Error Analyses of the Single Hidden Layer Neural Network Based on the Extreme Learning Machine for Solving Differential Equations

Abstract

The differential equations, especially the stiff differential equations with the large eigenvalue and instability of solutions, impose strict limitations on step size for traditional numerical methods. The single hidden layer neural network method based on the extreme learning machine is widely used to solve various differential equations due to its advantages of few parameters and high efficiency, and the optimal estimation of parameters is obtained. In practice, this does not mean that the solutions of differential equations converge well. This paper aims to establish theoretical results for solving non-stiff and stiff differential equations using the single hidden layer neural network based on the extreme learning machine. Under the classical Lipschitz condition and the one-sided Lipschitz condition, we derive convergence results for the approximate solutions of these equations. Numerical experiments validate the theoretical findings, and the results indicate that computational speed can be improved by converting deep neural networks into shallow ones when solving differential equations.