用聚类加权改进动态模式分解，让不同区域用不同模型，预测更准。
Cluster-Weighted EDMD
- 分区域建模：通过软聚类划分状态空间，每区独立训练动力学模型。
- 预测更准：在摆、杜芬和洛伦兹系统上，单步误差平均降低12至57倍。
- 适合多动态系统：尤其对非线性、多模态系统建模效果显著。
扩展动态模式分解（EDMD）从数据中近似柯普曼算子，但单一全局算子在状态空间不同区域呈现异质局部动力学时效率低下。本文提出聚类加权EDMD（CW-EDMD），联合学习软相空间划分与每簇对应的EDMD算子。其期望最大化（EM）目标函数根据几何邻近性和预测残差分配每条状态转移，使各簇专注于局部柯普曼模型准确的区域，而非数据密集区。在洛伦兹系统、阻尼摆和杜芬系统上，36种配置与10次随机种子测试中，CW-EDMD相较于匹配阶数的全局EDMD，在单步预测和5秒滚动预测中均表现更优。288组对比中，258次误差显著下降，4次上升，26次无差异。单步预测误差中位数分别降低57倍（摆）、2.7倍（杜芬）和12倍（洛伦兹）。
原文摘要 · Abstract (English)
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.
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