PINNsFailureRegion Localization and Refinement through White-box AdversarialAttack

TL;DR

This paper introduces WbAR, a white-box adversarial attack-based sampling strategy that effectively locates and refines failure regions in PINNs, improving their ability to solve complex PDEs with sharp or multi-scale solutions.

Contribution

The paper presents a novel failure region localization method for PINNs using white-box adversarial attacks, enhancing training focus on critical regions for complex PDEs.

Findings

WbAR successfully locates failure regions in various complex PDEs.

Refinement guided by WbAR improves PINNs' accuracy on challenging problems.

WbAR's effectiveness is independent of failure region size or distribution complexity.

Abstract

Physics-informed neural networks (PINNs) have shown great promise in solving partial differential equations (PDEs). However, vanilla PINNs often face challenges when solving complex PDEs, especially those involving multi-scale behaviors or solutions with sharp or oscillatory characteristics. To precisely and adaptively locate the critical regions that fail in the solving process we propose a sampling strategy grounded in white-box adversarial attacks, referred to as WbAR. WbAR search for failure regions in the direction of the loss gradient, thus directly locating the most critical positions. WbAR generates adversarial samples in a random walk manner and iteratively refines PINNs to guide the model's focus towards dynamically updated critical regions during training. We implement WbAR to the elliptic equation with multi-scale coefficients, Poisson equation with multi-peak solutions, high-dimensional Poisson equations, and Burgers equation with sharp solutions. The results demonstrate that WbAR can effectively locate and reduce failure regions. Moreover, WbAR is suitable for solving complex PDEs, since locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution.