Methods of solving higher order linear differential equations by neural networks

Abstract

This article present two methods to find approximate solutions for the Cauchy problem in th order linear differential equations by neural networks (NN). The first is designing NN that generates a function of one variable depending on the parameters of the network and proposing a cost function which the minimum of this function corresponds to the NN that approximates the solution of the Cauchy problem. The second is transforming a  order linear differential equation into a system of linear differential equations with n hidden functions and designing a NN that generates a vector function where each component corresponds to a hidden function to be found. Then, proposing a cost function to determine the set of parameters of the NN corresponding to the vector function approximating the solution of the system and an approximate solution of the Cauchy problem is obtained. The authors apply both methods to find the numerical solutions of some specific examples. Both methods work well, with high accuracy.