做非线性动力学建模或流体仿真的研究者，DeepMDMD用代数约束解决了Koopman学习中的字典选择难题，值得关注其如何在高维噪声下保持稳定预测。
Koopman理论将非线性动力学转化为线性谱问题，但计算中依赖硬性的有限维选择。DeepMDMD结合深度Koopman方法与结构保持方法，学习潜在空间及其划分，同时强制Koopman乘积规则作为精确代数约束。训练在精确乘法算子更新和可微潜在聚类步骤之间交替，后者促进Koopman封闭性。结果在哈密顿、混沌和流体示例中，学习到的字典比几何MDMD划分更紧凑且动态一致，减少谱污染，揭示更丰富的连续谱结构，并在高维流动中保持相干结构和长期谱统计。
Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning
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.