A Neural Network Method For A Class Of Elliptic Partial Differential Equation

Elliptic partial differential equations are mainly used to describe the equilibrium and stable state in physics,such as electromagnetic field,gravitational field and reaction diffusion phenomenon in steady state.Elliptic equations have applications in fluid mechanics,elastic mechanics and geometry.Because of the complexity of partial differential equations,it is difficult to find its analytical solution,even there is no analytical solution.Therefore,the numerical solution of partial differential equations becomes more and more important.Neural networks have the advantages of no mesh generation,and have good self-learning and adaptive ability,have been widely used to construct numerical solutions of partial differential equations,and can effectively solve problems.The main work of this paper is to give an improved neural network method for solving different types of elliptic equations.The first method will construct the experimental solution through the joint training of multiple neural networks,use multiple deep networks to approximate the boundary conditions and partial derivatives in the experimental solution,then select the optimization algorithm to solve the network weights,so as to find the numerical solution of the elliptic equation.The second method constructs a neural network model based on triangular basis function,uses triangular basis function as the basis function of hidden layer neurons,selects deep extreme learning machine algorithm for network weight training.The influence of the extreme learning machine algorithm with different depths on the numerical solution accuracy of the elliptic equation is discussed.The experimental results show that both methods can effectively solve the elliptic equation,and the numerical experiments also verify the effectiveness and superiority of the method.The structure and specific contents of this paper are as follows : The first chapter briefly describes the development history of neural networks,the research status of neural networks in solving differential equations and the significance of this paper.The second chapter introduces the basic knowledge involved in this article;the third chapter introduces the multi-network multi-layer perceptron model to solve the elliptic equation algorithm,which is used to solve the elliptic equation with mixed boundary.In the fourth chapter,the specific steps of solving elliptic equations by combining triangular basis function neural network and deep extreme learning machine algorithm are given,and the second-order and fourth-order elliptic equations are solved by this method.The fifth chapter summarizes this paper and looks forward to the future work.