Improving Algorithms for Learning Radial Basic Functions Networks to Solve the Boundary Value Problems

Abstract

Digital twins are widely used in modern industry. A digital twin is a computer model that copies the behavior of a physical object. Digital twins of objects with distributed parameters are mathematically boundary value problems for partial differential equation. Traditionally, such problems are solved by finite difference and finite element methods, which require a complex grid construction procedure. The numerical solution of boundary value problems employs mesh less methods that do not require grid construction. Among mesh fewer methods, projection methods that use radial basis functions (RBFs) as basic functions are popular. Methods using RBF allow us to obtain a differentiable solution at any point in the solution domain, applicable to problems of arbitrary dimension with complex computational domains. When solving the problem, the parameters of the basic functions are selected, and the weights are calculated so that the residuals obtained after the substitution of the approximate solution at the test points in the equation are zero.