Approximation of a fractal curve using feed-forward neural networks

Abstract

The approximation of fractal curves in the form of Brownian functions by two-layer feed-forward neural networks is studied. The network parameters are restricted within a finite range. For given realizations of the Brownian target function, all local minima in the output error measure with appreciable sizes of basins of attraction are located and found to be about a dozen in number in each case. The error follows a log-normal distribution which can be explained by a distribution of mean squared normal deviates. Its mean value exhibits simple scaling relationships with the number of hidden neurons and the number of training patterns. Our numerical findings are explained by comparison with a simple piecewise linear fit approach.