Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ReLU2 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 D, width W of the ReLU2 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.