Koopman Mode Decomposition in Dynamical Systems

• Koopman Mode Decomposition is a spectral framework that recasts nonlinear dynamics into a linear operator structure using eigenvalues, eigenfunctions, and modes.
• It employs data-driven algorithms like Dynamic Mode Decomposition (DMD) and Extended DMD (EDMD) to efficiently extract high-resolution spatiotemporal patterns and reduce complex models.
• The method is applied across fluid mechanics, power systems, and climate science, providing actionable insights into periodic, chaotic, and stochastic system behaviors.

Koopman Mode Decomposition (KMD) is a spectral methodology for analyzing dynamical systems by recasting nonlinear evolution as a linear process in the space of observables. This approach enables high-resolution spatiotemporal decompositions of complex signals and systems, including both deterministic and stochastic dynamics. KMD builds upon the Koopman operator, a (generally infinite-dimensional) linear operator that propagates scalar-valued functions (observables) defined on the system’s state space. By extracting a spectrum of operator eigenvalues, eigenfunctions, and corresponding modes, KMD provides a rigorous framework for modal analysis, model reduction, prediction, and control across highly nonlinear settings.

1. Koopman Operator Theory and KMD Foundations

At the core of Koopman Mode Decomposition lies the Koopman operator, 𝒦, which evolves an observable function $g : \mathcal{X} \rightarrow \mathbb{C}$ according to $[\mathcal{K}g](x) = g(F(x))$, where $F$ may be a nonlinear or stochastic flow on the state space $\mathcal{X}$. Despite the potentially nonlinear dynamics of the underlying system, $\mathcal{K}$ is linear, and thus amenable to spectral analysis.

The principle of KMD is that observables can, in favorable cases, be decomposed into a sum (or integral) of Koopman eigenfunctions multiplied by temporal exponentials and spatial modes: $$f(x_k) = \sum_{j=1}^{\infty} \lambda_j^k \psi_j(x_0) v_j,$$ where $\lambda_j$ are Koopman eigenvalues (generally complex), $\psi_j$ are eigenfunctions, and $v_j$ are Koopman modes in the observable space (Williams et al., 2014, Dey, 2022).

If the system exhibits measure-preserving or ergodic properties, the spectrum of $\mathcal{K}$ can contain both discrete (corresponding to strictly periodic or quasi-periodic dynamics) and continuous (associated to mixing/chaotic components) parts (Arbabi et al., 2017, Colbrook et al., 1 May 2024). In general, the eigenfunctions are functions on state space that evolve linearly in time, and the modes provide physical spatial structure.

2. Data-Driven KMD: Algorithms and Implementations

Traditional KMD requires global knowledge of the operator, but practical implementation leverages data-driven approximations—predominantly Dynamic Mode Decomposition (DMD) and its generalizations.

Dynamic Mode Decomposition (DMD): DMD approximates the action of 𝒦 by fitting a best-fit linear model to a sequence of vector observations (snapshots), yielding eigenvalues and modes corresponding to the principal directions of evolution in data. DMD is mathematically equivalent to KMD when the observable is the full state vector and the underlying system is linear.

Extended Dynamic Mode Decomposition (EDMD): EDMD generalizes DMD by allowing an arbitrary dictionary of observables—potentially nonlinear functions such as polynomials, Fourier modes, or radial basis functions—so that the span of the dictionary better approximates leading Koopman eigenfunctions and captures nonlinear phenomena. Given a snapshot dataset $\{x_m, y_m\}$ (with $y_m = F(x_m)$ or a stochastic realization), and dictionary $\mathcal{D} = \{\psi_1, \ldots, \psi_K\}$, EDMD computes matrices: $$G = \frac{1}{M} \sum_m \Psi(x_m)^* \Psi(x_m), \qquad A = \frac{1}{M} \sum_m \Psi(x_m)^* \Psi(y_m),$$ and constructs the Koopman approximation as $K=G^+A$. The eigenvalues and eigenvectors of $K$ yield approximations to Koopman spectral objects, and the resulting modes produce reconstructions of observables (Williams et al., 2014, Snyder et al., 2021).

Stochastic and Robust Variants: For systems with process or measurement noise, subspace DMD (subspace DMD) employs orthogonal projections of future snapshots onto the subspace of past snapshots, statistically averaging out uncorrelated noise and isolating the underlying stochastic Koopman operator (Takeishi et al., 2017). Ensemble Kalman filter approaches can provide real-time Bayesian updates to estimates of KMD spectra in the presence of temporal drift and observation noise, handling non-autonomous systems and delivering uncertainty quantification (Liu et al., 24 Sep 2024).

Sparsity-Promoting and Structure-Preserving Extensions: Sparsity-promoting DMD enforces an $\ell_1$ penalty on modal amplitudes to select a minimal (physically interpretable) subset of modes, facilitating dimensionality reduction and identification of dominant transient or coherent structures (Zhang et al., 17 Jun 2025). Multiplicative DMD (MultDMD) constrains the approximation to preserve the product structure of the Koopman operator, selecting indicator functions as observables and enforcing that $K(f \odot g) = (Kf) \odot (Kg)$ for Hadamard product, ensuring the spectral properties reflect group-theoretic structure (Boullé et al., 8 May 2024).

Koopman-Schur Decomposition: To address numerical instability associated with poorly conditioned eigenvector decompositions (such as in nonnormal or defective operators), the Koopman-Schur approach employs a unitary/orthogonal Schur decomposition, organizing the modal basis as a flag of invariant subspaces and ensuring computational stability (Drmač et al., 2023).

