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.

翻译：库普曼理论将非线性动力学转化为线性谱问题。然而在实际计算中，一切都依赖于一个艰难的有限维选择：可观测量必须具有表达能力、在动力学下近似不变，且理想情况下与复合运算兼容。深度库普曼方法学习灵活坐标，而结构保持方法则在固定字典上强制执行算子恒等式。我们通过引入深度嵌入乘法动态模态分解（DeepMDMD）结合了这些思想，该方法在将库普曼乘积规则作为精确代数约束强制执行的同时，学习潜在空间及其划分。训练在精确乘法算子更新与促进库普曼闭包的可微潜在聚类步骤之间交替进行。最终在学习到的潜在单元上得到有限转移映射。其非零谱位于单位圆上，字典由动力学而非环境几何塑造，预测在潜在坐标下进行，之后再解码到物理空间。在哈密顿、混沌和流体示例中，DeepMDMD学习到的字典比几何MDMD划分生成的字典更为紧凑且动力学一致性更强。它减少了谱污染，揭示了更丰富的连续谱结构，并在严重噪声下提供稳定的预测。在高维流动中，包括158,624维圆柱尾流和噪声条件下的$Re=20,000$顶盖驱动腔，它能在状态空间MDMD失效的情况下保持相干结构和长时间谱统计特性。这些结果揭示了库普曼学习的实用准则：学习坐标，约束代数。