Solving Differential Equations With Radial Basis Function Networks

This paper is concerned with the application of radial basis function networks (RBFNS) for numerical solution of high order differential equations (DES).This method is non-mesh method.Traditional numerical methods of ordinary differential equations have solutionsof the Euler method, Runge-Kutta method, Linear Multi-step method, Shooting method and Relaxation method. Based on radial basis function method for solving differential equations are the true non-mesh method, compared to other non-mesh method, its advantages are without numericalintegration, solving the direct line of thought, programming is relatively simple, and can be easily applied to the solution of boundary value problemsfor ordinary differential equations. radial basis function is a function of space as : Given a unary function (?)→R, in domain x∈(?) ,all such asΦ(x-c) =Φ(||x c||),And a linear combination of function space is called function derived space by the functionΦof the radial basis.Under certain conditions, people take as long as X_j different each,Φ(x - x_j) is linearly independent,thus to form a group of subspace-based of the composition of a radial basis function.A one-dimensional function y(x)can use the following form to interpolationor approximation:In which x is the independent variable, m is the number of radial basis function,)(?) is the radial basis function and the set (?) is the networkof the set weight we ask out. We use two types of non-symmetrical configuration of the RBF approximation of ordinary differential equations. Based on the differential form of direct approach, first of all, a linear combinationof RBF replace the alternative solutions we asking for, and then on the new RBF-based differential RBF, with a new linear basis function portfolio to replace the relative derivative; and based on the integral form of the indirect approach is a linear combination of RBF ,replace the most high-end derivative, and then on the new points-based RBF function, a new linear combination of basis functions have replaced the relative derivative. We obtained from the MQ function and Gauss function of a new analytical RBFs, if the RBFs in the new analytical rather not come out, we passed into integral (2.2), through the form of numerical integration to get a new RBFs. In practice, we selected the MQ radial basis function as a function of solving a second order ordinary differential equations. The best parameters values: m = 10,n = 17 have been obtained. Numerical results show that it is a good results. Compared with the usual direct approach, indirect approach may be appropriate for better accuracy and higher convergence order. And numerical results also showed that it can still get a better accuracy in the case of a small amount of configuration points ,using the same circumstances.The applicationRBFS approximation method have higher accuracy than the traditional finite difference method. The article also indirect approximation method for solving fourth-order boundary value problems for ordinary differential equations.We use MQ function to have a very good approximation. We have obtained the best parameters, a direct approach convergence order of 0(h^-1) and indirect approach convergence order of 0(h⁶),and the convergence order of approximation is 0(h²) after modified direct method.Numerical methods for solving partial differential divided into grid method and the non-mesh method by whether or not to use uniform discrete rules. At present, people have many methods for solving PDF. For examples, finite element method,Boundary element Method,Infinite element method,Domain decomposition method,Weighted residual method, Collocation method,Spectra method,Quasi-spectral method,Finite volume method and the Radial basis function methods, etc. Finite difference methods, finite element method, boundary element methods, spectral methods,but because of their meshstructure and aspects of the difficulties, we now have people looking for non- mesh method. The use of radial basis function to solve partial differential equation is concerned in recent years.Radial basis function based on partial differential equations to solve the same approximation method are also direct approximation method and indirect approximation method. Solving steps are as follows:(1): a linear combination of radial basis function alternative u.(2): partial differential radial basis function for new radial basis function in different direction, the linear combination of new radial basis functions to instead of the u different order partial derivatives in different direction.(3): substituting the linear combination of radial basis function for u and the linear combination of new radial basis function for u's different partial derivatives into the equation.(4):using inner points and boundary points, these points substituting account the composition of the equation by the RBF, we can obtain the linear equations: AW = B(5): Solving AW = B, be W, and get the approximate solution uThrough the indirect approach for solving partial differential equations .The steps are as follows:(l):Using the combination of radial basis function to replace the exact solution u's highest partial derivative in different direction.(2):Integer radial basis function in different directions, so that we can obtain new radial basis functions, and the combination of new radial basis functions instead each different partial derivative in different direction.(3):Instead linear combination of radial basis function and new radial basis functions into the equation.(4)Substitute interior points and boundary points into the equation combinated by the RBF.(5):Since integration, we have some integral constants, (4) also can not be linear equations. Here we can get the approximate solution u in differentformat in different directions ,from the points we can get interior point and boundary points into the equation to form a different format, such as (u_[x1](x) =μ[x₂](x)). these equations add to (4),can be the linear equations on the weights.: AW = B(6):Solving AW = B, be W, get the approximate solution u.Finally, the article with these two methods of solving the Dirichlet boundary value problem Possion equation. Radial Basis Function Solution of Partial Differential Equations is a good way.