3. Advanced Spectral Analysis and Continuous Spectrum

Standard DMD-type algorithms are effective when the Koopman spectrum is purely discrete. However, many physical systems—especially high-dimensional, turbulent, or chaotic flows—possess continuous or mixed spectra. In such cases, traditional DMD may generate spurious modes or fail to isolate continuous spectral components.

Rigged DMD: The Rigged Dynamic Mode Decomposition framework extends the modal analysis to arbitrary spectra by leveraging a rigged Hilbert space (Gelfand triple) approach. The methodology constructs wave-packet (smeared) approximations for generalized eigenfunctions via the Koopman operator’s resolvent, convolving with high-order smoothing kernels to recover the spectral measure associated with continuous spectrum. The convergence properties are made explicit for both discrete and continuous components, enabling the recovery of coherent features even when standard DMD is nonconvergent (Colbrook et al., 1 May 2024).

In fluid mechanic applications (e.g., high-Reynolds lid-driven cavity), Rigged DMD successfully recovers both point and continuous spectral components and provides a pathway toward reduced-order modeling and control in chaotic regimes.

4. Applications and Interpretability of Koopman Modes

KMD has demonstrated utility in a broad set of theoretical and applied contexts:

• Fluid Mechanics: By decomposing velocity or vorticity fields into Koopman modes, researchers can resolve dynamically coherent spatiotemporal patterns such as traveling waves, periodic/quasi-periodic cycles, and chaotic fluctuations. KMD enables low-dimensional modeling and accurate reconstruction of flows, outcompeting energy-optimal bases like POD for oscillatory and quasi-periodic regimes (Arbabi et al., 2017, Page et al., 2018).
• Thermodynamics of Noisy Oscillators: In nonlinear Langevin systems, KMD decomposes thermodynamic dissipation into contributions from oscillatory modes, showing that each mode’s energetic cost is proportional to its frequency squared and intensity. This frequency-resolved thermodynamic picture provides new insights into phenomena such as stochastic resonance and bifurcation in biological oscillators (Sekizawa et al., 24 Oct 2025).
• Power Systems and Control: KMD-derived participation factors generalize the modal participation concept to nonlinear settings—revealing both linear and nonlinear modal interactions in complex infrastructures like multi-area power grids and supporting real-time measurement-based monitoring (Netto et al., 2018, Masuda et al., 2019).
• Weather and Climate: Sparse KMD frameworks recover dominant transient or coherent weather features (e.g., thermal convection or vorticity “bubbles”) from high-dimensional short-term forecasts, enabling efficient surrogate models and improved physical diagnostics (Zhang et al., 17 Jun 2025).
• Manifold Learning: KMD can serve as a nonlinear parameterization tool, revealing intrinsic coordinates even for highly nonlinear data such as Swiss-Roll manifolds, and facilitating dynamical systems-based manifold learning (Williams et al., 2014).

These applications demonstrate that the modal decomposition not only summarizes dynamics but also links directly to physically interpretable phenomena.

5. Theoretical Guarantees, Convergence, and Limitations

Under standard assumptions—such as ergodicity, sufficient richness of the dictionary, and large data limits—data-driven KMD converges (almost surely) to a Galerkin projection of the true Koopman operator (Williams et al., 2014). Convergence rates are typically $\mathcal{O}(M^{-1/2})$ for Monte Carlo sampling, with additional functional-analytic regularity requirements for convergence of generalized eigenfunction approximations in continuous spectrum cases (Colbrook et al., 1 May 2024).

Limitations arise when the observable span fails to approximate the true Koopman eigenfunctions, when the system is highly nonnormal or lacks diagonalizability, or in the presence of complex continuous spectra. Structural degeneracies, as encountered in nonlinear PDEs such as Burgers equation, can render Koopman modes initial-condition dependent and nonidentifiable without precise knowledge of the correct eigenfunction family (Page et al., 2017).

Algorithmic choices such as dictionary selection, basis richness, and feature scaling (e.g., via Mahalanobis learning or kernel methods) are critical for robust application, particularly in high-dimensional or partially observed systems (Aristoff et al., 2023).

6. Interactions with Optimization and Tensor Decompositions

A theoretical connection has been established between KMD (as computed by DMD or EDMD) and tensor component analysis (TCA; CP decomposition): under generic conditions (full-rank Koopman eigenbasis and tensor rank matching the mode count), the decomposition of snapshots into temporal, spatial, and initial condition modes aligns structurally with canonical tensor decompositions (Redman, 2021). This duality bridges the operator-theoretic and optimization community approaches and opens the door to hybrid algorithmic strategies that scale to large, multi-indexed datasets.

By linking tensor approaches with functional-analytic operator decompositions, KMD forms a foundation for interpretable, computationally tractable model reduction and system identification in large-scale, nonlinear dynamical systems.

7. Future Directions and Open Challenges

Emerging research directions include extension of KMD to nonautonomous and controlled systems, further development of adaptive and robust filtering-based KMD variants for online and partially observed data, as well as integration with kernel methods and machine learning for scalable high-dimensional approximation (Kawashima et al., 2022, Liu et al., 24 Sep 2024).

Open problems include: systematically handling continuous and mixed spectra (e.g., via wavelet or time-frequency generalizations of Rigged DMD), robust extraction of Koopman eigenfunctions from partial or indirect observations, adaptive selection of dictionaries and feature spaces, and the integration of KMD with real-time control and estimation frameworks.

A plausible implication is that, as algorithms for enforcing operator-theoretic structure (e.g., multiplicativity, unitarity) and adaptive feature scaling mature, KMD will provide the backbone for a unified time-frequency-spatial modal analysis in complex, nonlinear, and stochastic systems across science and engineering.