Numerical analysis of the learning of fuzzified neural networks from fuzzy if-then rules

The main aim of this paper is to clearly show how fuzzified neural networks are trained by back-propagation-type learning algorithms for approximately realizing fuzzy if-then rules. Our fuzzified neural network is a three-layer feedforward neural network where connection weights are fuzzy numbers. A set of fuzzy if-then rules is used as training data for the learning of our fuzzified neural network. That is, inputs and targets are linguistic values such as "small" and "large". In this paper, we first demonstrate that the fuzziness in training data propagates backward in our fuzzified neural network. Next we examine the ability of our fuzzified neural network to approximately realize fuzzy if-then rules. In computer simulations, we compare four types of connection weights (i.e., real numbers, symmetric triangular fuzzy numbers, asymmetric triangular fuzzy numbers, and asymmetric trapezoidal fuzzy numbers) in terms of the fitting ability to training data and the computation time. We also examine a partially fuzzified neural network. In our partially fuzzified neural network, connection weights and biases to output units are fuzzy numbers while those to hidden units are real numbers. Simulation results show that such a partially fuzzified neural network is a good hybrid architecture of fully fuzzified neural networks and neural networks with non-fuzzy connection weights. (C) 2001 Elsevier Science B.V. All rights reserved.