@Article{NMTMA-17-181,
author = {Chen , MoJiao , YulingLu , XiliangSong , PengchengWang , Fengru and Yang , Jerry Zhijian},
title = {Analysis of Deep Ritz Methods for Semilinear Elliptic Equations},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2024},
volume = {17},
number = {1},
pages = {181--209},
abstract = {<p style="text-align: justify;">In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ${\rm ReLU}^2$ activations. Firstly, we present a comprehensive
formulation based on the penalized variational form of the elliptical equations. We
then apply the Deep Ritz Method, which works for a wide range of equations. We
obtain an upper bound on the errors between the acquired solutions and the true
solutions in terms of the depth $\mathcal{D},$ width $\mathcal{W}$ of the ${\rm ReLU}^2$ ResNet, and the number of training samples $n.$ Our simulation results demonstrate that our method can
effectively overcome the curse of dimensionality and validate the theoretical results.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2023-0058 },
url = {http://global-sci.org/intro/article_detail/nmtma/22915.html}
}