Research On The Method Of Solving Partial Differential Equations Numerically Based On Artificial Neural Network

In this paper,we focus on solving boundary value problems of the Helmholtz equation using artificial neural networks,which is commonly used to characterize the scattering of waves in acoustics and electromagnetism.Generally speaking,the difficulty of solving this equation is relatively large.For example,if the dimension of the equation is too large,it will lead to "dimension disaster".The grid points required by the grid-based method will surge,and the computing and storage costs will also become large.It is almost impossible to solve the equation defined on the unbounded field directly(except in some special cases,the method of infinite elements can be used).The artificial boundary method is usually used to truncate the unbounded field and simplify the problem on the unbounded field to the problem on the bounded field.However,it is difficult to design the appropriate artificial boundary conditions.In this paper,a method based on artificial neural network is proposed for solving boundary value problems of Helmholtz equation on bounded and unbounded domains.First,on the basis of the neural network method for solving partial differential equations in the regular region proposed by Lagaris et al in 1998,the original neural network for training real number solutions was extended to real part neural network and imaginary part neural network for simultaneous training according to the characteristic that the analytical solution of Helmholtz equation is a complex valued function.For the Helmholtz equation in the unit square region,a numerical solution expression(including the output of neural network)satisfying the boundary conditions is proposed and applied to the numerical solution of the equation.At the same time,this method is extended to the general rectangular region,that is,it is mapped into the second-order boundary value problem of partial differential equation in the unit square region by linear transformation,and then solved by using the neural network method above.Then,a neural network method for solving general Dirichlet boundary conditions and specific Dirichlet boundary conditions is proposed for partial differential equations with variable coefficients in two-dimensional bounded domain.The equation is mapped to the unit square region by linear transformation,and the numerical solution expression satisfying the boundary conditions is proposed.According to the partial differential equation of two subregions,two loss functions are constructed respectively for the numerical solution of the equation.Finally,a neural network method is proposed to solve the boundary value problem of Helmholtz equation in semi-infinite region.The perfect matching layer is used to truncate the semi-infinite region and transform the problem on the semi-infinite region into the problem on the bounded domain.Then the transformed problem on the bounded domain is mapped to the unit square region and the numerical solution expression satisfying the boundary conditions is proposed.According to the different equations in the two regions,two loss functions are constructed for the numerical solution of the equation.Compared with the traditional numerical method based on network,the grid-free neural network method does not have the problem of huge storage cost and computational cost caused by precise mesh subdivision.Moreover,the neural network method proposed in this paper has fewer adjustable parameters,so the calculation speed is faster and the overhead is lower.Finally,the feasibility of the algorithm is analyzed through an experimental example.