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The approaches and methods of radial basis functions networks learning to solve mathematical physic problems

Keywords:

V.I. Gorbachenko – Professor, Head of Chair «Computer technology», Penza State University. E-mail: gorvi@mail.ru M.V. Zhukov – Post-graduate Student, Penza State University. E-mail: maxim.zh@gmail.com


One of the most effective numerical methods of mathematical physics problems solving is a meshfree method, based on radial basis function networks (RBFNs). The main difficulty, related with it, is parameters and weights of radial basis functions (RBFs) definition. The purpose of this article is to explore RBFN parameters configuration approaches and their realization methods. There are two parameters configuration approaches: Adjust RBFs weights only and set special fixed values for the others RBFs parameters. Adjust all RBFs parameters. There are peculiarities of use of both approaches to solve linear, non-linear and non-stationary partial difference equations were examined; three methods of the former approach realization and five methods of the latter approach realization were considered in this article. So, when the former approach is used, the task of network learning reduces to a linear least squares problem. Its solution is found as the solution of an overdetermined system of linear equations. QR or SVD decomposition can be used to solve it or it can be converted to a normal system of linear equations. Moreover, to preserve linearity of the least squares problem quasi-linearization should be used. The network learning reduces to a quadratic functional minimization problem when the latter approach is used. Among the methods of this approach realization hybrid method, fast c cubic algorithm, Fast Cubic Algorithm, dense clouds method, trust region method and Levenberg-Marquardt were examined. To demonstrate how both approaches are practiced and to compare them and their methods of realization with one another two differential equations were solved: Poisson equation and semilinear elliptic equation with a first-type boundary condition and a third-type boundary condition accordingly. According to the exploration, the conclusion was made in the last part of the article: the former approach, realized with SVD decomposition, makes it possible to solve linear difference equations effectively, moreover, this approach can be used to solve nonlinear equations. The quasilinearization is required in such case, but it makes worse the qualitative characteristic of the approach sharply. The main difficulties, related with this approach, are optimal values selection of widths and RBFs’ amount, quasilinearization need. The latter approach is more commonly and can solve linear equations as well as nonlinear. From the point of view of effectiveness, according to the experiments result again, it is inferior to the former to a very little degree when it is used to solve linear equations and excels it in nonlinear equations solving. Among examined its realization methods, trust region is best. The main difficulty, related with it, is finding an initial approximation.
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