From 55e1638d20575822ab4e30fad34cb1fcc7848002 Mon Sep 17 00:00:00 2001 From: LM Date: Thu, 23 Nov 2023 23:50:45 +0100 Subject: [PATCH] update neural ode and dynamical systems --- dl/assets/references/dl-fundamentals.bib | 9 + dl/assets/references/general.bib | 9 + .../ode-exponential-growth.png | Bin .../ode_infection_model.png | Bin .../differential-equations/ode_receprocal.png | Bin .../differential-equations/ode_sin_t.png | Bin .../neural-ode/neural-ode-and-functions.jpg | Bin 0 -> 192401 bytes dl/dynamical-systems/index.md | 29 +++ dl/dynamical-systems/neural-ode.md | 243 ++++++++++++++++++ dl/neural-ode/index.md | 186 -------------- .../neural-ode/index.md | 1 - .../timeseries.neural-ode.md | 1 + mkdocs.yml | 9 +- 13 files changed, 296 insertions(+), 191 deletions(-) rename dl/{neural-ode => dynamical-systems}/assets/differential-equations/ode-exponential-growth.png (100%) rename dl/{neural-ode => dynamical-systems}/assets/differential-equations/ode_infection_model.png (100%) rename dl/{neural-ode => dynamical-systems}/assets/differential-equations/ode_receprocal.png (100%) rename dl/{neural-ode => dynamical-systems}/assets/differential-equations/ode_sin_t.png (100%) create mode 100644 dl/dynamical-systems/assets/neural-ode/neural-ode-and-functions.jpg create mode 100644 dl/dynamical-systems/index.md create mode 100644 dl/dynamical-systems/neural-ode.md delete mode 100644 dl/neural-ode/index.md delete mode 100644 dl/time-series-deep-learning/neural-ode/index.md create mode 100644 dl/time-series-deep-learning/timeseries.neural-ode.md diff --git a/dl/assets/references/dl-fundamentals.bib b/dl/assets/references/dl-fundamentals.bib index b016aa25..07f221e6 100644 --- a/dl/assets/references/dl-fundamentals.bib +++ b/dl/assets/references/dl-fundamentals.bib @@ -599,3 +599,12 @@ @online{Pascanu2012-qv primaryClass = "cs.LG", eprint = "1211.5063" } + + +@MISC{Jaeger2013-vx, + title = "A tutorial on training recurrent neural networks, covering {BPTT}, + {RTRL}, {EKF} and the ``echo state network'' approach", + author = "Jaeger, Herbert and Kirkpatrick, James", + url = "https://www.ai.rug.nl/minds/uploads/ESNTutorialRev.pdf", + file = "All Papers/JAEGER/Jaeger and Kirkpatrick - A tutorial on training recurrent neural networks, covering BPTT, RTRL, EKF and the 'echo state network' approach.pdf" +} diff --git a/dl/assets/references/general.bib b/dl/assets/references/general.bib index 76c1a200..5dd53aa1 100644 --- a/dl/assets/references/general.bib +++ b/dl/assets/references/general.bib @@ -130,3 +130,12 @@ @ARTICLE{Imai2021-kk issn = "1047-1987,1476-4989", language = "en" } + + +@misc{enwiki:1186188179, +author = "{Wikipedia contributors}", +title = "Lorenz system --- {Wikipedia}{,} The Free Encyclopedia", +year = "2023", +url = "https://en.wikipedia.org/w/index.php?title=Lorenz_system&oldid=1186188179", +note = "[Online; accessed 23-November-2023]" +} diff --git a/dl/neural-ode/assets/differential-equations/ode-exponential-growth.png b/dl/dynamical-systems/assets/differential-equations/ode-exponential-growth.png similarity index 100% rename from dl/neural-ode/assets/differential-equations/ode-exponential-growth.png rename to dl/dynamical-systems/assets/differential-equations/ode-exponential-growth.png diff --git a/dl/neural-ode/assets/differential-equations/ode_infection_model.png b/dl/dynamical-systems/assets/differential-equations/ode_infection_model.png similarity index 100% rename from dl/neural-ode/assets/differential-equations/ode_infection_model.png rename to dl/dynamical-systems/assets/differential-equations/ode_infection_model.png diff --git a/dl/neural-ode/assets/differential-equations/ode_receprocal.png