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On Dynamic Mode Decomposition: Theory and Applications
Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz
TL;DR
Existing DMD theory primarily targets sequential, full-rank time series, limiting its formal scope. This paper defines DMD as the eigendecomposition of an approximating operator for paired data, then uses that framework to generalize sampling, explain rank-deficient cases, and connect DMD with Koopman theory, ERA, and LIM. The resulting framework extends DMD to broader datasets while preserving or strengthening these theoretical connections.
Problem
Existing DMD theory primarily focuses on sequential time series and does not adequately cover nonsequential or rank-deficient datasets.
Method
The paper defines DMD as the eigendecomposition of A = YX+, an approximating linear operator constructed from paired data matrices.
Results
The framework generalizes DMD to paired datasets, supports nonuniform sampling and concatenated runs, and strengthens connections with Koopman theory, ERA, and LIM.
Takeaways & Limitations
DMD can be interpreted and applied within a broader operator-based framework that includes nonsequential data and established system-identification and climate-modeling connections.
Takeaways & Limitations
The Koopman interpretation depends on linear consistency and a full set of eigenvectors for A; without these assumptions, DMD modes may not approximate Koopman modes.
Abstract
from arXiv · showhide
Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature, but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.
1. Introduction.
The paper generalizes DMD from sequential time series to paired data vectors, establishing a framework for broader sampling strategies, rank-deficient datasets, and connections to Koopman, ERA, and LIM.
- Motivation: DMD analyzes nonlinear dynamics through modes and eigenvalues associated with a finite approximation to the Koopman operator.The Koopman operator is linear but infinite-dimensional, and its spectral decomposition captures the evolution of observables for nonlinear systems.
- Contributions: The framework defines DMD from pairs of n-dimensional data vectors and an approximating linear operator rather than only sequential time series.This generalization is implemented through a modification of an existing DMD algorithm.
- Limitations and theory: The framework explains how rank-deficient datasets can yield either meaningful or misleading DMD results through the condition of linear consistency.When linear consistency fails, the Koopman analogy can break down.
- Applications: Nonuniform temporal sampling can reduce computational cost with little effect on the accuracy of dominant DMD modes and eigenvalues.The paper also considers concatenating data from multiple experimental runs to address noise.
- Applications: Concatenating multiple experimental datasets produces sharper, more isolated spectral peaks and reveals higher-frequency modes obscured by traditional DMD.This strategy is presented as a way to mitigate noise effects.
- Connections: DMD eigenvalues reduce to ERA model poles, while under certain conditions DMD is equivalent to LIM.These connections link DMD with system identification and climate-science modeling methods.
2. Theory.
The theory defines exact DMD as the eigendecomposition of a least-squares approximating operator for paired datasets, relates it to standard SVD-based DMD, and characterizes consistency and projected modes.
- Comparing definitions: The generalized and standard algorithms produce the same DMD eigenvalues, while their modes differ by projection onto the range of X.The standard algorithm’s projected modes are eigenvectors of P_XA, and the two mode sets coincide when the Y columns lie in the span of X.
- New definition: The generalized formulation accepts data pairs without requiring sequential ordering, uniform sampling, or overlapping columns between X and Y.Standard DMD assumes sequential vectors, whereas the new formulation uses arbitrary paired datasets.
- New definition: Exact DMD for paired data defines A = YX+ and takes its eigenvectors and eigenvalues as the DMD modes and eigenvalues.The operator is the least-squares/minimum-norm solution to AX = Y.
- Algorithms: The reduced SVD algorithm computes the reduced operator ˜A = U*YVΣ^-1 and obtains DMD eigenvalues from its eigendecomposition without explicitly forming A.This avoids direct manipulation of potentially large n × n operators.
- Algorithms: Algorithm 2 identifies every nonzero eigenvalue of A and produces corresponding eigenvector/eigenvalue pairs.The result follows from the relationship between the reduced operator ˜A and the approximating operator A.
- Limitations: Zero-eigenvalue modes are generally not of interest because they do not contribute to the dynamics.The algorithm can nevertheless identify certain zero-eigenvalue eigenvectors in the images of X or Y.
- Linear consistency: Linear consistency means Xc = 0 implies Yc = 0, and it is equivalent to the exact relation AX = Y for A = YX+.This condition generalizes the requirement that identical inputs have identical outputs to linearly dependent input sets.
3. Applications.
The framework extends DMD to nonsequential and rank-deficient datasets, enabling flexible sampling while clarifying when results are meaningful or misleading. Applications show accurate dominant-mode recovery with reduced data and improved spectra from multiple experimental runs.
- Generalized data structure: DMD can analyze paired data vectors without requiring sequential sampling, treating sequential time series as a special case.Only pairwise correspondence between X and Y matters; jointly permuting their columns does not change the approximating operator or DMD results.
- Rank-deficient datasets: Rank-deficient datasets can yield meaningful DMD estimates when the paired data preserve the relevant relationship, but standing-wave examples expose failures caused by linear inconsistency.For one noisy rank-one time series, least-squares fitting accurately estimates the decay rate; for another oscillatory dataset, DMD may return only exponential growth or decay or fail to capture the correct dynamics.
