eJl2S3D8+oOhTSQha0qqRNJbNl1qCZMcPCmpkyWzRDYeI6ldYhGAtktD0Is3ixkwkmj7dl647nsyU3WP7B4raUv4n/x6BA+w3Sl6n3Kb582t6eaLgfIG3A+8Ytr2n0OfFN5ztGWuDWxFHfK1puHEq6YuWq8vyLL5Zrf/mr8tDDj6g/G1Vm7ra7zze56ntXlSOPONJfvvs24wY/mEAYGKMPifplObqjygGHJ1evL+kkZ/uy7s5Pv0YHJdcWPvmu33cIvmhr4LYQX1Wqx1EJaBJlE3al3Ed89NFHFqSgYfPGG29YeDJ79uzy85//olx51ffKjjvuqP5ijAqmX6TPlOkJtANFWrRREPq8++570p65T0KfO8rTTz/j+h555FE2WXbWWWeLljMtUP7dHb+zhs1HH35YDjrooPL9739fAptDy8eqA5o3T+qMGYQvp512uk2iHXPMEWXa9Gnl3b+9W26+9dbyxz/erb5uiTR+ppZTBfM95T9eZ8+MVj8VbUIYHqY2oBQ9N4L09b0ItrvKK6++Un55zS/LPXPnSqu1x/3gscr/3/7bfy9n6TycUQhs0IbUHy6fC2FiWvEVzYkPV70Leaf0tgvaBwZiA3c7vQk1FGgo0FCgocC2okCOieCLMasR2Gwr2jZ4ko+qU6L/mL61aRLvozkL82iJo80vwHMAYn5D81D+ckrvnPrhHs6QzZ3Mg3vFg2F+fY2sRcSZq6slvNFagtYX0MghjfUENi16s6Ln1SyCghuBzSj5WmOo5tvUr+2ybRU3RJKizF9SNwCpUztDAER0pFX8UPDlFWBFJLAHDn6rslqRwCqe8rKsKirjIzMAUYPIymbd2LhkxrfabBn4qTM8GZBcwqT65TMLbSDND4gDUnhdb+ciLnJ5C6f4YOyhjNLmSjS7wx/mzZYdY0eU8ePGaE7dKZNn47URUhteNecfpcUVr48IL5u/wM0sxQ/ddIpyaV+7jpSWJQPc35GCyzbEXaCMcNAm4wOuf1yktTAlaOM3FBhSCvAt5uU+Ri8rfiOwGdLHsP0L4yE3ApvtT+e/mxK2OB6ZJao1A8AAzuGKRA+Qjs7YgT5AwqXzY5JZIB852dtR8Q/GQ4cTpci39kwMwGjKoNHRLYZq9RqEN2slvFkuM1YcLLjaDFiPFhSlTCFGS4tjfaHavIMOFLQ6b8WIIARibU/8llx798gmj8xR72A+lMqtFgNjYS/8etpwqfMO1w6RzAXjE8IHdq3HLnOENWjmsFA2SgKI0Vro4pBBBDfW3HE4NHjiLBw0hDhgMBbTONhZFIJ8pour6RLprCGrKIgQRouDmLPxIiECFuLls0iI4KUXbRntnl6fwhjRijSbQSNddN24aWTp22E0rTBu04eGwKD6adFSU8Dp5MVZjVx5KHOjTI4RSxuCfqqn4BBkcB/10yKmGDgzeILmXCHbsNUOb4Q1qD0joEEVeqpUojF7BhOHJg0HEm6SqSFUqvtkqgkhzRqd+bBixbKydOliadAs1PWpzJjJlNmKVYrjzBntopJJM65uCXLQ8hkprSiEMRMnTpAZM5kz23GGD+eeMmWyhTGcM4Mps2lKAya1Z1JLBhy8VyN4v7RwaUrAsPKnNK7Pcjy7LbnPk39LeZv4fxwKtN8gdWqbvU+8h9FHJGMHZdxnqBPk/eV9br9r8X0bpiIhi/ibdBBo0fdKP7JUZryek0m0X/3q1+Xhhx/Vuz/CApvTTz+jpMAm+r74/uktlNm9RuKlmvmduIKkKpK/gCVX/LXgnFlxgov6VnkqYU22iT4NHMNVL75x2mfc4NcVY4EgkC7jaLLiXSN/skEzkqgOtcDRdgQ2nF9z++23yyTZm+pXlutcmb3Kz37+83LVVVfpMNQdpWU51uWEAJ2eW+ODulD6cYrh/CvOv7n++uttXg0+bN999/P5MpdeemmZvefs8tZf3ix33HGHhEN3WsB8zNHHlJ/+80/L4YcdVj7Q+TTkffjhh13OvvvurzN0zi0XXjin7L7bbuV1CZJ++cvrjBta7L3Pvkq/QFo43ymHHXqIx6Xog/ssoLbGo2iRFEeLak3XmvKyTLFdc8015Z6750qQ3y2h9Jhy9DFHl3/9H/+znCttnxDYqO+r9XNJq3iLTEzTrv8P9AyaEs8zabukfeZNvw3RhBoKNBRoKNBQYNtQgP43x3/CjYbNtqFrgyUoMHB8r9OlxjrUo6tw8GsVliou+YPgIMDNH+sHFhpU7EKwFBU3ojjm9uL8ND/V3F+waN3gbxRgt82vhxWPNTKVtlIbQZdrPWHVKllb8HmsrClo7cHWJbSuoI2g5paEgHL036obPGRwujH3tva6+D54Udpqax/1PIoM7RUqTt5wdbxVk5TcClVQ5Kgcac5ctZkUIandGVilVWiUTmbBxNybjUVAE0kaYhHVh7oSD7EqZwhAdHF2q34EFOcCB98PrytqK51NW/ha6iijFIfJ8rEyS9Ip0/ITpE0z0WbOOsv48ZwXzPnAbJhE61v4Va5NmxsHT0/4qGFVFVE46p/1Mn3iObfakZVOGHzVpe7ylra2XdxEv0i4X2IFljnTb+duQg0FhoIC/jb5PnXxrubVCGyGgvpDWAYPuBHYDCHBv+lFDTYm5ThUjZAxZNUBMwxgMhsRdnOdIZEQQ4QYA+yyytVTchB2QvWTg24wWjA7upSLM26wO4t6c5fOUFmja7UYrVU632bVag4Z1GGDqzdYWwQtEkx3RVUY8llYZ9iP4d+MiYRC+Dt44VJhAUcOcsXF95LhqIWyqAGIUmBIaIv5BXwYCSMJ3yyEAIbL/MzIfoIY5ceMlxkSGBt1uFRFvsPgNdIUllAHOeBcJqVGrfhNAQ3MFZo0VNlhGEtVxYIZMVimiejiBcUKgWEF31epfAd9YgAIZjcYXeAU6zrwfMARC5Oxg5zF0HgTeFICo20uo6KT6608ErhsXL9WMButeYTt2o4xI8uEcR0SlkijRcKZcZ1oK8HEjbQAByHNyhVLyyKZMVuIKbNlS3QmxAqfPbNq1Uqb8enCrNma1TLv1mUNGxZZ+/q0i2fMGAl8JPSZpPNupBkzSecxTJA5s/HjUb2e4PjJih+n+zGCxYxZB+cWybxRB/fSnqJt1hqoFriD+iZF/tBiP7N4bjS8cQ0Fti8Fql5BhfDBt+987/eR6GDq3J2p80BYgVZZ9Gv67hHKKKsFODb5qA9SecFm2A096kMw77UpBA4W2NxQHn7kEX3jw8tuu+9RzjzzTAlsvicNG5lEU076xejH1FO4XvSqYKQHCdz5vbgs11H10F8yng634HVX9WtOB0d13yutkKVLl5b33ntfGizvOn53mWn4VgW/AABAAElEQVQ74IADrfWiQu1oL3nx6SMJI0hRQNjSZahOS/WfOlPrww8/thDl9krDBoHN7Fmzyj/99J8lsLnSGnidYzvdBpocrdRkFpS6p4/ulvlJzJzddNPN5Q5p2SxZskSmzWZYQ+kKnQG05157yRzZn2167eGHH5J2S3c56eSTyj//8z+Xww+VwOaD+eXXv/51uV9aOgigJ0uwfPrpp0kD5yqZTtvFwqCbbr6lPP74E+q7xpZjjzuhfOe73ylnnHFGmTVrD9cpJuAI7Gg/7a7aqvphOnK1+lE0bGwSbe48n901ctToctRRCGz+R5kzB4GNTD6qLzT9KtL5+RoVzx/ERl6lpgdAVR4hCNVy9J/cZL70WwBNoKFAQ4GGAg0FthEFckwEncdF8QKNSbRtRNwGzVYokON++PXf/pnMVYgjgBdo8wPA877ix5ZG5pyRHmmBRRxf5IPPU5hZudhY5RMPqLCmiNrgKb4HSxPa0NijdQXWFtZJWIOmDUIbzm7tWrO2rJK/SmbDMRXuzY7e8AguHBufNKHn8qwXPjPLznqFXzE5gs34xKGsVVz4FXwtmsJcXhRKk1qu3f4AihZGcqYB38pi3gvteYQuICQxN7EqDH25IBSpJLNYw6ZJ+Gcu1kC4tNHSFjIIC44Nqmibj9OcfpzMxE/qlFBGWjNsWB2juX4Hc3v5ozW/R+MGQc1IbW5lTQS+2ZfL476Ko4pcckkmh/UT0ZkTAFJwrYBC7XD/FANu9tOGrgo1RIYzNf3NsjcRDQW2KwX8bVbfaM6Z8RuBzXYl+9Aj50E3Apuhp/s3tkTGoLxy/BnoC4A/XPj1DHXgDANYC5MLTQqzSTGMtgZQoapDUkY4xdIBKd2XoBjK2SnDRuoNeo/xYbZWremRRsVqLdzp8Phl3TKZpsPkZcqlW+cGrNeuYVSaJbsR8wEOFumCuQIX9+yC4fwDswouTz+qK784MwhiVoBOR/35Q0yBCyaCdF1iZmIncyyMuhRLL4AUHuNSngoffhtzhigvIFqpJgmxchQYhXJnZ5Nv7sQzAj+YMBivFLJQMrt7cNHZSzAkTqtP+BDSuATjjvJzd1CrXGqgavaJqODEzBsmxbw7hmesOlg05iqqHSqcnTfAYBJtY+9aa9Ngl3akdt50jh1lG7aTJ4/TOUCY29G7op39G7QYu369hHErl5UFH31Y5r//Xpk//30Jbj6VKvsKPeN1PhsiF6BpE+fFjMREnXaFc/YCpor22GNWYcGSM2imTZuu3T0TrWGDYKajQ+cNyUfzCaeqQx7ThRs0mGBqaTvaNLQVmnnBGCIYnrzQNC7dtBz9LY60xjUU2JYUiDerwli9Z3GX75zuFB/MXbyDMHRomq2SebOVWvTHfCDvZmfnOAk0J0uQOcGaKYpUXjT0evQNMLHbpP51eXn2uefKDTfcWB555DE+iLL7rFnlbJnzQsvkiCMOqz4gJel7t0DA5atP4EPx96F+R0GqS59LHHVCiEJC6xtS8fRDUfdoA20LYYAgGRQEQx8wf/788sQTT6hOjxr++OOPsybI/vvvpxx8k+QkD+Xom1a/H1p/8S1XyQFU+6Wfpp70L2jYcIYN2i+vv/F6nGGzxx7lR//lv5QrJWzZQ+FxneOjOwCH649giPKEh80D8j/55FPh+L3Osrmt/PWvf1V/NVKClWNt9gyBzVNPPV3m6eyYl156yf3YGWec7nNyDj304IKJtFtl8mzevHnlww8+ML7jlPe//PhH0hScWh577Ily9z33lNdff8Oag+fOOU+CtO+XI488XP3e1CC6ydYeg1xP1TVNoq1etVoaNi+X6351vevRpXO9RiOwkabPf//XfxVdz/0MgQ3PqCJAjZYRVOGmUHUHcVsun1M+jfRbAE2goUBDgYYCDQW2EQUYj5IvJcw43AhsthFxGzRboUDFD1a8QHABdV4gsmYMfKAZKvvBQTjNPzofxWsLQMV8FbYiWIvI57mumEC97bqCH4N9jHvBKOw1Bn8D8HtFWjWYYu/RpkAJa8QTcXbuUp1906U4NG441xWfTaT6bJQfvrbaFIppdvGYEU+JVL/iNWFGqQvFUp74XnwzqYrfQfNL6mXHfYRav4By2ZFIVnnws7hIC17TiaRthgVIhC4bBBJrFFGF2JjIqkbkpbDEj8896zjabCnWHSFLmn0fJhPlw7UhF8shaNKgQTNp0gTN6yeUyROkSaM4m4bHLLzyUt2KFPJjrpD3FOo2UU2XqZKrerg1bqpaZR/6Aefa9V92arXbgADJRZjfjOUJtWMN5J9MjxgKcUHyMyX9gGh+GwoMFQXoM/LKOTN+I7AZqicwROXwkBuBzRAR+++hmByH6mNRjkPpe6AKxiN/Y/ACIIEG+vXGk6takKuiExrU7bBCrZtA7eq5bgzI6pB0WWAjdESDFW0KDhZcu3ZDWd2lSUdXrzVvVkvrhvNZ0MZZqwvbtL06wyUENcOVF5M5gRchzvBhUl+Gm1AdYAL4VlhsM8PjeilsH2ZBcDA24vRit7crqThlr2BtUsgLkY4WkxL5AwW/4MaRt52fmGSyjEtAmUq9uHNnTbDlBGRk1A2TRDB+ODKrjdQTmgUC+dG5AwSzBwO5UQxX5hHfKDzswnYgMEXDHOaHtOGUY8ShGo1wBqaO0t1enpjiEOZgSqdDtmnHSv2Z82lGiHnbsKFbptqkFdO9RruZVopBXuara81KmTtbIW0anTezfJmeHzBx5gyLtmPGSCNHgpfJk6fomixNGWnPaMGZMNcEadWMtybNBC2kjpPGzFhr0CDMQXCUE1XazhkTvhcNrClEO6GTaVVRRFH6b+VTou641DAu6FS5xJ20zvi6nzD1uIHhwfJ/nnwD8TT3304KxJsZ7yUtzPeFCVDL6R0m3pciOVh+yZLF5UUJBJ6XtgzaKWOlGYJprsMPP7zss88+Mu3V6e8g+q9e9RtMTnvLJ58ulEDhmfKb3/xW56g85b5y9p57lbPPPrtcfvnlyn+YvnlKVvn65h2khybsvkP9TH5T+l4Q2BBvTT3lyXoCyncIKAyoUoRXfRGdkuNoH9cO7hcQ2Nx//wM6u+WPhjn77LPKRRdd5PNf6AvBB54NagO5hktIYmGRcTEBrX/XEQ5hDdCMLxsksFlQ5upMF0yVYTJs2bKlZXcJaa684spy2XcuK/uKbvQ/taehcpMGkCDGkiU6W+YRaSfdeecfRMunpB24WubKDi1X//CHZdas2eXBhx4ujz76qAVECJvPOuuscsUVlxeET4sXL5J2zf1lroQyL774goTWK8vBBx9sLRq0XhCi8UwXL1ladtPZQpdedpnPr+GZcrYaz8PPRD8eSav3xM9ExEaTaMWKleUFnVN03XW/Kvfdf792nK5X3k4Llf7lX/6lnKlnjdYhfWZLeGaqQqnAn2NXxKjtwh3PH3oGTUmLdyGhGLcI+6fmZ3rjNxRoKNBQoKHAtqJAu1+OvrgR2GwryjZ4tkwBZux1PgDurs0T9M8HL1DnBzIMhgiLa1FIJrZ9H/DwjcFbxH3FWLTzCMDzevOG7TyumfJaeKOfjZLGIJTh6ta1Tps/eySoYR0BLRyfkSr+iDCaOcR3dwdMr+DA43m4iqBOrkcwOW5xbB6CvxWAeFvztwp7c2kFD3i7bUmd5KPjnhYETfGZh5KZK11yZBFn1OLLh1W0M//tCtap7YLNz+dZu2FlYgfxfzpjVhoyWMEYq7N7x2hOjzWMMbrvlGBm7NjRTh+N9oysi4zQpICLzY5iG2mq8WY9mTNkO9t8YlV3VbaqWosOFQmzcS1Anh+kDLxBlXaLCGUcMDjuiU1a9U/vfwfMQLj+EGBsXEOBoaAA/VtefDN5NQKboaD+EJbBQ24ENkNI8G94Ua3BMMci6ttvHKoScnGnYgba4HXggeE2VA7OAVGLV7CdK0MJlXDc61LHRAyMEDtbMjXjEMZIk9m2abthpGSCBhv8XV09WhjTrvLVOmBQYRahzHiJqUItuuyAPX4ENVyIGmCIhF8Xwh1Fxn+NU+AQQlShqYw7TudKOCXJEU8t+Q0sChkkmQRScAEX4favGQwyyrlOVSifWeYO7Po1LJ23Fin1F+WTGoufJpxxBNKNLIYKWTKVfdqpBE5SwRX4on1RAzJHqcCwYIePBgpmk6xJAyOoBVpUpH2Gjxg2BDcbNojmErhskDm0srFHuFW28nR3ry3r1q7Sc9ElIQ1Cm3U6RwHNGtLJi+o0h2yPH69LfqfMlU2QCTMLZ2QWKDQDxtvMGRoCwLIIPRrNGS1kjpAgDmaTunpHE23gX21PGimJiKrR7fgwHQRNuQwFYAbkE8m7EInBcEZyP9wR1fqtw7Uia4Et5f2sfDUUTfBbToF8C/O1zHfGvWQ7svWe0xN1qz98//351tKYO29uWbhwob+fo48+upxzzrnl2GOP1Xc0Ud2g+gwJavgG9dVI0I3Q4mOb27rttt+Vp595VpOvEWUvCQPOPeccCw0OO0TnpPCReKzgA6OT5qKm6sHaH5DuqWDVn6sPDQ0bxfg7Jd7TWEVEQ/z1VqiNzviGWUNowYIF5dlnnyt/+tOftHtvWKEtJ554Ypk1a1Y1CWbX5EZf/sQRNlhY5GrpRw7c6atMl6cycPBLH6uM++69t9z5hz/YbNkSCbp2mzmzXHTxxYXzZxCccOYV9WUCDM20AdFtVlQ8A8WvWLGivPTiS9KEmVvmSQD0gbRm9t5773KlNJRm7rZbeeCBBy10wXzaAQceYNqefc7ZZfasWRJgry6vSPtl7tx79Pzmlo+l9YP5t5NPPc31fP75F8q7776vMoeVQyQEuuyy76h+F5VddtlFk2xM3ale+kcYFZSnnnkFjRDgPaNne/31v1ZdHvJzp3894YQTys9/8YtymkywocXI5FuYqvzyKkdM0C5jou3x7NtlZWq8s8pBRrsMpJ/xjd9QoKFAQ4GGAtuKAvS9OSYTbgQ224qyDZ6gAON93eX4nz7cx0CYOnxwE+2YvG/zBnprhaHadGN+JKDh89qYk5+LfKS1XRXXjnCI78HfBHh0sfVGywuVGTU0bDR3lQWPbp0L26MLQY2FNlqAYG0Bgc0G7Szlm+qVoIcNT+RZL16SDaakMf/mssjGwiPWHmxkTHWo+CtVzzWkHg5Rr6hTq4W6d6qZqDYv1eaphAOYaCrACjJHXy9fm5iURhI8v61liK8PKxVYqkAgM8r8o4Uv4q99Lq+EMRbU6MzZMQqPltDGJs8qM2fM/WPOrVqrjaagKkSdgv8MP9sQKxbULGJ4Yg7TVgXivmqAvRZEwOWvG+qb6qfK47vNw4GFEnDt9HYoUuJ3INzgUPUcTbihwPagQPZP+IzheTUCm+1B7a8RJw+4Edh8jQ/gG1Y0gzWuNfRkgITW+ESAS4n65x2qO8s0ahHtoRQWIxHWyjCLUuGUFxDgJpRXlOOiKKDqlEgmZ+xOCaYKboRFKmDYbwN7kA715nU6XHD1Gp11sEyqzTqEnoMFV6xcox3KXWK4pN47cowWH0fqu9C3oe0tCHHgMeraN4mfwqPmwVxlOe4wc6tIFQkc7XGzHI6ctBAuib/Axm0VjkyGBiftdFLSxeSpOuiA0q8j2+WQh0XQihDWkgFWuLNj5xbcILewSuTjjCHIiDMjrAVbzCH1adGWi8VXFvqiNAEpfaMYUA4lN9V1zxOwarTwjLYmzWirQK/RWTMfffhB+fjD98vSxQtE+2VaPF4nk3WomMvs0oZewcXuHezajtHOnSnSlNl7r9naXb5/2WuvvbRrfDeb9kFgM2bMaJvm8XkK2jGfNA5hlaqmesZ7WdHGjeL8DqWozbTTmjW0SO0ijsGOyzDKz+Pk/KERYmBhZsFkNBUN4o5YiFYRLiJbvwO/FRJ4Bltzg+VJ+M/Km3CN/+2nQK2r0DfYvovFeLWfyYvi4/0OeqxZ01XefPOtcutvbrVZrsWLFtlM4HHHnyBNjivKmWeeaUEo7zuH0PfJNCHfAd/E/A8+1IH3j5Tbb/+dTKM9r35zVNl3v/3KnHPPkzbHJRJaHBRfgj8Fvid1pHxr+l5sIlL+CPWzvMMb3XfEhNV9lb5Y4kdIewNH2cM8ORS8/vLbtJC46qQQziCI4TwZhDafylQiuxRn6FyYXXbeRe2YpD5FmpT+ppnU05/SFv1QljTtMIeJg0Z8lmlaAjwWPgl0g/rATz5ZIAHGAzKL9gdroCxetLjsLEEIJsIultAGIREaMTwG2kZgBH2GkILXTmlrdEbMOzKF9sB995Xf/Pa35dVXXys77ryzcewkH/q+/sab1gg89dRTypzz5pRjhHunHWe4r124cFG577555dfXX1/efustaRFOLHtLO4qu/r333tMYt7xMnz5Dz1FaRqrXSRJcTZ48MdoiGLSm/Exkd9zPpWo3be3VwLdI+BF8Xf/rG8ojqgtj4IwZO5ZTTjml/OSn/6QzdU42TU0bPU9cNK/+6+jWj/tbEwH4yNNK3CyQxEp/M4AmoqFAQ4GGAg0FviIF2v0yQ0Lwv41JtK9I1CZ7RYHBxvqIE0e6GQw83uaOuHp83ofPO6u3Vqa8Ay84SAnsWUZirVJbixXM+3jv4yIXLEpe4A3LEcQLVjynz3lVCWD2pR/zk2KpOCeWdQOmw15D0FoCgpp1Minb1bVWfF+XLEfonFWFMbXWg7l2ZfB5OHx7bgv8ojb/DRuliogvVV2pBy7qqfaqHLFsio8/V4R2U3fBBb+pezHtLVooM2E74ATMfH1YWSu+UEIb8akW1GgjDuZvMRE+TnPsiRM45zU2QnZKa2bUKMzHs/0zyhL7LR46TKNpmtzPUV/KDCsbJLWo5nDUh7iAc0C38UyzjrRfkAaLHEDzCH1Ho1t4wQCtyBGwUCixR0z8AhnxmT/u66kG2eJPQqa/RcAmoaHAdqGAvxP6DL5tvufqYq77/vvvl7vuusvz0e/pbNmBTufK8nmGe0XmIsiQE1gQ4fATrB4XuZrfoaIAz6AR2AwVtb/55bQ+3Kqq8bUSq6tfIjdK1X98xwFJtk2b7WogFpeMAkNowle4lRYDakDGLzAJS3+RaZmXssFqMCcGQ0Mk2MSGiLHiHJbMyoITC3Trezm7QTthdLETpmstBwr2inniIEEOHcSkWo/ipZUjDZz12imDAAdGTGuVZsxcNxYLXUbYq40aVnWmr9M/9eByP6c4930CNN1aDWjVMFC0asxtpoEv217zCVYgprDDlBOoXBuFTT/SEt4RbZisT9QT+gVzGCaVEF6w8MhCJoIauESYWO261yIpO+83SsiyYX2PtWG6ZbJsbddqadF0Ka5bC67VpTMwNiKUURzaNBt1L41q78bp7OzQguN4acR0ikHssNbMRNm8xfbtRJkzg1m0Fo3MnU3QPYKa0aNHa4cPGjOxGMqCbISzXbQ7iOP3oCIKJDAZlJbpLDoSjqtN6owjK7SxEIjMNVehVQwJycIGAPlN0wqe+7qrp9Xjm3BDgc9LAd6o6PGqHPmK+d3jhpdXvt9vvmV93bo4k+Rv77zjc1Ruv/328qE0PDrGjpU5syPKd77zHR8qzwI9mml9fOcSwCKaZoKJcOHue+ZZ8+JtCR06x43X2SZHlQsuuLCcLs2L2bNnaYJGkSoPQa8uzGxxj1kFviMmqD30GcLHPTv5Rkrwg3CGzHzPTEp7ehDk9goeQYsmhvrmx+jbD+2OMKVmWLWPc2ywM47ZRL6tsWoPF/DDNaNESLEBPCKHUEefrnqE1g39mNKUD00+dhRSBmfLMInl26UfXCazjM89/5zMkc21+TXMsO24007ltNNOs/m1kyXI2GWXXWPi7WcgYS+PQHXXLegdRnPm008+KY/J7BlCFwRfYzrGlsNkTm7ixMnljTfflOm5T4V7R2vHnH/+eWU/aTEhfKIdq3Tu0GOPPuK8L8l0Wbfo1CmNKHroNavXlB7RbJ999pXw7UoLkw44YD/3m5Qfc2r6O6BVqX5XmMtboDN2nnzySZ1TdJPOxHlMfeuIsueee5dzzj2nXPm9K3UezpF6TNH3Rv5oV/9fnljbZf9HHTJPO3VgyECKTH9genPfUKChQEOBhgJflQJ1PjXGueYMm69K0yZ/UiAWz/MufDhWXPxGXPxuPtrXYzLc3+edNRdTrT9kamDsX0ak6bcmsEkeA/7MdVKy4cw3xyY+5sSer5HVnHCNcxIuz4zxhQM8rBdkGH4NHhrBDdo1aOLgszEGDZvYHIofF/w5lkNgRzdudIHm1dA+Jy02YYbvtrvdAUftqDu8mTcbwW/rMq+GL16W+S58NmHY7VEjemWyTLy5eG42P+U1QhN0zN5iwhwfHt0wyhNsunCrrChPvgK+x6/CUNQEsc+7AKXkKhiCdkF8B51eRVcYddc/luR4siDin7uICa8KA4gLprOFpX9q4iaWNuW9c27lJ+HS3wpok9RQYDtQwN8//Z8u+qe8GoHNdiD214mSB9wIbL7OJ/DNKrtieVSpGHzilwFMV7/RrRqc5MUYW90DWTFMAzIIY+DpPxC2kUbZ0KONK8IxeBpSzFA15rpca4Uoh3f4mrECSzJXKlEMCxmyZLDjfG8GC6aqiHFiUbDPwhvbo5WpoDVdutZy5k2PVZxtk1ZnPuCvF7NlFWcQaReM9k+rPvjZCtpA+bpUDj6CHRYlaR40C40WwrX2qg21u83D5HVsHYoWgUdI++EiNjrv2AFtICJBUsHTAK5wbazExTk0pG+S1gzCmWC0RF/ttN8gwUtPd3fpWbdW9Oi2EKa3Z500ZUJYs3bNatGqSwxqtxjV9cYBS8vhhNizHSP7tghmpuhAQnZ9T5E5s6k6DHvqFJ1Bo/NouCZP0qV7BDQdY2TSTMxiS7ulXW1X3u2HOtkIv5gCkm+yiPZ+DoKON8rZqp9A5h1LzhfMLsjAV0XVMkAT4cnCWu88IHrOfkYBHvWqwSo644Bo4Qjw5rehwBemAG9vvefJT5r3LEwO6CXO/lETPptf0Exwnfq5jz76yDtwOPj+nXf+5sng3hIKnCPTZmiM7LPPfhKaTlJ+/jWZ7F2vc2+WSPPiqXLX3feUl156ReepLC2Tp06WoOYMn5VyxBGHSxNjuoqUcFwCmVWrVpSlykPfgIC1o6ND30CfzsxZZo2YbvUjaLcg9JmqfmDK1CkKj3Mfydksi6T5s1LnqXSrn+Hb6pRQd9q0adIeme6zq9gJSN9Ae7u6unwWzzJplzDppO6TJk6S0KbDvA51RwuHD3v06DHqT0Za8EEcedevXy8BzXALiKepP0JDBUExWnzQFYFNl4RB7773bnnwgQfLLbfcUl5//Y0yXZo8xxxzjARWF9h02e57zOr3TIa7r6DfoAePfgUhFO165umny3XXXVcef+IJ9Znr3bbRohH0QQiz1957lauv/kE5Txo2aAzRFs27Leh6VnlvuvFGPY8nCgKW9ZrlY7qTsQatpyMkVPmnf/onPZvTJETa2UJuqsKoKE+OCTRvEL6c6kcd12usw0Tbo488Wm6++VY/b+h1xBFHlUsuuaTMOX+ONaoYW6IPA4dfkwG/jm79gDtdbAbIu8H8qGFiHQyiiWso0FCgoUBDga9GAY9LHqMYAtiY0AhsvhpFm9xtCsBbVPxFO7Lij3KMjwR4o7aDV6jfZ8ogcfAtSm7P8RIW3ymtiFZu8UnmxhzRim3BRSDyBt9CWE6gyT1FauAhrnWvQIRbWcyzwYaj1N2nH31iwX1VcQhtNtqyh7RtlMgaA5tLrQVO+6rvMjdcce85q0uKcggGJdgkhECmEswoDE/oewQ2mof77EEENgqPGrlJAhvlqYQ1ijZfZ15R7YXf9KMQ/mwYMI6vis54U6ciZz5PKBM0VKNV78DbAgoMisc5Vj+RGr+tlAgEfP3XYFk5JSgYOQFqh+rBLaGKHOSp5SPSrp6rDjMYbOZp/IYC248C7gfoC/xdxXyMOVkjsNl+NP9aMPOAG4HN10L6b1yhDPIIO4LliOq1hyAGqbyT3xIM1OKyReJIYtwlT21wqxYM2/gzLXAEysQHMsLiCLbgWrkJCDRzRuele3VYFpgoxW2jUs6kn4oLMYOlW3bBhPaMGCbBsfMb5qlXi18sgCHMWSstnBVSY14uM2orV66yOjN2a9msvWHjCHWOmNsRBYXLzBzlq2osnm3iXBPfY/4HRo0U2kZaVXOlZxsIOZxJggQri5y0hXx1R5sdTSSMmf06PvLgIq8pojzOp7pAK2VzPu+41rNCfVncoNrHzngOG5fQRoKajTp/Zt06mZDTLvNlSxeXFTpwe83qVRLcoE3TY40anzdjxk+MoIQz48Z3yIzP9DJ79u5ljz12KzvvvKNMnE3SGTTa/a4DCxHGcI3WxU4edtB7Z7v8Uexu1y732MmtJtA4aOo2U08Y0TY9WFTmvA1rBKj+0N2mjdxAhU1nGlvPA23CGa8KMU6TrQ1H2ZGe0KAxNVvxytlOrIXqdRyIA7B6ei1bE2wo8JkU4DWNfkHvEdB+b+NdHVRgQ/8mjTjOokGQMnfuvPKb3/y2vPba6wUzKJOnTC0HHXSQNUZOPe10aVXsZSEGk7MVy5fq7JRXy/0yCfboo4+VxYuWus+bMn2aTHldUr7//e/rHJa9pMkx1sKd1dIC+ctf3ixPPC7YxYvKDMEhbFkr7R40sBEYobHDZBLhyj4SFh0twcde++wtLb21Eoa8rjNpni0ffQxcjyeZk9R37Cu4E044sRym81kQ9qJxBy/ziTRWnnvuOZ298oz7ELRAjjvuOJ8xg+Dn4YcfNj4ERLvsOrNMnDSl/PWv75Q33nhD/foKC5hGSQA0derUcqDOjTnppJNs3m266s1uRAQ2lIPQBu2Tf/u3fyvP6cwcBMv77bd/OUtnzKCdtN+++7lN8a1rBFD7oF/2LzwkTEiu06aAl//853LTTTeVBx96SObcPtF4IlOS6vN20Nk6Y8Z0lMMOO7T87Oc/L2effVb0j+pbqQu7Nl979dVyh4RtDz70gNrxt9KlDQWMMeTnDK/jTzzBApsTde7MVAnCEJipEn5JeFdCaOKR0PF9aE0qmZ2g0GXevffKXN7vyosvvCQh1sRy5hlnFVTrjz3hWJmB27nW/8ZL5/ev6gMjzAvZdvW+rxHYtOnShBoKNBRoKPB1UYB+OXlQwo3A5ut6Et/GchHWcA105kAUORinoOiKj+2Xi3lpOgf5qcV5XjcYPsH0g69g7NXgE8x1UrzxtfEH68S3QlykE4qLuWC7NoTrjlJcUpUAroQnzLKAvz3Bpe/8AAky4TPcTot0cCe+KIt5bFK3FRBUxMdmG2PxD/yp4fXjZhsyfwIOvBRi3yhz5kHJzH75A6Ze34CnfwkKGTQqC6xczFMIZQsIf15HHnJmXtVAawQ7bKqtH/Vr0JbwZv4tpbuQ/JFPS3Mtwa3WfeMaCgwtBegr8vI35u+3EdgM7VMYgtJ4yI3AZggI/XdQBMIaLSGppsFYxfBTDb6uPzHVZelKDlDpV42sbln4b4/IMRC2BusW85Z55auTiWG37Ud5Fd7NPBaZcIlDISFIHMRSzYQyA1GlRrYoJ2omOAWsqEsewq2Lb6SEmTTZm+2y1k23TaZ1axcy2jndPTvIZ5FLKs9WcQ7fAh/FeeeMyBHCoBAImTmr6NmqI/cql7qni7Ai5WyCLGopIJ5NGxKIuNNvMifE0ZgqLaPtKz7NACHcYCGSBUR2faM9g2AGnx3y66U5w9kymDDDnNn6Hp33I80azJ3JJprOohimHTpaXJSwZayELmjOYNqsUwKZ8ePHaiF2fJk+bbJM+0wv06ZP1f1E28VF02akFh45wJAFRswOteztujFqoQUyboJIUzGBwdm6/cEEVi1309VeBFuCoX9zW/nJi1wRGUj5NYmCgk5yivI4PnC3oip6ch94BEaRAVx7Is7R+klYIqhX3dXT6vFNuKHA56EAbxPvX+tNzdfL7z83vP/ydU+/jEC6V5oyfP9rpFXy/HMvlLulLfO4hCrvyYTtMAkKpkhos/fe+5QDJbjZbeZutmVNj7lcAtq/ygQawh1ge7UDcJw0Y/aUkIaD7S+97DIJZHeyYGCj+pLlEvA899yzOuvmtvLXt/8iU4fj3DewILRiBQKSXvW1m8qqlavNi0yT1gznVM3UGVWrJAhe8PECCXoWu1/iPBgESpgtQ7vmyCOOKCeffEo56qgjy6677mqBzvvvzy8PPfSghVDsFjzjjDPKRRdeaJyYfLv55putUYRQZMqUaRImT7LGzYrlK9T30IfLTKbMqtEnTpVg6XCZJ+PMlmOOPdraLZiF2CABdp9o+KLMkP3v/+9/i26Pa2ciAqBdXN7VP/xhOfigQ0xH+lWezsjhMt3mJ5QPR7GiQa8GjnckGMHOMMKRl19+payUSTcEMqNlHg2tmBN19szVP7xaAqoTFM9Tjn6N6egH8z8oDz1wv87TuVOCqOfL6i710aob5iIx03aqzLT96Ec/FI2OkoC8M8zNxRuh2sQY7ZFQ7wdjhRl/pWOu7SUJku644/cy/TZPz/xdn19zoUzefe/7PygHHXKghFSTXE/6r3z3ws87v3nC1nb1vq8R2LTp0oQaCjQUaCjwdVEgeOXotwk3Apuv60l8G8uFz+Cqu+QR8OvhCqbNJtUzKayE9kRf98HDtIGEqzWJS7xVqt5r528Dt4sGLxsB+9UFXADHWkErG1VISJWVNahjJ1x3WRP8DNfhXbzSMq6VvwLOPHWchN0k+a0m1+Pq8S2EiqxcK08WSrwKasUnYM1PSkR9mHUk4oiJR1OrLcm+DR6RYC01ciuiRc/NKVArfbBgvfJVfVwl8duyehLOBbQzU8lBXR3XIADOVs9LSxqBzSCUaqKGkAKes1VzN8/F9AHjNxo2Q/gQhqIoHnQjsBkKSn/zywiBjSQTrQGTQTT+ovYMTlxyFtiwXFTdO3LgbSz+JL4c6OO+Yt7AYxT8VEO278EdkA5s9hMDa5sFqAAUvVmc8ZEeeeohx9DROVWA6uQI4zKufp9CHHYwawOyNXB6ejDDgwYOZ99wBg5m03rKOu0aXyfzNus4VFC2ajlU0PZpUXmW1g5h1J6DSURApLKrTjcKT/pFbUhDOLKJM2RcVbWUjtkEjEY63GpvPSVawwIhi7Ro+KA5w2JhnCGBIIbDD7t9MOK6tWtsvqhH58xg9mw959BIWLPJmja9qtBGC1nGj+8s07VrG+2ZGTIfNE1hTPawG32KzJxN5vwZroljZQYpDhinNTiqGU8cMZn+XO+oJ+l1unOX99ZiEu1w1q4BU5WIVhALmtCl7UhUSa2oCCQ+31U3kDgGPGVxHKl1+MwVcFkG+ewEGtDK3orsDwvc1tICUfu3DtuOjVD/dg5Mbe7/kSiQr6DbXH/38uOwH30KfQCCCRg6BBcff/SxtEX+pEX/P+gclWelaRLCEzRNxnLgqEyKcfAoAmO06TA3xvk3vO2Yydpll5nlmOOOLRdfcnE57dTTy4SJ4/VtimFUH7N06RKZ+nqsXPfLX1pwgzA1TZXtvvseEu7sovvh5YMP5pf33nvfB7KOlWABIRBaLAgeZs2aVXYRHB8X58W8L0ER2je77rJLOeuss8pVV10lLZiD1Vf1lL+9+7dyj86W+f3vf29SzJlzbvme0g87/HAJN+aXf//3fy833niTBD+rJCQeJc2cCRI8TLGQib4Lfuijjz5UXd6zaTW0dzANd8UVl5fDZeptsrSAEHZRlzd1xgymzB568CELnzrHjy+nyfTYT3/6UwmTjpQge6T6+BDYoCkITfie6SP4dunVED5xjs1TMjH3x7v+WB4QrsWLlzi9Y2xnOeDAA3WW0LnlgosutKYPQm0LgdSH09etlNDrheeftWm2++57oCxfKU1H4ezsHGch2tnS+Ln88iskQDpQGoyjhddVd13ox3NjRcRHP8zrs2bNGp1b83i5SQKuxx9/sixZvKzwvC699DLTe8+9Z1uLalgliKJNNMx+9UsnSt9ed/X+7PMJbPrnr+Nqwg0FGgo0FGgo8NUp4HGJQUCOcCOw+eo0bTAkBZivVXN+R/Ge1cf1AffJzOLXwZyXHxJ0mc+VP+gCvDKyA6fuqmwBz03lXAb4dO9vIOoD92KonDzW6kKR1ecStSG70oHPq4WeCDlj1Q++Z7TkyQa24jfP78zVD3nrrkJdj+oXBn5gnn4A1Y1hXJ96RIRJM6ldW9VYEdSfNSNc8HjwtgmnOAH5ryKS04AVTNbHSdVNxhmz5gj4Qf3NACiycrSeCxd+0lXG3UJgw0MBRVUPg/p9yXyOqf3UcdaiCQYRapEgRmAz4D2rQTTBhgLbmwKM13nF+pW+PL3vjcBme1N+iPHzkBuBzRAT/RtbXAzBOfBRzRgq4zeqvaXwlhpVHxQHCRNllG28DNN5V88xWAkJ104bGBMdWTu9jTFD6WdrE9YDf95kjYSeMZsd4TZths1Zrcf16lBAtHAshFEEi6BcLIwinAkTa7E4GsIabNSCQ2kIcnRhgoZFVF9CxnfJhdZLCFnQGkELynpAwRBVHI87aS3ksfCZh2SP0I5v7rnIxwLrYpkEwgTRp58u9AJj15ouCZUQyKz3jnIWWNfL7FBfX4/w64BxtbejY4zPlNhpxx21aLqTBDLTykTOlNEZC52VJs2ECeNLp85VGKNDujncm0XWkdKY8WGFOrwbs2gjpEVDPfs50TKZsvZAExAegOrAmVV5/GzIW8fn+3ydasDGkfdthALfLIVn20a5eZ527ib0j0gBGCDeS7Qf6u8ecQPvkz7ED0wn7fPGJZ4v6oN/cxf9IWk2GciXpDD9wNtv/1Vmzu63ybA33njTWidgwM4158Nw4eif3EdoUjVGfQPmzY6Qpsv5559Xjj/heC/qjx41yvSgjGXLlpUndDbLf15zTXn6qafcJ8zec7bOezm6HHvMsWXWrD3cvyGEeerpp2Q67cmyUKbTxsv81syZM6Xhcnhl0mw3C5fRRvnTn/4kE10vqO590n45ufxEZ7QcdeRREjr3SuvnPWmE3CONnjs8hZtz7jkhsDnsMGuj/Od//me54YYby7IVy8sknU1z8KGHlBNl9uxQmVabPHGycVIXzL09rfNhVkmjB2HHJTL3duZZZ5Y9JLSgP0bzD42de+bOLfPmzZPpttc0Dmx0XX/845/IP9Znb9Ef876gYYOABXrTH9vxbkj4vEpm49555x1pOd2tet9uYRF0Hqd+9vjjj5fm0mXW8pm1xx7GFf1fPF80jl599RWZtPuNtIruLYuXLHN/znOhXRecf345WRpCu+++m/pk6sC7W/WTqgQm0BiDiGOs4IwwxqMlEhrde999ptVLL70kmD5rW10mgQ1n2MzcbVef69M6wwYEclvqNfN9rH8ndV7DmTf72RK2zQCbiIYCDQUaCjQU+JIUoH/OvplwI7D5koRssm2BAvArXION6YPFbQXcJQhXsEBCmQEnVD9bwQlEv+R6/npCzE7rWDfLqgjnhq+rANNv5csIoa5j3xI9EryVv4W5HfPZIbDU1lKq+rXKz8DmhQXqTM+CKriKzXPV4b+j1VU5TozFYmdP4BptArKNPEGymPCzUum34fun98+VL0TkUkkWtFV5B6L40jStlwnSzRDXAZpwQ4HtTgHPKfnGdDGG59UIbLY76Ye2AB5wI7AZWpo3pQ0lBWKBNUvkfd/c0cFtHktMwGd6+hFPWlywCRKKaOBGkEMcPvxAlmatEC3SWUCjHddxz6QIgY8ENRbYSFijBUfOlejV+RIpsGHxrC6wCayqCxXMzhnGTAtt7FxnV3dLaOMFuuFltXZLv/322+V1mTHiXALOeVijRVqERDiEO5wbM1K7wFnU6xgzXObMRulw6zHeXb/zzjvb5BALqAhuJmqXOYIchDOcOcNi7ggWJCv+xXRT45Os0DcHkTqtIZOhSOevag+xOOOJoNOqYOM1FPhaKJACm3xP8XHRD8TXnnFZQe5JHxj/eeMSzxf1699O5h0sjjT6l6VLOZvmlfKUhCovSBjywQcfSNNmpbQFuy2koU+jDfQvoySYxdzhjBk6l2rWLJskO/nkk8qes2erzxirvii+ZQQBy5at8Fkv10hg89RTT5dJEyfajNn5F5xXjpWZsZkyZYZZtAULPpYps4d8lsubb/2lTJXJs1NPObWgIXL00UeXHdXvoEHz3nvvlnkSTPzxj38Q7qUWjPz8Fz8vxx1znPtShC13S8Pmtttud1eJdow1bCSQmS/zYddc80sJIW4oyyWwwYzbHNVjznnnSWBzSBnX0ek2fiBBzL0yT3b3XXeXt956q+y8084SSJ1fLrroIp1Ts5/7bczGIYzifJ27JSDibJxly5dLqHGgz3hBSELbxkiwTd9oYUn1vvSxg5DXRa8Pz6RXQnK0ah599NFy5513WngD3SdIYIMZNMpGoIQQpuV49YQD823v/E3CHtXhkYcfEZ6lFtigLXT66adLA+lM15nza1LQSDXcV6sCTLoRDuFGqB+nXWuFEzre9ce7ZELuVpmye1taUxN9ns/FF18ss2+n+9kjlOedyMtItvCT797A72AL4E10Q4GGAg0FGgoMEQXon7NvJgyvw2YAtEhvueWWcuutt5aFCxdaqI/50SuuuKL8UKY/OXNuvDRLG9dQ4B+VAsnbDGz/luIHwuV3NzA+7mEUY24xePrgsfqE9T2307IuA+OBGCyunXPLIXAmXs/gW7wgeeAL23kDjojgF9spTaihQEOBr0KB/A7xcy6GzxjueZzMbTN/5uzRgU5msPn8w7EAQIaccGfHhJ9g9bjM1/hDQwGeQSOwGRpaN6V8HRQIZif7mq3XoM1gDA7fPx2YuIRVi1zadx9slYpUipkgykumBVgLapyvnaDbKh7hEjlgoKpAhadVju7DjIzqoj/+cXhKaYWTKSJGMiEvVN555x9luudBH2rdI40aFlYnahFu6tQ4BHziREyXTdRZM9O0GId5s4k6a2aCJ2McII55JGvwaPFxuIQ6CIUwz0PJDAy0k7NsqBftz01P2X76+Qy7oq3aVvWuEnM8aMNEqE0TaBp5BsI09w0FtjcF4luM7zPe6eBlEHrgrKXwBd/PfLe/6nudeLZEg3p6ve7Es2iPGawFCxZ4keaNN94o7777blkkrTw0QNDCGybBLqa2WPxHw2K/ffeRya4DCpofaNmNlZYdwt908BbLdTYM5tZ+KZNoCII4D+e73/2uhB8Xlt12m6n+ZZw1OlatWikTXI8K7tryusreY/aeZjDPlsmz6RI+oNFnwZIEG2i0cBbNgk8WlKN1fs3Pfv4zCX+O9bky8z9TYHNNueHXN+islzXSQDmx/OifflyOPe44MbQzLLCmB1uyZIm0a54pd0nj5cknnrTg5Lw5c8pFElYcoPN16F9HSLi9dl2XBDp/sQk2zMnNl5Br15m7lvMkADr77LNtpo3FLPrKURaeYz4hqSOf10i0p/9cKw1IFseeeeYZC8ugNwKbAw44wBpMCMs7ZZYOWFy+K+v13BYuWujzdF595VVpTq7U2WMbdD7PFOc7xIKeqc6Lhk+cqUNfHRo/4OM54WgTgn9M3v1Z59fceecfJBi7SybzPiqzZ8+yObQ5580phxxyiARvE1rver5L4Bj4jhGHy/isd8Q2vw0FGgo0FGgo8HVTgP45+2bCjAuNwObrfipN+d90CiRf81n1/LxwA/HEVCLXAwambu2+/3w7yw9/sLT+cVvDnGngSrwZB4Mb/CAxFU5VP9dEIq0xIdamVxNqKPDVKJDfIX58X/ENMoY3ApuvRttvVG4ecCOw+UY9kqYy25QCXhHbphhBlkxKdJSKwIZpbZF2YKnqPqu1uWC89NlVLuKDnWnHZajuR55KLONFv0rQUQeqwi30yoQ5toULF5fnn39BZnve8CHjLCBi1myCzqbAZ1ERjRmucVqQHafd852do6VpM8a76YH3ZE6ILYxRe4ZXu+iJaS8iRgWoGW1uu6hRm5HLlIBs3dVomHHpJ825d10yofEbCgwBBervH8XFt99mkFJggxZDpuMnA+XI2k/9HU7c9bga6OcKJo6tAddh6mXx/ZKGj+AAgQW7adG6QYhDHHzCDjsM9zko48aNK5MnTyrTpk+1gBehy0hp6CHANd7q00cItNwaNn8q1+mslz/LtNaee+5ZrtJOn4suvNBaGiGI2eByHn/8cWnAXFPelFbL3vvu5x28Z5xxhoQDk8pIafJhHpI63X/f/d71O1/n3hx22CHlZz/7WTnu2ONkamyTz7hB2yQ0bDb5/BnOuDns0DCJBn5MotGms889u/zXf/1v5VCZS+OMnuHqx6k/mjMvvvhSuUs7k9D6QVAyZ855Not2oM6UAWaEBDDQZIG0FR+QIPy3v/1teeXlV6yhcuSRR5QLJdxBaIOmi03KCZ4+U5mrnjEEfTZzWdF+uTR00H5koQzcmJZE8AIO6kA/jMtnx/PijBy0cRYtXmwzZj06Nw1NzjHqu9lVhVZOh86ucV4/F8YQ/UVV/MwpC4fAhnEGJh+h2Fxdf/7zy9Li6ZZpusPKD6++WubVTvXZQWPGhNk76pL1MZLP+PkisJ+BqkluKNBQoKFAQ4FtQAHG/+ybkxdoBDbbgLANin8oCtR57C/a8MHyJp/2RXENBj8Y/jZc8ITt+88OgW9wnFvCBQP6xfjFz65FA9FQ4B+bAvkd4ud8DJ/5IXM55rGNhs234B3hATNZ58FiMuk//uM/fPAwCxYsQKHujOozO0b33ntvLeSOcwfNLuJcmPoWkKFpwreWAggKQliwzZoodAgt0ukT8iJcrMJ5RcxJbYiETD9g+uGIzAkwqJ/4MjdAxOVelViGi6yG1Y+/bwlsOL9m2dJl2im/xgAskk6QkGasNWdGaSGRswuCkQJ/ta7YWtRzGwO1cQITA0Mr0jShPJzT+7Wpio/KV5kCMiaJmyXUYCKYuJ0LLrZxDQWGkAL5/vG+EuZi3MTFOxyVIUw8F+G2GarIl/D1PHXcgeWL/yaOz8pZh6MOA+vLPTCM/1y0I30E05jMQliB+UXk1JKSuEi0N+zoAxzcwVo7y5cuL0/qzJlrr72uvPbaq2Uf8RGoZp9/wfkW9oyWMAH8a7u6ymOPP1au+c9ryl/Ei+y3/wES2PyonH766RYoI/RAAITABk1BTLS8++7fykEyQfazn//cZ71wBur8+fNtHgyBDcKQOXPOLRbYSCiDSbRfCv+NEtis6+kuF0jL53/+P//LWiwjR46qzIQVn9/z55dfFqN7t4VD9Jdo2HCWzEEHHWSawQPhOOOGs25uu+02mzT79JOFZReZjLnk0kssmJo9GzNxHdLeqTQSRZuglN4T0c7n2YhkvCd1nmrgc+I5ZFzCQTfcMD0PPckYl6oDcnkM6fRII68ieU55lg4jCHgxjQkMbcI83ctq+4033iQzb49Y2wZzaKeeeqqex9US3Bwh4VGHaOWHb1pQTv19znIH8z8v3GB5m7iGAg0FGgo0FNj2FGBsyb6ZMONCI7DZ9nRuMH77KZB82pdp6cC88GVc28SJJ4x1B7jFDCdm5gIZ/nw+dR1Y38i5JVwUQNoXLOjzVaeBaijwD0mB/A7x+bbyYgxvBDbfoleCB9wIbL5FD7RpygAKsGpVW7kakPqFb1sMz4Cc4j/6lwIjk0xJplT36lARrnB2QNtFmrraiJKXudowhKpYGKUMK9b56Kirv8zL902njdmc7nU9WhjdoMW64T742zvitUCHCRwwRKePsCYW9OS5POJbRVGG412i06saA9yqE3EJ4YQKQeStYsALlCPrWCJ94K/rUUU2DN9A6jT3Q0GBfAfjWwnmiO+L+2SS8OtwGT+wfgPj6zhJ+6Iuy/w8+bK+A2GzDsQTpm0pFHCdvEgfaQRDY4SvWBc//qYJx3dNv4N5LcyKXXvttQUza5j4uuLyy8s5557TX2CzVgKbxx7zppG3/vJ22X//A8uPf/JjC2ww34jApk/C5+XaTHL//ffbJBoCm4MPOaj84he/KCccd4Lqu0kM6nyZKLu73HY7AptNlcDme9Ki0Rk2SvulNGxukjCiR+fGXHDxReV//r//S8Kh/W0Oze1QM9B0eemlP9sk2oMPPCANlQ7hmWNTbghscCmI6xae9+e/Xx7T+TO/+90d1mREG+bEk04ql0jAc9SRR5ZddAYYGil1c3H0llDO74/ojEu65fPP55T3wHJxn0KbKiOojLJ6BL6BHtxbSFO9U3iBj/LjGVMODsENAjEEULfccqvO53nOZvAOk1m1c3QW0Bk6u2b33fewwC5NqpEv60f4s9wXgf0sXE16Q4GGAg0FGgp8dQrkWAOmHPsbgc1Xp2uDoaFAUiD5rLz/fD5zi88HuVUosXj1NYPNYeHZN4/9rJjP16Y6bgr5EgV9VkWa9IYC/6AU4BvMi/lVXswVG4HNt+il4CE3Aptv0QNtmjKAArEoNSDyy90OyvAEI6Jlr0okkagBDqakP5OkuIorajM6wjGAgUlZD7WHzao7xD2swrXTqvx01AZsM0PRif9f9t7DzapiSxtf0HSTmyBZQjcgOYqKiiLhEkTJYrzqvXfG+X7zm5nn+09m7nPvnXnmJq85AYqSERVFQRQDOUmUoOTUTWi6+da7aq+z9zl9TtMNqA39Vvc+lVatqv3uc3bVrnfXqrBHjpne0fz6+mY8SJlMhzk7HHCWiw+k6V+I6yRh1BnYOWhe0OIDSp+oDkq8NaYQOn1TG0+A+qg+aLrSRF6MF3SFmhOqGCQCPwsC4TcVfn/2E8FvUQ+fQPfvabbvKNI83xuLtMz0bGVdPpefqTeXnKe7vNflbTACIf5hGkEQymg7dXUGfrJGNqsM4jEZoDnAwiTw66+nYwtdEZMibF6QrbpHSz/d82amrtodN26c3NK6lZn9uqREcmmCsNm2bYfujQPC5lkZ9cAo228LhA3Io+NKKCxXEgUbIQfCZoA898/Pyd26Dw0Iit0RYTM3g7AZrCtsMHjFCh4QNlit8/CUyfJv//ffpVfv3rani99VQNisU/NmC3UPm+XvLzfzYhOUsMDeOwMGDNDrFUynAbMyXaGMia1NmzbJ7DmzZdnS9y1e3L3YSI7x48ZLP10F1ETNTWKVjeGn7cSdFRjhOvi18GsD366LNsiQjO53kMP1gXPCyCJRfgV0QTfUa9mgN9SD6+TXGGVQBIfXDb+kpFR27txpK2vmz59vK506dOgoD+oK6/FqEu6223rqvmYtbXWVm7+zdlojrvxRE9kra6MEESACRIAIXA8EcP/3+zPC6GdI2FwPZKmDCAQEfKxVEzyyPTfXpLzJ2pAQg8KqHMaHVeVXnVf1ublur8D9qnUylwgQgSsjgN+eH+jD/SBhc2XsbigJXGQSNjfUJWNja4RAbEbGpr6yjRMqDWZ8cJGoqJKM57ksCJsrDYjCPJrNpNk0XKzDQ/ChBbqCNtyIPRU+Jtn0wSqqCsSOtgBzc5F80G46UFD/LRcTjPqH33v5ZTWjE5SqSZ4gY5N5vtJGH9TCvgpBJ27+MH9jk7PRRCUqDLVak6KPoCuZkgxnG3ha9VoMzjuZEAufaG+mgxwdEfglEPBBEerG99Djye8uBknJeLKd/t2FDA7Xkzb5nixQw3B1fy+QQ/3wve7MtiHPzwNhO+yG4r8//N7dxBYIjPD7D6QBzq1eIFh0RcynusLmxRdftBU2fXQ1C0yigQBp2bKFFBQU2Mq/EphE0xU2f/v732T7ju90JU5/+fXTkUm0Zs1thQ1IlqO6V8sHH4Q9Y3bt2q0ESn/dw+Z3Mnz43dqcCtm9O6ywiQmbCfLYo4/bChsQNn/769/ktddeE5BEMFuGFTY9lYjAykO/k548eUI2bNgoixcukmXLlhmpNF7bO236NBk0YFCQje5DvgLphx9+sBU58+bNk42bNtqeSB338wAAQABJREFUMSN0lc2TTzwpI+4bEUiniLCxa69wBRXh7u330+T3Iol/8vqgd4C8p0ERdOnlUCxxrZzUCSQ7rp05yETfOzNpp4WQFfIv22qoL7/8UhYvXiLYTwjE1SBdXfPUU0/JmDFjpbGudML1agBzeKHxqW9oZhwZ0JstPVWIASJABIgAEfjFEUjeqxFGP0TC5he/LGxALUUgjJnSG3elsU7Ny4Rxt46k0iuqMhbGghDJVl/uonG53DLVy6lcr+v2Zwf3q6ePUkSACORGAL83P3AP8gN9OJ55uYdNbuxuqBxcZBI2N9QlY2NrhECSsKluwWhwgYmsKgdKPgiB3sqEjeYmKgzhoE/DNv5y7ZFcUjxRMjlYCwOhePCWPkCMazTNkZjdyLUtvqoGN3GvGdVYKa/bz9mW+OhkHgT0wzoAvE2PzgDnhYk+ZETyXjxuAQqmu+oQNiiRPKdwvpl6vLb0dMaIwM+BgH8n3Ued9hvTvtTD8H3Q5HIe9+93Mt0K4msdVKT9BizvOn943e5DvbcLYaTbfUJ9pCNuB+SilRqhseEeoCJKYuivPzqHcG+IVtjo3lmffbZKXn75ZV2Fstn2xXv00Vlmqqx169ZKhhTYGOSsrlIBSfC8mk7bvn2HmSl7+mk1iaZmuJo1bWb7q2CsghU2IGzeeustXWGzy/aU+ad/+ifbwwZt3L17j5lEq0zYhBU2f9N9+kDYlFeUhxU2//Fv1ibs34KVMzgvEBUbNmwwwmbpsqXSqGEjM4k2XQmbgQOVsKkH2YCL42WrctavkyVLluqxWA4fPqLm2gYoWfSY/Grcr2zTR+yF49fY78F2N1MYXR/OAWEc7pAG52l2LRJplo72QLn+B1NoIHVQJr2c60E6iB3XBRNxGNwvWrTYDoSxUSWIqgcffNBWFhUUNLTr4O1LNBFqU+2ziH74uXicPhEgAkSACNQ+BJL3aoRJ2NS+a8QW1R4E8Bu5Xs7HdZX1hXF3atBYWSBHShiLp7cxGp9rifR0qIjzciisUXJl/V7cx7Xx2NZz6BMBInB1COD35kf8bBbMbZOwuTpMa2UpXGQSNrXy0rBR1wWBmhI28cAl96ADDYvlQjPTSRDkByn/9AFKmFALQ720KTstgtRYTmuI4vHAMLQpjkMklPByHke96lS0HJNyaibH36RHrdBj2sNsnskhDWdh6aITlzp56bLQg47ASJ/ELB3K4M/ytWToLDAgDJXD98FoLsLGJCM5l0UaXDjf4Hue+0GCn0Tg50fAvvf65cZ30X43SoJiguPixYty7tw5axBMeGE1QiAC9LdjK9jC78G/w/A9nNTpaT/VmfnvCvqtXv0B437j9XpbKsnpzcHajLsEbhQ24R9W02DFhnI24X5g2sTwOKaEzapVq+WVV14xs2G33XabPKIm0UAEtGlzSyBsdO+UM2dOG2HzwgsvqCmuBGEzapRuct/UTDmWK2EDYuTDDz40k2jffbfTCJvnnvvnSoTN2zCJpvcVrOR51FbYKGGze7ftkfP6668rYVMhk6dOln/99//fTH3lYYWNETYiWGGzUVfYLFKTaEuWLjGTaBPVJNj0GdNlkBE2em+M7nHAAwQJVggdPnxYPlv1mbVt3fr10uaWNjJRzYk9OEkJj/4DBAQVbp8xznYF8BEhFt/rYplwvzaZ6PuSvD52PYJSFdFGhX+7rgh7fSiD7yi+hzhA1mBPM6RhlRD2rvnqq69l7ty3baUT6rvnnntk1qxZcscdd0i7du1t1RAUQhcIOn8JALLuvN2I+/cnmeZy9IkAESACRKB2IIB7td+nva/gCpvacW3YitqHgI9trlfL/LeXrg+DzHj8l56XK4ZxfMiL2xineamq8lzmav1Yd6YGNCx+zsjMZZwIEIGaI4Dfmx+4j/iBZzsSNjXHs9aWwEUmYVNrLw8bds0IhAGPq6lqIOGDnNyylQc9kM2tM2gKA7FoBGWDL68hTKZhABMc2hrCToLEkjYPF3JhEg0ZKo4S2gB8hrD6lhedDMLQBRG/iafrjvKgIAMAaNUzRo66UEekPSR5VojFIh4334VQPmhEcsDEBLz5aWkhR0tE55ZZxvPpE4GfCwH/LuK7678hpPmBwRHMYu1WUgAT4Jicb9u2rRQWFppJLRA3kIUcHCbNncxB3PX47xRpP4VDPe7sXKJz8N9kVe3wkjj/YDYxIgBw74juS44N6oAJMxA2n36KFTYvybZt23UPm34yc+ZMGTt2rLRq1So2iVZ6VgmbT+T555+PTKLpHja6wuaBBx4wc2LACgTDCd0TBytssIfNjh07pH+//vLcvzxn5AKwxQqbxYsXKfEwF6AaMWSEzaB0wqZCCYup06bJv+kKmx5qEi24QFydOnVaNqtZs4ULFsqixYvt+k2cMEFmzJyRIGwCGo5hWdlFOXu2VLboPj3Y++ajFR/Jjz/+KN27dzfS5uHJk6VHjx62V47da614qM9usxoH9tmuv1+zZH5mmunUk7B0v8Z6XVJ34Og64zxDHZCtsPEf9q7ZvHmTrgxaovsDfSAHDx6ytmJlzWRtd7du3ew6oWwwoxleAHASEunu/HuEuLcR4WQ64nREgAgQASJQOxDwvgWtQRh9KQmb2nFt2Irah0BybHMtrbvyuAgDxTDWrGk9cRsrz11UlVeTerLpidMyNWE0irb4qDQzn3EiQARqigB+b37gt+UH+nASNjVFsxbL4yKTsKnFF4hNu0YE0gc72QcSlQczqLSybFVyuQYgXsbzcWMNp5R7zALZWC5IR4XgabZJREkhP+Mzqi5o8rzQFjsvawQUpStRCavAJl1tkOjtDzpiTIJkkHP9ST9qgOlDOurxuqAz1JMskS0c16dNzQ1YtqJMIwLXFYHkd9HD7qMikBMgEFatWiX79u2z1QiY6C4uLpZbb71VV5O0MeIBsvguZxI2SHf3U33Xk+31upI+6oUMDoST7UiVhQz+dLIfB2SwQ1bqN+5BTXPC5rPPPpOXXgJhs00Jm/66wmaWETa+h02FmicrLS2RlZ9+In//299lm+LYr68SNs8+K6MeGCUwJQbCJmkSDYTNzp2+wuY5W2GTJGzCChsQNhPCChvdiwVk2l/VJNrrahINd6OpauLs30HY9IwJG+Bx6tQpJWw22wqbRYsWScNGDY10mT59uppEGyh50Uoc3Eb19M2Vl1fo+V4y0u6rr76SFStWyLp16+w6Y4UK9r8ZMGCANGve3M4llNLyGsDdFG2HS+Ju91erI/t1SV4TK6wflhZdP9eHtJRsJIh24/rBFBpWLa1du1ZNoS2SjRs3KTnT0Aiw0aNHy+233657DbW0dkEH2gkf39+qCJvK9UVAeUPpEwEiQASIQK1AAPdr7+/9Pk/CplZcGjailiKQOca5mmb6by53WYxUcdTcxe3DWD69fFV56ZJVx4KedP2x7syy/kyR0ZhMMcaJABGoNgL4vfnhz4/wSdhUG8IbQxAXGW8D49i+fbtNZrz33nu6+exxS3PzJXjTEm+HNmvWzL4YmDzJ9rB+Y5w1W1l3EEgf7GQfSKQPNhybdNnsMpDVn5C69AHIlQdhKBNcej1hws7zkr7J2ekkBm+oVm/M+Mvq0DjLCvlhAhATbpDWUrB8ZjN3GBJqoma4WSAnc+L2pWPg6e6bvlQzEEhFNBwm+YIM9GCvCDoicOMhgO+7T1r74Ahncf78eZugx6bz2Kge5rHQfw4bNswmvfv372/EDciHBrr5vPehfq9I6rpWVOLfZPT7VoXJtGTY6/U0b09mG5CP20Z48MNvGzEfKIJswL3D69OAZoOwOa4rYlavXm2EzebNW6RPnz5qZutRGTdunLRoUajkQL7hee5cqXz62Ur5+9+VsNGVOH369M1K2Bw9elQ+WL5cZs+ebQRMfzU19rvf/VaGDx9uevbu3WvEwztvv233OZheQ30wZbZ7z27521//ZqtzcHuaPHWK/Ou//av01BU22JcGDiTGyZMnZYu2dbESGIt1hQ2uGUybTZ4y2Uyb2bULQBiuuJ8iCtNo589fsJU1IG1w3ocOHZIOHTvK6DGjZfDgwbYvTEF+gcprAXX4NGyhBHFN97zMa+Jxl7MC+hFKIqahKJLU4eWSaTBRie8xvrdYCfTFF1+YSTpcr+Li7jJmzBgjp0DWwLyfl0UZHE7YZOqGnKfB93Lue5vpEwEiQASIQO1AIHmvRhj3eBI2tePasBU3FgI+/sls9dWNgTCgi0d4mTqrisftSH92R5mq8qrSmZnnY9/K6Zltjse1YdSbWYJxIkAErgYB/Jb9wD3GD/ThXGFzNYj+jGXiG3HlSjM7DMiCrIEDYfO///u/AsIGAzXY4++pb54+/vjjtulsUVGRvSGMMj7ZVLkGphCB2oZA5sDh+rdPp6WuqPTKEldUEY3bspyP3qSzOohmZuHmHglbVrKs5pmzDJeKhHN48f0mOSj0SuPJu9D4kJ55H8qhmslEoFYhgO96/H0Pk+veQPSX2KgehA1MS2H1B/ZewcoarLApLi42c2CDdKUH+lJMhDdq1Mgmvv334L7X4XEMvOAQ97RMGW/HlXwv53Kuz+O5/FS51P1C7w8pPDAx7/eL8BvHrQSEDcyLff311zJnzhz57rudRmI99NBDMmLECMWniZIBDQzTc+dL5csvv1Qi5i3ZtWu3mhLroatSpss9d9+t446mRhr4HjZYxQSzYxiMghR77LHHZOjQoTbJ9P3338tHH31kpA3O7QFdoYP6QAAd2L9f3nprtpo6W2CnOXrsaHn8qSekqLhIxzQN9Fapbdd2nz59SsdD28z02ooVH0szvY4gMMb+aqz0uq2XkjtgulUFTtJcuPchilVApaXn5OjRI0qE/KBm0kqkiZ7nrZ07yy26pw3IH+yXg/JQYc7VaMSvRwrvSCTpuYynoTju6tBn5+AZke+6vFyIh1Jor++/c+jQD7biprCwhX1nb7kFeww1tO+o64APPX54uutOVl1VXlKOYSJABIgAEfh5EPD7Mmrz+7bf15GGMA7MA2zevNlecHjzzTeN2Mezf6dOnWxvs6eeesr63+a6apSOCBCBnwqBxADxp6riZ9ObGvX+bDWyIiLwcyHgfWuqX8UDZeLnC1Pcybg/R6Xk8RCpzuPJdif76Mx077OT+pywwbNyu3bt7Dk5WQ7helow1bz1uvEqHqphEqJ9+/apRkCpi3nD3M9UyHjNEHBcc5VK4gxZl4c5l//5n/8xwubs2bP24A77608++aRMmjRJunTpYoQNyvvblbnqYDoRqGsIpG56VZy4D1X8Dqk/pVrqcDbVOaNczfcTi+/zkEzee3KVZDoRqI0IeD/pPr7Lye8zCJtNmzbJu+++G5mW2miEBfpKTNJjwNS7d2+56667BKttunbtamOiFi1a2KS4r2KATq/DcfB4sk6k4fA2uO9lqvKT5aqSy5YX3xU0ZDcyb4fGgQmStSBWm4AMOHfunJJXu8xUHAiBjrra5M4777SVNg0b5qfIAOwBs3Pnd7Jq9Sr54cfD0l43ub9LV8306tXLiK38BmElDsYmW7dsla++WmurmKDvvvvuk+LiYsMDK3Aw7oSJL6zkgwkzjD9vvbWzrRxerWTPV2u/0jZelv4D+st9I++Xdu3b2UsojmFpaansV+IHJs0wYdW4UWMB0YYVMqgvF2HjeGGgbIetXg5m4/J0lUp9JWqsDr89agELGnSJRE33a+463fc2ehw+zgWYQ4NeASRVw6FEuHa+yhptxiVFOzExh8PrS37fPM39alRGESJABIgAEagFCCT7Fr+Hexrifq8/ffq07ckG86MkbGrBhWMTiAARIAJEoFYi4H2oNU77UTxMeVqyX0Wa97vwPZx2UniMwyOaOs93Xck0hJHuh+uDj+c5WJxYoC8okrABUrXYJS9urmbiorocCBussMEbwhioYaKluLhY8BYNTKJ11jdEsYEyHuIxCeVfoly6mU4E6hIC0b21ylOu7lRaupKk5qvTkK6vOjHUmay3OmWSMt5O95N5DBOBGwsBHwyh1d7vIc3DSAdhgw3nsUJ1/vz5RhqAsIBDf1lQUCB4E7Vt27ZSVFRkq22GDBkiffv2lQ4dOqh5sBYm432r98vJepL1QW+yXYhn5iPNXVV5LnMlP9wR8InftfqKAfzQRg1bMj7CuAIDRuztArNbWGkCjGAODqZVmzRpou2FCiU39EAYK3Kw2gN7wTRQgqNZ02ZS0FDNh+kfcIH+cs0DoVJSUqoD0nLbbwUrdYAvHHRg7AIZFEA9qC9fzZAh76y+NYy2oF6sfCls1cLqQllghNaDwICJtnO6UqbsYhnO0Ei15s2ah1VROVbYQIc7vzbw4XTXFzVBGQbnqMMw0lQLAzoAkHBeLpFkwUw5JIYaQiicgYle4QOl4pKBrMG1RFvC+M7wiNrl5wOl3gb3r1ARs4kAESACRKAWI5DZ36A/uFaTaJk6a/Hps2lEgAgQASJABK4JAX8mQt+HPhTOn6Myn6FcFjIum3z+9/4zWT5TDmVdL3yXhQ9ZLJgBYYMFM7BEkem4wiYTkV8w7he8qia4DEyi/fnPf7Y3hLHpLjajxQqbpEk0THzgC+Vfqqr0Mo8I1CUEMKmnt+bolH0iDHEPx7m5cIklkxLRpJ4lQZ/XkZS5Utg1J9vjepCXGUaal7mS7mz5rs/9bDJMIwI3DgLeT2Ig5OFk632FjRM2MI/mhA3k0GfiZQestoE5NJhJ69atmxQVFVk/i71diouLLc/NUKEc6kv6Fkl8oC3eHpdNZKeCVeWlhK4Q8HpisXCfQLo108yiaXsvhzYjHStt1DOHeJANxIXyF3qXgUx5SNeEPCNm6muq4qxlIYG2e/srlEyJB61YAaISUQWQQZ7nh3IgH8KeWV6/NQg6QaA4iRKarPUhEEzFuh6NWjJMmdn4xwkba2WqoApVdt621F0c9ZqYl1Pl+u/n5xq8nMfdz5RDOprnzrV6PLePUsmSGrProxrsP9aEOjPbk60duetiDhEgAkSACNRGBNAHJ7uC0A8Ek2h4CSVzhQ3GLo888oi9zAmTpFWZRMvsN2rj+bNNRIAIEAEiQASuFYHksxL6PsTtRUG1DIEXAfH8j+d7t17gfa3VGz0bJp+tvP9EWlI35F3OdXh9LovnVxI213pFf8byfrFzVYl8n5TAChsnbLDpLiabsIfNrFmzbIVNjx49zCY/dJGwyYUo0+smAvbIk+XUMelVeWIsi2AqKVOTTpVFD1PQhbfM44m0VKEqA8lJOW8PCiTD6QpCDcly6flXjnkb3b9yCUoQgRsVAfSjIGw2btwo77zzju2vAvNoGKj5oArnhjD6TqwywYFVIVhZg34We7rce++9RuK0atXKVnJgYAe5pI4kRp6O+nF4PCnj