b/dl/dynamical-systems/assets/differential-equations/ode_receprocal.png similarity index 100% rename from dl/neural-ode/assets/differential-equations/ode_receprocal.png rename to dl/dynamical-systems/assets/differential-equations/ode_receprocal.png diff --git 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With this limit, we expect the oscillation amplitude to be infinite. + + ```python + class SinMultiplyT(DiffEqn): + def fn(self, y: float, t: np.ndarray) -> np.ndarray: + lbd = self.fn_args["lbd"] + dydt = np.sin(lbd * t) * t + + return dydt + + def _formula(self): + return r"\frac{dh(t)}{d t} = \sin(\lambda t) t" + + y0_sin = 1 + t = np.linspace(0, 10, 101) + lbd = 2 + sin_multiply_t = SinMultiplyT(t, y0_sin, lbd=lbd) + sin_multiply_t.plot() + ``` + + ![ODE oscillatory](assets/differential-equations/ode_sin_t.png) + + === ":material-cursor-default-click: Receprocal" + + We design a system that grows according to the receprocal of its value, + + $$ + \frac{dh(t)}{d t} = \frac{1}{a * y + b}. + $$ + + ```python + class Receprocal(DiffEqn): + def fn(self, y: float, t: np.ndarray) -> np.ndarray: + shift = self.fn_args["shift"] + scale = self.fn_args["scale"] + dydt = 1/(shift + scale * y) + return dydt + + def _formula(self): + return r"\frac{dh(t)}{d t} = \frac{1}{shift + scale * y}" + + receprocal = Receprocal(t, 1, shift=-5, scale=-10) + receprocal.plot() + ``` + + ![receprocal](assets/differential-equations/ode_receprocal.png) + + +## Finite Difference Form of Differential Equations + +Eq. $\eqref{eq:1st-order-ode}$ can be rewritten in the finite difference form as + +$$ +\frac{h(t+\Delta t) - h(t)}{\Delta t} = f_\theta(h(t), t), +$$ + +with $\Delta t$ small enough. + +!!! note "Derivatives" + + The definition of the first-order derivative is + $$ + h'(t) = \lim_{\Delta t\to 0} \frac{h(t+\Delta t) - h(t)}{\Delta t}. + $$ + + In a numerical computation, $\lim$ is approached by taking a small $\Delta t$. + +If we take $\Delta t = 1$, the equation becomes + +$$ +\begin{equation} +h(t+1) = h(t) + f_\theta(h(t), t) +\label{eq:1st-order-ode-finite-difference-deltat-1} +\end{equation} +$$ + +!!! note "$\Delta t = 1$? Rescale of Time $t$." + + As there is no absolute measure of time $t$, we can always rescale $t$ to $\hat t$ so that $\Delta t = 1$. However, for the sake of clarity, we will keep $t$ as the time variable. + + +Coming back to neural networks, if $h(t)$ represents the state of a neural network block at depth $t$, Eq. $\eqref{eq:1st-order-ode-finite-difference-deltat-1}$ is nothing fancy but the **residual connection** in ResNet[@He2015-ie]. This connection between the finite difference form of a first-order differential equation and the residual connection leads to the new family of deep neural network models called neural ode[@Chen2018-mp]. + +In a neural ODE, we treat each layer of a neural network as a function $h$ of depth $t$, i.e., the state of the layer is equivalent to a function $h(t)$. + +![Neural ODE and Differential Equations](assets/neural-ode/neural-ode-and-functions.jpg) + + +??? note "Solving Differential Equations" + + The right side of the Eq. $\eqref{eq:1st-order-ode}$ is called the Jacobian. Given any Jacobian $f(h(t), \theta(t), t)$, we can, at least numerically, perform integration to find out the value of $h(t)$ at any $t=t_N$, + + $$ + h(t_N) = \int_{t_0}^{t_i} f(h(t), \theta(t), t) \mathrm dt. + $$ + + In this formalism, we find out the transformed input $h(t_0)$ by solving this differential equation. In traditional neural networks, we achieve this by stacking many layers of neural networks using skip connections. + + In the original Neural ODE paper, the authors