深度嵌入乘法動態模式分解於代數保持庫普曼學習

摘要

庫普曼理論將非線性動力學轉化為線性譜問題。然而在計算中，一切取決於一個困難的有限維度選擇：觀測量必須具有表達力、在動力學下近乎不變，且理想上需與複合運算相容。深度庫普曼方法學習靈活的坐標，而結構保持方法則在固定字典上強制執行算子恆等式。我們結合這些概念，提出深度嵌入乘性動態模態分解（Deep Embedded Multiplicative Dynamic Mode Decomposition，DeepMDMD），該方法學習一個潛在空間及其劃分，同時將庫普曼乘積規則作為精確代數約束強制執行。訓練過程在精確乘性算子更新與可微分潛在聚類步驟之間交替進行，後者促進庫普曼閉合性。其結果是在學習到的潛在單元上形成一個有限轉移映射。該映射的非零譜位於單位圓上，其字典由動力學而非環境幾何決定，預測在潛在坐標中進行後再解碼至物理空間。在哈密頓、混沌及流體實例中，DeepMDMD學習到的字典遠比幾何MDMD劃分產生的字典更為緊湊且動力學上更一致。它減少了頻譜污染，揭示了更豐富的連續譜結構，並在嚴重噪聲下提供穩定預測。在高維流場中，包括一個158,624維的圓柱尾流及噪聲Re=20,000的頂蓋驅動流，DeepMDMD在狀態空間MDMD失敗之處仍能保留相干結構與長時間譜統計量。這些結果表明了一條實用的庫普曼學習法則：學習坐標，約束代數。

English

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy Re=20,000 lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.