Using Machine Learning to predict extreme events in the Hénon map

Abstract

Machine Learning (ML) inspired algorithms provide a flexible set of tools for analyzing and forecasting chaotic dynamical systems. We here analyze the performance of one algorithm for the prediction of extreme events in the two-dimensional Hénon map at the classical parameters. The task is to determine whether a trajectory will exceed a threshold after a set number of time steps into the future. This task has a geometric interpretation within the dynamics of the Hénon map, which we use to gauge the performance of the neural networks that are used in this work. We analyze the dependence of the success rate of the ML models on the prediction time $T$ , the number of training samples $N_T$ and the size of the network $N_p$. We observe that in order to maintain a certain accuracy, $N_T\propto exp(2hT)$ and $N_p\propto exp(hT)$, where $h$ is the topological entropy. Similar relations between the intrinsic chaotic properties of the dynamics and ML parameters might be observable in other systems as well.

9 pages, 12 figures

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• Model-assisted deep learning of rare extreme events from partial observations
• A filtered Hénon map