used the so-called reverse-model derivative method[@Chen2018-mp]. + + We will not dive deep into solving differential equations in this section. Instead, we will show some applications of neural ODEs in section [Time Series Forecasting with Neural ODE](../time-series-deep-learning/timeseries.neural-ode.md). diff --git a/dl/neural-ode/index.md b/dl/neural-ode/index.md deleted file mode 100644 index 46578a10..00000000 --- a/dl/neural-ode/index.md +++ /dev/null @@ -1,186 +0,0 @@ -# Neural ODE - -Many of real world systems are dynamical systems. Differential equation is a handy tool to model a dynamical system. For example, the action potentials of a squid giant axon can be modeled by the famous [Hodgkin-Huxley model](https://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model). - -Many time series data are generated by dynamical systems. A naive philosophy is to come up with a set of differential equations to model the time series. - -In this section, we discuss the creative idea of neural ODE[@Chen2018-mp]. - -A first order ordinary differential equation is - -$$ -\frac{\mathrm d h(t)}{\mathrm dt} = f_\theta(h(t), t), -$$ - -where $h(t)$ is the function that describes a dynamical system. - -## Examples of First Order Differential Equations - -??? info ":material-code-json: Utility Code for ODE (Run this first)" - - ```python - from abc import ABC, abstractmethod - import numpy as np - from scipy.integrate import odeint - import inspect - - from typing import Optional, Any - - import matplotlib.pyplot as plt - import seaborn as sns; sns.set() - - class DiffEqn(ABC): - """A first order differential equation and the corresponding solutions. - - :param t: time steps to solve the differential equation for - :param y_0: initial condition of the ode - """ - def __init__(self, t: np.ndarray, y_0: float, **fn_args: Optional[Any]): - self.t = t - self.y_0 = y_0 - self.fn_args = fn_args - - @abstractmethod - def fn(self, y: float, t: np.ndarray) -> np.ndarray: - pass - - def solve(self) -> np.ndarray: - return odeint(self.fn, self.y_0, self.t) - - @abstractmethod - def _formula(self) -> str: - pass - - def __str__(self): - return f"{self._formula()}" - - def __repr__(self): - return f"{self._formula()}" - - def plot(self, ax: Optional[plt.Axes]=None): - - if ax is None: - _, ax = plt.subplots(figsize=(10, 6.18)) - - sns.lineplot( - x=self.t, y=self.solve()[:, 0], - ax=ax - ) - - ax.set_xlabel("t") - ax.set_ylabel("y") - ax.set_title(f"Solution for ${self}$ with {self.fn_args}") - - ``` - -=== ":material-cursor-default-click: Infection Model" - - The logistic model of infectious disease is - - $$ - \frac{dh(t)}{d t} = r * y * (1-y). - $$ - - ```python - class Infections(DiffEqn): - def fn(self, y: float, t: np.ndarray) -> np.ndarray: - r = self.fn_args["r"] - dydt = r * y * (1 - y) - return dydt - - def _formula(self): - return r"\frac{dh(t)}{d t} = r * y * (1-y)" - - infections_s = Infections(t, 0.1, r=0.9) - infections_s.plot() - ``` - - ![infectious model](assets/differential-equations/ode_infection_model.png) - - -=== ":material-cursor-default-click: Exponential Growth" - - The following equation describes an exponentially growing $h(t)$. - - $$ - \frac{\mathrm d h(t)}{\mathrm d t} = \lambda h(t), - $$ - - with $\lambda > 0$. - - ```python - class Exponential(DiffEqn): - def fn(self, y: float, t: np.ndarray) -> np.ndarray: - lbd = self.fn_args["lbd"] - dydt = lbd * y - - return dydt - - def _formula(self): - return r"\frac{dh(t)}{d t} = \lambda h(t)" - - y0_exponential = 1 - t = np.linspace(0, 10, 101) - lbd = 2 - - exponential = Exponential(t, y0_exponential, lbd=lbd) - exponential.plot() - ``` - - ![solution