Dynamic Mode Decomposition Overview

Updated 6 January 2026

Dynamic Mode Decomposition is a data-driven method that decomposes high-dimensional dynamics into coherent modes and frequencies by approximating the Koopman operator.
DMD employs singular value decomposition and reduced-order approximations to compute eigenvalues and modes, enabling reliable forecasting and model reduction in diverse applications.
Recent advancements in DMD include noise-robust, tensor-based, and structure-preserving variants that improve performance in real-time and high-dimensional data scenarios.

Dynamic Mode Decomposition (DMD) is a data-driven modal decomposition framework for analyzing high-dimensional dynamical systems by extracting a set of spatiotemporal coherent structures (modes) and associating each with a temporal behavior characterized by a single frequency and growth/decay rate. DMD is fundamentally rooted in spectral analysis of the Koopman operator and provides a unified framework for modal analysis, forecasting, reduction, and control in diverse fields spanning fluid mechanics, plasma physics, control, neuroscience, and beyond (Colbrook, 2023, Taylor et al., 2017).

1. Fundamental Principles and Mathematical Formulation

Given a sequence of measurements $\{x_1, x_2, \ldots, x_m\} \subset \mathbb{R}^n$ , DMD seeks a best-fit linear operator $A \in \mathbb{R}^{n \times n}$ such that

$x_{k+1} \approx A x_k.$

One constructs snapshot matrices: $X = [x_1,\, x_2,\, \ldots,\, x_{m-1}] \in \mathbb{R}^{n \times (m-1)}, \qquad Y = [x_2,\, x_3,\, \ldots,\, x_m] \in \mathbb{R}^{n \times (m-1)}.$ The minimum norm solution for $A$ is

$A = Y X^\dagger,$

where $X^\dagger$ denotes the Moore–Penrose pseudoinverse. For large-scale problems ( $n \gg m$ or $m \gg n$ ), one forms a reduced-order approximation via SVD: $X = U_r \Sigma_r V_r^*,$ with $U_r \in \mathbb{R}^{n \times r}$ , $\Sigma_r \in \mathbb{R}^{r \times r}$ , $V_r \in \mathbb{R}^{(m-1) \times r}$ , where $r$ is a prescribed or data-driven truncation rank, e.g., determined by Gavish–Donoho thresholding (Taylor et al., 2017). DMD modes and eigenvalues are then computed by projecting $A$ : $\tilde{A} = U_r^* Y V_r \Sigma_r^{-1} \in \mathbb{R}^{r \times r}, \qquad \tilde{A} W = W \Lambda,$ where columns of $W$ are eigenvectors and $\Lambda = \operatorname{diag}(\lambda_1,\ldots,\lambda_r)$ are DMD eigenvalues. Full-state DMD modes are given by

$\Phi = Y V_r \Sigma_r^{-1} W \in \mathbb{R}^{n \times r}.$

The DMD solution reconstructs the system for $k \geq 0$ as

$x_{k+1} \approx \Phi \Lambda^k b,$

where $b = \Phi^\dagger x_1$ are mode amplitudes (Colbrook, 2023, Dey, 2022).

2. Relation to Koopman Operator and Theoretical Underpinnings

DMD approximates the spectral decomposition of the (infinite-dimensional) Koopman operator $\mathcal{K}$ , defined for a dynamical system $x_{k+1} = F(x_k)$ as $(\mathcal{K}g)(x_k) = g(x_{k+1})$ for an observable $g$ (Colbrook, 2023, Dey, 2022). The DMD eigenvalues approximate Koopman eigenvalues, and DMD modes correspond to Koopman modes projected onto the measurement space.

This connection justifies the use of DMD for a broad class of nonlinear systems, with DMD extracting coherent spatiotemporal structures and frequencies even in the presence of nonlinearity, provided the dynamics admit a suitable embedding in the observable space (Dey, 2022). DMD thereby bridges data-driven and operator-theoretic perspectives.

