Neural Networks and Fuzzy Basis Functions for Functional Approximation

Abstract

Universal approximation capabilities of neural networks and fuzzy basis functions are given in this chapter using the Stone-Weierstrass theorem, Kolmogorov’s theorem and functional analysis methods. This study focuses on few commonly-used neural networks such as multilayered feedforward neural networks (MFNNs) with sigmoidal activation functions, trigonometric networks, higher-order neural networks, Gaussian radial basis function networks, and fuzzy basis function networks. The results show that an arbitrary continuous function on a compact set may be approximated to any degree of accuracy by such a neural network or fuzzy system. However, the accuracy of the approximations is strongly related to the design of the learning phases of the network parameters. The theory presented in this chapter provides a theoretical basis for applications to the fields of identification, control and pattern recognition.