@Article{CSIAM-AM-3-761,
author = {Duan , ChenguangJiao , YulingLai , YanmingLu , XiliangQuan , Qimeng and Yang , Jerry Zhijian},
title = {Deep Ritz Methods for Laplace Equations with Dirichlet Boundary Condition},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2022},
volume = {3},
number = {4},
pages = {761--791},
abstract = {<p style="text-align: justify;">Deep Ritz methods (DRM) have been proven numerically to be efficient in
solving partial differential equations. In this paper, we present a convergence rate in $H^1$ norm for deep Ritz methods for Laplace equations with Dirichlet boundary condition, where the error depends on the depth and width in the deep neural networks and
the number of samples explicitly. Further we can properly choose the depth and width
in the deep neural networks in terms of the number of training samples. The main idea
of the proof is to decompose the total error of DRM into three parts, that is approximation error, statistical error and the error caused by the boundary penalty. We bound the
approximation error in $H^1$ norm with ${\rm ReLU}^2$ networks and control the statistical error
via Rademacher complexity. In particular, we derive the bound on the Rademacher
complexity of the non-Lipschitz composition of gradient norm with ${\rm ReLU}^2$ network,
which is of immense independent interest. We also analyze the error inducing by the
boundary penalty method and give a prior rule for tuning the penalty parameter.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2021-0043},
url = {http://global-sci.org/intro/article_detail/csiam-am/21155.html}
}