Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning

TL;DR

This paper reviews the numerical analysis of physics-informed neural networks (PINNs), focusing on error components, approximation capabilities, and the impact of solution regularity, with numerical illustrations and discussion of training challenges.

Contribution

It provides a unified framework for analyzing PINNs, reviews existing theoretical results, and highlights training errors as a key bottleneck in physics-informed machine learning.

Findings

Error analysis framework for PINNs

Impact of solution regularity on approximation accuracy

Training errors significantly affect model performance

Abstract

Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed machine learning. We provide a unified framework in which analysis of the various components of the error incurred by PINNs in approximating PDEs can be effectively carried out. A detailed review of available results on approximation, generalization and training errors and their behavior with respect to the type of the PDE and the dimension of the underlying domain is presented. In particular, the role of the regularity of the solutions and their stability to perturbations in the error analysis is elucidated. Numerical results are also presented to illustrate the theory. We identify training errors as a key bottleneck which can adversely affect the overall performance of various models in physics-informed machine learning.