PyTorch Implementation of Differentiable SDE Solvers

This library provides stochastic differential equation (SDE) solvers with GPU support and efficient backpropagation.

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Installation

pip install git+https://github.com/google-research/torchsde.git

Requirements: Python >=3.6 and PyTorch >=1.6.0.

Documentation

Available here.

Examples

Quick example

import torch
import torchsde

batch_size, state_size, brownian_size = 32, 3, 2
t_size = 20

class SDE(torch.nn.Module):
    noise_type = 'general'
    sde_type = 'ito'

    def __init__(self):
        super().__init__()
        self.mu = torch.nn.Linear(state_size, 
                                  state_size)
        self.sigma = torch.nn.Linear(state_size, 
                                     state_size * brownian_size)

    def f(self, t, y):
        return self.mu(y)  # shape (batch_size, state_size)

    def g(self, t, y):
        return self.sigma(y).view(batch_size, 
                                  state_size, 
                                  brownian_size)

sde = SDE()
y0 = torch.full((batch_size, state_size), 0.1)
ts = torch.linspace(0, 1, t_size)
# Initial state y0, the SDE is solved over the interval [ts[0], ts[-1]].
# ys will have shape (t_size, batch_size, state_size)
ys = torchsde.sdeint(sde, y0, ts)

Notebook

examples/demo.ipynb gives a short guide on how to solve SDEs, including subtle points such as fixing the randomness in the solver and the choice of noise types.

Latent SDE

examples/latent_sde.py learns a latent stochastic differential equation, as in Section 5 of [1]. The example fits an SDE to data, whilst regularizing it to be like an Ornstein-Uhlenbeck prior process. The model can be loosely viewed as a variational autoencoder with its prior and approximate posterior being SDEs. This example can be run via

python -m examples.latent_sde --train-dir <TRAIN_DIR>

The program outputs figures to the path specified by <TRAIN_DIR>. Training should stabilize after 500 iterations with the default hyperparameters.

Citation

If you found this codebase useful in your research, please consider citing:

@article{li2020scalable,
  title={Scalable gradients for stochastic differential equations},
  author={Li, Xuechen and Wong, Ting-Kam Leonard and Chen, Ricky T. Q. and Duvenaud, David},
  journal={International Conference on Artificial Intelligence and Statistics},
  year={2020}
}

References

[1] Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud. "Scalable Gradients for Stochastic Differential Equations." International Conference on Artificial Intelligence and Statistics. 2020. [arXiv]

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This is a research project, not an official Google product.