4Wx5KOMuW77nuZ9eD8rG5c0WmsX1N58gbFAFpJymwGoaEDGoLy8PZIquaFHCBvvT4N6H882rr/vaaAkMcOEgV18FMVZBGkgglK8PxkddOI8QdxnHI4xVdM+YlFx8HSxNgsmyzPNH+dCucqmv9TfQPW7cQdblM+9w3haclztDyqDSxPCvWUkBy0zpRLmgxzXEvtcbpwR8k3EPJ2rwpIRvrUrEEfQS2Qma0MpQxCUzFDBKBIgAESACNxAC6GtweN/icd/DBubQkibRakLY3EAwsKlEgAgQASJABK4aAX9u8/4UfSoWQOzevdssTWBfUDzfY39bJ21QWeacOsrh8L7Y49ka5jLJOiHvhA0sfmCFDRZfZDqusMlE5BeM4wK6wwV0l0z3i71t2zYziYY3hEHYYLIJE0l4kwYm0bDaBl8yyMPePr5gdESACAABTDXit+a/scywoxT/Hj2laj9THr+56v7ukmUz25NsZ2YLfGo1WT5T5kpx6Pc6riTLfCJQ+xHwPtP7Ue83veW+wgYmRRctWmTkDV56gLzLOlnjZTBgc9IGZtGw4gZ726DfxX43xcXFZkIt+UYOyro+hL09CMNlxkNq9k8/J8+tTtm4DO4PiXuEra6J7jMRYQO9PgSB7+2Gj+FIGLCG9PIKNR+n6fVhcrVenmquH0ylqQ6QNUFXIGxcD+4xIG18sOvp7lshlck08YV8f1HFSR/cruz8tZ1GKkEG5tqUJIJ+rK5BvslEtzbcKaNgqEo/odtdnKffAXwPooyQHuemQPKCVfjZrpHrzSyWqCEzS+MoFUoGvML5B/3xdxYFU3VG1wHlKp85JOmIABEgAkTgRkLA+0u/z3uchM2NdBXZViJABIgAEfilEfD+0/vTnTt3mpn07777LrWXLfZmbd+hvbRs0VLyC/LteQryOJLlPc11+bkl4y4PPynvhA1NojlqtdjHxUu6zAuMPL/QuLAwifaXv/zF9rA5ceKErbApLi6WKVOmyLhx4yS5wgYTLSRskugyXLcRiCe/cuOQ/nusSi4pmT7pBrImPSW3nl8650Zp5y+NE+uv7Qgk+060FXH0neg3vZ91wgaDo/fff1/wAoSvEPF+1ib+te+EQ5oPrtCf4iUIvBDRpk0bKSoqEphIw4F+t0uXLtK6dWtbdQM5d14+GfdwdXxve1I281yTeQjHZTLuebkIG9yvVNQxMB36gbuDwmgfoDJAjkAw1B8T0yajWGHPmTCkCW8Dq7DpBBkTVuqEa4J0uFBfCAedYTDsebg2SEdZo9ujVTtIsziur9ULckKbCWIo2tvFb8FOWiDfXYxPOEekmwbTG6SCvJcCOF76yn44l3S59OLpMa8lbo2XdbmAJ7ANxBbyA5bAIcTMg4Cl+Keff5RLjwgQASJABG4gBLy/ct/6v6jfu9Y9bG4gGNhUIkAEiAARIALXjID3pa5o/fr1Nre+atUqe8YvKiqyFzKxCKK4uFg6depkljbcogbK4wjPp3n2YqM9c/ojmyrOfA70MqgTeThI2PgVqOV+8guTeWGTTcekRTAxUiEwifb8888bYXP06FEjbPC278iRI81cC976ha1a/yIl9TBMBOo2AjrZWC9sMPbT4YCptzBV9tPVETTXS70hH0/3/dR1Uj8RqK0IgFCBqS44n+hHP4iwD5Sw5xtImuXLlwsGZnv37k2t4nAZ9MUgbfxAOhzSXQZ5WHWDfW46duwoAwcOtD540KBBtqwZ+8j5wA6yrgM+9GT2956fmY56PQ9hd9nkPM/9UC4MKpGm1eqRGE3q/cNjICssbOca2oe7CsgQI2k0E3XqqaiLSBKQJ/pX31a1YEPGch2PwDxaMJkWzhvt900dYVoNuqEDDniG83NckOrXK8iGtkCXrajROuFCW8IqRhAWlod60M6ItFH1QdYDIZoRixLVs7bEjYszNGSqQtVp6YigLdVxXjwQLIjhAILuPByneA78zO+B1wt9KZ1RJYhDm16darcvWRfDRIAIEAEiUDsQ8Hu/+94qxEnYOBr0iQARIAJEgAhUHwHvU7/99lv54x//KEuXLrU+FekwhV5cXCzDhg2Te+65R/r16yewsoFnezyTwvm8A3w8k6EcfH8+S7YEeV6fyzhhs3DhQlvZ89hjjyWLWJgm0SpB8vMm+EXzWrNdXM/DBfW3hDHBNGfOHIFJNEw8YbCGSSO83QsGEGG83etfGv9SuS76RKBuIoCZLH37W//wX10Xz8X5JFoofDlFlmTRZKIunyW/RknQ4w3ODKdP91VX7ZXuPVfKr249lCMCPycC6EMxaILz/hJhpOM7jTSQAceOHbNN/g4cOCCnT5+2PP/Ow/e+GL6HXQ98yLg8SJsmTZoYSeMm0mAmDZv8du7cOWUHN3OPm6Re6HSXLd3rqkrG85J+KBe3FfeyJGGTvIdhet/vMiHgOCA96MDdJ2ASYWDkSSC3UC8IGydnMO4AXQDCRxNxo9KymYQNyoRrhfJWJrpW2c4ZaThAyODPwuGkLKxXGNWEa4b6LAbN6Q4ylRyaiUTUXykTutRly7AiWTVW0uLFgWdQFlKS7Qzh3Pqy4QJdQWeo0sPQdSV9lRrJBCJABIgAEahVCPh9331vHOJndaPkzZs3yxtvvJHawwbjDcwHwFT6k08+KTDtghc5c7lMvbnkmE4EiAARIAJE4EZFwPu68CwbniORBsLmD3/4gyxevNi2G4E1DhAz/lImrGjguR4HVt1gjzjfxxZz7slnfNedxMie0qIXDj0dcqh7z549tqduu3bthISNo1NLfP/CJJuDC5fNJWURPnjwoCxbtswImzVr1sjhw4dtksovfHKiyidAsullGhGoWwikT5Klzt2JF7x5HubPfC9uE7FfJX6bUR427baJTo+nFCUCKJTj95yQqmbQ7wuoMA6HiThNqqodOWpI3lMgknnvyczPJpNDNZOJwC+KAL7Lyb4QjXESx7/X8L2f9DTIedh/D5lxyPgLEMjzA+ko06hRI3v7pn///vZGDlbb+MCusLAwtfrH9buP8u6ypXk7IJMt38tm+qFcaGcoi/LhhhE+4/taOmGDMioLYsTuY5DTNP3DHciSND/odxnElTBRAW9jEmOkZRI2qKNCB7B+fsDW8UV7Pd193OusDdYALasKrC7UiXahkH6oWAgjXsmFXJP1vEivlTLdIG0gFzuTT09KZfr5phJyBLx40I1YSFFkUiVCOI6nMhKBFB5IU1FIJ9vrYeiqjr6EagaJABEgAkSgliHg93z43t94mq+wef3116+KsHE9teyU2RwiQASIABEgAtcVAe/vwjNpOmHz+9//XpYsWSKnTp2y/eG9r8VzaePGjc2aBl7IHDp0qAwYMECKi4ttVQye75GPuQbIum5vOJ7JLuNZF8980YOgy6A9JGwcqVro+xfGm4a4fzHcRx7Sk7KYAAFh88EHH9jmSCBsjhw5YmWR5xMkSR1eB30iUNcRwDTclVxuCZ9ECxL608zubAItmkXLLlGDVK8zWxHPQ0NyNaZyOZOMPsJkXmUZT/F7D+8njgj9GwGB5PfVv8NXandmX5vU4SRCUpfLZ8r5Gzlt27aVvn37yv333y933323YPUNVuJAPnkkdXobkzo9rTp+Uhd0hHjG/cEIG6Th/uH3EMh6DSHNJUwmInlc2vapscEncsMKm7g+TYsF9c7kYxict9cBP8RBfmPcApfEGeWg210KE20Y8uBSaS4W5XnbU/muxEqq3lA8leoBYABSCQ21OkwujM28LUmMUa5yHa6tsp9eLWLpKaEETiackJ8W0jPrtbgKxPVn11VZU6iFn0SACBABInBjIOD3f/jxPT/0C07YYIUNjh9//NFeDkmusMFbwVhhkyyLM3e9NwYKbCURIAJEgAgQgeuLAJ5Bv/76a/nP//xPI2ywavXSpUtp/SWeT/FSJvawxcoamELHMz6e7/FyJuIwoYbVNuhn/UBL0c9mmvT2fOSBsMG+uu3bt+cKm+t7aa9dW+YgCXEfSLmPNDvwUB89iyMOwmbFihWyaNEi+fzzz1OEDUy94HAHWToiQAQCAsnJL/s9pSVURqmmv57kzy19YrKy7iuneOPcz1HCstHSmrY2oS8q6vcd5PDekcCHwRsGAR8AocHJ73BmGHFPQxkPZ56o5zmR4MSCy6Fc8neDt2vwlg3IGpgoxb42GMzhbRzsM4c8b2OyXGb9yTyv60p+Nh0hLeP+kCBstPUJUiRJ2qC2xL1Hg6Aw4Ewn8ENc/+NVM0kcg6wJWKlcH2FQi3tntvan6tOAtTQiUiAP3Y4TfC8ftSySh5yfBwol2uVBiKQcThTyOFsViGSQhPrdeV1ev6dfyU+vErH0lFA+1J+mK5xwepLFtLy1N+iJW+ii3urKOS5BnwgQASJABGo3At7nwE/2O4hfLWGT1Fm7z56tIwJEgAgQASJw/RDwftT7wa+++soIG5hEKykpSSNsXMZrx7M+iBuYPcdzPggbmB0tKioy0gXEDZ73bU9dffzC6hpbUBE9V2JOAfXjgG4SNo5sLfQzLz7i/uVJNhfpLot8EDI//PCDbZb8/vvvyxdffCFHjx61sk7YQB7MIL4c+FL4ZFNSL8NEoK4hoD8Lm4bLfd7xpBamv/TnVg0XhKA7c/LNTQ8FJdVSllHflcvYpGJGqauJek3eeVyNDpYhAr80Avj+2gBJG2KDo6j/RNj7V4TLysrsQJ+JuDvIuJz3u56XGUe6y8JHP4sBGt6yweBtxIgRcscdd0hRUZG0bt3abOEm5T2cTX9mnsvk8nO1LaTj5mR3NG1vRGcAF1OG88XGiak7QGq8EeqKJvyjbFtZo/vEoHD9aD8au6uGG6Dh4bhDSGEx/fDhvJ04P4RRLNQf8v1a+GDWy0DWVJgihFA2OoNIucWR5ZUFldGZhTLWcIhEeZU9EDUhN0YIUhEOlQtkTfG2WclEe0KLvQhi6Skhx04iEtL8DBH/boTSMQ4QtJKJ+kK7c5+tt4Q+ESACRIAI1F4EvE+B730AWos43gbesmWLwCRathU2TzzxRGoPGy+LctbHoYvxcO09fbaMCBABIkAEiMA1IYAnOfR3cD43jjj6xW+++Sa1wgYvQWAe3Z3LIJ4sD+IGcw4dOnSQwYMHmyn0IUOGWH+LNMwJeD3xs3Go259zoQ+Ezfz5843sefzxx73alF9PhVKPguvXr7cCmGDAkhzv1OG7WDItpYWBa0bA8YUixxjhZLoPrPAFwibJ+/fvt+uF1TalpaUQT01QeRh+Uh/idESACKRuewkoEpNaiWBKIFuRaGIvJZOM18MksE8Epyu0abUs+tLm2ZK64goSoSwKErk1CqqqzPtE8t5TI10UJgK/IAI+eML32V9g8EESvtM4kH7o0CHZuHGj7Nixw1aoQibzN5A8Df89JGUw2MJRUFBgy6OLiorMpi1Mj3Tr1s0OXyLtgzZvA/S4LveT9V3fcOa9wkYTURXh3pT8zJSGoN/BQp5+6n+4X3lOeosDXpDGeWbmeVmMcUKeY+A4I9XTEPZ0UxUp9DSXTcUzKkyvPqoQhbK6IJ0pla4ja8GrTMysCWrS21BV3aF0ZR1xmTh0lQ1kMSJABIgAEfiFEUj1b9oOD6OPxNgFk0tbt261/WtA2GBfW4xNMGE0bdo0efTRRwUbJsMkGtLhXAd8D1sGP4gAESACRIAI3KQI+LM7Ts/D6EdhEu1Pf/qT7RGPeXa82Ik+1vtHfyb1OMp6Gp7x27RpY30uTJFiDgB72eLlzVtvvVWaNWuW2mseZTBXgQNhHCBs3CQa+utMR8ImE5FaHscXCpNN+BJduHBBLl68mFpJ418gPwX/EnmcPhEgAvqQkjIHlIkGJrayT27l2u8gXYOXha8PQGmEDSQ9H9mJMLLc6U27eg4TdD5Jpzf7XPqqp4xSROCmQQADKLztgv4PfSVecMCB/hEH0tF/YnIDy55hWnTnzp0mg7xkP5rsQ70sgPI6MEDDRoMwf4bBGQZmMH8GwsY3IERbcKAMHPpwONcN38OWwQ8iQASIABEgAkSgziOQHI/4GATjBYQ9jjEFzLds377dCBussoEVDrhbbrlFRo8eLePGjbNJI+yhh0kid67ffU+nTwSIABEgAkTgZkTAn8u9D0WfiHmCTZs2yZw5c2yrERA2SEs673s9zZ/rEU/qwj43IG8GDBgg9957r+1zA9Np2PcGZtSwzy0O7HUDHdD7/fffG2GD+YRZs2Z5FSmfhE0KitofwJfBDwzQ/PAvSXLSB2l0RIAIZCBgfEhVv40qCJOqilk1KJsob4SN7yeVkZeUy2hi9aOJBpGwqT5slLypEUA/6AMg9JEgZ0DceD+JPLzsgIEZlh8vW7bMyBvIwXk/6n2ox5M+BlqwT4sBWL9+/Wx/mt69e9ueNRik+aRIkqhBedfhFyBbGvK87qSch+kTASJABIgAESACNz8CybGAj2EwbshMP3/+vOzRN3Rnz54tr7zyiuzdu9fGPnipBHvp+f55PsGUOfZI6rv5UeUZEgEiQASIQF1EAH2fr25J9nsgZ06cOCG7d++WH3/80eYJMIeQy3kf6v0y5BBGHwv9sLzhq26Ki4tttQ1MpuGFTqx8bdmypYDYwTwBdIGwwUukIGxmzJhRqVoSNpUgqb0J+CL4gYuLMJx/aTwM3+UQhoMMHREgAvrbiP4QSneZpErIDak1+f24bIXWkLzZe3rQmF53dWKZ7Y1+/0b+uO7q6KEMEag7CICswaALfSIGUk7YwATsvHnzZMmSJWb7PUnYJPtP71/h+1sz3bt3t0EXBmEwM9KzZ0/btwaravDGjDuUQX3Q53qQlxl3ec9Lxj2M8nREgAgQASJABIhA3UAAYwV3Hk6OBRD2F1NgHh1vB7/44ou2ahgkDiaD8AIJJo4wFnGX1OFp9IkAESACRIAI3OwIoP/z/tD7VfSjsFp17tw5I2swd5B0Ludp0IED6d6feth1QxZzAiBnYBYN8wWYN8CcAcylIYx5A5A7IGwwHwHCZubMmV5Nyidhk4Ki9gfwRUh+YfwL4r6fASak7T8a6CE/U8Zl6ROBuoYASBT7jVQ68exEik+T6q+oUomqE/CgFT9sxbJ6g7fI9dMXNNVUX9wihojAzYJAsp/0vi+z38SgbN26dfL222+nCBukeT/p8ohj4IW3ZTDowuqZgQMHyqhRo2TEiBH25ioGYsjzupI6MtMcY+h3OU+D7/Um0zycTd7z6BMBIkAEiAARIAI3FwLJMQHCPnbwsYWnYbLpyJEjMnfuXHn++edl27ZtNvHkE0f+0kpmuZsLLZ4NESACRIAIEIHqI1BVn+j9p/ezSa25nslRBgfKwEHOyR+89AkzpTCfPmnSJBk5cqRZ6oCZtH379tkKm/bt29MkWhLoGzXsXwD/EsBPpvl5eVquL5TL0ScCdQ4B3ZCmSsJG77HJfNAgNSdrHFXcsMNN21OgTe/glVLj/Fyh7LpSbcusJpcaphOBOoIA+j/7tVeE3/xl9eGw9BkrbJKEja+wcWhA0uDN1NatW5s5EZg869Wrl2B1TVFRka2owSaCeDMGgzOrKyJivN/1dOhEn4wDeZ7vdbnv/bbHc8l5Pn0iQASIABEgAkTg5kfAxxA4Uxsb6KMExjROxsCcC1YNg7DBHn2lpaWpcYePLXwyycsgDpfUbQn8IAJEgAgQASJwkyHgz9Xw/Rnd+z/43jd6n4nT97CXTcaR5umu0yFLyoGsweoZrK4ZNmyY7W2DF0DxIijyQNgsXLhQ2rVrR8LGAbyRfZtI1jknfCk8nHk+yS+IyekXkI4IEAFHQH85StpUdoGaCUyK/bpMJKTis6YOdeSup+bbzmTXV88UXU37ano+lCcCNwYCPnjCrxgTGt4nug9yZsOGDfLOO+/YGy1btmwxEgfl3IQIiBrsUdOtWzcjajCwwkALAy68DQNCxx3K+YE6EPaBIMJwXjfCnoZw0lVHJinPMBEgAkSACBABInDzI+DjAx9j4IwR9gmmM2fOyPLly22Vza5du2yFjZfBe2L164UXSzA28XQPI+5pNz+SPEMiQASIABGoiwj4szqe4dH/wfk+tzCHhpWq6Es9DfnoG72c95Ve1p/n4SPN5TwOc6SYT4BJNMwn4OVP7H0L82gdO3YUvPgJKx1O2GCOYdasWag2zdEkWhoctT/iXxT/gvgAy78gHvczQXpmmufRJwJ1E4HsxIdOo0ZwuJ8bneqQLXpzVQW5CBvovnI96S2oSl9NdaVrZowI3EwIJPtHTGaY077QfyWw7b5x40Z59913U4QN+kmQNS1atLAVNdgc8K677pI+ffrYGy9Ix4obDKww0IM8iB/oR9wHf0hH/T5wQ93eB3u7QoPSP10GqVXJpZdijAgQASJABIgAEbiZEfBxBc7Rxwru+xgHk02bN2+Wb775Ro4dO2YTTj4Oge9h6PA4wijvOhCnIwJEgAgQASJwMyKA52s8r+N5H32oHyBo9u/fLytXrjSTopgnQL8Iee9rk3igD/WykPE+1ecCMFcAMgYvfmIuYfjw4an9bt06B9oAebikSbRHHnkkWZWFSdhUgqR2J2T70niLfZKnKhmXpU8E6i4CVREfQMWndXMjRMImNzbMIQK/NAI+wPK+0PtGDL6Qhv1qQNjAfMjSpUttoIQ3YPD2C1bRuPmz4uJiW1GDN2ScqHFd0APTaj6gS06GQCZ5AA/Ie9lMfLydSM8lk1mGcSJABIgAESACRODmRsDHBz42QDyZ5nGMR44fPy6HDx+2TZOR7uMQIJQZxpgFzstbhB9EgAgQASJABG5iBJxc8X4UPggbmBJ9/fXXZfXq1XL27Fl7xvd+FHC4fLIvRbrrwzwBXuzEPjUgarCKBi99YkUN5hOQ7hY6vD9HeTgQNosWLbIXREnYBEzqxCe+VMkvWZ04aZ4kEagWAlURNlcma1DFtRM21asn/XSuvd3p+hgjAjcnAhgI+RujPrBCf+hLnOFv2rRJFixYYG/TnDx50sye3XnnnWZbFnvVYGDlb754X5qa4MDKOfwc1aGezLqQ7vX6oAw63CEtGU+me5g+ESACRIAIEAEiULcRwFjBxyDJ8UQyzWWAlMngESMeclQab0DGddVtdHn2RIAIEAEiUNcQQJ8J5/0gnuOxOvWPf/yjvciJeYELFy6kySTLeB+KeQEQNdiHprCw0F7yBFEzZMgQueOOO6Rv3742n4DVNChjfbVqrZ/RBydNopGwMdhvrg//8uQ6K/8i5spnOhGoawhcvgwTSYknmRQAeMKp3kNMttIpNamASplgpnRlsib37xRlM8unKogCod2ZqYwTgbqKQLZ+MZmGN1H37Nkj3377rb2NijdisLoGNmZB1DRv3lwKCgpscOUDLGDpv1MbcOlgLzPuMp7u8Uz5pE7IuEuW8zSUzZbu+fSJABEgAkSACBCBmxcBH7/4WMDjOONsaUjPJoP0bM51ZMtjGhEgAkSACBCBmwkB7x+970MccwJ/+tOfzFQ6VqvCGgfSkzII+4oavNQJoqZVq1bSvXt3GTRoUGo1TYcOHaRNmzZG4mA+wV/4dAwz63eTaNjDZubMmS6W8mkSLQXFjRnwC56r9f4ly5XPdCJQ9xAIZpEqn7eTNZUJlUzZK1EoKXm90V/JVf0bJWFzJfyYTwQyEcjWLyYHXSBsYOP9xx9/tNUxeCsGAyssVcagyn+T8PXd1jTO1NISv2uXhX6vIynjYejRO4w11dMy2+26kul+LtnyknIMEwEiQASIABEgAjcPAj6mwBlljgU87mfrY4RkuoV12IGxh+cndXlZ+kSACBABIkAE6jIC69atkz/84Q9phI3jgf7TD6yWwYudMKXeqVMnKS4uNqJm4MCBZla9Xbt2RuSYlY6o/0Vf7H1zsi9GGOkgbJYsWWIrdEjYOOo3sJ/tYmc7nerKZSvLNCJwcyNQewib5E07O+YkbLLjwlQiUDUC6ANx+ADLpZEGk2iXynX/mfJgzgwyeAPGTaB5Ge9H3YcOz3N9ST8p52HIw8FPpnk4Wd5lkZaZn8xLlmGYCBABIkAEiAARuLkQwBgAB/r+zPEDztTz/axdzuNJ3/UgDWG45JgiU5cJ8IMIEAEiQASIwE2GgPeV3u95/4gVNr///e+NsDl9+rSUlZWl+mBfVQMfZA2scsDcGcyeDRs2zIgapDds2DBtLgHQuel0N8uOuQZ/OdT74X3ffy9LFi82wmbGjBmVEOcKm0qQMIEIEIGbHoHogSV1nmFOVaOpQJQVHmxCBOGQbzRK6qEHuclysVwoF/LSJDLqx5xurocozTH9Nu+rQbypj/osbhXEmjN1eGfknVNoz7V8xnVdixaWJQI/BwKZvwev0/ecQTwp4wMnpHvY85EG5+khlvvTy1VXPrcm5hABIkAEiAARIAJ1DYErjSM8H7hUd6zhZZLynlbX8OX5EgEiQASIQN1DwPs/zAeg/0M8G2EDZECuwPQZzJUVFxdL79697YAZtC5dutgqm5YtWwpW3jgRk4mokzZeV6bc3r17ZeHChYLVObNmzcosLiRsKkHCBCJABG56BMB5uMvJQUAoFkx/oIlNJdlNP6sOJOqhHYFRLNoZuBh0uT4QL0gPUmhUTMaYjBVGetLFMslU1+k6QkekMdRtR5AGX4R6a+a8gPs1K01pIkAEiAARIAJEgAgQASJABIgAESACRIAIEAEi8HMiEM+VhVoxV5aNsIEcVsw0a9bMzKbfdtttcuedd9qqmp49exqBg3wnXyDvYWhG3OYIo5ND3I94Xq6ercDBvroLFiyQ9u3by2OPPRaViD0SNjEWDBEBIlBXEIh5GHAbORyEYkHcZCv0gLMbsJaLKRhLzviACaSwFBIZ9SPSxIX8Jn75coUmxfWE/HRCxm/wyKtXD3tsoCNAsaicnUNoTUQPmQw6ofQykE/XbQLV+nCg3K9WIQoRASJABIgAESACRIAIEAEiQASIABEgAkSACBCBnx0BzKdhH1vMj2FFDMyTIQ0OK2z+67/+y/aSOX/+vBE1Xbt2NdNn2J+mT58+ZgoNe942adpETamr+TNdfQPnc3pJwsYyEh8+lwefhE0CGAaJABEgAlkRiHiO3GQNSkHIBcGNxIQNyBeXsIB+GF2CZC0C0gRxfFaAkNE0KwPSBgX0I+SrpJEqMQsfOg7NDVUEzRoON3p0CghDSWibaUzJIh0ulDddEEsps0gcNdnqfngl7le3HOWIABEgAkSACBABIkAEiAARIAJEgAgQASJABIjAz4sA5sWwlwwIG9tLJk8JlzCdJhs2bJA///nPsmbNGiksLBSQNVhJ40fnzp0tHXvewjlJ4z7m/urbS9Xpe9b6GYZ5PJUiYeOQ0CcCRIAIXCsCuINHd3Go0iBuxiBLAmkSbrohHggS3LQRD6tmdCWMMe+B4KhXL8hDVbi5QxabnpejgC2j1ByrCDpQG+T8QAraAAIorJypp52N1qF/oV2hbRDzMimeBkVVKfQiLdQPyZo4J2rcr0lZyhIBIkAEiAARIAJEgAgQASJABIgAESACRIAIEIGfFwEnTlAr5sOcQNm+fbu89dZbcuDAAenfv78MHjzYVtQ0adLE9rABUYNVOck5NA9Dh+tDWmzhJszjIc/r9fog57I0iWbw8YMIEAEiUFMEcPMNN2CU9Juxkx6e5jdgve0jSR3K4CYM02X1Iw2BrAGxgpuzr7YBgwLSBiWRDl3l5UkTaeFmHm7qQQY1xB2Bm0cLbwygfFiOGdJDO6ADpaLOokLbp3E3rRZyqvPp5+d+dcpQhggQASJABIgAESACRIAIEAEiQASIABEgAkSACPxyCPicns+9oSUnTpyQrVu3SklJibRr105at24tzZs3l/z8fFuNYyty9EVszLOlnE6JXdZ5NdfnJtEQxxHm78K8macl05GPOT0SNilEGSACRIAI1ASB7ISNa9D7cHSDjm/UuAnHJImSNUrYmMONWypUHkQJCBuIKZEScvVTb+r6iZu2kzHIgnwgdxDTclrQOxfXBTIGZWCTE/Xn5TVIyYVSKoC61EN+0A9dTupAqjoOLYRzP8T4SQSIABEgAkSACBABIkAEiAARIAJEgAgQASJABGorAmG+LrTO5tV0Hq78Urlg7xo4X0kDOSdhbP7O5s7S58F83g75frh+xN0hzQ+Xg4/yJGwcJfpEgAgQgRohAPIFR7rDjRU3XCc+zCxZdEMOeeBHdLUMjst+8450qDq7WatekDZY5aLUvMbAwkNGPyCDP61DP0wXckJ9YoQMZC0fGUagxCtsnLBBetBpQqkPqIXLlhdycn1aA1EylwDTiQARIAJEgAgQASJABIgAESACRIAIEAEiQASIQK1CwObiogkxJ1U8zef5nKixdMzT2bxamANDWpJ0SZ6c60umIez6M8uiPhI2mWgxTgSIABGoFgJgNgK7YcsdNRZuwjFDHtREpIxGsL8MXDCFpqy5xQLFYStlNB76h3DjD9xHIIBMNziaqFMw0kaXWYJYQV4gbKKVMdBjbYtIGS2XrDtUi7LRKhrLx7nggIvaXCPuxYXdD5r4SQSIABEgAkSACBABIkAEiAARIAJEgAgQASJABGozAiBOks7jTqwk8xDGXFyYB8RcnpZFPJHu5V0WftK5Xviuy+f3SNgkkWKYCBABIlBtBGKCI7Dtap4sMknmKuzmG+0JgzRn5euZfcv6RthURKRLg7y8YN7MC5sP8ifQOtCF8iBe8urlgVOxDgHdQag3mFeDXCBe0HFABorswzTiI5A7aC8IG5SLdKv+0EkEG5woX33nwu5XvyQliQARIAJEgAgQASJABIgAESACRIAIEAEiQASIwC+BQJhLCySM1+/zcIg7kYK0JLniYS+fzEfY013OdcP3/GQZr4eETRIphokAESACioDTG1VTDxExol5YzRJutmBSAtGB0pFpNMgo0XKx7KJcvHhBysrKlXhRybx826wsvyBfCnTTMtyY4eJ6wwqaiopyuXDhgh3YiwYkDWSxKqdhw4Z6NLINz6wlxtB4G0KbysrK5Ny5c1r3RdMPoiY/v0AaN25kPqq1TgSnpJHQkZhoDT681e7XoChFiQARIAJEgAgQASJABIgAESACRIAIEAEiQASIwC+IAOb3zIpORLaE+bH0eS6fu/NmetzJGaRnS0umI0zCBijQEQEiQASqiQB4C7j0W3JIiz+NhYnIGlAoRpfYDTeU0z1q8AcCRY/zF87LsaNH5cDBA/LDDz/KufMXpUnzQrm1063SqVMnadGyheQ3yDf19ep5C3RPGlVWUlJi5Q4cOCDHjx8XkDYNCwqkVctW0qVLFyvfqFEj6xBCBxHag9U4IGlQZufOnXLw4EHTX1io9d56q5Vt1aqVpsUEjwlUiofUqj8dLferlmYuESACRIAIEAEiQASIABEgAkSACBABIkAEiAARqA0IOOECP2mZBm1Dmu9hk2yrl0kSNB52OZdBPJmHdD+Q7gfq5gobR48+ESACRCBCwOmSqqmHsLdMNtDshoslNErYwCEO0mT9+nXy8ccr5LPPVsuPR45J567dZOzYsTJy5Ei57bbbpFmzZibrhI3drFXHwQP75ZNPVspHH30k6zesl5KzJdKyVUsZNGCQTJgwQe677z5ppfG8vAZ6gw/1oc7y8nI5efKk1rte3nvvPVm1apXp79Gjh4waNVrrHmP1JveyCSRT6ESgq/oOwjUqUH3VlCQCRIAIEAEiQASIABEgAkSACBABIkAEiAARIAI/AQJOnEB1klTxqpDvhArCLufEjue5PPykXDLdw15nUjf0kLBxhOgTASJABBIIgLSAC/RDLhIiWj2TKOdBmD+7pGbIbL+Z+rrfjOo7cfKEbN68SZYtXSrzFyyQPfsOSMdOXYw4mfjgRBk+/C4lXVrpDb3cWPsGarasPvaXUV07tm+Xd95+R5YuWyabNm+2FTfNdZVM/wEDZNr06TJp0iTp1LGDNFLzaEayoPPQA6trvt9/QFZ++pnMnj3HCBuszBl+990yfcYMeWDk/dJNSSPrjPQ04zPNXHHjZ1aVH5euSop5RIAIEAEiQASIABEgAkSACBABIkAEbn4EMK8Q5hai+d1oIjj92bnypG5iriH5nI7H/Ag0e4ZPPMFnYhnVqhLpdWXKxXHXHKfEIc+Drurqi0szRARuFAT8t4j2ht9YaHlmusch4+HMMqFk1Z8o6wd0+UHCpmrcmEsEiECdRAB0TRiQhMFN9Qck4UaNpZPYc+ai+fkN6ku9+vV0VcxZ2btvj7z//jJ57bU3ZOv2nVLYoo0MHTpUJj38kDw4aYK0b9dOb9YVahqtgRTooSVFlcj6b9fLSy+9LO8vXy579u5Vc2oXpGHjxtK9Zw+ZNmO6TJ8+TboXF0nzZk21DNquzL/e+M/rvjVbtu6QhUuWyTvvzJN1336rK3NaycQHH5SnnnpShg4ZIm1uaR06BQzljO1J75jq5FeAJ00EiAARIAJEgAgQASJABIgAESACROCqEcBzeYUeTr5gYhfP3PqUr4/5SQLHJ3zD87iWwyRu9FyvD+DBqRcAADKbSURBVPaqAwXieQmUDXpCGiQynZU3OZSO5KJK/bk/LgMNetg/wnBoK/woz3QgQV8qpSMCROC6IOBkDXwna+CTsLku8FIJESACNxcCPjjygU0Y3FTnHJ2wKS+/ZPvM4CbbQDehgZ3Li2UXdQ+bI7Li44/kxRdflq+/2SBll0Q6dOyoZM1E+fXTv5YePbpLAyVq8vP10DKXtfyF0vOy9ssv5cUXXpKVKz+VHw4flvMXL0iD/Hzp3K2rTJkyRWY+MlP69O4lrVoU6qocbSlu9jqwKi0tlc+/WCtvvjVXPvjgQ9m3b5+079BBpkyeLE8++aT06dNbCgub20qe+hFZkxwsVh7IVQcFyhABIkAEiAARIAJEgAgQASJABIgAEajLCIDoiAgbwJAiXHx+Ab6HTQAf6pwg0VzL1vkJI1qS8l4ukCqwxg4Z/OG5HmROcMiIgpGX/Rnf60gKk7BJR44xInD9EbDfLX67mMOz3y5+dyRsrj/S1EgEiMBNgEAY6OBE9FZpn9U9qTCQCpuTYf8YM2+mKuori1Ku5EtJyVn5/PPV8vLLr8pnq7+Qo0dPSsNGDWXU6FHym9/+RgYPHqQESqE0bFggDXSQdUlNmp08fkJWf7ZKXtYVNl98uVZOnj4lFy9d0lU79aVt+/YybsJ4manmzYYNGyLt2ra1FTYYnqHlZ86e0VU5H8rzL4SypSUlUty9u0ybNk0emTlTiouLpEmTxjaoC4RN6Bi0aKqzQJiOCBABIkAEiAARIAJEgAgQASJABIgAEaguAiA/QNjo3rd4Ok8RNigf5hmMHtGPQMwE0gUMi8dNEkWNjEGpUA7pXhZhU480TPoiiolfhBAxQfUR1Hw4J23ctzZGeSYQpKJ2uAKvGz4dESAC1wMB+23j943fLn630cEVNtcDXeogAkTgJkPg6ggbH/yEG20gbbBCBoMfHwiV6b4269ev0xUvs+XDjz6RXXv26lLHy3L7sNvlsccelXtH3CtdunSRxkri5Ckhc76kVA5+f0A+XrFC3njjDS27Qc4piYMy0Ix9bIbfc7dMnTZFxijpU9Sti47CKiRPyR6stAHZ8/a89+S///xX2bp1m5IzTZQUGixTp061fW9u7dRRCnSlDgaE1kYfiyWuqLc9kZQ16B1M1kwmEgEiQASIABEgAkSACBABIkAEiAARqDMIYF6hXM9WJ2KVrMGjdjCJFiZlwY/4HAKeuRHGgZc9USZ+vvZ49Myu0VAWvpJBqgdlQrlIp6ZBJ57zUac7TAKbXpW3PwiYs9a5WOSH8nGiy7of5zBEBIjA1SHgv3v/vYffLVfYXB2aLEUEiMBNjsDVETYABTdZ/bQDgyEcGJrZ4EnHNRW66mav7kGzXM2TvTd/oaxe86Wc031m+vXrpyTKFBn7q7FqpqyPrXrJ08HT6ROnZNuWLbJc965ZoPLYvyZPTabhvZkLamINrrhHDxk/YZzMnDldBg0cYGQNyp4/Vyq7du6Subp3zcuvvan75+yVTh07yZixY42sueeee6RNm1t0v5w802PNRlvNpi4HYQEUfhIBIkAEiAARIAJEgAgQASJABIgAEagZAoFMURvo6my/GfXxHB9eiAzP24F4wfxBmEvAfIK+txnJ6PwBbJ2pC2VQHn8aT+ixiV5oBvnij/E2L2El8WF6wlxF0OWTwpZpH2EOI44jRMImHQ/GiMD1RwC/Sz/8dwmfK2yuP9bUSASIwA2PgA+DwkAoHvVU78TCQKhCzCSaEjYga+rh0JsuBm0ndNXLt+u+lTdnvy0LFi62eFFRNxk3fpwRKcN0tU3zZs2s3LEjR+SLz7+QpYuXyCefrJSTp05JYcuWUj8vT86cOSun1DxaM92DZsR998lvfvOMjLj3HmncUFfnaF1HdK+bL9askXnvzpeFy96Xw4eP6D43veXRxx6TcePGSW/d86aweTNbyaON01FcGAxivx20lY4IEAEiQASIABEgAkSACBABIkAEiAARqDkCePavqABhA4sb/oztz9nBiLlP1JqPKoywCXMH9ogerYjBMzpcuU3uBtIlftEyzF9As034mqQTQCFSXh5eJoUef95Pf+YnYRPBRo8I/KwIJO8B9vvVHz58EjY/62VgZUSACNwYCCQHKzbsqWGzwZCHARFuvkbYGAGiZsx0hc2Fixdk377vZc7cd+St2XM0vM/2rcGKl2nTpsrIkfdLq1atpEFeAzl08KAsW7pUFurqmk2bNkt+Qb507tJV8vML5LCSObt279b9bMpk6O1D5Xe/fVZ+pSt0bmnVWkmYerJzxw6ZP3++LFqyTNZt2KT755TIsDuGyXPPPScPPPCAtG/fzsyhYe8aH7TZ6zrR2VqTo/d3kKSnktWlD/SyijCRCBABIkAEiAARIAJEgAgQASJABIhAnUHACRvsa4tHaXtuVvNkWDUDAqd+/TyblK3QuQOdog35EZ+DOQQ8p4dHcCutz+Mwu66kje2VG0gggAm94HMwB4AwdIdn92CmPchjjiI896MMdOXpS6BOBAU98QM/rKihTZZun8lQ1MhUOgNEgAhcLQIkbK4WOZYjAkSgDiKAgYoPVjAYufKABDdZdyA6MDhDmpWMiiPtUvkl3WKmQo4dPy7Lli2X2W/NlW+++cZInMGDB8ojjzwiEyZMlLZt2yopky979+yRt958U+a/N99WyMCk2YCBA6VZ80I1cbZPPv/8czl6/Jj06ddHfv3UUzLpwYnSVffAyc+rL+t0Fc+rr7yi5tc+kkM/HpH8ho3k/vvvV8Lmn+Wu4XdJU93PBoMwtBfkEAaEcH4u9er5OWFlkJ9dZZ+ETWVMmEIEiAARIAJEgAgQASJABIgAESACdRcBPP+XV5QpyVKuJMslKbtYJqWl56TkbKkat7isz/sNpVGjRnY01D1sQeDoZjdGovizOSYUQLhcOH9BSkpLpVT3uL2oevDo3lCf75s0aWrm1AsKGugzvSbaMz2e6/U5X5/hMf9w/vx5tc5xRl/g1Hp1LgKETkO1yoGXRJupZQ938fO/zglAFSpXFz6ToTjFBPhBBIjAVSNAwuaqoWNBIkAE6h4CYCecocBg5MoDEgx84MIbLXh7BeUDGYIl0IiXq0y5roZBDla7rF37tbz37gIlbpbJ/v37pedtPeSxxx6VaVOnSYcOHaSgoEC2b98uf/3LX1XuXWiTQYMHy3333S/tNX/rtu3yzjvvyJ59e6R79+4yfcZ0efjhh6V/3z66cqaBmkP73Mp+/PEnugrnstzatauMHz9ennrqSenXv194YwcDNm1PA90Xx5Zpa+vQdLQXY72Iw9F4bgxI2CiAdESACBABIkAEiAARIAJEgAgQASJABCIEsLKmvLxM5wHK5eKFC3Ly5Ck5dOgHOXjwkJEoTZs2k3bt2umzf0dpfcst0kCf4TEPgWdzEDZYeYPJgwta9uixo3LwwEH54cfDSvqUqky+tG59i3S+tbN06tRBmjZtZCtswuQv5iHCUVZ2SU6cOC579EXQ/fsP6P655+3FUNTbq1cv6dixo8mG+Y94DiNJ2OB0wmyAzwm4H50oPSJABK4agfCbDXOI/ruFT5NoVw0pCxIBInDzIoCBCg44DEaqNyDBjdZK6KssIYxBEsKBzEFuIHYuyzl9s2bLlm2yRPemmTv3bdm8ZYt07dpFZijpMn36dOnWtZu+OXNBvvzyS3nttdfl4xUf65s3jc2U2eQpU+TWW2+Vb9evkxdffFE2b96sBE5725dm8uTJcoeaPYOF25UrP5F/PP8P+XzNF9KkWXMZfvc98tBDD8no0aOkc+fOeorhPDEYdJNolc8UMjgPnBmcB9yHmnDelhsLmjQ/iAARIAJEgAgQASJABIgAESACRIAI1DUEKkDYRCtsQLJ8v2+/fP3117Jq1Wo5oKbPC9VqRp8+feWuu+6SwYOHSMtWLc0aB8yhwVwZDjx+Hz92XNavXy+frVol337zre1ri5U1xUXF+jLnfXLvvfcq8aOEj66wqdC9apTtsed7zD2cPXtWdmz/Tj744AObWziv5E/7du3ljjvvkFGjRuu+tn0CYaP1pJ7wLQBrIakUu3RxPD29rl1Xni8RuJ4IkLC5nmhSFxEgAjc5AoHICCeJwUhNByRO0qCchrHJYISY3Yw1jDdsvv9+v3z04Ufy0suv6OBprZpBayMTJ06QqbrCpnfv3nL69CkdWH2oq2veswFa69atBYTMY48/boTN2rVr5S9//Yuu1Fmrb9Q0leHDhwvInPt10HaprEw+/PADeeWVV2Xd+g3SodOtMkXzcPTX1TUtW7bQwWDYXydsVhifZWD1k0QMCZvo8tEjAkSACBABIkAEiAARIAJEgAgQASJwRQRShI2+wFlytkR26/6zeP5/+515snXLVnuGHzhwkEyYOFHnASbaahmsqsELlQ0a5AUrGEq+fP/992aVY8H8BbL68zVyQs2rN9HVOb1u6yVTp0yVadOnSXFRFzVzlm+Ej7I1pgPm0I4cOSpffrFW3lQz6x98+KGZV+vbr6+9yPnQQ2qdo/+AQNhEZ2PzFvYSZmTePeMsA2njsxsZmYwSASJQYwRI2NQYMhYgAkSg7iJwdYQNbrQgO0DSGFETrWABZeNDGgzAIHfp0iU5eeKU7kGzRp7/+/Py8SefSCO1QXvHncPkwQcflDvvuNOWLi9YsNBIG5A73bp1k1mPzpInnnhC2rRpI2u/+kpefOlF+fTTT9Wm7Xnp07evmUQbO3asnFUbtcuWLpV58+bJdzt3qbm1XvLrp59WMmiqLnvuII0bN7J25GEwhx0K1dmWhtr0JGGDU/C4CaXOxM8oJnb8/AMGQZqfRIAIEAEiQASIABEgAkSACBABIkAE6hoCIGwulV+0FzgvXrgoP6g5tBVqrvzVV17TfWy/lXr6HN5F95+dMGGCPPnEk9KzZ099NtfVNUrW5OmetOHFynqybds2I1wWL1mi4e36rH9Wzafly62dOtvz/6OPPiJ9evWUwhbNw6ocBRrzDpculeuqnu9lhVrreOutOTr3sFqwV86IESPkkVmzbH/bLl26ZhA2+pxvj/rxHAauW5jTQAYom3guAHl0RIAIXD0CJGyuHjuWJAJEoE4iANIFrnqDEQxg8G8lMkgby4vyYR4NcSxVPnfunHzz9bfyj3/8Q5Yv/0CwPBmmzkaNekDG/WqcnDp1yvaoWbNmjeZdlNuHDpVHHnlEHtZVNi1btpQNGzfK7Nmz5X3dA+fAwQNmfxZ71IwZM0YOHz4s77+/TFZ9tlpOqK3cQUOGyLPPPiuTJk2SFi0LJV/t4qJjMHNoaK8eYfiFk9CwRpAfXIh7ON0PMe9kArlTPcxCSX4SASJABIgAESACRIAIEAEiQASIABG4uRBwwgYTBTBPdub0GTNXDisYqz5bpXvanJSmzZqpyfIx8rvf/lbNog3WlTNNdT/afCNuMG9QrqTLOphCf+FFnTNYLgd0H5vz+rIm9p+95ZY28iudNwBhM+z2IWoWrU2KsMGTPKxugODBS6DvvjtfNm3aKK1at7J9bZ948gkZMmSotG51S2rKw2YEUo/y0ODzAX5dIBH+PIU+ESAC14aAz6XB9/k0+NzD5tpwZWkiQASIgCHgN9lwg0WSDq9sjINhVhjoQCYMenTApsRN2cUy2bxps7z66iuydOky3YDwR6mvb9Jgdc2M6TOlpKRE5sydK5s2b5KWLVrIr8aNsxUysFHbvHlz2bVrlyxatEjee+89WbdunS2pvv/+++1Nmb379spHK1bInl27Jb9hgdx9z73y618/rYPB0ba6Bm/uwCQa/rC+JrmHDcia3M4z3Y8lvYOJUxgiAkSACBABIkAEiAARIAJEgAgQASJQ9xBwwgbP1yBYytXKxoaNm3S1zFv6cuX78t2O76Ss7JI+q98jz/3zc7ryBXvRtLNVMHkqX1ZeJqU6J/CF7mv7d7XK8Ymuzjl+4oRZ6wBt0qxpc9u/ZsbM6TLqgfulqFsXBVlz9Fkfz/dY1fP1N99YfcuWvS/f79+vL4h2MjPpTzz5pJlUa9S4sV0Ye7pPmwjwuQvN1iDmDZyswbnQEQEicH0Q8LlEn09z0oaEzfXBl1qIABEgAokVKQAjDHACSRPACeGwwgbhch2c7d69x0iXxYsX6z41G21TwD59+sg4JWdKS87J+x8sN5u1RUVFMkuXLcNcWr9+/dQ+bUM5qBsVwhza22+/LZ+oSTXovP3222XYsGGya/cufXtnjZlG66irdkaPHiszZsywDQ0L1LYtVtYYYaNlMDizQZ2mWdhGa6HNPjgLMc1PDeKSQpEsPSJABIgAESACRIAIEAEiQASIABEgAkRAX9Is17fkL+nDdnhzXh+9Zc+evWq6fJm+dLlATZR9LiWlJTJgwEB5VJ/1MQeAvWwbN2msz90iF9QCB8yo4Vn/1ddela/WfqVWOs6rXqVPVFejgkY2NzDxwYkyZfIkGdC/L5ghW2UDUqW09Jx8rC9xPq8WPVDXBSVwevfpLVOmTtEXRGdIl67dJF9X88DZ033qWR8pYT4DodilKJs4iSEiQASuCQHM4/nhZA18EjbXBCsLEwEiQAQcgTBoCgMbpOEdFB3kYJwTuRAP6XZD1mXRR9R02Zf6xsyihYvtLZuDhw5Jxw4dZejQYWoG7bx8qytnTut+NIMGDpSnn3nGzJ11VTu3GFgd180Gv/nma5kzZ44sXrxEl1iftn1uet52m+w/cEC2bt1mRMxALTt5yhQZr7Zx+yoZ1EDNoRlhgxZqG+DcNFrmGM3aHLUfw7iQnyRrkuGUIANEgAgQASJABIgAESACRIAIEAEiQATqLAJqVyMQNvrc7S8+Hj16TNas+VLmqiWNJfoMf/zESSkqLpKJEybqfjQPyR133CHNC5tJXv08I3N27NhhptDwkuZ2NW9WP6+B6bqkL39ij5p27drLyJH3yzNPPyX33H2XkTX18/JsNc/hI0dtnuD5fzyv5tA2mZWOu4YP17mByTJ+3Hhp376DrcTBBbJXN9Me7eOJjJCMeCBsInoHxeiIABG4RgScrIFPwuYawWRxIkAEiEAmAhiMpcgN42niAU5K1pJUKmLQQX6c131ssBHgYjVt9vLLr8hWHYQ1atRIOrTvKGWXyuTw0aNGzgzXgdUzzz4jI9QcWps2baRhQUNbPbNp80ZbYTP/vflmz7agoED3qGkpZ86U6N41J6SJvp0z8oEH5KmndACnS6w7dewYdwI64MKeOiCV0DHoymn1U621dsYxDwWBdOImUcjF6BMBIkAEiAARIAJEgAgQASJABIgAEaizCGCGoFzPPp4bKDlbYvvKzJ491168PKQraNq0baPm0EaYqbKRD4zUvWlusX1ssKctTJ8vXrJYTagvtX1qW7ZspfMDBToXcFbjR5QQuqwkzzD5j//4N3lw4gQp0HkCmEM7ffqUbNmyRebPXyCz58yWvXv3SucuXWXixIny0MMPC4ibVjpv4ERStif66MkfswbRObiU+3X2wvLEicB1Q8DnB0nYXDdIqYgIEAEiECOQRtggWcc0KQInFovTQZIoyQM7thiIrdClyi+98JKs1WXOJSWl0lBtyV7S1S+XNL9tu7a2L80TTzxh5s4KCwslX9+sOX/+nOzetVPee/c9mffOPNmx8zs5pxsQYgCHt23KtTwGexN0ifQzzzwtAwcOksLC5hmEDUyiOZNf+V0ZdBqVna+0QQ4GaxywVcaIKUSACBABIkAEiAARIAJEgAgQASJQdxHAs3RM2CB2Ufex3b//gCyYv1D3lnlTyZttagEjX4YOu12mT58uE8aPlw4d2ivxUiBH9eXNT1Z+Ing5c7WaNMPcQFFRsbRu1VpO6F42GzduVP+k9FeT6f/fv/4fmawrdFppHqxx/PDDITOfvmDhQptrALnTp29fwd41E9TyRo+ePaVpkyYpwgbXKPOpPjzp4xMtx+ES7msSHREgAteEAAmba4KPhYkAESACVSMAwibpjKzBmAYuOZ7RNLshY8Bjq1suS9nFi2qPdq28/trrOpj6xPasKStXMkfJEtivxWBq7NixNoCDfdvGugIHJszKdQXOId3HZqnufzNXl0ivX79ejh47pvpRZ31p3LSpFBUX65s6k+XxJx6X7sXddeAXbNRas7BMRmVjwgZLnFE43VUmbUjYpCPEGBEgAkSACBABIkAEiAARIAJEgAgQgSQCeLZOEDb6oK6P+Wra/IR89OEKmT17tu0tc0ZNoPfr30+f92fI5MkPS9euXVJ71s6fP98saoDYKSxsIUN1z9quulLm8JHDquND2avWOnp07yFP//ope+7v1q2rNNaXP3ft2iXvadklS5aoObTNup9NqQy740557l+ek1GjRknraBWPr7BBq5PTFh7GDEEga2ySAWLqPDfE+EkEiMDVI0DC5uqxY0kiQASIQA4EYnIj22oaJ22whsWGND6u0WLIwwobsCsXdTPBbVu3yqJFi82O7fr1G6T0/AVbIdNcB2WDhgyWCVi6/NAkue22XlKge9CAWLlcUS4nlKD57NNPbZXNSvX3ff+9bkKoapWwadu+vdx9z91qC/dh+dXYMWqjtr3ZtMXJxAOzy1I/Gdey4HEyXSBtkkSNS0A4FIBMrNfz6RMBIkAEiAARIAJEgAgQASJABIgAEahrCGC+oDz1wmY0K6Dmy8+qZY2v5d1331VCZans27dPX7QskimTpyhpM0169+4tTZs2kZ07d8qrr76qptPm2ouZPfVFzjFjRksvzd+nRM3st96SLTqP0En3wH3wwQeV7JmsFjlul2bNm8vGTRvlzTfelPeXL5cjR47Y/jUj7x8pTz39tJpQu9NMp8N0WiBj/Ik+XJ/whB+e8ZES5gdwLp7mfpDnJxEgAlePAAmbq8eOJYkAESACORDAoAVH+AwkTPrgBatdUkObNCZEyQ2ssNHcsrIy28fm05UrddD2nnz22So5efqMlOvrN23atpV777tPHlSyBvZsu3bpIg10YGX7zWjZUn0bZ/PGTWrTdpm9QYO3Z2BK7bLWVawDOgzaMHgbNGigtGzRwggVkCpoF+pGk1KETZSGgVhaUyGqDmUqp+N8wzmTsDGY+EEEiAARIAJEgAgQASJABIgAESACdR4BPGCnEzaXL9eTc7qP7bZtO2TZsmW2j82GDRvs5cpgVWOa7UnTtElT27/mhRdflAULFpg5NLyMOWPGDF2N01927Nghf/vb3+Xrr76SwuaFgj1v8aLmmDFjjJz5cu2X8tLLL8snn3xihBFIoPHjJ8ikSQ8Z4dNAXwI1viZxjXwmIzzhewwCYd4giIbcRDEGiQARuAYESNhcA3gsSgSIABHIjgAGYMZy6KqW4GeuMHHCxsvbWzXR2MdW2Gj5Cl0pc+zoMfnqq2/knbnvyPIPPtS3YI5KmdqovbVzZ9uDZqISNgOVdGmv+9mAsMlT5gRES9nFC3Lkhx/l088+k5deeklWrVot59UuLkZfAwcPlmee/Y2aUxsjnTp1VFNqjc2UmhE2aJA2rl49XWHjLIzG0V7ke5K3O7cfD9hI2ORGiTlEgAgQASJABIgAESACRIAIEAEiUJcQwBwBTKKp0yCetTFvcEEtbBw8eEhWrvxUXlZSZdWqVUay3HXXXfbC5ejRo9SsWSPbg+bVV1+Tjz/+2MycTXroQfnNb34r/ZWw2bRpk/zpv/9bPlUdeA7v1au3PPzQQ2oWbYqaVW+iulfa/MCaL9bovjatZKwSOZM1b+jtw5Qc6iD18/QlUH0L1OcCME/hcwA2XYGmp5zOG9i8R/Ts74KpfAaIABG4WgRI2FwtcixHBIgAEciJAEYxYSQTQmmjmjAos7KBoUmSNRitVVRc0twKDI30LZvzamd2t9myXb36c7Nre1EJm/YdOsgDY0bLiPtGSEclXZo3a2pkjRM2FeXlcl7f0NmyZYu8O2+efPHFWinReJ5uNNhXB3IP6/41Q4ferqtrCqVA04yM0frc2cBLSZvoNMwP4694wOay2f1YV5yfLS3OZYgIEAEiQASIABEgAkSACBABIkAEiMDNjQDmB4JVDTxvm+ly9cv1Gf7kyZP27P7CCy/IBx98YGRJjx49ZMKECTJt2lTdw6ZAFi5caBY4Nql5s7b64ubUKVPl6WeeUXLmNjWpvk2e/8cLslxNnh0/dlw66LwBVtBMnTpV8gsKlMhZKXPmzjXTaJ07d5GpqhN5MKvWXFfkwDpIXl6e5OmLns6/+FO8+Wh6ypGwSUHBABG4zgiQsLnOgFIdESACRCCwHGEkg08MetKcRkMKKBkd9vgICEJK2JSX60qYy+X6ZkuevW1z/NgJ2bx5sy2PPnP2rFy8VC4tWrbUPWyGSO++vaSJbh6Y3yDPzKHlBY1QpKbVRH48fFi+WLNGiZttUlJ6ThroIK1rcZEMVRu23bp2lUYNG9pgDFXDhZVAaFU8gLR0yw0hH7jhJMK5XYnE8RN0P6WMASJABIgAESACRIAIEAEiQASIABEgAnUIAcwGRM/bGtIpAD1gnvyyvbAJU2gvvfSy7mOzRAmcE9JCTZiPGvWAPK37zOTnN5A333xTPvzwQzmsz/o9enZXIme6PProo1LUrUj3t9ll5tSwD+727dt1T5qmWna0ykyTekrCfKr72y5ZukRNr++TXn36yGOPP6575EyWjh07qO58I41gFg2Hu7CKJpq2QNMjF9KRgOd8PVITBS5BnwgQgatFgITN1SLHckSACBCBnAhg0BKPZOJQRHBoAtJgvMxXrbiqy7p/zWUla0T9eroUGX9lZeVSUlIqpUq4XNK3bir0FZz6DfKlafNm0kQ3HcTbL1iyjAFTPR3kmY/xkiq9qGbQTp0+LWdOK9Gj4XoN6kvjZs2ksLCFmkJrlCBrQiutRmuUDiBVF/5CK7yFuXyVQoVZHTJyZmYtwUQiQASIABEgAkSACBABIkAEiAARIAI3HwJ49g6EjT2FK1njz8sXLlyU7777TlfQzJfFixfJ1q1b1VTaebnzzjvln/7pd1JQkC9vvPGmrNGXMtU2h9x1151q7myqjB83wUiX/fsP2B448+a9K6tXr9Y9birkPt37dtr0aRouV8JmpaZ/rvMLJXL7HcPksccek3HjxklrNY8GU2iXdL/cQNjkGexxy8JViJ/qteVgmrQVoe2ak3tCIBTmJxEgAtVGgIRNtaGiIBEgAkSgugjYsCslnDWGQVnWAQ2kA1mCgY+veAnKwvDIhnauNB4xqbQmRoSNlY1aYBqV5KnQwReW4eDNmrAfTVwYnQGcpWi7/G0ZpCMW2qH0jdcb6U56QSaZ4uG4Hk+hTwSIABEgAkSACBABIkAEiAARIAJEoG4ioC9q4g/P19FDNp69L10qk0OHDhnZglUyH320Qn7QvWl79eqlpsumGJmybNkyXT2zw/agGT9+vEzSfW2HK3HTpk0bOXr0qO19M0/NosMs2okTp3XP20G6f+1YOX3mtJpb+0L27N4jzfQlzpG6amfmzJkyYsQIada0qT3zYy+d+jpfgJdC4TKf5NPj2nhru6ZmndswFfwgAkTgKhAgYXMVoLEIESACRIAIEAEiQASIABEgAkSACBABIkAEiAARIAJEoKYIgOcIhA0sbOhRUW4+9rM9ffqUkiq7ZfGSZfLaq2/Ijh07pW3btka8oJ5Nm7bIMd2fprh7d3nkkVkyYfw46d3nNmnVsoWUnD0jGzasl3ffe1cWLlgo+/cfkg6dOsuAAf3lxx9/0LIbpUz3xO11220yadIkmTx5igwZOkT3xmloVjsqUzS5zyydvMktxxwiQARqjgAJm5pjxhJEgAgQASJABIgAESACRIAIEAEiQASIABEgAkSACBCBGiMQCBsUq1BLGDB7ribR61VIuRI2F86fs71rPlj+ofz1r3+X9es3Sp6aRG/Tpq3K15Ojx46pmbQyJWEGyG9+81tdPTNaOukeNE2bNlYz6Bdk966dMn/+uzJn7jtK9uxSvXnSqnUr3R+nVFfcnNBVOnly+7Bhtu8NVt707NlT98TNj84hWNiozooZEjYRZPSIwE+AAAmbnwBUqiQCRIAIEAEiQASIABEgAkSACBABIkAEiAARIAJEgAgkEUhYQdNkmC6/pKtryo0jqVC/XM2iXbxwQU2brVbC5m9qHm2NnDp1Wk2bY1+ZekrqXJaC/Ia6r81d8ty/PCcj779PWrYstP1tKnSlzg8/HJIluv/Nm2++JevWbZTjJ06pbjV6Xg9kjEjTZk3lgZEPyLO/eVbuuftuadeuvTTIy9N8a04wg3YFE2ckaxQrOiLwEyJAwuYnBJeqiQARIAJEgAgQASJABIgAESACRIAIEAEiQASIABEgAkAgO2FTIfXzsHusrrnRVTZYKbN+/QaZM2eu7kXzoezQPWuwquayEikga7Da5t4R98kzzzwtdw+/S5o3b2orZ2Be7fjxo/LxihVado58uupzOXjwkBI29c0EG1bStG3XVh6cOFF++9vfyoCBA6VpkyZqDk33rCFJwy8oEag1CJCwqTWXgg0hAkSACBABIkAEiAARIAJEgAgQASJABIgAESACROBmRsD3sDH6BnvYaEI9XQEjODRcVlYm3323U1fKLJEFCxfKV199LSUl55R0qSdNmjST4uJiGT16jMx65BEZMmSwNG7cUPLylJRRXWfOnJZvv/nayi1ctFi2bt2BhTlWVUvd5wYkzUOTHpKpU6dIkeopyC+IuBpdhWM1APnKa2gqp9zMV4jnRgR+WQRI2Pyy+LN2IkAEiAARIAJEgAgQASJABIgAESACRIAIEAEiQATqCALxHja23kbJlAo71ECaESsVGj944IB8/PEnMm/ePPlk5ady6vQZNYcm0qKwhQwaNFgmTJggkyZNkj59einpki/16yvhouWwV83ePbvlw48+kldefV3WfPGl6gQZVE+6dO4s48aPt3LDhw+X9u3aSR5MrUWra4w4wjXIstqGhE0d+XLyNGsFAiRsasVlYCOIABEgAkSACBABIkAEiAARIAJEgAgQASJABIgAEbjZEUgSNjCBBkIlcCRK2eg+NP+vvXv5kaO44wBe+wCbQJAfaxZEOMRYIhIyAsMphks4EAlO4f0HcObPSXJGyhEJ2SCURJiAIhFBBOYhYnxgzUPCOGCHg4E1O079qrZmejyzs7vjNUbsp+2eru569Xz8OOxX1X0xv6fm7Nmv0zvvHE8vvPBC+uvf/p5Onz6Tvl++kBb27kv3P/BAeuSRR9L9hw+n2277VXmc2mweIFbp/JBX55z737n0xhv/Sn/805/TK6+8WoKcmdnZdODAgfToHx5Nv8+PRLvjjjty+HNjCWfq3KORzOiVn/ufjO9H4KchILD5afw5uAsCBAgQIECAAAECBAgQIECAAAECBH7mAhHYxBaPIOut/FAClVghU670ejm06aXz58+npaWldOzYq+kfr72Wvjj9Zfruu+W0d+9CeiAHNg/+7sG8uuY3ac/e3SXsiTfgRGATgc+FC8vpww//k1fY/CW9/vo/00qeYzYHNvtvvz09/PDD6fBvD6ebb7457bxuZ0xZtnGhjcCm2vgk8GMLCGx+bHHzESBAgAABAgQIECBAgAABAgQIECCwLQVaYFMCmvwYs3iUWZRraJJyiBOhy4W8yuZcOnHio3TioxPp3DffpOW8wuaXN9yYDh68K915551p9+5daceOHf03zsQ4Fy/mFTr5eObMf9Obb/47nTh5sowVK3AWFxfz49TuSr/evz/dcP31aX5+PrctdxFxz+r8g5hmUNqWf0y+NIGrJiCwuWr0JiZAgAABAgQIECBAgAABAgQIECBAYDsJxKKW+IFsCWnKF4+1NpGc1L2c5VU23y8vp6++Opu+Pns2hzXLqZfrr712R9q7ZyGvtNmbduawZm4uv4NmdZlMBD4R2MQ7cb799rv0xZdnSujzQw6AYr5fXHddCW127dqVw5prStATdxH9IrCJwsZDmujZ3Tbes9tLmQCBUQGBzaiJKwQIECBAgAABAgQIECBAgAABAgQIENhygYg6erGqJocoMzkkiaeh1aymhi0lPqkpSn6cWTzmLLeNMCU/1mxmZra0jYwmHnMWzWKLQy3XFTsxXu/iTGkbc8Wqndjmr7kmzZeQZ6aMGz8YruPUgVaHK20nf8S3iL31aMfJvdQSILC+gMBmfSMtCBAgQIAAAQIECBAgQIAAAQIECBAgcNkCsVImfkXEUfb8UR9nFtciBIktX8y/V1Zy2JIvRbCTE5t8Kf+K8Cb2aJB/ly0SmlrInxHa5KAnBzYxVwl8YoycDM3mMSKgiUekRcIT7WIrw5Qxy+kGPqJf7O0G2nEDXTUhQGCigMBmIo9KAgQIECBAgAABAgQIECBAgAABAgQIbI1AiWtKTrIa2uSsowQnOTyJHKVGKHWuEu6UUGW2BCwlpolgJRrmrRuT1PAlxsgzRFBTVthEYJNHzA27QU8LbGKMEh5Fk9Ux49raW/fuotzuoB3X7qmGAIGNCQhsNuakFQECBAgQIECAAAECBAgQIECAAAECBC5LoEUeNSipZ5OykroIpiQuI/OOxiQxXg1SVkcugUy3Y+0z6DkodVuNK7exu3Wtdzt265QJEJhGQGAzjZo+BAgQIECAAAECBAgQIECAAAECBAgQ2KRADVJap/oum1ipsnZo0x6UNhqKjF5poUqtqQ9ZG56x1nQ/272sd2xjd9u1O2jHbp0yAQLTCAhsplHThwABAgQIECBAgAABAgQIECBAgAABApsUGMQnqwFIXUKTR1krtKlhSH2A2vBkozFJC1WipgY9sZKnu9U+3c9u7aRyG7vbpt1BO3brlAkQmEZAYDONmj4ECBAgQIAAAQIECBAgQIAAAQIECBDYpEANUDohSj+wKZFNfIzdao9W2Y616fDZoHtnlv6j0aJtvAtnY9u4kKbbs43Tjt06ZQIEphEQ2Eyjpg8BAgQIECBAgAABAgQIECBAgAABAgQ2KbBWYFMjj/y5RvaxFYFNDL11YU374nXUduZIgMDlCQhsLs9PbwIECBAgQIAAAQIECBAgQIAAAQIECGxIoBvYrJHNjB1nUmATHcaN1V1hk8qj0TYe10T7+MFxGXvc4LmiVsej3NZoUHr7IEBgMwICm81oaUuAAAECBAgQIECAAAECBAgQIECAAIEpBS6mXnk8WUQnwzHH8FkM339a2kw3eqnvpon62qP7GVcHW7dXXB2dYdB2UGq9WmATgcygtlsS2HQ1lAlsjYDAZmscjUKAAAECBAgQIECAAAECBAgQIECAAIEJArG+pv4aH9gMkpG2uqU/2GpoE3FKi1TqGOsHNoNR+6OtUeiOHoFRnK8R2OSqeh9Rv/EZ1pjYZQIEVgUENv4qECBAgAABAgQIECBAgAABAgQIECBA4IoL1LCmTTMcc8TZ4MpPPbBpYU4NdAb33b6bIwEC0wkIbKZz04sAAQIECBAgQIAAAQIECBAgQIAAAQKbEGiBTX9tSqdvCz3qcSiwKZdqn/ispRKVxPqXMkbr3RmwtBt3vdtmMFq72kafvMJGYNO8HAlsrYDAZms9jUaAAAECBAgQIECAAAECBAgQIECAAIExAhHY9PL1CEXqA81ao/Y+mHY+/JixFvTUni1SqSOsHdi0sSYfY7Q24nDLQWgUjz0bVxcXo+6SyuGmzggQ2ISAwGYTWJoSIECAAAECBAgQIECAAAECBAgQIEBgOoHJgU2ENi38aMc6z3BgM5h7EPpMH5kIbAaeSgSuvsCWBTbvvfdeOnXqVLr33nvT4uJihKslJ44Jhv+DGfzHc/W/vjsgQIAAAQIECBAgQIAAAQIECBAgQIDAjyXQy+tZ8s9L44enZY9585WyyGUQuwz/PDV6DLZWrq27n4M2Gy+10S7t0e6pXh+5n9X7Hb5+6RjOCRDYrEBb2TYuV1laWkovvvhiuummm9ITTzwxMvRM7tT/F/3++++nTz/9NB06dKgGNrl5rxdL/GpA0/3H2+k2EuaUDj4IECBAgAABAgQIECBAgAABAgQIECBAgAABAttAoGUm3Rylfe2oi/2zzz5LL730UlpYWEiPPfZYq+4fhwKbDz74oAQ299xzT0l4WqsYKCYZN1Fr40iAAAECBAgQIECAAAECBAgQIECAAAECBAgQ2G4CkaH0LvbKyrtxWUrUxxYLZmKFzb59+9YPbLorbGJJTncT1nQ1lAkQIECAAAECBAgQIECAAAECBAgQIECAAAEC9UllLZSZnZ0dIolspdV98skn6ejRo2XBzOOPPz7ULk6GVtjEO2yiQ3uHTQtp2mDt2BKi7vnIyC4QIECAAAECBAgQIECAAAECBAgQIECAAAECBLaBQMtL4qvGq2ZWVlbS3Nxc2VvdqVOnSmATK2yefPLJEZWhwObdd99N0eG+++7rPxKthTatZztvx5iolVsbRwIECBAgQIAAAQIECBAgQIAAAQIECBAgQIDAdhFooUx83whsYo/VNt38JPKXI0eOlPzlqaeeGqEZCmyOHz+elpaW+itsWhjTBmyDt/M2WvdG2jVHAgQIECBAgAABAgQIECBAgAABAgQIECBAgMDPXSAykwhouplKlGPvXo932LTA5umnnx5hGRvYxAqbxcXFfvITk106UYzUgpuosxEgQIAAAQIECBAgQIAAAQIECBAgQIAAAQIEtpNAy0niO7eApn3/qGv1UddW2ET+sqEVNh9//HE6dOhQuuWWW/rLdbqBTCvHJLHiJs7btXYTjgQIECBAgAABAgQIECBAgAABAgQIECBAgACB7SDQgpnISuLdNS0ziQwl9tjiWjzh7OjRo2XBzLqBzdtvv51OnjyZDh48WAKb+fn5ocEabAtrWmATS3psBAgQIECAAAECBAgQIECAAAECBAgQIECAAIHtJNDykvjOEcp0H4EW57FHlhJBTqywefnll9Ott966/gqbY8eOpdj379+f9uzZk+bm5vquMWls3WOU24T9hgoECBAgQIAAAQIECBAgQIAAAQIECBAgQIAAgW0g0FbRtLCmLXCJ/KRbDorTp0+nt956K919993pmWeeGdEZeofN888/n5577rm0sLCQdu7cWRrHJLG1SaPcDWpafVy3ESBAgAABAgQIECBAgAABAgQIECBAgAABAgS2g0BkJS07iXAmVtG0kCauxxYZSrSL/fz58+nzzz9PDz30UHr22WdHiIYCm5FaFwgQIECAAAECBAgQIECAAAECBAgQIECAAAECBK64gMDmihObgAABAgQIECBAgAABAgQIECBAgAABAgQIECAwWUBgM9lHLQECBAgQIECAAAECBAgQIECAAAECBAgQIEDgigv8HypO1gR1/9m4AAAAAElFTkSuQmCC" style="max-width:700px;"/></div></div></div>
+</div>
+<p>To know how to change <span class="math notranslate nohighlight">\(w_0\)</span> and <span class="math notranslate nohighlight">\(w_1\)</span> to reduce the loss, we compute the derivatives (or gradients):</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+\frac{\partial L_{MSE}}{\partial w_0} = \frac{1}{n}\sum_{i=1}^{n} -2\cdot [y_i - (w_0 + \mathbf{w_1}\cdot x_i)]
+\end{equation*}\]</div>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+\frac{\partial L_{MSE}}{\partial \mathbf{w_1}} = \frac{1}{n}\sum_{i=1}^{n} -2\cdot [y_i - (w_0 + \mathbf{w_1}\cdot x_i)] \cdot x_i
+\end{equation*}\]</div>
