Robust training and rigorous error analysis of physics-informed neural networks for the $p$-Laplace equation

Abstract

With the rise of scientific machine learning, physics-informed neural networks (PINNs) have been extensively applied to a wide range of problems. Nevertheless, most theoretical analyses of PINNs remain confined to linear equations, and a substantial gap persists between PINNs and classical numerical analysis for nonlinear problems. To address this issue, we propose a robust training framework for PINNs solving nonlinear partial differential equations, together with a rigorous error analysis. Specifically, for the $p$-Laplace equation, we introduce a novel loss formulation that combines a dual residual loss measured in $W^{-1,p^{\prime }}$ with a boundary loss measured in a fractional Sobolev norm $W^{1-\frac{1}{p},p}$. This formulation is designed to accommodate the limited regularity of weak solutions and enables us to establish rigorous \textit{a priori} and \textit{a posteriori} error estimates. Moreover, the proposed framework and its analysis are extended to a parametric setting in which the exponent, the source term, and the boundary condition may all vary with the parameters. Finally, we present numerical experiments that substantiate our theoretical findings.