The Multiverse of Dynamic Mode Decomposition Algorithms
It serves as a practical guide and theoretical reference for experts and newcomers in data-driven analysis of complex systems, but is incremental as it reviews existing methods.
This review tackles the problem of understanding and applying Dynamic Mode Decomposition (DMD) algorithms by providing a comprehensive examination of their theory and practice, categorizing methods into linear regression-based, Galerkin approximations, and structure-preserving techniques, and including a MATLAB package for practical use.
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.