The Constructive Methods and Numerical Results for Approximation of Neural Networks

Neural networks are widely used in many applications including astronomical physics, image processing, recognition, robotics, and automated target tracking, etc. Their ability to approximate arbitrary continuous functions is the main reason for this popularity. The authors of this paper show by constructive methods that for any continuous function f on [a, b], the function can be approximated by a feedforward network with one hidden layer of sigmoidal neurons, and the first order modulus of continuity is used to estimate the approximation error. In particular, when the activation functions of hidden layer are satlin function, we give the constructive proofs for the neural network to reach the second order modulus of continuity result. Using the univariate constructive technique of the neural network, we present a multivariate interpolation neural network. Some numerical results are given to demonstrate the theoretical results.