# The Multiverse of Dynamic Mode Decomposition Algorithms

arXiv: https://arxiv.org/abs/2312.00137 · HTML: https://mlsanity.com/paper/2312.00137/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2023.

## Built on by (most cited first, 26 of 26)

- [Adversarial dynamical systems characterize when data-driven learning succeeds or fails](https://mlsanity.com/paper/2407.06312/) (2024)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators](https://mlsanity.com/paper/2405.00782/) (2024)
- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Multiplicative Dynamic Mode Decomposition](https://mlsanity.com/paper/2405.05334/) (2024)
- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [On the Convergence of Hermitian Dynamic Mode Decomposition](https://mlsanity.com/paper/2401.03192/) (2024)
- [Weighted Birkhoff Averages Accelerate Data-Driven Methods](https://mlsanity.com/paper/2511.17772/) (2025)
- [Toeplitz Based Spectral Methods for Data-driven Dynamical Systems](https://mlsanity.com/paper/2602.09791/) (2026)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [A Novel Use of Pseudospectra in Mathematical Biology: Understanding HPA Axis Sensitivity](https://mlsanity.com/paper/2408.00845/) (2024)
- [Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators](https://mlsanity.com/paper/2505.06806/) (2025)
- [Online Meal Detection Based on CGM Data Dynamics](https://mlsanity.com/paper/2507.00080/) (2025)
- [Deficiency of equation-finding approach to data-driven modeling of dynamical systems](https://mlsanity.com/paper/2509.03769/) (2025)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)
- [A Dynamic Mode Decomposition Approach to Morphological Component Analysis](https://mlsanity.com/paper/2510.05977/) (2025)
- [Finding Koopman Invariant Subspaces via Personalized PageRank](https://mlsanity.com/paper/2605.24666/) (2026)
- [Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning](https://mlsanity.com/paper/2606.05131/) (2026)
- [Integral Formulation of QENDy for Robust Nonlinear System Identification](https://mlsanity.com/paper/2606.11629/) (2026)
- [Advances in Scientific Machine Learning for Coupled Fluid Flow and Transport](https://mlsanity.com/paper/2606.19562/) (2026)
- [fSRD: Fuzzy Spectral Region Decomposition -- Automated Multi Operator Koopman Representations via an Adaptive Spectral Learning Architecture](https://mlsanity.com/paper/2607.17990/) (2026)
- [Data-driven Koopman mode approximation: A neural power iteration algorithm](https://mlsanity.com/paper/2608.26943/) (2026)
- [Using Composition Operators to Linearize LLM Semantic Transformations](https://mlsanity.com/paper/2609.22143/) (2026)
- [Functional dynamic mode decomposition: Learning infinite-dimensional systems from data](https://mlsanity.com/paper/2609.29159/) (2026)
- [Test-Time Forecasting: Recurrent Depth Enables In-Context Dynamics Prediction](https://mlsanity.com/paper/2610.19552/) (2026)

## Abstract (arXiv)

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.