Solving Linear Fredholm Integro-differential Equations Based On Feedforward Neural Network

Differential equations,integral equations and integro-differential equations are widely used to simulate the physical world,and there have been plentiful research results in the related field.The linear Fredholm integro-differential equations,which are often viewed as part of calculus,play important roles in biological mathematics,atomic physics,neural networks,and transportation.This paper mainly studies the numerical methods for solving linear Fredholm integro-differential equations,as well as the proof of the unique existence of the solution.Based on different kinds of polynomials,two types of feedforward neural networks are proposed for solving the equations.From the viewpoint of function approximation,neural networks have powerful numerical approximate capabilities.According to this property,feedforward neural network based on Taylor expansion is constructed to solve linear Fredholm integro-differential equations.Firstly,Taylor's polynomials are used to approximate the unknown function in Fredholm integro-differential equation.The n-th order derivative of the unknown function at the initial value are used as the input of the neural network.Secondly,the loss function of the network which consists of internal errors and external errors is proposed,due to the existence of initial value conditions.Finally,the approximate solution is obtained through the gradient descent method by adjusting the weights.Furthermore,this paper presents a neural network method based on Legendre polynomial to solve linear Fredholm integro-differential equation.The n-th order Gaussian integration points calculated by finding the roots of the n-th power Legendre polynomial are utilized as the input of the neural network.Gaussian integral is used to calculate the error of function in the whole range,which is used as the loss function of neural network,and the error is also composed of internal error and external error.The approximate solutions are obtained through neural network learning algorithm.Two numerical examples are presented to show the efficiency of the both types neural network methods.By comparison with the existing methods,numerical experiments suggests that the proposed method has better numerical effects for the solving the Fredholm integro-differential equation.