Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

2026-06-03

AI Summary

Constructing finite-dimensional dictionaries of observables for nonlinear dynamical systems that simultaneously exhibit expressiveness, dynamical invariance, and algebraic closure remains a significant challenge. This work proposes a novel approach that, for the first time, integrates deep coordinate learning with strict algebraic constraints derived from the Koopman product rule. By alternately optimizing multiplicative Koopman operators and differentiable latent clustering, the method achieves dynamics-driven partitioning of the latent space while preserving spectral structure. Evaluated on high-dimensional chaotic and fluid systems, the approach substantially mitigates spectral pollution, yielding compact and dynamically consistent observable dictionaries. These dictionaries enable stable long-term predictions under strong noise and accurately retain coherent structures and spectral statistical properties.

Abstract

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.

Problem

• Koopman operator
• nonlinear dynamics
• algebra-preserving learning
• spectral pollution
• coherent structures

Innovation

• Koopman operator
• deep learning
• multiplicative DMD
• algebra-preserving
• latent dynamics