Solving direct and inverse boundary value problems for piecewise homogeneous media on radial basis functions networks

The application of physics-informed radial basis function networks for solving boundary value problems describing piecewise homogeneous media is considered. A meshless algorithm for solving boundary value problems for piecewise homogeneous media is proposed, using the solution of individual problems for each region with different properties of the medium, and the conditions for the conjugation of media. To solve the coefficient inverse problem of determining the properties of a piecewise inhomogeneous medium, a parametric optimization algorithm is proposed that uses separate networks to approximate the properties of the medium and solve the direct problem. To train networks, a fast algorithm developed by the authors based on the Levenberg–Marquardt method was applied. The work of the proposed algorithms is demonstrated on model problems.