深度嵌入乘法动态模式分解用于代数保持的库普曼学习

摘要

Koopman理论将非线性动力学转化为线性谱问题。然而，在计算中，一切都取决于一个困难的有限维选择：可观测量必须具有表达性，在动力学下近乎不变，并且理想情况下与复合运算兼容。深度Koopman方法学习灵活的坐标，而结构保持方法则在固定字典上强制满足算子恒等式。我们通过引入深度嵌入乘性动态模态分解（DeepMDMD）来结合这些思想，该方法在强制Koopman乘积规则作为精确代数约束的同时，学习潜空间及其划分。训练过程交替进行精确的乘性算子更新和可微的潜聚类步骤，后者促进Koopman封闭性。最终在学习的潜单元上得到一个有限转移映射。其非零谱位于单位圆上，字典由动力学而非背景几何塑造，预测在潜坐标中完成后再解码到物理空间。在哈密顿、混沌和流体实例中，DeepMDMD学习到的字典比几何MDMD划分产生的字典更紧凑且动态上更连贯。它减少了谱污染，揭示了更丰富的连续谱结构，并在强噪声下给出稳定的预测。在高维流动中，包括一个158,624维的圆柱绕流和噪声下的Re=20,000顶盖驱动腔流，DeepMDMD保留了连贯结构，并在状态空间MDMD失效的情况下保持了长时间谱统计特性。这些结果提出了Koopman学习的一个实用准则：学习坐标，约束代数。

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.