PINNs failure region localization and refinement through white-box adversarial attack

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 searches 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 on 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, as locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution. Code link: https://github.com/yaoli90/WbAR