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Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning
Kelan Gray, Finlay Brown, Nicolas Boullé, Matthew J. Colbrook
TL;DR
Koopman learning must balance expressive, nearly invariant dictionaries with algebraic structure, but geometry-based partitions can be poorly adapted to nonlinear dynamics. DeepMDMD learns a latent partition while enforcing multiplicativity, producing compact, dynamically coherent dictionaries and stable forecasts across Hamiltonian, chaotic, and noisy high-dimensional flow examples.
Problem
Choosing expressive, low-dimensional observables that are nearly Koopman-closed remains difficult, especially when geometric partitions are poorly adapted to dynamics.
Method
DeepMDMD jointly refines an autoencoder latent representation and its partition while enforcing the Koopman product rule exactly through multiplicative operator updates.
Results
Across pendulum, Lorenz–96, cylinder-wake, and lid-driven-cavity experiments, DeepMDMD yields compact dictionaries, reduced spectral pollution, richer continuous-spectrum structure, and stable accurate flow-statistics rollouts.
Takeaways & Limitations
DeepMDMD is most useful when ambient geometry poorly guides dynamics, because it learns the partition while preserving Koopman structure.
Takeaways & Limitations
In high-dimensional systems, POD preprocessing may discard nonlinear features important to complex or chaotic dynamics.
Abstract
from arXiv · showhide
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.
1 Introduction
Koopman methods replace difficult nonlinear state-space analysis with linear evolution of observables, but finite-dimensional computation depends on choosing an expressive, compact, nearly closed dictionary. DeepMDMD learns such a dictionary in autoencoder latent space while enforcing multiplicativity as a hard algebraic constraint.
- Motivation: For high-dimensional nonlinear systems, snapshot data support data-driven inference, while Koopman analysis shifts the problem from states to observables that evolve linearly.The Koopman spectrum encodes growth rates, frequencies, and coherent structures.
- Motivation: EDMD approximates the Koopman operator in finite dimensions, but its dictionary must be expressive, low-dimensional, and nearly closed under Koopman action.These requirements are difficult to reconcile, motivating structure-preserving alternatives.
- DeepMDMD: DeepMDMD partitions the latent space of a pretrained autoencoder instead of the state space, then jointly refines representation and clustering to promote Koopman closure.This replaces a geometry-first partition with a learned one.
- DeepMDMD: The method produces a compact, interpretable dictionary adapted to the dynamics while retaining multiplicativity of the Koopman approximation as a hard algebraic constraint.The paper subsequently evaluates spectral accuracy, forecasting, flow statistics, and noise robustness.
2 Preliminaries and related work
The preliminaries establish Koopman evolution as a linear spectral framework whose multiplicativity and unitary setting impose strong algebraic structure. They review EDMD, KMD, and MDMD, emphasizing dictionary-design limitations and the need for dynamically adapted, structure-preserving coordinates.
- Koopman framework: Koopman theory converts nonlinear dynamics into generally infinite-dimensional linear evolution of observables, enabling spectral approaches.
- Koopman framework: Multiplicativity preserves admissible pointwise products, making products of eigenfunctions generate products of eigenvalues and inducing group structure in Koopman spectra.
- Koopman framework: Under the unitary assumption, induced by invertible measure-preserving dynamics, the spectrum lies on the unit circle and comprises unions of subgroups of T.
- EDMD and KMD: EDMD approximates Koopman dynamics on a fixed dictionary, but faithful spectral recovery requires near-invariance, which is difficult to design for complex or high-dimensional systems.
- EDMD and KMD: Full-state KMD can become expensive as dictionary size grows with state dimension and is sensitive to snapshot noise, which can destabilize spectral data and forecasts.
- MDMD: MDMD uses indicator functions on measurable partitions to preserve multiplication, yielding finite transition maps whose nonzero eigenvalues are roots of unity on the unit circle.The MDMD constraint produces {0,1} matrices with at most one nonzero per row; iterations enter cycles or terminate, and the indicator-dictionary minimizer costs O(M+N^2).
- MDMD: Geometric k-means partitions favor compact Voronoi cells rather than dynamical adaptation, while high-dimensional clustering suffers from the curse of dimensionality and POD may discard nonlinear features.
