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Abstract

Making accurate predictions of chaotic dynamical systems is an essential but challenging task with many practical applications in various disciplines. However, dynamical methods can currently obtain short-term accurate predictions, while deep learning methods, being fully statistical, can accurately predict longer periods of time, but suffer from problems such as modeling complexity and model interpretability ("black-box"). Here, we propose a new dynamics-based deep learning method, named as Dynamical System Deep Learning (DSDL), to achieve long-term precise predictions by reconstructing the nonlinear dynamics of chaotic systems. Based on multivariate observed time series, the DSDL can take full advantage of nonlinear interactions among those variables and build the prediction model by a diffeomorphism map between two reconstructed attractors according to embedding theorems. One of the attractors is reconstructed by time-lagged coordinates of the single target variable, and the other is reconstructed by multiple key variables which are constructed and selected by the multi-layers nonlinear network of the DSDL framework. As validated by three chaotic dynamical systems with different complexities, the DSDL significantly outperforms other existing methods used for comparison in this paper. In addition, the DSDL method not only improves the model predictive capability, but also realizes the dimensionality reduction and increases the interpretability of the prediction model. And we further believe that satisfactory performances of the DSDL make it potentially promising for comprehending and predicting chaotic systems in the real world.