to the exponential growth differential equation](assets/differential-equations/ode-exponential-growth.png) - -=== ":material-cursor-default-click: Oscillations" - - We construct an oscillatory system using [sinusoid](https://de.wikipedia.org/wiki/Sinusoid), - - $$ - \frac{dh(t)}{d t} = \sin(\lambda t) t. - $$ - - Naively, we expect the oscillations to be large for larger $t$. - Taking the limit $t\to\infty$, the first order derivative $\frac{\mathrm dy}{\mathrm d t}\to\infty$. With this limit, we expect the oscillation amplitude to be infinite. - - ```python - class SinMultiplyT(DiffEqn): - def fn(self, y: float, t: np.ndarray) -> np.ndarray: - lbd = self.fn_args["lbd"] - dydt = np.sin(lbd * t) * t - - return dydt - - def _formula(self): - return r"\frac{dh(t)}{d t} = \sin(\lambda t) t" - - y0_sin = 1 - t = np.linspace(0, 10, 101) - lbd = 2 - sin_multiply_t = SinMultiplyT(t, y0_sin, lbd=lbd) - sin_multiply_t.plot() - ``` - - ![ODE oscillatory](assets/differential-equations/ode_sin_t.png) - -=== ":material-cursor-default-click: Receprocal" - - We design a system that grows according to the receprocal of its value, - - $$ - \frac{dh(t)}{d t} = \frac{1}{a * y + b}. - $$ - - ```python - class Receprocal(DiffEqn): - def fn(self, y: float, t: np.ndarray) -> np.ndarray: - shift = self.fn_args["shift"] - scale = self.fn_args["scale"] - dydt = 1/(shift + scale * y) - return dydt - - def _formula(self): - return r"\frac{dh(t)}{d t} = \frac{1}{shift + scale * y}" - - receprocal = Receprocal(t, 1, shift=-5, scale=-10) - receprocal.plot() - ``` - - ![receprocal](assets/differential-equations/ode_receprocal.png) diff --git a/dl/time-series-deep-learning/neural-ode/index.md b/dl/time-series-deep-learning/neural-ode/index.md deleted file mode 100644 index 54b2bb94..00000000 --- a/dl/time-series-deep-learning/neural-ode/index.md +++ /dev/null @@ -1 +0,0 @@ -# Neural ODE for Time Series Forecasting diff --git a/dl/time-series-deep-learning/timeseries.neural-ode.md b/dl/time-series-deep-learning/timeseries.neural-ode.md new file mode 100644 index 00000000..d6f05db2 --- /dev/null +++ b/dl/time-series-deep-learning/timeseries.neural-ode.md @@ -0,0 +1 @@ +# Time Series Forecasting with Neural ODE diff --git a/mkdocs.yml b/mkdocs.yml index 9f1cd059..b7769040 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -217,8 +217,9 @@ nav: - "Recurrent Neural Networks": deep-learning-fundamentals/recurrent-neural-networks.md - "Transformers": - "Vanilla Transformers": transformers/transformers.vanilla.md - - "Neural ODE": - - "Dynamical System": neural-ode/index.md + - "Dynamical Systems": + - "Why Dynamical Systems": "dynamical-systems/index.md" + - "Neural ODE": "dynamical-systems/neural-ode.md" - "Energy-based Models": - "Introduction": energy-based-models/intro.md - "Diffusion Models": energy-based-models/ebm.diffusion.md @@ -244,8 +245,8 @@ nav: - "Pendulum Dataset": "time-series-deep-learning/timeseries.dataset.pendulum.md" - "Forecasting with Feedforward Neural Networks": "time-series-deep-learning/timeseries.feedforward.md" - "Forecasting with RNN": time-series-deep-learning/timeseries.rnn.md - - "Transformers for Time Series Forecasting": "time-series-deep-learning/timeseries.transformer.md" - - "Neural ODE for Time Series Forecasting": "time-series-deep-learning/neural-ode/index.md" + - "Time Series Forecasting with Transformers": "time-series-deep-learning/timeseries.transformer.md" + - "Time Series Forecasting with Neural ODE": time-series-deep-learning/timeseries.neural-ode.md - "TimeGrad": time-series-deep-learning/timeseries.deep-learning.timegrad.md - "Supplementary": - "About Supplementary Materials": supplementary.md