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arxiv: 2606.05131 · v1 · pith:JI4VDJDSnew · submitted 2026-06-03 · 💻 cs.LG · cs.NA· math.DS· math.NA· math.OC· math.SP

Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

Kelan Gray , Finlay Brown , Nicolas Boull\'e , Matthew J. Colbrook This is my paper

Pith reviewed 2026-06-28 06:59 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.DSmath.NAmath.OCmath.SP
keywords Koopman learningDynamic mode decompositionLatent spaceAlgebraic constraintsNonlinear dynamicsDeep learningSpectral methods
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The pith

DeepMDMD learns Koopman dictionaries by enforcing the product rule exactly on partitions of a learned latent space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Deep Embedded Multiplicative Dynamic Mode Decomposition to combine flexible latent-space learning with exact algebraic structure preservation. It alternates an exact multiplicative update of the transition operator with a differentiable clustering step that shapes the partition to promote Koopman closure. The resulting finite map on latent cells has nonzero spectrum confined to the unit circle and is decoded back to physical space for forecasts. Across Hamiltonian, chaotic, and high-dimensional fluid examples the learned dictionaries prove more compact and dynamically coherent than those from fixed geometric partitions, yielding reduced spectral pollution and stable long-time behavior even under severe noise.

Core claim

DeepMDMD produces a finite transition map on learned latent cells whose nonzero spectrum lies on the unit circle by enforcing the Koopman product rule as an exact algebraic constraint during training that alternates between multiplicative operator updates and differentiable latent clustering.

What carries the argument

Alternating exact multiplicative operator update with differentiable latent-clustering to enforce the Koopman product rule on a learned partition.

If this is right

  • Dictionaries become far more compact and dynamically coherent than those from geometric MDMD partitions.
  • Spectral pollution is reduced while richer continuous-spectrum structure is revealed.
  • Forecasts remain stable under severe noise where state-space MDMD fails.
  • Coherent structures and long-time spectral statistics are preserved in flows up to 158624 dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same alternation pattern could be applied to enforce other algebraic identities such as Lie-bracket closure in learned operator models.
  • The approach suggests that dynamic shaping of the partition matters more for Koopman quality than ambient geometric regularity of the dictionary.
  • One could test whether the learned cells align with known invariant manifolds or ergodic components on analytically solvable systems.

Load-bearing premise

The alternating procedure between exact multiplicative operator update and differentiable latent-clustering step will converge to a partition that actually promotes Koopman closure rather than merely fitting the training trajectory.

What would settle it

If the learned latent partitions produce no measurable improvement in long-time spectral statistics or forecast stability over geometric or random partitions on the 158624-dimensional cylinder wake, the central advantage claim is false.

Figures

Figures reproduced from arXiv: 2606.05131 by Finlay Brown, Kelan Gray, Matthew J. Colbrook, Nicolas Boull\'e.

Figure 1
Figure 1. Figure 1: Schematic of the DeepMDMD pipeline: (a) the latent encodings [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of hard and soft cell assignments in the latent space [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The nonlinear pendulum system in Eq. (4.1). Left: Example trajectories. Middle: DeepMDMD [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Nonlinear pendulum. relative L 2 error of one-step predictions for the Hamiltonian h as a function of dictionary size N, comparing MDMD and DeepMDMD averaged over 50 random seeds. 4.1 Nonlinear pendulum We begin with the nonlinear pendulum, a two-dimensional test problem used here to assess partition optimization rather than dimension reduction. The state x= (x1,x2) sat￾isfies x˙1 =x2, x˙2 =−sin(3x1), (x1,… view at source ↗
Figure 5
Figure 5. Figure 5: Eigenfunctions for the pendulum system along with the corresponding eigenvalues calculated using [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Top row: eigenvalues of the finite-dimensional Koopman approximation for the pendulum system [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Koopman eigenfunctions of the Lorenz-96 system at successive bifurcations (f =2.0,3.5,4.2), visualized on the DeepMDMD latent space. As f increases, the latent embeddings develop increasingly complex structure, reflecting the transition from periodic to chaotic dynamics. with indices interpreted modulo d. We select d = 9 and consider the forcing values f ∈ {2.0,3.5,4.2}, corresponding respectively to perio… view at source ↗
Figure 8
Figure 8. Figure 8: DeepMDMD training losses for the cylinder wake following the autoencoder warm-up, across alternat [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Cylinder wake with noise added after training. Left: DeepMDMD latent-space forecasts at different [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cylinder wake. One-step forecasts from DeepMDMD and MDMD with [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Noisy cavity flow experiment. Top: vorticity profiles predicted at timestep [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), which learns a latent space and its partition while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates an exact multiplicative DMD operator update with a differentiable latent-clustering step. The resulting finite transition map on learned cells has nonzero spectrum on the unit circle; forecasts are performed in latent space before decoding. Experiments on Hamiltonian, chaotic, and high-dimensional fluid systems (including a 158k-dimensional cylinder wake and noisy Re=20,000 lid-driven cavity) report more compact, dynamically coherent dictionaries than geometric MDMD, reduced spectral pollution, richer continuous spectra, and stable long-time forecasts where state-space DMD fails.

