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

• Published in: Advances in difference equations Vol. 2018; no. 1; pp. 1 - 24
• Main Authors: Yang, Yunlei, Hou, Muzhou, Luo, Jianshu
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 19.12.2018; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Analysis; Basis functions; Classic Emden–Fowler equation; Comparative studies; Difference and Functional Equations; Functional Analysis; Improved extreme learning machine; Legendre neural network; Legendre polynomial; Machine learning; Mathematical analysis; Mathematics; Mathematics and Statistics; Neural networks; Neurons; Nonlinear differential equations; Nonlinear equations; Numerical analysis; ODEs; Ordinary Differential Equations; Partial Differential Equations; Polynomials; Test procedures; Legendre neural network; Legendre polynomial; ODEs; Classic Emden–Fowler equation; Improved extreme learning machine
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-018-1927-x

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.