A predictor-corrector deep learning algorithm for high dimensional stochastic partial differential equations

TL;DR

This paper introduces a deep learning-based predictor-corrector algorithm for efficiently approximating high-dimensional stochastic partial differential equations, combining stochastic and deterministic PDE solutions with neural networks.

Contribution

It develops a novel predictor-corrector method that decomposes SPDEs and employs neural networks for the deterministic part, with proven error estimates and convergence rates.

Findings

The algorithm achieves high accuracy in numerical examples.

Error estimates and convergence rates are established.

The method effectively handles high-dimensional SPDEs.

Abstract

In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely, we decompose the original SPDE into a degenerate SPDE and a deterministic PDE. Then in the prediction step, we solve the degenerate SPDE with the Euler scheme, while in the correction step we solve the second-order deterministic PDE by deep neural networks via its equivalent backward stochastic differential equation (BSDE). Under standard assumptions, error estimates and the rate of convergence of the proposed algorithm are presented. The efficiency and accuracy of the proposed algorithm are illustrated by numerical examples.