Multiplicative Dynamic Mode Decomposition

TL;DR

This paper introduces Multiplicative Dynamic Mode Decomposition (MultDMD), a novel method for approximating Koopman operators that preserves their multiplicative structure, leading to more accurate spectral analysis of nonlinear dynamical systems.

Contribution

MultDMD enforces the Koopman operator's multiplicative structure in finite-dimensional approximations, improving spectral fidelity and robustness to noise compared to existing methods.

Findings

MultDMD accurately captures spectral properties of nonlinear systems.

The method demonstrates robustness to noise in fluid dynamics data.

MultDMD outperforms traditional DMD in several nonlinear examples.

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.