Discrete- vs. Continuous-Time Nonlinear Signal Processing: Attractors, Transitions and Parallel Implementation Issues

Artificial neural networks (ANNs) are often used for short term discrete time prediction of experimental data. In this paper we focus on the capability of such networks to identify long term behavior and, in particular, observed bifurcations correctly. The usual discrete time mapping approach is (precisely because of its discrete nature) often incapable of reproducing observed bifurcation sequences. If the interest is only in periodic or temporally more complicated behavior, a Poincaré map extracted from the experimental time series can be used to circumvent this problem. A complete dynamic picture including bifurcations of steady states can, however, only be captured by a continuous-time model. We present ANN configurations which couple a "nonlinear principal component" network for data preprocessing with (a) a composite ANN based on a simple explicit integrator scheme and (b) a recurrent ANN based on an implicit integrator scheme. These ANNs are able to correctly reconstruct bifurcation diagrams based on experimental data from the electrodissolution of metals in acidic solutions. We also discuss some issues of parallel implementation of the training algorithms.