3. Algorithmic Workflow and Computational Strategies

Standard DMD Algorithm (with Truncation)

Snapshot Pairing: Construct $X$ , $Y$ as above using consecutive (or appropriately paired) snapshots.
SVD/Economy SVD: Compute $X = U_r \Sigma_r V_r^*$ , selecting $r$ by singular-value energy, information threshold, or principled thresholding.
Reduced Operator: Form $\tilde{A} = U_r^* Y V_r \Sigma_r^{-1}$ .
Spectral Decomposition: Diagonalize $\tilde{A}$ to obtain eigenvalues $\Lambda$ and eigenvectors $W$ .
Mode Construction: Compute $\Phi = Y V_r \Sigma_r^{-1} W$ .
Prediction/Forecasting: Use $x_{k+1} \approx \Phi \Lambda^{k} b$ for reconstruction and future-state prediction (Taylor et al., 2017, Colbrook, 2023).

Computational Considerations

The dominant cost is the SVD of the $n \times (m-1)$ snapshot matrix; for small $n,m$ (e.g., $n=16$ in HIT-SI plasma data), computation is trivial (Taylor et al., 2017).
Efficient online/streaming DMD algorithms with rank-1 (Sherman–Morrison) or rank-2 (Woodbury) updates allow real-time mode extraction for time-varying systems, including weighted DMD with exponential forgetting and windowed DMD (Zhang et al., 2017).
For high-dimensional or big-data scenarios, projection-assisted DMDs combine time-delay embedding, random or Krylov subspace projections to achieve substantial storage and runtime reduction while retaining spectral accuracy (Murshed et al., 2020).

4. Methodological Extensions and Recent Developments

DMD's flexibility has led to a multiverse of algorithms addressing different challenges (Colbrook, 2023):

Noise-Robust and Structure-Preserving Variants

Total Least Squares DMD (tlsDMD): Accounts for noise in both $X$ and $Y$ (Colbrook, 2023).
Forward-Backward (fbDMD): Averages forward and inverse propagators to reduce sensor-noise bias (Colbrook, 2023).
Optimized DMD (optDMD): Uses nonlinear least squares with variable projection to directly fit exponential time dynamics, dramatically reducing bias at higher computational cost (Colbrook, 2023).
Consistent/Optimized Noise Separation: Recent algorithms explicitly model noise, use global error minimization, and combine forward-backward consistency for robust separation of coherent dynamics and noise (Weiner et al., 2024).

Galerkin and Koopman-Theoretic Approaches

Extended DMD (EDMD): Projects dynamics onto a nonlinear dictionary of observables, capturing nonlinear phenomena and enabling convergence to Koopman spectra (Colbrook, 2023).
Hankel-DMD: Uses delay-embedding via Hankel matrices for richer spectral information in scalar observables (Colbrook, 2023).
Residual DMD (ResDMD/DDMD_RRR): Quantifies spectral pollution, computes a posteriori residuals for Ritz pairs, and refines Ritz vectors to identify accurate Koopman modes (Drmač et al., 2017).

Structure-Preserving DMD

Physics-Informed DMD (piDMD, mpEDMD): Enforces constraints reflecting physical symmetries (e.g., energy conservation, measure-preservation) during spectral computation (Colbrook, 2023).

Tensor and Projection-based Methods

Tensor-Based DMD: For intrinsically high-dimensional multidimensional data, tensor-train (TT) formats enable DMD without explicit vectorization, vastly reducing memory and computational requirements when data is low-rank in tensor format (Klus et al., 2016).
Projection-assisted DMD: Employs random projections, sparsity-promoting maps, or Krylov subspaces to reduce computational cost for large $n$ (Murshed et al., 2020).

Parametric and Adaptive Variants

Parametric DMD: Interpolates DMD eigenpairs or operators over parameter space, supporting efficient multi-query reduced-order modeling and uncertainty quantification (Huhn et al., 2022).
DMD for Adaptive Mesh/Coarsening Simulations: Projects snapshots with spatially/temporally adaptive mesh onto a reference space prior to DMD, maintaining coherence for time-varying grids (Barros et al., 2021).

5. Interpretability, Visualization, and Component Selection

Phasor Notation: Interpretation of DMD conjugate-pair modes as real, strictly positive spatial envelopes modulated by real spatiotemporal waveforms—addressing common interpretability questions with standard DMD modes (Lapo et al., 3 Sep 2025).
Amplitudes & Error-Scaling: Modified scaling factors and error-scaling in "exact DMD" variants ensure error-free or error-bounded reconstruction and clarify how to attribute physical meaning to components (Krake et al., 2019).

