Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit

TL;DR

This paper introduces neural stochastic differential equations (SDEs) as a deep latent Gaussian model in the diffusion limit, enabling flexible stochastic modeling with neural network parameterizations and variational inference via stochastic automatic differentiation.

Contribution

It develops a variational inference framework for neural SDEs using stochastic automatic differentiation and Girsanov transformations, extending deep latent models to continuous-time stochastic processes.

Findings

Effective inference with synthetic data demonstrated.

Flexible neural SDE models successfully implemented.

End-to-end automatic differentiation enabled.

Abstract

In deep latent Gaussian models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. This work considers the diffusion limit of such models, where the number of layers tends to infinity, while the step size and the noise variance tend to zero. The limiting latent object is an It\^o diffusion process that solves a stochastic differential equation (SDE) whose drift and diffusion coefficient are implemented by neural nets. We develop a variational inference framework for these \textit{neural SDEs} via stochastic automatic differentiation in Wiener space, where the variational approximations to the posterior are obtained by Girsanov (mean-shift) transformation of the standard Wiener process and the computation of gradients is based on the theory of stochastic flows. This permits the use of black-box SDE solvers and automatic differentiation for end-to-end inference. Experimental results with synthetic data are provided.