Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data

TL;DR

Weak-PDE-LEARN is a robust neural network-based method that accurately discovers nonlinear PDEs from noisy, limited data by leveraging weak form loss functions.

Contribution

It introduces a novel weak form based loss function for PDE discovery that enhances robustness to noise and limited data scenarios.

Findings

Successfully learns benchmark PDEs from noisy data

Demonstrates robustness to measurement noise

Effective with limited solution measurements

Abstract

We introduce Weak-PDE-LEARN, a Partial Differential Equation (PDE) discovery algorithm that can identify non-linear PDEs from noisy, limited measurements of their solutions. Weak-PDE-LEARN uses an adaptive loss function based on weak forms to train a neural network, $U$, to approximate the PDE solution while simultaneously identifying the governing PDE. This approach yields an algorithm that is robust to noise and can discover a range of PDEs directly from noisy, limited measurements of their solutions. We demonstrate the efficacy of Weak-PDE-LEARN by learning several benchmark PDEs.