A Neural Network Solution To The Wave Equation

Wave equation is a hyperbolic partial differential equation to describe various wave phenomena in nature.It has been developed and applied in various fields,but its analytical solution is difficult to be obtain,so the numerical solution of wave equation becomes more and more important.Artificial neural network is an important tool for solving numerical solutions of wave equations,which can approximate analytic solutions of partial differential equations with arbitrary precision.The artificial neural network algorithm to solve the wave equation and its improvement are mainly studied in this paper as follows:In the first chapter,the background knowledge of neural network and its development at home and abroad are introduced,and i briefly describe the work done in this paper.In the second chapter,the neuron modeling in human brain and the single hidden layer neural network modeling are introduced.Taking the gradient descent method as an example,the forward learning process of the neural network and the back propagation process of error as well as the mathematical theorems needed in the whole process are introduced.The mathematical model of wave equation is mainly given in the third chapter.Based on the principle of basis function,the construction of approximate solution and loss function containing neural network are given for this model.By substituting the approximate solution into the defined optimization problem,the wave equation solving problem is successfully transformed into the optimization problem of the loss function.Finally,a proper optimization method is used to solve the problem of wave equation.In chapter 4,the neural network algorithm introduced in Chapter 3 is used to solve numerical examples of wave equation,and the advantages,disadvantages and effectiveness of the algorithm are analyzed.In chapter 5,after analyzing the numerical example results of Chapter 4 and the problems in the programming process,in order to simplify the construction of approximate solution of neural network and improve the accuracy of neural network algorithm,by using the"black box"principle of neural network,a second single hidden layer neural networkN₂(x,t,p)is used to replace the boundary condition satisfied by the neural network in the original approximate solution.A third single hidden layer neural networkN₃(x,t,p)is used to replace the partial derivatives of the neural network at the boundary of the original approximate solution.A new approximate solution of the synergistic effect of the three neural networks is constructed,and the improved approximate solution is used to solve the numerical example in Chapter 4.The advantages and disadvantages of the improved approximate solution and the original approximate solution are analyzed.The work has done in this paper and the future research direction are summarized in the sixth chapter.