CLUSTER-WEIGHTED EDMD

Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method for approximating the Koopman operator, which linearizes nonlinear dynamical systems. However, a single global operator often struggles to capture complex dynamics across an entire state space. This paper introduces Cluster-Weighted EDMD (CW-EDMD) to address this by partitioning the state space into regions, each governed by its own local Koopman operator.

CW-EDMD uses an Expectation-Maximization (EM) framework to jointly learn both the partition and the local operators. Unlike standard mixture models that partition based solely on data density, CW-EDMD is 'residual-aware.' During training, a cluster's responsibility for a data point is determined by both its geometric proximity to the cluster center and the accuracy of its local Koopman prediction. This ensures that the partition boundaries align with the regions where each local model performs best. Each local operator is then fit using responsibility-weighted least squares on a recentered polynomial basis.

The authors evaluated CW-EDMD against standard EDMD across three classical dynamical systems: the Lorenz attractor, a damped pendulum, and a Duffing oscillator. Across 288 paired comparisons, CW-EDMD significantly outperformed EDMD in 258 cases, with median error reductions ranging from 2.7x to 57x depending on the system. The results demonstrate that partitioning the state space allows for more accurate local approximations, particularly in systems where global polynomial bases struggle to capture nonlinearities.

By allowing for local linear approximations of global nonlinear dynamics, CW-EDMD provides a flexible way to scale Koopman-based modeling to more complex systems. It offers a clear alternative to methods that rely on global basis enrichment (like deep autoencoders or kernel methods), suggesting that in many cases, a well-partitioned set of simple local models can be more effective than a single, highly complex global model.