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. We present a detailed review of available results on approximation, generalization and training errors and their behaviour with respect to the type of the PDE and the dimension of the underlying domain. In particular, we elucidate the role of the regularity of the solutions and their stability to perturbations in the error analysis. 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.

中文翻译：

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物理信息机器学习中物理信息神经网络及相关模型的数值分析


近年来，物理信息神经网络 (PINN) 及其变体作为偏微分方程正向和反演问题的数值模拟算法非常流行。本文旨在对构成物理信息机器学习支柱的 PINN 和相关模型的当前可用数值分析结果进行全面回顾。我们提供了一个统一的框架，可以有效地分析 PINN 在逼近偏微分方程时产生的误差的各个组成部分。我们详细回顾了近似、泛化和训练误差的可用结果及其相对于偏微分方程类型和基础域维度的行为。特别是，我们阐明了解的规律性及其对误差分析中扰动的稳定性的作用。还提供了数值结果来说明该理论。我们认为训练错误是一个关键瓶颈，它可能会对物理信息机器学习中各种模型的整体性能产生不利影响。