New Neural Network Technique to the Numerical Solution of Mathematical Physics Problems. Ⅰ: Simple Problems

摘要

Neural networks are considered to be the new universal approach to the construction of mathematical models of systems with distributed parameters. Networks effectively allow us to find the approximate solutions of initial and boundary problems for the partial differential equations and to take into account nonlinear effects and coefficient perturbations. Neural networks of known and new architectures are applied to the solution of Laplace, Helmholtz, and Schroedinger; heat conduction equations in domains with fixed, free, and controlled boundaries; and original training algorithms of these networks are given. This work reviews our work in 2003 to 2004 and consists of two parts. In Part Ⅰ, simple problems with well-known analytical solutions are considered from a neural network point of view. The solution technique is extended over practically important problems.