Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion

TL;DR

This paper introduces a neural network-based numerical method for solving stochastic partial differential equations (SPDEs) using Wiener chaos expansion, providing approximation rates and demonstrating effectiveness on several example SPDEs.

Contribution

It presents a novel approach combining neural networks with Wiener chaos expansion to solve SPDEs, including theoretical approximation rates and practical numerical applications.

Findings

Successful approximation of the stochastic heat equation

Effective solution of the Heath-Jarrow-Morton equation

Accurate results for the Zakai equation

Abstract

In this paper, we solve stochastic partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their corresponding solution. Moreover, we provide some approximation rates for learning the solution of SPDEs with additive and/or multiplicative noise. Finally, we apply our results in numerical examples to approximate the solution of three SPDEs: the stochastic heat equation, the Heath-Jarrow-Morton equation, and the Zakai equation.