# Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

arXiv: https://arxiv.org/abs/2606.05131 · HTML: https://mlsanity.com/paper/2606.05131/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 67 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 24 of 24)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Deep Clustering: A Comprehensive Survey](https://mlsanity.com/paper/2210.04142/) (2022)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [The Multiverse of Dynamic Mode Decomposition Algorithms](https://mlsanity.com/paper/2312.00137/) (2023)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [Port-Hamiltonian Dynamic Mode Decomposition](https://mlsanity.com/paper/2204.13474/) (2022)
- [Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size](https://mlsanity.com/paper/2403.05891/) (2024)
- [Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators](https://mlsanity.com/paper/2405.00782/) (2024)
- [Multiplicative Dynamic Mode Decomposition](https://mlsanity.com/paper/2405.05334/) (2024)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [Modes of Homogeneous Gradient Flows](https://mlsanity.com/paper/2007.01534/) (2020)
- [On the Convergence of Hermitian Dynamic Mode Decomposition](https://mlsanity.com/paper/2401.03192/) (2024)
- [On the algebra of Koopman eigenfunctions and on some of their infinities](https://mlsanity.com/paper/2604.21825/) (2026)

## Abstract (arXiv)

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.