Two Kinds Of Fredholm Integro-differential Equations Are Solved Based On Feedforward Neural Networks

Fredholm integro-differential equation has been wildly applied in many scientific fields such as mechanics,physics,circuitry,economics,engineering,etc.,but there is relatively little research on the numerical solutions of two-dimensional nonlinear Fredholm integro-differential equation and fractional Fredholm integro-differential equation with variable coefficients.With the rapid development of artificial intelligence in recent years,using neural network to solve some partial differential and integro-differential equations has become a new popular algorithm.Compared with traditional numerical methods,neural network can achieve meshless approximate solution,reduce the calculation amount and improve the fitness.Therefore,this paper mainly uses feedforward neural networks based on polynomials to solve fractional Fredholm integro-differential equations with variable coefficients and two-dimensional nonlinear Fredholm integro-differential equations.The main structure of the paper is presented as follows:Chapter 1 describes the background and research significance of this paper,and reviews and analyzes the numerical solution methods of fractional order integro-differential equations and two-dimensional integro-differential equations in recent years.Chapter 2 gives the definition and properties of fractional derivative,Bernstein polynomial and Legendre polynomial,and briefly introduces the structure and algorithm of feedforward neural network.In Chapter 3,a numerical method for solving feedforward neural networks based on Bernstein polynomials is proposed.Firstly,according to the definition of Caputo fractional derivative,the variable coefficient fractional integro-differential equations are transformed into a matrix form on Bernstein polynomial space.The coefficients of Bernstein polynomial are used as weights to construct a feedforward neural network,and the gradient descent method is used to learn the weights to obtain an approximate solution.It is proved theoretically that the feedforward neural network algorithm is convergent.Finally,the numerical examples show that the method in this chapter is feasible and effective.In Chapter 4,a new network structure based on a two-dimensional Lengendre polynomial feedforward network is constructed to solve the two-dimensional nonlinear Fredholm integro-differential equations on the interval [0,1]×[0,1].First,the two-dimensional nonlinear Fredholm integro-differential equations are transformed into matrix form in the space of two-dimensional Legendre polynomials using matrix operations and the GaussLegendre quadrature,and the coefficients of the two-dimensional Legendre polynomials are the parameters to be solved.Then a new network structure is constructed to solve this type of equation,which differs from the neural network constructed in Chapter 3 in that it is a network structure with variable number of hidden layers and neurons and each hidden layer has weights and activation functions,and the coefficients of the two-dimensional Legendre polynomials are determined by learning these weights.Finally,numerical simulations using Py Torch are performed to verify the high accuracy and effectiveness of the method in this chapter.Chapter 5 summarizes the work accomplished,points out the innovations and shortcomings of this paper,and looks forward to the future research work.