Neural network approach for solving nonlocal boundary value problems
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ORIGINAL ARTICLE
Neural network approach for solving nonlocal boundary value problems V. Palade1 • M. S. Petrov2 • T. D. Todorov3 Received: 17 January 2019 / Accepted: 22 February 2020 Springer-Verlag London Ltd., part of Springer Nature 2020
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 L2 -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 nonLipschitz 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. Keywords RBF neural network Variable learning rate Nonlocal nonlinear problem Two-point step size gradient method
1 Introduction Many mathematical models in engineering and science are usually represented by differential equations. Partial differential equations have been used for describing heat propagation, mass transfer, mass concentration, etc. Various boundary value problems were employed when modeling mechanical quantities, such as velocity, stress, strain & T. D. Todorov [email protected] V. Palade [email protected] M. S. Petrov [email protected] 1
Faculty of Engineering and Computing, Coventry University, Priory Street, Coventry, UK
2
Department of Technical Mechanics, Technical University of Gabrovo, 5300 Gabrovo, Bulgaria
3
Department of Mathematics and Informatics, Technical University of Gabrovo, 5300 Gabrovo, Bulgaria
and displacements. The effective solving of complicated engineering problems requires stable mathematical models, which can be implemented in real time. The finite-element method, the finite difference method and the finite volume methods have been widely used for solving various boundary and eigenvalue problems. The meshfree methods can be applied for solving elliptic partial differential equations beyond the area of application of the classical mesh methods. The multigrid version of the FEM is the most po
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