+<p>Since we know that we can iteratively take little steps down along the gradient to reduce the loss, aka, <em>gradient descent</em>, the size of the step is determined by the learning rate (<span class="math notranslate nohighlight">\(\eta\)</span>):</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+w_0^{new} = w_0^{current} -   \eta \cdot \frac{\partial L_{MSE}}{\partial w_0}
+\end{equation*}\]</div>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+\mathbf{w_1^{new}} = \mathbf{w_1^{current}} -  \eta \cdot \frac{\partial L_{MSE}}{\partial \mathbf{w_1}}
+\end{equation*}\]</div>
+<p>The gradients needed above are computed using the ML library’s <a class="reference external" href="https://pytorch.org/tutorials/beginner/blitz/autograd_tutorial.html">autograd feature</a>, and described more <a class="reference external" href="https://blog.paperspace.com/pytorch-101-understanding-graphs-and-automatic-differentiation/">here</a>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">step</span><span class="p">(</span><span class="n">lr</span><span class="p">):</span>
+    <span class="n">y_hat</span> <span class="o">=</span> <span class="n">x</span> <span class="o">@</span> <span class="n">w</span>
+    <span class="n">loss</span> <span class="o">=</span> <span class="n">mse</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_hat</span><span class="p">)</span>
+    <span class="c1"># calculate the gradient of a tensor! It is now stored at w.grad</span>
+    <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+
+    <span class="c1"># To prevent tracking history and using memory</span>
+    <span class="c1"># (code block where we don&#39;t need to track the gradients but only modify the values of tensors)</span>
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="c1"># lr is the learning rate. Good learning rate is a key part of Neural Networks.</span>
+        <span class="n">w</span><span class="o">.</span><span class="n">sub_</span><span class="p">(</span><span class="n">lr</span> <span class="o">*</span> <span class="n">w</span><span class="o">.</span><span class="n">grad</span><span class="p">)</span>
+        <span class="c1"># We want to zero the gradient before we are re-evaluate it.</span>
+        <span class="n">w</span><span class="o">.</span><span class="n">grad</span><span class="o">.</span><span class="n">zero_</span><span class="p">()</span>
+
+    <span class="k">return</span> <span class="n">loss</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">item</span><span class="p">(),</span> <span class="n">y_hat</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<blockquote>
+<div><p>Note: In PyTorch, we need to set the gradients to zero before starting to do back propragation because PyTorch accumulates the gradients on subsequent backward passes. This is convenient while training Recurrent Neural Networks (RNNs). So, the default action is to accumulate or sum the gradients on every <code class="docutils literal notranslate"><span class="pre">loss.backward()</span></code> call.
+Because of this, when you start your training loop, ideally you should zero out the gradients so that you do the parameter update correctly. Else the gradient would point in some other direction than the intended direction towards the minimum (or maximum, in case of maximization objectives).</p>
+</div></blockquote>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">w</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">as_tensor</span><span class="p">([</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">])</span>
+<span class="n">w</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
+<span class="n">lr</span> <span class="o">=</span> <span class="mf">0.1</span>
+<span class="n">losses</span> <span class="o">=</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="s2">&quot;inf&quot;</span><span class="p">)]</span>
+<span class="n">y_hats</span> <span class="o">=</span> <span class="p">[]</span>
+<span class="n">epoch</span> <span class="o">=</span> <span class="mi">100</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Train model and perform gradient descent</span>
+<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">epoch</span><span class="p">):</span>
+    <span class="n">loss</span><span class="p">,</span> <span class="n">y_hat</span> <span class="o">=</span> <span class="n">step</span><span class="p">(</span><span class="n">lr</span><span class="p">)</span>
+    <span class="n">losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">loss</span><span class="p">)</span>
+    <span class="n">y_hats</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y_hat</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">losses</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Iteration&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Loss vs Iteration&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/72d4a252b717b8767ddf229ed3a66122efd314fc010fa4050b4c15560c7816fe.png" src="../_images/72d4a252b717b8767ddf229ed3a66122efd314fc010fa4050b4c15560c7816fe.png" />
+</div>
+</div>
+<p>As seen above the loss function is reduced as more optimization steps are taken. The animation below shows how the prediction gets closer to the data as more steps are taken.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">80</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
+<span class="n">scatter_yhat</span> <span class="o">=</span> <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">y_hat</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$</span><span class="se">\\</span><span class="s2">tilde</span><span class="si">{y}</span><span class="s2">$&quot;</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$x$&quot;</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+
+<span class="p">(</span><span class="n">line</span><span class="p">,)</span> <span class="o">=</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">losses</span><span class="p">)),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">losses</span><span class="p">))</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Iteration&quot;</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Loss&quot;</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Loss vs Iteration&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+
+
+<span class="k">def</span> <span class="nf">animate</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
+    <span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Loss = </span><span class="si">%.2f</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">losses</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+    <span class="n">scatter_yhat</span><span class="o">.</span><span class="n">set_offsets</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[[],</span> <span class="p">[]])</span>
+    <span class="n">scatter_yhat</span><span class="o">.</span><span class="n">set_offsets</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">y_hats</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
+    <span class="n">line</span><span class="o">.</span><span class="n">set_data</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">losses</span><span class="p">[:</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]))</span>
+    <span class="k">return</span> <span class="n">scatter_yhat</span><span class="p">,</span> <span class="n">line</span>
+
+
+<span class="c1"># plt.show()</span>
+
+<span class="n">animation</span> <span class="o">=</span> <span class="n">FuncAnimation</span><span class="p">(</span><span class="n">fig</span><span class="p">,</span> <span class="n">animate</span><span class="p">,</span> <span class="n">frames</span><span class="o">=</span><span class="n">epoch</span><span class="p">,</span> <span class="n">interval</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">blit</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="c1"># let animation load</span>
+<span class="n">time</span><span class="o">.</span><span class="n">sleep</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>In case you have some difficulties running the cell below without importing certain packages. Run the following code in a terminal in <code class="docutils literal notranslate"><span class="pre">L96M2lines</span></code> environment.</p>
+<div class="highlight-shell notranslate"><div class="highlight"><pre><span></span>conda<span class="w"> </span>install<span class="w"> </span>-c<span class="w"> </span>conda-forge<span class="w"> </span>ffmpeg
+</pre></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">display</span><span class="p">(</span><span class="n">HTML</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;&lt;div style=&quot;text-align:center;&quot;&gt;</span><span class="si">{</span><span class="n">animation</span><span class="o">.</span><span class="n">to_html5_video</span><span class="p">()</span><span class="si">}</span><span class="s1">&lt;/div&gt;&#39;</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
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+">
+  Your browser does not support the video tag.
+</video></div></div></div>
+</div>
+</div>
+</div>
+<div class="section" id="neural-networks">
+<h2>Neural Networks<a class="headerlink" href="#neural-networks" title="Permalink to this headline">#</a></h2>
+<p>The above discussion focussed on fitting a simple function, a line, through our data. However, data in many problems correspond to underlying functions that are much more complex and their exact form is not known. For these problem we need functions that are much more flexible; neural networks are these more flexible functions.</p>
+<p>This is shown by the <strong>Universal Approximation Theorm</strong>, which states that neural networks can approximate any continuous function. A visual demonstration that neural nets can compute any function can be seen in <a class="reference external" href="http://neuralnetworksanddeeplearning.com/chap4.html">this page</a>.</p>
+<p>Now we give a brief overview of neural networks and how to build them using PyTorch. If you want to go through it in depth, check out these resources: <a class="reference external" href="https://pytorch.org/tutorials/beginner/deep_learning_60min_blitz.html">Deep Learning With Pytorch: A 60 Minute Blitz</a> or <a class="reference external" href="https://pytorch.org/tutorials/beginner/blitz/neural_networks_tutorial.html">Neural Networks</a>.</p>
+<div class="section" id="generate-some-data">
+<h3>Generate some data<a class="headerlink" href="#generate-some-data" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># We create many data points, as neural networks need lot of data to train from scratch.</span>
+<span class="n">n</span> <span class="o">=</span> <span class="mi">10000</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+<span class="c1"># uniformly sample x points between -1 and 1.</span>
+<span class="c1"># Underscore functions in pytorch means replace the value (update)</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">uniform_</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+<span class="n">y</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.1</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>In this problem we create a more wigly function (sin), which can not be approximated as a line.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">y</span><span class="p">,</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[&lt;matplotlib.lines.Line2D at 0x7f9f3d3e6ca0&gt;]
+</pre></div>
+</div>
+<img alt="../_images/58c9ad2b74d1dbfbc15789948bc52714357fcb5d18eb6dd6d176f91bd6d7e0b7.png" src="../_images/58c9ad2b74d1dbfbc15789948bc52714357fcb5d18eb6dd6d176f91bd6d7e0b7.png" />
+</div>
+</div>
+</div>
+<div class="section" id="define-the-neural-network">
+<h3>Define the neural network<a class="headerlink" href="#define-the-neural-network" title="Permalink to this headline">#</a></h3>
+<p>There are many many flavors of neural networks, as can be seen at the <a class="reference external" href="https://www.asimovinstitute.org/neural-network-zoo/">neural network zoo</a>. Here we use the simplest - a fully connected neural net.</p>
+<p>Visually the fully connected neural networks may be represented as:</p>
+<div class="figure align-default" id="neural-network">
+<a class="reference internal image-reference" href="https://miro.medium.com/max/720/1*VHOUViL8dHGfvxCsswPv-Q.png"><img alt="https://miro.medium.com/max/720/1*VHOUViL8dHGfvxCsswPv-Q.png" src="https://miro.medium.com/max/720/1*VHOUViL8dHGfvxCsswPv-Q.png" style="width: 600px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 2 </span><span class="caption-text">A neural network with 4 hidden layers and an output layer.</span><a class="headerlink" href="#neural-network" title="Permalink to this image">#</a></p>
+</div>
+<p>Our particular design consists of 3 hidden dense layers, two of which have a non-linear Relu activation function.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Simple_Neural_Net</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dense1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dense2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">Dense3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+        <span class="bp">self</span><span class="o">.</span><span class="n">relu</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">ReLU</span><span class="p">()</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="c1"># This method is automatically executed when</span>
+        <span class="c1"># we call a object of this class</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dense1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dense2</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Dense3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">neural_net</span> <span class="o">=</span> <span class="n">Simple_Neural_Net</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">net_input</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">out</span> <span class="o">=</span> <span class="n">neural_net</span><span class="p">(</span><span class="n">net_input</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span>
+    <span class="sa">f</span><span class="s2">&quot;The output of the random input </span><span class="si">{</span><span class="n">net_input</span><span class="o">.</span><span class="n">item</span><span class="p">()</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> from untrained network is: </span><span class="si">{</span><span class="n">out</span><span class="o">.</span><span class="n">item</span><span class="p">()</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The output of the random input 1.6324 from untrained network is: -0.3811
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="updating-the-weights-using-an-optimizer">
+<h3>Updating the Weights using an Optimizer<a class="headerlink" href="#updating-the-weights-using-an-optimizer" title="Permalink to this headline">#</a></h3>
+<p>In the linear regression problem above we used a very simple optimization procedure, just adjusting the weights manually. However, in more complex problems it is sometimes more convenient to use the optimization module available in PyTorch as it can allow the use of many advanced options. The implementation of almost every optimizer that we’ll ever need can be found in PyTorch itself. The choice of which optimizer we choose might be very important as it will determine how fast the network will be able to learn.</p>
+<p>In the example below, we show one of the popular optimizers <code class="docutils literal notranslate"><span class="pre">Adam</span></code>.</p>
+<div class="section" id="adam-optimizer">
+<h4>Adam Optimizer<a class="headerlink" href="#adam-optimizer" title="Permalink to this headline">#</a></h4>
+<p>A popular optimizer that is used in many neural networks is the Adam optimizer. It is an adaptive learning rate method that computes individual learning rates for different parameters. For further reading, check out this <a class="reference external" href="https://towardsdatascience.com/adam-latest-trends-in-deep-learning-optimization-6be9a291375c">post</a> about Adam, and this <a class="reference external" href="https://www.ruder.io/optimizing-gradient-descent/">post</a> about other optimizers.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Here we use the Adam optimizer.</span>
+<span class="n">learning_rate</span> <span class="o">=</span> <span class="mf">0.005</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">neural_net</span><span class="o">.</span><span class="n">parameters</span><span class="p">())</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="id1">
+<h3>Loss function<a class="headerlink" href="#id1" title="Permalink to this headline">#</a></h3>
+<p>We use the mean square error as loss function again.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># MSE loss function</span>
+<span class="n">loss_fn</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="training-the-network">
+<h3>Training the network<a class="headerlink" href="#training-the-network" title="Permalink to this headline">#</a></h3>
+<p>The training step is defined below, where <code class="docutils literal notranslate"><span class="pre">optimizer.step()</span></code> essentially does the job of updating the parameters.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train_step</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">loss_fn</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+    <span class="c1"># Set the model to training mode - important for batch normalization and dropout layers</span>
+    <span class="c1"># Unnecessary in this situation but added for best practices</span>
+
+    <span class="n">pred</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+    <span class="n">loss</span> <span class="o">=</span> <span class="n">loss_fn</span><span class="p">(</span><span class="n">pred</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+
+    <span class="c1"># Backpropagation</span>
+    <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+    <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>
+    <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>
+
+    <span class="n">loss</span> <span class="o">=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+    <span class="k">return</span> <span class="n">loss</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>When it comes to real training problems, often the data sets are very large and can not be fit into memory. In these cases the data might be broken up into smaller chunks called batches, and the training is done on one batch at a time. A pass over all the batches in the data set is referred to as an epoch. Since our data set is very small we do not break it up into batches here, but still refer to each training step as an epoch.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">epochs</span> <span class="o">=</span> <span class="mi">1000</span>
+<span class="n">Loss</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">epochs</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">epochs</span><span class="p">):</span>
+    <span class="n">Loss</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">=</span> <span class="n">train_step</span><span class="p">(</span><span class="n">neural_net</span><span class="p">,</span> <span class="n">loss_fn</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+    <span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">mod</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Loss at Epoch </span><span class="si">{</span><span class="n">t</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s2"> is &quot;</span><span class="p">,</span> <span class="n">Loss</span><span class="p">[</span><span class="n">t</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Loss at Epoch 1 is  0.5445301532745361
+Loss at Epoch 201 is  0.22029677033424377
+Loss at Epoch 401 is  0.06729889661073685
+Loss at Epoch 601 is  0.04869914427399635
+Loss at Epoch 801 is  0.029005618765950203
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Loss</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Iteration&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Loss vs Iteration&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/f4fcb282a21d1af2d9a79da00db8a0c72964b3b3ad51fd435cc48ffab690b551.png" src="../_images/f4fcb282a21d1af2d9a79da00db8a0c72964b3b3ad51fd435cc48ffab690b551.png" />
+</div>
+</div>
+</div>
+<div class="section" id="compare-predictions-with-ground-truth">
+<h3>Compare Predictions with Ground Truth<a class="headerlink" href="#compare-predictions-with-ground-truth" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Generate some points where the predictions of the model will be tested.</span>
+<span class="c1"># Here we pick the testing domain to be larger than the training domain to check if the model</span>
+<span class="c1"># has any skill at extrapolation.</span>
+<span class="n">x_test</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">501</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">501</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+<span class="c1"># Generate the predictions from the trained model.</span>
+<span class="n">pred</span> <span class="o">=</span> <span class="n">neural_net</span><span class="p">(</span><span class="n">x_test</span><span class="p">)</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s2">&quot;.&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Data&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_test</span><span class="p">,</span> <span class="n">pred</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Predicted fit&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7f9f07cd4130&gt;
+</pre></div>
+</div>
+<img alt="../_images/41d16af90a8cf38c5d85fa9874ab0fbd03a8d2187f4023e40c9f7d7af4f0bf86.png" src="../_images/41d16af90a8cf38c5d85fa9874ab0fbd03a8d2187f4023e40c9f7d7af4f0bf86.png" />
+</div>
+</div>
+<p>As seen above, the neural network is able to approximate the sine function relatively well over the range over which the training data is available.
+However, it should also be noted that the neural network possesses no skill to extrapolate beyond the range of the data that it was trained on.</p>
+</div>
+</div>
+<div class="section" id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline">#</a></h2>
+<p>In this notebook we introduced the basics of machine learning - fitting flexible functions to data. In the next few notebooks we will show how neural networks can be used to model sub-grid forcing in the Lorenz 96 model, and also some more useful topics related to neural network training and interpretation.</p>
+</div>
+</div>
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+    <i class="fa-solid fa-list"></i> Contents
+  </div>
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline:</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction-to-machine-learning">Introduction to machine learning</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#pytorch">PyTorch</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#linear-regression">Linear regression</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#data-generation-for-linear-regression">Data generation for linear regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#loss-function">Loss function</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-descent-and-optimization">Gradient descent and optimization</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#neural-networks">Neural Networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-some-data">Generate some data</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-neural-network">Define the neural network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#updating-the-weights-using-an-optimizer">Updating the Weights using an Optimizer</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#adam-optimizer">Adam Optimizer</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Loss function</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#training-the-network">Training the network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#compare-predictions-with-ground-truth">Compare Predictions with Ground Truth</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
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+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
+  
+</a></div>
+        <div class="sidebar-primary-item"><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="closing_remarks.html">Outlook</a></li>
+<li class="toctree-l1"><a class="reference internal" href="citing.html">Citing this book</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../bibliography.html">References</a></li>
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+              
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>CNNs and Polynomial Maps</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">CNNs and Polynomial Maps</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#i-introduction">I. Introduction</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#one-variable-l96-equation">One-variable L96 equation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#local-networks">Local networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-feedforward-neural-network-ann">1. Simple feedforward neural network (ANN)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#polynomial-stencil">2. Polynomial stencil</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#global-networks">Global networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-convolutional-neural-network-global-cnn">1. Global convolutional neural network (global CNN)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-feedforward-neural-network-global-ann">2. Global feedforward neural network (global ANN)</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-gcm-objects">Define GCM objects</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-neural-network-objects">Define neural network objects</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simulate-1-variable-l96-equation-as-ground-truth">Simulate 1-variable L96 equation as ground truth</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#train-neural-networks-with-local-and-global-setup-and-evaluate-their-performance">Train neural networks with local and global setup, and evaluate their performance</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-a-global-cnn">1. Train a global CNN</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-a-global-ann">2. Train a global ANN</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-a-local-polynomial-map">3. Train a local polynomial map</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-a-local-ann-map">4. Train a local ANN map</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#display-training-error">Display training error</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#ii-test-predictions-of-the-networks">II. Test predictions of the networks</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#instantaneous-predictions">Instantaneous predictions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#time-dependent-initial-value-simulations">Time-dependent initial-value simulations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#show-parameters-learned-by-the-polynet">Show parameters learned by the polynet</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#iii-energy-conserved-case">III. Energy-conserved case</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-networks">Train networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-simulation-data-with-networks-trained-on-the-energy-conserved-case">Generate simulation data with networks trained on the energy-conserved case</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-instantaneous-prediction">Plot instantaneous prediction</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#iii-a-more-realistic-model-the-burger-s-equation">III. A more realistic model: the Burger’s equation</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#problem-with-the-1-variable-l96-equation">Problem with the 1-variable L96 equation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-burger-s-equation">The Burger’s equation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-training-data-from-burger-s-equation">Generate training data from Burger’s equation</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-the-same-set-of-networks-with-the-burger-s-equation">Train the same set of networks with the Burger’s equation</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#conclusions">Conclusions</a></li>
+</ul>
+
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="cnns-and-polynomial-maps">
+<h1>CNNs and Polynomial Maps<a class="headerlink" href="#cnns-and-polynomial-maps" title="Permalink to this headline">#</a></h1>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">matplotlib</span>
+<span class="kn">import</span> <span class="nn">os</span>
+<span class="kn">from</span> <span class="nn">IPython</span> <span class="kn">import</span> <span class="n">display</span>
+
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">from</span> <span class="nn">torch.autograd</span> <span class="kn">import</span> <span class="n">Variable</span>
+<span class="kn">import</span> <span class="nn">torch.utils.data</span> <span class="k">as</span> <span class="nn">Data</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">EulerFwd</span><span class="p">,</span> <span class="n">RK2</span><span class="p">,</span> <span class="n">RK4</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">GCM_no_param</span>  <span class="c1"># The GCM object based on prior tutorials</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">GCM_discrete</span>  <span class="c1"># New GCM with tendency by global network</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">GCM_local_discrete</span>  <span class="c1"># New GCM with tendency by local network</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">Net_ANN</span>  <span class="c1"># Import a simple neural network</span>
+
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="p">(</span>
+    <span class="n">train_model</span><span class="p">,</span>
+<span class="p">)</span>  <span class="c1"># Train model for each epoch (Adapted from Yani&#39;s tutorial)</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">test_model</span>  <span class="c1"># Calculate loss (Adapted from Yani&#39;s tutorial)</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">train_network</span>  <span class="c1"># Wrapper function for training a neural network</span>
+
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="p">(</span>
+    <span class="n">gen_local_training_data</span><span class="p">,</span>
+<span class="p">)</span>  <span class="c1"># Generate individual patches as training data</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="p">(</span>
+    <span class="n">get_poly_features</span><span class="p">,</span>
+<span class="p">)</span>  <span class="c1"># Generate 2nd-order polynomial features from patches</span>
+
+<span class="c1"># Import Burger&#39;s equation class</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">Burgers</span>
+
+<span class="o">%</span><span class="k">matplotlib</span> notebook
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="tex2jax_ignore mathjax_ignore section" id="i-introduction">
+<h1>I. Introduction<a class="headerlink" href="#i-introduction" title="Permalink to this headline">#</a></h1>
+<p>The petagogical model we use to illustrate different methods at M2LInES is the [Lorenz-96 model](file:///Users/ziweili/Downloads/10829-predictability-problem-partly-solved.pdf) by Edward Lorenz published in 1996. However, Ed Lorenz did not provide much physical reasoning behind the equation, and the dynamics of the slow, large-scale variables and fast, small-scale variables are seemingly ad-hoc. To see how, consider the original L96 equation:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-1f75fe6e-19c7-4493-8756-04e1e3ed3ee8">
+<span class="eqno">(41)<a class="headerlink" href="#equation-1f75fe6e-19c7-4493-8756-04e1e3ed3ee8" title="Permalink to this equation">#</a></span>\[\begin{align}
+\dot{X_k} &amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k - (hc/b)\sum_jY_{j, k}\\
+\dot{Y_{k,j}} &amp;= - cb Y_{j+1, k} \left( Y_{j+2, k} - Y_{j-1, k} \right) - cY_{j, k} + (hc/b)X_k. 
