Differential Equations Solving Method Based On Support Vector Machines And Reproducing Kernels Theory

Differential equations have always been a hot topic studied by scholars,among which,the study of discrete differential equations and their inverse problems have important application value.However,there are relatively few researches on discrete differential equations at present.Support vector machines(SVMs)and reproducing kernels theory show excellent effectiveness and feasibility in solving approximation problems.By combining reproducing kernels theory,Tikhonov regularization and SVMs,the discrete differential equations and their inverse problems are put into the framework of SVMs,then the corresponding reproducing kernels are constructed,and finally a new method for solving discrete linear differential equations and their inverse problems is proposed.This method can be used to solve general discrete linear differential equations and their inverse problems.The solutions obtained by this method has analytical expressions and certain sparsity,which is convenient for subsequent applications.The effectiveness of the proposed method is verified by two numerical experiments.This thesis is divided into three chapters:In chapter 1,the research background and significance for inverse problems of differential equations and discrete differential equations are described,and the research status for discrete differential equations and inverse problems of differential equations are analyzed.Then the problems considered in this paper are briefly described.In chapter 2,we introduce the basic knowledge of reproducing kernels theory,statistical learning theory and support vector machines.In chapter 3,a new method for solving discrete differential equations and their corresponding inverse problems is proposed by combining reproducing kernels theory,Tikhonov regularization and SVMs.Then error analysis of the proposed algorithm is given.Finally,the effectiveness of the proposed method is verified by two numerical experiments.