Physics Informed Neural Networks (PINNs) as intelligent computing technique for solving partial differential equations: Limitation and Future prospects

TL;DR

This paper reviews the limitations of Physics-Informed Neural Networks (PINNs) in solving PDEs, analyzes their root causes, and discusses future directions to enhance their accuracy, convergence, and integration with classical methods.

Contribution

It systematically identifies key limitations of PINNs and proposes future research directions to overcome these challenges in solving PDEs.

Findings

PINNs have issues with multiscale approximation and ill-conditioning.

Weak mathematical rigor due to limited convergence and error analysis.

Inadequate physical information integration causes residual and error mismatch.

Abstract

In recent years, Physics-Informed Neural Networks (PINNs) have become a representative method for solving partial differential equations (PDEs) with neural networks. PINNs provide a novel approach to solving PDEs through optimization algorithms, offering a unified framework for solving both forward and inverse problems. However, some limitations in terms of solution accuracy and generality have also been revealed. This paper systematically summarizes the limitations of PINNs and identifies three root causes for their failure in solving PDEs: (1) Poor multiscale approximation ability and ill-conditioning caused by PDE losses; (2) Insufficient exploration of convergence and error analysis, resulting in weak mathematical rigor; (3) Inadequate integration of physical information, causing mismatch between residuals and iteration errors. By focusing on addressing these limitations in PINNs, we outline the future directions and prospects for the intelligent computing of PDEs: (1) Analysis of ill-conditioning in PINNs and mitigation strategies; (2) Improvements to PINNs by enforcing temporal causality; (3) Empowering PINNs with classical numerical methods.