# DeepMDMD：面向代数保持的Koopman学习的深度嵌入乘性动态模式分解

- 来源：HuggingFace Daily Papers（社区热门论文）
- 发布时间：2026-06-03 08:00
- AIHOT 分数：63
- AIHOT 链接：https://aihot.virxact.com/items/cmpz8zc4601dnslkpcijh6orh
- 原文链接：https://arxiv.org/abs/2606.05131

## AI 摘要

DeepMDMD结合深度Koopman方法与结构保持方法，学习潜空间并分区，同时将Koopman乘积规则作为精确代数约束强制执行。训练交替进行精确乘法算子更新和可微潜聚类步骤，得到非零谱位于单位圆上的有限转移图，字典由动力学而非环境几何塑造。在哈密顿、混沌和流体示例中，比几何MDMD更紧凑且动态一致，减少谱污染，揭示更丰富的连续谱结构，并在严重噪声下稳定预测。在高维流（包括158,624维圆柱尾流和噪声Re=20,000顶盖驱动空腔）中，保持相干结构和长期谱统计，而状态空间MDMD失效。

## 正文

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.