Computing Iterative Roots with Neural Networks

Many real processes are composed of a n-fold repetition ofsome simpler process. If the whole process can be modelledwith a neural network, we present a method to derive a modelof the basic process, too, thus performing not only a systemidentificationbut also a decomposition into basic blocks.Mathematically this is equivalent to the problem of computingiterative or functional roots: Given the equation F(x)=f(f(x))and an arbitrary function F(x) we seek a solution for f(x). Solvingthis functional equation in a closed form is an exceptionallyhard problem and often impossible, even for simplefunctions. Furthermore there are no standard numerical methodsavailable yet. But a special topology of multilayer perceptronsand a simple addition to the delta rule of backpropagationwill allow most NN tools to compute good approximationseven of higher order iterative roots.Applications range from data analysis within chaos theory(many chaotic systems are derived from iterated functions) tothe optimization of industrial processes, where productionlines like steel mills often consist of several identical machinesin a row.