Solving nonlinear ordinary differential equations using neural networks

Abstract

This paper present a novel framework for the numerical solution of nonlinear differential equations using neural networks. The benefit of this method is the trial solution can be used to solve any ordinary differential equations such as high order nonlinear differential equations relies upon the function approximation capabilities of recurrent neural networks. The approach represents a smooth approximation function on the domain that can be evaluated and differentiated continuously. Calculating and constructing the initial/boundary conditions are one of the issues which is faced for solving these equations that it has been satisfied by this method and the network is trained to satisfy the differential equation. The advantages of this method are high accuracy and high convergence speed compared with the same works which are used only for solving linear differential equations. We illustrate the method by solving a class of nonlinear differential equations by the method, analysis the solution and compared with the analytical solutions.