+\end{align}\]</div>
+<p>The advection terms of <span class="math notranslate nohighlight">\(X\)</span> and <span class="math notranslate nohighlight">\(Y\)</span> are similar to the usual 1D advection form:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-2bcdfe15-2c13-4a1b-b762-8abd1b0f7043">
+<span class="eqno">(42)<a class="headerlink" href="#equation-2bcdfe15-2c13-4a1b-b762-8abd1b0f7043" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\dot{X_k} \sim - X_{k-1} \left( X_{k-2} - X_{k+1}\right) \sim X\nabla X,
+\end{equation}\]</div>
+<p>but the advection term in <span class="math notranslate nohighlight">\(X\)</span> is negative compared with the usual momentum conservation in fluid dynamics: <span class="math notranslate nohighlight">\(Du = 0\Rightarrow \partial_t u = - u\nabla u\)</span>.</p>
+<p>To simplify the problem a little bit, and understand the skill and stability of machine-learning algorithms when they are applied to physical systems, we consider the 1-variable version of the L96 system. This approach is similar to Brandon Reichl’s approach (see notebook at this <a class="reference external" href="https://github.com/m2lines/L96_demo/blob/main/08-Implementation/Neural%20Network%20Advection.ipynb">link</a>). Here we use this view to consider the performance of the following networks:</p>
+<ol class="arabic simple">
+<li><p>Local ANN</p></li>
+<li><p>Local polynomial stencils</p></li>
+<li><p>Global CNN</p></li>
+<li><p>Global ANN</p></li>
+</ol>
+<div class="section" id="one-variable-l96-equation">
+<h2>One-variable L96 equation<a class="headerlink" href="#one-variable-l96-equation" title="Permalink to this headline">#</a></h2>
+<p>The one-variable L96 equation writes</p>
+<div class="amsmath math notranslate nohighlight" id="equation-432eb12b-02b7-4b3a-8f6b-79a1c97be43f">
+<span class="eqno">(43)<a class="headerlink" href="#equation-432eb12b-02b7-4b3a-8f6b-79a1c97be43f" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\dot{X_k} = - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F
+\end{equation}\]</div>
+<p>To see how machine learning helps us with simulating Earth’s dynamics, we use three approaches to simulate the 1-variable L96 system. Our approach is based on the following principles:</p>
+<ol class="arabic simple">
+<li><p>The networks’ perception fields must be local, as required by physical laws and the practical constraints of implementation.</p></li>
+<li><p>The network structure must obey symmetries. They should be translational invariant, and is compatible with periodic boundary conditions.</p></li>
+<li><p>The complexity of the networks must be small for interpretability.</p></li>
+</ol>
+</div>
+<div class="section" id="local-networks">
+<h2>Local networks<a class="headerlink" href="#local-networks" title="Permalink to this headline">#</a></h2>
+<p>In one-dimensional setting, we define local maps as the following approximation problem: given a receptive field of the network of width <span class="math notranslate nohighlight">\((2m+1)\)</span>: <span class="math notranslate nohighlight">\(\mathbf{X^{(m)}_k} = (X_{k-m}, ..., X_{k}, ..., X_{k+m})\)</span>, we approximate function <span class="math notranslate nohighlight">\(f(\mathbf{X^{(m)}_k})\)</span> such that</p>
+<div class="amsmath math notranslate nohighlight" id="equation-b8e8d736-5e9c-49e3-8366-e00a1ed7222a">
+<span class="eqno">(44)<a class="headerlink" href="#equation-b8e8d736-5e9c-49e3-8366-e00a1ed7222a" title="Permalink to this equation">#</a></span>\[\begin{equation}
+f: \mathbf{X^{(m)}_k} \rightarrow dX_k/dt
+\end{equation}\]</div>
+<p>In this tutorial, we set <span class="math notranslate nohighlight">\(m = 2\)</span>, and the local map is illustrated in the following figure:</p>
+<div class="figure align-default">
+<a class="reference internal image-reference" href="../_images/local_map.png"><img alt="../_images/local_map.png" src="../_images/local_map.png" style="width: 400px;" /></a>
+</div>
+<p>we choose the functional form, <span class="math notranslate nohighlight">\(f(\cdot)\)</span>, to be one of the following two choices, simple feedforward neural network, and polynomial stencils.</p>
+<div class="section" id="simple-feedforward-neural-network-ann">
+<h3>1. Simple feedforward neural network (ANN)<a class="headerlink" href="#simple-feedforward-neural-network-ann" title="Permalink to this headline">#</a></h3>
+<p>The feedforward neural network is embedded in the GCM, replacing all terms on the right-hand-side. For this tutorial, the network is fully connected, has two hidden layers with 10 neurons each, and has <span class="math notranslate nohighlight">\(tanh()\)</span> as its activation function. The structure can be seen in the following figure:</p>
+<div class="figure align-default">
+<a class="reference internal image-reference" href="../_images/local_map_ANN.png"><img alt="../_images/local_map_ANN.png" src="../_images/local_map_ANN.png" style="width: 400px;" /></a>
+</div>
+</div>
+<div class="section" id="polynomial-stencil">
+<h3>2. Polynomial stencil<a class="headerlink" href="#polynomial-stencil" title="Permalink to this headline">#</a></h3>
+<p>Physical dynamical systems are usually written as ODEs and PDEs, and when they are solved numerically, they are discretized in space and time into regular ODE sets (whether it’s in the spectral space or in the real space). Take the real space discretization for example, the terms on the right-hand side usually take the form of polynomials of the information from adjacent points. The one-variable L96 system is a nice example:</p>
+<div class="figure align-default">
+<a class="reference internal image-reference" href="../_images/local_map_polynet.png"><img alt="../_images/local_map_polynet.png" src="../_images/local_map_polynet.png" style="width: 400px;" /></a>
+</div>
+</div>
+</div>
+<div class="section" id="global-networks">
+<h2>Global networks<a class="headerlink" href="#global-networks" title="Permalink to this headline">#</a></h2>
+<div class="section" id="global-convolutional-neural-network-global-cnn">
+<h3>1. Global convolutional neural network (global CNN)<a class="headerlink" href="#global-convolutional-neural-network-global-cnn" title="Permalink to this headline">#</a></h3>
+<p>We apply a simple global convolutional neural network to predict future tendencies. The network has a filter width of 5 (consistent with <span class="math notranslate nohighlight">\(m = 2\)</span> as above), 2 covolutional layers and 2 fully connected-layers with 100 neurons each. The first convolution layer has 1 channel as input, 2 channels as output, and the second convolution layer has 2 channels as input, 4 channels as output.</p>
+<div class="figure align-default">
+<a class="reference internal image-reference" href="../_images/diagram_CNN.png"><img alt="../_images/diagram_CNN.png" src="../_images/diagram_CNN.png" style="width: 400px;" /></a>
+</div>
+</div>
+<div class="section" id="global-feedforward-neural-network-global-ann">
+<h3>2. Global feedforward neural network (global ANN)<a class="headerlink" href="#global-feedforward-neural-network-global-ann" title="Permalink to this headline">#</a></h3>
+<p>As a point of comparison with the global CNN, we also train a global ANN with takes in global field and predicts global tendencies.</p>
+</div>
+</div>
+<div class="section" id="define-gcm-objects">
+<h2>Define GCM objects<a class="headerlink" href="#define-gcm-objects" title="Permalink to this headline">#</a></h2>
+<p>Here, we define GCM objects for</p>
+<ol class="arabic simple">
+<li><p>the 1-variable L96 equation</p></li>
+<li><p>the L96 system with tendency represented by global networks (global ANN, global CNN)</p></li>
+<li><p>the L96 system with tendency represented by local networks (local ANN, polynet)</p></li>
+</ol>
+</div>
+<div class="section" id="define-neural-network-objects">
+<h2>Define neural network objects<a class="headerlink" href="#define-neural-network-objects" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define a convolutional neural network object</span>
+<span class="k">class</span> <span class="nc">Net_CNN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
+        <span class="bp">self</span><span class="p">,</span>
+        <span class="n">inout_size</span><span class="o">=</span><span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span>
+        <span class="n">kernel_size</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
+        <span class="n">hidden_neurons</span><span class="o">=</span><span class="p">[</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">],</span>
+        <span class="n">strides</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
+        <span class="n">channels</span><span class="o">=</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>
+    <span class="p">):</span>
+        <span class="c1"># Class of a very simple 1-dimensional convolutional neural network. This network contains</span>
+        <span class="c1"># 2 convolutional layers with different channels channel each, and 2 fully-connected layers</span>
+        <span class="c1"># after the convolutional layers.</span>
+
+        <span class="c1"># Input parameters:</span>
+        <span class="c1">#</span>
+        <span class="c1"># inout_size:     A 2-element list, the first element contains the input size;</span>
+        <span class="c1">#                 the second element contains the output size.</span>
+        <span class="c1">#</span>
+        <span class="c1"># kernel_size:    The size of the convolutional kernel. Same across all conv layers</span>
+        <span class="c1">#</span>
+        <span class="c1"># hidden_neurons: List of hidden neurons of the fully-connected layers after the conv layers</span>
+
+        <span class="c1"># Inherit from default class</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_CNN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+
+        <span class="c1"># Set padding to half the size of the kernel, and padding mode to periodic (circular)</span>
+        <span class="n">padding</span> <span class="o">=</span> <span class="n">kernel_size</span> <span class="o">//</span> <span class="mi">2</span>
+        <span class="n">padding_mode</span> <span class="o">=</span> <span class="s2">&quot;circular&quot;</span>
+
+        <span class="c1"># Set dilation to 1, i.e., no dilation</span>
+        <span class="n">dilation</span> <span class="o">=</span> <span class="mi">1</span>
+
+        <span class="c1"># Define two convolutional layers</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">conv1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">Conv1d</span><span class="p">(</span>
+            <span class="mi">1</span><span class="p">,</span>
+            <span class="n">channels</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
+            <span class="n">kernel_size</span><span class="p">,</span>
+            <span class="n">stride</span><span class="o">=</span><span class="n">strides</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
+            <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span>
+            <span class="n">padding_mode</span><span class="o">=</span><span class="n">padding_mode</span><span class="p">,</span>
+        <span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">conv2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">Conv1d</span><span class="p">(</span>
+            <span class="n">channels</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
+            <span class="n">channels</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
+            <span class="n">kernel_size</span><span class="p">,</span>
+            <span class="n">stride</span><span class="o">=</span><span class="n">strides</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
+            <span class="n">padding</span><span class="o">=</span><span class="n">padding</span><span class="p">,</span>
+            <span class="n">padding_mode</span><span class="o">=</span><span class="n">padding_mode</span><span class="p">,</span>
+        <span class="p">)</span>
+
+        <span class="c1"># Define a function to calculate the dimensionality of the input to the fully-connected layer</span>
+        <span class="c1"># (see definition of L_out in https://pytorch.org/docs/stable/generated/torch.nn.Conv1d.html)</span>
+        <span class="n">output_size</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">L_in</span><span class="p">,</span> <span class="n">padding</span><span class="p">,</span> <span class="n">dilation</span><span class="p">,</span> <span class="n">kernel_size</span><span class="p">,</span> <span class="n">stride</span><span class="p">:</span> <span class="nb">int</span><span class="p">(</span>
+            <span class="p">(</span><span class="n">L_in</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">padding</span> <span class="o">-</span> <span class="n">dilation</span> <span class="o">*</span> <span class="p">(</span><span class="n">kernel_size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">stride</span> <span class="o">+</span> <span class="mi">1</span>
+        <span class="p">)</span>
+        <span class="c1"># Use the function to calculate the flattened output size of the convolutional layers</span>
+        <span class="n">output_size1</span> <span class="o">=</span> <span class="n">output_size</span><span class="p">(</span>
+            <span class="n">inout_size</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">padding</span><span class="p">,</span> <span class="n">dilation</span><span class="p">,</span> <span class="n">kernel_size</span><span class="p">,</span> <span class="n">strides</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+        <span class="p">)</span>
+        <span class="n">output_size2</span> <span class="o">=</span> <span class="n">output_size</span><span class="p">(</span>
+            <span class="n">output_size1</span><span class="p">,</span> <span class="n">padding</span><span class="p">,</span> <span class="n">dilation</span><span class="p">,</span> <span class="n">kernel_size</span><span class="p">,</span> <span class="n">strides</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+        <span class="p">)</span>
+
+        <span class="c1"># Assign information to be used in the fully connected layer and self.forward()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">output_size2</span> <span class="o">=</span> <span class="n">output_size2</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">channels</span> <span class="o">=</span> <span class="n">channels</span>
+
+        <span class="c1"># Define fully-connected layers after the convolutional layers</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">output_size2</span> <span class="o">*</span> <span class="n">channels</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">hidden_neurons</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">hidden_neurons</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">inout_size</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">conv1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>  <span class="c1"># Convolution layer 1</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">conv2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>  <span class="c1"># Convolution layer 2</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_size2</span><span class="p">)</span>  <span class="c1"># Flatten</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>  <span class="c1"># Fully-connected layer 1</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>  <span class="c1"># Fully-connected layer 2</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="simulate-1-variable-l96-equation-as-ground-truth">
+<h2>Simulate 1-variable L96 equation as ground truth<a class="headerlink" href="#simulate-1-variable-l96-equation-as-ground-truth" title="Permalink to this headline">#</a></h2>
+<p>The 1-variable simulation has 100 grid points of <span class="math notranslate nohighlight">\(X\)</span>, and is run 20000 timesteps with <span class="math notranslate nohighlight">\(dt = 0.001\)</span>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>  <span class="c1"># For reproducibility</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>  <span class="c1"># For reproducibility</span>
+<span class="n">plot_path</span> <span class="o">=</span> <span class="s2">&quot;./figs/&quot;</span>
+<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">isdir</span><span class="p">(</span><span class="n">plot_path</span><span class="p">):</span>
+    <span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">plot_path</span><span class="p">)</span>
+
+<span class="c1"># Default case, 1-variable L96</span>
+<span class="n">K</span> <span class="o">=</span> <span class="mi">100</span>  <span class="c1"># The number of slow variables, and</span>
+<span class="n">J</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># The number of fast variables per slow variable, set to 0</span>
+<span class="n">F</span> <span class="o">=</span> <span class="mi">18</span>  <span class="c1"># Constant external forcing</span>
+<span class="n">h</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># No interaction with subgrid scale</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.001</span>  <span class="c1"># Timestep</span>
+
+<span class="c1"># Define an 1-variable L96 object</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="n">h</span><span class="p">)</span>
+
+<span class="c1"># Define total timesteps, time, and sampling interval</span>
+<span class="n">Nt</span> <span class="o">=</span> <span class="mi">20000</span>  <span class="c1"># Total timesteps</span>
+<span class="n">T</span> <span class="o">=</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">Nt</span>  <span class="c1"># Total time of simulation</span>
+<span class="n">si</span> <span class="o">=</span> <span class="n">dt</span>  <span class="c1"># Sampling interval</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Run the L96 model as the ground truth</span>
+<span class="n">X_true_temp</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">si</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">X_true</span> <span class="o">=</span> <span class="n">X_true_temp</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">dX_true</span> <span class="o">=</span> <span class="n">X_true_temp</span><span class="p">[</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">X_true_temp</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+<span class="k">del</span> <span class="n">X_true_temp</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="train-neural-networks-with-local-and-global-setup-and-evaluate-their-performance">
+<h2>Train neural networks with local and global setup, and evaluate their performance<a class="headerlink" href="#train-neural-networks-with-local-and-global-setup-and-evaluate-their-performance" title="Permalink to this headline">#</a></h2>
+<p>First, we define some training parameters. These training parameters are the same across all training procedures in this notebook.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">train_size</span> <span class="o">=</span> <span class="mi">2000</span>  <span class="c1"># Number of training data</span>
+<span class="n">test_size</span> <span class="o">=</span> <span class="mi">400</span>  <span class="c1"># Number of test data</span>
+<span class="n">batch_size</span> <span class="o">=</span> <span class="mi">100</span>  <span class="c1"># Number of training data per batch</span>
+<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">1000</span>  <span class="c1"># Total number of epochs</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="train-a-global-cnn">
+<h3>1. Train a global CNN<a class="headerlink" href="#train-a-global-cnn" title="Permalink to this headline">#</a></h3>
+<p>The global CNN has 2 convolution layers with ReLU activation, and 2 fully connected layers with tanh activation. The number of input/output channels of the two convolutional layers are all 1. The fully-connected layers has 100 hidden neurons each.</p>
+<p>First, we define training data for the global network.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define a random index to select training and test data</span>
+<span class="n">ind</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">train_size</span> <span class="o">+</span> <span class="n">test_size</span><span class="p">)</span>
+
+<span class="c1"># Add one extra dimension to represent channels</span>
+<span class="n">X_feature_train_global</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="n">ind</span><span class="p">[:</span><span class="n">train_size</span><span class="p">],</span> <span class="p">:][:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">X_target_train_global</span> <span class="o">=</span> <span class="n">dX_true</span><span class="p">[</span><span class="n">ind</span><span class="p">[:</span><span class="n">train_size</span><span class="p">],</span> <span class="p">:][:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">X_feature_test_global</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="n">ind</span><span class="p">[</span><span class="n">train_size</span><span class="p">:],</span> <span class="p">:][:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">X_target_test_global</span> <span class="o">=</span> <span class="n">dX_true</span><span class="p">[</span><span class="n">ind</span><span class="p">[</span><span class="n">train_size</span><span class="p">:],</span> <span class="p">:][:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define CNN parameters</span>
+<span class="n">inout_size</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="p">,</span> <span class="n">K</span><span class="p">]</span>  <span class="c1"># Input and output dimensions, global field</span>
+<span class="n">kernel_size</span> <span class="o">=</span> <span class="mi">5</span>  <span class="c1"># Kernel width</span>
+<span class="n">channels</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>  <span class="c1"># Number of channels of the first and second layer</span>
+<span class="n">hidden_neurons</span> <span class="o">=</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">]</span>  <span class="c1"># Number of hidden neurons of the fully-connected layers</span>
+
+<span class="c1"># Define a global CNN</span>
+<span class="n">net_CNN_global</span> <span class="o">=</span> <span class="n">Net_CNN</span><span class="p">(</span>
+    <span class="n">inout_size</span><span class="o">=</span><span class="n">inout_size</span><span class="p">,</span>
+    <span class="n">kernel_size</span><span class="o">=</span><span class="n">kernel_size</span><span class="p">,</span>
+    <span class="n">hidden_neurons</span><span class="o">=</span><span class="n">hidden_neurons</span><span class="p">,</span>
+    <span class="n">channels</span><span class="o">=</span><span class="n">channels</span><span class="p">,</span>
+<span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+
+<span class="c1"># Train the global CNN using global data</span>
+<span class="n">net_CNN_global</span><span class="p">,</span> <span class="n">train_loss_CNN</span><span class="p">,</span> <span class="n">test_loss_CNN</span> <span class="o">=</span> <span class="n">train_network</span><span class="p">(</span>
+    <span class="n">net_CNN_global</span><span class="p">,</span>
+    <span class="n">X_feature_train_global</span><span class="p">,</span>
+    <span class="n">X_target_train_global</span><span class="p">,</span>
+    <span class="n">X_feature_test_global</span><span class="p">,</span>
+    <span class="n">X_target_test_global</span><span class="p">,</span>
+    <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
+    <span class="n">n_epochs</span><span class="o">=</span><span class="n">n_epochs</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>epoch = 1000, traing loss = 1.820713e-05, test loss = 3.095299e-05
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="train-a-global-ann">
+<h3>2. Train a global ANN<a class="headerlink" href="#train-a-global-ann" title="Permalink to this headline">#</a></h3>
+<p>The global ANN has 2 hidden layers with 100 hidden neurons each.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define a global ANN</span>
+<span class="n">net_ANN_global</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">([</span><span class="n">K</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="n">K</span><span class="p">],</span> <span class="n">filter_loc</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+
+<span class="c1"># Train the global ANN using global data</span>
+<span class="n">net_ANN_global</span><span class="p">,</span> <span class="n">train_loss_ANN_global</span><span class="p">,</span> <span class="n">test_loss_ANN_global</span> <span class="o">=</span> <span class="n">train_network</span><span class="p">(</span>
+    <span class="n">net_ANN_global</span><span class="p">,</span>
+    <span class="n">X_feature_train_global</span><span class="p">,</span>
+    <span class="n">X_target_train_global</span><span class="p">,</span>
+    <span class="n">X_feature_test_global</span><span class="p">,</span>
+    <span class="n">X_target_test_global</span><span class="p">,</span>
+    <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
+    <span class="n">n_epochs</span><span class="o">=</span><span class="n">n_epochs</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>epoch = 1000, traing loss = 2.221758e-04, test loss = 2.898207e-04
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="train-a-local-polynomial-map">
+<h3>3. Train a local polynomial map<a class="headerlink" href="#train-a-local-polynomial-map" title="Permalink to this headline">#</a></h3>
+<p>To train local maps, we first generate training data as local information around a grid point. Here, we define a local stencil that extends 2 grid points to the left and right.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define the local map location</span>
+<span class="n">filter_start</span> <span class="o">=</span> <span class="o">-</span><span class="mi">2</span>
+<span class="n">filter_end</span> <span class="o">=</span> <span class="mi">2</span>  <span class="c1"># 5 points: k+2, k+1, k, k-1, k-2</span>
+<span class="n">filter_loc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">filter_start</span><span class="p">,</span> <span class="n">filter_end</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># Filter location list</span>
+
+<span class="c1"># Generate local maps consistent with the geometry of the stencil</span>
+<span class="n">X_local_train</span><span class="p">,</span> <span class="n">dX_train</span><span class="p">,</span> <span class="n">X_local_test</span><span class="p">,</span> <span class="n">dX_test</span> <span class="o">=</span> <span class="n">gen_local_training_data</span><span class="p">(</span>
+    <span class="n">X_true</span><span class="p">,</span> <span class="n">dX_true</span><span class="p">,</span> <span class="n">filter_loc</span><span class="p">,</span> <span class="n">train_size</span><span class="p">,</span> <span class="n">test_size</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Generate polynomial features from data</span>
+<span class="n">X_feature_train_poly</span> <span class="o">=</span> <span class="n">get_poly_features</span><span class="p">(</span><span class="n">X_local_train</span><span class="p">)</span>  <span class="c1"># Training data</span>
+<span class="n">X_feature_test_poly</span> <span class="o">=</span> <span class="n">get_poly_features</span><span class="p">(</span><span class="n">X_local_test</span><span class="p">)</span>  <span class="c1"># Test data</span>
+
+<span class="c1"># Define a polynet as a regular ANN, with no hidden layers and a single bias</span>
+<span class="n">net_poly</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">([</span><span class="n">X_feature_train_poly</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">],</span> <span class="n">filter_loc</span><span class="o">=</span><span class="n">filter_loc</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+
+<span class="c1"># Train the polynet</span>
+<span class="n">net_poly</span><span class="p">,</span> <span class="n">train_loss_poly</span><span class="p">,</span> <span class="n">test_loss_poly</span> <span class="o">=</span> <span class="n">train_network</span><span class="p">(</span>
+    <span class="n">net_poly</span><span class="p">,</span>
+    <span class="n">X_feature_train_poly</span><span class="p">,</span>
+    <span class="n">dX_train</span><span class="p">,</span>
+    <span class="n">X_feature_test_poly</span><span class="p">,</span>
+    <span class="n">dX_test</span><span class="p">,</span>
+    <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
+    <span class="n">n_epochs</span><span class="o">=</span><span class="n">n_epochs</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>epoch = 1000, traing loss = 2.317021e-07, test loss = 1.979235e-07
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="train-a-local-ann-map">
+<h3>4. Train a local ANN map<a class="headerlink" href="#train-a-local-ann-map" title="Permalink to this headline">#</a></h3>
+<p>Using the local information from above, we train a local ANN as another point of comparison. This local ANN has two hidden layers with 10 neurons each, and has <span class="math notranslate nohighlight">\(tanh()\)</span> as activation function.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define a regular ANN consistent with the local map</span>
+<span class="n">net_ANN_local</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">([</span><span class="nb">len</span><span class="p">(</span><span class="n">filter_loc</span><span class="p">),</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">filter_loc</span><span class="o">=</span><span class="n">filter_loc</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+
+<span class="c1"># Train the local ANN</span>
+<span class="n">net_ANN_local</span><span class="p">,</span> <span class="n">train_loss_ANN</span><span class="p">,</span> <span class="n">test_loss_ANN</span> <span class="o">=</span> <span class="n">train_network</span><span class="p">(</span>
+    <span class="n">net_ANN_local</span><span class="p">,</span>
+    <span class="n">X_local_train</span><span class="p">,</span>
+    <span class="n">dX_train</span><span class="p">,</span>
+    <span class="n">X_local_test</span><span class="p">,</span>
+    <span class="n">dX_test</span><span class="p">,</span>
+    <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
+    <span class="n">n_epochs</span><span class="o">=</span><span class="n">n_epochs</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>epoch = 1000, traing loss = 1.124198e-05, test loss = 1.063997e-05
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="display-training-error">
+<h3>Display training error<a class="headerlink" href="#display-training-error" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">dX_var</span> <span class="o">=</span> <span class="n">dX_train</span><span class="o">.</span><span class="n">var</span><span class="p">()</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_ANN_global</span> <span class="o">/</span> <span class="n">dX_var</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:blue&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global ANN&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">test_loss_ANN_global</span> <span class="o">/</span> <span class="n">dX_var</span><span class="p">,</span> <span class="s2">&quot;:&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:blue&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_CNN</span> <span class="o">/</span> <span class="n">dX_var</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:red&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global CNN&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">test_loss_CNN</span> <span class="o">/</span> <span class="n">dX_var</span><span class="p">,</span> <span class="s2">&quot;:&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:red&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_ANN</span> <span class="o">/</span> <span class="n">dX_var</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:grey&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local ANN&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">test_loss_ANN</span> <span class="o">/</span> <span class="n">dX_var</span><span class="p">,</span> <span class="s2">&quot;:&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:grey&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss_poly</span> <span class="o">/</span> <span class="n">dX_var</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local polynet&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">test_loss_poly</span> <span class="o">/</span> <span class="n">dX_var</span><span class="p">,</span> <span class="s2">&quot;:&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:orange&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Normalized MSE loss of different models&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Epochs&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Normalized MSE&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<script type="application/javascript">/* Put everything inside the global mpl namespace */
+/* global mpl */
+window.mpl = {};
+
+mpl.get_websocket_type = function () {
+    if (typeof WebSocket !== 'undefined') {
+        return WebSocket;
+    } else if (typeof MozWebSocket !== 'undefined') {
+        return MozWebSocket;
+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
+mpl.figure = function (figure_id, websocket, ondownload, parent_element) {
+    this.id = figure_id;
+
+    this.ws = websocket;
+
+    this.supports_binary = this.ws.binaryType !== undefined;
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+    if (!this.supports_binary) {
+        var warnings = document.getElementById('mpl-warnings');
+        if (warnings) {
+            warnings.style.display = 'block';
+            warnings.textContent =
+                'This browser does not support binary websocket messages. ' +
+                'Performance may be slow.';
+        }
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+
+    this.imageObj = new Image();
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+    this.context = undefined;
+    this.message = undefined;
+    this.canvas = undefined;
+    this.rubberband_canvas = undefined;
+    this.rubberband_context = undefined;
+    this.format_dropdown = undefined;
+
+    this.image_mode = 'full';
+
+    this.root = document.createElement('div');
+    this.root.setAttribute('style', 'display: inline-block');
+    this._root_extra_style(this.root);
+
+    parent_element.appendChild(this.root);
+
+    this._init_header(this);
+    this._init_canvas(this);
+    this._init_toolbar(this);
+
+    var fig = this;
+
+    this.waiting = false;
+
+    this.ws.onopen = function () {
+        fig.send_message('supports_binary', { value: fig.supports_binary });
+        fig.send_message('send_image_mode', {});
+        if (fig.ratio !== 1) {
+            fig.send_message('set_device_pixel_ratio', {
+                device_pixel_ratio: fig.ratio,
+            });
+        }
+        fig.send_message('refresh', {});
+    };
+
+    this.imageObj.onload = function () {
+        if (fig.image_mode === 'full') {
+            // Full images could contain transparency (where diff images
+            // almost always do), so we need to clear the canvas so that
+            // there is no ghosting.
+            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
+        }
+        fig.context.drawImage(fig.imageObj, 0, 0);
+    };
+
+    this.imageObj.onunload = function () {
+        fig.ws.close();
+    };
+
+    this.ws.onmessage = this._make_on_message_function(this);
+
+    this.ondownload = ondownload;
+};
+
+mpl.figure.prototype._init_header = function () {
+    var titlebar = document.createElement('div');
+    titlebar.classList =
+        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';
+    var titletext = document.createElement('div');
+    titletext.classList = 'ui-dialog-title';
+    titletext.setAttribute(
+        'style',
+        'width: 100%; text-align: center; padding: 3px;'
+    );
+    titlebar.appendChild(titletext);
+    this.root.appendChild(titlebar);
+    this.header = titletext;
+};
+
+mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._root_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._init_canvas = function () {
+    var fig = this;
+
+    var canvas_div = (this.canvas_div = document.createElement('div'));
+    canvas_div.setAttribute('tabindex', '0');
+    canvas_div.setAttribute(
+        'style',
+        'border: 1px solid #ddd;' +
+            'box-sizing: content-box;' +
+            'clear: both;' +
+            'min-height: 1px;' +
+            'min-width: 1px;' +
+            'outline: 0;' +
+            'overflow: hidden;' +
+            'position: relative;' +
+            'resize: both;' +
+            'z-index: 2;'
+    );
+
+    function on_keyboard_event_closure(name) {
+        return function (event) {
+            return fig.key_event(event, name);
+        };
+    }
+
+    canvas_div.addEventListener(
+        'keydown',
+        on_keyboard_event_closure('key_press')
+    );
+    canvas_div.addEventListener(
+        'keyup',
+        on_keyboard_event_closure('key_release')
+    );
+
+    this._canvas_extra_style(canvas_div);
+    this.root.appendChild(canvas_div);
+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='b491fe24-910d-405e-82f5-a5f82c0d5358'></div></div></div>
+</div>
+</div>
+</div>
+</div>
+<div class="tex2jax_ignore mathjax_ignore section" id="ii-test-predictions-of-the-networks">
+<h1>II. Test predictions of the networks<a class="headerlink" href="#ii-test-predictions-of-the-networks" title="Permalink to this headline">#</a></h1>
+<p>global ANN, global CNN, local ANN, and polynomial stencils</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Prediction parameters</span>
+<span class="n">temp_dt</span> <span class="o">=</span> <span class="n">si</span>  <span class="c1"># Timestep same as the sampling interval (same as training data)</span>
+<span class="n">t_plot</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">temp_dt</span><span class="p">)</span>  <span class="c1"># Total testing timesteps</span>
+<span class="n">t_series_plot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">t_plot</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">temp_dt</span>
+
+<span class="c1"># Define GCM objects with tendencies represented by networks</span>
+<span class="n">gcm_ANN</span> <span class="o">=</span> <span class="n">GCM_discrete</span><span class="p">(</span><span class="n">net_ANN_global</span><span class="p">)</span>
+<span class="n">gcm_CNN</span> <span class="o">=</span> <span class="n">GCM_discrete</span><span class="p">(</span><span class="n">net_CNN_global</span><span class="p">)</span>
+<span class="n">gcm_local_ANN</span> <span class="o">=</span> <span class="n">GCM_local_discrete</span><span class="p">(</span><span class="n">net_ANN_local</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+<span class="n">gcm_local_poly</span> <span class="o">=</span> <span class="n">GCM_local_discrete</span><span class="p">(</span><span class="n">net_poly</span><span class="p">,</span> <span class="n">get_poly_features</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="instantaneous-predictions">
+<h2>Instantaneous predictions<a class="headerlink" href="#instantaneous-predictions" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Instataneous predictions by polynomial network and tanh network</span>
+<span class="n">X_predict_poly</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="n">X_predict_local_ANN</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="n">X_predict_global_ANN</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="n">X_predict_global_CNN</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+    <span class="n">X_predict_poly</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">gcm_local_poly</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:],</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">X_predict_local_ANN</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">gcm_local_ANN</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:],</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">X_predict_global_ANN</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">gcm_ANN</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:],</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">X_predict_global_CNN</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">gcm_CNN</span><span class="p">(</span><span class="n">X_true</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:],</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_plot</span> <span class="o">=</span> <span class="mi">9</span>  <span class="c1"># Example variable, the 9th grid point</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+<span class="n">t_ind</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">11000</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Instantaneous predictions (shifted)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">dX_true</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="mf">0.03</span>
+    <span class="o">+</span> <span class="n">X_predict_global_ANN</span><span class="p">[</span><span class="n">t_ind</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">]</span>
+    <span class="o">-</span> <span class="n">X_predict_global_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">dX_true</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="mf">0.02</span>
+    <span class="o">+</span> <span class="n">X_predict_global_CNN</span><span class="p">[</span><span class="n">t_ind</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">]</span>
+    <span class="o">-</span> <span class="n">X_predict_global_CNN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global CNN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">dX_true</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="mf">0.01</span> <span class="o">+</span> <span class="n">X_predict_poly</span><span class="p">[</span><span class="n">t_ind</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_predict_poly</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local polynet&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">dX_true</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">X_predict_local_ANN</span><span class="p">[</span><span class="n">t_ind</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_predict_local_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;Ground truth $dX_</span><span class="si">{0}</span><span class="s2">$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_plot</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;Predicted $dX_</span><span class="si">{0}</span><span class="s2">$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_plot</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<script type="application/javascript">/* Put everything inside the global mpl namespace */
+/* global mpl */
+window.mpl = {};
+
+mpl.get_websocket_type = function () {
+    if (typeof WebSocket !== 'undefined') {
+        return WebSocket;
+    } else if (typeof MozWebSocket !== 'undefined') {
+        return MozWebSocket;
+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
+mpl.figure = function (figure_id, websocket, ondownload, parent_element) {
+    this.id = figure_id;
+
+    this.ws = websocket;
+
+    this.supports_binary = this.ws.binaryType !== undefined;
+
+    if (!this.supports_binary) {
+        var warnings = document.getElementById('mpl-warnings');
+        if (warnings) {
+            warnings.style.display = 'block';
+            warnings.textContent =
+                'This browser does not support binary websocket messages. ' +
+                'Performance may be slow.';
+        }
+    }
+
+    this.imageObj = new Image();
+
+    this.context = undefined;
+    this.message = undefined;
+    this.canvas = undefined;
+    this.rubberband_canvas = undefined;
+    this.rubberband_context = undefined;
+    this.format_dropdown = undefined;
+
+    this.image_mode = 'full';
+
+    this.root = document.createElement('div');
+    this.root.setAttribute('style', 'display: inline-block');
+    this._root_extra_style(this.root);
+
+    parent_element.appendChild(this.root);
+
+    this._init_header(this);
+    this._init_canvas(this);
+    this._init_toolbar(this);
+
+    var fig = this;
+
+    this.waiting = false;
+
+    this.ws.onopen = function () {
+        fig.send_message('supports_binary', { value: fig.supports_binary });
+        fig.send_message('send_image_mode', {});
+        if (fig.ratio !== 1) {
+            fig.send_message('set_device_pixel_ratio', {
+                device_pixel_ratio: fig.ratio,
+            });
+        }
+        fig.send_message('refresh', {});
+    };
+
+    this.imageObj.onload = function () {
+        if (fig.image_mode === 'full') {
+            // Full images could contain transparency (where diff images
+            // almost always do), so we need to clear the canvas so that
+            // there is no ghosting.
+            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
+        }
+        fig.context.drawImage(fig.imageObj, 0, 0);
+    };
+
+    this.imageObj.onunload = function () {
+        fig.ws.close();
+    };
+
+    this.ws.onmessage = this._make_on_message_function(this);
+
+    this.ondownload = ondownload;
+};
+
+mpl.figure.prototype._init_header = function () {
+    var titlebar = document.createElement('div');
+    titlebar.classList =
+        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';
+    var titletext = document.createElement('div');
+    titletext.classList = 'ui-dialog-title';
+    titletext.setAttribute(
+        'style',
+        'width: 100%; text-align: center; padding: 3px;'
+    );
+    titlebar.appendChild(titletext);
+    this.root.appendChild(titlebar);
+    this.header = titletext;
+};
+
+mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._root_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._init_canvas = function () {
+    var fig = this;
+
+    var canvas_div = (this.canvas_div = document.createElement('div'));
+    canvas_div.setAttribute('tabindex', '0');
+    canvas_div.setAttribute(
+        'style',
+        'border: 1px solid #ddd;' +
+            'box-sizing: content-box;' +
+            'clear: both;' +
+            'min-height: 1px;' +
+            'min-width: 1px;' +
+            'outline: 0;' +
+            'overflow: hidden;' +
+            'position: relative;' +
+            'resize: both;' +
+            'z-index: 2;'
+    );
+
+    function on_keyboard_event_closure(name) {
+        return function (event) {
+            return fig.key_event(event, name);
+        };
+    }
+
+    canvas_div.addEventListener(
+        'keydown',
+        on_keyboard_event_closure('key_press')
+    );
+    canvas_div.addEventListener(
+        'keyup',
+        on_keyboard_event_closure('key_release')
+    );
+
+    this._canvas_extra_style(canvas_div);
+    this.root.appendChild(canvas_div);
+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='896c1a90-33c3-4caf-8872-290eba1c8165'></div></div><script type="application/javascript">/* Put everything inside the global mpl namespace */
+/* global mpl */
+window.mpl = {};
+
+mpl.get_websocket_type = function () {
+    if (typeof WebSocket !== 'undefined') {
+        return WebSocket;
+    } else if (typeof MozWebSocket !== 'undefined') {
+        return MozWebSocket;
+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
+mpl.figure = function (figure_id, websocket, ondownload, parent_element) {
+    this.id = figure_id;
+
+    this.ws = websocket;
+
+    this.supports_binary = this.ws.binaryType !== undefined;
+
+    if (!this.supports_binary) {
+        var warnings = document.getElementById('mpl-warnings');
+        if (warnings) {
+            warnings.style.display = 'block';
+            warnings.textContent =
+                'This browser does not support binary websocket messages. ' +
+                'Performance may be slow.';
+        }
+    }
+
+    this.imageObj = new Image();
+
+    this.context = undefined;
+    this.message = undefined;
+    this.canvas = undefined;
+    this.rubberband_canvas = undefined;
+    this.rubberband_context = undefined;
+    this.format_dropdown = undefined;
+
+    this.image_mode = 'full';
+
+    this.root = document.createElement('div');
+    this.root.setAttribute('style', 'display: inline-block');
+    this._root_extra_style(this.root);
+
+    parent_element.appendChild(this.root);
+
+    this._init_header(this);
+    this._init_canvas(this);
+    this._init_toolbar(this);
+
+    var fig = this;
+
+    this.waiting = false;
+
+    this.ws.onopen = function () {
+        fig.send_message('supports_binary', { value: fig.supports_binary });
+        fig.send_message('send_image_mode', {});
+        if (fig.ratio !== 1) {
+            fig.send_message('set_device_pixel_ratio', {
+                device_pixel_ratio: fig.ratio,
+            });
+        }
+        fig.send_message('refresh', {});
+    };
+
+    this.imageObj.onload = function () {
+        if (fig.image_mode === 'full') {
+            // Full images could contain transparency (where diff images
+            // almost always do), so we need to clear the canvas so that
+            // there is no ghosting.
+            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
+        }
+        fig.context.drawImage(fig.imageObj, 0, 0);
+    };
+
+    this.imageObj.onunload = function () {
+        fig.ws.close();
+    };
+
+    this.ws.onmessage = this._make_on_message_function(this);
+
+    this.ondownload = ondownload;
+};
+
+mpl.figure.prototype._init_header = function () {
+    var titlebar = document.createElement('div');
+    titlebar.classList =
+        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';
+    var titletext = document.createElement('div');
+    titletext.classList = 'ui-dialog-title';
+    titletext.setAttribute(
+        'style',
+        'width: 100%; text-align: center; padding: 3px;'
+    );
+    titlebar.appendChild(titletext);
+    this.root.appendChild(titlebar);
+    this.header = titletext;
+};
+
+mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._root_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._init_canvas = function () {
+    var fig = this;
+
+    var canvas_div = (this.canvas_div = document.createElement('div'));
+    canvas_div.setAttribute('tabindex', '0');
+    canvas_div.setAttribute(
+        'style',
+        'border: 1px solid #ddd;' +
+            'box-sizing: content-box;' +
+            'clear: both;' +
+            'min-height: 1px;' +
+            'min-width: 1px;' +
+            'outline: 0;' +
+            'overflow: hidden;' +
+            'position: relative;' +
+            'resize: both;' +
+            'z-index: 2;'
+    );
+
+    function on_keyboard_event_closure(name) {
+        return function (event) {
+            return fig.key_event(event, name);
+        };
+    }
+
+    canvas_div.addEventListener(
+        'keydown',
+        on_keyboard_event_closure('key_press')
+    );
+    canvas_div.addEventListener(
+        'keyup',
+        on_keyboard_event_closure('key_release')
+    );
+
+    this._canvas_extra_style(canvas_div);
+    this.root.appendChild(canvas_div);
+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='1e0d4c9a-4346-4096-bb1b-fd74cef3f904'></div></div></div>
+</div>
+</div>
+<div class="section" id="time-dependent-initial-value-simulations">
+<h2>Time-dependent initial-value simulations<a class="headerlink" href="#time-dependent-initial-value-simulations" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Perform simulations with the GCM objects</span>
+<span class="n">init_cond</span> <span class="o">=</span> <span class="n">X_true</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">X_simulation_global_ANN</span> <span class="o">=</span> <span class="n">gcm_ANN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_global_CNN</span> <span class="o">=</span> <span class="n">gcm_CNN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_local_ANN</span> <span class="o">=</span> <span class="n">gcm_local_ANN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_poly</span> <span class="o">=</span> <span class="n">gcm_local_poly</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+
+<span class="c1"># Perform the groud truth simulation</span>
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM_no_param</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">temp_dt</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Show trajectories of a single variable</span>
+<span class="n">temp_t</span> <span class="o">=</span> <span class="nb">min</span><span class="p">([</span><span class="n">X_no_param</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1200</span><span class="p">])</span>
+<span class="n">t_ind</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Initial-value simulations, $X_9$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">X_no_param</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span> <span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">X_simulation_global_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Global ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">X_simulation_global_CNN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Global CNN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">X_simulation_poly</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Local polynet&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">X_simulation_local_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Local ANN&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$X_</span><span class="si">{0:d}</span><span class="s2">$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_plot</span><span class="p">))</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1"># Energy conservation</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Initial-value simulations, energy&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_no_param</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_simulation_global_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Global ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_simulation_global_CNN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Global CNN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_simulation_poly</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;--&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Local polynet&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_simulation_local_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;--&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Local ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">init_cond</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;Energy ($|\mathbf</span><span class="si">{X}</span><span class="s2">|^2$)&quot;</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
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+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='8223440e-aca4-40a7-8cd5-930b239ba6a6'></div></div></div>
+</div>
+</div>
+<div class="section" id="show-parameters-learned-by-the-polynet">
+<h2>Show parameters learned by the polynet<a class="headerlink" href="#show-parameters-learned-by-the-polynet" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define a function to show parameters in the polynet</span>
+<span class="k">def</span> <span class="nf">plot_parameters</span><span class="p">(</span><span class="n">net_poly</span><span class="p">,</span> <span class="n">si</span><span class="p">,</span> <span class="n">filter_loc</span><span class="p">,</span> <span class="n">var_str</span><span class="p">,</span> <span class="n">weights_true</span><span class="p">):</span>
+    <span class="c1"># Store the weights of the polynomial networks and normalize by sampling interval</span>
+    <span class="n">weights_poly</span> <span class="o">=</span> <span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">net_poly</span><span class="o">.</span><span class="n">named_parameters</span><span class="p">())[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span> <span class="o">/</span> <span class="n">si</span><span class="p">)</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
+
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">filter_loc</span><span class="p">)</span>
+    <span class="n">n_feature</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span>
+    <span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">combinations</span>
+
+    <span class="n">index_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">combinations</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="n">filter_loc</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">x_plot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_feature</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.8</span>
+
+    <span class="k">def</span> <span class="nf">label_part</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">f</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="k">return</span> <span class="sa">r</span><span class="s2">&quot;+</span><span class="si">{0:d}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">f</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="k">return</span> <span class="sa">r</span><span class="s2">&quot;-</span><span class="si">{0:d}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="o">-</span><span class="n">f</span><span class="p">)</span>
+        <span class="k">elif</span> <span class="n">f</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="k">return</span> <span class="sa">r</span><span class="s2">&quot;&quot;</span>
+
+    <span class="n">xlabels</span> <span class="o">=</span> <span class="p">(</span>
+        <span class="p">[</span><span class="sa">r</span><span class="s2">&quot;$&quot;</span> <span class="o">+</span> <span class="n">var_str</span> <span class="o">+</span> <span class="s2">&quot;_{k&quot;</span> <span class="o">+</span> <span class="n">label_part</span><span class="p">(</span><span class="o">-</span><span class="n">f</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;}$&quot;</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">filter_loc</span><span class="p">]</span>
+        <span class="o">+</span> <span class="p">[</span><span class="sa">r</span><span class="s2">&quot;$&quot;</span> <span class="o">+</span> <span class="n">var_str</span> <span class="o">+</span> <span class="s2">&quot;^2_{k&quot;</span> <span class="o">+</span> <span class="n">label_part</span><span class="p">(</span><span class="o">-</span><span class="n">f</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;}$&quot;</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">filter_loc</span><span class="p">]</span>
+        <span class="o">+</span> <span class="p">[</span>
+            <span class="sa">r</span><span class="s2">&quot;$&quot;</span>
+            <span class="o">+</span> <span class="n">var_str</span>
+            <span class="o">+</span> <span class="s2">&quot;_{k&quot;</span>
+            <span class="o">+</span> <span class="n">label_part</span><span class="p">(</span><span class="o">-</span><span class="n">index</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+            <span class="o">+</span> <span class="s2">&quot;}&quot;</span>
+            <span class="o">+</span> <span class="n">var_str</span>
+            <span class="o">+</span> <span class="s2">&quot;_{k&quot;</span>
+            <span class="o">+</span> <span class="n">label_part</span><span class="p">(</span><span class="o">-</span><span class="n">index</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+            <span class="o">+</span> <span class="s2">&quot;}$&quot;</span>
+            <span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="n">index_list</span>
+        <span class="p">]</span>
+    <span class="p">)</span>
+    <span class="c1"># Scatter plot</span>
+    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+    <span class="n">cmap</span> <span class="o">=</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">colormaps</span><span class="p">[</span><span class="s2">&quot;bone_r&quot;</span><span class="p">]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span>
+        <span class="n">x_plot</span> <span class="o">-</span> <span class="mf">0.10</span><span class="p">,</span>
+        <span class="n">weights_poly</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Polynet&quot;</span><span class="p">,</span>
+        <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:blue&quot;</span><span class="p">,</span>
+        <span class="n">edgecolor</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span>
+        <span class="n">width</span><span class="o">=</span><span class="mf">0.15</span><span class="p">,</span>
+    <span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span>
+        <span class="n">x_plot</span> <span class="o">+</span> <span class="mf">0.10</span><span class="p">,</span>
+        <span class="n">weights_true</span><span class="p">,</span>
+        <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth&quot;</span><span class="p">,</span>
+        <span class="n">color</span><span class="o">=</span><span class="s2">&quot;tab:brown&quot;</span><span class="p">,</span>
+        <span class="n">edgecolor</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span>
+        <span class="n">width</span><span class="o">=</span><span class="mf">0.15</span><span class="p">,</span>
+    <span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">x_plot</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">x_plot</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">5</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="s2">&quot;--k&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">x_plot</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=-</span><span class="mi">30</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">(</span><span class="n">xlabels</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="n">x_plot</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x_plot</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
+    <span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+
+<span class="c1"># Show parameters learned by the polynet</span>
+<span class="n">weights_true</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">])</span>
+<span class="n">plot_parameters</span><span class="p">(</span><span class="n">net_poly</span><span class="p">,</span> <span class="n">si</span><span class="p">,</span> <span class="n">filter_loc</span><span class="p">,</span> <span class="s2">&quot;X&quot;</span><span class="p">,</span> <span class="n">weights_true</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<script type="application/javascript">/* Put everything inside the global mpl namespace */
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+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
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+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='7b41fe1f-db6b-4929-8ad0-b937ea92f30c'></div></div></div>
+</div>
+<p>Compare with the original 1-variable L96 equation:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-2cebb843-2498-40bb-9e05-72afac58ebf6">
+<span class="eqno">(45)<a class="headerlink" href="#equation-2cebb843-2498-40bb-9e05-72afac58ebf6" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\dot{X_k} = - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F
+\end{equation}\]</div>
+<p>we see that the polynomial stencil can reconstruct the advection term and the exponential decay term.</p>
+</div>
+<div class="section" id="iii-energy-conserved-case">
+<h2>III. Energy-conserved case<a class="headerlink" href="#iii-energy-conserved-case" title="Permalink to this headline">#</a></h2>
+<p>It would be interesting to see the performance of ANN, CNN, and polynomial stencils when the underlying system conserves energy:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-3acc5b72-025a-43e0-9061-9c065fe2f93a">
+<span class="eqno">(46)<a class="headerlink" href="#equation-3acc5b72-025a-43e0-9061-9c065fe2f93a" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\dot{X_k} = - X_{k-1} \left( X_{k-2} - X_{k+1} \right)
+\end{equation}\]</div>
+<div class="section" id="train-networks">
+<h3>Train networks<a class="headerlink" href="#train-networks" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># no forcing</span>
+<span class="n">a</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># no exponential decay</span>
+<span class="n">h</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># no interaction with subgrid scale</span>
+
+<span class="n">Nt</span> <span class="o">=</span> <span class="mi">20000</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.001</span>
+<span class="n">si</span> <span class="o">=</span> <span class="n">dt</span>  <span class="c1"># Sampling interval</span>
+<span class="n">T</span> <span class="o">=</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">Nt</span>  <span class="c1"># Total time of simulation</span>
+
+<span class="n">K</span> <span class="o">=</span> <span class="mi">100</span>  <span class="c1"># The number of slow variables, and</span>
+<span class="n">J</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># The number of fast variables, set to 0</span>
+
+<span class="c1"># Define the ground truth model to generate training data</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="n">h</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">run_analysis</span>
+
+<span class="p">(</span>
+    <span class="n">net_ANN_global_2</span><span class="p">,</span>
+    <span class="n">net_CNN_global_2</span><span class="p">,</span>
+    <span class="n">net_ANN_local_2</span><span class="p">,</span>
+    <span class="n">net_poly_2</span><span class="p">,</span>
+    <span class="n">X_true_2</span><span class="p">,</span>
+    <span class="n">dX_true_2</span><span class="p">,</span>
+<span class="p">)</span> <span class="o">=</span> <span class="n">run_analysis</span><span class="p">(</span><span class="n">W</span><span class="p">,</span> <span class="n">si</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n_epochs</span><span class="o">=</span><span class="n">n_epochs</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>training global CNN
+
+epoch = 1000, traing loss = 2.475079e-08, test loss = 1.048361e-07
+
+training global ANN
+
+epoch = 1000, traing loss = 2.270146e-08, test loss = 2.260663e-06
+
+training local polynet
+
+epoch = 1000, traing loss = 4.540985e-09, test loss = 2.419624e-08
+
+training local ANN
+
+epoch = 1000, traing loss = 1.515881e-06, test loss = 2.457820e-05
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="generate-simulation-data-with-networks-trained-on-the-energy-conserved-case">
+<h3>Generate simulation data with networks trained on the energy-conserved case<a class="headerlink" href="#generate-simulation-data-with-networks-trained-on-the-energy-conserved-case" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#### Test predictions</span>
+<span class="n">temp_dt</span> <span class="o">=</span> <span class="n">si</span>
+<span class="n">t_plot</span> <span class="o">=</span> <span class="n">X_true_2</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span>
+<span class="n">t_series_plot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">t_plot</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">temp_dt</span>
+
+<span class="c1"># Define GCM objects using the trained networks</span>
+<span class="n">gcm_ANN</span> <span class="o">=</span> <span class="n">GCM_discrete</span><span class="p">(</span><span class="n">net_ANN_global_2</span><span class="p">)</span>
+<span class="n">gcm_CNN</span> <span class="o">=</span> <span class="n">GCM_discrete</span><span class="p">(</span><span class="n">net_CNN_global_2</span><span class="p">)</span>
+<span class="n">gcm_local_ANN</span> <span class="o">=</span> <span class="n">GCM_local_discrete</span><span class="p">(</span><span class="n">net_ANN_local_2</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+<span class="n">gcm_local_poly</span> <span class="o">=</span> <span class="n">GCM_local_discrete</span><span class="p">(</span><span class="n">net_poly_2</span><span class="p">,</span> <span class="n">get_poly_features</span><span class="p">)</span>
+
+<span class="c1"># Instataneous predictions by polynomial network and tanh network</span>
+<span class="n">X_predict_poly</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X_true_2</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="n">X_predict_local_ANN</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X_true_2</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="n">X_predict_global_ANN</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X_true_2</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="n">X_predict_global_CNN</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X_true_2</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">X_true_2</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+    <span class="n">X_predict_poly</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">gcm_local_poly</span><span class="p">(</span><span class="n">X_true_2</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:],</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">X_predict_local_ANN</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">gcm_local_ANN</span><span class="p">(</span><span class="n">X_true_2</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:],</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">X_predict_global_ANN</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">gcm_ANN</span><span class="p">(</span><span class="n">X_true_2</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:],</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">X_predict_global_CNN</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">gcm_CNN</span><span class="p">(</span><span class="n">X_true_2</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:],</span> <span class="mi">1</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]</span>
+
+<span class="c1"># Perform simulations with the GCM objects</span>
+<span class="n">init_cond</span> <span class="o">=</span> <span class="n">X_true_2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
+<span class="n">X_simulation_global_ANN</span> <span class="o">=</span> <span class="n">gcm_ANN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_global_CNN</span> <span class="o">=</span> <span class="n">gcm_CNN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_local_ANN</span> <span class="o">=</span> <span class="n">gcm_local_ANN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_poly</span> <span class="o">=</span> <span class="n">gcm_local_poly</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+
+<span class="c1"># Perform the groud truth simulation with different forward-stepping schemes</span>
+<span class="c1"># 1. RK4</span>
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM_no_param</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">temp_dt</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="c1"># 2. RK2</span>
+<span class="n">gcm_no_param_RK2</span> <span class="o">=</span> <span class="n">GCM_no_param</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK2</span><span class="p">)</span>
+<span class="n">X_no_param_RK2</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm_no_param_RK2</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">temp_dt</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="c1"># 3. Euler</span>
+<span class="n">gcm_no_param_Euler</span> <span class="o">=</span> <span class="n">GCM_no_param</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">EulerFwd</span><span class="p">)</span>
+<span class="n">X_no_param_Euler</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gcm_no_param_Euler</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">temp_dt</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="plot-instantaneous-prediction">
+<h3>Plot instantaneous prediction<a class="headerlink" href="#plot-instantaneous-prediction" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_plot</span> <span class="o">=</span> <span class="mi">9</span>  <span class="c1"># Example variable, the 9th grid point</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+<span class="n">t_ind</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">11000</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Instantaneous predictions (shifted)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">dX_true_2</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="mf">0.18</span>
+    <span class="o">+</span> <span class="n">X_predict_global_ANN</span><span class="p">[</span><span class="n">t_ind</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">]</span>
+    <span class="o">-</span> <span class="n">X_predict_global_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">dX_true_2</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="mf">0.12</span>
+    <span class="o">+</span> <span class="n">X_predict_global_CNN</span><span class="p">[</span><span class="n">t_ind</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">]</span>
+    <span class="o">-</span> <span class="n">X_predict_global_CNN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global CNN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">dX_true_2</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="mf">0.06</span> <span class="o">+</span> <span class="n">X_predict_poly</span><span class="p">[</span><span class="n">t_ind</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_predict_poly</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local polynet&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">dX_true_2</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">X_predict_local_ANN</span><span class="p">[</span><span class="n">t_ind</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">]</span> <span class="o">-</span> <span class="n">X_predict_local_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="c1"># plt.ylim([-0.3, 0.2])</span>
+<span class="c1"># plt.xlim([-0.3, 0.2])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;Ground truth $dX_</span><span class="si">{0}</span><span class="s2">$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_plot</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;Predicted $dX_</span><span class="si">{0}</span><span class="s2">$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_plot</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<script type="application/javascript">/* Put everything inside the global mpl namespace */
+/* global mpl */
+window.mpl = {};
+
+mpl.get_websocket_type = function () {
+    if (typeof WebSocket !== 'undefined') {
+        return WebSocket;
+    } else if (typeof MozWebSocket !== 'undefined') {
+        return MozWebSocket;
+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
+mpl.figure = function (figure_id, websocket, ondownload, parent_element) {
+    this.id = figure_id;
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+    this.ws = websocket;
+
+    this.supports_binary = this.ws.binaryType !== undefined;
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+    if (!this.supports_binary) {
+        var warnings = document.getElementById('mpl-warnings');
+        if (warnings) {
+            warnings.style.display = 'block';
+            warnings.textContent =
+                'This browser does not support binary websocket messages. ' +
+                'Performance may be slow.';
+        }
+    }
+
+    this.imageObj = new Image();
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+    this.context = undefined;
+    this.message = undefined;
+    this.canvas = undefined;
+    this.rubberband_canvas = undefined;
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+    this.image_mode = 'full';
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+    this.root.setAttribute('style', 'display: inline-block');
+    this._root_extra_style(this.root);
+
+    parent_element.appendChild(this.root);
+
+    this._init_header(this);
+    this._init_canvas(this);
+    this._init_toolbar(this);
+
+    var fig = this;
+
+    this.waiting = false;
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+    this.ws.onopen = function () {
+        fig.send_message('supports_binary', { value: fig.supports_binary });
+        fig.send_message('send_image_mode', {});
+        if (fig.ratio !== 1) {
+            fig.send_message('set_device_pixel_ratio', {
+                device_pixel_ratio: fig.ratio,
+            });
+        }
+        fig.send_message('refresh', {});
+    };
+
+    this.imageObj.onload = function () {
+        if (fig.image_mode === 'full') {
+            // Full images could contain transparency (where diff images
+            // almost always do), so we need to clear the canvas so that
+            // there is no ghosting.
+            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
+        }
+        fig.context.drawImage(fig.imageObj, 0, 0);
+    };
+
+    this.imageObj.onunload = function () {
+        fig.ws.close();
+    };
+
+    this.ws.onmessage = this._make_on_message_function(this);
+
+    this.ondownload = ondownload;
+};
+
+mpl.figure.prototype._init_header = function () {
+    var titlebar = document.createElement('div');
+    titlebar.classList =
+        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';
+    var titletext = document.createElement('div');
+    titletext.classList = 'ui-dialog-title';
+    titletext.setAttribute(
+        'style',
+        'width: 100%; text-align: center; padding: 3px;'
+    );
+    titlebar.appendChild(titletext);
+    this.root.appendChild(titlebar);
+    this.header = titletext;
+};
+
+mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._root_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._init_canvas = function () {
+    var fig = this;
+
+    var canvas_div = (this.canvas_div = document.createElement('div'));
+    canvas_div.setAttribute('tabindex', '0');
+    canvas_div.setAttribute(
+        'style',
+        'border: 1px solid #ddd;' +
+            'box-sizing: content-box;' +
+            'clear: both;' +
+            'min-height: 1px;' +
+            'min-width: 1px;' +
+            'outline: 0;' +
+            'overflow: hidden;' +
+            'position: relative;' +
+            'resize: both;' +
+            'z-index: 2;'
+    );
+
+    function on_keyboard_event_closure(name) {
+        return function (event) {
+            return fig.key_event(event, name);
+        };
+    }
+
+    canvas_div.addEventListener(
+        'keydown',
+        on_keyboard_event_closure('key_press')
+    );
+    canvas_div.addEventListener(
+        'keyup',
+        on_keyboard_event_closure('key_release')
+    );
+
+    this._canvas_extra_style(canvas_div);
+    this.root.appendChild(canvas_div);
+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='d659d28e-888c-4675-a0d7-d935043c314f'></div></div><script type="application/javascript">/* Put everything inside the global mpl namespace */
+/* global mpl */
+window.mpl = {};
+
+mpl.get_websocket_type = function () {
+    if (typeof WebSocket !== 'undefined') {
+        return WebSocket;
+    } else if (typeof MozWebSocket !== 'undefined') {
+        return MozWebSocket;
+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
+mpl.figure = function (figure_id, websocket, ondownload, parent_element) {
+    this.id = figure_id;
+
+    this.ws = websocket;
+
+    this.supports_binary = this.ws.binaryType !== undefined;
+
+    if (!this.supports_binary) {
+        var warnings = document.getElementById('mpl-warnings');
+        if (warnings) {
+            warnings.style.display = 'block';
+            warnings.textContent =
+                'This browser does not support binary websocket messages. ' +
+                'Performance may be slow.';
+        }
+    }
+
+    this.imageObj = new Image();
+
+    this.context = undefined;
+    this.message = undefined;
+    this.canvas = undefined;
+    this.rubberband_canvas = undefined;
+    this.rubberband_context = undefined;
+    this.format_dropdown = undefined;
+
+    this.image_mode = 'full';
+
+    this.root = document.createElement('div');
+    this.root.setAttribute('style', 'display: inline-block');
+    this._root_extra_style(this.root);
+
+    parent_element.appendChild(this.root);
+
+    this._init_header(this);
+    this._init_canvas(this);
+    this._init_toolbar(this);
+
+    var fig = this;
+
+    this.waiting = false;
+
+    this.ws.onopen = function () {
+        fig.send_message('supports_binary', { value: fig.supports_binary });
+        fig.send_message('send_image_mode', {});
+        if (fig.ratio !== 1) {
+            fig.send_message('set_device_pixel_ratio', {
+                device_pixel_ratio: fig.ratio,
+            });
+        }
+        fig.send_message('refresh', {});
+    };
+
+    this.imageObj.onload = function () {
+        if (fig.image_mode === 'full') {
+            // Full images could contain transparency (where diff images
+            // almost always do), so we need to clear the canvas so that
+            // there is no ghosting.
+            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
+        }
+        fig.context.drawImage(fig.imageObj, 0, 0);
+    };
+
+    this.imageObj.onunload = function () {
+        fig.ws.close();
+    };
+
+    this.ws.onmessage = this._make_on_message_function(this);
+
+    this.ondownload = ondownload;
+};
+
+mpl.figure.prototype._init_header = function () {
+    var titlebar = document.createElement('div');
+    titlebar.classList =
+        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';
+    var titletext = document.createElement('div');
+    titletext.classList = 'ui-dialog-title';
+    titletext.setAttribute(
+        'style',
+        'width: 100%; text-align: center; padding: 3px;'
+    );
+    titlebar.appendChild(titletext);
+    this.root.appendChild(titlebar);
+    this.header = titletext;
+};
+
+mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._root_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._init_canvas = function () {
+    var fig = this;
+
+    var canvas_div = (this.canvas_div = document.createElement('div'));
+    canvas_div.setAttribute('tabindex', '0');
+    canvas_div.setAttribute(
+        'style',
+        'border: 1px solid #ddd;' +
+            'box-sizing: content-box;' +
+            'clear: both;' +
+            'min-height: 1px;' +
+            'min-width: 1px;' +
+            'outline: 0;' +
+            'overflow: hidden;' +
+            'position: relative;' +
+            'resize: both;' +
+            'z-index: 2;'
+    );
+
+    function on_keyboard_event_closure(name) {
+        return function (event) {
+            return fig.key_event(event, name);
+        };
+    }
+
+    canvas_div.addEventListener(
+        'keydown',
+        on_keyboard_event_closure('key_press')
+    );
+    canvas_div.addEventListener(
+        'keyup',
+        on_keyboard_event_closure('key_release')
+    );
+
+    this._canvas_extra_style(canvas_div);
+    this.root.appendChild(canvas_div);
+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='28051494-3698-4b3e-a938-dceef516329f'></div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Evolution of one variable</span>
+<span class="n">temp_t</span> <span class="o">=</span> <span class="nb">min</span><span class="p">([</span><span class="n">X_no_param</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1200</span><span class="p">])</span>
+<span class="n">X_plot</span> <span class="o">=</span> <span class="mi">9</span>
+<span class="n">t_ind</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2000</span><span class="p">)</span>
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Initial-value simulations, $X_9$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">X_no_param</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span> <span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth (RK4)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">X_no_param_Euler</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span> <span class="s2">&quot;grey&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth (Euler)&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">X_simulation_global_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">X_simulation_global_CNN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span>
+    <span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global CNN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">X_simulation_poly</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local polynet&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">X_simulation_local_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="n">X_plot</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local ANN&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$X_</span><span class="si">{0:d}</span><span class="s2">$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_plot</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mi">80</span><span class="p">,</span> <span class="mi">80</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+
+<span class="c1"># Energy evolution</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Initial-value simulations, energy&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_no_param</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth (RK4)&quot;</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_no_param_Euler</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;grey&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Truth (Euler)&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_simulation_global_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_simulation_global_CNN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;-.&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;global CNN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_simulation_poly</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;--&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local polynet&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">t_series_plot</span><span class="p">[</span><span class="n">t_ind</span><span class="p">],</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X_simulation_local_ANN</span><span class="p">[</span><span class="n">t_ind</span><span class="p">,</span> <span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
+    <span class="s2">&quot;--&quot;</span><span class="p">,</span>
+    <span class="n">label</span><span class="o">=</span><span class="s2">&quot;local ANN&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">init_cond</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;Energy ($|\mathbf</span><span class="si">{X}</span><span class="s2">|^2$)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<script type="application/javascript">/* Put everything inside the global mpl namespace */
+/* global mpl */
+window.mpl = {};
+
+mpl.get_websocket_type = function () {
+    if (typeof WebSocket !== 'undefined') {
+        return WebSocket;
+    } else if (typeof MozWebSocket !== 'undefined') {
+        return MozWebSocket;
+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
+mpl.figure = function (figure_id, websocket, ondownload, parent_element) {
+    this.id = figure_id;
+
+    this.ws = websocket;
+
+    this.supports_binary = this.ws.binaryType !== undefined;
+
+    if (!this.supports_binary) {
+        var warnings = document.getElementById('mpl-warnings');
+        if (warnings) {
+            warnings.style.display = 'block';
+            warnings.textContent =
+                'This browser does not support binary websocket messages. ' +
+                'Performance may be slow.';
+        }
+    }
+
+    this.imageObj = new Image();
+
+    this.context = undefined;
+    this.message = undefined;
+    this.canvas = undefined;
+    this.rubberband_canvas = undefined;
+    this.rubberband_context = undefined;
+    this.format_dropdown = undefined;
+
+    this.image_mode = 'full';
+
+    this.root = document.createElement('div');
+    this.root.setAttribute('style', 'display: inline-block');
+    this._root_extra_style(this.root);
+
+    parent_element.appendChild(this.root);
+
+    this._init_header(this);
+    this._init_canvas(this);
+    this._init_toolbar(this);
+
+    var fig = this;
+
+    this.waiting = false;
+
+    this.ws.onopen = function () {
+        fig.send_message('supports_binary', { value: fig.supports_binary });
+        fig.send_message('send_image_mode', {});
+        if (fig.ratio !== 1) {
+            fig.send_message('set_device_pixel_ratio', {
+                device_pixel_ratio: fig.ratio,
+            });
+        }
+        fig.send_message('refresh', {});
+    };
+
+    this.imageObj.onload = function () {
+        if (fig.image_mode === 'full') {
+            // Full images could contain transparency (where diff images
+            // almost always do), so we need to clear the canvas so that
+            // there is no ghosting.
+            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
+        }
+        fig.context.drawImage(fig.imageObj, 0, 0);
+    };
+
+    this.imageObj.onunload = function () {
+        fig.ws.close();
+    };
+
+    this.ws.onmessage = this._make_on_message_function(this);
+
+    this.ondownload = ondownload;
+};
+
+mpl.figure.prototype._init_header = function () {
+    var titlebar = document.createElement('div');
+    titlebar.classList =
+        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';
+    var titletext = document.createElement('div');
+    titletext.classList = 'ui-dialog-title';
+    titletext.setAttribute(
+        'style',
+        'width: 100%; text-align: center; padding: 3px;'
+    );
+    titlebar.appendChild(titletext);
+    this.root.appendChild(titlebar);
+    this.header = titletext;
+};
+
+mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._root_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._init_canvas = function () {
+    var fig = this;
+
+    var canvas_div = (this.canvas_div = document.createElement('div'));
+    canvas_div.setAttribute('tabindex', '0');
+    canvas_div.setAttribute(
+        'style',
+        'border: 1px solid #ddd;' +
+            'box-sizing: content-box;' +
+            'clear: both;' +
+            'min-height: 1px;' +
+            'min-width: 1px;' +
+            'outline: 0;' +
+            'overflow: hidden;' +
+            'position: relative;' +
+            'resize: both;' +
+            'z-index: 2;'
+    );
+
+    function on_keyboard_event_closure(name) {
+        return function (event) {
+            return fig.key_event(event, name);
+        };
+    }
+
+    canvas_div.addEventListener(
+        'keydown',
+        on_keyboard_event_closure('key_press')
+    );
+    canvas_div.addEventListener(
+        'keyup',
+        on_keyboard_event_closure('key_release')
+    );
+
+    this._canvas_extra_style(canvas_div);
+    this.root.appendChild(canvas_div);
+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='52bbc79e-cd4a-4867-b348-48d1cf39ac6f'></div></div></div>
+</div>
+</div>
+</div>
+</div>
+<div class="tex2jax_ignore mathjax_ignore section" id="iii-a-more-realistic-model-the-burger-s-equation">
+<h1>III. A more realistic model: the Burger’s equation<a class="headerlink" href="#iii-a-more-realistic-model-the-burger-s-equation" title="Permalink to this headline">#</a></h1>
+<div class="section" id="problem-with-the-1-variable-l96-equation">
+<h2>Problem with the 1-variable L96 equation<a class="headerlink" href="#problem-with-the-1-variable-l96-equation" title="Permalink to this headline">#</a></h2>
+<p>One problem with the L96 model is that it sometimes lacks the physical basis. If we look at the temporal evolution of the 1-variable L96 equation, we see that its dynamics look quite unstable, and most of its variabilities can be numerical artifacts.</p>
+<p>Start from a bump initial condition</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Energy conserving case, 1-variable L96</span>
+<span class="n">K</span> <span class="o">=</span> <span class="mi">100</span>  <span class="c1"># The number of slow variables, and</span>
+<span class="n">J</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># The number of fast variables per slow variable, set to 0</span>
+<span class="n">F</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># no external forcing</span>
+<span class="n">a</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># no exponential decay</span>
+<span class="n">h</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># no interaction with subgrid scale</span>
+
+<span class="c1"># Set a bump initial condition</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+<span class="n">X_init</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(((</span><span class="n">x</span> <span class="o">-</span> <span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="o">/</span> <span class="n">K</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span> <span class="o">*</span> <span class="mi">10</span>
+
+<span class="n">W_conserve</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">F</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="n">h</span><span class="p">)</span>
+<span class="p">(</span>
+    <span class="n">X_true_3</span><span class="p">,</span>
+    <span class="n">_</span><span class="p">,</span>
+    <span class="n">_</span><span class="p">,</span>
+<span class="p">)</span> <span class="o">=</span> <span class="n">W_conserve</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">si</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Animate conservative 1-variable L96 equation</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ioff</span><span class="p">()</span>  <span class="c1"># close notebook mode</span>
+<span class="kn">from</span> <span class="nn">matplotlib.animation</span> <span class="kn">import</span> <span class="n">FuncAnimation</span>
+
+<span class="n">interval</span> <span class="o">=</span> <span class="mi">10</span>
+<span class="n">frames</span> <span class="o">=</span> <span class="mi">200</span>
+
+<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">(</span><span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">ylim</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span> <span class="mi">25</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Energy-conserved initial value problem&quot;</span><span class="p">)</span>
+<span class="p">(</span><span class="n">line</span><span class="p">,)</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">([],</span> <span class="p">[],</span> <span class="n">lw</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$k$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$X$&quot;</span><span class="p">)</span>
+
+
+<span class="k">def</span> <span class="nf">init</span><span class="p">():</span>
+    <span class="n">line</span><span class="o">.</span><span class="n">set_data</span><span class="p">([],</span> <span class="p">[])</span>
+    <span class="k">return</span> <span class="p">(</span><span class="n">line</span><span class="p">,)</span>
+
+
+<span class="k">def</span> <span class="nf">animate</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+    <span class="n">y</span> <span class="o">=</span> <span class="n">X_true_3</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="mi">15</span><span class="p">,</span> <span class="p">:]</span>
+    <span class="n">line</span><span class="o">.</span><span class="n">set_data</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+    <span class="k">return</span> <span class="p">(</span><span class="n">line</span><span class="p">,)</span>
+
+
+<span class="n">anim</span> <span class="o">=</span> <span class="n">FuncAnimation</span><span class="p">(</span><span class="n">fig</span><span class="p">,</span> <span class="n">animate</span><span class="p">,</span> <span class="n">init_func</span><span class="o">=</span><span class="n">init</span><span class="p">,</span> <span class="n">frames</span><span class="o">=</span><span class="n">frames</span><span class="p">,</span> <span class="n">interval</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">blit</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">gif_name</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">plot_path</span><span class="p">,</span> <span class="s2">&quot;L96_one_variable.gif&quot;</span><span class="p">)</span>
+<span class="n">anim</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="n">gif_name</span><span class="p">,</span> <span class="n">writer</span><span class="o">=</span><span class="s2">&quot;pillow&quot;</span><span class="p">)</span>
+<span class="c1"># reopen inline mode</span>
+<span class="o">%</span><span class="k">matplotlib</span> notebook
+</pre></div>
+</div>
+</div>
+</div>
+<p><img alt="SegmentLocal" src="../_images/L96_one_variable.gif" /></p>
+<p>The 1-variable energy-conserved case with the L96 system is just an advection equation with the sign of the sign advection term flipped:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-5f5b4cc9-7924-4652-a661-b95995ba8c2c">
+<span class="eqno">(47)<a class="headerlink" href="#equation-5f5b4cc9-7924-4652-a661-b95995ba8c2c" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\dot{X_k} = - X_{k-1} \left( X_{k-2} - X_{k+1} \right)
+\end{equation}\]</div>
+<p>We can find much similarity with the Burger’s equation.</p>
+</div>
+<div class="section" id="the-burger-s-equation">
+<h2>The Burger’s equation<a class="headerlink" href="#the-burger-s-equation" title="Permalink to this headline">#</a></h2>
+<p>To see the performance of the neural networks on a more realistic model, we use the Burger’s equation as an example. The Burger’s equation is an advection-diffusion equation. It has a diffusive term which reduces the tendency to develop shock waves and numerical instabilities.</p>
+<div class="amsmath math notranslate nohighlight" id="equation-45006733-688c-40a9-ba9c-645e86d05bf7">
+<span class="eqno">(48)<a class="headerlink" href="#equation-45006733-688c-40a9-ba9c-645e86d05bf7" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\frac{\partial u}{\partial t} = -u\cdot\nabla u + \nu\nabla^2 u. 
