A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

Yunlei Yang,Muzhou Hou,Jianshu Luo

doi:10.1186/s13662-018-1927-x

Abstract

This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden–Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A single hidden layer Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials. The improved extreme learning machine (IELM) algorithm is used for network weights training when solving algebraic equation systems, and several algorithm steps are summed up. Convergence was analyzed theoretically to support the proposed method. In order to demonstrate the performance of the method, various testing problems are solved by the proposed approach. A comparative study with other approaches such as conventional methods and latest research work reported in the literature are described in detail to validate the superiority of the method. Experimental results show that the proposed Legendre network with IELM algorithm requires fewer neurons to outperform the numerical algorithm in the latest literature in terms of accuracy and execution time.

Highlights

Many problems encountered in science and engineering, for example, physics, chemistry, biology, mechanics, astronomy, population, resources, economics, and so on, are related to a mathematical model in the form of differential equations
Many numerical methods were proposed for solving ordinary differential equations (ODEs) [1], including single- and multi-step methods; single-step methods include Euler first order method (EM) [2], second order Runge–Kutta (R–K) method inspired by Taylor’s expansion, Suen third order R–K method (Suen-R-K3), the classic fourth-order R–K method (R-K4) [3
6.1 Experimental results Example 1 First, we consider the initial value problem of ODEs expressed as cos(x) y(1) =

Summary

Introduction

Many problems encountered in science and engineering, for example, physics, chemistry, biology, mechanics, astronomy, population, resources, economics, and so on, are related to a mathematical model in the form of differential equations. This paper presents a novel Legendre neural network method with improved extreme learning machine algorithm for solving several types of linear or nonlinear differential equations. We believe that the proposed method may be the first to use Legendre neural network model with IELM algorithm in solving differential equations.

Results

Conclusion