Visualization and Clustering Techniques

Improved component definitions and spatiotemporal clustering methods (e.g., argument-magnitude scatter plots, dominance bar charts, distance-based and harmonic clustering) support physically relevant segmentation of DMD modes, direct mode selection, and identification of underlying temporal regimes (transient, steady, harmonic) (Krake et al., 2020).

Sparse and Transient Mode Extraction

Regularized DMD with time-varying amplitudes under sparsity and smoothness-promoting penalties enables detection of dynamically significant modes and their transient activities, facilitating interpretable analysis of non-steady flow and regime shifts (Tanaka et al., 14 Aug 2025).

6. Applications and Case Studies

DMD has been successfully applied across domains:

Plasma Physics: For diagnostic analysis, mode identification (e.g., spheromak, injector-driven modes), and low-rank surrogates in magnetohydrodynamics (Taylor et al., 2017).
Fluid Mechanics: Extraction of Kármán vortex street modes, reduced-order modeling, and disturbance prediction; DMD modes recover analytically known eigenfrequencies and spatial structures in vibrating membranes and nonlinear convection with sub-percent accuracy (Rot et al., 2022, Klus et al., 2016).
Source Separation, Change-point Detection: DMD provides a second-order blind source separation framework, outperforming PCA and some ICA methods in mixtures of stationary time series; regime changes can be detected by tracking DMD spectral evolution (Prasadan et al., 2019).
Time-varying/Streaming Data: Online and windowed DMD enable real-time tracking of system dynamics, e.g., in wind-tunnel measurements or variable-frequency oscillators (Zhang et al., 2017).
Partial Observations and Memory-Dependent Dynamics: Incorporation of Mori–Zwanzig formalism enables DMD to account for unresolved degrees of freedom and capture ensemble-averaged decay rates when only partial state measurements are available (Curtis et al., 2020).

7. Best Practices, Limitations, and Future Directions

Best Practices (Taylor et al., 2017, Hirsh et al., 2019, Colbrook, 2023):

Use principled rank selection (e.g., Gavish–Donoho thresholding) and ensure sampling meets Nyquist criteria for relevant frequencies.
Center data when modeling around nonzero equilibria or with constant bias components (Hirsh et al., 2019).
Apply mode refinement (e.g., DDMD_RRR) and a posteriori residual checks in high-noise or ill-conditioned contexts (Drmač et al., 2017).
Leverage online/streaming or projection-based algorithms for large-scale or real-time problems.
Use regularized and phasor-form interpretations for improved physical clarity.

Limitations:

Vanilla DMD is inherently noise-sensitive and can suffer from spectral bias or mode pollution under high-noise, partial observation, or insufficient sampling.
Strong nonlinearity or high intrinsic dimension may require extended DMD (e.g., EDMD) or delay-embedding strategies (Colbrook, 2023).
Tensor-based DMD's efficiency depends on low TT-rank; otherwise, it can be less effective (Klus et al., 2016).
Structured physical constraints (e.g., conservation laws) require tailored DMD variants for accurate long-time behavior.

Future Directions:

Integration with deep learning for automatic dictionary/basis selection or nonlinear embedding.
Development of structure-preserving and adaptive DMD for high-fidelity reduced-order models across parameter spaces.
Probabilistic and uncertainty-aware DMD variants, including Bayesian frameworks or explicit uncertainty quantification (Weiner et al., 2024, Huhn et al., 2022).
Theoretical advances in Koopman operator theory for extensions beyond $L^2$ observables (Colbrook, 2023).

Dynamic Mode Decomposition remains a central mathematical tool for spatiotemporal data analysis, model reduction, and interpretable spectral analysis of complex dynamical systems, supported by an extensive and rapidly evolving ecosystem of algorithmic enhancements and theoretical developments (Colbrook, 2023, Taylor et al., 2017, Dey, 2022, Drmač et al., 2017, Lapo et al., 3 Sep 2025, Tanaka et al., 14 Aug 2025).