Basic Structure of Fuzzy Neural Networks

In this chapter we shall discuss the structure of fuzzy neural networks. We start with general definitions of multifactorial functions. And we show that a fuzzy neu-ron can be formulated by means of standard multifactorial function. We also give definitions of a fuzzy neural network based on fuzzy relationship and fuzzy neurons. Finally, we describe a learning algorithm for a fuzzy neural network based on V and A operations. 6.1 Definition of Fuzzy Neurons Neural networks alone have demonstrated their ability to classify, recall, and associate information [l]. In this chapter, we shall incorporate fuzziness to the networks. The objective to include the fuzziness is to extend the capability of the neural networks to handle " vague " information than " crisp " information only. Previous work has shown that fuzzy neural networks have achieved some level of success both fundamentally and practically [l-lo]. As indicated in reference [l], there are several ways to classify fuzzy neural networks: (1) a fuzzy neuron with crisp signals used to evaluate fuzzy weights, (2) a fuzzy neuron with fuzzy signals which is combined with fuzzy weights, and (3) a fuzzy neuron described by fuzzy logic equations. In this chapter, we shall discuss a fuzzy neural network where both inputs and outputs can be either a crisp value or a fuzzy set. To do this we shall first introduce multifactorial function [ll, 121. We have pointed out from Chapter 4 that one of the basic functions of neurons is that the input to a neuron is synthesized first, then activated, where the basic operators to be used as synthesizing are " + " and ". " denoted by (+, .) and called synthetic operators. However, there are divers styles operators will be multifactorial functions, so we now briefly introduce the concept of multifactorial functions. In [0, lIm, a natural partial ordering " 5 " is defined as follows: A multifactorial function is actually a projective mapping from an rn-ary space to a