LEGENDRE DEEP NEURAL NETWORK (LDNN) AND ITS APPLICATION FOR APPROXIMATION OF NON-LINEAR VOLTERRA-FREDHOLM-HAMMERSTEIN IN-TEGRAL EQUATIONS

Abstract

Various phenomena in biology, physics, and engineering are modeled by differential equations. These differential equations including partial differential equations and ordinary differential equations can be converted and represented as integral equations. In particular, Volterra-Fredholm-Hammerstein integral equations are the main type of these integral equations and researchers are interested in investigating and solving these equations. In this paper, we propose Legendre Deep Neural Network (LDNN) for solving nonlinear Volterra-Fredholm-Hammerstein integral equations (V-F-H-IEs). LDNN utilizes Legendre orthogonal polynomials as activation functions of the Deep structure. We present how LDNN can be used to solve nonlinear V-F-H-IEs. We show using the Gaussian quadrature collocation method in combination with LDNN results in a novel numerical solution for nonlinear V-F-H-IEs. Several examples are given to verify the performance and accuracy of LDNN.

1. INTRODUCTION

Deep neural networks are a main and beneficial part of machine learning family which are applied in various areas including speech processing, computer vision, natural language processing and image processing (LeCun et al., 2015; Krizhevsky et al., 2012) . Also, the approximation of the functions is a significant branch in scientific computational and achieving success in this area is considered by some research (Tang et al., 2019; Hanin, 2019) . Solving differential equations is the other main branch of scientific computational which neural networks and deep learning have been shown success in this area. (Lample & Charton, 2019; Berg & Nyström, 2018; Raissi et al., 2019) . Various phenomena in biology, physics, finance, neuroscience and engineering are modeled by differential equations (Courant & Hilbert, 2008; Davis, 1961) . In recent years, several researchers studied the solving differential equations via deep learning or neural networks. differential equations consists of ordinary differential equations, partial differential equations and integral equations. (Sirignano & Spiliopoulos, 2018; Lu et al., 2019; Meng et al., 2020) . It is notable that the various numerical methods are applied for solving differential equations. Homotopy analysis method (HAM) (Liao, 2012) and variational iteration method (VIM) (He & Wu, 2007) are known as analytical/semi-analytical methods. Usually, spectral methods (Canuto et al., 2012) , Runge-Kutta methods (Hairer et al., 2006) , the finite difference methods (FDM) (Smith, 1985) and the finite element methods (FEM) (Johnson, 2012) are considered as the popular numerical methods. When the complexity of the model does not allow us to obtain the solution explicitly, numerical methods are a proper selection for finding the approximate solution for the models. Recently, some of the machine learning methods are applied for solving differential equations. Chakraverty & Mall (2017) introduced orthogonal neural networks which used orthogonal polynomials in the structure of the network. Raja et al. ( 2019) applied meta-heuristic optimization algorithm to neural network for obtaining the solution of differential equations. Moreover, other methods of machine learning such as support vector machine (Vapnik, 2013) are used to approximate the solution of the models. Least squares support vector machines are considered in these researches (Hajimohammadi et al., 2020; Mehrkanoon & Suykens, 2015) . Baker et al. (2019) selected deep neural networks for solving the differential equations. Pang et al.

