[Paper Review] The Multiverse of Dynamic Mode Decomposition Algorithms

Matthew J. Colbrook|arXiv (Cornell University)|Nov 30, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper presents a comprehensive, theory-driven review of Dynamic Mode Decomposition (DMD) algorithms, unifying their diverse variants under the Koopman operator framework. It systematically categorizes DMD methods into regression-based, Galerkin-based, and structure-preserving approaches, offering theoretical insights, algorithmic details, and practical MATLAB implementations for spectral analysis of nonlinear dynamical systems.

ABSTRACT

Dynamic Mode Decomposition (DMD) is a popular data-driven analysis technique used to decompose complex, nonlinear systems into a set of modes, revealing underlying patterns and dynamics through spectral analysis. This review presents a comprehensive and pedagogical examination of DMD, emphasizing the role of Koopman operators in transforming complex nonlinear dynamics into a linear framework. A distinctive feature of this review is its focus on the relationship between DMD and the spectral properties of Koopman operators, with particular emphasis on the theory and practice of DMD algorithms for spectral computations. We explore the diverse "multiverse" of DMD methods, categorized into three main areas: linear regression-based methods, Galerkin approximations, and structure-preserving techniques. Each category is studied for its unique contributions and challenges, providing a detailed overview of significant algorithms and their applications as outlined in Table 1. We include a MATLAB package with examples and applications to enhance the practical understanding of these methods. This review serves as both a practical guide and a theoretical reference for various DMD methods, accessible to both experts and newcomers, and enabling readers to delve into their areas of interest in the expansive field of DMD.

Motivation & Objective

• To provide a unified theoretical and practical framework for understanding the diverse landscape of Dynamic Mode Decomposition (DMD) algorithms.
• To clarify the role of Koopman operators in enabling linear spectral analysis of nonlinear dynamical systems.
• To categorize and compare DMD variants across three main paradigms: regression-based, Galerkin-based, and structure-preserving methods.
• To identify and discuss open problems in convergence theory, dictionary selection, and verified control for Koopman-based methods.
• To offer a MATLAB package with examples to bridge theory and application for both newcomers and experts.

Proposed method

• Categorizes DMD methods into three main classes: linear regression-based, Galerkin approximation-based, and structure-preserving techniques.
• Uses Koopman operator theory as a unifying theoretical foundation, linking DMD to spectral analysis of nonlinear systems.
• Analyzes regression-based variants such as fbDMD, tlsDMD, optDMD, rDMD, and mrDMD, emphasizing noise robustness and compression.
• Examines Galerkin-based methods including Extended DMD (EDMD) with dictionary selection, Hankel-DMD via time-delay embedding, and ResDMD for error control.
• Introduces structure-preserving methods like piDMD (physics-informed) and mpEDMD (measure-preserving), with convergence theory.
• Applies compactification techniques to continuous-time systems and discusses connections to transfer operators and spectral measures.

Experimental results

Research questions

• RQ1 How do different DMD variants relate to the spectral properties of Koopman operators?
• RQ2 What are the theoretical and practical trade-offs between regression-based, Galerkin-based, and structure-preserving DMD algorithms?
• RQ3 What are the key open problems in convergence theory, dictionary selection, and verified control for Koopman-based DMD?
• RQ4 How can Koopman operator theory be extended to handle continuous spectra, stochastic systems, and transfer operators?
• RQ5 What are the limitations of current DMD methods in capturing transient and off-attractor dynamics?

Key findings

• The Koopman operator framework enables linear spectral analysis of nonlinear systems, providing a theoretical basis for DMD's effectiveness.
• Regression-based DMD variants like optDMD and rDMD improve robustness to noise and enable efficient computation via randomized linear algebra.
• EDMD with well-chosen dictionaries converges to the Koopman spectrum in the limit of large data and rich observables, with convergence theory established.
• Structure-preserving methods such as piDMD and mpEDMD maintain physical or measure-theoretic constraints, improving interpretability and accuracy.
• Hankel-DMD and HAVOK methods effectively recover Koopman modes from time-delayed data, especially for systems with weakly nonlinear or oscillatory behavior.
• Open problems include establishing lower bounds for computational impossibility, improving dictionary selection, and verifying Koopman models for control.