Physics-Informed Radial Basis Function Networks: Solving Inverse Problems for Partial Differential Equations

Abstract

AbstractThe advantages of using neural network models as digital twins of objects with distributed parameters are shown. The analysis of using physically-informed neural networks to solve direct and inverse boundary value problems is presented. It is proposed to use radial basis function neural networks (RBFNNs) as physically-informed neural networks, which have a simple structure and the ability to adjust the non-linear parameters of the basis functions. The approach to the solving of the coefficient inverse boundary value problems, which allows determining the unknown function describing the physical environment, is proposed. The algorithm uses a unified approach to solve direct and inverse problems on RBFNN. RBFNN training is proposed to be performed using the fast algorithm of the Levenberg-Marquardt method developed by the authors. The analytical calculation of the Jacobi matrix in the Levenberg-Marquardt method is performed. An example of the solving of the inverse coefficient problem for a piecewise homogeneous medium is given. To solve the direct problem for a piecewise homogeneous medium we have used the algorithm developed by the authors which is based on solving separate problems for each region with different properties of the medium and on using the general error functional taking into account errors on the border of regions.KeywordsPartial differential equationsInverse problemsPhysics-informed neural networksRadial basis function networksLevenberg-Marquardt method