Dynamics-based predictions of infinite-dimensional complex systems using dynamical system deep learning method

Abstract Predicting complex nonlinear chaotic dynamical systems constitutes a critical and formidable challenge across various disciplines. A novel methodology termed dynamical system deep learning (DSDL) has recently been introduced and utilized for the prediction of nonlinear chaotic dynamical systems. This method not only exceeds current techniques but also extracts key variables of the target dynamical systems, thereby providing a feasible resolution to the ‘black box’ problem. Nonetheless, the present focus of this method has chiefly been on predicting finite-dimensional dynamical systems, but real-world phenomena are mainly governed by partial differential equations (PDEs). This research selects the Lorenz’ 96 system, a set of coupled ordinary differential equations with spatiotemporal dynamics, and the Kuramoto–Sivashinsky PDE as representative examples of infinite-dimensional dynamical systems to evaluate the effectiveness of the DSDL. We also conduct comparisons with several mainstream methods including the ANN, RC-ESN, LSTM, NG-RC and SINDy. The findings demonstrate that the DSDL method exhibits outstanding prediction performance for PDEs. In long-term predictions, DSDL’s results align most accurately with the statistical characteristics of the reference true values, outperforming the other methods mentioned above. Finally, this study discusses the efficacy, efficiency and superiority of the DSDL in predicting dynamical systems, as well as its significant contributions to key variable extraction.