🧠 Flexible Inference for Predictive Coding Networks in JAX ⚡️

JPC is a JAX library for training neural networks with Predictive Coding (PC). It is built on top of three main libraries:

• Equinox, to define neural networks with PyTorch-like syntax,
• Diffrax, to solve the PC inference (activity) dynamics, and
• Optax, for parameter optimisation.

JPC provides a simple, fast and flexible API for training of a variety of PCNs including discriminative, generative and hybrid models.

• Like JAX, JPC is completely functional, and the core library is <1000 lines of code.
• Unlike existing implementations, JPC leverages ordinary differential equation (ODE) solvers to integrate the inference dynamics of PC networks (PCNs), which we find can provide significant speed-ups compared to standard optimisers, especially for deeper models.
• JPC also provides some analytical tools that can be used to study and diagnose issues with PCNs.

If you're new to JPC, we recommend starting from the example notebooks and checking the documentation.

Overview

• Installation
• Documentation
• Quick example
• Advanced usage
• Citation

️💻 Installation

pip install jpc

Requires Python 3.9+, JAX 0.4.23+, Equinox 0.11.2+, Diffrax 0.6.0+, and Optax 0.2.4+.

For GPU usage, upgrade jax to the appropriate cuda version (12 as an example here).

pip install --upgrade "jax[cuda12]"

📖 Documentation

Available at https://thebuckleylab.github.io/jpc/.

⚡️ Quick example

Use jpc.make_pc_step to update the parameters of any neural network compatible with PC updates (see the notebook examples )

import jax.random as jr
import jax.numpy as jnp
import equinox as eqx
import optax
import jpc

# toy data
x = jnp.array([1., 1., 1.])
y = -x

# define model and optimiser
key = jr.PRNGKey(0)
model = jpc.make_mlp(key, layer_sizes=[3, 5, 5, 3], act_fn="tanh")
optim = optax.adam(1e-3)
opt_state = optim.init(
    (eqx.filter(model, eqx.is_array), None)
)

# perform one training step with PC
result = jpc.make_pc_step(
    model=model,
    optim=optim,
    opt_state=opt_state,
    output=y,
    input=x
)

# updated model and optimiser
model = result["model"]
optim, opt_state = result["optim"], result["opt_state"]

Under the hood, jpc.make_pc_step

1. integrates the inference (activity) dynamics using a diffrax ODE solver, and
2. updates model parameters at the numerical solution of the activities with a given optax optimiser.

See the documentation for more details.

NOTE: All convenience training and test functions such as make_pc_step are already "jitted" (for optimised performance) for the user's convenience.

🚀 Advanced usage

Advanced users can access all the underlying functions of jpc.make_pc_step as well as additional features. A custom PC training step looks like the following:

import jpc

# 1. initialise activities with a feedforward pass
activities = jpc.init_activities_with_ffwd(model=model, input=x)

# 2. run inference to equilibrium
equilibrated_activities = jpc.solve_inference(
    params=(model, None), 
    activities=activities, 
    output=y, 
    input=x
)

# 3. update parameters at the activities' solution with PC
result = jpc.update_params(
    params=(model, None), 
    activities=equilibrated_activities,
    optim=optim,
    opt_state=opt_state,
    output=y, 
    input=x
)

which can be embedded in a jitted function with any other additional computations. Again, see the docs for details.

📄 Citation

If you found this library useful in your work, please cite (arXiv link):

@article{innocenti2024jpc,
    title={JPC: Flexible Inference for Predictive Coding Networks in JAX},
    author={Innocenti, Francesco and Kinghorn, Paul and Yun-Farmbrough, Will 
    and Singh, Ryan and De Llanza Varona, Miguel and Buckley, Christopher},
    journal={arXiv preprint},
    year={2024}
}

Also consider starring the repo! ⭐️

🙏 Acknowledgements

We are grateful to Patrick Kidger for early advice on how to use Diffrax.

See also: other PC libraries

JAX-based:

• pcx
• pyhgf

PyTorch-based:

• Torch2PC
• pypc
• pybrid