Significance. If the algebra preservation generalizes, the method would usefully combine the flexibility of deep embeddings with the structural guarantees of multiplicative DMD. Explicit enforcement of the product rule as an identity (rather than a soft penalty) and the ability to operate on very high-dimensional flows are concrete strengths. The reported preservation of coherent structures and spectral statistics under noise would be a practical advance for Koopman-based analysis in fluids and control if the generalization claim holds.

major comments (2)
  1. [Abstract / training alternation paragraph] Abstract and training-procedure description: the alternating exact multiplicative update + differentiable clustering is presented without a convergence argument, uniqueness result, or explicit penalty ensuring the product rule holds for held-out initial conditions or longer horizons. This is load-bearing for the central claim that the method enforces Koopman closure rather than merely interpolating the training trajectory.
  2. [Numerical experiments section] High-dimensional fluid experiments (cylinder wake, lid-driven cavity): performance gains over state-space MDMD are reported, yet no ablation isolates the contribution of the multiplicative constraint versus the deep embedding alone, nor is a direct check of the product rule on unseen data provided. Without this, the improvements could arise from overfitting the same noisy snapshots.
minor comments (2)
  1. [Method section] Notation for the latent partition and multiplicative operator should be introduced with explicit definitions before the alternation is described.
  2. [Figures] Figure captions for the fluid examples should state the precise noise level and forecast horizon used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. Both major points identify substantive gaps in theoretical justification and experimental controls. We address each below and commit to revisions that clarify limitations and add supporting analyses without overstating the current results.

read point-by-point responses

  1. Referee: [Abstract / training alternation paragraph] Abstract and training-procedure description: the alternating exact multiplicative update + differentiable clustering is presented without a convergence argument, uniqueness result, or explicit penalty ensuring the product rule holds for held-out initial conditions or longer horizons. This is load-bearing for the central claim that the method enforces Koopman closure rather than merely interpolating the training trajectory.

    Authors: We agree that the manuscript provides no convergence analysis, uniqueness result, or theoretical guarantee that the product rule holds outside the training trajectories or for longer horizons. The alternation enforces the algebraic constraint exactly on the observed data, but generalization remains an empirical claim supported only by the reported experiments. In revision we will (i) rewrite the abstract and training section to state this limitation explicitly, (ii) add a dedicated limitations paragraph, and (iii) include a quantitative check of product-rule violation on held-out initial conditions. revision: yes

  2. Referee: [Numerical experiments section] High-dimensional fluid experiments (cylinder wake, lid-driven cavity): performance gains over state-space MDMD are reported, yet no ablation isolates the contribution of the multiplicative constraint versus the deep embedding alone, nor is a direct check of the product rule on unseen data provided. Without this, the improvements could arise from overfitting the same noisy snapshots.

    Authors: The referee is correct: the present experiments contain neither an ablation that isolates the multiplicative constraint from the deep embedding nor an explicit verification of the product rule on unseen data. Consequently we cannot yet rule out that gains arise from overfitting. We will add (i) an ablation comparing DeepMDMD to a deep-embedding baseline that omits the multiplicative DMD step and (ii) a direct product-rule evaluation on held-out fluid trajectories, both to be reported in the revised numerical-experiments section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with external comparisons

full rationale

The abstract and described procedure introduce DeepMDMD via an alternating exact multiplicative operator update (enforcing the product rule on the current partition) and a differentiable clustering step. No load-bearing step reduces a reported performance metric, spectrum property, or forecast accuracy to a quantity defined by the same fitted parameters or by a self-citation chain. All quantitative claims are presented as empirical outcomes against external geometric MDMD baselines on held-out or noisy data. No uniqueness theorems, ansatzes smuggled via prior work, or renamings of known results appear in the provided text. The algebra preservation is an explicit constraint of the method rather than a derived prediction that collapses to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated beyond standard Koopman assumptions and the new training alternation.

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Reference graph

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