@Article{CiCP-31-1272,
author = {Jiao , YulingLai , YanmingLi , DingweiLu , XiliangWang , FengruWang , Yang and Yang , Jerry Zhijian},
title = {A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs},
journal = {Communications in Computational Physics},
year = {2022},
volume = {31},
number = {4},
pages = {1272--1295},
abstract = {<p style="text-align: justify;">In recent years, physical informed neural networks (PINNs) have been
shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for
the second order elliptic equations with Dirichlet boundary condition, by establishing
the upper bounds on the number of training samples, depth and width of the deep
neural networks to achieve desired accuracy. The error of PINNs is decomposed into
approximation error and statistical error, where the approximation error is given in $C^2$ norm with ReLU$^3$ networks (deep network with activation function max$\{0,x^3\}$) and
the statistical error is estimated by Rademacher complexity. We derive the bound on
the Rademacher complexity of the non-Lipschitz composition of gradient norm with
ReLU$^3$ network, which is of immense independent interest.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2021-0186},
url = {http://global-sci.org/intro/article_detail/cicp/20384.html}
}