Discovering stochastic partial differential equations from limited data using variational Bayes inference

TL;DR

This paper introduces a new variational Bayes-based framework for discovering stochastic partial differential equations from limited data, combining stochastic calculus, sparse learning, and an extended Kramers-Moyal expansion.

Contribution

It is the first method to successfully identify SPDEs from data, integrating variational Bayes inference with sparse learning for accurate discovery.

Findings

Accurately identified SPDEs from limited data

Applied to stochastic heat, Allen-Cahn, and Nagumo equations

Demonstrated potential for scientific applications like climate and finance

Abstract

We propose a novel framework for discovering Stochastic Partial Differential Equations (SPDEs) from data. The proposed approach combines the concepts of stochastic calculus, variational Bayes theory, and sparse learning. We propose the extended Kramers-Moyal expansion to express the drift and diffusion terms of an SPDE in terms of state responses and use Spike-and-Slab priors with sparse learning techniques to efficiently and accurately discover the underlying SPDEs. The proposed approach has been applied to three canonical SPDEs, (a) stochastic heat equation, (b) stochastic Allen-Cahn equation, and (c) stochastic Nagumo equation. Our results demonstrate that the proposed approach can accurately identify the underlying SPDEs with limited data. This is the first attempt at discovering SPDEs from data, and it has significant implications for various scientific applications, such as climate modeling, financial forecasting, and chemical kinetics.