+\end{equation}\]</div>
+<p>When discretized with simple finite difference, the tendency of <span class="math notranslate nohighlight">\(u\)</span> at a particular location be written as functions of adjacent variables</p>
+<div class="amsmath math notranslate nohighlight" id="equation-44f1b879-736c-49f1-9137-2b1b8f82ba06">
+<span class="eqno">(49)<a class="headerlink" href="#equation-44f1b879-736c-49f1-9137-2b1b8f82ba06" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\frac{d u_k}{d t} = -\frac{1}{2\Delta x}u_k (u_{k+1} - u_{k-1}) + \frac{\nu}{\Delta x^2} (u_{k+1} - 2u_k + u_{k-1}), 
+\end{equation}\]</div>
+<p>which makes it an ideal target for learning local maps with machine-learning algorithms.</p>
+<div class="section" id="generate-training-data-from-burger-s-equation">
+<h3>Generate training data from Burger’s equation<a class="headerlink" href="#generate-training-data-from-burger-s-equation" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Define parameters of the Burger&#39;s equation</span>
+<span class="n">nu</span> <span class="o">=</span> <span class="mf">0.3</span>
+<span class="n">dx</span> <span class="o">=</span> <span class="mi">1</span>
+<span class="n">K</span> <span class="o">=</span> <span class="mi">100</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.01</span>
+<span class="n">T</span> <span class="o">=</span> <span class="mi">100</span>
+<span class="n">si</span> <span class="o">=</span> <span class="n">dt</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+
+<span class="c1"># Use two sets of initial-value simulations for training</span>
+<span class="n">W1</span> <span class="o">=</span> <span class="n">Burgers</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">)</span>
+<span class="n">init_cond1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(((</span><span class="n">x</span> <span class="o">-</span> <span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="o">/</span> <span class="n">K</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">W1</span><span class="o">.</span><span class="n">set_init</span><span class="p">(</span><span class="n">init_cond1</span><span class="p">)</span>
+
+<span class="n">init_cond2</span> <span class="o">=</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="o">/</span> <span class="n">K</span> <span class="o">*</span> <span class="mi">15</span><span class="p">))</span>
+<span class="n">W2</span> <span class="o">=</span> <span class="n">Burgers</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">)</span>
+<span class="n">W2</span><span class="o">.</span><span class="n">set_init</span><span class="p">(</span><span class="n">init_cond2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">init_cond1</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;initial condition 1&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">init_cond2</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;initial condition 2&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;k&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;u&quot;</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<script type="application/javascript">/* Put everything inside the global mpl namespace */
+/* global mpl */
+window.mpl = {};
+
+mpl.get_websocket_type = function () {
+    if (typeof WebSocket !== 'undefined') {
+        return WebSocket;
+    } else if (typeof MozWebSocket !== 'undefined') {
+        return MozWebSocket;
+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
+mpl.figure = function (figure_id, websocket, ondownload, parent_element) {
+    this.id = figure_id;
+
+    this.ws = websocket;
+
+    this.supports_binary = this.ws.binaryType !== undefined;
+
+    if (!this.supports_binary) {
+        var warnings = document.getElementById('mpl-warnings');
+        if (warnings) {
+            warnings.style.display = 'block';
+            warnings.textContent =
+                'This browser does not support binary websocket messages. ' +
+                'Performance may be slow.';
+        }
+    }
+
+    this.imageObj = new Image();
+
+    this.context = undefined;
+    this.message = undefined;
+    this.canvas = undefined;
+    this.rubberband_canvas = undefined;
+    this.rubberband_context = undefined;
+    this.format_dropdown = undefined;
+
+    this.image_mode = 'full';
+
+    this.root = document.createElement('div');
+    this.root.setAttribute('style', 'display: inline-block');
+    this._root_extra_style(this.root);
+
+    parent_element.appendChild(this.root);
+
+    this._init_header(this);
+    this._init_canvas(this);
+    this._init_toolbar(this);
+
+    var fig = this;
+
+    this.waiting = false;
+
+    this.ws.onopen = function () {
+        fig.send_message('supports_binary', { value: fig.supports_binary });
+        fig.send_message('send_image_mode', {});
+        if (fig.ratio !== 1) {
+            fig.send_message('set_device_pixel_ratio', {
+                device_pixel_ratio: fig.ratio,
+            });
+        }
+        fig.send_message('refresh', {});
+    };
+
+    this.imageObj.onload = function () {
+        if (fig.image_mode === 'full') {
+            // Full images could contain transparency (where diff images
+            // almost always do), so we need to clear the canvas so that
+            // there is no ghosting.
+            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
+        }
+        fig.context.drawImage(fig.imageObj, 0, 0);
+    };
+
+    this.imageObj.onunload = function () {
+        fig.ws.close();
+    };
+
+    this.ws.onmessage = this._make_on_message_function(this);
+
+    this.ondownload = ondownload;
+};
+
+mpl.figure.prototype._init_header = function () {
+    var titlebar = document.createElement('div');
+    titlebar.classList =
+        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';
+    var titletext = document.createElement('div');
+    titletext.classList = 'ui-dialog-title';
+    titletext.setAttribute(
+        'style',
+        'width: 100%; text-align: center; padding: 3px;'
+    );
+    titlebar.appendChild(titletext);
+    this.root.appendChild(titlebar);
+    this.header = titletext;
+};
+
+mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._root_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._init_canvas = function () {
+    var fig = this;
+
+    var canvas_div = (this.canvas_div = document.createElement('div'));
+    canvas_div.setAttribute('tabindex', '0');
+    canvas_div.setAttribute(
+        'style',
+        'border: 1px solid #ddd;' +
+            'box-sizing: content-box;' +
+            'clear: both;' +
+            'min-height: 1px;' +
+            'min-width: 1px;' +
+            'outline: 0;' +
+            'overflow: hidden;' +
+            'position: relative;' +
+            'resize: both;' +
+            'z-index: 2;'
+    );
+
+    function on_keyboard_event_closure(name) {
+        return function (event) {
+            return fig.key_event(event, name);
+        };
+    }
+
+    canvas_div.addEventListener(
+        'keydown',
+        on_keyboard_event_closure('key_press')
+    );
+    canvas_div.addEventListener(
+        'keyup',
+        on_keyboard_event_closure('key_release')
+    );
+
+    this._canvas_extra_style(canvas_div);
+    this.root.appendChild(canvas_div);
+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='7d183b4e-f97a-48be-b82f-babafbd7fac8'></div></div></div>
+</div>
+</div>
+<div class="section" id="train-the-same-set-of-networks-with-the-burger-s-equation">
+<h3>Train the same set of networks with the Burger’s equation<a class="headerlink" href="#train-the-same-set-of-networks-with-the-burger-s-equation" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Import a wrapper function to perform all of the training above</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">run_analysis</span>
+
+<span class="p">(</span>
+    <span class="n">net_ANN_global_burger</span><span class="p">,</span>
+    <span class="n">net_CNN_global_burger</span><span class="p">,</span>
+    <span class="n">net_ANN_local_burger</span><span class="p">,</span>
+    <span class="n">net_poly_burger</span><span class="p">,</span>
+    <span class="n">X_true_burger</span><span class="p">,</span>
+    <span class="n">dX_true_burger</span><span class="p">,</span>
+<span class="p">)</span> <span class="o">=</span> <span class="n">run_analysis</span><span class="p">([</span><span class="n">W1</span><span class="p">,</span> <span class="n">W2</span><span class="p">],</span> <span class="n">si</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n_epochs</span><span class="o">=</span><span class="n">n_epochs</span><span class="p">,</span> <span class="n">Poly_init</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>training global CNN
+
+epoch = 1000, traing loss = 1.443418e-08, test loss = 1.438685e-08
+
+training global ANN
+
+epoch = 1000, traing loss = 4.980171e-09, test loss = 5.035909e-09
+
+training local polynet
+
+epoch = 1000, traing loss = 6.165268e-12, test loss = 5.961172e-12
+
+training local ANN
+
+epoch = 1000, traing loss = 2.302375e-08, test loss = 2.148764e-08
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#### Test predictions with a different initial condition</span>
+<span class="n">gcm_ANN</span> <span class="o">=</span> <span class="n">GCM_discrete</span><span class="p">(</span><span class="n">net_ANN_global_burger</span><span class="p">)</span>
+<span class="n">gcm_CNN</span> <span class="o">=</span> <span class="n">GCM_discrete</span><span class="p">(</span><span class="n">net_CNN_global_burger</span><span class="p">)</span>
+<span class="n">gcm_local_ANN</span> <span class="o">=</span> <span class="n">GCM_local_discrete</span><span class="p">(</span><span class="n">net_ANN_local_burger</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
+<span class="n">gcm_local_poly</span> <span class="o">=</span> <span class="n">GCM_local_discrete</span><span class="p">(</span><span class="n">net_poly_burger</span><span class="p">,</span> <span class="n">get_poly_features</span><span class="p">)</span>
+
+<span class="c1"># Define a different initial condition</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+<span class="n">t_plot</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">T</span> <span class="o">/</span> <span class="n">si</span><span class="p">)</span>
+<span class="n">init_cond</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="mf">0.1</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="o">/</span> <span class="n">K</span> <span class="o">*</span> <span class="mi">20</span><span class="p">))</span>
+
+<span class="c1"># Calculate ground truth using Burger&#39;s equation</span>
+<span class="n">W1</span><span class="o">.</span><span class="n">set_init</span><span class="p">(</span><span class="n">init_cond</span><span class="p">)</span>
+<span class="n">X_simulation_truth</span> <span class="o">=</span> <span class="n">W1</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">si</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
+
+<span class="c1"># Calculate initial-value simulations with the trained networks</span>
+<span class="n">X_simulation_global_ANN</span> <span class="o">=</span> <span class="n">gcm_ANN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_global_CNN</span> <span class="o">=</span> <span class="n">gcm_CNN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_local_ANN</span> <span class="o">=</span> <span class="n">gcm_local_ANN</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+<span class="n">X_simulation_poly</span> <span class="o">=</span> <span class="n">gcm_local_poly</span><span class="p">(</span><span class="n">init_cond</span><span class="p">,</span> <span class="n">t_plot</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">ioff</span><span class="p">()</span>
+<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">animate_Burgers</span>
+
+<span class="n">animate_Burgers</span><span class="p">(</span>
+    <span class="n">X_simulation_truth</span><span class="p">,</span>
+    <span class="n">X_simulation_global_ANN</span><span class="p">,</span>
+    <span class="n">X_simulation_global_CNN</span><span class="p">,</span>
+    <span class="n">X_simulation_local_ANN</span><span class="p">,</span>
+    <span class="n">X_simulation_poly</span><span class="p">,</span>
+    <span class="n">plot_path</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="o">%</span><span class="k">matplotlib</span> notebook
+</pre></div>
+</div>
+</div>
+</div>
+<p><img alt="SegmentLocal" src="../_images/Burgers_simulation.gif" /></p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">weights_true</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
+    <span class="p">[</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="n">nu</span> <span class="o">/</span> <span class="n">dx</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span>
+        <span class="o">-</span><span class="mi">2</span> <span class="o">*</span> <span class="n">nu</span> <span class="o">/</span> <span class="n">dx</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span>
+        <span class="n">nu</span> <span class="o">/</span> <span class="n">dx</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="o">-</span><span class="mi">1</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">/</span> <span class="n">dx</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">1</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">/</span> <span class="n">dx</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+        <span class="mi">0</span><span class="p">,</span>
+    <span class="p">]</span>
+<span class="p">)</span>
+<span class="n">filter_start</span> <span class="o">=</span> <span class="o">-</span><span class="mi">2</span>
+<span class="n">filter_end</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="c1"># this includes k+2, k+1, k, k-1, k-2, note that roll by +2 means looking at location k-2</span>
+<span class="n">filter_loc</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">filter_start</span><span class="p">,</span> <span class="n">filter_end</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">plot_parameters</span><span class="p">(</span><span class="n">net_poly_burger</span><span class="p">,</span> <span class="n">si</span><span class="p">,</span> <span class="n">filter_loc</span><span class="p">,</span> <span class="s2">&quot;u&quot;</span><span class="p">,</span> <span class="n">weights_true</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<script type="application/javascript">/* Put everything inside the global mpl namespace */
+/* global mpl */
+window.mpl = {};
+
+mpl.get_websocket_type = function () {
+    if (typeof WebSocket !== 'undefined') {
+        return WebSocket;
+    } else if (typeof MozWebSocket !== 'undefined') {
+        return MozWebSocket;
+    } else {
+        alert(
+            'Your browser does not have WebSocket support. ' +
+                'Please try Chrome, Safari or Firefox ≥ 6. ' +
+                'Firefox 4 and 5 are also supported but you ' +
+                'have to enable WebSockets in about:config.'
+        );
+    }
+};
+
+mpl.figure = function (figure_id, websocket, ondownload, parent_element) {
+    this.id = figure_id;
+
+    this.ws = websocket;
+
+    this.supports_binary = this.ws.binaryType !== undefined;
+
+    if (!this.supports_binary) {
+        var warnings = document.getElementById('mpl-warnings');
+        if (warnings) {
+            warnings.style.display = 'block';
+            warnings.textContent =
+                'This browser does not support binary websocket messages. ' +
+                'Performance may be slow.';
+        }
+    }
+
+    this.imageObj = new Image();
+
+    this.context = undefined;
+    this.message = undefined;
+    this.canvas = undefined;
+    this.rubberband_canvas = undefined;
+    this.rubberband_context = undefined;
+    this.format_dropdown = undefined;
+
+    this.image_mode = 'full';
+
+    this.root = document.createElement('div');
+    this.root.setAttribute('style', 'display: inline-block');
+    this._root_extra_style(this.root);
+
+    parent_element.appendChild(this.root);
+
+    this._init_header(this);
+    this._init_canvas(this);
+    this._init_toolbar(this);
+
+    var fig = this;
+
+    this.waiting = false;
+
+    this.ws.onopen = function () {
+        fig.send_message('supports_binary', { value: fig.supports_binary });
+        fig.send_message('send_image_mode', {});
+        if (fig.ratio !== 1) {
+            fig.send_message('set_device_pixel_ratio', {
+                device_pixel_ratio: fig.ratio,
+            });
+        }
+        fig.send_message('refresh', {});
+    };
+
+    this.imageObj.onload = function () {
+        if (fig.image_mode === 'full') {
+            // Full images could contain transparency (where diff images
+            // almost always do), so we need to clear the canvas so that
+            // there is no ghosting.
+            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);
+        }
+        fig.context.drawImage(fig.imageObj, 0, 0);
+    };
+
+    this.imageObj.onunload = function () {
+        fig.ws.close();
+    };
+
+    this.ws.onmessage = this._make_on_message_function(this);
+
+    this.ondownload = ondownload;
+};
+
+mpl.figure.prototype._init_header = function () {
+    var titlebar = document.createElement('div');
+    titlebar.classList =
+        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';
+    var titletext = document.createElement('div');
+    titletext.classList = 'ui-dialog-title';
+    titletext.setAttribute(
+        'style',
+        'width: 100%; text-align: center; padding: 3px;'
+    );
+    titlebar.appendChild(titletext);
+    this.root.appendChild(titlebar);
+    this.header = titletext;
+};
+
+mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._root_extra_style = function (_canvas_div) {};
+
+mpl.figure.prototype._init_canvas = function () {
+    var fig = this;
+
+    var canvas_div = (this.canvas_div = document.createElement('div'));
+    canvas_div.setAttribute('tabindex', '0');
+    canvas_div.setAttribute(
+        'style',
+        'border: 1px solid #ddd;' +
+            'box-sizing: content-box;' +
+            'clear: both;' +
+            'min-height: 1px;' +
+            'min-width: 1px;' +
+            'outline: 0;' +
+            'overflow: hidden;' +
+            'position: relative;' +
+            'resize: both;' +
+            'z-index: 2;'
+    );
+
+    function on_keyboard_event_closure(name) {
+        return function (event) {
+            return fig.key_event(event, name);
+        };
+    }
+
+    canvas_div.addEventListener(
+        'keydown',
+        on_keyboard_event_closure('key_press')
+    );
+    canvas_div.addEventListener(
+        'keyup',
+        on_keyboard_event_closure('key_release')
+    );
+
+    this._canvas_extra_style(canvas_div);
+    this.root.appendChild(canvas_div);
+
+    var canvas = (this.canvas = document.createElement('canvas'));
+    canvas.classList.add('mpl-canvas');
+    canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'pointer-events: none;' +
+            'position: relative;' +
+            'z-index: 0;'
+    );
+
+    this.context = canvas.getContext('2d');
+
+    var backingStore =
+        this.context.backingStorePixelRatio ||
+        this.context.webkitBackingStorePixelRatio ||
+        this.context.mozBackingStorePixelRatio ||
+        this.context.msBackingStorePixelRatio ||
+        this.context.oBackingStorePixelRatio ||
+        this.context.backingStorePixelRatio ||
+        1;
+
+    this.ratio = (window.devicePixelRatio || 1) / backingStore;
+
+    var rubberband_canvas = (this.rubberband_canvas = document.createElement(
+        'canvas'
+    ));
+    rubberband_canvas.setAttribute(
+        'style',
+        'box-sizing: content-box;' +
+            'left: 0;' +
+            'pointer-events: none;' +
+            'position: absolute;' +
+            'top: 0;' +
+            'z-index: 1;'
+    );
+
+    // Apply a ponyfill if ResizeObserver is not implemented by browser.
+    if (this.ResizeObserver === undefined) {
+        if (window.ResizeObserver !== undefined) {
+            this.ResizeObserver = window.ResizeObserver;
+        } else {
+            var obs = _JSXTOOLS_RESIZE_OBSERVER({});
+            this.ResizeObserver = obs.ResizeObserver;
+        }
+    }
+
+    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {
+        var nentries = entries.length;
+        for (var i = 0; i < nentries; i++) {
+            var entry = entries[i];
+            var width, height;
+            if (entry.contentBoxSize) {
+                if (entry.contentBoxSize instanceof Array) {
+                    // Chrome 84 implements new version of spec.
+                    width = entry.contentBoxSize[0].inlineSize;
+                    height = entry.contentBoxSize[0].blockSize;
+                } else {
+                    // Firefox implements old version of spec.
+                    width = entry.contentBoxSize.inlineSize;
+                    height = entry.contentBoxSize.blockSize;
+                }
+            } else {
+                // Chrome <84 implements even older version of spec.
+                width = entry.contentRect.width;
+                height = entry.contentRect.height;
+            }
+
+            // Keep the size of the canvas and rubber band canvas in sync with
+            // the canvas container.
+            if (entry.devicePixelContentBoxSize) {
+                // Chrome 84 implements new version of spec.
+                canvas.setAttribute(
+                    'width',
+                    entry.devicePixelContentBoxSize[0].inlineSize
+                );
+                canvas.setAttribute(
+                    'height',
+                    entry.devicePixelContentBoxSize[0].blockSize
+                );
+            } else {
+                canvas.setAttribute('width', width * fig.ratio);
+                canvas.setAttribute('height', height * fig.ratio);
+            }
+            /* This rescales the canvas back to display pixels, so that it
+             * appears correct on HiDPI screens. */
+            canvas.style.width = width + 'px';
+            canvas.style.height = height + 'px';
+
+            rubberband_canvas.setAttribute('width', width);
+            rubberband_canvas.setAttribute('height', height);
+
+            // And update the size in Python. We ignore the initial 0/0 size
+            // that occurs as the element is placed into the DOM, which should
+            // otherwise not happen due to the minimum size styling.
+            if (fig.ws.readyState == 1 && width != 0 && height != 0) {
+                fig.request_resize(width, height);
+            }
+        }
+    });
+    this.resizeObserverInstance.observe(canvas_div);
+
+    function on_mouse_event_closure(name) {
+        /* User Agent sniffing is bad, but WebKit is busted:
+         * https://bugs.webkit.org/show_bug.cgi?id=144526
+         * https://bugs.webkit.org/show_bug.cgi?id=181818
+         * The worst that happens here is that they get an extra browser
+         * selection when dragging, if this check fails to catch them.
+         */
+        var UA = navigator.userAgent;
+        var isWebKit = /AppleWebKit/.test(UA) && !/Chrome/.test(UA);
+        if(isWebKit) {
+            return function (event) {
+                /* This prevents the web browser from automatically changing to
+                 * the text insertion cursor when the button is pressed. We
+                 * want to control all of the cursor setting manually through
+                 * the 'cursor' event from matplotlib */
+                event.preventDefault()
+                return fig.mouse_event(event, name);
+            };
+        } else {
+            return function (event) {
+                return fig.mouse_event(event, name);
+            };
+        }
+    }
+
+    canvas_div.addEventListener(
+        'mousedown',
+        on_mouse_event_closure('button_press')
+    );
+    canvas_div.addEventListener(
+        'mouseup',
+        on_mouse_event_closure('button_release')
+    );
+    canvas_div.addEventListener(
+        'dblclick',
+        on_mouse_event_closure('dblclick')
+    );
+    // Throttle sequential mouse events to 1 every 20ms.
+    canvas_div.addEventListener(
+        'mousemove',
+        on_mouse_event_closure('motion_notify')
+    );
+
+    canvas_div.addEventListener(
+        'mouseenter',
+        on_mouse_event_closure('figure_enter')
+    );
+    canvas_div.addEventListener(
+        'mouseleave',
+        on_mouse_event_closure('figure_leave')
+    );
+
+    canvas_div.addEventListener('wheel', function (event) {
+        if (event.deltaY < 0) {
+            event.step = 1;
+        } else {
+            event.step = -1;
+        }
+        on_mouse_event_closure('scroll')(event);
+    });
+
+    canvas_div.appendChild(canvas);
+    canvas_div.appendChild(rubberband_canvas);
+
+    this.rubberband_context = rubberband_canvas.getContext('2d');
+    this.rubberband_context.strokeStyle = '#000000';
+
+    this._resize_canvas = function (width, height, forward) {
+        if (forward) {
+            canvas_div.style.width = width + 'px';
+            canvas_div.style.height = height + 'px';
+        }
+    };
+
+    // Disable right mouse context menu.
+    canvas_div.addEventListener('contextmenu', function (_e) {
+        event.preventDefault();
+        return false;
+    });
+
+    function set_focus() {
+        canvas.focus();
+        canvas_div.focus();
+    }
+
+    window.setTimeout(set_focus, 100);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'mpl-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'mpl-button-group';
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'mpl-button-group';
+            continue;
+        }
+
+        var button = (fig.buttons[name] = document.createElement('button'));
+        button.classList = 'mpl-widget';
+        button.setAttribute('role', 'button');
+        button.setAttribute('aria-disabled', 'false');
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+
+        var icon_img = document.createElement('img');
+        icon_img.src = '_images/' + image + '.png';
+        icon_img.srcset = '_images/' + image + '_large.png 2x';
+        icon_img.alt = tooltip;
+        button.appendChild(icon_img);
+
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    var fmt_picker = document.createElement('select');
+    fmt_picker.classList = 'mpl-widget';
+    toolbar.appendChild(fmt_picker);
+    this.format_dropdown = fmt_picker;
+
+    for (var ind in mpl.extensions) {
+        var fmt = mpl.extensions[ind];
+        var option = document.createElement('option');
+        option.selected = fmt === mpl.default_extension;
+        option.innerHTML = fmt;
+        fmt_picker.appendChild(option);
+    }
+
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+};
+
+mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {
+    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,
+    // which will in turn request a refresh of the image.
+    this.send_message('resize', { width: x_pixels, height: y_pixels });
+};
+
+mpl.figure.prototype.send_message = function (type, properties) {
+    properties['type'] = type;
+    properties['figure_id'] = this.id;
+    this.ws.send(JSON.stringify(properties));
+};
+
+mpl.figure.prototype.send_draw_message = function () {
+    if (!this.waiting) {
+        this.waiting = true;
+        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    var format_dropdown = fig.format_dropdown;
+    var format = format_dropdown.options[format_dropdown.selectedIndex].value;
+    fig.ondownload(fig, format);
+};
+
+mpl.figure.prototype.handle_resize = function (fig, msg) {
+    var size = msg['size'];
+    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {
+        fig._resize_canvas(size[0], size[1], msg['forward']);
+        fig.send_message('refresh', {});
+    }
+};
+
+mpl.figure.prototype.handle_rubberband = function (fig, msg) {
+    var x0 = msg['x0'] / fig.ratio;
+    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;
+    var x1 = msg['x1'] / fig.ratio;
+    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;
+    x0 = Math.floor(x0) + 0.5;
+    y0 = Math.floor(y0) + 0.5;
+    x1 = Math.floor(x1) + 0.5;
+    y1 = Math.floor(y1) + 0.5;
+    var min_x = Math.min(x0, x1);
+    var min_y = Math.min(y0, y1);
+    var width = Math.abs(x1 - x0);
+    var height = Math.abs(y1 - y0);
+
+    fig.rubberband_context.clearRect(
+        0,
+        0,
+        fig.canvas.width / fig.ratio,
+        fig.canvas.height / fig.ratio
+    );
+
+    fig.rubberband_context.strokeRect(min_x, min_y, width, height);
+};
+
+mpl.figure.prototype.handle_figure_label = function (fig, msg) {
+    // Updates the figure title.
+    fig.header.textContent = msg['label'];
+};
+
+mpl.figure.prototype.handle_cursor = function (fig, msg) {
+    fig.canvas_div.style.cursor = msg['cursor'];
+};
+
+mpl.figure.prototype.handle_message = function (fig, msg) {
+    fig.message.textContent = msg['message'];
+};
+
+mpl.figure.prototype.handle_draw = function (fig, _msg) {
+    // Request the server to send over a new figure.
+    fig.send_draw_message();
+};
+
+mpl.figure.prototype.handle_image_mode = function (fig, msg) {
+    fig.image_mode = msg['mode'];
+};
+
+mpl.figure.prototype.handle_history_buttons = function (fig, msg) {
+    for (var key in msg) {
+        if (!(key in fig.buttons)) {
+            continue;
+        }
+        fig.buttons[key].disabled = !msg[key];
+        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);
+    }
+};
+
+mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {
+    if (msg['mode'] === 'PAN') {
+        fig.buttons['Pan'].classList.add('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    } else if (msg['mode'] === 'ZOOM') {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.add('active');
+    } else {
+        fig.buttons['Pan'].classList.remove('active');
+        fig.buttons['Zoom'].classList.remove('active');
+    }
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Called whenever the canvas gets updated.
+    this.send_message('ack', {});
+};
+
+// A function to construct a web socket function for onmessage handling.
+// Called in the figure constructor.
+mpl.figure.prototype._make_on_message_function = function (fig) {
+    return function socket_on_message(evt) {
+        if (evt.data instanceof Blob) {
+            var img = evt.data;
+            if (img.type !== 'image/png') {
+                /* FIXME: We get "Resource interpreted as Image but
+                 * transferred with MIME type text/plain:" errors on
+                 * Chrome.  But how to set the MIME type?  It doesn't seem
+                 * to be part of the websocket stream */
+                img.type = 'image/png';
+            }
+
+            /* Free the memory for the previous frames */
+            if (fig.imageObj.src) {
+                (window.URL || window.webkitURL).revokeObjectURL(
+                    fig.imageObj.src
+                );
+            }
+
+            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(
+                img
+            );
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        } else if (
+            typeof evt.data === 'string' &&
+            evt.data.slice(0, 21) === 'data:image/png;base64'
+        ) {
+            fig.imageObj.src = evt.data;
+            fig.updated_canvas_event();
+            fig.waiting = false;
+            return;
+        }
+
+        var msg = JSON.parse(evt.data);
+        var msg_type = msg['type'];
+
+        // Call the  "handle_{type}" callback, which takes
+        // the figure and JSON message as its only arguments.
+        try {
+            var callback = fig['handle_' + msg_type];
+        } catch (e) {
+            console.log(
+                "No handler for the '" + msg_type + "' message type: ",
+                msg
+            );
+            return;
+        }
+
+        if (callback) {
+            try {
+                // console.log("Handling '" + msg_type + "' message: ", msg);
+                callback(fig, msg);
+            } catch (e) {
+                console.log(
+                    "Exception inside the 'handler_" + msg_type + "' callback:",
+                    e,
+                    e.stack,
+                    msg
+                );
+            }
+        }
+    };
+};
+
+function getModifiers(event) {
+    var mods = [];
+    if (event.ctrlKey) {
+        mods.push('ctrl');
+    }
+    if (event.altKey) {
+        mods.push('alt');
+    }
+    if (event.shiftKey) {
+        mods.push('shift');
+    }
+    if (event.metaKey) {
+        mods.push('meta');
+    }
+    return mods;
+}
+
+/*
+ * return a copy of an object with only non-object keys
+ * we need this to avoid circular references
+ * https://stackoverflow.com/a/24161582/3208463
+ */
+function simpleKeys(original) {
+    return Object.keys(original).reduce(function (obj, key) {
+        if (typeof original[key] !== 'object') {
+            obj[key] = original[key];
+        }
+        return obj;
+    }, {});
+}
+
+mpl.figure.prototype.mouse_event = function (event, name) {
+    if (name === 'button_press') {
+        this.canvas.focus();
+        this.canvas_div.focus();
+    }
+
+    // from https://stackoverflow.com/q/1114465
+    var boundingRect = this.canvas.getBoundingClientRect();
+    var x = (event.clientX - boundingRect.left) * this.ratio;
+    var y = (event.clientY - boundingRect.top) * this.ratio;
+
+    this.send_message(name, {
+        x: x,
+        y: y,
+        button: event.button,
+        step: event.step,
+        modifiers: getModifiers(event),
+        guiEvent: simpleKeys(event),
+    });
+
+    return false;
+};
+
+mpl.figure.prototype._key_event_extra = function (_event, _name) {
+    // Handle any extra behaviour associated with a key event
+};
+
+mpl.figure.prototype.key_event = function (event, name) {
+    // Prevent repeat events
+    if (name === 'key_press') {
+        if (event.key === this._key) {
+            return;
+        } else {
+            this._key = event.key;
+        }
+    }
+    if (name === 'key_release') {
+        this._key = null;
+    }
+
+    var value = '';
+    if (event.ctrlKey && event.key !== 'Control') {
+        value += 'ctrl+';
+    }
+    else if (event.altKey && event.key !== 'Alt') {
+        value += 'alt+';
+    }
+    else if (event.shiftKey && event.key !== 'Shift') {
+        value += 'shift+';
+    }
+
+    value += 'k' + event.key;
+
+    this._key_event_extra(event, name);
+
+    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });
+    return false;
+};
+
+mpl.figure.prototype.toolbar_button_onclick = function (name) {
+    if (name === 'download') {
+        this.handle_save(this, null);
+    } else {
+        this.send_message('toolbar_button', { name: name });
+    }
+};
+
+mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {
+    this.message.textContent = tooltip;
+};
+
+///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////
+// prettier-ignore
+var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError("Constructor requires 'new' operator");i.set(this,e)}function h(){throw new TypeError("Function is not a constructor")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line
+mpl.toolbar_items = [["Home", "Reset original view", "fa fa-home", "home"], ["Back", "Back to previous view", "fa fa-arrow-left", "back"], ["Forward", "Forward to next view", "fa fa-arrow-right", "forward"], ["", "", "", ""], ["Pan", "Left button pans, Right button zooms\nx/y fixes axis, CTRL fixes aspect", "fa fa-arrows", "pan"], ["Zoom", "Zoom to rectangle\nx/y fixes axis", "fa fa-square-o", "zoom"], ["", "", "", ""], ["Download", "Download plot", "fa fa-floppy-o", "download"]];
+
+mpl.extensions = ["eps", "jpeg", "pgf", "pdf", "png", "ps", "raw", "svg", "tif", "webp"];
+
+mpl.default_extension = "png";/* global mpl */
+
+var comm_websocket_adapter = function (comm) {
+    // Create a "websocket"-like object which calls the given IPython comm
+    // object with the appropriate methods. Currently this is a non binary
+    // socket, so there is still some room for performance tuning.
+    var ws = {};
+
+    ws.binaryType = comm.kernel.ws.binaryType;
+    ws.readyState = comm.kernel.ws.readyState;
+    function updateReadyState(_event) {
+        if (comm.kernel.ws) {
+            ws.readyState = comm.kernel.ws.readyState;
+        } else {
+            ws.readyState = 3; // Closed state.
+        }
+    }
+    comm.kernel.ws.addEventListener('open', updateReadyState);
+    comm.kernel.ws.addEventListener('close', updateReadyState);
+    comm.kernel.ws.addEventListener('error', updateReadyState);
+
+    ws.close = function () {
+        comm.close();
+    };
+    ws.send = function (m) {
+        //console.log('sending', m);
+        comm.send(m);
+    };
+    // Register the callback with on_msg.
+    comm.on_msg(function (msg) {
+        //console.log('receiving', msg['content']['data'], msg);
+        var data = msg['content']['data'];
+        if (data['blob'] !== undefined) {
+            data = {
+                data: new Blob(msg['buffers'], { type: data['blob'] }),
+            };
+        }
+        // Pass the mpl event to the overridden (by mpl) onmessage function.
+        ws.onmessage(data);
+    });
+    return ws;
+};
+
+mpl.mpl_figure_comm = function (comm, msg) {
+    // This is the function which gets called when the mpl process
+    // starts-up an IPython Comm through the "matplotlib" channel.
+
+    var id = msg.content.data.id;
+    // Get hold of the div created by the display call when the Comm
+    // socket was opened in Python.
+    var element = document.getElementById(id);
+    var ws_proxy = comm_websocket_adapter(comm);
+
+    function ondownload(figure, _format) {
+        window.open(figure.canvas.toDataURL());
+    }
+
+    var fig = new mpl.figure(id, ws_proxy, ondownload, element);
+
+    // Call onopen now - mpl needs it, as it is assuming we've passed it a real
+    // web socket which is closed, not our websocket->open comm proxy.
+    ws_proxy.onopen();
+
+    fig.parent_element = element;
+    fig.cell_info = mpl.find_output_cell("<div id='" + id + "'></div>");
+    if (!fig.cell_info) {
+        console.error('Failed to find cell for figure', id, fig);
+        return;
+    }
+    fig.cell_info[0].output_area.element.on(
+        'cleared',
+        { fig: fig },
+        fig._remove_fig_handler
+    );
+};
+
+mpl.figure.prototype.handle_close = function (fig, msg) {
+    var width = fig.canvas.width / fig.ratio;
+    fig.cell_info[0].output_area.element.off(
+        'cleared',
+        fig._remove_fig_handler
+    );
+    fig.resizeObserverInstance.unobserve(fig.canvas_div);
+
+    // Update the output cell to use the data from the current canvas.
+    fig.push_to_output();
+    var dataURL = fig.canvas.toDataURL();
+    // Re-enable the keyboard manager in IPython - without this line, in FF,
+    // the notebook keyboard shortcuts fail.
+    IPython.keyboard_manager.enable();
+    fig.parent_element.innerHTML =
+        '<img src="' + dataURL + '" width="' + width + '">';
+    fig.close_ws(fig, msg);
+};
+
+mpl.figure.prototype.close_ws = function (fig, msg) {
+    fig.send_message('closing', msg);
+    // fig.ws.close()
+};
+
+mpl.figure.prototype.push_to_output = function (_remove_interactive) {
+    // Turn the data on the canvas into data in the output cell.
+    var width = this.canvas.width / this.ratio;
+    var dataURL = this.canvas.toDataURL();
+    this.cell_info[1]['text/html'] =
+        '<img src="' + dataURL + '" width="' + width + '">';
+};
+
+mpl.figure.prototype.updated_canvas_event = function () {
+    // Tell IPython that the notebook contents must change.
+    IPython.notebook.set_dirty(true);
+    this.send_message('ack', {});
+    var fig = this;
+    // Wait a second, then push the new image to the DOM so
+    // that it is saved nicely (might be nice to debounce this).
+    setTimeout(function () {
+        fig.push_to_output();
+    }, 1000);
+};
+
+mpl.figure.prototype._init_toolbar = function () {
+    var fig = this;
+
+    var toolbar = document.createElement('div');
+    toolbar.classList = 'btn-toolbar';
+    this.root.appendChild(toolbar);
+
+    function on_click_closure(name) {
+        return function (_event) {
+            return fig.toolbar_button_onclick(name);
+        };
+    }
+
+    function on_mouseover_closure(tooltip) {
+        return function (event) {
+            if (!event.currentTarget.disabled) {
+                return fig.toolbar_button_onmouseover(tooltip);
+            }
+        };
+    }
+
+    fig.buttons = {};
+    var buttonGroup = document.createElement('div');
+    buttonGroup.classList = 'btn-group';
+    var button;
+    for (var toolbar_ind in mpl.toolbar_items) {
+        var name = mpl.toolbar_items[toolbar_ind][0];
+        var tooltip = mpl.toolbar_items[toolbar_ind][1];
+        var image = mpl.toolbar_items[toolbar_ind][2];
+        var method_name = mpl.toolbar_items[toolbar_ind][3];
+
+        if (!name) {
+            /* Instead of a spacer, we start a new button group. */
+            if (buttonGroup.hasChildNodes()) {
+                toolbar.appendChild(buttonGroup);
+            }
+            buttonGroup = document.createElement('div');
+            buttonGroup.classList = 'btn-group';
+            continue;
+        }
+
+        button = fig.buttons[name] = document.createElement('button');
+        button.classList = 'btn btn-default';
+        button.href = '#';
+        button.title = name;
+        button.innerHTML = '<i class="fa ' + image + ' fa-lg"></i>';
+        button.addEventListener('click', on_click_closure(method_name));
+        button.addEventListener('mouseover', on_mouseover_closure(tooltip));
+        buttonGroup.appendChild(button);
+    }
+
+    if (buttonGroup.hasChildNodes()) {
+        toolbar.appendChild(buttonGroup);
+    }
+
+    // Add the status bar.
+    var status_bar = document.createElement('span');
+    status_bar.classList = 'mpl-message pull-right';
+    toolbar.appendChild(status_bar);
+    this.message = status_bar;
+
+    // Add the close button to the window.
+    var buttongrp = document.createElement('div');
+    buttongrp.classList = 'btn-group inline pull-right';
+    button = document.createElement('button');
+    button.classList = 'btn btn-mini btn-primary';
+    button.href = '#';
+    button.title = 'Stop Interaction';
+    button.innerHTML = '<i class="fa fa-power-off icon-remove icon-large"></i>';
+    button.addEventListener('click', function (_evt) {
+        fig.handle_close(fig, {});
+    });
+    button.addEventListener(
+        'mouseover',
+        on_mouseover_closure('Stop Interaction')
+    );
+    buttongrp.appendChild(button);
+    var titlebar = this.root.querySelector('.ui-dialog-titlebar');
+    titlebar.insertBefore(buttongrp, titlebar.firstChild);
+};
+
+mpl.figure.prototype._remove_fig_handler = function (event) {
+    var fig = event.data.fig;
+    if (event.target !== this) {
+        // Ignore bubbled events from children.
+        return;
+    }
+    fig.close_ws(fig, {});
+};
+
+mpl.figure.prototype._root_extra_style = function (el) {
+    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.
+};
+
+mpl.figure.prototype._canvas_extra_style = function (el) {
+    // this is important to make the div 'focusable
+    el.setAttribute('tabindex', 0);
+    // reach out to IPython and tell the keyboard manager to turn it's self
+    // off when our div gets focus
+
+    // location in version 3
+    if (IPython.notebook.keyboard_manager) {
+        IPython.notebook.keyboard_manager.register_events(el);
+    } else {
+        // location in version 2
+        IPython.keyboard_manager.register_events(el);
+    }
+};
+
+mpl.figure.prototype._key_event_extra = function (event, _name) {
+    // Check for shift+enter
+    if (event.shiftKey && event.which === 13) {
+        this.canvas_div.blur();
+        // select the cell after this one
+        var index = IPython.notebook.find_cell_index(this.cell_info[0]);
+        IPython.notebook.select(index + 1);
+    }
+};
+
+mpl.figure.prototype.handle_save = function (fig, _msg) {
+    fig.ondownload(fig, null);
+};
+
+mpl.find_output_cell = function (html_output) {
+    // Return the cell and output element which can be found *uniquely* in the notebook.
+    // Note - this is a bit hacky, but it is done because the "notebook_saving.Notebook"
+    // IPython event is triggered only after the cells have been serialised, which for
+    // our purposes (turning an active figure into a static one), is too late.
+    var cells = IPython.notebook.get_cells();
+    var ncells = cells.length;
+    for (var i = 0; i < ncells; i++) {
+        var cell = cells[i];
+        if (cell.cell_type === 'code') {
+            for (var j = 0; j < cell.output_area.outputs.length; j++) {
+                var data = cell.output_area.outputs[j];
+                if (data.data) {
+                    // IPython >= 3 moved mimebundle to data attribute of output
+                    data = data.data;
+                }
+                if (data['text/html'] === html_output) {
+                    return [cell, data, j];
+                }
+            }
+        }
+    }
+};
+
+// Register the function which deals with the matplotlib target/channel.
+// The kernel may be null if the page has been refreshed.