- Nonuniform sampling: Nonuniform sampling preserves dominant DMD eigenvalues with less than 10% error while using 90% less data and reducing computation time by 85.8%.Larger errors occur mainly for modes with norms around 10^-5, two orders of magnitude below those of the dominant modes, and therefore have negligible contribution to the impulse response.
- Combining multiple trajectories: Combining five experimental runs produces more isolated harmonic spectral peaks, reveals a fourth harmonic obscured in a single run, and yields smoother, more symmetric modal structures.The multi-run results more closely match computational results for the turbulent bluff-body wake experiment.
- Nonuniform sampling: Nonuniformly sampled DMD recovers dominant flow modes that match those obtained from uniformly sampled data at the same dominant frequencies.The agreement is observed in the real parts of the displayed modes and similarly in their imaginary parts.
4. Connections to other methods.
The paper establishes theoretical and algorithmic connections between DMD, Koopman spectral analysis, ERA, and LIM, clarifying when these methods share interpretations or computational strategies.
- Koopman spectral analysis: The Koopman interpretation provides a theoretical foundation for applying DMD to nonlinear dynamics, but depends on linear consistency and a diagonalizable approximating operator.Without these assumptions, DMD modes need not closely approximate Koopman modes, including near attractors or for dynamics with Jordan blocks.
- Linear inverse modeling (LIM): The framework links DMD to climate-science LIM and supports cross-method interpretation of the low-order dynamics generated from data.The paper presents LIM equivalence as another way to interpret DMD analysis.
- Koopman spectral analysis: DMD extends Koopman connections from sequential time series to more general datasets using exact DMD and the weaker requirement of linear consistency.For linearly consistent data with diagonalizable A, DMD modes and eigenvalues correspond to Koopman modes and eigenvalues.
- The eigensystem realization algorithm (ERA): DMD and ERA use low-order linear operators related by a similarity transform, motivating ERA-inspired data augmentation to address rank problems in DMD.Appending time-shifted data to X and Y is suggested as a strategy for overcoming rank deficiencies.
- The eigensystem realization algorithm (ERA): Given matrices H and H′, ERA produces a reduced-order operator that can compute DMD modes and eigenvalues, although ERA targets linear input-output systems while DMD permits nonlinear or forced dynamics.The methods are algorithmically closely related but are designed for different dynamical-system settings.
- Linear inverse modeling (LIM): Under certain conditions, LIM and DMD are equivalent and share the same low-dimensional approximating linear dynamics.With the data mean removed, this map yields the statistically most likely future state under the stated LIM interpretation.
5. Conclusions.
The framework extends DMD from full-rank sequential time series to general pairs of data vectors while preserving connections to Koopman theory, ERA, and LIM. It supports nonuniform sampling, noise mitigation, and analysis of rank-deficient datasets.
- DMD is defined as the eigendecomposition of an approximating linear operator for pairs of data vectors.
- The generalized framework permits pairwise-related data without uniform sampling, a single time series, or overlapping data matrices.
- Nonuniform trajectory sampling reduces computational costs, while concatenating datasets can average out noise effects.
- Rank-deficient DMD computations may be meaningful or misleading depending on the dataset.
- DMD is closely related to ERA and LIM, and is equivalent to LIM under certain conditions.
Appendix A. Choices for mode scaling.
Mode norms are not intrinsically reliable indicators of dynamical importance because DMD modes permit arbitrary scalings. Scaling choices should therefore reflect the application and computational constraints.
- DMD mode norms require an appropriate scaling before they can identify dynamically interesting modes.Definition 1 permits arbitrary rescaling of DMD modes.
- Scaling options include general linear-algebraic choices and choices tailored to sequential time series.The computational cost of scaling factors matters for high-dimensional datasets.
A.1. Generic data.
For generic data, DMD modes can be scaled using normalized eigenvectors or biorthogonality with adjoint modes. These choices provide algebraically defined alternatives to norm-based selection.
- Unit-norm eigenvectors yield projected DMD modes with unit norm.The normalization follows from the orthonormality of the matrix U.
- Exact DMD modes are orthogonal to adjoint DMD modes associated with different eigenvalues.
- Scaling adjoint eigenvectors so z∗w = 1 makes exact and adjoint DMD modes a biorthogonal set.No additional scaling is needed after computing the DMD modes via the stated procedure.
A.2. Sequential time series.
For sequential time series, initial-state-based mode amplitudes can be computed without identifying zero-eigenvalue modes, but pseudoinverse-based scaling has substantial computational and conditioning costs.
- Projected and exact DMD modes are often scaled so their sum equals the first data vector x0.
- Representing x0 may require zero-eigenvalue modes that Algorithm 2 does not identify, but amplitudes can still be computed without them.
- When data matrices are linearly consistent, y0 lies in the range of the DMD modes and the amplitude system has a solution.QR decomposition can be used to solve for the mode amplitudes.
- A pseudoinverse can compute amplitudes without explicitly forming the full mode matrix.
- For generic data, computing Y∗Y is nearly as costly as the DMD algorithm and squaring the condition number can reduce accuracy.For sequential time series, Y∗Y requires only m new inner products after X∗X is available.