Prediction of Chaotic Time Series With Neural Networks and the Issue of Dynamic Modeling

This paper deals with the role of neural-network based prediction for the modeling of nonlinear dynamical systems. We show experimentally that the backpropagation learning rule to train neural networks and the prediction error, so widely utilized in teaching and comparing nonlinear predictors, do not consistently indicate that the neural network based model has indeed captured the dynamics of the system that produced the time series. Frequently, but not always, the neural network when used as an autonomous system in a feedback configuration was able to generate a time series that has dynamical invariants similar to the original time series. We show that the estimation of the dynamical invariants (correlation dimension, largest Lyapunov exponent) of the predicted and original time series are an appropriate tool to validate the predictive model.