Navigating PINNs via maximum residual-based continuous distribution

Abstract

Physics-informed neural networks (PINNs) constitute a powerful framework that seamlessly integrate neural networks with underlying physical principles through the physics-informed loss function. This framework proficiently addresses both forward and inverse problems of partial differential equations (PDEs), while encounters challenges in abrupt spatio-temporal domains and sharp solutions. The prevailing consensus underscores the importance of data quality in influencing the convergence rate, predictive accuracy and other performance metrics. Nevertheless, the generation and selection of high-quality training data remain an open problem. In this study, we devise a sampling-enhanced framework to unify existing sampling methods for PINNs from the perspective of deep Galerkin method. Furthermore, we propose MRD (\textbf{M}aximum \textbf{R}esidual-based continuous \textbf{D}istribution) to navigate PINNs and move training points towards the high-residual region. It not only extends existing residual-based adaptive sampling to a continuous form for more precise residual indication, but also effectively generates a high-quality training dataset. Our method is generic, straightforward, and easily extensible to any scenarios of PINNs. Experimental results across all four equations demonstrate a significant improvement relative to baselines, validating the generality and efficacy of our MRD method. Mathematics Subject Classification (2020) 65M50 · 65C10 · 68T07