Quantum mechanics

A fuzzified neural network copes with fuzzy signals and/or weights so that the information about the uncertainty of input and output can be served in the training process. Since learning process is the main function of fuzzy neural networks, in this study, we focus on review and comparison of the existing learning algorithms, so that the theoretical achievement and the application agenda of each considered algorithm can be clarified from the aspects of computation complexity and accuracy. Two numerical examples of nonlinear mapping of fuzzy numbers and realization of fuzzy IF-THEN rules are used for illustration and analysis. Many issues related to fuzzy neural network have been discussed extensively in the literatures. On theoretical studies, most research focused on how a fuzzy neural network is developed to approximate a fuzzy function [2, 6, 20]. Among them, Buckley and Hayashi [2] evaluated its approximation capability and concluded that fuzzy neural networks can not be used as a universal approximator, for which Liu [20] proposed some extended functions to support the argument. Besides, Feuring and Lippe [6] also defined a class of fuzzy functions and proved that they can be approximated by a certain fuzzy neural networks As regards the applications, developing an effective learning algorithm has been the core topic. In 1994, Buckley and Hayashi [1] have given a thorough survey on fuzzy neural networks and suggested to use more general fuzzy sets in order to facilitate wider applications; for instance, using generalized fuzzy numbers for fuzzy signals/weights. However, this would mean to deal with complex computations of the fuzzy arithmetic. Thus, the research on fuzzy neural networks has put forth for investigating learning algorithms with fuzzy arithmetic and obtained many significant results