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 -

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.