Dynamic Mode Decomposition (DMD)

• Dynamic Mode Decomposition (DMD) is a data-driven modal analysis technique that approximates nonlinear systems using a best-fit linear operator derived from paired snapshot data.
• The method employs snapshot matrices, SVD-based rank truncation, and eigendecomposition to extract spatial modes along with their temporal growth, decay, and oscillation rates.
• Advanced DMD variants—such as DMD with control, tensor DMD, and online algorithms—enhance noise robustness, computational efficiency, and applicability to high-dimensional or streaming data.

Searching arXiv for recent and foundational papers on Dynamic Mode Decomposition to ground the article. Dynamic Mode Decomposition (DMD) is a data-driven modal analysis technique that approximates the evolution of a generally nonlinear dynamical system by a best-fit linear operator inferred from snapshot data. In its standard form, DMD extracts coherent spatial structures together with their temporal behavior—growth or decay rates and oscillation frequencies—from time-resolved measurements, and it is commonly interpreted as a finite-dimensional approximation to Koopman spectral analysis (Rot et al., 2022, Dey, 2022, Colbrook, 2023).

1. Definition and conceptual setting

DMD is formulated from paired data matrices rather than from an explicit governing equation. Given two sets of data vectors,

$$X=\bigl[x_1\;x_2\;\cdots\;x_m\bigr],\qquad Y=\bigl[y_1\;y_2\;\cdots\;y_m\bigr],$$

the method defines an approximating linear operator

$A=Y\,X^+,$

where $X^+$ is the Moore–Penrose pseudoinverse. In this formulation, DMD is the eigendecomposition of $A$: each eigenvalue is a DMD eigenvalue and each corresponding eigenvector is a DMD mode (Tu et al., 2013).

This operator-theoretic definition is broader than the original sequential-time-series setting. It accommodates arbitrary paired samples $(x_k,y_k)$, including non-sequential time series, multi-trajectory concatenations, and nonuniform subsampling strategies, provided the columns of $X$ are paired with those of $Y$ in a meaningful way (Tu et al., 2013). For sequential data, one typically interprets $x_k$ and $y_k$ as consecutive snapshots of the same observable.

The modern viewpoint places DMD within the spectral theory of the Koopman operator. For a nonlinear map $x_{n+1}=F(x_n)$, the Koopman operator acts linearly on observables $A=Y\,X^+,$0 by composition, $A=Y\,X^+,$1. DMD then serves as a practical finite-dimensional procedure for approximating Koopman eigenvalues, eigenfunctions, and modes from data (Dey, 2022, Colbrook, 2023). This connection explains why a linear decomposition can remain informative even when the underlying state dynamics are nonlinear.

2. Standard formulation and reconstruction

For a discrete-time state sequence $A=Y\,X^+,$2, the standard snapshot matrices are

$A=Y\,X^+,$3

DMD seeks a linear operator $A=Y\,X^+,$4 satisfying

$A=Y\,X^+,$5

The least-squares solution is

$A=Y\,X^+,$6

after computing a rank-$A=Y\,X^+,$7 truncation of the singular value decomposition

$A=Y\,X^+,$8

Rather than form $A=Y\,X^+,$9 explicitly, one projects onto the POD subspace and defines the reduced operator

$X^+$0

The small eigenproblem

$X^+$1

yields eigenvalues $X^+$2, and the lifted DMD modes are

$X^+$3

Collecting $X^+$4, the modal amplitudes are obtained from the initial condition,

$X^+$5

and the discrete-time reconstruction is

$X^+$6

With sampling interval $X^+$7, the continuous-time form uses

$X^+$8

These formulas constitute the standard exact-DMD pipeline (Dey, 2022, Rot et al., 2022).

The spectrum has an immediate dynamical interpretation. If $X^+$9, the corresponding mode grows; if $A$0, it decays. Its oscillation frequency is

$A$1

Thus DMD decomposes data into spatial patterns, each governed by a single exponential or oscillatory temporal law (Rot et al., 2022).

Rank truncation is central. In practice, $A$2 may be chosen by an energy threshold on singular values; one reported criterion is to keep

$A$3

Too small a rank may discard dynamically important modes; too large a rank may retain spurious or noise-dominated components (Dey, 2022, Rot et al., 2022).

3. Koopman-theoretic generalizations and the algorithmic “multiverse”

The Koopman extension replaces the state by an observable $A$4. Writing $A$5, one seeks

$A$6

where $A$7 is a finite-dimensional approximation of the infinite-dimensional Koopman operator $A$8. All standard DMD steps carry over to the observable data, and in Extended DMD (EDMD) one builds a dictionary of basis functions $A$9 and approximates $(x_k,y_k)$0 on their span (Dey, 2022).

A recent review organizes DMD algorithms into three broad classes: linear regression-based methods, Galerkin approximations, and structure-preserving techniques (Colbrook, 2023). Regression-based methods include exact DMD, forward-backward DMD, Total-Least-Squares DMD, optimized DMD, compressed and randomized DMD, multiresolution DMD, and DMD with control. Galerkin-based methods include EDMD, Kernel EDMD, Delay-Embedding and Hankel-DMD, HAVOK, and Residual DMD. Structure-preserving methods include physics-informed DMD, measure-preserving EDMD, and generator compactification (Colbrook, 2023).

