TY - JOUR
T1 - Convergence Rate Analysis for Deep Ritz Method
AU - Duan , Chenguang
AU - Jiao , Yuling
AU - Lai , Yanming
AU - Li , Dingwei
AU - Lu , Xiliang
AU - Yang , Jerry Zhijian
JO - Communications in Computational Physics
VL - 4
SP - 1020
EP - 1048
PY - 2022
DA - 2022/03
SN - 31
DO - http://doi.org/10.4208/cicp.OA-2021-0195
UR - https://global-sci.org/intro/article_detail/cicp/20375.html
KW - Deep Ritz method, deep neural networks, convergence rate analysis.
AB - <p style="text-align: justify;">Using deep neural networks to solve PDEs has attracted a lot of attentions
recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz
method (DRM) [47] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM
using deep networks with ${\rm ReLU}^2$ activation functions. In addition to providing a
theoretical justification of DRM, our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of
number of training samples. Technically, we derive bound on the approximation error
of deep ${\rm ReLU}^2$ network in $C^1$ norm and bound on the Rademacher complexity of the
non-Lipschitz composition of gradient norm and ${\rm ReLU}^2$ network, both of which are of
independent interest.</p>