Physics-informed radial basis function network based on Hausdorff fractal distance for solving Hausdorff derivative elliptic problems

Abstract

This paper proposes a physics-informed radial basis function network (RBFN) based on Hausdorff fractal distance to resolve Hausdorff derivative elliptic problems. In the proposed scheme, we improve the performance of RBFN via setting the source points outside the computational domain, and allocating distinct shape parameter values to each RBF. Furthermore, on the basis of the modified RBFN, we take full advantage of the physical laws described by Hausdorff derivative partial differential equations and the constraints imposed by the boundary conditions, and establish a physics-informed optimization system for Hausdorff derivative elliptic problems. Utilizing MATLAB optimization toolbox function lsqnonlin, we solve the optimization system and then obtain the optimized network parameters including coordinates of source points, values of shape parameters and unknown RBF weights simultaneously, with which we deal with Hausdorff derivative elliptic problems successfully. Numerical experiments associated with acoustic, anisotropic heat conduction and fourth order problems are carried out to demonstrate the performance of the developed methodology.