A general framework for functional networks

In this paper, we introduce functional networks as a generalization and ext ension of the standard neural networks in the sense that every problem that can be solved by a neural network can also be formulated by a functional n etwork. But, more importantly, we give examples of problems that cannot be solved using neural networks but can be naturally formulated using function al networks. Functional networks are defined as a collection of connected f unctional units on a set of nodes. A functional unit or neuron connects inp ut nodes to output nodes. The values of the output nodes are calculated fro m the values of the input nodes by given functions of one or several argume nts. The main differences with neural networks are that (a) the neural func tions can be multivariate and can be different from neuron to neuron tin wh ich case, no weights are necessary, because they subsume by the different f unctions) and (b) the neuron outputs can be coupled, that is, coincident. T his mathematical model of functional networks parallels printed circuit boa rds with electronic components, thus giving an intuitive interpretation to functional networks and an interesting and natural additional application. The existence of functional units with common outputs leads to functional e quations whose solution can lead to substantial simplification of the initi al topology of the network and the neural functions involved. Two types of functional networks (the one-layer and serial functional networks) are disc ussed in detail. For the one-layer functional networks, a very simple simpl ification algorithm is given. For the serial functional networks, systems o f functional equations are obtained. The methods are illustrated by several examples of applications. (C) 2000 John Wiley & Sons, Inc.