Deep Radial Basis Function Networks

Abstract

The use of radial basis functions networks as physics-informed neural networks for solving direct and inverse boundary value problems is demonstrated. On the Levenberg-Marquardt basis optimization method, algorithms have been developed for solving partial differential equations. Comparison of the gradient descent method and the Levenberg-Marquardt method for solving the Poisson equation is given. To solve a direct boundary value problem describing processes in a piecewise homogeneous environment, an algorithm is proposed based on solving individual problems for each region with different properties of the environment associated with the conjugation conditions. It removes restrictions on the radial basis functions used. To solve the coefficient inverse problem of recovering the properties of the piecewise inhomogeneous medium, an algorithm based on parametric optimization is proposed. An algorithm uses two networks of radial basis functions. The first network approximates the solution to the direct problem. And another network approximates a function which describes the properties of the environment. Network learning is performed using an algorithm developed by the authors based on the Levenberg-Marquardt method. Expressions are obtained for the analytical calculation of the Jacobi matrix elements in the Levenberg-Marquardt method and the residual gradient vector elements. The application of the developed algorithms is demonstrated by the example of model direct boundary value problems and inverse coefficient boundary value problems for piecewise homogeneous media.