Neural network theory, Dynamical Systems, Approximation Theory
Networks with sigmoid node functions have been shown to be universal approximators, and can use straightforward implementations of learning algorithms. Mathematically, what is common to different sigmoid functions used by different researchers? We establish a common representation of inverse sigmoid functions in terms of the Guass Hypergeometric function, generalizing different node function formulations. We also show that the continuous Hopfield network equation can be transformed into a Legendre differential equation, without assuming the specific form of the node function; this establishes a link between Hopfield nets and the method of function approximation using Legendre polynomials
Menon, Anil Ravindran; Mehrotra, Kishan; Mohan, Chilukuri K.; and Ranka, Sanjay, "On Inverse Sigmoid Functions" (1993). Electrical Engineering and Computer Science Technical Reports. 160.