Chaotic behavior learning via information tracking

Chaotic time series are often encountered in real-world applications, where modeling and understanding such time series present a significant challenge. Existing simulation-based methods for characterizing chaotic behavior may be sensitive to the respective model settings, while data-driven methods cannot adapt well to irregularities and aperiodicity in chaotic time series during the prediction stage, leading to nonlinearly aggregated forecasting errors after a finite number of time steps. In our preliminary work (Ren et al., 2023), we introduced a data-driven method that incorporates the idea of information tracking to capture and adapt to the chaotic changes over time, and tested its effectiveness on a real-world task—decadal temperature prediction. While the practical usefulness of the method has been initially validated on one domain-specific application scenario, a set of open questions is yet to be answered: Is the concept of information tracking generalizable to model and solve a broad spectrum of learning tasks that involve chaotic behavior? If yes, what are the underlying principles and learning behaviors that make the general method work well? Moreover, how can we assess the efficacy of the method in different scenarios? This study constitutes a significant step forward by systematically investigating and addressing the questions mentioned above. Specifically, we first provide a rigorous definition of the formalism and computational mechanism of the general method, with all necessary variables, symbols, and formulations. More importantly, we theoretically characterize and understand the generalizability of the method by mathematically uncovering the fundamental principles and behavior of information tracking-based chaotic behavioral learning, as prescribed by two chaotic behavioral indexes, namely, the Lyapunov and the Hurst exponents. Further, we comprehensively demonstrate the generalizability and robustness of the method by empirically examining a variety of chaotic behavioral learning problems generated by three representative chaotic systems: The logistic map, the double pendulum, and the Lorenz system.