Solving of Inverse Coefficient Problems on Networks of Radial Basis Functions

Abstract

AbstractThe prospects of using radial basis function networks as physics-informed neural networks for solving direct and inverse boundary value problems described by partial differential equations are shown. To solve the coefficient inverse problem of recovering the properties of a piecewise inhomogeneous medium, an algorithm based on parametric optimization is proposed. The algorithm uses two radial basis function networks, one of which approximates the solution of direct problems, and the second approximates a function that describes the properties of the medium. Network training is performed using an algorithm developed by the authors based on the Levenberg-Marquardt method. Expressions for the analytical calculation of the elements of the Jacobi matrix in the Levenberg-Marquardt method are obtained. The application of the developed algorithm is demonstrated by the example of a model coefficient inverse problem for a piecewise homogeneous medium. To solve the direct problem on radial basis function networks for a piecewise homogeneous medium, an algorithm developed by the authors was used, based on solving individual problems for each area with different properties of the medium and using a common error functional that takes into account errors on the border of areas.KeywordsPartial differential equationsInverse problemsPhysics Informed neural-networksRadial basis functions networksNeural network learningLevenberg-Marquardt method