Control enters naturally through the regression

$(x_k,y_k)$1

where $(x_k,y_k)$2 collects input snapshots. This produces DMD with control (DMDc), and the same operator-theoretic logic extends to Koopman with inputs and control (Heersink et al., 2017). For interconnected systems with graph structure, Network DMDc localizes the regression to directly connected components and returns block-sparse global operators $(x_k,y_k)$3 and $(x_k,y_k)$4. Reported consequences include lower sample complexity, reduced computational cost, and better numerical conditioning than a full dense DMDc fit (Heersink et al., 2017).

For parameter-dependent models, parametric DMD extends the reduced model across a training set of parameter values. Two non-intrusive approaches are reduced eigenpair interpolation (rEPI) and reduced Koopman operator interpolation (rKOI). The reported advantage of rKOI is a smoother interpolation assumption, monotonic accuracy versus rank, and linear cost in the number of interpolation neighbors (Huhn et al., 2022).

4. Exact reconstruction, rank pathologies, and preprocessing

Theoretical work distinguishes several DMD variants. Companion DMD (CDMD), SVD-DMD (SDMD), and Exact DMD (EXDMD) are related through low-dimensional similarity transformations, but they do not treat reconstruction identically. A central result is the exact reconstruction property: a DMD variant has this property if, for suitably chosen amplitudes and modes, the original snapshots $(x_k,y_k)$5 are recovered exactly, up to a residual term for $(x_k,y_k)$6 in CDMD and SDMD. EXDMD achieves exact high-dimensional eigenvectors through

$(x_k,y_k)$7

together with amplitudes

$(x_k,y_k)$8

and an additional error-scaling term when $(x_k,y_k)$9 (Krake et al., 2019).

A separate foundational issue is linear consistency. The pair $X$0 is linearly consistent if

$X$1

equivalently $X$2. This condition is necessary and sufficient for $X$3 with $X$4 exactly. Rank deficiency alone is therefore not the problem; the problem is rank deficiency combined with failure of linear consistency (Tu et al., 2013).

Standing-wave data exemplify the pathology. If snapshots remain confined to a one-dimensional subspace, standard DMD may return a real eigenvalue and miss the oscillation. Time-shift augmentation, in which multiple shifted snapshots are stacked, is a standard remedy and was explicitly recommended for standing-wave phenomena to boost rank and improve robustness to noise (Tu et al., 2013, Rot et al., 2022).

Preprocessing also matters. Centering data,

$X$5

is equivalent to fitting an affine model

$X$6

with

$X$7

This is not equivalent to computing a discrete Fourier transform. The reported conclusion is stronger: centered DMD can always be used to compute eigenvalue spectra of the dynamics, whereas in many cases DMD without centering cannot model the corresponding dynamics, most notably if the dynamics have full effective rank (Hirsh et al., 2019).

Accuracy diagnostics can also be internalized. Residual-based DMD computes

$X$8

for each Ritz pair, enabling selection of reliable modes. DDMD_RRR then refines Ritz vectors by minimizing the residual within the trial subspace, and it generalizes naturally to weighted inner-product spaces (Drmač et al., 2017).

5. Noise robustness, online updates, and large-scale computation

Several DMD developments address noise, streaming data, and high-dimensional state representations. For time-varying systems, online DMD updates the exact least-squares operator by rank-1 recursions and does not require storage of past data. With a forgetting factor $X$9, older data receive exponentially small weight; a windowed variant performs exact least-squares fits on the last $Y$0 snapshots. The reported per-update cost is $Y$1, and for problems in which the state dimension is less than about $Y$2, the method was reported as the most efficient for real-time computation (Zhang et al., 2017).

For noisy measurements, one recent approach embeds Koopman modes, eigenvalues, and amplitudes into an Ensemble Kalman filter. In a low-dimensional autonomous ODE with observable noise $Y$3, standard noisy DMD gave eigenvalues $Y$4, whereas EnKF-DMD gave $Y$5, close to the noise-free values $Y$6. On the same example, reconstruction RMSE decreased from $Y$7 to $Y$8, and prediction RMSE from $Y$9 to $x_k$0 (Liu et al., 2024).

High-dimensional tensor data motivate tensor-native DMD formulations. Tensor-Train-based DMD replaces explicit matrices by low-rank tensor decompositions and performs the heavy linear-algebra operations at the level of TT cores. On the two-merging-vortices example at $x_k$1, reported runtimes were approximately $x_k$2 versus $x_k$3 for full DMD at $x_k$4, and approximately $x_k$5 at $x_k$6, with leading eigenvalue and mode errors below $x_k$7 (Klus et al., 2016). A later $x_k$8-product tensor framework reported equal or better accuracy for the same storage than standard DMD and, in a streaming cylinder-flow test, batch-wise reconstruction errors that were consistently $x_k$9–$y_k$0 lower than Streaming DMD (Saibaba et al., 13 Aug 2025).

