Neural Network Method Based On Random Sampling For Solving Singularly Perturbed Differential Equations

The singular perturbation problems(SPPs)often appear in many applications such as fluid mechanics,biological sciences,control theory,economics and engineering.It is a differential equation with a small parameter in the highest derivative term.The parameter is called the singular perturbation parameter.It causes the solution of the differential equation to have a large gradient in some areas near the boundary.This phenomenon is called the boundary layer phenomenon.Due to the existence of this phenomenon,it is difficult to achieve good results using traditional differential equation numerical solutions to solve this problem.Neural networks have gradually developed into powerful techniques for solving various real-world problems with their excellent function fitting capabilities.In recent years,scholars have begun to focus on solving differential equations using neural networks and deep learning.The key technology is to use the neural network to construct the approximate solution of the differential equation,and then convert the problem of solving the differential equation into a linear equation system or a nonlinear optimization problem about the parameters of the neural network.This paper starts from the characteristics of the solution of the singular perturbed differential equation,improve and innovate the existing model,and propose a new neural network algorithm for solving this problem.In Chapter 1,the author introduces the background knowledge of singular perturbation problems,briefly describes the development of singular perturbation differential equations and two mainstream traditional algorithms for solving singular perturbation problems.Then lists some researches on using neural network to solve differential equations.Finally,by comparing the neural network method with traditional numerical methods,the advantages of the neural network for solving differential equations are obtained,which leads to the research motivation and starting point of this paper.In Chapter 2,the author introduces the relevant theoretical knowledge of this article,including the theory of neural network,the definition and some properties of Legendre polynomial.In Chapter 3,a Legendre neural network based on projection and piecewise optimization is proposed for solving linear singularly perturbed initial boundary value problems.Firstly,generate uniform grid points,and perform projection transformation on the uniform grid points,so that most grid points are distributed in the fast-changing interval.Then use the single hidden layer Legendre neural network to construct the approximate solution of the equation,and substitute the projected grid point and the approximate solution into the differential equation and the initial boundary value condition.So far,the linear singular perturbation problem has been transformed into a linear equation solving problem about neural network parameters,follow the idea of regional decomposition in the traditional numerical methods and propose corresponding strategies for piecewise optimization for initial value problems and boundary value problems.Numerical experiments show that the method proposed in this paper is suitable for solving linear singularly perturbed initial boundary value problems,and can achieve good results.In Chapter 4,a multi-layer neural network algorithm based on truncated lognormal sampling is proposed to solve the nonlinear singular perturbation initial and boundary value problem.Firstly,use a multi-layer feed-forward neural network to construct the approximate solution of the differential equation,and transform the approximate solution according to the specific initial boundary value condition,so that the approximate solution after the transformation can naturally meet the initial boundary value condition.Then the approximate solution is substituted into the differential equation,and the square of the error on both sides of the equation is used as the target loss function.Next,the objective function is optimized using a small batch stochastic gradient descent algorithm.In order to make the sample points used in the optimization process gather in the fast changing interval,the sample points used in each iteration are sampled from the truncated lognormal distribution.Finally,numerical experiments show that the method proposed in this paper is suitable for solving nonlinear singular perturbation initial boundary value problems,and it can also achieve good results when ? is very small.In Chapter 5,the author summarizes the main work done in this paper,points out the advantages and disadvantages of the neural network algorithm proposed in this paper for solving singular perturbation differential equations,and plans and prospects for the next work.