The Neural Network Approximation Problems

Neural network has been considered to be a main branch of artificial intelligence, and its significance has been acknowledged by many scientists. The theories and methods of neural network have been applied in many areas, such as engineering, computer science, physics, biology, economy and managements, etc. Many problems in these fields can be converted into the problems of approximating multivariate functions by neural network. Based on the nonlinear approximate ability of neural network, we investigate a feedforward neural network with one hidden layer about its ability of approximating integral function and continuous function. Two forms of neural network have been studied in this article:Here and the activation function g is typically the Gaussian function. (This network is also referred as Radial basis neural network.), and g is the activation function, usually is a Sigmoid function.This article includes four parts:In the first chapter, we introduce the history of neural network briefly, and summarize the results on neural network in recent years. In the second part, we provide some relevant mathematical foundations of this problem. Our main results are summarized in chapter 3 and chapter 4. We extend some results on the approximation of neural network focusing on the approximate ability of radial neural network andfeedforward neural network in . The condition of activation function g has been weakened; this has great mathematical significance.