Algorithm For Solving High Dimensional Poisson Equation Based On Radial Basis Function Neural Network

Poisson equation,as one of the most widely used equations in electrostatics,its numerical solution has very important research value.Basing on the mesh,the traditional numerical method which relies on the mesh division precision,faced with the contradiction between the accuracy and amount of calculation.It is also difficult to solve the high-dimensional problems.In this paper,we propose a meshless numerical algorithm,which is an efficient algorithm for solving Poisson's equation based on radial basis function neural network.Firstly,a certain number of selected coordinate points from the solution space and its boundary are used to construct the differential operator in the equation through the automatic differentiation technique.Then the loss function is constructed by introducing the initial boundary condition.Iterating the optimization algorithm helps to get the final numerical solution.Furthermore,the performance of the proposed algorithm is also strongly related to the neural network used in training.after the comparison of supervised learning method and self-organizing learning method in determining the parameters of radial basis function neural network,the supervised learning method is selected for subsequent numerical experiments.In the numerical experiment part,firstly,the two-dimensional and three-dimensional Poisson equations with exact solutions are solved.It is proved that the algorithm proposed in this paper can be realized and can be used to solve the Poisson equation numerically.When solving the three-dimensional Poisson equation,the Poisson equation given the first kind of boundary conditions and the second kind of boundary conditions is considered.Through the final result analysis,It is proved that the algorithm proposed in this paper has better effect in solving Poisson equation with the first kind of boundary conditions.Secondly,the algorithm proposed in this paper is used to solve a practical problem of two-dimensional Poisson equation without exact solution,and the results are compared with those obtained in references;Finally,the algorithm proposed in this paper is used to solve the four-dimensional Poisson equation with the first kind of boundary conditions,and good fitting results are obtained.The algorithm is successfully extended to solve the high-dimensional Poisson equation.It is verified that the algorithm can ignore the restriction of dimension when solving the high-dimensional Poisson equation,and has certain feasibility and efficiencyCompared with the traditional methods for solving high-dimensional partial differential equations,the algorithm proposed in this paper breaks the previous operation of solving partial differential equations based on grid and avoids the limitations of traditional methods in solving some problems.Secondly,compared with feedforward network,radial basis function neural network also has better properties because of its unique physical structure.