+if (IPython.notebook.kernel !== null) {
+    IPython.notebook.kernel.comm_manager.register_target(
+        'matplotlib',
+        mpl.mpl_figure_comm
+    );
+}
+</script><div class="output text_html"><div id='c45cbde0-0a29-4870-bb9a-85b97282859a'></div></div></div>
+</div>
+<p>The polynet appropriately recovers the terms in the Burger’s equation:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-9081e0cf-c58a-4170-a62e-beb74d78368f">
+<span class="eqno">(50)<a class="headerlink" href="#equation-9081e0cf-c58a-4170-a62e-beb74d78368f" title="Permalink to this equation">#</a></span>\[\begin{equation}
+\frac{d u_k}{d t} = 0.5 u_k (u_{k+1} - u_{k-1}) + 0.3 (u_{k+1} - 2u_k + u_{k-1}), 
+\end{equation}\]</div>
+</div>
+</div>
+</div>
+<div class="tex2jax_ignore mathjax_ignore section" id="conclusions">
+<h1>Conclusions<a class="headerlink" href="#conclusions" title="Permalink to this headline">#</a></h1>
+<ul class="simple">
+<li><p>Convolutional neural networks has better error performance during training, partially due to its added complexity</p></li>
+<li><p>Local maps is potentially more useful than global maps in the following aspects:</p>
+<ul>
+<li><p>It achieves better performance in initial-value simulations with smaller number of parameters, energy-conservation properties, and less training data.</p></li>
+<li><p>It is not constrained by training data geometry – can be easily generalized to larger simulation domains.</p></li>
+<li><p>It does not need access to global data</p></li>
+<li><p>Local maps are more explanable due to its simplified structure and parameters</p></li>
+</ul>
+</li>
+<li><p>Local polynomial stencils might be useful as an alternative functional class than off-the-shelf ML algorithms.</p></li>
+</ul>
+</div>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
+        codeMirrorConfig: {
+            theme: "abcdef",
+            mode: "python"
+        },
+        kernelOptions: {
+            name: "python3",
+            path: "./notebooks"
+        },
+        predefinedOutput: true
+    }
+    </script>
+    <script>kernelName = 'python3'</script>
+
+                </article>
+              
+
+              
+              
+                <footer class="bd-footer-article">
+                  
+<div class="footer-article-items footer-article__inner">
+  
+    <div class="footer-article-item"><!-- Previous / next buttons -->
+<div class="prev-next-area">
+    <a class="left-prev"
+       href="GP_lorenz96.html"
+       title="previous page">
+      <i class="fa-solid fa-angle-left"></i>
+      <div class="prev-next-info">
+        <p class="prev-next-subtitle">previous</p>
+        <p class="prev-next-title">Gaussian Processes</p>
+      </div>
+    </a>
+    <a class="right-next"
+       href="L96_ResNet_RNN.html"
+       title="next page">
+      <div class="prev-next-info">
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+
+  <div class="sidebar-secondary-item">
+  <div class="page-toc tocsection onthispage">
+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">CNNs and Polynomial Maps</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#i-introduction">I. Introduction</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#one-variable-l96-equation">One-variable L96 equation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#local-networks">Local networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-feedforward-neural-network-ann">1. Simple feedforward neural network (ANN)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#polynomial-stencil">2. Polynomial stencil</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#global-networks">Global networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-convolutional-neural-network-global-cnn">1. Global convolutional neural network (global CNN)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#global-feedforward-neural-network-global-ann">2. Global feedforward neural network (global ANN)</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-gcm-objects">Define GCM objects</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-neural-network-objects">Define neural network objects</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simulate-1-variable-l96-equation-as-ground-truth">Simulate 1-variable L96 equation as ground truth</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#train-neural-networks-with-local-and-global-setup-and-evaluate-their-performance">Train neural networks with local and global setup, and evaluate their performance</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-a-global-cnn">1. Train a global CNN</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-a-global-ann">2. Train a global ANN</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-a-local-polynomial-map">3. Train a local polynomial map</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-a-local-ann-map">4. Train a local ANN map</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#display-training-error">Display training error</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#ii-test-predictions-of-the-networks">II. Test predictions of the networks</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#instantaneous-predictions">Instantaneous predictions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#time-dependent-initial-value-simulations">Time-dependent initial-value simulations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#show-parameters-learned-by-the-polynet">Show parameters learned by the polynet</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#iii-energy-conserved-case">III. Energy-conserved case</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-networks">Train networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-simulation-data-with-networks-trained-on-the-energy-conserved-case">Generate simulation data with networks trained on the energy-conserved case</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#plot-instantaneous-prediction">Plot instantaneous prediction</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#iii-a-more-realistic-model-the-burger-s-equation">III. A more realistic model: the Burger’s equation</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#problem-with-the-1-variable-l96-equation">Problem with the 1-variable L96 equation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-burger-s-equation">The Burger’s equation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generate-training-data-from-burger-s-equation">Generate training data from Burger’s equation</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-the-same-set-of-networks-with-the-burger-s-equation">Train the same set of networks with the Burger’s equation</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#conclusions">Conclusions</a></li>
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+    <p class="title logo__title">Learning Machine Learning with Lorenz-96</p>
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+        <div class="sidebar-primary-item"><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
+    <div class="bd-toc-item navbar-nav active">
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
+
+
+
+
+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="closing_remarks.html">Outlook</a></li>
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+    <h1>Random Forest</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#starting-with-a-single-regression-tree">Starting with a single regression tree</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#offline-training-fitting-and-testing">Offline training / fitting and testing</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fit-and-test-scores">Fit and test scores</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#overfitting">Overfitting</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#online-implementation-of-the-parameterization">Online implementation of the parameterization</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#multiple-features">Multiple features</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forest-ensemble-of-decision-trees">Random Forest: ensemble of decision trees</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting">Gradient boosting</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-features-using-past-state">More features: using past state</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparison-with-nn-parameterization-based-on-yani-s-code">Comparison with NN parameterization (based on Yani’s code)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#which-model-model-parameters-should-we-go-for">Which model / model parameters should we go for?</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#standard-machine-learning-approach-validation">Standard Machine Learning approach: validation</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="random-forest">
+<h1>Random Forest<a class="headerlink" href="#random-forest" title="Permalink to this headline">#</a></h1>
+<p>In this notebook we present some basic application of random forest to
+the parameterization of the L96 model previously introduced.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">L96_model</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">tree</span>
+<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">RandomForestRegressor</span><span class="p">,</span> <span class="n">GradientBoostingRegressor</span>
+<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
+<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">deque</span>
+<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">RandomizedSearchCV</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">RK4</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span>
+
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">import</span> <span class="nn">torch.nn</span> <span class="k">as</span> <span class="nn">nn</span>
+<span class="kn">import</span> <span class="nn">torch.nn.functional</span> <span class="k">as</span> <span class="nn">FF</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Main script parameters</span>
+<span class="n">K</span> <span class="o">=</span> <span class="mi">8</span>
+<span class="n">J</span> <span class="o">=</span> <span class="mi">32</span>
+<span class="n">FORCING</span> <span class="o">=</span> <span class="mi">18</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.01</span>
+<span class="n">T_SPINUP</span> <span class="o">=</span> <span class="mi">50</span>
+<span class="n">T_SIMULATIONS</span> <span class="o">=</span> <span class="mi">200</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model</span> <span class="o">=</span> <span class="n">L96_model</span><span class="o">.</span><span class="n">L96</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">FORCING</span><span class="p">)</span>
+<span class="c1"># First run for spin up</span>
+<span class="n">model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T_SPINUP</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="c1"># The data from the run below will be used both for offline training and testing</span>
+<span class="n">X_history</span><span class="p">,</span> <span class="n">Y_history</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">closure</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span>
+    <span class="n">dt</span><span class="p">,</span> <span class="n">T_SIMULATIONS</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># We will use the quantity below in our online test</span>
+<span class="n">var_X</span> <span class="o">=</span> <span class="n">X_history</span><span class="o">.</span><span class="n">var</span><span class="p">()</span>
+<span class="n">var_X</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>20.64021181455364
+</pre></div>
+</div>
+</div>
+</div>
+<p>Scatter-plot of subgrid tendency vs large-scale state X</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">_</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">X_history</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="n">closure</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="n">bins</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;binary&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;State X&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Closure&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/2c8133dc5df1d40c56c0b670e04699756bcfc258ba3d5c598da8286a673c7907.png" src="../_images/2c8133dc5df1d40c56c0b670e04699756bcfc258ba3d5c598da8286a673c7907.png" />
+</div>
+</div>
+<div class="section" id="starting-with-a-single-regression-tree">
+<h2>Starting with a single regression tree<a class="headerlink" href="#starting-with-a-single-regression-tree" title="Permalink to this headline">#</a></h2>
+<p>We start with a very simple approach: we use a single value of the state as our feature (i.e. our feature is a scalar), and the subgrid tendency for the same large-scale variable as the target.</p>
+<div class="section" id="offline-training-fitting-and-testing">
+<h3>Offline training / fitting and testing<a class="headerlink" href="#offline-training-fitting-and-testing" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">closure</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">X_history</span><span class="o">.</span><span class="n">shape</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>((20001, 8), (20001, 8))
+</pre></div>
+</div>
+</div>
+</div>
+<p>First we split the data into training and test. Since the relation between <span class="math notranslate nohighlight">\(X_k\)</span> and its associated subgrid tendency is not expected to depend on <span class="math notranslate nohighlight">\(k\)</span>, we transform the recorded history into a column vector before splitting.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">,</span> <span class="n">xy_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span>
+    <span class="n">X_history</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">closure</span><span class="o">.</span><span class="n">flatten</span><span class="p">(),</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.33</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;X_train has shape: </span><span class="si">{</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;xy_train has shape: </span><span class="si">{</span><span class="n">xy_train</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>X_train has shape: (107205, 1)
+xy_train has shape: (107205,)
+</pre></div>
+</div>
+</div>
+</div>
+<p>Then we define our decision tree. For the sake of the example, we limit the number of leaves to 15. Feel free to increase that number and see how the plot below, showing the true and fitted data, changes. In particular, for larger values, we can see that we overfit the data.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">single_tree</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_leaf_nodes</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
+<span class="n">single_tree</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><style>#sk-container-id-1 {color: black;background-color: white;}#sk-container-id-1 pre{padding: 0;}#sk-container-id-1 div.sk-toggleable {background-color: white;}#sk-container-id-1 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-1 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-1 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-1 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-1 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-1 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-1 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-1 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-1 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-1 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-1 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-1 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-1 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-1 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-1 div.sk-item {position: relative;z-index: 1;}#sk-container-id-1 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-1 div.sk-item::before, #sk-container-id-1 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-1 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-1 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-1 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-1 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-1 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-1 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-1 div.sk-label-container {text-align: center;}#sk-container-id-1 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-1 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-1" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>DecisionTreeRegressor(max_leaf_nodes=15)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-1" type="checkbox" checked><label for="sk-estimator-id-1" class="sk-toggleable__label sk-toggleable__label-arrow">DecisionTreeRegressor</label><div class="sk-toggleable__content"><pre>DecisionTreeRegressor(max_leaf_nodes=15)</pre></div></div></div></div></div></div></div>
+</div>
+<p>Here we plot the true training data (the large scale state values on the x-axis, and the parameterization value on the y-axis) and the fitted data.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">,</span> <span class="s2">&quot;b*&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">single_tree</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="s2">&quot;r*&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;X&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s2">&quot;True&quot;</span><span class="p">,</span> <span class="s2">&quot;Fitted&quot;</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Fit of a regression tree with 15 leaves&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/d559a834ae9e9ebe7c47363d4bda7459af71e797e8607573969b8e1400d273f6.png" src="../_images/d559a834ae9e9ebe7c47363d4bda7459af71e797e8607573969b8e1400d273f6.png" />
+</div>
+</div>
+<p>Now, same plot but for the test data. The test data has not been seen during the fitting procedure, so in some sense this is the true test. While we may fit very well the training data, overfitting will result in poor performance on the test dataset.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">xy_test</span><span class="p">,</span> <span class="s2">&quot;b*&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">single_tree</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">),</span> <span class="s2">&quot;r*&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;X&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s2">&quot;True&quot;</span><span class="p">,</span> <span class="s2">&quot;Predicted&quot;</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Test of a regression tree with 15 leaves&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/20c48ec84661b304848cc284be5a45735138974fdf55584798bc59459296988c.png" src="../_images/20c48ec84661b304848cc284be5a45735138974fdf55584798bc59459296988c.png" />
+</div>
+</div>
+</div>
+<div class="section" id="fit-and-test-scores">
+<h3>Fit and test scores<a class="headerlink" href="#fit-and-test-scores" title="Permalink to this headline">#</a></h3>
+<p>We can compute the <span class="math notranslate nohighlight">\(R^2\)</span> score for both the training data (fit score) and test data.</p>
+<div class="amsmath math notranslate nohighlight">
+\[\begin{equation*}
+R^2 = 1 - \frac{\sum_{i}{(\hat{S}_i - S_i)^2}}{\sum_{i}{(S_i - S_\text{mean})^2}}
+\end{equation*}\]</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fit_score</span> <span class="o">=</span> <span class="n">single_tree</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+<span class="n">test_score</span> <span class="o">=</span> <span class="n">single_tree</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">xy_test</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">fit_score</span><span class="p">,</span> <span class="n">test_score</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.85559718734831 0.8608891051149465
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xy_test</span><span class="p">[:</span><span class="mi">8000</span><span class="p">:</span><span class="n">K</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">single_tree</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)[:</span><span class="mi">8000</span><span class="p">:</span><span class="n">K</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;time step&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$xy_0$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s2">&quot;True&quot;</span><span class="p">,</span> <span class="s2">&quot;Predicted&quot;</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/b7ac09dc6850309ad41b8081ce1b7cb2e9b086651fdf1a6a90d99b5eeb704cce.png" src="../_images/b7ac09dc6850309ad41b8081ce1b7cb2e9b086651fdf1a6a90d99b5eeb704cce.png" />
+</div>
+</div>
+</div>
+<div class="section" id="overfitting">
+<h3>Overfitting<a class="headerlink" href="#overfitting" title="Permalink to this headline">#</a></h3>
+<p>To illustrate what is meant by overfitting, we will increase the number of leaves, and store the fit loss and test loss for plotting</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nb_leaves</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">/</span> <span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">20</span><span class="p">)))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fit_scores</span><span class="p">,</span> <span class="n">test_scores</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
+<span class="k">for</span> <span class="n">nb</span> <span class="ow">in</span> <span class="n">nb_leaves</span><span class="p">:</span>
+    <span class="n">current_tree</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_leaf_nodes</span><span class="o">=</span><span class="n">nb</span><span class="p">)</span>
+    <span class="n">current_tree</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+    <span class="n">fit_scores</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">current_tree</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">))</span>
+    <span class="n">test_scores</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">current_tree</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">xy_test</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span><span class="n">nb_leaves</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">fit_scores</span><span class="p">),</span> <span class="s2">&quot;*&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span><span class="n">nb_leaves</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">test_scores</span><span class="p">),</span> <span class="s2">&quot;*&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s2">&quot;fit_score&quot;</span><span class="p">,</span> <span class="s2">&quot;test_scores&quot;</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Number of leaf nodes&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Score&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/e188611f8f6341e1e0d95b1a99e40b513e333dcecdeb595dbac6dfbab0e9f21a.png" src="../_images/e188611f8f6341e1e0d95b1a99e40b513e333dcecdeb595dbac6dfbab0e9f21a.png" />
+</div>
+</div>
+</div>
+<div class="section" id="online-implementation-of-the-parameterization">
+<h3>Online implementation of the parameterization<a class="headerlink" href="#online-implementation-of-the-parameterization" title="Permalink to this headline">#</a></h3>
+<p>Now let’s implement this as a parameterization for L96.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Parameterization</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">predictor</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">predictor</span> <span class="o">=</span> <span class="n">predictor</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">predictor</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Here we use the GCM class defined by Yani in his notebook</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GCM</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">parameterization</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        GCM with paramerization</span>
+<span class="sd">        Args:</span>
+<span class="sd">            F: forcing</span>
+<span class="sd">            parameterization: function that takes parameters and returns a tendency</span>
+<span class="sd">            time_stepping: time stepping method</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">parameterization</span> <span class="o">=</span> <span class="n">parameterization</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            X: state vector</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">parameterization</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            X0: initial conditions</span>
+<span class="sd">            dt: time increment</span>
+<span class="sd">            nt: number of forward steps to take</span>
+<span class="sd">            param: parameters of our closure</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="n">Parameterization</span><span class="p">(</span><span class="n">single_tree</span><span class="p">))</span>
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">0</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_steps</span> <span class="o">=</span> <span class="mi">2000</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">online_test</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">n_runs</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">):</span>
+    <span class="n">simulations</span> <span class="o">=</span> <span class="p">[]</span>
+    <span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_runs</span><span class="p">):</span>
+        <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">10</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;run &quot;</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
+        <span class="n">X_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+        <span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+        <span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span> <span class="o">*</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+        <span class="n">simulations</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">X_true</span><span class="p">,</span> <span class="n">X_param</span><span class="p">))</span>
+    <span class="n">squared_errors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span>
+        <span class="p">[((</span><span class="n">true</span> <span class="o">-</span> <span class="n">sim</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">true</span><span class="p">,</span> <span class="n">sim</span> <span class="ow">in</span> <span class="n">simulations</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span>
+    <span class="p">)</span>
+    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">median</span><span class="p">(</span><span class="n">squared_errors</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="o">/</span> <span class="n">var_X</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="n">Parameterization</span><span class="p">(</span><span class="n">single_tree</span><span class="p">))</span>
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">0</span><span class="p">)</span>
+<span class="n">X_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span> <span class="o">*</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;true&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_no_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with paramerization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$X_0$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/ab2d061f7f27ff2e653b69d12465b8b5fd5c04f8a6aafa1863c3f76912b3f8c6.png" src="../_images/ab2d061f7f27ff2e653b69d12465b8b5fd5c04f8a6aafa1863c3f76912b3f8c6.png" />
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="multiple-features">
+<h2>Multiple features<a class="headerlink" href="#multiple-features" title="Permalink to this headline">#</a></h2>
+<p>In this section, we use the global state.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">single_treeV2</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_leaf_nodes</span><span class="o">=</span><span class="mi">300</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">,</span> <span class="n">xy_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span>
+    <span class="n">X_history</span><span class="p">,</span> <span class="n">closure</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.33</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">single_treeV2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><style>#sk-container-id-2 {color: black;background-color: white;}#sk-container-id-2 pre{padding: 0;}#sk-container-id-2 div.sk-toggleable {background-color: white;}#sk-container-id-2 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-2 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-2 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-2 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-2 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-2 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-2 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-2 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-2 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-2 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-2 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-2 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-2 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-2 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-2 div.sk-item {position: relative;z-index: 1;}#sk-container-id-2 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-2 div.sk-item::before, #sk-container-id-2 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-2 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-2 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-2 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-2 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-2 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-2 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-2 div.sk-label-container {text-align: center;}#sk-container-id-2 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-2 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-2" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>DecisionTreeRegressor(max_leaf_nodes=300)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-2" type="checkbox" checked><label for="sk-estimator-id-2" class="sk-toggleable__label sk-toggleable__label-arrow">DecisionTreeRegressor</label><div class="sk-toggleable__content"><pre>DecisionTreeRegressor(max_leaf_nodes=300)</pre></div></div></div></div></div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">single_treeV2</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9461126584860248
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">single_treeV2</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">xy_test</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.8729996071244494
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xy_test</span><span class="p">[:</span><span class="mi">1000</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">single_treeV2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)[:</span><span class="mi">1000</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;time steps&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$xy_0$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s2">&quot;True&quot;</span><span class="p">,</span> <span class="s2">&quot;Predicted&quot;</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/8a0fc844ea03d4f64739a99a846b1f7ffa13b6319ebed0d27c9dd6a85c9a4c10.png" src="../_images/8a0fc844ea03d4f64739a99a846b1f7ffa13b6319ebed0d27c9dd6a85c9a4c10.png" />
+</div>
+</div>
+<p>We need to adapt our parameterization so that it makes use of the homogeneity of the problem</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">ParameterizationV2</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">predictor</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">predictor</span> <span class="o">=</span> <span class="n">predictor</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+        <span class="n">new_X</span> <span class="o">=</span> <span class="p">[]</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">K</span><span class="p">):</span>
+            <span class="n">new_X</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="o">-</span><span class="n">i</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">))</span>
+        <span class="n">new_X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">(</span><span class="n">new_X</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">predictor</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">new_X</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="n">ParameterizationV2</span><span class="p">(</span><span class="n">single_treeV2</span><span class="p">))</span>
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">0</span><span class="p">)</span>
+<span class="n">X_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span> <span class="o">*</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;true&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_no_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with paramerization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$X_0$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/2443dcc242cbf12308f66a8200dcdd726f4e40646e071f01eba2475d9ba2aaf2.png" src="../_images/2443dcc242cbf12308f66a8200dcdd726f4e40646e071f01eba2475d9ba2aaf2.png" />
+</div>
+</div>
+</div>
+<div class="section" id="random-forest-ensemble-of-decision-trees">
+<h2>Random Forest: ensemble of decision trees<a class="headerlink" href="#random-forest-ensemble-of-decision-trees" title="Permalink to this headline">#</a></h2>
+<p>Here we use a random forest instead of a single decision tree. The idea is simple: to avoid overfitting, one trains several decision trees on the same data.  The prediction of the random forest is then the average prediction of all decision trees within the ensemble.</p>
+<p>There are two main ways in which the fitted trees within the forest might differ:</p>
+<ul class="simple">
+<li><p>the selected features for the splits: at each iteration of the fitting procedure, the algorithm does not consider all features to select the best split, at least not when the number of features is large. Instead, a finite number of features is considered at random.</p></li>
+<li><p>the data on which they are trained. Below, by setting bootstrap to True and max_samples to 0.5, we specify that each tree within the forest will only be trained on half of the training data</p></li>
+</ul>
+<p>Random forest are an efficient way to avoid overfitting compared to using a single tree. However they do come with a computational and memory cost.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># We start with an ensemble of 10 decision trees</span>
+<span class="n">rf</span> <span class="o">=</span> <span class="n">RandomForestRegressor</span><span class="p">(</span>
+    <span class="n">n_estimators</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">max_leaf_nodes</span><span class="o">=</span><span class="mi">300</span><span class="p">,</span> <span class="n">bootstrap</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">max_samples</span><span class="o">=</span><span class="mf">0.5</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>We fit to the same training data as in the previous section</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><style>#sk-container-id-3 {color: black;background-color: white;}#sk-container-id-3 pre{padding: 0;}#sk-container-id-3 div.sk-toggleable {background-color: white;}#sk-container-id-3 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-3 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-3 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-3 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-3 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-3 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-3 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-3 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-3 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-3 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-3 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-3 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-3 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-3 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-3 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-3 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-3 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-3 div.sk-item {position: relative;z-index: 1;}#sk-container-id-3 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-3 div.sk-item::before, #sk-container-id-3 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-3 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-3 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-3 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-3 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-3 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-3 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-3 div.sk-label-container {text-align: center;}#sk-container-id-3 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-3 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-3" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>RandomForestRegressor(max_leaf_nodes=300, max_samples=0.5, n_estimators=50)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-3" type="checkbox" checked><label for="sk-estimator-id-3" class="sk-toggleable__label sk-toggleable__label-arrow">RandomForestRegressor</label><div class="sk-toggleable__content"><pre>RandomForestRegressor(max_leaf_nodes=300, max_samples=0.5, n_estimators=50)</pre></div></div></div></div></div></div></div>
+</div>
+<p>We can actually get the prediction of each tree within the forest:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">tree_predictions</span> <span class="o">=</span> <span class="p">[]</span>
+<span class="k">for</span> <span class="n">est</span> <span class="ow">in</span> <span class="n">rf</span><span class="o">.</span><span class="n">estimators_</span><span class="p">:</span>
+    <span class="n">tree_predictions</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">est</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">[:</span><span class="mi">1</span><span class="p">,</span> <span class="p">:]))</span>
+<span class="n">tree_predictions</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">tree_predictions</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;tree_predictions = </span><span class="si">{</span><span class="n">tree_predictions</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;mean of tree_predictions = </span><span class="si">{</span><span class="n">tree_predictions</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;prediction on first row = </span><span class="si">{</span><span class="n">rf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">[:</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="p">:])[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>tree_predictions = [-6.4573486  -6.06461396 -4.88866112 -7.84598495 -4.02069189 -6.11670094
+ -6.14333155 -6.25748197 -5.13250645 -6.16887507 -7.60661857 -7.32986544
+ -6.61873039 -7.20379449 -6.68960421 -5.84444954 -7.4095241  -6.96928775
+ -5.26970853 -7.05451154 -7.01396758 -8.07039345 -7.35832294 -5.94846774
+ -7.57849707 -6.82908975 -7.00526204 -6.19203361 -5.90930418 -7.03509526
+ -6.73553096 -6.54034233 -6.50990201 -6.44768472 -6.19145528 -7.70578284
+ -6.22230973 -5.58215999 -6.29828815 -7.29972659 -5.6206472  -6.91029985
+ -7.04004174 -6.57132794 -7.44427363 -6.72832008 -5.52538    -5.9918895
+ -5.39751423 -7.27894348]
+mean of tree_predictions = -6.521490899090696
+prediction on first row = -6.521490899090695
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">),</span> <span class="n">rf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">xy_test</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(0.9589042178623464, 0.9065738774066192)
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xy_test</span><span class="p">[:</span><span class="mi">1000</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">rf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)[:</span><span class="mi">1000</span><span class="p">])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;time steps&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$xy_0$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s2">&quot;True&quot;</span><span class="p">,</span> <span class="s2">&quot;Predicted&quot;</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/f8916c8346561ac6eb3d64ec465d37c857f5d53384dc4de0fbddd324c1c701a3.png" src="../_images/f8916c8346561ac6eb3d64ec465d37c857f5d53384dc4de0fbddd324c1c701a3.png" />
+</div>
+</div>
+<p>Again, we can test this online.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="n">ParameterizationV2</span><span class="p">(</span><span class="n">rf</span><span class="p">))</span>
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">0</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span> <span class="o">*</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;true&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_no_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with paramerization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$X_0$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/f498c9bfe2620cd508e5ba5fbb5ec599b33e01571e6afe49c0febae6d0db314c.png" src="../_images/f498c9bfe2620cd508e5ba5fbb5ec599b33e01571e6afe49c0febae6d0db314c.png" />
+</div>
+</div>
+</div>
+<div class="section" id="gradient-boosting">
+<h2>Gradient boosting<a class="headerlink" href="#gradient-boosting" title="Permalink to this headline">#</a></h2>
+<p>Gradient boosting is a type of machine learning <a class="reference external" href="https://en.wikipedia.org/wiki/Boosting_(machine_learning)">boosting</a>. It relies on the intuition that the best possible next model, when combined with previous models, minimizes the overall prediction error. The key idea is to set the target outcomes for this next model in order to minimize the error.</p>
+<p>Further reading <a class="reference external" href="https://www.displayr.com/gradient-boosting-the-coolest-kid-on-the-machine-learning-block/">here</a>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">gbr</span> <span class="o">=</span> <span class="n">GradientBoostingRegressor</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">200</span><span class="p">)</span>
+<span class="n">gbr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+<span class="n">score_train</span> <span class="o">=</span> <span class="n">gbr</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+<span class="n">score_test</span> <span class="o">=</span> <span class="n">gbr</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">xy_test</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">score_train</span><span class="p">,</span> <span class="n">score_test</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9287011503795917 0.8978127923974935
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="more-features-using-past-state">
+<h2>More features: using past state<a class="headerlink" href="#more-features-using-past-state" title="Permalink to this headline">#</a></h2>
+<p>Here we further increase the complexity of our approach by incorporating past state into the features used to carry out our prediction.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_times</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">new_X</span> <span class="o">=</span> <span class="n">X_history</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_times</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+    <span class="n">new_X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">new_X</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X_history</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">new_X</span> <span class="o">=</span> <span class="n">new_X</span><span class="p">[</span><span class="n">n_times</span><span class="p">:]</span>
+<span class="n">new_closure</span> <span class="o">=</span> <span class="n">closure</span><span class="p">[</span><span class="n">n_times</span><span class="p">:]</span>
+
+<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">,</span> <span class="n">xy_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span>
+    <span class="n">new_X</span><span class="p">,</span> <span class="n">new_closure</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.33</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">False</span>
+<span class="p">)</span>
+<span class="n">X_train</span><span class="o">.</span><span class="n">shape</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(13399, 16)
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rf</span><span class="o">.</span><span class="n">max_leaf_nodes</span> <span class="o">=</span> <span class="mi">350</span>
+<span class="n">rf</span><span class="o">.</span><span class="n">max_samples</span> <span class="o">=</span> <span class="mf">0.99</span>
+<span class="n">rf</span><span class="o">.</span><span class="n">n_estimators</span> <span class="o">=</span> <span class="mi">100</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><style>#sk-container-id-4 {color: black;background-color: white;}#sk-container-id-4 pre{padding: 0;}#sk-container-id-4 div.sk-toggleable {background-color: white;}#sk-container-id-4 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-4 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-4 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-4 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-4 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-4 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-4 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-4 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-4 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-4 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-4 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-4 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-4 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-4 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-4 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-4 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-4 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-4 div.sk-item {position: relative;z-index: 1;}#sk-container-id-4 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-4 div.sk-item::before, #sk-container-id-4 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-4 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-4 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-4 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-4 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-4 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-4 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-4 div.sk-label-container {text-align: center;}#sk-container-id-4 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-4 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-4" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>RandomForestRegressor(max_leaf_nodes=350, max_samples=0.99)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-4" type="checkbox" checked><label for="sk-estimator-id-4" class="sk-toggleable__label sk-toggleable__label-arrow">RandomForestRegressor</label><div class="sk-toggleable__content"><pre>RandomForestRegressor(max_leaf_nodes=350, max_samples=0.99)</pre></div></div></div></div></div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9680265002935357
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">xy_test</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9110472454347986
+</pre></div>
+</div>
+</div>
+</div>
+<p>Again we need to adapt the form of the parameterization, here so that it records past values of the state for later use.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Parameterization</span><span class="p">:</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">predictor</span><span class="p">,</span> <span class="n">n_times</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">5</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">predictor</span> <span class="o">=</span> <span class="n">predictor</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">past_states</span> <span class="o">=</span> <span class="n">deque</span><span class="p">(</span><span class="n">maxlen</span><span class="o">=</span><span class="n">n_times</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">n_times</span> <span class="o">=</span> <span class="n">n_times</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
+        <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
+        <span class="n">new_X</span> <span class="o">=</span> <span class="p">[]</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">K</span><span class="p">):</span>
+            <span class="n">new_X</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="o">-</span><span class="n">i</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">))</span>
+        <span class="n">new_X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">(</span><span class="n">new_X</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
+        <span class="n">shape</span> <span class="o">=</span> <span class="n">new_X</span><span class="o">.</span><span class="n">shape</span>
+        <span class="n">save_X</span> <span class="o">=</span> <span class="n">new_X</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_times</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
+            <span class="k">try</span><span class="p">:</span>
+                <span class="n">new_X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">new_X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">past_states</span><span class="p">[</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+            <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
+                <span class="n">new_X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">new_X</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="p">)),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">past_states</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">save_X</span><span class="p">)</span>
+        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">predictor</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">new_X</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_steps</span> <span class="o">=</span> <span class="mi">2000</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="n">Parameterization</span><span class="p">(</span><span class="n">rf</span><span class="p">,</span> <span class="n">n_times</span><span class="p">))</span>
+<span class="n">gcm_no_param</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">0</span><span class="p">)</span>
+<span class="n">X_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span> <span class="o">*</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;true&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_no_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without parameterization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with paramerization&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$X_0$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/483901aad6f659091f8f1cbb01e95399785f9738deaa4493254dadca89c64d89.png" src="../_images/483901aad6f659091f8f1cbb01e95399785f9738deaa4493254dadca89c64d89.png" />
+</div>
+</div>
+</div>
+<div class="section" id="comparison-with-nn-parameterization-based-on-yani-s-code">
+<h2>Comparison with NN parameterization (based on Yani’s code)<a class="headerlink" href="#comparison-with-nn-parameterization-based-on-yani-s-code" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Net_ANN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_ANN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 8 inputs, 16 neurons for first hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>  <span class="c1"># 16 neurons for second hidden layer</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>  <span class="c1"># 8 outputs</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">FF</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">FF</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GCM_network</span><span class="p">:</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    GCM class including a neural network parameterization in rhs of equation for tendency</span>
+<span class="sd">    Args:</span>
+<span class="sd">        F: forcing</span>
+<span class="sd">        parameterization: function that takes parameters and returns a tendency</span>
+<span class="sd">        time_stepping: time stepping method</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">network</span><span class="p">,</span> <span class="n">time_stepping</span><span class="o">=</span><span class="n">RK4</span><span class="p">):</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">network</span> <span class="o">=</span> <span class="n">network</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span> <span class="o">=</span> <span class="n">time_stepping</span>
+
+    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            X: state vector</span>
+<span class="sd">            param: parameters of our closure, unused here but maintained for consistency of interface</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="o">.</span><span class="n">linear1</span><span class="o">.</span><span class="n">in_features</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">X_torch</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">X_torch</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+        <span class="k">return</span> <span class="n">L96_eq1_xdot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">F</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span>
+            <span class="bp">self</span><span class="o">.</span><span class="n">network</span><span class="p">(</span><span class="n">X_torch</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+        <span class="p">)</span>  <span class="c1"># Adding NN parameterization</span>
+
+    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X0</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">nt</span><span class="p">,</span> <span class="n">param</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+<span class="w">        </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">        Args:</span>
+<span class="sd">            X0: initial conditions</span>
+<span class="sd">            dt: time increment</span>
+<span class="sd">            nt: number of forward steps to take</span>
+<span class="sd">            param: parameters of our closure</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">time</span><span class="p">,</span> <span class="n">hist</span><span class="p">,</span> <span class="n">X</span> <span class="o">=</span> <span class="p">(</span>
+            <span class="n">dt</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
+            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X0</span><span class="p">)))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span>
+            <span class="n">X0</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span>
+        <span class="p">)</span>
+        <span class="n">hist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span>
+
+        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nt</span><span class="p">):</span>
+            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">time_stepping</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">param</span><span class="p">)</span>
+            <span class="n">hist</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">time</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="k">return</span> <span class="n">hist</span><span class="p">,</span> <span class="n">time</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Load network</span>
+<span class="n">path_load</span> <span class="o">=</span> <span class="s2">&quot;networks/network_3_layers_100_epoches.pth&quot;</span>
+<span class="n">nn_3l</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">()</span><span class="o">.</span><span class="n">double</span><span class="p">()</span>
+<span class="n">nn_3l</span><span class="o">.</span><span class="n">load_state_dict</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">path_load</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;All keys matched successfully&gt;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n_steps</span> <span class="o">=</span> <span class="mi">2000</span>
+<span class="n">gcm</span> <span class="o">=</span> <span class="n">GCM</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="n">ParameterizationV2</span><span class="p">(</span><span class="n">single_treeV2</span><span class="p">))</span>
+<span class="n">gcm_network</span> <span class="o">=</span> <span class="n">GCM_network</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">F</span><span class="p">,</span> <span class="n">nn_3l</span><span class="p">)</span>
+<span class="n">X_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_network</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_network</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_no_param</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">gcm_no_param</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">X</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">n_steps</span> <span class="o">*</span> <span class="n">model</span><span class="o">.</span><span class="n">dt</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_true</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;true&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_no_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM without parameterization (tree)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_param</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;GCM with paramerization (network)&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$X_0$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/22148e8b9dbc1ca4294b2cf2b19964c3b03486083341a69c32aa4432bb227f77.png" src="../_images/22148e8b9dbc1ca4294b2cf2b19964c3b03486083341a69c32aa4432bb227f77.png" />
+</div>
+</div>
+</div>
+<div class="section" id="which-model-model-parameters-should-we-go-for">
+<h2>Which model / model parameters should we go for?<a class="headerlink" href="#which-model-model-parameters-should-we-go-for" title="Permalink to this headline">#</a></h2>
+<div class="section" id="standard-machine-learning-approach-validation">
+<h3>Standard Machine Learning approach: validation<a class="headerlink" href="#standard-machine-learning-approach-validation" title="Permalink to this headline">#</a></h3>
+<p>The Machine Learning approach to this problem is to separate the data into three data sets:</p>
+<ul class="simple">
+<li><p>training dataset</p></li>
+<li><p>validation dataset</p></li>
+<li><p>test dataset</p></li>
+</ul>
+<p>Only data from the training dataset is used for fitting. The validation dataset is then used for model selection and hyperparameters tuning: we select the model whose performance on the validation dataset is best.</p>
+<p>In order to choose hyperparameters, one needs to make a decision on what set of values to try. One approach is to try a grid of values. Another approach is to try random values. This is recommanded when the possible number of hyperparameter combinations is large. Finally Bayesian approaches to this problem are also popular. Further reading on grid search, random search and bayesian approach <a class="reference external" href="https://towardsdatascience.com/a-practical-introduction-to-grid-search-random-search-and-bayes-search-d5580b1d941d">here</a>.</p>
+<p>Here we shall implement the random approach: for each model, we will define a range of values for each parameter we wish to tune. We will randomly select a combination of parameter values, fit the model to the training dataset, and assess its performance on the validation dataset. The combination of values leading to the best validation score will be retained.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">single_tree_params_dist</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;max_leaf_nodes&quot;</span><span class="p">:</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">11</span><span class="p">)))),</span>
+<span class="p">}</span>
+
+<span class="n">rf_params_dist</span> <span class="o">=</span> <span class="p">{</span>
+    <span class="s2">&quot;max_leaf_nodes&quot;</span><span class="p">:</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">13</span><span class="p">)))),</span>
+    <span class="s2">&quot;n_estimators&quot;</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="o">*</span> <span class="mi">50</span><span class="p">,</span>
+<span class="p">}</span>
+
+<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">xy_train</span><span class="p">,</span> <span class="n">xy_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span>
+    <span class="n">X_history</span><span class="p">,</span> <span class="n">closure</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.33</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>The cells below are very slow to execute. Uncomment them to perform the parameter search.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># best_single_treeV2 = RandomizedSearchCV(</span>
+<span class="c1">#     single_treeV2, single_tree_params_dist, n_iter=10, verbose=True</span>
+<span class="c1"># )</span>
+<span class="c1"># best_rf = RandomizedSearchCV(rf, rf_params_dist, n_iter=20, verbose=True)</span>
+
+<span class="c1"># best_single_treeV2.fit(X_train, xy_train)</span>
+<span class="c1"># best_rf.fit(X_train, xy_train)</span>
+
+<span class="c1"># print(&quot;single tree&quot;)</span>
+<span class="c1"># print(f&quot;best score = {best_single_treeV2.best_score_}&quot;)</span>
+<span class="c1"># print(f&quot;best parameters = {best_single_treeV2.best_params_}&quot;)</span>
+<span class="c1"># print(&quot;\nrandom forest&quot;)</span>
+<span class="c1"># print(f&quot;best score = {best_rf.best_params_}&quot;)</span>
+<span class="c1"># print(f&quot;best parameters = {best_rf.score(X_test, xy_test)}&quot;)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+</div>
+
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+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#starting-with-a-single-regression-tree">Starting with a single regression tree</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#offline-training-fitting-and-testing">Offline training / fitting and testing</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fit-and-test-scores">Fit and test scores</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#overfitting">Overfitting</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#online-implementation-of-the-parameterization">Online implementation of the parameterization</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#multiple-features">Multiple features</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forest-ensemble-of-decision-trees">Random Forest: ensemble of decision trees</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting">Gradient boosting</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-features-using-past-state">More features: using past state</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparison-with-nn-parameterization-based-on-yani-s-code">Comparison with NN parameterization (based on Yani’s code)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#which-model-model-parameters-should-we-go-for">Which model / model parameters should we go for?</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#standard-machine-learning-approach-validation">Standard Machine Learning approach: validation</a></li>
+</ul>
+</li>
+</ul>
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+<p class="component-author">
+By The M<sup>2</sup>LInES Community
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
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+    <h1>Applying SINDy equation identification to L96</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Applying SINDy equation identification to L96</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#import">Import</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#routines-for-plotting">Routines for plotting</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#l96-two-scale-model">L96 two-scale model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-sindy-model">Define SINDy model</a></li>
+</ul>
+</li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#optimize-coefficients">Optimize coefficients</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#compare">Compare</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#same-initial-conditions">Same initial conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-initial-conditions">Other initial conditions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#case-of-strong-coupling">Case of strong coupling</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#build-the-sindy-model">Build the SINDy model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#run-the-predicted-model">Run the predicted model</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conclusions">Conclusions</a></li>
+</ul>
+</li>
+</ul>
+
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+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="applying-sindy-equation-identification-to-l96">
+<h1>Applying SINDy equation identification to L96<a class="headerlink" href="#applying-sindy-equation-identification-to-l96" title="Permalink to this headline">#</a></h1>
+<div class="section" id="outline">
+<h2>Outline<a class="headerlink" href="#outline" title="Permalink to this headline">#</a></h2>
+<p>This notebook provides an example of SINDy equation identification in application to the two-scale L96 model with the use of <code class="docutils literal notranslate"><span class="pre">pysindy</span></code> library. The objective is to try to identify the governing ODEs for the large-scale variable (<span class="math notranslate nohighlight">\(X_i\)</span>) if we know only their time measurements. In other words, we want to see how well the SINDy model can capture the form of the governing equations for large-scale variables in the presence of the subgrid forcing by small-scale variables (<span class="math notranslate nohighlight">\(Y\)</span>) when there is no measurements of the small scales.</p>
+</div>
+<div class="section" id="import">
+<h2>Import<a class="headerlink" href="#import" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pysindy</span> <span class="k">as</span> <span class="nn">ps</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span><span class="p">,</span> <span class="n">L96_eq1_xdot</span><span class="p">,</span> <span class="n">integrate_L96_2t</span><span class="p">,</span> <span class="n">integrate_L96_1t</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="routines-for-plotting">
+<h2>Routines for plotting<a class="headerlink" href="#routines-for-plotting" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">colors</span> <span class="o">=</span> <span class="p">[</span>
+    <span class="s2">&quot;#1f77b4&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#ff7f0e&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#2ca02c&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#d62728&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#9467bd&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#8c564b&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#e377c2&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#7f7f7f&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#bcbd22&quot;</span><span class="p">,</span>
+    <span class="s2">&quot;#17becf&quot;</span><span class="p">,</span>
+<span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_coefficients</span><span class="p">(</span>
+    <span class="n">coefficients</span><span class="p">,</span> <span class="n">input_features</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">feature_names</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">heatmap_kws</span>
+<span class="p">):</span>
+    <span class="k">if</span> <span class="n">input_features</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="n">input_features</span> <span class="o">=</span> <span class="p">[</span><span class="sa">f</span><span class="s2">&quot;$\dot x_</span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s2">$&quot;</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">coefficients</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])]</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="n">input_features</span> <span class="o">=</span> <span class="p">[</span><span class="sa">f</span><span class="s2">&quot;$\dot </span><span class="si">{</span><span class="n">fi</span><span class="si">}</span><span class="s2">$&quot;</span> <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">input_features</span><span class="p">]</span>
+
+    <span class="k">if</span> <span class="n">feature_names</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="n">feature_names</span> <span class="o">=</span> <span class="p">[</span><span class="sa">f</span><span class="s2">&quot;f</span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s2">&quot;</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">coefficients</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])]</span>
+
+    <span class="k">with</span> <span class="n">sns</span><span class="o">.</span><span class="n">axes_style</span><span class="p">(</span><span class="n">style</span><span class="o">=</span><span class="s2">&quot;white&quot;</span><span class="p">,</span> <span class="n">rc</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;axes.facecolor&quot;</span><span class="p">:</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)}):</span>
+        <span class="k">if</span> <span class="n">ax</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+            <span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+        <span class="n">max_mag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">coefficients</span><span class="p">))</span>
+        <span class="n">heatmap_args</span> <span class="o">=</span> <span class="p">{</span>
+            <span class="s2">&quot;xticklabels&quot;</span><span class="p">:</span> <span class="n">input_features</span><span class="p">,</span>
+            <span class="s2">&quot;yticklabels&quot;</span><span class="p">:</span> <span class="n">feature_names</span><span class="p">,</span>
+            <span class="s2">&quot;center&quot;</span><span class="p">:</span> <span class="mf">0.0</span><span class="p">,</span>
+            <span class="s2">&quot;cmap&quot;</span><span class="p">:</span> <span class="n">sns</span><span class="o">.</span><span class="n">color_palette</span><span class="p">(</span><span class="s2">&quot;vlag&quot;</span><span class="p">,</span> <span class="n">n_colors</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">as_cmap</span><span class="o">=</span><span class="kc">True</span><span class="p">),</span>
+            <span class="s2">&quot;ax&quot;</span><span class="p">:</span> <span class="n">ax</span><span class="p">,</span>
+            <span class="s2">&quot;linewidths&quot;</span><span class="p">:</span> <span class="mf">0.1</span><span class="p">,</span>
+            <span class="s2">&quot;linecolor&quot;</span><span class="p">:</span> <span class="s2">&quot;whitesmoke&quot;</span><span class="p">,</span>
+        <span class="p">}</span>
+        <span class="n">heatmap_args</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="o">**</span><span class="n">heatmap_kws</span><span class="p">)</span>
+
+        <span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">coefficients</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="o">**</span><span class="n">heatmap_args</span><span class="p">)</span>
+
+        <span class="n">ax</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s2">&quot;y&quot;</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">ax</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">signed_sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">compare_coefficient_plots</span><span class="p">(</span><span class="n">coefficients</span><span class="p">,</span> <span class="n">input_features</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">feature_names</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
+    <span class="n">n_cols</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coefficients</span><span class="p">)</span>
+    <span class="k">with</span> <span class="n">sns</span><span class="o">.</span><span class="n">axes_style</span><span class="p">(</span><span class="n">style</span><span class="o">=</span><span class="s2">&quot;white&quot;</span><span class="p">,</span> <span class="n">rc</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;axes.facecolor&quot;</span><span class="p">:</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)}):</span>
+        <span class="n">max_mag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">coefficients</span><span class="p">))</span>
+        <span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">sharex</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+        <span class="n">plot_coefficients</span><span class="p">(</span>
+            <span class="n">signed_sqrt</span><span class="p">(</span><span class="n">coefficients</span><span class="p">),</span>
+            <span class="n">input_features</span><span class="o">=</span><span class="n">input_features</span><span class="p">,</span>
+            <span class="n">feature_names</span><span class="o">=</span><span class="n">feature_names</span><span class="p">,</span>
+            <span class="n">ax</span><span class="o">=</span><span class="n">axs</span><span class="p">,</span>
+            <span class="n">cbar</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
+            <span class="n">vmax</span><span class="o">=</span><span class="n">max_mag</span><span class="p">,</span>
+            <span class="n">vmin</span><span class="o">=-</span><span class="n">max_mag</span><span class="p">,</span>
+        <span class="p">)</span>
+        <span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="l96-two-scale-model">
+<h2>L96 two-scale model<a class="headerlink" href="#l96-two-scale-model" title="Permalink to this headline">#</a></h2>
+<div class="figure align-default" id="l96-equation-figure-sindy2scale">
+<a class="reference internal image-reference" href="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png"><img alt="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" src="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" style="width: 400px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 7 </span><span class="caption-text"><em>Visualisation of a two-scale Lorenz ‘96 system with J = 8 and K = 6. Global-scale variables (<span class="math notranslate nohighlight">\(X_k\)</span>) are updated based on neighbouring variables and on the local-scale variables (<span class="math notranslate nohighlight">\(Y_{j,k}\)</span>) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from <a class="reference external" href="https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436">Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.</a></em>.</span><a class="headerlink" href="#l96-equation-figure-sindy2scale" title="Permalink to this image">#</a></p>
+</div>
+<p>Two-time-scale Lorenz 96 model:
+$<span class="math notranslate nohighlight">\( \frac{d}{dt} X_k = - X_{k-2} X_{k-1} + X_{k-1} X_{k+1} - X_k + F - \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k} \)</span><span class="math notranslate nohighlight">\(
+\)</span><span class="math notranslate nohighlight">\( \frac{d}{dt} Y_{j,k} = - c b Y_{j+1,k} (Y_{j+2,k} - Y_{j-1,k}) - c Y_{j,k} + \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} X_{k} \)</span>$</p>
+<p>We will put K=8 large-scale variables (<span class="math notranslate nohighlight">\(X\)</span>), for each of them there is J=32 small-scale variables (<span class="math notranslate nohighlight">\(Y\)</span>). For the first example, we will consider the case of <strong>weak</strong> interaction between the scales, which is defined by the parameter <span class="math notranslate nohighlight">\(h\)</span> in the L96 class.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">K</span> <span class="o">=</span> <span class="mi">8</span>  <span class="c1"># Number of degrees of freedom for X</span>
+<span class="n">J</span> <span class="o">=</span> <span class="mi">32</span>  <span class="c1"># Number of degrees of freedom for Y per X</span>
+<span class="n">Forcing</span> <span class="o">=</span> <span class="mf">18.0</span>  <span class="c1"># Model parameter</span>
+<span class="n">h</span> <span class="o">=</span> <span class="mf">0.1</span>  <span class="c1"># WEAK coupling</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;A non-dimension coordinate from -1..+1 corresponding to k=0..K&quot;&quot;&quot;</span>
+    <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="n">K</span> <span class="o">-</span> <span class="mi">1</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Initial conditions</span>
+<span class="n">k</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>  <span class="c1"># For coordinate in plots</span>
+<span class="n">j</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">)</span>  <span class="c1"># For coordinate in plots</span>
+<span class="c1"># Initial conditions</span>
+<span class="n">X_init</span> <span class="o">=</span> <span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+<span class="n">Y_init</span> <span class="o">=</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">s</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">J</span> <span class="o">*</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">steps</span> <span class="o">=</span> <span class="mi">1000</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="mf">0.005</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create a &quot;real world&quot; with K=8 and J=32</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="n">h</span><span class="p">)</span>
+<span class="n">W</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">Y_init</span><span class="p">)</span>
+<span class="c1"># Run the true state</span>
+<span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">steps</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">contours</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">contours</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/e37cf216aaaec972b8c6f59e547a4da3866f28e031fa801e4260d16e1b7dcc40.png" src="../_images/e37cf216aaaec972b8c6f59e547a4da3866f28e031fa801e4260d16e1b7dcc40.png" />
+</div>
+</div>
+<p>The data array <span class="math notranslate nohighlight">\(X\)</span> will be used for the system identification by SINDy method. We suppose that we do not dispose measurements for the small-scale variables <span class="math notranslate nohighlight">\(Y\)</span>, so their contribution to the large-scale dymanics will basically treated as noise.</p>
+</div>
+<div class="section" id="define-sindy-model">
+<h2>Define SINDy model<a class="headerlink" href="#define-sindy-model" title="Permalink to this headline">#</a></h2>
+<p>The goal is to identify the governing equations for the large-scale variables <span class="math notranslate nohighlight">\(X\)</span>, having the measurement for them.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">differentiation_method</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">SINDyDerivative</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="s2">&quot;finite_difference&quot;</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">feature_library</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">PolynomialLibrary</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># sequential thresholded least squares optimizer</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">STLSQ</span><span class="p">(</span><span class="n">threshold</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">SINDy</span><span class="p">(</span>
+    <span class="n">differentiation_method</span><span class="o">=</span><span class="n">differentiation_method</span><span class="p">,</span>
+    <span class="n">feature_library</span><span class="o">=</span><span class="n">feature_library</span><span class="p">,</span>
+    <span class="n">optimizer</span><span class="o">=</span><span class="n">optimizer</span><span class="p">,</span>
+    <span class="n">feature_names</span><span class="o">=</span><span class="p">[(</span><span class="s2">&quot;x&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">k</span><span class="p">],</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="tex2jax_ignore mathjax_ignore section" id="optimize-coefficients">
+<h1>Optimize coefficients<a class="headerlink" href="#optimize-coefficients" title="Permalink to this headline">#</a></h1>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><style>#sk-container-id-1 {color: black;background-color: white;}#sk-container-id-1 pre{padding: 0;}#sk-container-id-1 div.sk-toggleable {background-color: white;}#sk-container-id-1 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-1 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-1 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-1 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-1 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-1 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-1 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-1 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-1 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-1 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-1 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-1 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-1 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-1 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-1 div.sk-item {position: relative;z-index: 1;}#sk-container-id-1 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-1 div.sk-item::before, #sk-container-id-1 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-1 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-1 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-1 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-1 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-1 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-1 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-1 div.sk-label-container {text-align: center;}#sk-container-id-1 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-1 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-1" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>SINDy(differentiation_method=SINDyDerivative(k=1, kind=&#x27;finite_difference&#x27;),
+      feature_library=PolynomialLibrary(),
+      feature_names=[&#x27;x0&#x27;, &#x27;x1&#x27;, &#x27;x2&#x27;, &#x27;x3&#x27;, &#x27;x4&#x27;, &#x27;x5&#x27;, &#x27;x6&#x27;, &#x27;x7&#x27;],
+      optimizer=STLSQ(threshold=0.3))</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-1" type="checkbox" ><label for="sk-estimator-id-1" class="sk-toggleable__label sk-toggleable__label-arrow">SINDy</label><div class="sk-toggleable__content"><pre>SINDy(differentiation_method=SINDyDerivative(k=1, kind=&#x27;finite_difference&#x27;),
+      feature_library=PolynomialLibrary(),
+      feature_names=[&#x27;x0&#x27;, &#x27;x1&#x27;, &#x27;x2&#x27;, &#x27;x3&#x27;, &#x27;x4&#x27;, &#x27;x5&#x27;, &#x27;x6&#x27;, &#x27;x7&#x27;],
+      optimizer=STLSQ(threshold=0.3))</pre></div></div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-2" type="checkbox" ><label for="sk-estimator-id-2" class="sk-toggleable__label sk-toggleable__label-arrow">feature_library: PolynomialLibrary</label><div class="sk-toggleable__content"><pre>PolynomialLibrary()</pre></div></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-3" type="checkbox" ><label for="sk-estimator-id-3" class="sk-toggleable__label sk-toggleable__label-arrow">PolynomialLibrary</label><div class="sk-toggleable__content"><pre>PolynomialLibrary()</pre></div></div></div></div></div></div><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-4" type="checkbox" ><label for="sk-estimator-id-4" class="sk-toggleable__label sk-toggleable__label-arrow">optimizer: STLSQ</label><div class="sk-toggleable__content"><pre>STLSQ(threshold=0.3)</pre></div></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-5" type="checkbox" ><label for="sk-estimator-id-5" class="sk-toggleable__label sk-toggleable__label-arrow">STLSQ</label><div class="sk-toggleable__content"><pre>STLSQ(threshold=0.3)</pre></div></div></div></div></div></div></div></div></div></div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">feature_names</span> <span class="o">=</span> <span class="n">feature_library</span><span class="o">.</span><span class="n">get_feature_names</span><span class="p">()</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">feature_names</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[&#39;1&#39;, &#39;x0&#39;, &#39;x1&#39;, &#39;x2&#39;, &#39;x3&#39;, &#39;x4&#39;, &#39;x5&#39;, &#39;x6&#39;, &#39;x7&#39;, &#39;x0^2&#39;, &#39;x0 x1&#39;, &#39;x0 x2&#39;, &#39;x0 x3&#39;, &#39;x0 x4&#39;, &#39;x0 x5&#39;, &#39;x0 x6&#39;, &#39;x0 x7&#39;, &#39;x1^2&#39;, &#39;x1 x2&#39;, &#39;x1 x3&#39;, &#39;x1 x4&#39;, &#39;x1 x5&#39;, &#39;x1 x6&#39;, &#39;x1 x7&#39;, &#39;x2^2&#39;, &#39;x2 x3&#39;, &#39;x2 x4&#39;, &#39;x2 x5&#39;, &#39;x2 x6&#39;, &#39;x2 x7&#39;, &#39;x3^2&#39;, &#39;x3 x4&#39;, &#39;x3 x5&#39;, &#39;x3 x6&#39;, &#39;x3 x7&#39;, &#39;x4^2&#39;, &#39;x4 x5&#39;, &#39;x4 x6&#39;, &#39;x4 x7&#39;, &#39;x5^2&#39;, &#39;x5 x6&#39;, &#39;x5 x7&#39;, &#39;x6^2&#39;, &#39;x6 x7&#39;, &#39;x7^2&#39;]
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model</span><span class="o">.</span><span class="n">print</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(x0)&#39; = 17.900 1 + -1.019 x0 + 0.998 x1 x7 + -0.998 x6 x7
+(x1)&#39; = 17.980 1 + -1.026 x1 + 1.000 x0 x2 + -0.999 x0 x7
+(x2)&#39; = 17.996 1 + -1.020 x2 + -0.999 x0 x1 + 0.999 x1 x3
+(x3)&#39; = 17.992 1 + -1.023 x3 + -1.001 x1 x2 + 1.000 x2 x4
+(x4)&#39; = 17.948 1 + -1.022 x4 + -1.000 x2 x3 + 1.000 x3 x5
+(x5)&#39; = 17.949 1 + -1.017 x5 + -1.001 x3 x4 + 1.000 x4 x6
+(x6)&#39; = 18.009 1 + -1.016 x6 + -1.001 x4 x5 + 0.999 x5 x7
+(x7)&#39; = 17.951 1 + -1.016 x7 + 0.998 x0 x6 + -0.998 x5 x6
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9999919207350484
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model</span><span class="o">.</span><span class="n">coefficients</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[17.90003772, -1.01862501,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.99829818,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        , -0.99760226,  0.        ],
+       [17.97958814,  0.        , -1.02560838,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.9995848 ,  0.        ,  0.        ,  0.        ,
+         0.        , -0.99916774,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
+       [17.99602273,  0.        ,  0.        , -1.02037387,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+        -0.99938584,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.99879915,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
+       [17.99204463,  0.        ,  0.        ,  0.        , -1.02324178,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        , -1.0005416 ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  1.00030224,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
+       [17.94751299,  0.        ,  0.        ,  0.        ,  0.        ,
+        -1.02190284,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+        -1.00012659,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  1.00033709,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
+       [17.94900238,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        , -1.01701087,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        , -1.00096398,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  1.00008416,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
+       [18.00933399,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        , -1.0160708 ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        , -1.00062155,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.99871963,  0.        ,  0.        ,  0.        ],
+       [17.95111469,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        , -1.01591188,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.9975044 ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+         0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
+        -0.99829913,  0.        ,  0.        ,  0.        ,  0.        ]])
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">with</span> <span class="n">sns</span><span class="o">.</span><span class="n">axes_style</span><span class="p">(</span><span class="n">style</span><span class="o">=</span><span class="s2">&quot;white&quot;</span><span class="p">,</span> <span class="n">rc</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;axes.facecolor&quot;</span><span class="p">:</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)}):</span>
+    <span class="n">max_mag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">coefficients</span><span class="p">()))</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span>
+        <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">sharex</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">tight_layout</span><span class="o">=</span><span class="kc">True</span>
+    <span class="p">)</span>
+    <span class="n">plot_coefficients</span><span class="p">(</span>
+        <span class="n">model</span><span class="o">.</span><span class="n">coefficients</span><span class="p">(),</span>
+        <span class="n">input_features</span><span class="o">=</span><span class="p">[(</span><span class="s2">&quot;x&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">k</span><span class="p">],</span>
+        <span class="n">feature_names</span><span class="o">=</span><span class="n">feature_names</span><span class="p">,</span>
+        <span class="n">ax</span><span class="o">=</span><span class="n">axs</span><span class="p">,</span>
+        <span class="n">cbar</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+        <span class="n">vmax</span><span class="o">=</span><span class="n">max_mag</span><span class="p">,</span>
+        <span class="n">vmin</span><span class="o">=-</span><span class="n">max_mag</span><span class="p">,</span>
+        <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;seismic&quot;</span><span class="p">,</span>
+    <span class="p">)</span>
+    <span class="n">axs</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">title</span><span class="o">=</span><span class="s2">&quot;SINDy coefficients&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/77586c6d07cfa1b8ecb389af29adade10e2e45a4bd56512c696df8cab5e3ba75.png" src="../_images/77586c6d07cfa1b8ecb389af29adade10e2e45a4bd56512c696df8cab5e3ba75.png" />
+</div>
+</div>
+<p>We can see that the SINDy model captures correctly the dominant terms and gives quite accurate predictions for the model coefficients and the value of the forcing. The subgrid forcing does not appear in this prediction.</p>
+<div class="section" id="compare">
+<h2>Compare<a class="headerlink" href="#compare" title="Permalink to this headline">#</a></h2>
+<div class="section" id="same-initial-conditions">
+<h3>Same initial conditions<a class="headerlink" href="#same-initial-conditions" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sindy_simulation</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">simulate</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">component</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)):</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="n">component</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+        <span class="n">t</span><span class="p">,</span>
+        <span class="n">sindy_simulation</span><span class="p">[:,</span> <span class="n">component</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
+        <span class="n">label</span><span class="o">=</span><span class="n">model</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="n">component</span><span class="p">],</span>
+    <span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">&quot;t&quot;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$x_i$&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/390efb536a9f3fcb4a5ee648acabb97f79eefbd54dfca5adc9eda054028a610d.png" src="../_images/390efb536a9f3fcb4a5ee648acabb97f79eefbd54dfca5adc9eda054028a610d.png" />
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="other-initial-conditions">
+<h2>Other initial conditions<a class="headerlink" href="#other-initial-conditions" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_init_new</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
+<span class="n">W</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">X_init_new</span><span class="p">,</span> <span class="n">Y_init</span><span class="p">)</span>
+<span class="c1"># Run the true state</span>
+<span class="n">X_new</span><span class="p">,</span> <span class="n">Y_new</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">steps</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sindy_simulation_new</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">simulate</span><span class="p">(</span><span class="n">X_init_new</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">component</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)):</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_new</span><span class="p">[:,</span> <span class="n">component</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+        <span class="n">t</span><span class="p">,</span>
+        <span class="n">sindy_simulation_new</span><span class="p">[:,</span> <span class="n">component</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
+        <span class="n">label</span><span class="o">=</span><span class="n">model</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="n">component</span><span class="p">],</span>
+    <span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">&quot;t&quot;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$x_i$&quot;</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">&quot;New initial conditions&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/7f0e3dbc3ebe31e3b438fa96b007b2fce9304b30277b46913f37382b08d316d8.png" src="../_images/7f0e3dbc3ebe31e3b438fa96b007b2fce9304b30277b46913f37382b08d316d8.png" />
+</div>
+</div>
+<p>Similar to the case with L63 model, the system predicted by SINDy gives an accurate prediction of trajectories for a certain time, then the prediction strts deviating from the original model. It is related to the error in the coefficients obtained in optimization anf the chaotical nature of the Lorenz system.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_init_new</span> <span class="o">=</span> <span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">s</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>  <span class="c1"># Initial conditions</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X_new</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">integrate_L96_1t</span><span class="p">(</span><span class="n">X_init_new</span><span class="p">,</span> <span class="n">Forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">steps</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sindy_simulation_new</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">simulate</span><span class="p">(</span><span class="n">X_init_new</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">component</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)):</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_new</span><span class="p">[:,</span> <span class="n">component</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+        <span class="n">t</span><span class="p">,</span>
+        <span class="n">sindy_simulation_new</span><span class="p">[:,</span> <span class="n">component</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
+        <span class="n">label</span><span class="o">=</span><span class="n">model</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="n">component</span><span class="p">],</span>
+    <span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">&quot;t&quot;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$x_i$&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/78d3f85d354e1b5df230555d293d47c03bcfa8d3d90b40bfd6798b5d6b29ac1f.png" src="../_images/78d3f85d354e1b5df230555d293d47c03bcfa8d3d90b40bfd6798b5d6b29ac1f.png" />
+</div>
+</div>
+</div>
+<div class="section" id="case-of-strong-coupling">
+<h2>Case of strong coupling<a class="headerlink" href="#case-of-strong-coupling" title="Permalink to this headline">#</a></h2>
+<p>Now, let us consider the L96 two-scale model with a stronger coupling between the large and small scales. The coupling coefficient here is put to be 10 times higher.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create a &quot;real world&quot; with K=8 and J=32</span>
+<span class="n">W_strong</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">Forcing</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
+<span class="n">W_strong</span><span class="o">.</span><span class="n">set_state</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">Y_init</span><span class="p">)</span>
+<span class="c1"># Run the true state</span>
+<span class="n">X_s</span><span class="p">,</span> <span class="n">Y_s</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">W_strong</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">dt</span> <span class="o">*</span> <span class="n">steps</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">contours</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">X_s</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">contours</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">&quot;k&quot;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="s2">&quot;t&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/c2569f64702e4c7e0832d3c68435534714804bfb904d941a72f8c40beba27c97.png" src="../_images/c2569f64702e4c7e0832d3c68435534714804bfb904d941a72f8c40beba27c97.png" />
+</div>
+</div>
+<div class="section" id="build-the-sindy-model">
+<h3>Build the SINDy model<a class="headerlink" href="#build-the-sindy-model" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model_s</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">SINDy</span><span class="p">(</span>
+    <span class="n">differentiation_method</span><span class="o">=</span><span class="n">differentiation_method</span><span class="p">,</span>
+    <span class="n">feature_library</span><span class="o">=</span><span class="n">feature_library</span><span class="p">,</span>
+    <span class="n">optimizer</span><span class="o">=</span><span class="n">optimizer</span><span class="p">,</span>
+    <span class="n">feature_names</span><span class="o">=</span><span class="p">[(</span><span class="s2">&quot;x&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">k</span><span class="p">],</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model_s</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_s</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><style>#sk-container-id-2 {color: black;background-color: white;}#sk-container-id-2 pre{padding: 0;}#sk-container-id-2 div.sk-toggleable {background-color: white;}#sk-container-id-2 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-2 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-2 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-2 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-2 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-2 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-2 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-2 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-2 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-2 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-2 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-2 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-2 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-2 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-2 div.sk-item {position: relative;z-index: 1;}#sk-container-id-2 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-2 div.sk-item::before, #sk-container-id-2 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-2 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-2 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-2 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-2 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-2 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-2 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-2 div.sk-label-container {text-align: center;}#sk-container-id-2 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-2 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-2" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>SINDy(differentiation_method=SINDyDerivative(k=1, kind=&#x27;finite_difference&#x27;),
+      feature_library=PolynomialLibrary(),
+      feature_names=[&#x27;x0&#x27;, &#x27;x1&#x27;, &#x27;x2&#x27;, &#x27;x3&#x27;, &#x27;x4&#x27;, &#x27;x5&#x27;, &#x27;x6&#x27;, &#x27;x7&#x27;],
+      optimizer=STLSQ(threshold=0.3))</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-6" type="checkbox" ><label for="sk-estimator-id-6" class="sk-toggleable__label sk-toggleable__label-arrow">SINDy</label><div class="sk-toggleable__content"><pre>SINDy(differentiation_method=SINDyDerivative(k=1, kind=&#x27;finite_difference&#x27;),
+      feature_library=PolynomialLibrary(),
+      feature_names=[&#x27;x0&#x27;, &#x27;x1&#x27;, &#x27;x2&#x27;, &#x27;x3&#x27;, &#x27;x4&#x27;, &#x27;x5&#x27;, &#x27;x6&#x27;, &#x27;x7&#x27;],
+      optimizer=STLSQ(threshold=0.3))</pre></div></div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-7" type="checkbox" ><label for="sk-estimator-id-7" class="sk-toggleable__label sk-toggleable__label-arrow">feature_library: PolynomialLibrary</label><div class="sk-toggleable__content"><pre>PolynomialLibrary()</pre></div></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-8" type="checkbox" ><label for="sk-estimator-id-8" class="sk-toggleable__label sk-toggleable__label-arrow">PolynomialLibrary</label><div class="sk-toggleable__content"><pre>PolynomialLibrary()</pre></div></div></div></div></div></div><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-9" type="checkbox" ><label for="sk-estimator-id-9" class="sk-toggleable__label sk-toggleable__label-arrow">optimizer: STLSQ</label><div class="sk-toggleable__content"><pre>STLSQ(threshold=0.3)</pre></div></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-10" type="checkbox" ><label for="sk-estimator-id-10" class="sk-toggleable__label sk-toggleable__label-arrow">STLSQ</label><div class="sk-toggleable__content"><pre>STLSQ(threshold=0.3)</pre></div></div></div></div></div></div></div></div></div></div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model_s</span><span class="o">.</span><span class="n">print</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(x0)&#39; = 17.721 1 + -1.812 x0 + -0.348 x1 + 1.053 x1 x7 + -1.019 x6 x7
+(x1)&#39; = 16.639 1 + -1.877 x1 + 1.008 x0 x2 + -0.994 x0 x7
+(x2)&#39; = 16.293 1 + -1.746 x2 + -1.010 x0 x1 + 1.010 x1 x3
+(x3)&#39; = 16.817 1 + -1.801 x3 + -1.009 x1 x2 + 1.002 x2 x4
+(x4)&#39; = 17.244 1 + -1.874 x4 + -1.016 x2 x3 + 1.018 x3 x5
+(x5)&#39; = 16.967 1 + -1.785 x5 + -1.004 x3 x4 + 1.002 x4 x6
+(x6)&#39; = 17.114 1 + -1.946 x6 + -1.011 x4 x5 + 1.022 x5 x7
+(x7)&#39; = 17.387 1 + -1.808 x7 + 0.991 x0 x6 + -1.004 x5 x6
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">with</span> <span class="n">sns</span><span class="o">.</span><span class="n">axes_style</span><span class="p">(</span><span class="n">style</span><span class="o">=</span><span class="s2">&quot;white&quot;</span><span class="p">,</span> <span class="n">rc</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;axes.facecolor&quot;</span><span class="p">:</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)}):</span>
+    <span class="n">max_mag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">model_s</span><span class="o">.</span><span class="n">coefficients</span><span class="p">()))</span>
+    <span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span>
+        <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">sharex</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">tight_layout</span><span class="o">=</span><span class="kc">True</span>
+    <span class="p">)</span>
+    <span class="n">plot_coefficients</span><span class="p">(</span>
+        <span class="n">model_s</span><span class="o">.</span><span class="n">coefficients</span><span class="p">(),</span>
+        <span class="n">input_features</span><span class="o">=</span><span class="p">[(</span><span class="s2">&quot;x&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">k</span><span class="p">],</span>
+        <span class="n">feature_names</span><span class="o">=</span><span class="n">feature_library</span><span class="o">.</span><span class="n">get_feature_names</span><span class="p">(),</span>
+        <span class="n">ax</span><span class="o">=</span><span class="n">axs</span><span class="p">,</span>
+        <span class="n">cbar</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+        <span class="n">vmax</span><span class="o">=</span><span class="n">max_mag</span><span class="p">,</span>
+        <span class="n">vmin</span><span class="o">=-</span><span class="n">max_mag</span><span class="p">,</span>
+        <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;seismic&quot;</span><span class="p">,</span>
+    <span class="p">)</span>
+    <span class="n">axs</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">title</span><span class="o">=</span><span class="s2">&quot;SINDy coefficients&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/bd564aef15a80c2f34ca76d4aa669ea0af4b738a3bc50aacc12f19a897011d42.png" src="../_images/bd564aef15a80c2f34ca76d4aa669ea0af4b738a3bc50aacc12f19a897011d42.png" />
+</div>
+</div>
+<p>We can see that despite the fact that in general SINDy captures the structure of ODEs, the coefficients become different from the original model, due to the influence of the interaction with the small scale. Interestingly, the coefficients in front of linear terms in the ODEs are now all close to 1.7-1.8 (instead of 1.0 in the true model), and the predicted forcing value is systematically smaller than the true value 18.</p>
+</div>
+<div class="section" id="run-the-predicted-model">
+<h3>Run the predicted model<a class="headerlink" href="#run-the-predicted-model" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">sindy_simulation_s</span> <span class="o">=</span> <span class="n">model_s</span><span class="o">.</span><span class="n">simulate</span><span class="p">(</span><span class="n">X_init</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">component</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)):</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">X_s</span><span class="p">[:,</span> <span class="n">component</span><span class="p">],</span> <span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+        <span class="n">t</span><span class="p">,</span>
+        <span class="n">sindy_simulation_s</span><span class="p">[:,</span> <span class="n">component</span><span class="p">],</span>
+        <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
+        <span class="n">label</span><span class="o">=</span><span class="n">model</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="n">component</span><span class="p">],</span>
+    <span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">&quot;t&quot;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$x_i$&quot;</span><span class="p">)</span>
+<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/a9c98831fdf9a2f0a4cd1ee886829a9bcb15dc3fd0b0af3f8b944a71ae1a1db5.png" src="../_images/a9c98831fdf9a2f0a4cd1ee886829a9bcb15dc3fd0b0af3f8b944a71ae1a1db5.png" />
+</div>
+</div>
+<p>The resulting trajectories start deviating fast from the true trajectories.</p>
+</div>
+</div>
+<div class="section" id="conclusions">
+<h2>Conclusions<a class="headerlink" href="#conclusions" title="Permalink to this headline">#</a></h2>
+<p>Sparse system identification method works well in conditions when we have already have some knowledgement about the equations (about sparsity, the type on non-lineartity) and optimal data representation (good coordinates).</p>
+</div>
+</div>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
+        codeMirrorConfig: {
+            theme: "abcdef",
+            mode: "python"
+        },
+        kernelOptions: {
+            name: "python3",
+            path: "./notebooks"
+        },
+        predefinedOutput: true
+    }
+    </script>
+    <script>kernelName = 'python3'</script>
+
+                </article>
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+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Applying SINDy equation identification to L96</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#outline">Outline</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#import">Import</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#routines-for-plotting">Routines for plotting</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#l96-two-scale-model">L96 two-scale model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-sindy-model">Define SINDy model</a></li>
+</ul>
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+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#optimize-coefficients">Optimize coefficients</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#compare">Compare</a><ul class="nav section-nav flex-column">
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#case-of-strong-coupling">Case of strong coupling</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#build-the-sindy-model">Build the SINDy model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#run-the-predicted-model">Run the predicted model</a></li>
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="random_forest_parameterization.html">Random Forest</a></li>
+<li class="toctree-l1"><a class="reference internal" href="GP_lorenz96.html">Gaussian Processes</a></li>
+<li class="toctree-l1"><a class="reference internal" href="local_map_L96_and_Burgers.html">CNNs and Polynomial Maps</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#gplearn">gplearn</a><ul class="nav section-nav flex-column">
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+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#interpret-results">Interpret results</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sequentially-thresholded-least-squares-stlsq-from-pysindy">Sequentially Thresholded Least Squares (STLSQ) from pysindy</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#tweaking-the-feature-library">Tweaking the Feature Library</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#handling-noise">Handling noise</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-methods-for-symbolic-regression">Other Methods for Symbolic Regression</a></li>
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+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="introduction-to-equation-discovery-comparing-symbolic-regression-methods">
+<h1>Introduction to Equation Discovery - Comparing Symbolic Regression Methods<a class="headerlink" href="#introduction-to-equation-discovery-comparing-symbolic-regression-methods" title="Permalink to this headline">#</a></h1>
+<p>Upto now we have seen that the climate models we developed are using physical equations that are based on our understanding of the physical processes that govern the climate. However, these equations are often complex and difficult to solve, and they can only be used to model the climate at a coarse resolution.</p>
+<p><strong>Equation discovery is a different approach that uses machine learning to automatically discover equations that can be used to model the climate.</strong> Specifically, equation discovery in climate modeling can be used to:</p>
+<ul class="simple">
+<li><p><strong>Automate the process of developing subgrid parameterizations.</strong> Subgrid parameterizations are mathematical equations that are used to represent the effects of processes that are too small to be resolved by climate models. These equations are often complex and difficult to develop, but they can be automatically discovered using equation discovery.</p></li>
+<li><p><strong>Identify relationships between variables that are important for understanding climate change.</strong> Machine learning algorithms can learn from large datasets of climate data to identify relationships between variables that are important for understanding climate change.</p></li>
+<li><p><strong>Develop models that are more interpretable and that can be used to make better decisions about how to mitigate and adapt to climate change.</strong> Machine learning algorithms can generate equations that are easier to understand than traditional physical equations. This can help scientists and policymakers to understand how climate models work and to make better decisions about how to mitigate and adapt to climate change.</p></li>
+</ul>
+<p>In this notebook, we’ll review some common techniques for <strong>Symbolic Regression (SR)</strong>, a family of methods of discovering (simple) equations that relate inputs to outputs.</p>
+<p>In particular, we’ll cover the following methods:</p>
+<ul class="simple">
+<li><p>Symbolic regression based on <a class="reference internal" href="#genetic-programming-section"><span class="std std-ref">Genetic Programming</span></a></p>
+<ul>
+<li><p><a class="reference internal" href="#gplearn-sec"><span class="std std-ref">gplearn</span></a></p></li>
+<li><p><a class="reference internal" href="#pysr-sec"><span class="std std-ref">PySR</span></a></p></li>
+</ul>
+</li>
+<li><p>Symbolic regression based on <a class="reference internal" href="#sparse-regression-sec"><span class="std std-ref">Manually Constructed Library + Sparse Regression</span></a></p>
+<ul>
+<li><p><a class="reference internal" href="#linear-regression-sec"><span class="std std-ref">Linear regression</span></a></p></li>
+<li><p><a class="reference internal" href="#lasso-sec"><span class="std std-ref">L1-regularized linear regression (LASSO)</span></a></p></li>
+<li><p><a class="reference internal" href="#rvm-sec"><span class="std std-ref">Relevance vector machines (RVM)</span></a></p></li>
+<li><p><a class="reference internal" href="#stlsq-sec"><span class="std std-ref">Sequentially Thresholded Least Squares (STLSQ) from pysindy</span></a></p></li>
+</ul>
+</li>
+<li><p>Brief overview of other techniques</p></li>
+</ul>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pysindy</span> <span class="k">as</span> <span class="nn">ps</span>
+<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">Lasso</span><span class="p">,</span> <span class="n">LinearRegression</span>
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Number</span><span class="p">,</span> <span class="n">latex</span>
+
+<span class="kn">from</span> <span class="nn">rvm</span> <span class="kn">import</span> <span class="n">RVR</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="data">
+<h2>Data<a class="headerlink" href="#data" title="Permalink to this headline">#</a></h2>
+<p>When explaining the methods, we’ll consider the following bivariate system:</p>
+<div class="math notranslate nohighlight">
+\[
+y = x_0^2 - \frac{1}{2}x_1^2 + \sin\left(\frac{1}{2} x_0 x_1\right)
+\]</div>
+<p>Although the equation is relatively simple, discovering it from <span class="math notranslate nohighlight">\(x_0\)</span> and <span class="math notranslate nohighlight">\(x_1\)</span> directly is a bit tough, since the <span class="math notranslate nohighlight">\(\sin\)</span> term requires going at least three levels deep in term combinations, and also involves an inner multiplication by a constant (which not all symbolic regression frameworks support).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">toy_function</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+    <span class="n">x0</span> <span class="o">=</span> <span class="n">x</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
+    <span class="n">x1</span> <span class="o">=</span> <span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span>
+    <span class="k">return</span> <span class="n">x0</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">x0</span> <span class="o">*</span> <span class="n">x1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">contour</span><span class="p">(</span>
+    <span class="n">f</span><span class="p">,</span> <span class="n">lines</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span> <span class="n">linestyles</span><span class="o">=</span><span class="s2">&quot;solid&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kw</span>
+<span class="p">):</span>
+    <span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">150</span><span class="p">)</span>
+    <span class="n">xyg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span> <span class="n">grid</span><span class="p">))</span>
+    <span class="n">X</span> <span class="o">=</span> <span class="n">xyg</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
+    <span class="n">zg</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
+    <span class="n">xg</span><span class="p">,</span> <span class="n">yg</span> <span class="o">=</span> <span class="n">xyg</span>
+    <span class="n">vlim</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">zg</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">xg</span><span class="p">,</span> <span class="n">yg</span><span class="p">,</span> <span class="n">zg</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xg</span><span class="o">.</span><span class="n">shape</span><span class="p">),</span> <span class="mi">300</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=-</span><span class="n">vlim</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="n">vlim</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;bwr&quot;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">y</span> <span class="o">=</span> <span class="n">toy_function</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">contour</span><span class="p">(</span><span class="n">toy_function</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Function value&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Toy domain for illustrating</span><span class="se">\n</span><span class="s2">symbolic regression&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x_0$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$x_1$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/d500949fe2d1e400e2ab002527fb1a76c012fd88ddb886fe827b47d6acab3360.png" src="../_images/d500949fe2d1e400e2ab002527fb1a76c012fd88ddb886fe827b47d6acab3360.png" />
+</div>
+</div>
+</div>
+<div class="section" id="genetic-programming">
+<span id="genetic-programming-section"></span><h2>Genetic Programming<a class="headerlink" href="#genetic-programming" title="Permalink to this headline">#</a></h2>
+<p>Genetic programming is a classic technique in AI, and interest in the idea dates back at least to Alan Turing (who proposes the idea in the <a class="reference external" href="https://doi.org/10.1093/mind/LIX.236.433">same paper</a> that introduces the imitation game, or Turing test).</p>
+<p>The general idea is to have a:</p>
+<ul class="simple">
+<li><p>“Population” of programs (in our case, possible symbolic expressions, usually represented as trees)</p></li>
+<li><p>Some notion of individual “fitness” (in our case, how well they model the data, as well as their simplicity)</p></li>
+<li><p>Some means of applying random “mutations” (e.g. adding a term, deleting a term)</p></li>
+</ul>
+<div class="figure align-default" id="genetic-programming-mutation-types">
+<img alt="../_images/gp_mutation_types.png" src="../_images/gp_mutation_types.png" />
+<p class="caption"><span class="caption-number">Fig. 6 </span><span class="caption-text">Some examples for different mutation types</span><a class="headerlink" href="#genetic-programming-mutation-types" title="Permalink to this image">#</a></p>
+</div>
+<p>“Crossover” and “subtree” mutations combine features from multiple candidate equations, imitating the process of genetic recombination from sexual reproduction. “Hoist” and “point” mutations modify individual equations, imitating the process of genetic mutation from, e.g., random errors in DNA replication.</p>
+<p>Specific mutations aside, programs can have a limited lifespan, and those with high “fitness” produce more “offspring” (possibly-mutated versions of themselves). However, everything is probabilistic, so programs with lower fitness can still persist. This is critically important, because it allows genetic programming to overcome local minima during optimization.</p>
+<p>Note that although genetic programming can be a robust way to solve extremely difficult non-convex optimization problems (as perhaps evidenced by life itself), it also tends to be slow and resource-intensive.</p>
+<div class="section" id="gplearn">
+<span id="gplearn-sec"></span><h3>gplearn<a class="headerlink" href="#gplearn" title="Permalink to this headline">#</a></h3>
+<p>One nicely-documented library for “pure” genetic programming-based symbolic regression is <a class="reference external" href="https://gplearn.readthedocs.io/en/stable/">gplearn</a>, which provides a scikit-learn style API and a number of configuration options:</p>
+<div class="section" id="run-gplearn">
+<h4>Run gplearn<a class="headerlink" href="#run-gplearn" title="Permalink to this headline">#</a></h4>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">gplearn</span>
+<span class="kn">import</span> <span class="nn">gplearn.genetic</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">gplearn_sr</span> <span class="o">=</span> <span class="n">gplearn</span><span class="o">.</span><span class="n">genetic</span><span class="o">.</span><span class="n">SymbolicRegressor</span><span class="p">(</span>
+    <span class="n">population_size</span><span class="o">=</span><span class="mi">5000</span><span class="p">,</span>
+    <span class="n">generations</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span>
+    <span class="n">p_crossover</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span>
+    <span class="n">p_subtree_mutation</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
+    <span class="n">p_hoist_mutation</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span>
+    <span class="n">p_point_mutation</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
+    <span class="n">max_samples</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span>
+    <span class="n">verbose</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+    <span class="n">parsimony_coefficient</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span>
+    <span class="n">function_set</span><span class="o">=</span><span class="p">(</span><span class="s2">&quot;add&quot;</span><span class="p">,</span> <span class="s2">&quot;mul&quot;</span><span class="p">,</span> <span class="s2">&quot;sin&quot;</span><span class="p">),</span>
+<span class="p">)</span>
+<span class="n">gplearn_sr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>    |   Population Average    |             Best Individual              |
+---- ------------------------- ------------------------------------------ ----------
+ Gen   Length          Fitness   Length          Fitness      OOB Fitness  Time Left
+   0    18.23          1.93622        6         0.536217         0.582676      2.27m
+   1     7.14          1.04392       11         0.399333         0.352389      1.80m
+   2     6.06         0.982334       11         0.385689         0.475179      1.65m
+   3     4.27         0.883136       11          0.37803         0.544117      1.58m
+   4     5.67         0.881956       10         0.351756         0.422697      1.56m
+   5     9.36         0.840405       10          0.35068         0.432386      1.66m
+   6    10.67         0.822111        9         0.283554         0.308482      1.72m
+   7    10.22         0.836667        9         0.282104         0.321529      1.60m
+   8     9.95         0.865733        9         0.279664         0.343494      1.52m
+   9     9.92         0.859199        9         0.276768          0.36956      1.52m
+  10     9.50         0.863652        9         0.276001         0.376458      1.45m
+  11     9.00         0.845543        9         0.275458         0.381348      1.35m
+  12     8.94         0.824171        9         0.275452         0.381396      1.35m
+  13     8.98         0.853988        9          0.27615         0.375116      1.27m
+  14     8.98         0.822575        9         0.276182          0.37483      1.24m
+  15     8.99         0.835989        9         0.277461         0.363321      1.25m
+  16     9.10         0.851439        9          0.27544         0.381508      1.18m
+  17     8.90         0.843436        9         0.277732         0.360881      1.12m
+  18     9.02         0.832375        9         0.276722         0.369974      1.14m
+  19     9.05         0.829508       12         0.207436         0.253782      1.06m
+  20     8.99         0.831832       12         0.208956         0.240096      1.03m
+  21     9.13         0.837242       12         0.207388         0.254212     59.94s
+  22     9.28         0.844509       13        0.0465518        0.0697833      1.01m
+  23    10.05         0.858256       13        0.0419453         0.111242     56.88s
+  24    11.52         0.884959       13        0.0431528         0.100375     56.73s
+  25    12.15         0.860891       13        0.0406882         0.122556     57.38s
+  26    12.72         0.821942       13         0.041453         0.115673     53.96s
+  27    12.90         0.795284       13        0.0412741         0.117283     50.80s
+  28    12.86         0.780633       13        0.0393452         0.134643     49.16s
+  29    12.92         0.771492       13        0.0385569         0.141738     45.30s
+  30    12.93         0.769627       13        0.0419824         0.110908     44.64s
+  31    12.79         0.795298       13        0.0404993         0.124256     40.82s
+  32    12.81         0.772848       13        0.0401964         0.126982     38.69s
+  33    12.79         0.773815       13         0.041688         0.113557     36.32s
+  34    12.91         0.768662       13        0.0394116         0.134045     35.33s
+  35    13.01         0.788808       13        0.0382859         0.144176     31.70s
+  36    12.81          0.78538       13        0.0412858         0.117178     29.43s
+  37    12.80         0.768888       13        0.0402093         0.126866     28.18s
+  38    12.81         0.777798       13        0.0423059        0.0918826     24.97s
+  39    13.00         0.777302       13        0.0391069         0.120674     22.72s
+  40    12.87         0.830627       13        0.0392723         0.119185     21.15s
+  41    12.78         0.794545       13        0.0394023         0.118015     18.22s
+  42    12.70         0.785239       13        0.0402226         0.110632     16.66s
+  43    12.91         0.792994       13        0.0383406         0.127571     13.65s
+  44    12.96         0.784547       13        0.0396515         0.115772     11.38s
+  45    12.87         0.787953       13        0.0368546         0.140945      9.20s
+  46    12.97         0.788778       13        0.0384182         0.126872      7.12s
+  47    12.93         0.786967       13        0.0396641         0.115659      4.58s
+  48    12.87          0.78805       13        0.0409886         0.100143      2.29s
+  49    12.92         0.794259       13        0.0380632         0.130067      0.00s
+</pre></div>
+</div>
+<div class="output text_html"><style>#sk-container-id-1 {color: black;background-color: white;}#sk-container-id-1 pre{padding: 0;}#sk-container-id-1 div.sk-toggleable {background-color: white;}#sk-container-id-1 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-1 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-1 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-1 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-1 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-1 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-1 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-1 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-1 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-1 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-1 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-1 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-1 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-1 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-1 div.sk-item {position: relative;z-index: 1;}#sk-container-id-1 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-1 div.sk-item::before, #sk-container-id-1 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-1 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-1 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-1 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-1 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-1 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-1 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-1 div.sk-label-container {text-align: center;}#sk-container-id-1 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-1 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-1" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>add(mul(X0, X0), mul(mul(-0.496, X1), add(mul(-0.890, X0), X1)))</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-1" type="checkbox" checked><label for="sk-estimator-id-1" class="sk-toggleable__label sk-toggleable__label-arrow">SymbolicRegressor</label><div class="sk-toggleable__content"><pre>add(mul(X0, X0), mul(mul(-0.496, X1), add(mul(-0.890, X0), X1)))</pre></div></div></div></div></div></div></div>
+</div>
+</div>
+<div class="section" id="interpret-results">
+<h4>Interpret results<a class="headerlink" href="#interpret-results" title="Permalink to this headline">#</a></h4>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">gplearn_sr</span><span class="o">.</span><span class="n">_program</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>add(mul(X0, X0), mul(mul(-0.496, X1), add(mul(-0.890, X0), X1)))
+</pre></div>
+</div>
+</div>
+</div>
+<p>This looks very close to the right answer, though the constants are slightly off.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">const</span> <span class="k">for</span> <span class="n">const</span> <span class="ow">in</span> <span class="n">gplearn_sr</span><span class="o">.</span><span class="n">_program</span><span class="o">.</span><span class="n">program</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">const</span><span class="p">,</span> <span class="nb">float</span><span class="p">)]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[-0.496430992770164, -0.8899106568576463]
+</pre></div>
+</div>
+</div>
+</div>
+<p>This is likely because gplearn mutations which add or update constants always pick values by drawing random uniform values (within pre-specified ranges, by default -1 to 1). In my limited experience so far, this is one of the major inefficiencies.</p>
+</div>
+</div>
+<div class="section" id="pysr">
+<span id="pysr-sec"></span><h3>PySR<a class="headerlink" href="#pysr" title="Permalink to this headline">#</a></h3>
+<p><a class="reference external" href="https://pysr.readthedocs.io/en/latest/">PySR</a> is a different library for symbolic regression which, though still based on genetic programming, includes simulated annealing and gradient-free optimization to set the values of constants as well as performance improvements (although accessible via a Python API, the main library is written in Julia for speed). This seems to make it a bit more reliable and precise.</p>
+<div class="section" id="run-pysr">
+<h4>Run PySR<a class="headerlink" href="#run-pysr" title="Permalink to this headline">#</a></h4>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pysr</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equations</span> <span class="o">=</span> <span class="n">pysr</span><span class="o">.</span><span class="n">PySRRegressor</span><span class="p">(</span>
+    <span class="n">niterations</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
+    <span class="n">binary_operators</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;+&quot;</span><span class="p">,</span> <span class="s2">&quot;*&quot;</span><span class="p">],</span>  <span class="c1"># operators that can combine two terms</span>
+    <span class="n">unary_operators</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;sin&quot;</span><span class="p">],</span>  <span class="c1"># operators that modify a single term</span>
+<span class="p">)</span>
+<span class="n">equations</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Compiling Julia backend...
