Cluster-Weighted EDMD Improves Dynamic System Prediction.

Extended Dynamic Mode Decomposition (EDMD) is a powerful technique for approximating Koopman operators, which are used to model complex dynamical systems from data. However, a limitation of traditional EDMD is its reliance on a single global operator, which can be inefficient when different regions of a system's state space exhibit distinct local behaviors. Researchers have introduced Cluster-Weighted EDMD (CW-EDMD) to address this. This novel method simultaneously learns a soft partition of the phase space and a specialized EDMD operator for each cluster. The core idea is an Expectation-Maximization (EM) objective that assigns each data transition to a cluster based on both its geometric proximity and how well the cluster's local Koopman model predicts its behavior. This allows clusters to specialize in areas where their local models are most accurate, rather than simply where data is most dense. Extensive experiments on well-known chaotic systems, including Lorenz, damped pendulum, and Duffing systems, demonstrated that CW-EDMD consistently outperforms matched-degree EDMD. The method showed significant reductions in one-step and 5-second rollout prediction errors, with median one-step error reductions ranging from 2.7x to 57x across different systems. This indicates that CW-EDMD provides a more robust and accurate way to model systems with heterogeneous dynamics.