3 Methodology
DeepMDMD learns a low-dimensional latent space and dynamics-adapted partition, then alternates exact multiplicative Koopman updates with differentiable partition optimization. Forecasts are performed in latent coordinates and decoded only for physical-space outputs, reducing dependence on ambient dimension.
- Latent-space construction: DeepMDMD places the MDMD partition in a learned latent space Z⊂R^k with k≪d, avoiding direct partition construction in the high-dimensional state space X⊂R^d.The latent partition is shaped by the dynamics rather than fixed purely by ambient geometry.
- Latent forecasting: Forecasts evolve in latent space and are decoded only when physical-space output is required, while mode formation costs O(MNk) rather than O(MNd).The dictionary size is governed by latent dimension k, mitigating the curse of dimensionality when k≪d.
- Latent-space construction: The latent space is initialized by reconstruction-pretrained autoencoding and k-means, then adapted because reconstruction-based cells need not be dynamically coherent.The reconstruction loss ignores snapshot pairing and can produce indicator dictionaries with poor closure properties.
- Alternating optimization: With the partition fixed, MDMD computes K exactly, preserving the multiplicative constraint and defining the latent cluster-to-cluster transition map.The operator update uses the current latent indicator dictionary and forward latent snapshots.
- Alternating optimization: With K fixed, gradient descent updates the encoder and centroids using Student’s t-kernel soft assignments, enabling differentiable optimization before restoring hard assignments.Using α=1 gives a Cauchy kernel whose heavy tails make assignments less local than Gaussian assignments, helping alleviate crowding in low-dimensional embeddings.
4 Numerical Experiments
Numerical experiments compare DeepMDMD with MDMD across pendulum, Lorenz–96, cylinder-wake, and lid-driven-cavity systems. DeepMDMD achieves compact, dynamically organized dictionaries, richer spectral resolution, and more robust noisy forecasts in the reported tests.
- Nonlinear pendulum: DeepMDMD achieves comparable pendulum Hamiltonian accuracy with roughly an order of magnitude fewer basis functions than MDMD using k-means++ partitions.The comparison averages DeepMDMD over 50 random seeds.
- Nonlinear pendulum: DeepMDMD respects the unit circle while resolving more of the pendulum’s continuous spectrum than MDMD, whose eigenvalues concentrate on low-order cyclic subgroups.EDMD with the same learned dictionary produces spurious eigenvalues inside the unit disk, isolating the role of the multiplicative constraint.
- Nonlinear pendulum: DeepMDMD resolves more algebraically independent pendulum eigenfunctions than MDMD, as indicated by substantially slower singular-value decay.The numerical rank of the sample matrix provides an empirical measure of resolved principal eigenfunctions.
- Lorenz–96: As Lorenz–96 forcing increases, the learned latent geometry changes from a closed curve to a torus-like set and then to a chaotic attractor.The latent spaces and selected Koopman eigenfunctions are shown together in Fig. 7.
5 Conclusion
DeepMDMD learns a dynamics-shaped latent partition while exactly preserving the Koopman operator’s multiplicative structure. Experiments show compact, coherent dictionaries and stable latent-space forecasting across nonlinear, chaotic, and high-dimensional systems, with training as the main cost.
- Method: DeepMDMD moves the MDMD partition into a learned latent representation shaped by the dynamics while exactly preserving Koopman multiplicative structure.The Koopman matrix is constrained algebraically rather than merely fitted to data.
- Numerical results: DeepMDMD learns compact dictionaries with reduced spectral pollution and richer continuous-spectrum approximation on the nonlinear pendulum.Its partitions align with invariant energy levels.
- Numerical results: On Lorenz–96, learned latent spaces recover transitions from periodic to quasiperiodic and chaotic dynamics.
- Numerical results: For the cylinder wake and lid-driven cavity, DeepMDMD produces stable latent-space forecasts in high-dimensional systems.Forecasting occurs in latent coordinates, with decoding needed only for output after training.
- Limitations and outlook: Training is DeepMDMD’s main cost, but forecasting is cheaper after learning because it occurs in latent space and requires decoding only for output.This makes the method attractive for high-dimensional systems where full-state Koopman mode decompositions are expensive and noise-sensitive.