Chaos Prediction: Machine Learning Versus Dynamical Models

We address the question of whether machine learning can improve the predictability of chaos. Due to the presence of chaos, chaotic time series are “contaminated” beyond the horizon of predictability as numerical errors accumulate. Therefore, evaluations should be based on error-free computation of chaos because learning the “contaminated” time series may not even be related to learning the chaotic dynamics. We use error-free computation of two isomorphic chaotic systems, namely the Logistic map and the Tent map, to investigate the ability of Echo State Networks (ESNs) to learn and predict chaos. ESNs exhibit significantly different predictive performance on the two isomorphic dynamical systems, suggesting that learning relies more on the arithmetic representation of the dynamical data than on the complexity or entropy production of the underlying dynamics. ESNs can achieve long horizons of predictability, but they do not improve the predictability of the corresponding dynamical models. Moreover, even when optimal predictive performance is achieved, this provides no guidance for other predictions within the same or its isomorphic dynamical system. The optimal hyperparameters depend strongly on the input time series and cannot be transferred, while increasing training length does not necessarily improve predictive performance.