+Started!
+</pre></div>
+</div>
+<div class="output text_html"><style>#sk-container-id-2 {color: black;background-color: white;}#sk-container-id-2 pre{padding: 0;}#sk-container-id-2 div.sk-toggleable {background-color: white;}#sk-container-id-2 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-2 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-2 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-2 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-2 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-2 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-2 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-2 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-2 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-2 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-2 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-2 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-2 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-2 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-2 div.sk-item {position: relative;z-index: 1;}#sk-container-id-2 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-2 div.sk-item::before, #sk-container-id-2 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-2 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-2 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-2 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-2 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-2 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-2 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-2 div.sk-label-container {text-align: center;}#sk-container-id-2 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-2 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-2" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>PySRRegressor.equations_ = [
+	    pick         score                                           equation  \
+	0         0.000000e+00                                         0.48876795   
+	1         6.306835e-07                                    sin(0.51097894)   
+	2         1.048097e+00                                          (x0 * x0)   
+	3         1.712557e-01                           (-0.5235571 + (x0 * x0))   
+	4         8.942267e-08                      ((x0 * x0) + sin(-0.5510205))   
+	5         9.991993e-05           (((x0 * x0) + -0.52071977) * 0.99427116)   
+	6         4.062558e-02              ((x0 * (x0 + sin(x1))) + -0.51387244)   
+	7         1.416445e+00            ((x0 * x0) + ((x1 * x1) * -0.49461997))   
+	8         1.982723e-04       ((x0 * x0) + ((x1 * x1) * sin(-0.51387244)))   
+	9         1.303444e-03  (((x0 * x0) * 1.008712) + ((x1 * x1) * -0.4946...   
+	10        3.366265e-01  ((x0 * (x0 + sin(x1))) + ((x1 * x1) * -0.51387...   
+	11        4.537356e-01  ((x0 * (x0 + sin(sin(x1)))) + ((x1 * x1) * -0....   
+	12        8.790939e-01  ((x0 * (x0 + (sin(x1) * 0.70106995))) + ((x1 *...   
+	13        3.062261e-01  ((x0 * (x0 + (sin(sin(x1)) * 0.70106995))) + (...   
+	14  &gt;&gt;&gt;&gt;  3.254359e-01  ((x0 * x0) + (((x1 * x1) + (-1.0297319 * (x1 *...   
+	15        1.633921e-02  ((0.26510233 * ((-1.1019801 * 1.7545131) * ((s...   
+	
+	        loss  complexity  
+	0   2.695489           1  
+	1   2.695488           2  
+	2   0.945049           3  
+	3   0.670971           5  
+	4   0.670971           6  
+	5   0.670904           7  
+	6   0.644194           8  
+	7   0.156265           9  
+	8   0.156234          10  
+	9   0.156031          11  
+	10  0.111433          12  
+	11  0.070788          13  
+	12  0.029388          14  
+	13  0.021636          15  
+	14  0.015626          16  
+	15  0.014879          19  
+]</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-2" type="checkbox" checked><label for="sk-estimator-id-2" class="sk-toggleable__label sk-toggleable__label-arrow">PySRRegressor</label><div class="sk-toggleable__content"><pre>PySRRegressor.equations_ = [
+	    pick         score                                           equation  \
+	0         0.000000e+00                                         0.48876795   
+	1         6.306835e-07                                    sin(0.51097894)   
+	2         1.048097e+00                                          (x0 * x0)   
+	3         1.712557e-01                           (-0.5235571 + (x0 * x0))   
+	4         8.942267e-08                      ((x0 * x0) + sin(-0.5510205))   
+	5         9.991993e-05           (((x0 * x0) + -0.52071977) * 0.99427116)   
+	6         4.062558e-02              ((x0 * (x0 + sin(x1))) + -0.51387244)   
+	7         1.416445e+00            ((x0 * x0) + ((x1 * x1) * -0.49461997))   
+	8         1.982723e-04       ((x0 * x0) + ((x1 * x1) * sin(-0.51387244)))   
+	9         1.303444e-03  (((x0 * x0) * 1.008712) + ((x1 * x1) * -0.4946...   
+	10        3.366265e-01  ((x0 * (x0 + sin(x1))) + ((x1 * x1) * -0.51387...   
+	11        4.537356e-01  ((x0 * (x0 + sin(sin(x1)))) + ((x1 * x1) * -0....   
+	12        8.790939e-01  ((x0 * (x0 + (sin(x1) * 0.70106995))) + ((x1 *...   
+	13        3.062261e-01  ((x0 * (x0 + (sin(sin(x1)) * 0.70106995))) + (...   
+	14  &gt;&gt;&gt;&gt;  3.254359e-01  ((x0 * x0) + (((x1 * x1) + (-1.0297319 * (x1 *...   
+	15        1.633921e-02  ((0.26510233 * ((-1.1019801 * 1.7545131) * ((s...   
+	
+	        loss  complexity  
+	0   2.695489           1  
+	1   2.695488           2  
+	2   0.945049           3  
+	3   0.670971           5  
+	4   0.670971           6  
+	5   0.670904           7  
+	6   0.644194           8  
+	7   0.156265           9  
+	8   0.156234          10  
+	9   0.156031          11  
+	10  0.111433          12  
+	11  0.070788          13  
+	12  0.029388          14  
+	13  0.021636          15  
+	14  0.015626          16  
+	15  0.014879          19  
+]</pre></div></div></div></div></div></div></div>
+</div>
+</div>
+<div class="section" id="id1">
+<h4>Interpret results<a class="headerlink" href="#id1" title="Permalink to this headline">#</a></h4>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">round_expr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">num_digits</span><span class="o">=</span><span class="mi">4</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">n</span><span class="p">:</span> <span class="nb">round</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">num_digits</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Number</span><span class="p">)})</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">round_expr</span><span class="p">(</span><span class="n">equations</span><span class="o">.</span><span class="n">sympy</span><span class="p">())</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_latex math notranslate nohighlight">
+\[\displaystyle x_{0}^{2} - 0.5139 x_{1}^{2} + 0.5292 x_{1} \sin{\left(x_{0} \right)}\]</div>
+</div>
+</div>
+<p>It looks like PySR is able to not only discover the correct equation, but also format it for us nicely.</p>
+<p>In addition, it saves a Pareto frontier of possible expressions of varying complexities, which had the lowest error of any other equations of equal or lesser complexity (with a configurable trade-off rule):</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equations</span><span class="o">.</span><span class="n">equations_</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><div>
+<style scoped>
+    .dataframe tbody tr th:only-of-type {
+        vertical-align: middle;
+    }
+
+    .dataframe tbody tr th {
+        vertical-align: top;
+    }
+
+    .dataframe thead th {
+        text-align: right;
+    }
+</style>
+<table border="1" class="dataframe">
+  <thead>
+    <tr style="text-align: right;">
+      <th></th>
+      <th>complexity</th>
+      <th>loss</th>
+      <th>score</th>
+      <th>equation</th>
+      <th>sympy_format</th>
+      <th>lambda_format</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <th>0</th>
+      <td>1</td>
+      <td>2.695489</td>
+      <td>0.000000e+00</td>
+      <td>0.48876795</td>
+      <td>0.488767950000000</td>
+      <td>PySRFunction(X=&gt;0.488767950000000)</td>
+    </tr>
+    <tr>
+      <th>1</th>
+      <td>2</td>
+      <td>2.695488</td>
+      <td>6.306835e-07</td>
+      <td>sin(0.51097894)</td>
+      <td>0.489031377338912</td>
+      <td>PySRFunction(X=&gt;0.489031377338912)</td>
+    </tr>
+    <tr>
+      <th>2</th>
+      <td>3</td>
+      <td>0.945049</td>
+      <td>1.048097e+00</td>
+      <td>(x0 * x0)</td>
+      <td>x0**2</td>
+      <td>PySRFunction(X=&gt;x0**2)</td>
+    </tr>
+    <tr>
+      <th>3</th>
+      <td>5</td>
+      <td>0.670971</td>
+      <td>1.712557e-01</td>
+      <td>(-0.5235571 + (x0 * x0))</td>
+      <td>x0**2 - 0.5235571</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.5235571)</td>
+    </tr>
+    <tr>
+      <th>4</th>
+      <td>6</td>
+      <td>0.670971</td>
+      <td>8.942267e-08</td>
+      <td>((x0 * x0) + sin(-0.5510205))</td>
+      <td>x0**2 - 0.523556957885906</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.523556957885906)</td>
+    </tr>
+    <tr>
+      <th>5</th>
+      <td>7</td>
+      <td>0.670904</td>
+      <td>9.991993e-05</td>
+      <td>(((x0 * x0) + -0.52071977) * 0.99427116)</td>
+      <td>0.99427116*x0**2 - 0.517736649752833</td>
+      <td>PySRFunction(X=&gt;0.99427116*x0**2 - 0.517736649...</td>
+    </tr>
+    <tr>
+      <th>6</th>
+      <td>8</td>
+      <td>0.644194</td>
+      <td>4.062558e-02</td>
+      <td>((x0 * (x0 + sin(x1))) + -0.51387244)</td>
+      <td>x0*(x0 + sin(x1)) - 0.51387244</td>
+      <td>PySRFunction(X=&gt;x0*(x0 + sin(x1)) - 0.51387244)</td>
+    </tr>
+    <tr>
+      <th>7</th>
+      <td>9</td>
+      <td>0.156265</td>
+      <td>1.416445e+00</td>
+      <td>((x0 * x0) + ((x1 * x1) * -0.49461997))</td>
+      <td>x0**2 - 0.49461997*x1**2</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.49461997*x1**2)</td>
+    </tr>
+    <tr>
+      <th>8</th>
+      <td>10</td>
+      <td>0.156234</td>
+      <td>1.982723e-04</td>
+      <td>((x0 * x0) + ((x1 * x1) * sin(-0.51387244)))</td>
+      <td>x0**2 - 0.491553228879802*x1**2</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.491553228879802*x1**2)</td>
+    </tr>
+    <tr>
+      <th>9</th>
+      <td>11</td>
+      <td>0.156031</td>
+      <td>1.303444e-03</td>
+      <td>(((x0 * x0) * 1.008712) + ((x1 * x1) * -0.4946...</td>
+      <td>1.008712*x0**2 - 0.49461997*x1**2</td>
+      <td>PySRFunction(X=&gt;1.008712*x0**2 - 0.49461997*x1...</td>
+    </tr>
+    <tr>
+      <th>10</th>
+      <td>12</td>
+      <td>0.111433</td>
+      <td>3.366265e-01</td>
+      <td>((x0 * (x0 + sin(x1))) + ((x1 * x1) * -0.51387...</td>
+      <td>x0*(x0 + sin(x1)) - 0.51387244*x1**2</td>
+      <td>PySRFunction(X=&gt;x0*(x0 + sin(x1)) - 0.51387244...</td>
+    </tr>
+    <tr>
+      <th>11</th>
+      <td>13</td>
+      <td>0.070788</td>
+      <td>4.537356e-01</td>
+      <td>((x0 * (x0 + sin(sin(x1)))) + ((x1 * x1) * -0....</td>
+      <td>x0*(x0 + sin(sin(x1))) - 0.51387244*x1**2</td>
+      <td>PySRFunction(X=&gt;x0*(x0 + sin(sin(x1))) - 0.513...</td>
+    </tr>
+    <tr>
+      <th>12</th>
+      <td>14</td>
+      <td>0.029388</td>
+      <td>8.790939e-01</td>
+      <td>((x0 * (x0 + (sin(x1) * 0.70106995))) + ((x1 *...</td>
+      <td>x0*(x0 + 0.70106995*sin(x1)) - 0.51387244*x1**2</td>
+      <td>PySRFunction(X=&gt;x0*(x0 + 0.70106995*sin(x1)) -...</td>
+    </tr>
+    <tr>
+      <th>13</th>
+      <td>15</td>
+      <td>0.021636</td>
+      <td>3.062261e-01</td>
+      <td>((x0 * (x0 + (sin(sin(x1)) * 0.70106995))) + (...</td>
+      <td>x0*(x0 + 0.70106995*sin(sin(x1))) - 0.51387244...</td>
+      <td>PySRFunction(X=&gt;x0*(x0 + 0.70106995*sin(sin(x1...</td>
+    </tr>
+    <tr>
+      <th>14</th>
+      <td>16</td>
+      <td>0.015626</td>
+      <td>3.254359e-01</td>
+      <td>((x0 * x0) + (((x1 * x1) + (-1.0297319 * (x1 *...</td>
+      <td>x0**2 - 0.51387244*x1**2 + 0.529150843998836*x...</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.51387244*x1**2 + 0.5...</td>
+    </tr>
+    <tr>
+      <th>15</th>
+      <td>19</td>
+      <td>0.014879</td>
+      <td>1.633921e-02</td>
+      <td>((0.26510233 * ((-1.1019801 * 1.7545131) * ((s...</td>
+      <td>x0**2 - 0.512559056932061*x1*(x1 - sin(0.89952...</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.512559056932061*x1*(...</td>
+    </tr>
+  </tbody>
+</table>
+</div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equations</span><span class="o">.</span><span class="n">equations_</span><span class="o">.</span><span class="n">loss</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0     2.695489
+1     2.695488
+2     0.945049
+3     0.670971
+4     0.670971
+5     0.670904
+6     0.644194
+7     0.156265
+8     0.156234
+9     0.156031
+10    0.111433
+11    0.070788
+12    0.029388
+13    0.021636
+14    0.015626
+15    0.014879
+Name: loss, dtype: float64
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">equations</span><span class="o">.</span><span class="n">equations_</span><span class="p">)),</span>
+    <span class="n">equations</span><span class="o">.</span><span class="n">equations_</span><span class="o">.</span><span class="n">loss</span><span class="p">,</span>
+    <span class="n">width</span><span class="o">=</span><span class="mf">0.33</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+
+
+<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">&quot;log&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Mean squared error&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span>
+    <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">equations</span><span class="o">.</span><span class="n">equations_</span><span class="p">)),</span>
+    <span class="p">[</span><span class="sa">f</span><span class="s2">&quot;$</span><span class="si">{</span><span class="n">latex</span><span class="p">(</span><span class="n">round_expr</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="mi">7</span><span class="p">))</span><span class="si">}</span><span class="s2">$&quot;</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">equations</span><span class="o">.</span><span class="n">equations_</span><span class="o">.</span><span class="n">sympy_format</span><span class="p">],</span>
+    <span class="n">rotation</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span>
+    <span class="n">ha</span><span class="o">=</span><span class="s2">&quot;right&quot;</span><span class="p">,</span>
+    <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;PySR Pareto frontier&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Expression (more complex $</span><span class="se">\\</span><span class="s2">to$)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+
+<span class="n">ax2</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">twinx</span><span class="p">()</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">equations</span><span class="o">.</span><span class="n">equations_</span><span class="p">))</span> <span class="o">+</span> <span class="mf">0.33</span><span class="p">,</span>
+    <span class="n">equations</span><span class="o">.</span><span class="n">equations_</span><span class="o">.</span><span class="n">score</span><span class="p">,</span>
+    <span class="n">width</span><span class="o">=</span><span class="mf">0.33</span><span class="p">,</span>
+    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;orange&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;PySR score&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;orange&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/95fce3daa97c9ad0db5a4e2419aa317286dfbb63a80cbfef45b6791b86fa6b6c.png" src="../_images/95fce3daa97c9ad0db5a4e2419aa317286dfbb63a80cbfef45b6791b86fa6b6c.png" />
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="manually-constructed-library-sparse-regression">
+<span id="sparse-regression-sec"></span><h2>Manually Constructed Library + Sparse Regression<a class="headerlink" href="#manually-constructed-library-sparse-regression" title="Permalink to this headline">#</a></h2>
+<p>Another approach to performing symbolic regression is to</p>
+<ul class="simple">
+<li><p>Hand-construct a library of basis functions (e.g. by recursively combining all terms with various operations)</p></li>
+<li><p>Run sparse linear regression to extract a small set of basis functions</p></li>
+</ul>
+<p>This approach works well for problems where any constants in the underlying formula can be extracted out to the outermost level (so they can be determined by linear regression). It also works well when we have domain knowledge about which terms could potentially appear in the final sum.</p>
+<p>It does not work well in cases where constants can’t be pulled outside certain operations (e.g. in this case, with the sine operator, though powers and derivatives are generally fine). This method can also struggle when basis function values are highly correlated, though it depends on the sparse regression method.</p>
+<p>Let’s create an augmentation function which takes an array of features and adds additional basis functions by combining them with binary multiplication and unary sine operations:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">augment</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">names</span><span class="p">):</span>
+    <span class="n">existing</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">names</span><span class="p">)</span>
+    <span class="n">new_feats</span> <span class="o">=</span> <span class="p">[]</span>
+    <span class="n">new_names</span> <span class="o">=</span> <span class="p">[]</span>
+
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">feats</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+        <span class="n">name</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">&quot;sin(</span><span class="si">{</span><span class="n">names</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="si">}</span><span class="s2">)&quot;</span>
+        <span class="k">if</span> <span class="n">name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">existing</span><span class="p">:</span>
+            <span class="n">new_feats</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">feats</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]))</span>
+            <span class="n">new_names</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
+        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">feats</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
+            <span class="n">name</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">&quot;mul(</span><span class="si">{</span><span class="n">names</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="si">}</span><span class="s2">, </span><span class="si">{</span><span class="n">names</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="si">}</span><span class="s2">)&quot;</span>
+            <span class="k">if</span> <span class="n">name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">existing</span><span class="p">:</span>
+                <span class="n">new_feats</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">feats</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">feats</span><span class="p">[:,</span> <span class="n">j</span><span class="p">])</span>
+                <span class="n">new_names</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">feats</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">new_feats</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="p">)),</span> <span class="n">names</span> <span class="o">+</span> <span class="n">new_names</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">repeatedly_augment</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">depth</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">names</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="n">names</span> <span class="o">=</span> <span class="p">[</span><span class="sa">f</span><span class="s2">&quot;x</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">feats</span><span class="p">))]</span>
+    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">depth</span><span class="p">):</span>
+        <span class="n">feats</span><span class="p">,</span> <span class="n">names</span> <span class="o">=</span> <span class="n">augment</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">names</span><span class="p">)</span>
+    <span class="k">return</span> <span class="n">feats</span><span class="p">,</span> <span class="n">names</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">feats</span><span class="p">,</span> <span class="n">names</span> <span class="o">=</span> <span class="n">repeatedly_augment</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="s2">&quot;x0&quot;</span><span class="p">,</span> <span class="s2">&quot;x1&quot;</span><span class="p">],</span> <span class="n">depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>Applying this twice brings us from 2 features up to 37:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">feats</span><span class="o">.</span><span class="n">shape</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(1000, 37)
+</pre></div>
+</div>
+</div>
+</div>
+<p>We can see that the complexity of this approach starts to blow up exponentially, though, as we increase the maximum depth of an expression:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">augment</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">names</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(1000, 742)
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">augment</span><span class="p">(</span><span class="o">*</span><span class="n">augment</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">names</span><span class="p">))[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(1000, 276397)
+</pre></div>
+</div>
+</div>
+</div>
+<p>If we need arbitrary polynomials of up to 4th order, we end up with orders of magnitude more features than samples, which will cause problems.</p>
+<p>Note that since this method lacks the ability to apply operations with constants within other operations, <span class="math notranslate nohighlight">\(\sin(0.5 x_0 x_1)\)</span> isn’t present in any of these possible sets, though <span class="math notranslate nohighlight">\(\sin(x_0 x_1)\)</span> is.</p>
+<p>Let’s now see how different methods for (sparse) linear regression perform on this task:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">learned_expr</span><span class="p">(</span><span class="n">coef</span><span class="p">,</span> <span class="n">names</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">):</span>
+    <span class="n">coef_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argwhere</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">cutoff</span><span class="p">)[:,</span> <span class="mi">0</span><span class="p">]</span>
+    <span class="n">sort_order</span> <span class="o">=</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">coef</span><span class="p">[</span><span class="n">coef_idx</span><span class="p">])))</span>
+    <span class="k">return</span> <span class="s2">&quot; + &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
+        <span class="p">[</span><span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">coef</span><span class="p">[</span><span class="n">coef_idx</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">*</span><span class="si">{</span><span class="n">names</span><span class="p">[</span><span class="n">coef_idx</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="si">}</span><span class="s2">&quot;</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sort_order</span><span class="p">]</span>
+    <span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="linear-regression">
+<span id="linear-regression-sec"></span><h3>Linear regression<a class="headerlink" href="#linear-regression" title="Permalink to this headline">#</a></h3>
+<p>We can start with completely unregularized linear regression, which will find the linear combination of basis terms that minimizes mean squared error (to high precision, due to the convexity of the problem and an analytic way of solving it).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linreg</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
+<span class="n">linreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">linreg</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">names</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;1.0404*mul(x0, x0) + -0.8475*mul(sin(x0), sin(x1)) + 0.6921*mul(x0, sin(x1)) + 0.6754*mul(x1, sin(x0)) + -0.5292*mul(x1, x1) + -0.5269*x1 + 0.4580*sin(x1) + -0.3849*x0 + 0.3470*sin(x0) + 0.0699*sin(sin(x1)) + -0.0635*mul(x0, x1) + 0.0542*mul(x1, mul(x1, x1)) + -0.0540*mul(x0, sin(x0)) + 0.0408*mul(sin(x1), mul(x1, x1)) + 0.0397*mul(x0, mul(x0, x0)) + 0.0394*sin(sin(x0)) + 0.0376*mul(x1, sin(x1)) + 0.0344*sin(mul(x0, x1)) + 0.0285*mul(sin(x0), mul(x0, x0)) + -0.0196*mul(mul(x0, x0), mul(x0, x1)) + -0.0176*mul(mul(x0, x1), mul(x1, x1))&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<p>Unfortunately, this doesn’t seem to give us a very sparse solution in this case.</p>
+</div>
+<div class="section" id="l1-regularized-linear-regression-lasso">
+<span id="lasso-sec"></span><h3>L1-regularized linear regression (LASSO)<a class="headerlink" href="#l1-regularized-linear-regression-lasso" title="Permalink to this headline">#</a></h3>
+<p>A common way to make linear regression return more sparse solutions is to minimize a linear combination of the mean squared error and the sum of the magnitudes of the weights. This is called applying an “L1 penalty” since it is the L1-norm of the weight vector, and is also referred to as the “Least Absolute Shrinkage and Selection Operator”, or LASSO. From a Bayesian perspective, this corresponds to doing Bayesian linear regression with a Laplace or double-exponential prior on the weights.</p>
+<p>Note that although this penalty does tend to encourage sparsity, it’s also meant as a regularizer, which can help with the presence of noise.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">lasso</span> <span class="o">=</span> <span class="n">Lasso</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">lasso</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">lasso</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">names</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;0.9620*mul(x0, x0) + -0.4842*mul(x1, x1) + 0.2276*mul(x0, x1) + 0.1274*mul(x1, sin(x0)) + 0.1011*sin(mul(x0, x1)) + 0.0930*mul(x0, sin(x1))&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<p>Here, LASSO ends up doing a pretty good job finding the first two terms of the ground-truth model (<span class="math notranslate nohighlight">\(x_0^2 - 0.5x_1^2\)</span>), with an approximation of the remaining <span class="math notranslate nohighlight">\(\sin(0.5 x_0 x_1)\)</span> term as <span class="math notranslate nohighlight">\( 0.2(x_0\sin(x_1)+x_1\sin(x_0)) + 0.1(\sin(x_0 x_1) + x_0x_1)\)</span>:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">x0</span> <span class="o">=</span> <span class="n">x</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
+<span class="n">x1</span> <span class="o">=</span> <span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span>
+<span class="n">sin</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
+    <span class="n">sin</span><span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">x0</span> <span class="o">*</span> <span class="n">x1</span><span class="p">),</span>
+    <span class="mf">0.2</span> <span class="o">*</span> <span class="p">(</span><span class="n">x0</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span> <span class="o">+</span> <span class="n">x1</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span> <span class="o">+</span> <span class="mf">0.1</span> <span class="o">*</span> <span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x0</span> <span class="o">*</span> <span class="n">x1</span><span class="p">)</span> <span class="o">+</span> <span class="n">x0</span> <span class="o">*</span> <span class="n">x1</span><span class="p">),</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$\sin(0.5 x_0 x_1)$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span>
+    <span class="sa">r</span><span class="s2">&quot;$0.2(x_0\sin(x_1)+x_1\sin(x_0))$&quot;</span> <span class="o">+</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span> <span class="o">+</span> <span class="sa">r</span><span class="s2">&quot;$+ 0.1(\sin(x_0 x_1) + x_0x_1)$&quot;</span><span class="p">,</span>
+    <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/8459a9ba6bfd36d76cf05773faff380d44a3f86faada576a0d0d110031c862ea.png" src="../_images/8459a9ba6bfd36d76cf05773faff380d44a3f86faada576a0d0d110031c862ea.png" />
+</div>
+</div>
+</div>
+<div class="section" id="relevance-vector-machines-rvm">
+<span id="rvm-sec"></span><h3>Relevance vector machines (RVM)<a class="headerlink" href="#relevance-vector-machines-rvm" title="Permalink to this headline">#</a></h3>
+<p>Relevance vector machines, used by <a class="reference external" href="https://laurezanna.github.io/files/Zanna-Bolton-2020.pdf">Zanna and Bolton 2020</a> for equation discovery, provide another way of learning sparse linear models in a Bayesian manner, though with a different prior that prioritizes sparsity over shrinkage when the data permits it. RVMs can also be used with a kernel trick to learn nonlinear decision boundaries which are sparse in a kernelized feature space, though in this case, we use a linear kernel on top of our manually computed library features.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rvm</span> <span class="o">=</span> <span class="n">RVR</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">rvm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">names</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">rvm</span><span class="o">.</span><span class="n">m_</span><span class="p">,</span> <span class="n">rvm</span><span class="o">.</span><span class="n">labels</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;1.0032*mul(x0, x0) + -0.8532*mul(sin(x0), sin(x1)) + 0.6930*mul(x1, sin(x0)) + 0.6830*mul(x0, sin(x1)) + -0.5051*mul(x1, x1) + -0.0669*mul(x0, x1) + 0.0344*sin(mul(x0, x1)) + 0.0193*sin(x0) + -0.0192*sin(sin(x0)) + -0.0192*mul(mul(x0, x1), mul(x1, x1)) + -0.0178*mul(mul(x0, x0), mul(x0, x1))&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<p>RVM does discover the <span class="math notranslate nohighlight">\(x_0^2\)</span> and <span class="math notranslate nohighlight">\(-0.5x_1^2\)</span> terms almost exactly, but ends up with an even more complex approximate expression for the missing <span class="math notranslate nohighlight">\(\sin(0.5x_0x_1)\)</span> term.</p>
+</div>
+<div class="section" id="sequentially-thresholded-least-squares-stlsq-from-pysindy">
+<span id="stlsq-sec"></span><h3>Sequentially Thresholded Least Squares (STLSQ) from pysindy<a class="headerlink" href="#sequentially-thresholded-least-squares-stlsq-from-pysindy" title="Permalink to this headline">#</a></h3>
+<p>The PySINDy library provides a number of efficient Python implementations of sparse regression algorithms. Their default regression method is sequentially thresholded least squares (STLSQ), which (as its name might suggest) repeatedly runs L2-regularized linear regression with penalty strength <code class="docutils literal notranslate"><span class="pre">alpha</span></code> and prunes features with weight magnitudes below a given <code class="docutils literal notranslate"><span class="pre">threshold</span></code>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">stlsq</span><span class="o">.</span><span class="n">STLSQ</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.0001</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+<span class="n">stlsq</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">stlsq</span><span class="o">.</span><span class="n">coef_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">names</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;1.0070*mul(x0, x0) + -0.4991*mul(x1, x1) + 0.3110*mul(x1, sin(x0)) + 0.2824*mul(x0, sin(x1))&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<p>This method ends up finding a sparser solution than Lasso, though I did tweak the parameters a bit to make that happen :)</p>
+<p>It’s worth checking out <a class="reference external" href="https://pysindy.readthedocs.io/en/latest/api/pysindy.optimizers.html">PySINDy’s full suite of sparse regression methods</a> for other approaches!</p>
+<p>Overall, all of these methods were able to find the <span class="math notranslate nohighlight">\(x_0^2\)</span> and <span class="math notranslate nohighlight">\(-0.5x_1^2\)</span> terms, but all of them also needed to find approximations for the remaining term because it wasn’t in the feature library. Meaning the expressions are both less accurate and more complex.</p>
+</div>
+<div class="section" id="tweaking-the-feature-library">
+<h3>Tweaking the Feature Library<a class="headerlink" href="#tweaking-the-feature-library" title="Permalink to this headline">#</a></h3>
+<p>Now, we can potentially resolve some of these issues by pre-multiplying <span class="math notranslate nohighlight">\(x_1\)</span> by 0.5 before the augmentation process (which we’ll call <span class="math notranslate nohighlight">\(x_{1h}\)</span>).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">feats2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+<span class="n">feats2</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">feats2</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="mf">0.5</span>
+<span class="n">names2</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;x0&quot;</span><span class="p">,</span> <span class="s2">&quot;x1h&quot;</span><span class="p">]</span>
+<span class="n">depth</span> <span class="o">=</span> <span class="mi">2</span>
+
+<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">depth</span><span class="p">):</span>
+    <span class="n">feats2</span><span class="p">,</span> <span class="n">names2</span> <span class="o">=</span> <span class="n">augment</span><span class="p">(</span><span class="n">feats2</span><span class="p">,</span> <span class="n">names2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>In this case, the correct model should be expressed as</p>
+<div class="math notranslate nohighlight">
+\[
+y = x_0^2 - 2x_{1h}^2 + \sin(x_0 x_{1h})
+\]</div>
+<p>which is now a linear combination of our basis terms. Let’s see if these regression techniques can find it:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linreg</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
+<span class="n">linreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">linreg</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">names2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-2.0000*mul(x1h, x1h) + 1.0000*sin(mul(x0, x1h)) + 1.0000*mul(x0, x0)&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">lasso</span> <span class="o">=</span> <span class="n">Lasso</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">lasso</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">lasso</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">names2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-1.8129*mul(x1h, x1h) + 0.9724*mul(x0, x0) + 0.8848*sin(mul(x0, x1h)) + -0.0722*mul(mul(x1h, x1h), mul(x1h, x1h)) + 0.0161*mul(x0, x1h)&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rvm</span> <span class="o">=</span> <span class="n">RVR</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">rvm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">names2</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">rvm</span><span class="o">.</span><span class="n">m_</span><span class="p">,</span> <span class="n">rvm</span><span class="o">.</span><span class="n">labels</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-2.0000*mul(x1h, x1h) + 1.0000*sin(mul(x0, x1h)) + 1.0000*mul(x0, x0)&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">stlsq</span><span class="o">.</span><span class="n">STLSQ</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.0001</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+<span class="n">stlsq</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">stlsq</span><span class="o">.</span><span class="n">coef_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">names2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-2.0000*mul(x1h, x1h) + 1.0000*mul(x0, x0) + 1.0000*sin(mul(x0, x1h))&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<p>In this case, every method except Lasso finds the true model exactly (which held true across many choices of regularization parameter).</p>
+</div>
+</div>
+<div class="section" id="handling-noise">
+<h2>Handling noise<a class="headerlink" href="#handling-noise" title="Permalink to this headline">#</a></h2>
+<p>What if we add noise to the data?</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">noise</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">feats2</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.01</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linreg</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
+<span class="n">linreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span> <span class="o">+</span> <span class="n">noise</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">linreg</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">names2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;0.9509*sin(mul(x0, x1h)) + 0.8659*mul(x0, x0) + -0.6626*mul(x1h, x1h) + -0.5179*mul(sin(x1h), sin(x1h)) + -0.4456*mul(x1h, sin(x1h)) + -0.3426*sin(mul(x1h, x1h)) + -0.3061*mul(mul(x1h, x1h), mul(x1h, x1h)) + 0.2311*mul(x0, sin(x0)) + -0.1164*mul(sin(x0), sin(x0)) + -0.0813*mul(x1h, mul(x0, x1h)) + 0.0761*mul(x0, sin(x1h)) + 0.0761*mul(x0, mul(x1h, x1h)) + -0.0478*mul(sin(x0), sin(x1h)) + -0.0456*sin(sin(x1h)) + 0.0441*mul(x1h, sin(x0)) + -0.0403*mul(x1h, mul(x0, x0)) + 0.0388*mul(x0, mul(x0, x1h)) + -0.0381*mul(mul(x0, x0), mul(x1h, x1h)) + 0.0371*mul(mul(x0, x1h), mul(x0, x1h)) + 0.0340*x1h + 0.0270*x0 + -0.0165*mul(sin(x1h), mul(x1h, x1h)) + -0.0159*mul(x0, x1h) + -0.0146*sin(x0) + 0.0137*mul(mul(x0, x1h), sin(x1h)) + 0.0136*mul(mul(x0, x0), mul(x0, x0)) + -0.0129*sin(sin(x0)) + 0.0125*sin(x1h) + -0.0100*mul(sin(x0), mul(x1h, x1h))&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">lasso</span> <span class="o">=</span> <span class="n">Lasso</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">lasso</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span> <span class="o">+</span> <span class="n">noise</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">lasso</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">names2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-1.8028*mul(x1h, x1h) + 0.9740*mul(x0, x0) + 0.8730*sin(mul(x0, x1h)) + -0.0772*mul(mul(x1h, x1h), mul(x1h, x1h)) + 0.0284*mul(x0, x1h)&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rvm</span> <span class="o">=</span> <span class="n">RVR</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">rvm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span> <span class="o">+</span> <span class="n">noise</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">names2</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">rvm</span><span class="o">.</span><span class="n">m_</span><span class="p">,</span> <span class="n">rvm</span><span class="o">.</span><span class="n">labels</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;0.9758*sin(mul(x0, x1h)) + 0.8902*mul(x0, x0) + -0.6695*mul(x1h, x1h) + -0.5104*mul(sin(x1h), sin(x1h)) + -0.4483*mul(x1h, sin(x1h)) + -0.3418*sin(mul(x1h, x1h)) + -0.3037*mul(mul(x1h, x1h), mul(x1h, x1h)) + 0.1889*mul(x0, sin(x0)) + -0.0943*mul(sin(x0), sin(x0)) + 0.0283*mul(x0, sin(x1h)) + 0.0110*mul(mul(x0, x0), mul(x0, x0))&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">stlsq</span><span class="o">.</span><span class="n">STLSQ</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.0001</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+<span class="n">stlsq</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span> <span class="o">+</span> <span class="n">noise</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">stlsq</span><span class="o">.</span><span class="n">coef_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">names2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;1.0013*sin(mul(x0, x1h)) + 0.9994*mul(x0, x0) + -0.6759*mul(x1h, x1h) + -0.5243*mul(sin(x1h), sin(x1h)) + -0.4429*mul(x1h, sin(x1h)) + -0.3313*sin(mul(x1h, x1h)) + -0.3016*mul(mul(x1h, x1h), mul(x1h, x1h))&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<p>In this case, it actually becomes <em>Lasso</em> which gets closest to the true model (i.e. is the only regression technique whose first three leading terms match the ground-truth model). This reversal illustrates how noise can strongly impact the effectiveness of different methods for learning symbolic models.</p>
+<p>Let’s see how genetic programming-based techniques deal with noise.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">noisy_equations</span> <span class="o">=</span> <span class="n">pysr</span><span class="o">.</span><span class="n">PySRRegressor</span><span class="p">(</span>
+    <span class="n">niterations</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
+    <span class="n">binary_operators</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;+&quot;</span><span class="p">,</span> <span class="s2">&quot;*&quot;</span><span class="p">],</span>  <span class="c1"># operators that can combine two terms</span>
+    <span class="n">unary_operators</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;sin&quot;</span><span class="p">],</span>  <span class="c1"># operators that modify a single term</span>
+<span class="p">)</span>
+<span class="n">noisy_equations</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">y</span><span class="o">.</span><span class="n">shape</span><span class="p">));</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Started!
