TY - JOUR
T1 - Analysis of Deep Ritz Methods for Semilinear Elliptic Equations
AU - Chen , Mo
AU - Jiao , Yuling
AU - Lu , Xiliang
AU - Song , Pengcheng
AU - Wang , Fengru
AU - Yang , Jerry Zhijian
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 181
EP - 209
PY - 2024
DA - 2024/02
SN - 17
DO - http://doi.org/10.4208/nmtma.OA-2023-0058 
UR - https://global-sci.org/intro/article_detail/nmtma/22915.html
KW - Semilinear elliptic equations, Deep Ritz method, ReLU$^2$ ResNet, convergence rate.
AB - <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>