Robust prediction of chaotic systems with random errors using dynamical system deep learning

Abstract To predict nonlinear dynamical systems, a novel method called the dynamical system deep learning (DSDL), which is based on the state space reconstruction (SSR) theory and utilizes time series data for model training, was recently proposed. In the real world, observational data of chaotic systems are subject to random errors. Given the high nonlinearity and sensitivity of chaotic systems, the impact of random errors poses a significant challenge to the prediction. Mitigating the impact of random errors in the prediction of chaotic systems is a significant practical challenge. Traditional data-driven methods exhibit insufficient robustness against superimposed random errors, due to little consideration for temporal dynamic evolutionary of chaotic systems. Therefore, reducing the impact of random errors in the prediction of chaotic systems remains a difficult issue. In previous work, the DSDL demonstrated superiority in the noise-free scenario. This study primarily introduces the delay embedding theorem under noisy conditions and investigates the predictive capability of the DSDL in the presence of random errors in the training data. The performance of the DSDL is tested on three example systems, namely the Lorenz system, hyperchaotic Lorenz system and conceptual ocean–atmosphere coupled Lorenz system. The results show that the DSDL exhibits high accuracy and stability compared to various traditional machine learning methods and previous dynamic methods. Notably, as the magnitude of errors decreases, the advantage of the DSDL over traditional machine learning methods becomes more pronounced, highlighting the DSDL’s capacity to effectively extract the temporal evolution characteristics of chaotic systems from time series and to identify the true system state within observational error bands, significantly mitigating the impact of random errors. Moreover, unlike other contemporary deep learning methods, the DSDL requires faster hyperparameter tuning by using fewer parameters for improving accuracy, and based on the advantage of the SSR theoretical framework, the DSDL does not require prior knowledge of the original governing equations. Our work extends the theoretical applicability of the DSDL under random error conditions and points to the new and superior data-driven method DSDL based on the dynamic framework, holding significant potential for mitigating the impact of random errors and achieving robust predictions of real-world systems.