物理信息神经网络及相关模型的数值分析
数值分析
2024-11-20 v1 机器学习
数值分析
摘要
物理信息神经网络 (PINNs) 及其变体在近年来作为求解偏微分方程正问题和逆问题的数值仿真算法非常受欢迎。本文旨在提供当前关于 PINNs 及构成物理信息机器学习基础框架的各类模型数值分析结果的综合综述。在统一的框架下, 可有效进行对 PINNs 近似偏微分方程过程中所产生误差各组成部分的分析。present 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.
引用
@article{arxiv.2402.10926,
title = {Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning},
author = {Tim De Ryck and Siddhartha Mishra},
journal= {arXiv preprint arXiv:2402.10926},
year = {2024}
}