Numerical solution of differential equations using multiquadric radial basis function networks

By Nam Mai-Duy, Thanh Tran-Cong. Neural Networks, 2001

347 citations recorded by the source corpus.

This paper presents mesh-free procedures for solving linear differential equations (ODEs and elliptic PDEs) based on multiquadric (MQ) radial basis function networks (RBFNs). Based on our study of approximation of function and its derivatives using RBFNs that was reported in an earlier paper (Mai-Duy, N. & Tran-Cong, T. (1999). Approximation of function and its derivatives using radial basis function networks. Neural networks, submitted), new RBFN approximation procedures are developed in this paper for solving DEs, which can also be classified into two types: a direct (DRBFN) and an indirect (IRBFN) RBFN procedure. In the present procedures, the width of the RBFs is the only adjustable parameter according to a(i)=βd(i),whered(i) is the distance from the ith centre to the nearest centre. The IRBFN method is more accurate than the DRBFN one and experience so far shows that β can be chosen in the range 7≤β≤10 for the former. Different combinations of RBF centres and collocation points (uniformly and randomly distributed) are tested on both regularly and irregularly shaped domains. The results for a 1D Poisson's equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1.0×10−4) and O(1.0×10−8), respectively, with a centre density of 50. Similarly, the results for a 2D Poisson's equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1.0×10−3) and O(1.0×10−6) respectively, with a centre density of 12×12.

Subfields: Electrical and Electronic Engineering, Mechanics of Materials, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/04351b235395e9ea897ef64b1e8aba0f821e61c5

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Citations in the corpus

A volumetric integral radial basis function method for time-dependent partial differential equations. I. Formulation (2004, background): “Mai-Duy and Tran-Cong [31], [32] obtained very impressive computational results with singly or doubly integrated MQ RBFs that they cast as a minimization of the least-squares residual errors to find the expansion coefficients.”
Solving high order ordinary differential equations with radial basis function networks (2005, methodology): “There are two basic approaches for obtaining new basis functions from RBFs, namely the direct approach (DRBF) based on a differential process (Kansa [12]) and the indirect approach (IRBF) based on an integration process (Mai-Duy and Tran-Cong [15,18]).”
Trefftz and Collocation Methods (2008, methodology): “Hon and Schaback [212] used unsymmetrical collocation by radial basis functions, May-Duy and Tran-Cong [338, 339] used the radial basis function network methods for Poisson’s equations, and Chen et al.”
A novel application of radial basis functions for solving a model of first-order integro-ordinary differential equation (2011, methodology): “As a result, many further interesting developments and applications have been reported [21–28].”
A collocation method based on one-dimensional RBF interpolation scheme for solving PDEs (2007, neutral): “…equations has received a great deal of attention over the past 15 years (see, for example, Kansa, 1990; Fasshauer, 1997; Zerroukat et al, 1998; Mai-Duy and Tran-Cong, 2001; Fedoseyev et al, 2002; Power and Barraco, 2002; Cheng et al, 2003; Larsson and Fornberg, 2003; Sarler et al, 2004).”
Radial basis collocation methods for elliptic boundary value problems (2005, methodology): “May-Duy and Tran-Cong [12,13] used the radial basis function network methods for Poisson's equations.”
Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks (2001, methodology): “The results obtained [18,19] show that the IRBFN method achieves a better accuracy than the DRBFN method over a wide range of .”