Nonuniform discretizations require additional machinery. In adaptive mesh refinement/coarsening simulations, snapshots may live in different finite-element spaces. One reported strategy projects each adaptive snapshot onto a common reference function space using an $y_k$1-projection, after which a standard DMD pipeline is applied to the projected coefficient vectors. Reported projection costs were $y_k$2–$y_k$3 of the original solver time, with speedups reaching $y_k$4 in SEIRD simulations and $y_k$5–$y_k$6 in a bubble-rising example (Barros et al., 2021).

Parametric multi-query settings introduce a different computational problem. In radiative diffusion tests, rKOI achieved similar accuracy to stacked DMD while reducing runtime from about $y_k$7–$y_k$8 to $y_k$9–$x_{n+1}=F(x_n)$0, and the paper characterized this as approximately $x_{n+1}=F(x_n)$1 faster (Huhn et al., 2022). This suggests that interpolation of reduced operators is often preferable to aggregating all parameter instances into a single stacked regression.

6. Interpretation, applications, and emerging representations

DMD has been used extensively as an analysis tool for time-dependent partial differential equations. In a two-dimensional damped wave equation on a square membrane, DMD recovered the first approximately $x_{n+1}=F(x_n)$2 eigenfrequencies with less than $x_{n+1}=F(x_n)$3 relative error, and the DMD modes matched the analytic $x_{n+1}=F(x_n)$4 structures. On an irregular “duck”-shaped membrane, where no analytic solution exists, DMD still extracted well-defined oscillatory modes and a frequency-versus-power spectrum (Rot et al., 2022).

In resistive magnetohydrodynamics for the HIT-SI spheromak device, sliding-window DMD with window length $x_{n+1}=F(x_n)$5 snapshots consistently yielded three dynamically significant modes: a zero-frequency “DC” mode identified as the spheromak’s Taylor-state profile, and a complex-conjugate pair at the SIHI injector frequency $x_{n+1}=F(x_n)$6. Using the Gavish–Donoho hard threshold yielded $x_{n+1}=F(x_n)$7 almost universally, model error was reported as a few percent at most, and a rank-3 model built from $x_{n+1}=F(x_n)$8 experimental snapshots predicted the next $x_{n+1}=F(x_n)$9 snapshots with small error (Taylor et al., 2017).

Interpretation of DMD components has itself become a subject of methodological development. One proposed refinement replaces separate modes and amplitudes by scaled modes

$A=Y\,X^+,$00

arguing that $A=Y\,X^+,$01 simultaneously carries spatial pattern and true magnitude. The same work introduced a refined eigenvalue plot in $A=Y\,X^+,$02-space and two clustering approaches—distance-based clustering and harmonic clustering—for selecting physically relevant groups of DMD components. In the superposed quadgyre example, harmonic clustering separated two base frequencies, each cluster reconstructing one quadgyre with approximately $A=Y\,X^+,$03 error (Krake et al., 2020).

Standard DMD assumes constant modal amplitudes, which can obscure transient activation and deactivation. A recent extension introduces time-varying amplitudes $A=Y\,X^+,$04 together with $A=Y\,X^+,$05 sparsity and $A=Y\,X^+,$06 smoothness penalties. On transient flow over a NACA0012 airfoil, the method separated modes active only in the early transient from those that persist in the vortex-shedding regime. Reported reconstruction error remained below $A=Y\,X^+,$07 in the limit-cycle regime $A=Y\,X^+,$08, but rose to $A=Y\,X^+,$09 during the very initial stage; increasing the sparsity parameter produced fewer active modes and sharper interpretability at the expense of higher error (Tanaka et al., 14 Aug 2025).

A separate interpretive development is phasor notation for conjugate-pair DMD modes. For a complex mode $A=Y\,X^+,$10 and eigenvalue $A=Y\,X^+,$11, the pair contribution can be rewritten in real form as

$A=Y\,X^+,$12

where

$A=Y\,X^+,$13

is a strictly positive real spatial pattern and

$A=Y\,X^+,$14

is a spatially varying phase shift. This resolves the ambiguity of interpreting complex spatial modes directly and extends to windowed multiresolution settings through bandwise weighted averages (Lapo et al., 3 Sep 2025).

Beyond fluid and plasma applications, DMD has also been used for source separation, change-point detection, video background modeling, and neural line-noise removal. One source-separation result showed that when latent time series are uncorrelated at a chosen lag, DMD modes approximate the columns of the mixing matrix in the large-sample limit; this was contrasted with kurtosis-based ICA, which fundamentally cannot unmix mixed stationary, ergodic Gaussian time series (Prasadan et al., 2019). In a different line of work, fixed-frequency subtraction within centered DMD suppressed a $A=Y\,X^+,$15 ECoG line-noise peak by an order of magnitude while leaving the remaining neural modes intact (Hirsh et al., 2019).

Taken together, these developments present DMD not as a single algorithm but as a family of operator-inference procedures: least-squares regression on snapshots, Koopman approximations in observable space, and increasingly structured or adaptive variants for noise, control, sparsity, tensors, and transient behavior. A plausible implication is that the enduring core of DMD is less the specific Schmid-era algorithm than the spectral decomposition of an empirically fitted evolution operator.