Neural network approach for solving nonlocal boundary value problems

Abstract

This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the differential equation needs a completely different approach from the up-to-now-known methods for solving boundary value problems by using neural networks. A numerical integration procedure is developed for computing the local $$L^2$$ -inner product. It is known that the non-variational methods are not effective in solving nonlocal problems. In this paper, the weak formulation of the nonlocal problem is reduced to the minimization of a nonlinear functional. Unlike many previous works, we use an integral objective functional for implementing the learning procedure. Well-distributed nodes are used as the centers of the RBF neural network. The weights of the RBF network are determined by a two-point step size gradient method. The neural network method proposed in this paper is an alternative to the finite-element method (FEM) for solving nonlocal boundary value problems in non-Lipschitz domains. A new variable learning rate strategy has been developed and implemented in order to avoid the divergence of the training process. A comparison between the proposed neural network approach and the FEM is illustrated by challenging examples, and the performance of both methods is thoroughly analyzed.