Cluster-Weighted EDMD

2026-07-13

AI Summary

This work addresses the limitation of traditional Extended Dynamic Mode Decomposition (EDMD), which employs a single global Koopman operator and struggles to capture systems with heterogeneous local dynamics across the state space. The authors propose a joint learning framework that partitions the state space into dynamically consistent regions via soft clustering and learns a distinct local EDMD operator for each cluster. A novel expectation–maximization (EM) objective function is introduced, incorporating both prediction residuals and geometric proximity to prioritize clustering in regions where local models achieve high accuracy rather than merely high data density. Experiments on the Lorenz, damped pendulum, and Duffing systems demonstrate that the proposed method significantly outperforms standard EDMD in 258 out of 288 comparisons, achieving median one-step prediction error reductions by factors of 57, 2.7, and 12, respectively.

Abstract

Extended Dynamic Mode Decomposition (EDMD) approximates Koopman operators from data, but a single global operator is inefficient when different state-space regions exhibit distinct local dynamics. We introduce Cluster-Weighted EDMD (CW-EDMD), which jointly learns a soft phase-space partition and a per-cluster EDMD operator. Its Expectation-Maximization (EM) objective assigns each transition based on both geometric proximity and prediction residuals, so clusters specialize where local Koopman models are accurate rather than where the data are dense. On Lorenz, damped pendulum, and Duffing systems, across 36 configurations and 10 seeds, CW-EDMD improves matched-degree EDMD in one-step and 5s-rollout prediction. Across 288 paired comparisons, there are significant error reductions in 258 cases, increases in 4, and no differences in 26. Median one-step error reductions are 57x, 2.7x, and 12x on pendulum, Duffing, and Lorenz, respectively.

Problem

• Koopman operator
• Extended Dynamic Mode Decomposition
• local dynamics
• state-space partitioning
• nonlinear dynamical systems

Innovation

• Cluster-Weighted EDMD
• Koopman operator
• Expectation-Maximization
• local dynamics
• phase-space partitioning