# Multiplicative Dynamic Mode Decomposition

arXiv: https://arxiv.org/abs/2405.05334 · HTML: https://mlsanity.com/paper/2405.05334/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 64 references that are in the map.
- First posted on arXiv 8 May 2024.

## Builds on (most cited first, 13 of 13)

- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [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)
- [Optimal Permutation Recovery in Permuted Monotone Matrix Model](https://mlsanity.com/paper/1911.10604/) (2019)
- [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)
- [Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems](https://mlsanity.com/paper/2009.11544/) (2020)
- [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)

## Built on by (most cited first, 4 of 4)

- [One-Step Offline Distillation of Diffusion-based Models via Koopman Modeling](https://mlsanity.com/paper/2505.13358/) (2025)
- [Sequence Modeling with Spectral Mean Flows](https://mlsanity.com/paper/2510.15366/) (2025)
- [Avoiding spectral pollution for transfer operators using residuals](https://mlsanity.com/paper/2507.16915/) (2025)
- [Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning](https://mlsanity.com/paper/2606.05131/) (2026)

## Abstract (arXiv)

Koopman operators are infinite-dimensional operators that linearize nonlinear dynamical systems, facilitating the study of their spectral properties and enabling the prediction of the time evolution of observable quantities. Recent methods have aimed to approximate Koopman operators while preserving key structures. However, approximating Koopman operators typically requires a dictionary of observables to capture the system's behavior in a finite-dimensional subspace. The selection of these functions is often heuristic, may result in the loss of spectral information, and can severely complicate structure preservation. This paper introduces Multiplicative Dynamic Mode Decomposition (MultDMD), which enforces the multiplicative structure inherent in the Koopman operator within its finite-dimensional approximation. Leveraging this multiplicative property, we guide the selection of observables and define a constrained optimization problem for the matrix approximation, which can be efficiently solved. MultDMD presents a structured approach to finite-dimensional approximations and can more accurately reflect the spectral properties of the Koopman operator. We elaborate on the theoretical framework of MultDMD, detailing its formulation, optimization strategy, and convergence properties. The efficacy of MultDMD is demonstrated through several examples, including the nonlinear pendulum, the Lorenz system, and fluid dynamics data, where we demonstrate its remarkable robustness to noise.