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">noisy_equations</span><span class="o">.</span><span class="n">equations_</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><div>
+<style scoped>
+    .dataframe tbody tr th:only-of-type {
+        vertical-align: middle;
+    }
+
+    .dataframe tbody tr th {
+        vertical-align: top;
+    }
+
+    .dataframe thead th {
+        text-align: right;
+    }
+</style>
+<table border="1" class="dataframe">
+  <thead>
+    <tr style="text-align: right;">
+      <th></th>
+      <th>complexity</th>
+      <th>loss</th>
+      <th>score</th>
+      <th>equation</th>
+      <th>sympy_format</th>
+      <th>lambda_format</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <th>0</th>
+      <td>1</td>
+      <td>3.653608</td>
+      <td>0.000000e+00</td>
+      <td>0.46193257</td>
+      <td>0.461932570000000</td>
+      <td>PySRFunction(X=&gt;0.461932570000000)</td>
+    </tr>
+    <tr>
+      <th>1</th>
+      <td>2</td>
+      <td>3.653607</td>
+      <td>5.474042e-08</td>
+      <td>sin(0.4801726)</td>
+      <td>0.461932263986018</td>
+      <td>PySRFunction(X=&gt;0.461932263986018)</td>
+    </tr>
+    <tr>
+      <th>2</th>
+      <td>3</td>
+      <td>1.996533</td>
+      <td>6.043029e-01</td>
+      <td>(x0 * x0)</td>
+      <td>x0**2</td>
+      <td>PySRFunction(X=&gt;x0**2)</td>
+    </tr>
+    <tr>
+      <th>3</th>
+      <td>5</td>
+      <td>1.693676</td>
+      <td>8.225541e-02</td>
+      <td>((x0 * x0) + -0.55031127)</td>
+      <td>x0**2 - 0.55031127</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.55031127)</td>
+    </tr>
+    <tr>
+      <th>4</th>
+      <td>6</td>
+      <td>1.693675</td>
+      <td>7.675615e-07</td>
+      <td>((x0 * x0) + sin(-0.5827264))</td>
+      <td>x0**2 - 0.550302428935197</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.550302428935197)</td>
+    </tr>
+    <tr>
+      <th>5</th>
+      <td>7</td>
+      <td>1.693393</td>
+      <td>1.663976e-04</td>
+      <td>((x0 * (x0 + 0.024190227)) + -0.5257622)</td>
+      <td>x0*(x0 + 0.024190227) - 0.5257622</td>
+      <td>PySRFunction(X=&gt;x0*(x0 + 0.024190227) - 0.5257...</td>
+    </tr>
+    <tr>
+      <th>6</th>
+      <td>8</td>
+      <td>1.676705</td>
+      <td>9.903830e-03</td>
+      <td>((x0 * (x0 + sin(x1))) + -0.56431687)</td>
+      <td>x0*(x0 + sin(x1)) - 0.56431687</td>
+      <td>PySRFunction(X=&gt;x0*(x0 + sin(x1)) - 0.56431687)</td>
+    </tr>
+    <tr>
+      <th>7</th>
+      <td>9</td>
+      <td>1.095261</td>
+      <td>4.258376e-01</td>
+      <td>((x0 + (x1 * -0.43476593)) * (x0 + x1))</td>
+      <td>(x0 - 0.43476593*x1)*(x0 + x1)</td>
+      <td>PySRFunction(X=&gt;(x0 - 0.43476593*x1)*(x0 + x1))</td>
+    </tr>
+    <tr>
+      <th>8</th>
+      <td>10</td>
+      <td>1.060223</td>
+      <td>3.251314e-02</td>
+      <td>((x0 + (x1 * sin(-0.67482126))) * (x0 + x1))</td>
+      <td>(x0 - 0.624757763185892*x1)*(x0 + x1)</td>
+      <td>PySRFunction(X=&gt;(x0 - 0.624757763185892*x1)*(x...</td>
+    </tr>
+    <tr>
+      <th>9</th>
+      <td>11</td>
+      <td>1.041317</td>
+      <td>1.799367e-02</td>
+      <td>((x0 + (x1 * sin(sin(-0.67482126)))) * (x0 + x1))</td>
+      <td>(x0 - 0.584900810653589*x1)*(x0 + x1)</td>
+      <td>PySRFunction(X=&gt;(x0 - 0.584900810653589*x1)*(x...</td>
+    </tr>
+    <tr>
+      <th>10</th>
+      <td>12</td>
+      <td>1.036215</td>
+      <td>4.911030e-03</td>
+      <td>((x0 + (x1 * sin(sin(sin(-0.67482126))))) * (x...</td>
+      <td>(x0 - 0.552116684257055*x1)*(x0 + x1)</td>
+      <td>PySRFunction(X=&gt;(x0 - 0.552116684257055*x1)*(x...</td>
+    </tr>
+    <tr>
+      <th>11</th>
+      <td>13</td>
+      <td>1.016703</td>
+      <td>1.900931e-02</td>
+      <td>((x0 + (x1 * sin(sin(-0.67482126)))) * (x0 + (...</td>
+      <td>(x0 - 0.584900810653589*x1)*(x0 + 0.90857804*x1)</td>
+      <td>PySRFunction(X=&gt;(x0 - 0.584900810653589*x1)*(x...</td>
+    </tr>
+    <tr>
+      <th>12</th>
+      <td>15</td>
+      <td>1.012223</td>
+      <td>2.208117e-03</td>
+      <td>((x0 * x0) + ((-0.53690857 * x1) * (x1 + sin(s...</td>
+      <td>x0**2 - 0.53690857*x1*(x1 - sin(0.646188040102...</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.53690857*x1*(x1 - si...</td>
+    </tr>
+    <tr>
+      <th>13</th>
+      <td>17</td>
+      <td>1.008989</td>
+      <td>1.599981e-03</td>
+      <td>((((sin(-0.58534735) * x1) * x1) + (x0 * x0)) ...</td>
+      <td>x0**2 - 0.552488938905451*x1**2 + sin(x0)*sin(...</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.552488938905451*x1**...</td>
+    </tr>
+    <tr>
+      <th>14</th>
+      <td>18</td>
+      <td>1.005661</td>
+      <td>3.303900e-03</td>
+      <td>((((sin(sin(-0.58534735)) * x1) * x1) + (x0 * ...</td>
+      <td>x0**2 - 0.52480748921569*x1**2 + sin(x0)*sin(s...</td>
+      <td>PySRFunction(X=&gt;x0**2 - 0.52480748921569*x1**2...</td>
+    </tr>
+  </tbody>
+</table>
+</div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">round_expr</span><span class="p">(</span><span class="n">noisy_equations</span><span class="o">.</span><span class="n">sympy</span><span class="p">())</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_latex math notranslate nohighlight">
+\[\displaystyle \left(x_{0} - 0.4348 x_{1}\right) \left(x_{0} + x_{1}\right)\]</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">noisy_equations</span><span class="o">.</span><span class="n">equations_</span><span class="o">.</span><span class="n">sympy_format</span><span class="o">.</span><span class="n">values</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_latex math notranslate nohighlight">
+\[\displaystyle \left(x_{0} - 0.584900810653589 x_{1}\right) \left(x_{0} + 0.90857804 x_{1}\right)\]</div>
+</div>
+</div>
+<p>It looks like PySR was still able to discover the true expression as part of its Pareto frontier, though with the noise it’s only rated third-best for the default tradeoff between complexity and performance (with a sin-less version taking first).</p>
+</div>
+<div class="section" id="other-methods-for-symbolic-regression">
+<h2>Other Methods for Symbolic Regression<a class="headerlink" href="#other-methods-for-symbolic-regression" title="Permalink to this headline">#</a></h2>
+<p>To finish the notebook, here are a few other techniques for symbolic regression that don’t fit under either previously described methods (and all developed in the last several years):</p>
+<ul class="simple">
+<li><p><a class="reference external" href="https://arxiv.org/pdf/2102.08351.pdf">STreCH</a> (Julia implementation <a class="reference external" href="https://github.com/jongeunkim/STreCH">here</a>) formulates symbolic regression as a non-convex mixed-integer nonlinear programming (MINLP) technique, which it solves using a heuristic-guided cutting plane formulation.</p></li>
+<li><p><a class="reference external" href="http://proceedings.mlr.press/v80/sahoo18a/sahoo18a.pdf">EQL</a> (Python implementation <a class="reference external" href="https://github.com/martius-lab/EQL">here</a>) develops “equation layers” for neural networks which can be readily interpreted and also chained together, and are regularized with a thresholded L1 penalty to encourage sparsity.</p></li>
+<li><p><a class="reference external" href="https://arxiv.org/abs/2006.10782">AI Feynman</a> (hybrid Python-Fortran implementation <a class="reference external" href="https://github.com/SJ001/AI-Feynman">here</a>) starts by training a neural network, then computes <em>input gradients</em> of the learned network, and then uses attributes of those gradients to decompose the symbolic regression problem into something more tractable using graph theory.</p></li>
+<li><p><a class="reference external" href="https://papers.nips.cc/paper/2019/hash/567b8f5f423af15818a068235807edc0-Abstract.html">Symbolic Metamodeling</a> (Python implementation <a class="reference external" href="https://github.com/ahmedmalaa/Symbolic-Metamodeling">here</a>) uses gradient descent to learn compositions of Meijer-G functions, a flexible family that can be programmatically projected back onto a wide set of analytic, closed-form, or even algebriac expressions.</p></li>
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+    <h1>Symbolic Regression vs. Neural Networks on Multiscale L96</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dataset-generation">Dataset Generation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#obtaining-the-ground-truth">Obtaining the Ground Truth</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#split-the-data-into-training-and-test-sets">Split the Data into Training and Test Sets</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#build-the-dataset-and-the-dataloaders">Build the Dataset and the Dataloaders</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#creating-and-training-the-neural-network">Creating and Training the Neural Network</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-functions-to-train-and-evaluate-the-model">Define Functions to Train and Evaluate the Model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#create-the-neural-network">Create the Neural Network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-the-network">Train the Network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#visualize-results">Visualize Results</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparing-with-linear-regression">Comparing with Linear Regression</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#symbolic-regression-with-genetic-programming">Symbolic regression with genetic programming</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#explore-the-pareto-frontier">Explore the Pareto frontier</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#symbolic-regression-with-sparse-linear-regression-and-polynomial-features">Symbolic Regression with Sparse Linear Regression and Polynomial Features</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#non-sparse-linear-regression">Non-sparse linear regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sequentially-thresholded-least-squares">Sequentially thresholded least squares</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#add-threshold-features">Add threshold features</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sparse-regression">Sparse regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#genetic-programming">Genetic programming</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article" role="main">
+                  
+  <div class="tex2jax_ignore mathjax_ignore section" id="symbolic-regression-vs-neural-networks-on-multiscale-l96">
+<h1>Symbolic Regression vs. Neural Networks on Multiscale L96<a class="headerlink" href="#symbolic-regression-vs-neural-networks-on-multiscale-l96" title="Permalink to this headline">#</a></h1>
+<p>To recap from the previous notebooks, <a class="reference external" href="https://www.ecmwf.int/en/elibrary/10829-predictability-problem-partly-solved">Lorenz (1996)</a> describes a “two time-scale” model in two equations which are:</p>
+<div class="amsmath math notranslate nohighlight" id="equation-a5cf3018-cfb4-473c-bc76-0a7099a3fcd9">
+<span class="eqno">(20)<a class="headerlink" href="#equation-a5cf3018-cfb4-473c-bc76-0a7099a3fcd9" title="Permalink to this equation">#</a></span>\[\begin{align}
+\frac{d}{dt} X_k
+&amp;= - X_{k-1} \left( X_{k-2} - X_{k+1} \right) - X_k + F - \left( \frac{hc}{b} \right) \sum_{j=0}^{J-1} Y_{j,k}
+\\
+\frac{d}{dt} Y_{j,k}
+&amp;= - cbY_{j+1,k} \left( Y_{j+2,k} - Y_{j-1,k} \right) - c Y_{j,k} + \frac{hc}{b} X_k
+\end{align}\]</div>
+<p>This model can be visualized as follows:</p>
+<div class="figure align-default" id="l96-equation-figure-symbolicvnn">
+<a class="reference internal image-reference" href="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png"><img alt="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" src="https://www.researchgate.net/publication/319201436/figure/fig1/AS:869115023589376&#64;1584224577926/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values.png" style="width: 400px;" /></a>
+<p class="caption"><span class="caption-number">Fig. 8 </span><span class="caption-text"><em>Visualisation of a two-scale Lorenz ‘96 system with J = 8 and K = 6. Global-scale variables (<span class="math notranslate nohighlight">\(X_k\)</span>) are updated based on neighbouring variables and on the local-scale variables (<span class="math notranslate nohighlight">\(Y_{j,k}\)</span>) associated with the corresponding global-scale variable. Local-scale variabless are updated based on neighbouring variables and the associated global-scale variable. The neighbourhood topology of both local and global-scale variables is circular. Image from <a class="reference external" href="https://www.researchgate.net/figure/Visualisation-of-a-two-scale-Lorenz-96-system-with-J-8-and-K-6-Global-scale-values_fig1_319201436">Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures - Scientific Figure on ResearchGate.</a></em>.</span><a class="headerlink" href="#l96-equation-figure-symbolicvnn" title="Permalink to this image">#</a></p>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
+<span class="kn">import</span> <span class="nn">time</span>
+<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span><span class="p">,</span> <span class="n">SVG</span>
+
+<span class="kn">import</span> <span class="nn">gplearn</span>
+<span class="kn">import</span> <span class="nn">gplearn.genetic</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">pydot</span>
+<span class="kn">import</span> <span class="nn">pysindy</span> <span class="k">as</span> <span class="nn">ps</span>
+<span class="kn">import</span> <span class="nn">torch</span>
+<span class="kn">import</span> <span class="nn">torch.nn.functional</span> <span class="k">as</span> <span class="nn">F</span>
+<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
+<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">r2_score</span>
+<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
+<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span><span class="p">,</span> <span class="n">optim</span>
+<span class="kn">from</span> <span class="nn">torch.utils.data</span> <span class="kn">import</span> <span class="n">DataLoader</span><span class="p">,</span> <span class="n">TensorDataset</span>
+
+<span class="kn">from</span> <span class="nn">L96_model</span> <span class="kn">import</span> <span class="n">L96</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Ensuring reproducibility</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">14</span><span class="p">);</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="dataset-generation">
+<h2>Dataset Generation<a class="headerlink" href="#dataset-generation" title="Permalink to this headline">#</a></h2>
+<p>Let’s generate some data using the same conditions as in the previous notebooks <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/Universal_approximation.html">Introduction to Neural Networks</a> and <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/Neural_network_for_Lorenz96.html">Using Neural Networks for L96 Parameterization</a>.</p>
+<div class="section" id="obtaining-the-ground-truth">
+<h3>Obtaining the Ground Truth<a class="headerlink" href="#obtaining-the-ground-truth" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">time_steps</span> <span class="o">=</span> <span class="mi">20000</span>
+<span class="n">forcing</span><span class="p">,</span> <span class="n">dt</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="mi">18</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</span> <span class="o">*</span> <span class="n">time_steps</span>
+
+<span class="n">W</span> <span class="o">=</span> <span class="n">L96</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">32</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">forcing</span><span class="p">)</span>
+
+<span class="n">X_true</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">subgrid_tend</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">store</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">return_coupling</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">X_true</span><span class="p">,</span> <span class="n">subgrid_tend</span> <span class="o">=</span> <span class="n">X_true</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">),</span> <span class="n">subgrid_tend</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;X_true shape: </span><span class="si">{</span><span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;subgrid_tend shape: </span><span class="si">{</span><span class="n">subgrid_tend</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>X_true shape: (20001, 8)
+subgrid_tend shape: (20001, 8)
+</pre></div>
+</div>
+</div>
+</div>
+<p>In this case, although we have 8 different dimensions in <span class="math notranslate nohighlight">\(X\)</span>, each is derived in the exact same way from neighboring values of <span class="math notranslate nohighlight">\(X\)</span> (and local subdimensions <span class="math notranslate nohighlight">\(Y\)</span>). Therefore, we can actually reduce this problem from an 8-to-8D regression problem down to an 8-to-1D regression problem:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Dx</span> <span class="o">=</span> <span class="n">X_true</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">inputs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">([</span><span class="n">X_true</span><span class="p">[:,</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="n">i</span><span class="p">,</span> <span class="n">Dx</span> <span class="o">-</span> <span class="n">i</span><span class="p">)]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Dx</span><span class="p">)])</span>
+<span class="n">targets</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">([</span><span class="n">subgrid_tend</span><span class="p">[:,</span> <span class="o">-</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Dx</span><span class="p">)])</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;inputs shape: </span><span class="si">{</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;targets shape: </span><span class="si">{</span><span class="n">targets</span><span class="o">.</span><span class="n">shape</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>inputs shape: (160008, 8)
+targets shape: (160008,)
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell tag_hide-input docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell source</span>
+<span class="expanded">Hide code cell source</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Multiscale L96 trajectory&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Dx</span><span class="p">):</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">inputs</span><span class="p">[:,</span> <span class="n">i</span><span class="p">][:</span><span class="mi">500</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$x&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">ncol</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Value&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</details>
+<div class="cell_output docutils container">
+<img alt="../_images/f855d7d3566848851979c3e0d905fbdc1630b0abd51fe7f48c1ea397e63749fd.png" src="../_images/f855d7d3566848851979c3e0d905fbdc1630b0abd51fe7f48c1ea397e63749fd.png" />
+</div>
+</div>
+<div class="cell tag_hide-input docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell source</span>
+<span class="expanded">Hide code cell source</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">&quot;2D histograms of features vs. forcing&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Dx</span><span class="p">):</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">inputs</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">targets</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;$x_</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span>
+    <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">4</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Forcing&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</details>
+<div class="cell_output docutils container">
+<img alt="../_images/20def59700276516a64c0c1f0a0251f1700800791732d644369260d14404e5cf.png" src="../_images/20def59700276516a64c0c1f0a0251f1700800791732d644369260d14404e5cf.png" />
+</div>
+</div>
+<p>Looking at these marginal histograms of each large-scale variable vs. the subgrid forcing, we can see there’s a mostly linear relationship between <span class="math notranslate nohighlight">\(x_0\)</span> and its forcing, suggesting that the subgrid terms tend to force small values higher and large values lower. This suggests that an extremely simple linear model only containing <span class="math notranslate nohighlight">\(x_0\)</span> with negative slope will do reasonably well. However, the relationship isn’t <em>completely</em> linear, and also contains a bifurcation at larger <span class="math notranslate nohighlight">\(x_0\)</span>.</p>
+<p>For the other features, the relationships are less straightforward, but there’s clearly high mutual information.</p>
+</div>
+<div class="section" id="split-the-data-into-training-and-test-sets">
+<h3>Split the Data into Training and Test Sets<a class="headerlink" href="#split-the-data-into-training-and-test-sets" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">data_idxs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">))</span>
+<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">data_idxs</span><span class="p">)</span>
+
+<span class="n">num_train_samples</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.75</span><span class="p">)</span>
+<span class="n">train_idxs</span> <span class="o">=</span> <span class="n">data_idxs</span><span class="p">[:</span><span class="n">num_train_samples</span><span class="p">]</span>
+<span class="n">test_idxs</span> <span class="o">=</span> <span class="n">data_idxs</span><span class="p">[</span><span class="n">num_train_samples</span><span class="p">:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="build-the-dataset-and-the-dataloaders">
+<h3>Build the Dataset and the Dataloaders<a class="headerlink" href="#build-the-dataset-and-the-dataloaders" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">1024</span>
+
+<span class="c1"># Training Data</span>
+<span class="n">train_dataset</span> <span class="o">=</span> <span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">]),</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">targets</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">])</span>
+<span class="p">)</span>
+<span class="n">loader_train</span> <span class="o">=</span> <span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">train_dataset</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># Test Data</span>
+<span class="n">test_dataset</span> <span class="o">=</span> <span class="n">TensorDataset</span><span class="p">(</span>
+    <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]),</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">])</span>
+<span class="p">)</span>
+<span class="n">loader_test</span> <span class="o">=</span> <span class="n">DataLoader</span><span class="p">(</span><span class="n">dataset</span><span class="o">=</span><span class="n">test_dataset</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="creating-and-training-the-neural-network">
+<h2>Creating and Training the Neural Network<a class="headerlink" href="#creating-and-training-the-neural-network" title="Permalink to this headline">#</a></h2>
+<p>Now let’s create a neural network and train it on the dataset we generated above.</p>
+<div class="section" id="define-functions-to-train-and-evaluate-the-model">
+<h3>Define Functions to Train and Evaluate the Model<a class="headerlink" href="#define-functions-to-train-and-evaluate-the-model" title="Permalink to this headline">#</a></h3>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The code in this section is taken from the notebook <a class="reference external" href="https://m2lines.github.io/L96_demo/notebooks/Universal_approximation.html">Introduction to Neural Networks</a>.</p>
+</div>
+<div class="cell tag_hide-cell docutils container">
+<details class="hide above-input">
+<summary aria-label="Toggle hidden content">
+<span class="collapsed">Show code cell content</span>
+<span class="expanded">Hide code cell content</span>
+</summary>
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train the network for one epoch&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
+
+    <span class="n">train_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+        <span class="c1"># Get predictions</span>
+        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+            <span class="c1"># This if block is needed to add a dummy dimension if our inputs are 1D</span>
+            <span class="c1"># (where each number is a different sample)</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="n">prediction</span> <span class="o">=</span> <span class="n">network</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>
+
+        <span class="c1"># Compute the loss</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span>
+        <span class="n">train_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Clear the gradients</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">zero_grad</span><span class="p">()</span>
+
+        <span class="c1"># Backpropagation to compute the gradients and update the weights</span>
+        <span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
+        <span class="n">optimizer</span><span class="o">.</span><span class="n">step</span><span class="p">()</span>
+
+    <span class="k">return</span> <span class="n">train_loss</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+
+
+<span class="k">def</span> <span class="nf">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">loader</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Test the network&quot;&quot;&quot;</span>
+    <span class="n">network</span><span class="o">.</span><span class="n">eval</span><span class="p">()</span>  <span class="c1"># Evaluation mode (important when having dropout layers)</span>
+
+    <span class="n">test_loss</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
+        <span class="k">for</span> <span class="n">batch_x</span><span class="p">,</span> <span class="n">batch_y</span> <span class="ow">in</span> <span class="n">loader</span><span class="p">:</span>
+            <span class="c1"># Get predictions</span>
+            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">batch_x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
+                <span class="c1"># This if block is needed to add a dummy dimension if our inputs are 1D</span>
+                <span class="c1"># (where each number is a different sample)</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">network</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="n">batch_x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">prediction</span> <span class="o">=</span> <span class="n">network</span><span class="p">(</span><span class="n">batch_x</span><span class="p">)</span>
+
+            <span class="c1"># Compute the loss</span>
+            <span class="n">loss</span> <span class="o">=</span> <span class="n">criterion</span><span class="p">(</span><span class="n">prediction</span><span class="p">,</span> <span class="n">batch_y</span><span class="p">)</span>
+            <span class="n">test_loss</span> <span class="o">+=</span> <span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span>
+
+        <span class="c1"># Get an average loss for the entire dataset</span>
+        <span class="n">test_loss</span> <span class="o">/=</span> <span class="nb">len</span><span class="p">(</span><span class="n">loader</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">test_loss</span>
+
+
+<span class="k">def</span> <span class="nf">fit_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">val_loader</span><span class="p">,</span> <span class="n">n_epochs</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;Train and validate the network&quot;&quot;&quot;</span>
+    <span class="n">train_losses</span><span class="p">,</span> <span class="n">val_losses</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
+    <span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_epochs</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+        <span class="n">train_loss</span> <span class="o">=</span> <span class="n">train_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">train_loader</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">)</span>
+        <span class="n">val_loss</span> <span class="o">=</span> <span class="n">test_model</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">val_loader</span><span class="p">)</span>
+        <span class="n">train_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">train_loss</span><span class="p">)</span>
+        <span class="n">val_losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">val_loss</span><span class="p">)</span>
+    <span class="n">end_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+    <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Training completed in </span><span class="si">{</span><span class="nb">int</span><span class="p">(</span><span class="n">end_time</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">start_time</span><span class="p">)</span><span class="si">}</span><span class="s2"> seconds.&quot;</span><span class="p">)</span>
+
+    <span class="k">return</span> <span class="n">train_losses</span><span class="p">,</span> <span class="n">val_losses</span>
+</pre></div>
+</div>
+</div>
+</details>
+</div>
+</div>
+<div class="section" id="create-the-neural-network">
+<h3>Create the Neural Network<a class="headerlink" href="#create-the-neural-network" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Net_ANN</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
+    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
+        <span class="nb">super</span><span class="p">(</span><span class="n">Net_ANN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear1</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear2</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear1</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">linear2</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear3</span><span class="p">(</span><span class="n">x</span><span class="p">)[:,</span> <span class="mi">0</span><span class="p">]</span>
+        <span class="k">return</span> <span class="n">x</span>
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="train-the-network">
+<h3>Train the Network<a class="headerlink" href="#train-the-network" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">nn_3l</span> <span class="o">=</span> <span class="n">Net_ANN</span><span class="p">()</span>
+
+<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">20</span>
+<span class="n">criterion</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">MSELoss</span><span class="p">()</span>
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">optim</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">nn_3l</span><span class="o">.</span><span class="n">parameters</span><span class="p">(),</span> <span class="n">lr</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">train_loss</span><span class="p">,</span> <span class="n">validation_loss</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span>
+    <span class="n">nn_3l</span><span class="p">,</span> <span class="n">criterion</span><span class="p">,</span> <span class="n">optimizer</span><span class="p">,</span> <span class="n">loader_train</span><span class="p">,</span> <span class="n">loader_test</span><span class="p">,</span> <span class="n">n_epochs</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training completed in 17 seconds.
+</pre></div>
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="visualize-results">
+<h3>Visualize Results<a class="headerlink" href="#visualize-results" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">train_loss</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Training loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">validation_loss</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Validation loss&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/7f3fa80426ba0bae2a1796f89d9ff7b721f67a90c659d0e07c900c6d36819f9c.png" src="../_images/7f3fa80426ba0bae2a1796f89d9ff7b721f67a90c659d0e07c900c6d36819f9c.png" />
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">preds_nn</span> <span class="o">=</span> <span class="n">nn_3l</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;R2 Score:&quot;</span><span class="p">,</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">preds_nn</span><span class="p">,</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>R2 Score: 0.9223949649031744
+</pre></div>
+</div>
+</div>
+</div>
+<p>The neural network gets an <span class="math notranslate nohighlight">\(R^2\)</span> of ~ 0.916, which seems pretty good given the non-determinism of the problem.</p>
+</div>
+</div>
+<div class="section" id="comparing-with-linear-regression">
+<h2>Comparing with Linear Regression<a class="headerlink" href="#comparing-with-linear-regression" title="Permalink to this headline">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linear_reg</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
+<span class="n">linear_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">])</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;R2 Score:&quot;</span><span class="p">,</span> <span class="n">linear_reg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>R2 Score: 0.8160184487760795
+</pre></div>
+</div>
+</div>
+</div>
+<p>Linear regression also does fairly well, though about 0.1 lower in terms of <span class="math notranslate nohighlight">\(R^2\)</span>. Lets look at its coefficients:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">),</span> <span class="n">linear_reg</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">),</span> <span class="p">[</span><span class="sa">f</span><span class="s2">&quot;x</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">)],</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Coefficient value&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/05b4325b1a0283a53cf754fa3c07e7fd4eed7718ef854cf168d176a68f2ffc2c.png" src="../_images/05b4325b1a0283a53cf754fa3c07e7fd4eed7718ef854cf168d176a68f2ffc2c.png" />
+</div>
+</div>
+<p>This matches the 2D histogram plots from earlier that showed a negative linear relationship between forcing and <span class="math notranslate nohighlight">\(x_0\)</span>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Neural Network vs. Linear Regression&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+
+<span class="n">preds_nn</span> <span class="o">=</span> <span class="n">nn_3l</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">inputs</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+<span class="n">preds_lr</span> <span class="o">=</span> <span class="n">linear_reg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
+<span class="n">random_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">targets</span> <span class="o">-</span> <span class="mi">1000</span><span class="p">))</span>
+<span class="n">random_slice</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="n">random_idx</span><span class="p">,</span> <span class="n">random_idx</span> <span class="o">+</span> <span class="mi">1000</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_nn</span><span class="p">[</span><span class="n">random_slice</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;NN Predicted values&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_lr</span><span class="p">[</span><span class="n">random_slice</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;LR Predicted values&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">targets</span><span class="p">[</span><span class="n">random_slice</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.667</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Time (starting at idx </span><span class="si">{</span><span class="n">random_idx</span><span class="si">}</span><span class="s2">)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Subgrid forcing&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/92e8f274f380afbbff5642988990f667d73ecf3a2b6cc79702182989e470dd25.png" src="../_images/92e8f274f380afbbff5642988990f667d73ecf3a2b6cc79702182989e470dd25.png" />
+</div>
+</div>
+</div>
+<div class="section" id="symbolic-regression-with-genetic-programming">
+<h2>Symbolic regression with genetic programming<a class="headerlink" href="#symbolic-regression-with-genetic-programming" title="Permalink to this headline">#</a></h2>
+<p>Let’s try using one of the genetic programming libraries to discover a parameterization:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">symbolic_reg</span> <span class="o">=</span> <span class="n">gplearn</span><span class="o">.</span><span class="n">genetic</span><span class="o">.</span><span class="n">SymbolicRegressor</span><span class="p">(</span>
+    <span class="n">population_size</span><span class="o">=</span><span class="mi">5000</span><span class="p">,</span>
+    <span class="n">generations</span><span class="o">=</span><span class="mi">25</span><span class="p">,</span>
+    <span class="n">p_crossover</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span>
+    <span class="n">p_subtree_mutation</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
+    <span class="n">p_hoist_mutation</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span>
+    <span class="n">p_point_mutation</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
+    <span class="n">max_samples</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span>
+    <span class="n">verbose</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+    <span class="n">parsimony_coefficient</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span>
+    <span class="n">const_range</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
+    <span class="n">function_set</span><span class="o">=</span><span class="p">(</span><span class="s2">&quot;add&quot;</span><span class="p">,</span> <span class="s2">&quot;mul&quot;</span><span class="p">,</span> <span class="s2">&quot;sub&quot;</span><span class="p">,</span> <span class="s2">&quot;div&quot;</span><span class="p">,</span> <span class="s2">&quot;sin&quot;</span><span class="p">,</span> <span class="s2">&quot;log&quot;</span><span class="p">),</span>
+<span class="p">)</span>
+
+<span class="n">symbolic_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">][:</span><span class="mi">20000</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">][:</span><span class="mi">20000</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>    |   Population Average    |             Best Individual              |
+---- ------------------------- ------------------------------------------ ----------
+ Gen   Length          Fitness   Length          Fitness      OOB Fitness  Time Left
+   0    15.09          2250.77        3          1.64883          1.66775      3.16m
+   1    10.32          6.53204        5          1.62545          1.63651      3.10m
+   2    12.32          16.9512        5          1.62268          1.66148      3.00m
+   3     7.04          5.26195       20          1.57546           1.6044      2.22m
+   4     5.07          9.58891       24          1.55363          1.55098      1.81m
+   5     5.79          5.96614       21          1.51949          1.49863      1.89m
+   6     9.63          4.65507       17          1.50515          1.50598      2.67m
+   7    16.11          2.78142       28          1.47296          1.45776      3.66m
+   8    19.77          2.69051       28           1.4555          1.46709      3.94m
+   9    21.42          2.80303       41          1.45297          1.48639      3.84m
+  10    22.63          2.61995       37          1.44204          1.47566      3.94m
+  11    23.78          2.29202       41          1.42739           1.4853      3.91m
+  12    24.47           6.6118        8          1.41676          1.43687      3.60m
+  13    25.15          2.55741       22           1.4201          1.47976      3.27m
+  14    25.67          2.27797       28          1.41595          1.43681      2.96m
+  15    24.77          2.45272       42          1.35228          1.38372      2.59m
+  16    20.61          3.97112       23          1.34902          1.34875      2.04m
+  17    12.16          3.66793       24          1.33076          1.35901      1.29m
+  18    14.17          3.62681       24          1.32406          1.35392      1.29m
+  19    27.37          2.86559       29          1.31132          1.33767      1.82m
+  20    25.82          2.40423       31          1.30332          1.33496      1.41m
+  21    23.41          2.45246       38          1.30026          1.32775     59.00s
+  22    23.45          2.57487       38          1.29615          1.31448     38.09s
+  23    24.97          2.56582       33          1.29189          1.30837     19.95s
+  24    24.11          2.33584       37          1.29417          1.32678      0.00s
+</pre></div>
+</div>
+<div class="output text_html"><style>#sk-container-id-1 {color: black;background-color: white;}#sk-container-id-1 pre{padding: 0;}#sk-container-id-1 div.sk-toggleable {background-color: white;}#sk-container-id-1 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-1 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-1 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-1 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-1 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-1 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-1 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-1 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-1 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-1 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-1 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-1 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-1 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-1 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-1 div.sk-item {position: relative;z-index: 1;}#sk-container-id-1 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-1 div.sk-item::before, #sk-container-id-1 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-1 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-1 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-1 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-1 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-1 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-1 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-1 div.sk-label-container {text-align: center;}#sk-container-id-1 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-1 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-1" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>sub(sin(mul(-0.447, X0)), sub(X0, sin(sub(-1.107, log(add(X2, sub(sub(sub(-1.107, log(add(X2, sub(sin(sub(-1.107, log(mul(-0.447, X0)))), sub(X0, -0.220))))), X0), log(sin(mul(-0.447, X0))))))))))</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-1" type="checkbox" checked><label for="sk-estimator-id-1" class="sk-toggleable__label sk-toggleable__label-arrow">SymbolicRegressor</label><div class="sk-toggleable__content"><pre>sub(sin(mul(-0.447, X0)), sub(X0, sin(sub(-1.107, log(add(X2, sub(sub(sub(-1.107, log(add(X2, sub(sin(sub(-1.107, log(mul(-0.447, X0)))), sub(X0, -0.220))))), X0), log(sin(mul(-0.447, X0))))))))))</pre></div></div></div></div></div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">symbolic_reg</span><span class="o">.</span><span class="n">_program</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>sub(sin(mul(-0.447, X0)), sub(X0, sin(sub(-1.107, log(add(X2, sub(sub(sub(-1.107, log(add(X2, sub(sin(sub(-1.107, log(mul(-0.447, X0)))), sub(X0, -0.220))))), X0), log(sin(mul(-0.447, X0))))))))))
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span>
+    <span class="s2">&quot;R2 Score:&quot;</span><span class="p">,</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">symbolic_reg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]),</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">])</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>R2 Score: 0.8499308040527297
+</pre></div>
+</div>
+</div>
+</div>
+<p>While the program does slightly better than linear regression, it’s extremely complicated (graph visualization below for slightly easier inspection):</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Build the graph</span>
+<span class="n">graph</span> <span class="o">=</span> <span class="n">pydot</span><span class="o">.</span><span class="n">graph_from_dot_data</span><span class="p">(</span><span class="n">symbolic_reg</span><span class="o">.</span><span class="n">_program</span><span class="o">.</span><span class="n">export_graphviz</span><span class="p">())[</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">graph</span><span class="o">.</span><span class="n">write_svg</span><span class="p">(</span><span class="s2">&quot;figs/symbolic_regression.svg&quot;</span><span class="p">)</span>
+
+<span class="c1"># Display the graph</span>
+<span class="n">display</span><span class="p">(</span><span class="n">SVG</span><span class="p">(</span><span class="s2">&quot;figs/symbolic_regression.svg&quot;</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/3c9cfd7f89e0fc228bba3dc3d6f7a497d61892eac02cc639cd4d85a7221e639e.svg" src="../_images/3c9cfd7f89e0fc228bba3dc3d6f7a497d61892eac02cc639cd4d85a7221e639e.svg" /></div>
+</div>
+<div class="section" id="explore-the-pareto-frontier">
+<h3>Explore the Pareto frontier<a class="headerlink" href="#explore-the-pareto-frontier" title="Permalink to this headline">#</a></h3>
+<p>We can explore the Pareto frontier of the final population of programs as well, to see if there’s a simpler program that still does well</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="o">.</span><span class="n">from_dict</span><span class="p">(</span>
+    <span class="p">[</span>
+        <span class="nb">dict</span><span class="p">(</span>
+            <span class="n">idx</span><span class="o">=</span><span class="n">i</span><span class="p">,</span>
+            <span class="n">expr</span><span class="o">=</span><span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">),</span>
+            <span class="n">fitness</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">fitness_</span><span class="p">,</span>
+            <span class="n">length</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">length_</span><span class="p">,</span>
+            <span class="n">r2_score</span><span class="o">=</span><span class="n">r2_score</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">execute</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]),</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]),</span>
+        <span class="p">)</span>
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">symbolic_reg</span><span class="o">.</span><span class="n">_programs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
+    <span class="p">]</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pareto_frontier</span> <span class="o">=</span> <span class="p">[]</span>
+<span class="n">used_programs</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
+
+<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">40</span><span class="p">):</span>
+    <span class="n">programs</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="n">df</span><span class="o">.</span><span class="n">length</span> <span class="o">&lt;=</span> <span class="n">l</span><span class="p">]</span>
+    <span class="n">best</span> <span class="o">=</span> <span class="n">programs</span><span class="o">.</span><span class="n">r2_score</span><span class="o">.</span><span class="n">argmax</span><span class="p">()</span>
+    <span class="n">idx</span> <span class="o">=</span> <span class="n">programs</span><span class="o">.</span><span class="n">idx</span><span class="o">.</span><span class="n">values</span><span class="p">[</span><span class="n">best</span><span class="p">]</span>
+    <span class="n">r2</span> <span class="o">=</span> <span class="n">programs</span><span class="o">.</span><span class="n">r2_score</span><span class="o">.</span><span class="n">values</span><span class="p">[</span><span class="n">best</span><span class="p">]</span>
+    <span class="n">program</span> <span class="o">=</span> <span class="n">symbolic_reg</span><span class="o">.</span><span class="n">_programs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="n">idx</span><span class="p">]</span>
+    <span class="k">if</span> <span class="n">program</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">used_programs</span><span class="p">:</span>
+        <span class="n">pareto_frontier</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">length</span><span class="o">=</span><span class="n">l</span><span class="p">,</span> <span class="n">r2</span><span class="o">=</span><span class="n">r2</span><span class="p">,</span> <span class="n">expr</span><span class="o">=</span><span class="nb">str</span><span class="p">(</span><span class="n">program</span><span class="p">)))</span>
+        <span class="n">used_programs</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">program</span><span class="p">)</span>
+
+<span class="n">pd</span><span class="o">.</span><span class="n">set_option</span><span class="p">(</span><span class="s2">&quot;max_colwidth&quot;</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
+<span class="n">pareto_frontier</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="o">.</span><span class="n">from_dict</span><span class="p">(</span><span class="n">pareto_frontier</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pareto_frontier</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_html"><div>
+<style scoped>
+    .dataframe tbody tr th:only-of-type {
+        vertical-align: middle;
+    }
+
+    .dataframe tbody tr th {
+        vertical-align: top;
+    }
+
+    .dataframe thead th {
+        text-align: right;
+    }
+</style>
+<table border="1" class="dataframe">
+  <thead>
+    <tr style="text-align: right;">
+      <th></th>
+      <th>length</th>
+      <th>r2</th>
+      <th>expr</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <th>0</th>
+      <td>1</td>
+      <td>0.000000</td>
+      <td>-0.220</td>
+    </tr>
+    <tr>
+      <th>1</th>
+      <td>2</td>
+      <td>0.000000</td>
+      <td>sin(-1.107)</td>
+    </tr>
+    <tr>
+      <th>2</th>
+      <td>3</td>
+      <td>0.787718</td>
+      <td>sub(-0.220, X0)</td>
+    </tr>
+    <tr>
+      <th>3</th>
+      <td>8</td>
+      <td>0.837378</td>
+      <td>sub(sin(mul(-0.447, X0)), sub(X0, -0.220))</td>
+    </tr>
+    <tr>
+      <th>4</th>
+      <td>11</td>
+      <td>0.841150</td>
+      <td>sub(sin(mul(-0.447, X0)), sub(X0, sin(mul(-0.447, X0))))</td>
+    </tr>
+    <tr>
+      <th>5</th>
+      <td>12</td>
+      <td>0.842545</td>
+      <td>sub(sin(sin(mul(-0.447, X0))), sub(X0, sin(mul(-0.447, X0))))</td>
+    </tr>
+    <tr>
+      <th>6</th>
+      <td>13</td>
+      <td>0.843272</td>
+      <td>sub(sin(sin(mul(-0.447, X0))), sub(X0, sin(sin(mul(-0.447, X0)))))</td>
+    </tr>
+    <tr>
+      <th>7</th>
+      <td>16</td>
+      <td>0.854240</td>
+      <td>sub(sin(mul(-0.447, X0)), sub(X0, sin(sub(-1.107, log(add(X2, div(X0, -1.107)))))))</td>
+    </tr>
+    <tr>
+      <th>8</th>
+      <td>17</td>
+      <td>0.855131</td>
+      <td>sub(sin(sub(-1.107, log(add(X2, sub(sin(X0), X0))))), sub(X0, sin(mul(-0.447, X0))))</td>
+    </tr>
+    <tr>
+      <th>9</th>
+      <td>19</td>
+      <td>0.856080</td>
+      <td>sub(sin(sub(-1.107, log(add(X2, sub(sub(sin(X0), X0), -1.107))))), sub(X0, sin(mul(-0.447, X0))))</td>
+    </tr>
+    <tr>
+      <th>10</th>
+      <td>21</td>
+      <td>0.856531</td>
+      <td>sub(sin(sub(-1.107, log(add(X2, sub(sub(-0.447, X0), log(sub(X0, -0.220))))))), sub(X0, sin(mul(-0.307, X0))))</td>
+    </tr>
+    <tr>
+      <th>11</th>
+      <td>23</td>
+      <td>0.856728</td>
+      <td>sub(sin(sub(-1.107, log(add(X2, sub(sub(sin(X0), X0), log(sin(mul(-0.447, X0)))))))), sub(X0, sin(mul(-0.447, X0))))</td>
+    </tr>
+    <tr>
+      <th>12</th>
+      <td>35</td>
+      <td>0.856754</td>
+      <td>sub(sin(sub(-1.107, log(add(X2, sub(sin(sub(-1.107, log(add(X2, sub(sub(sin(-0.605), X0), log(sub(X0, -0.220))))))), sub(X0, sin(mul(-0.447, X0)))))))), sub(X0, sin(mul(-0.447, X0))))</td>
+    </tr>
+    <tr>
+      <th>13</th>
+      <td>39</td>
+      <td>0.857035</td>
+      <td>sub(sin(mul(-0.447, X0)), sub(X0, sin(sub(-1.107, log(add(X2, sub(sub(sin(sub(-1.107, log(add(X2, sub(sub(-0.605, X0), log(sub(-0.447, X0))))))), sub(X0, sin(mul(-0.447, X0)))), log(mul(-0.447, X0...</td>
+    </tr>
+  </tbody>
+</table>
+</div></div></div>
+</div>
+<p>Unfortunately, nothing really jumps out here as both simple and performant.</p>
+</div>
+</div>
+<div class="section" id="symbolic-regression-with-sparse-linear-regression-and-polynomial-features">
+<h2>Symbolic Regression with Sparse Linear Regression and Polynomial Features<a class="headerlink" href="#symbolic-regression-with-sparse-linear-regression-and-polynomial-features" title="Permalink to this headline">#</a></h2>
+<p>We can also apply sparse regression methods to this problem with polynomial features</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">learned_expr</span><span class="p">(</span><span class="n">coef</span><span class="p">,</span> <span class="n">names</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">):</span>
+    <span class="n">coef_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argwhere</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">cutoff</span><span class="p">)[:,</span> <span class="mi">0</span><span class="p">]</span>
+    <span class="n">sort_order</span> <span class="o">=</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">coef</span><span class="p">[</span><span class="n">coef_idx</span><span class="p">])))</span>
+    <span class="k">return</span> <span class="s2">&quot; + &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
+        <span class="p">[</span>
+            <span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">coef</span><span class="p">[</span><span class="n">coef_idx</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">*</span><span class="si">{</span><span class="n">names</span><span class="p">[</span><span class="n">coef_idx</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s1">&#39; &#39;</span><span class="p">,</span><span class="s1">&#39;*&#39;</span><span class="p">)</span><span class="si">}</span><span class="s2">&quot;</span>
+            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sort_order</span>
+        <span class="p">]</span>
+    <span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pf</span> <span class="o">=</span> <span class="n">PolynomialFeatures</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">feats</span> <span class="o">=</span> <span class="n">pf</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
+<span class="n">names</span> <span class="o">=</span> <span class="n">pf</span><span class="o">.</span><span class="n">get_feature_names_out</span><span class="p">([</span><span class="sa">f</span><span class="s2">&quot;x_</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">)])</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="section" id="non-sparse-linear-regression">
+<h3>Non-sparse linear regression<a class="headerlink" href="#non-sparse-linear-regression" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linear_reg</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
+<span class="n">linear_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">])</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">linear_reg</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">names</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-0.6913*x_0 + 0.4333*x_1 + 0.3393*x_7 + 0.2112*x_5 + 0.1420*x_6 + 0.1118*x_2 + -0.0399*x_0*x_1 + -0.0333*x_0*x_2 + -0.0323*x_6*x_7 + -0.0288*x_0*x_7 + -0.0283*x_1*x_2 + -0.0248*x_1*x_5 + -0.0242*x_3 + -0.0219*x_1*x_6 + -0.0209*x_5*x_7 + 0.0201*x_0^2 + -0.0185*x_1*x_4 + -0.0135*x_0*x_5 + 0.0124*x_0*x_3 + 0.0107*x_4 + 0.0099*x_1*x_3 + 0.0090*x_2*x_5 + -0.0088*x_4*x_7 + -0.0081*x_5^2 + -0.0076*x_2*x_3 + -0.0074*x_2*x_7 + 0.0071*x_3*x_4 + 0.0071*x_4*x_5 + -0.0069*x_3*x_5 + 0.0061*x_3*x_6 + 0.0059*x_1*x_7 + -0.0057*x_7^2 + 0.0050*x_3^2 + -0.0048*x_0*x_4 + -0.0045*x_1^2 + -0.0036*x_0*x_6 + -0.0036*x_3*x_7 + 0.0034*x_2^2 + 0.0024*x_4*x_6 + 0.0023*x_2*x_4 + 0.0015*x_2*x_6 + -0.0011*x_5*x_6&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;R2 Score:&quot;</span><span class="p">,</span> <span class="n">linear_reg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">feats</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>R2 Score: 0.8872654640415502
+</pre></div>
+</div>
+</div>
+</div>
+<p>Adding polynomial features brings our <span class="math notranslate nohighlight">\(R^2\)</span> much closer to that of the neural network, but we have a ton of terms. Let’s try making the regression sparser.</p>
+</div>
+<div class="section" id="sequentially-thresholded-least-squares">
+<h3>Sequentially thresholded least squares<a class="headerlink" href="#sequentially-thresholded-least-squares" title="Permalink to this headline">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">stlsq</span><span class="o">.</span><span class="n">STLSQ</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+<span class="n">stlsq</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">])</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">stlsq</span><span class="o">.</span><span class="n">coef_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">names</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-3.7208*1 + -0.7842*x_0 + 0.3574*x_1 + 0.2431*x_7 + 0.1900*x_2 + 0.1778*x_5 + 0.1522*x_6 + -0.0390*x_0*x_1 + -0.0356*x_0*x_2 + -0.0335*x_6*x_7 + 0.0267*x_0^2 + -0.0265*x_1*x_5 + -0.0257*x_0*x_7 + -0.0227*x_1*x_2 + 0.0189*x_4 + -0.0164*x_1*x_6 + -0.0156*x_5*x_7 + 0.0145*x_3 + -0.0120*x_0*x_5 + 0.0110*x_0*x_3&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">feats</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.882294259449451
+</pre></div>
+</div>
+</div>
+</div>
+<p>Looks like we can get a much sparser expression without sacrificing too much performance, though if we increase the threshold slightly, we will see a trade-off:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq2</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">stlsq</span><span class="o">.</span><span class="n">STLSQ</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.02</span><span class="p">)</span>
+<span class="n">stlsq2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">])</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">stlsq2</span><span class="o">.</span><span class="n">coef_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">names</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-1.5860*1 + -0.9857*x_0 + 0.1324*x_2 + 0.0728*x_3 + -0.0725*x_7 + -0.0436*x_1 + -0.0322*x_0*x_2 + 0.0307*x_0^2&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq2</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">feats</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.861645175881121
+</pre></div>
+</div>
+</div>
+</div>
+<p>This is much simpler, but less performant. It still strongly outperformed the particluar genetic programming method we used, though. Let’s visualize some of the predictions:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Neural network vs. quadratic regressions&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+
+<span class="n">preds_nn</span> <span class="o">=</span> <span class="n">nn_3l</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">inputs</span><span class="p">))</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+<span class="n">preds_lr_quad</span> <span class="o">=</span> <span class="n">linear_reg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">feats</span><span class="p">)</span>
+<span class="n">preds_st_quad</span> <span class="o">=</span> <span class="n">stlsq</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">feats</span><span class="p">)</span>
+<span class="n">preds_st_quad2</span> <span class="o">=</span> <span class="n">stlsq2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">feats</span><span class="p">)</span>
+<span class="n">random_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">targets</span> <span class="o">-</span> <span class="mi">1000</span><span class="p">))</span>
+<span class="n">random_slice</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="n">random_idx</span><span class="p">,</span> <span class="n">random_idx</span> <span class="o">+</span> <span class="mi">1000</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_nn</span><span class="p">[</span><span class="n">random_slice</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;NN predicted values&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_lr_quad</span><span class="p">[</span><span class="n">random_slice</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;LR + quadratic&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_st_quad</span><span class="p">[</span><span class="n">random_slice</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;STLSQ + quadratic, threshold=$0.01$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">preds_st_quad2</span><span class="p">[</span><span class="n">random_slice</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;STLSQ + quadratic, threshold=$0.02$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
+    <span class="n">targets</span><span class="p">[</span><span class="n">random_slice</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True values&quot;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.667</span>
+<span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Time (starting at idx </span><span class="si">{</span><span class="n">random_idx</span><span class="si">}</span><span class="s2">)&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Subgrid forcing&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/e323c198f2fa33e54b62a4be82530a22531fdaa42a2bec813ca38070e6976499.png" src="../_images/e323c198f2fa33e54b62a4be82530a22531fdaa42a2bec813ca38070e6976499.png" />
+</div>
+</div>
+</div>
+</div>
+<div class="section" id="add-threshold-features">
+<h2>Add threshold features<a class="headerlink" href="#add-threshold-features" title="Permalink to this headline">#</a></h2>
+<p>The 2D histogram of <span class="math notranslate nohighlight">\(x_0\)</span> looks like it might be potentially modelable by piecewise linear regression:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">inputs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">targets</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x_0$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Forcing&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/2eb0980f89a1b692ee578497b5bb6be9d3c2443fcff29895c866de77b2c47c73.png" src="../_images/2eb0980f89a1b692ee578497b5bb6be9d3c2443fcff29895c866de77b2c47c73.png" />
+</div>
+</div>
+<div class="section" id="sparse-regression">
+<h3>Sparse regression<a class="headerlink" href="#sparse-regression" title="Permalink to this headline">#</a></h3>
+<p>Let’s see if sparse regression will discover that if we provide it with thresholding features and cross-terms:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x0_thresholds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
+    <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">inputs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])]</span> <span class="o">+</span> <span class="p">[</span><span class="n">inputs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)]</span>
+<span class="p">)</span>
+<span class="n">x0_threshold_names</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;&quot;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="sa">f</span><span class="s2">&quot;*(x0&gt;</span><span class="si">{</span><span class="n">t</span><span class="si">}</span><span class="s2">)&quot;</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)]</span>
+
+<span class="n">feats2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
+    <span class="p">[</span>
+        <span class="n">feats</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">x0_thresholds</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">feats</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x0_thresholds</span><span class="p">))</span>
+    <span class="p">]</span>
+<span class="p">)</span><span class="o">.</span><span class="n">T</span>
+<span class="n">names2</span> <span class="o">=</span> <span class="p">[</span>
+    <span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">names</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="si">}{</span><span class="n">x0_threshold_names</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">feats</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x0_thresholds</span><span class="p">))</span>
+<span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq3</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">stlsq</span><span class="o">.</span><span class="n">STLSQ</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
+<span class="n">stlsq3</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">feats2</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">])</span>
+<span class="n">learned_expr</span><span class="p">(</span><span class="n">stlsq3</span><span class="o">.</span><span class="n">coef_</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">names2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/usr/share/miniconda/envs/L96M2lines/lib/python3.9/site-packages/sklearn/utils/extmath.py:189: RuntimeWarning: invalid value encountered in matmul
+  ret = a @ b
+</pre></div>
+</div>
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&#39;-2.8051*1*(x0&gt;4) + -0.9871*x_0 + -0.6571*1*(x0&gt;2) + 0.6440*x_0*(x0&gt;4) + -0.5681*1*(x0&gt;0) + -0.5286*1*(x0&gt;1) + -0.2734*1*(x0&gt;5)&#39;
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stlsq3</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">feats2</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.8590951075317006
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">150</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">hist2d</span><span class="p">(</span><span class="n">inputs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">targets</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
+
+<span class="n">fn</span> <span class="o">=</span> <span class="p">(</span>
+    <span class="k">lambda</span> <span class="n">x0</span><span class="p">:</span> <span class="o">-</span><span class="mf">2.1200</span> <span class="o">*</span> <span class="mi">1</span> <span class="o">*</span> <span class="p">(</span><span class="n">x0</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">)</span>
+    <span class="o">+</span> <span class="o">-</span><span class="mf">1.2695</span> <span class="o">*</span> <span class="mi">1</span> <span class="o">*</span> <span class="p">(</span><span class="n">x0</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span>
+    <span class="o">+</span> <span class="o">-</span><span class="mf">0.9488</span> <span class="o">*</span> <span class="n">x0</span>
+    <span class="o">+</span> <span class="o">-</span><span class="mf">0.6694</span> <span class="o">*</span> <span class="mi">1</span> <span class="o">*</span> <span class="p">(</span><span class="n">x0</span> <span class="o">&gt;</span> <span class="mi">4</span><span class="p">)</span>
+    <span class="o">+</span> <span class="mf">0.5175</span> <span class="o">*</span> <span class="n">x0</span> <span class="o">*</span> <span class="p">(</span><span class="n">x0</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">)</span>
+<span class="p">)</span>
+
+<span class="n">xt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
+<span class="n">yt</span> <span class="o">=</span> <span class="n">fn</span><span class="p">(</span><span class="n">xt</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xt</span><span class="p">,</span> <span class="n">yt</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$x_0$&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Forcing&quot;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/053e8c3c37b9a00cf8d20a19e31c676f31bf368a97da87912a091f99036bbf85.png" src="../_images/053e8c3c37b9a00cf8d20a19e31c676f31bf368a97da87912a091f99036bbf85.png" />
+</div>
+</div>
+<p>It looks like the model learns a piecewise linear function in a single variable, even though we didn’t explicitly require it!</p>
+</div>
+<div class="section" id="genetic-programming">
+<h3>Genetic programming<a class="headerlink" href="#genetic-programming" title="Permalink to this headline">#</a></h3>
+<p>We can also try using genetic programming here, but with a different function set that removes some of the nonlinear operators we included before and replaces them with min and max, which allow for piecewise behavior:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">symbolic_reg_pw</span> <span class="o">=</span> <span class="n">gplearn</span><span class="o">.</span><span class="n">genetic</span><span class="o">.</span><span class="n">SymbolicRegressor</span><span class="p">(</span>
+    <span class="n">population_size</span><span class="o">=</span><span class="mi">5000</span><span class="p">,</span>
+    <span class="n">generations</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span>
+    <span class="n">p_crossover</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span>
+    <span class="n">p_subtree_mutation</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
+    <span class="n">p_hoist_mutation</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span>
+    <span class="n">p_point_mutation</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
+    <span class="n">max_samples</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span>
+    <span class="n">verbose</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
+    <span class="n">parsimony_coefficient</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span>
+    <span class="n">const_range</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
+    <span class="n">function_set</span><span class="o">=</span><span class="p">(</span><span class="s2">&quot;add&quot;</span><span class="p">,</span> <span class="s2">&quot;mul&quot;</span><span class="p">,</span> <span class="s2">&quot;max&quot;</span><span class="p">,</span> <span class="s2">&quot;min&quot;</span><span class="p">),</span>
+<span class="p">)</span>
+<span class="n">symbolic_reg_pw</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">][:</span><span class="mi">20000</span><span class="p">],</span> <span class="n">targets</span><span class="p">[</span><span class="n">train_idxs</span><span class="p">][:</span><span class="mi">20000</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>    |   Population Average    |             Best Individual              |
+---- ------------------------- ------------------------------------------ ----------
+ Gen   Length          Fitness   Length          Fitness      OOB Fitness  Time Left
+   0    27.65      3.78741e+06        7          2.45249          2.54746      5.54m
+   1     8.00          9.51443        5          1.85843          1.82615      3.97m
+   2    10.82          7.30098        3           1.6483          1.63764      4.37m
+   3     8.34          27.6089        3          1.64321          1.68342      4.06m
+   4     4.92          26.8323       15          1.50401          1.50866      3.78m
+   5     3.87          16.3644       15          1.44304          1.44976      3.44m
+   6     6.13           148.18       11          1.40594          1.42375      3.70m
+   7    12.76          15.4382       13          1.37424          1.38711      4.63m
+   8    14.48          10.2082       13           1.3658          1.36791      4.85m
+   9    15.67          12.0612       21          1.35567          1.37664      4.91m
+  10    15.53          17917.1       23           1.3514          1.40418      4.85m
+  11    15.51          7.61133       29          1.35019          1.38838      4.66m
+  12    16.27          4.11042       19          1.34782          1.38404      4.70m
+  13    15.22          12.9581       23          1.34734          1.38836      4.46m
+  14    14.50          1540.79       17          1.34519          1.40768      4.24m
+  15    13.97          5.27408       17          1.34517          1.40786      4.04m
+  16    14.09          13.3842       17          1.34172          1.43891      4.04m
+  17    14.21          3.51828       13          1.34363          1.37425      3.84m
+  18    14.46          84.8708       19          1.34315          1.43713      3.73m
+  19    14.50           8.1359       15          1.33915          1.37526      3.60m
+  20    14.49          4.71403       15          1.33942          1.41217      3.46m
+  21    14.12          4.65016       15          1.33662          1.39804      3.43m
+  22    13.77          38.3175       19          1.33467          1.42346      3.17m
+  23    13.67          9.76407       17          1.33539          1.41711      3.04m
+  24    13.87          9.07149       15          1.33255          1.43467      2.93m
+  25    14.14          114.952       15          1.33405          1.42116      2.82m
+  26    14.53          110.894       15          1.33412          1.42058      2.72m
+  27    14.70          3.25037       19          1.33509          1.41981      2.68m
+  28    14.72          10.7036       13          1.33059          1.37007      2.52m
+  29    14.88          3.91762       25          1.32809          1.37427      2.42m
+  30    14.62          6.29094       15          1.32687          1.39063      2.28m
+  31    14.57           3.5343       17           1.3249          1.40442      2.18m
+  32    14.12          3.90761       21          1.32425          1.41418      2.04m
+  33    13.83          5.69476       13          1.32439          1.42588      1.96m
+  34    13.48          8.32606       15          1.32568          1.40132      1.77m
+  35    13.25          79.5995       17          1.32476          1.40569      1.65m
+  36    13.24          4.45101       13          1.32515          1.41905      1.53m
+  37    13.04          7.14949       13          1.32524           1.4182      1.40m
+  38    13.15          5.93211       15          1.32344          1.42146      1.29m
+  39    13.15          9.69874       17          1.32478          1.40562      1.21m
+  40    13.18          5.74077       13          1.32703          1.40212      1.06m
+  41    12.99          9.94645       13          1.32421          1.42748     56.52s
+  42    12.99          4.36878       13          1.32517          1.41883     49.42s
+  43    13.12          12.7419       13          1.32593          1.41204     42.62s
+  44    13.26          3.95264       13          1.32416          1.42798     36.77s
+  45    13.06          13.1207       21          1.32404          1.42906     28.35s
+  46    13.04          6.56359       13          1.32464          1.42364     21.24s
+  47    13.19          6.23439       21          1.32029          1.34429     14.26s
+  48    13.10          27.2948       21          1.32101          1.33781      7.11s
+  49    13.26          5.91966       21          1.31816          1.36342      0.00s
+</pre></div>
+</div>
+<div class="output text_html"><style>#sk-container-id-2 {color: black;background-color: white;}#sk-container-id-2 pre{padding: 0;}#sk-container-id-2 div.sk-toggleable {background-color: white;}#sk-container-id-2 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-2 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-2 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-2 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-2 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-2 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-2 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-2 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-2 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-2 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-2 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-2 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-2 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-2 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-2 div.sk-item {position: relative;z-index: 1;}#sk-container-id-2 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-2 div.sk-item::before, #sk-container-id-2 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-2 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-2 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-2 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-2 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-2 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-2 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-2 div.sk-label-container {text-align: center;}#sk-container-id-2 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-2 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-2" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>add(max(min(mul(max(X0, -1.442), -0.938), X1), -1.937), mul(min(X0, add(max(X5, X3), max(X6, X4))), -0.720))</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-2" type="checkbox" checked><label for="sk-estimator-id-2" class="sk-toggleable__label sk-toggleable__label-arrow">SymbolicRegressor</label><div class="sk-toggleable__content"><pre>add(max(min(mul(max(X0, -1.442), -0.938), X1), -1.937), mul(min(X0, add(max(X5, X3), max(X6, X4))), -0.720))</pre></div></div></div></div></div></div></div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">symbolic_reg_pw</span><span class="o">.</span><span class="n">_program</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>add(max(min(mul(max(X0, -1.442), -0.938), X1), -1.937), mul(min(X0, add(max(X5, X3), max(X6, X4))), -0.720))
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span>
+    <span class="s2">&quot;R2 Score:&quot;</span><span class="p">,</span>
+    <span class="n">r2_score</span><span class="p">(</span><span class="n">symbolic_reg_pw</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">inputs</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]),</span> <span class="n">targets</span><span class="p">[</span><span class="n">test_idxs</span><span class="p">]),</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>R2 Score: 0.8354791875984894
+</pre></div>
+</div>
+</div>
+</div>
+<p>While this leads to slightly higher <span class="math notranslate nohighlight">\(R^2\)</span> than before, the model is still pretty complex:</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Build the graph</span>
+<span class="n">graph</span> <span class="o">=</span> <span class="n">pydot</span><span class="o">.</span><span class="n">graph_from_dot_data</span><span class="p">(</span><span class="n">symbolic_reg_pw</span><span class="o">.</span><span class="n">_program</span><span class="o">.</span><span class="n">export_graphviz</span><span class="p">())[</span><span class="mi">0</span><span class="p">]</span>
+<span class="n">graph</span><span class="o">.</span><span class="n">write_svg</span><span class="p">(</span><span class="s2">&quot;figs/symbolic_regression_pw.svg&quot;</span><span class="p">)</span>
+
+<span class="c1"># Display the graph</span>
+<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span><span class="p">,</span> <span class="n">display</span>
+
+<span class="n">display</span><span class="p">(</span><span class="n">SVG</span><span class="p">(</span><span class="s2">&quot;figs/symbolic_regression_pw.svg&quot;</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/7705f159d34ac422757b9c58470031c3eeae039846e3c65a3f5e4b446ba66c28.svg" src="../_images/7705f159d34ac422757b9c58470031c3eeae039846e3c65a3f5e4b446ba66c28.svg" /></div>
+</div>
+<p>Tuning the hyperparameters could help, but requires more compute.</p>
+</div>
+</div>
+</div>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
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+            theme: "abcdef",
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+    <script>kernelName = 'python3'</script>
+
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+        <p class="prev-next-title">Applying SINDy equation identification to L96</p>
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+  <div class="sidebar-secondary-item">
+  <div class="page-toc tocsection onthispage">
+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dataset-generation">Dataset Generation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#obtaining-the-ground-truth">Obtaining the Ground Truth</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#split-the-data-into-training-and-test-sets">Split the Data into Training and Test Sets</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#build-the-dataset-and-the-dataloaders">Build the Dataset and the Dataloaders</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#creating-and-training-the-neural-network">Creating and Training the Neural Network</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#define-functions-to-train-and-evaluate-the-model">Define Functions to Train and Evaluate the Model</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#create-the-neural-network">Create the Neural Network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#train-the-network">Train the Network</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#visualize-results">Visualize Results</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparing-with-linear-regression">Comparing with Linear Regression</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#symbolic-regression-with-genetic-programming">Symbolic regression with genetic programming</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#explore-the-pareto-frontier">Explore the Pareto frontier</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#symbolic-regression-with-sparse-linear-regression-and-polynomial-features">Symbolic Regression with Sparse Linear Regression and Polynomial Features</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#non-sparse-linear-regression">Non-sparse linear regression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sequentially-thresholded-least-squares">Sequentially thresholded least squares</a></li>
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+        <div class="sidebar-primary-item"><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Lorenz-96 and General Circulation Models</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96-two-scale-description.html">The Lorenz-96 Two-Timescale System</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/gcm-analogue.html">The Lorenz-96 and its GCM Analog</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/gcm-parameterization-problem.html">GCM parameterizations, skill metrics, and other sources of uncertainity</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/estimating-gcm-parameters.html">Tuning GCM Parameterizations</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Neural Networks with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/intro_ML_and_NNs.html">Introduction to Machine Learning and Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_offline_training_NN.html">Using Neural Networks for L96 Parameterization: Offline Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_online_implement_NN.html">Using Neural Networks for L96 Parameterization: Online Testing</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_online_training_NN.html">Using Neural Networks for L96 Parameterization: Online Training</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Improving_Neural_networks.html">Improving Performance of Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Neural-Network-Interpretation.html">Interpreting Neural Networks</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/constraints.html">Adding constraints to Neural Networks</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Data Assimilation with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/DA_demo_L96.html">Data Assimilation demo in the Lorenz 96 (L96) two time-scale model</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Learning-DA-increments.html">Learning Data Assimilation Increments</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Equation Discovery with Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/symbolic_methods_comparison.html">Introduction to Equation Discovery - Comparing Symbolic Regression Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/sindy_L96_2scale.html">Applying SINDy equation identification to L96</a></li>
+
+<li class="toctree-l1"><a class="reference internal" href="notebooks/symbolic_vs_nn_multiscale_L96.html">Symbolic Regression vs. Neural Networks on Multiscale L96</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Other ML approaches for Lorenz-96</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/random_forest_parameterization.html">Random Forest</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="notebooks/L96_ResNet_RNN.html">ResNet and Recurrent Neural Networks</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">End Matter</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="notebooks/closing_remarks.html">Outlook</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/citing.html">Citing this book</a></li>
+<li class="toctree-l1"><a class="reference internal" href="bibliography.html">References</a></li>
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Assimilation demo in the Lorenz 96 (L96) two time-scale model","Gaussian Processes","Improving Performance of Neural Networks","The Lorenz-96 Two-Timescale System","ResNet and Recurrent Neural Networks","Using Neural Networks for L96 Parameterization: Offline Training","Using Neural Networks for L96 Parameterization: Online Testing","Using Neural Networks for L96 Parameterization: Online Training","Learning Data Assimilation Increments","Interpreting Neural Networks","Citing this book","Outlook","Adding constraints to Neural Networks","Tuning GCM Parameterizations","The Lorenz-96 and its GCM Analog","GCM parameterizations, skill metrics, and other sources of uncertainity","Introduction to Machine Learning and Neural Networks","CNNs and Polynomial Maps","Random Forest","Applying SINDy equation identification to L96","Introduction to Equation Discovery - Comparing Symbolic Regression Methods","Symbolic Regression vs. Neural Networks on Multiscale 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