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.

本文提出了一种基于径向基函数（RBF）网络的方法来求解带狄利克雷边界条件的非线性二阶椭圆方程。微分方程中涉及的非局部项需要与使用神经网络解决边值问题的最新方法完全不同的方法。开发了用于计算局部\（L ^ 2 \）的数值积分程序-内部产品。众所周知，非变分方法不能有效解决非局部问题。在本文中，非局部问题的弱公式被简化为非线性函数的最小化。与以前的许多作品不同，我们使用积分目标函数来实现学习过程。分布良好的节点用作RBF神经网络的中心。RBF网络的权重通过两点步长梯度法确定。本文提出的神经网络方法是解决非Lipschitz域中非局部边值问题的有限元方法（FEM）的替代方法。为了避免训练过程的分歧，已经开发并实施了一种新的可变学习率策略。