Approximate solution of elliptic equations on Hopfield neural networks

Background. Solution of mathematical physics' problems on artificial neural networks is an actively developing concept combining methods of calculus mathematics and computer science. Application of neural networks is especially effective in solution of reverse and incorrect problems and equations with inaccurately set parameters. At the present time the main method of solution of mathematical physics' problems on artificial neural networks is minimization of functional error. The study is aimed at development of a stable and quick method of solving mathematical physics' problems on artificial neural networks, based on the theory of differential equation solution stability. Materials and methods. The article describes an approximate method of elliptic equations' solution on Hopfield neural networks. The method consists in approximation of the source boundary problem of difference scheme and formation of a system of regular differential equations, the solution of which is reduced to solution of the difference scheme. Results. The authors suggest a method of boundary problem solution for linear and non-linear elliptic equations, based on the methods of the stability theory. Effectivenes of the method is demonstrated by the model examples. Conclusions. The results of the study may be used for solution of a wide class of boundary problems for linear and non-linear elliptic equations, determined in sectionally smooth areas.