Title: Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

Abstract: Koopman theory transforms nonlinear dynamics into a linear spectral problem. However, computational implementations face a critical bottleneck: the selection of observables requires a rigid, finite-dimensional choice. These observables must be expressive, nearly invariant under the dynamics, and ideally compatible with composition. While deep Koopman approaches learn flexible coordinate systems and structure-preserving methods enforce operator identities on fixed dictionaries, this work integrates both strategies. We introduce Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), which learns a latent space along with a partition of that space, while strictly enforcing the Koopman product rule as an exact algebraic constraint.

The training process alternates between an exact multiplicative operator update and a differentiable latent-clustering step designed to promote Koopman closure. This approach yields a finite transition map defined on learned latent cells. The resulting dictionary is shaped by the dynamics rather than ambient geometry, its nonzero spectrum resides on the unit circle, and forecasts are generated in latent coordinates before being decoded into physical space.

Evaluated across Hamiltonian, chaotic, and fluid systems, DeepMDMD produces dictionaries that are significantly more compact and dynamically coherent than those derived from geometric MDMD partitions. The method mitigates spectral pollution, uncovers richer continuous-spectrum structures, and ensures stable forecasting even under severe noise. In high-dimensional flow scenarios, including a 158,624-dimensional cylinder wake and a noisy lid-driven cavity at $Re=20,000$, DeepMDMD successfully preserves coherent structures and long-time spectral statistics, areas where state-space MDMD typically fails. These findings suggest a practical guideline for Koopman learning: learn the coordinates, but constrain the algebra.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC