PHYSICS-INFORMED RADIAL BASIS FUNCTION NETWORKS AND KOLMOGOROV-ARNOLD NETWORKS
Keywords: partial differential equations, radial basis function networks, Kolmogorov-Arnold networks, physics-informed neural networks, piecewise homogeneous medium, coefficient inverse problem, incorrect problem, Navier-Stokes equations, Kovasznay flow
TL;DR: Solving differential equations on neural networks.
Abstract: Physics-informed neural networks are trained by minimizing the loss function, which is the sum of the squares of the residuals of the equation or system of equations being solved. Such networks do not require grid construction, which is especially important when solving inverse boundary value problems and problems with a complex solution domain. We use radial basis function networks with a Gaussian function. Physics-informed radial basis function networks are easier to train than fully connected networks. They allow one to analytically obtain formulas for the gradient of the loss function. A special feature of our approach to training networks based on radial basis functions is the adjustment of not only the weights, but also the parameters of the radial basis functions, which does not require the selection of parameters of the radial basis functions and accelerates the training process. Algorithms for solving direct and inverse boundary value problems, an algorithm for solving a system of differential equations for modeling the Kovasznay flow have been developed. Programs have been developed that use various algorithms for training physics-informed radial basis function networks.
Submission Number: 25
Loading