FRACTIONAL ITERATIONS

Many real processes are composed of a n-fold repetition of some simpler process. If the whole process can be modelled with a neural network, we present a method to derive a model of the basic process, too, thus performing not only a systemidentification but also a decomposition into basic blocks. Mathematically this is equivalent to the problem of computing iterative or functional roots: Given the equation F(x)=f(f(x)) and an arbitrary function F(x) we seek a solution for f(x). Solving this functional equation in a closed form is an exceptionally hard problem and often impossible, even for simple functions. Furthermore there are no standard numerical methods available yet. But a special topology of multilayer perceptrons and a simple addition to the delta rule of backpropagation will allow most NN tools to compute good approximations even of higher order iterative roots. Applications range from data analysis within chaos theory (many chaotic systems are derived from iterated functions) to the optimization of industrial processes, where production lines like steel mills often consist of several identical machines in a row