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

Abstract

This paper discusses the Δ-rule and training of min-max neural networks by developing a differentiation theory for min-max functions, the functions containing min (∧) and/or max (∨) operations. We first prove that under certain conditions all min-max functions are continuously differentiable almost everywhere in the real number field R-fraktur sign and derive the explicit formulas for the differentiation. These results are the basis for developing the Δ-rule for the training of min-max neural networks. The convergence of the new Δ-rule is proved theoretically using the stochastic theory, and is demonstrated with a simulation example. © 1996 IEEE.