The min-max function differentiation and training of fuzzy neural networks

This paper discusses the /spl Delta/-rule and training of min-max neural networks by developing a differentiation theory for min-max functions, the functions containing min (/spl and/) and/or max (V) operations. We first prove that under certain conditions all min-max functions are continuously differentiable almost everywhere in the real number field /spl Rfr/ and derive the explicit formulas for the differentiation. These results are the basis for developing the /spl Delta/-rule for the training of min-max neural networks. The convergence of the new /spl Delta/-rule is proved theoretically using the stochastic theory, and is demonstrated with a simulation example.