Preconditioning for Physics-Informed Neural Networks

• DOI: 10.48550/arxiv.2402.00531
• Publication Date: 2024-02-01
• Authors: Songming Liu; Chang Su; Jiachen Yao; Zhongkai Hao; Hang Su; Youjia Wu; Jun Zhu

Physics-informed neural networks (PINNs) have shown promise in solving various partial differential equations (PDEs). However, training pathologies negatively affected the convergence and prediction accuracy of PINNs, which further limits their practical applications. In this paper, we propose to use condition number as a metric diagnose mitigate PINNs. Inspired by classical numerical analysis, where measures sensitivity stability, highlight its pivotal role dynamics We prove theorems reveal how is related both error control Subsequently, present an algorithm that leverages preconditioning improve number. Evaluations 18 PDE problems showcase superior performance our method. Significantly, 7 these problems, method reduces errors order magnitude. These empirical findings verify critical PINNs' training.