Training RBF neural networks for the solution of elliptic boundary value problems

Abstract

We propose a radial basis function (RBF) neural network method for solving two- and three–dimensional second and fourth order elliptic boundary value problems (BVPs). The neural network in question is trained by minimizing a nonlinear least squares functional, thus determining the optimal values of the various RBF parameters involved. The functional minimization is carried out using standard MATLAB® software efficiently. Several numerical experiments are presented to demonstrate the efficacy of the proposed method.

Introduction

In recent decades, various meshless methods have been developed for solving boundary value problems (BVPs) in complicated geometries in two (2D) and three dimensions (3D). The main attraction of meshless methods is the simplicity of the solution process as no tedious domain or boundary discretization is required. Among all meshless methods, radial basis function collocation methods (RBFCMs) [4], [8], [18], [32] have become particularly popular. Traditionally, the radial basis function (RBF) centres are placed inside the domain. In [5], [25], however, the domain in which the RBF centres were placed was extended outside the domain of the BVP in question resulting in better performance of RBFCMs. In general, for elliptic partial differential equations (PDEs), in the RBFCM solution process both the PDE and the boundary conditions (BCs) are collocated at selected collocation points to form a linear system of equations $\mathbf{Ax}=\mathbf{b}$. Once the RBF weights x are determined by a linear solver, the approximate solution can be calculated anywhere in the domain. The above solution process seems simple and straightforward, but the determination of an appropriate value of the RBF shape parameter (or appropriate values in case shape parameters are varied), which greatly affects the accuracy of the approximation, is a challenge and needs to be addressed prior the RBFCM solution process.

In an alternative approach, RBF neural networks have, in recent years, been used extensively for function approximation and the solution of BVPs [1], [2], [3], [6], [7], [9], [10], [12], [13], [16], [17], [23], [26], [27], [30], see also [31, Sections 4.2-4.3] and [14, Section 5.3].

It is the purpose of this paper to use an RBF neural network method for the solution of second and fourth order BVPs in 2D and 3D. The formulation of the proposed method is motivated by the approach of Gorbachenko and his co-workers [1], [2], [3], [7], [10], [11], [12] and the training of the network is achieved by minimizing a nonlinear cost functional. The minimization is carried out using the MATLAB® optimization toolbox function lsqnonlin which is a nonlinear least–squares solver. In contrast to traditional RBFCMs in which the (troublesome) appropriate shape parameter value needs to be determined prior to the solution procedure, the proposed approach allows us to find the optimal locations of the RBF centres, shape parameter(s) values, and RBF centre weights simultaneously through the use of lsqnonlin. In the application of this function we provide the Jacobian of the corresponding nonlinear system which leads to substantial computational time savings enabling us to solve 3D problems in complex geometries efficiently and accurately.

The paper is organized as follows. In Section 2, we briefly describe how to use the nonlinear least-squares solver lsqnonlin. In Section 3, the formulation of the proposed neural network method for BVPs for second and fourth order PDEs is given. Four numerical examples in 2D and 3D with irregular or non–smooth domains are studied in Section 4, illustrating the effectiveness of the current approach. Finally, in Section 5, some conclusions and ideas for future work are provided.