Numerical solution of differential equations using multiquadric radial basis function networks
Article
Article Title | Numerical solution of differential equations using multiquadric radial basis function networks |
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ERA Journal ID | 18090 |
Article Category | Article |
Authors | Mai-Duy, Nam (Author) and Tran-Cong, Thanh (Author) |
Journal Title | Neural Networks |
Journal Citation | 14 (2), pp. 185-199 |
Number of Pages | 15 |
Year | 2001 |
Place of Publication | Oxford, United Kingdom |
ISSN | 0893-6080 |
1879-2782 | |
Digital Object Identifier (DOI) | https://doi.org/10.1016/S0893-6080(00)00095-2 |
Web Address (URL) | https://www.sciencedirect.com/science/article/pii/S0893608000000952 |
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)=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. |
Keywords | radial basis function networks; multiquadric function; global approximation; mesh-free method; solution of differential equation |
ANZSRC Field of Research 2020 | 490409. Ordinary differential equations, difference equations and dynamical systems |
490101. Approximation theory and asymptotic methods | |
Public Notes | File reproduced in accordance with the copyright policy of the publisher/author. |
Byline Affiliations | Faculty of Engineering and Surveying |
https://research.usq.edu.au/item/9y41w/numerical-solution-of-differential-equations-using-multiquadric-radial-basis-function-networks
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