Multiplicative dynamic mode decomposition
File(s)
Accepted version
Author(s)
Boulle, Nicolas
Colbrook, Matthew J
Type
Journal Article
Abstract
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.
Date Issued
2025-06-01
Date Acceptance
2025-02-15
Citation
SIAM Journal on Applied Dynamical Systems, 2025, 24 (2), pp.1945-1968
ISSN
1536-0040
Publisher
Society for Industrial and Applied Mathematics
Start Page
1945
End Page
1968
Journal / Book Title
SIAM Journal on Applied Dynamical Systems
Volume
24
Issue
2
Copyright Statement
© 2025 Nicolas Boullé and Matthew Colbrook. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Published
Date Publish Online
2025-06-11