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.

翻译：动态模态分解（DMD）是一种流行的数据驱动分析技术，用于将复杂非线性系统分解为一系列模态，通过谱分析揭示其潜在模式与动力学特性。本综述对DMD进行了全面且具有教学意义的探究，重点阐释了Koopman算子在将复杂非线性动力学转化为线性框架中的关键作用。本综述的鲜明特色在于聚焦DMD与Koopman算子谱性质之间的关联，特别关注面向谱计算任务的DMD算法理论与实践。我们探索了DMD方法的多元"多重宇宙"，将其归纳为三大领域：基于线性回归的方法、Galerkin近似方法以及保持结构的技术。针对每个类别，我们研究其独特贡献与挑战，并依据表1所列内容系统梳理了重要算法及其应用案例。我们还附带了包含示例与应用的MATLAB工具包，以增强对这些方法的实践理解。本综述既可作为DMD各类方法的实用指南，也可作为理论参考，适用于专家与初学者，便于读者深入探索DMD广阔领域中各自感兴趣的方向。