A Study On Interpolation And Approximation Problem Of Some Neural Network Operators

The function approximation theory is an important part of function the-ory,involving in the basic problem is the function of approximate represen-tation problem.The application of neural network for function approxima-tion provides a new idea for the development of function approximation.The essence of neural network approximation is to express one or more functions by nonlinear parameter expansion.This thesis mainly consists of the following parts.The first part studies high-dimensional scattered da-ta approximation,where we propose new model and method for the error estimates.While the second part the approximate error of a single hidden layer feedforward neural network with interpolation property in L~Pspace is discussed,and some new results are obtained.In addition,this paper al-so studies convergence and approximation order of a class of multivariate operators in different spaces,and constructs a class of deep networks.The details are as follows.Chapter 2 discusses the interpolation for scattered data with B-spline neural network operators.Using the modulus of smoothness of function and the mesh norm scattered data as a measure of approximation,we have proved the uniform approximation theorem and estimated the approxima-tion errors.Finally,we have demonstrate some numerical results that show our theorem.Chapter 3 adopts the construction method studied the interpolation and approximation of neural network operators.First,utilizing two types of sig-moidal functions construct feedforward neural networks with fixed weight-s,respectively.Then,by means of Steklov mean function and using the modulus of smoothness of function as a measure of approximation,the er-ror of L~papproximation for the two kinds of neural network operators are estimated.Chapter 4 constructs a family of multivariate max-product operators of the Kantorovich type is introduced and their convergence is studied.When the above multivariate operators approximates the continuous functions,using the generalized absolute moment and the modulus of smoothness of function as a measure of approximation,the error of approximation for the operators are estimated and the pointwise approximation theorems are es-tablished.Moreover,the L~papproximations are also considered.Chapter 5 introduces a kind of deep network.First of all,we investigate the relationship between rectified linear unit(Re LU)function and continu-ous piecewise linear function with M breakpoints.Then,according to the principle of the fold line can approximate the continuous function,a class of deep neural network is constructed.Finally,We also prove the deep neural network can approximate squaring function with arbitrary precision.