Koopman Operator Framework

Updated 11 June 2026

Koopman Operator Framework is a linear, operator-theoretic approach that reformulates nonlinear dynamics by mapping state evolution onto function spaces.
Finite-dimensional approximations using EDMD, neural autoencoders, and kernel methods enable practical system identification and effective model reduction.
The framework facilitates advanced control strategies, including convex MPC and feedback linearization, for complex, multi-agent, and high-dimensional systems.

The Koopman operator framework provides a linear, operator-theoretic perspective for analyzing, modeling, and controlling nonlinear dynamical systems by shifting focus from state-space trajectories to the evolution of observables. Originating from Koopman’s 1931 construction, the framework encodes nonlinear dynamics as (infinite-dimensional) linear operators acting on function spaces, with practical deployments relying on finite-dimensional approximations, neural and kernel-based representations, and advancements in subspace selection and control synthesis. The Koopman approach enables powerful connections to spectral theory, model reduction, machine learning, and optimal control, with broad applications ranging from fluid flows and power systems to neuromodulation and algorithm analysis.

1. Foundations of Koopman Operator Theory

Let $(M,T)$ be a general discrete-time dynamical system on a state space $M$ (e.g., $M \subset \mathbb{R}^n$ ), with $T: M \to M$ . The Koopman operator $U$ acts on a Banach space $\mathcal{F}$ of observables $f: M \to \mathbb{C}$ as

$(Uf)(x) = f(T(x)).$

In continuous time with flow $\Phi^t$ , $U^t f(x) = f(\Phi^t(x))$ . The Koopman operator is always linear (though typically infinite-dimensional), allowing the dynamics to be analyzed via spectral theory: $M$ 0 where $M$ 1 are eigenvalue-eigenfunction pairs, and $M$ 2 is expanded in the eigenbasis. The spectrum decomposes the system into modal contributions: point spectrum (discrete dynamics), continuous spectrum (mixing/chaos), and associated Koopman modes for vector-valued observables (Budišić et al., 2012, Mezić, 2023).

Key concepts in the theory include:

Koopman Mode Analysis: Modal decomposition of observables into temporal/spatial structures, executed via data-driven algorithms such as Dynamic Mode Decomposition (DMD) or its generalizations, Extended DMD (EDMD). This enables extraction of coherent patterns, growth, decay rates, and phase information in nonlinear systems (Budišić et al., 2012, Mezić, 2023).
Koopman Eigenquotients and Invariant Sets: Eigenfunction level sets partition state space into invariant sets that correspond to conserved quantities, ergodic components, and isochrons (Budišić et al., 2012, Mezic, 2020).
Continuous Indicators of Ergodicity and Mixing: Comparing empirical measures along trajectories to invariant measures quantifies mixing and coverage rates (Budišić et al., 2012).

2. Finite-Dimensional Approximations, Learning, and Subspace Selection

While the Koopman operator is formally infinite-dimensional, practical computations require finite-dimensional surrogates. Common approaches include:

Extended Dynamic Mode Decomposition (EDMD): Constructing a matrix representation $M$ 3 by projecting the action of $M$ 4 onto the span of a chosen finite dictionary of observables:

$M$ 5

where $M$ 6 and $M$ 7 are data-driven Gram and cross-correlation matrices over snapshot pairs, and $M$ 8 the Moore–Penrose pseudoinverse (Mezić, 2023, Mauroy et al., 2017).

Neural/Autoencoder-Based Embeddings: Employing deep autoencoder architectures to learn nonlinear lifting and decoding functions, with a linear Koopman evolution in the latent space (Liang et al., 2021, Mezic, 2020, Estornell et al., 25 Apr 2025).
Kernel and RKHS Methods: Utilizing reproducing kernel Hilbert spaces to build expressive, possibly infinite-dimensional dictionaries with nonparametric, sparsity-controlled online learning via conditional mean embeddings (Hou et al., 27 Jan 2025, Giannakis et al., 20 Mar 2026).

Crucially, the success of Koopman-based modeling depends on the choice of lifting/observable subspace. Mis-specified dictionaries can result in poor invariance or expressive gap. Recent advances address this via:

Principal Angle/Subspace Pruning: Systematic removal of “leaky” basis directions with large principal angles between subspaces and their images under $M$ 9, guaranteeing minimal one-step invariance error and yielding low-complexity models with provable computational efficiency via rank-one SVD updates (Shah et al., 30 Mar 2026).
Hybrid Learning Frameworks: Combining convex optimization (semidefinite programming for latent-space discovery) with supervised learning of lifting and decoding maps, integrating model stability and spectral structure into neural architectures (Estornell et al., 25 Apr 2025).

These innovations facilitate tractable and accurate approximations, with applications to high-dimensional, stiff, and hybrid systems.

