A Bayesian Framework for learning governing Partial Differential Equation from Data

TL;DR

This paper introduces a Bayesian method combining variational Bayes and sparse regression to accurately discover PDEs from noisy data, demonstrated on classical equations in physics and engineering.

Contribution

It presents a novel Bayesian framework for PDE discovery that improves accuracy and sparsity in noisy data environments, outperforming existing methods.

Findings

Effective in identifying PDEs from noisy data

Successfully applied to classical physics equations

Demonstrates improved sparsity and accuracy

Abstract

The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to note that existing methods often struggle to identify the underlying equation accurately in the presence of noise. In this study, we present a new approach to discovering PDEs by combining variational Bayes and sparse linear regression. The problem of PDE discovery has been posed as a problem to learn relevant basis from a predefined dictionary of basis functions. To accelerate the overall process, a variational Bayes-based approach for discovering partial differential equations is proposed. To ensure sparsity, we employ a spike and slab prior. We illustrate the efficacy of our strategy in several examples, including Burgers, Korteweg-de Vries, Kuramoto Sivashinsky, wave equation, and heat equation (1D as well as 2D). Our method offers a promising avenue for discovering PDEs from data and has potential applications in fields such as physics, engineering, and biology.