Toward a physics-guided machine learning approach for predicting chaotic systems dynamics.

Predicting the dynamics of chaotic systems is crucial across various practical domains, including the control of infectious diseases and responses to extreme weather events. Such predictions provide quantitative insights into the future behaviors of these complex systems, thereby guiding the decision-making and planning within the respective fields. Recently, data-driven approaches, renowned for their capacity to learn from empirical data, have been widely used to predict chaotic system dynamics. However, these methods rely solely on historical observations while ignoring the underlying mechanisms that govern the systems' behaviors. Consequently, they may perform well in short-term predictions by effectively fitting the data, but their ability to make accurate long-term predictions is limited. A critical challenge in modeling chaotic systems lies in their sensitivity to initial conditions; even a slight variation can lead to significant divergence in actual and predicted trajectories over a finite number of time steps. In this paper, we propose a novel Physics-Guided Learning (PGL) method, aiming at extending the scope of accurate forecasting as much as possible. The proposed method aims to synergize observational data with the governing physical laws of chaotic systems to predict the systems' future dynamics. Specifically, our method consists of three key elements: a data-driven component (DDC) that captures dynamic patterns and mapping functions from historical data; a physics-guided component (PGC) that leverages the governing principles of the system to inform and constrain the learning process; and a nonlinear learning component (NLC) that effectively synthesizes the outputs of both the data-driven and physics-guided components. Empirical validation on six dynamical systems, each exhibiting unique chaotic behaviors, demonstrates that PGL achieves lower prediction errors than existing benchmark predictive models. The results highlight the efficacy of our design of data-physics integration in improving the precision of chaotic system dynamics forecasts.