Numerical solution of differential equations using multiquadric radial basis functions networks

Neural Networks : the Official Journal of the International Neural Network Society
N Mai-Duy, T Tran-Cong

Abstract

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) = betad(i), where d(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 beta can be chosen in the range 7 < or = beta 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 x 10...Continue Reading

References

Sep 23, 1998·Neural Computation· Van Hulle MM

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Citations

Jul 10, 2003·Neural Networks : the Official Journal of the International Neural Network Society·Li JianyuHuang Yaping
Jun 6, 2009·IEEE Transactions on Neural Networks·Kevin Stanley McFall, James Robert Mahan
Dec 12, 2007·Journal of Digital Imaging·Akmal YounisNigel John
Nov 17, 2007·Journal of Cutaneous Pathology·Galina Y StetsenkoZsolt Argenyi

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