Numerical solution for initial and boundary value problems of high-order ordinary differential equations using Hermite neural network algorithm with improved extreme learning machine
Abstract In this paper we put forth Hermite neural network (HNN) algorithm with improved extreme learning machine (IELM) to solve initial/boundary value problems of high-order ordinary differential equation(ODEs) and high-order system of ordinary differential equations (SODEs). The model function was expressed as a sum of two terms, where the first term contains no adjustable parameters but satisfies the initial/boundary conditions, the second term involved a single-layered neural network structure with IELM and Hermite basis functions to be trained. The approximate solution is presented in closed form by means of HNN algorithm, whose parameters are obtained by solving a system of linear equations utilizing IELM, which reduces the complexity of the problem. Numerical results demonstrate that the method is effective and reliable for solving high-order ODEs and high-order SODEs with initial and boundary conditions.Mathematics Subject Classification (2020) 34A30 ; 65D15