3. Koopman Operator in Control, Optimization, and Multi-Agent Systems

The linearity of the Koopman operator offers unique advantages for nonlinear control:

Koopman-MPC and Convex Control Synthesis: Nonlinear systems are lifted to latent coordinates with (approximately) linear time evolution, enabling the application of convex model predictive control (MPC) frameworks that yield global optima and real-time feasibility (Liang et al., 2021, Loya et al., 30 Mar 2026, Uchida et al., 2023).
- For instance, in closed-loop electrical neurostimulation for epilepsy, a deep-encoder-based Koopman-MPC suppresses seizures more effectively and with greater computational efficiency than RNN-based MPC, directly due to the convexity and online adaptability of the latent linear dynamics (Liang et al., 2021).
- Off-road vehicle tracking on deformable terrain is achieved by recursively updating a linear model in a lifted space, robustly integrating physics-based and data-driven information, and supporting real-time constrained tracking (Loya et al., 30 Mar 2026).
Feedback Linearization and Bilinearization: The Koopman framework provides global, coordinate-free feedback linearization for control-affine systems by embedding the dynamics into an infinite-dimensional bilinear system, leading to operator-theoretic controllability criteria (infinite-dimensional LARC) and relaxation of classical controllability requirements (Zhang et al., 2022).
Multi-Agent and Game-Theoretic Control: Koopman-based lifting extends naturally to multi-agent systems, enabling structured modeling of inter-agent couplings and hierarchical/time-scale separated dynamics. The framework supports both centralized (social-optimum) and Nash equilibrium solutions, with explicit characterization of the cross-agent coupling in the KKT conditions (Bakker, 18 Jun 2025).
Oblique Projection and Controller Generalization: Weak formulations with learned test functions (oblique projections) trade-off invariance and expressive power, allow more robust extraction of controllable subspaces, and yield superior closed-loop control and tracking performance in nonlinear systems (Uchida et al., 2023).

4. Infinite-Dimensional, Nonparametric, and High-Dimensional Perspectives

The Koopman framework generalizes naturally to infinite-dimensional settings, such as PDEs and spatially extended systems:

Koopman Semigroup on Function Spaces: For infinite-dimensional state spaces $M \subset \mathbb{R}^n$ $M \subset R^{n}$ 0, the Koopman semigroup $M \subset \mathbb{R}^n$ $M \subset R^{n}$ 1 (and its generator) are defined on the Banach space of bounded continuous functionals $M \subset \mathbb{R}^n$ $M \subset R^{n}$ 2, supporting spectral analysis and PDE structure identification purely from data (Mauroy, 2021).
- E.g., accurate recovery of leading spectral features and generator coefficients in nonlinear PDEs (e.g., Burgers’ equation) can be achieved via EDMD generalizations (Mauroy, 2021).
Kernel- and RKHS-Based Koopman Operators: Embedding the Koopman operator into an RKHS enables nonparametric modeling, with conditional mean embedding providing a consistent estimator. Streaming, sparse, and online algorithms balance model complexity and statistical efficiency, with convergence guarantees for high-dimensional, streaming data (Hou et al., 27 Jan 2025).
Tensor and Fock Space Amplifications, Quantum Connections: By amplifying the Koopman framework into Fock spaces of RKHSs (reproducing-kernel Hilbert algebras), one achieves structure preservation and efficient approximations, even allowing (for pure-point spectrum systems) efficient quantum-circuit implementations of Koopman evolution and observables (Giannakis et al., 20 Mar 2026).
Singular Perturbations and Multi-Scale Analysis: In systems with multiple timescales, Koopman eigenfunction geometry directly mirrors underlying slow/fast foliations and can be leveraged for asymptotic perturbation theory and reduced-order modeling (Katayama et al., 2024).

5. Applications across Domains

The operator-theoretic approach has been successfully applied to a broad range of scientific and engineering problems:

Fluid Mechanics: DMD and its Koopman-theoretic variants extract coherent structures, resonances, and instability precursors in complex flows (Budišić et al., 2012).
Epidemiology: Koopman-EDMD with epidemiologically informed dictionaries delivers accurate outbreak timing/magnitude predictions and spectral decomposition of dominant growth/decay modes in SIRSD models (Zinihi et al., 22 Aug 2025).
Neuromodulation: Real-time, adaptive control of epileptic dynamics via Koopman-MPC surpasses classical machine learning models and nonconvex controllers (Liang et al., 2021).
Robotics and Vehicle Control: Physics-informed, data-driven lifted models allow aggressive, robust, constrained tracking on challenging terrain (Loya et al., 30 Mar 2026).
Computational and Symbolic Dynamics: The Koopman framework recasts the halting/reachability problem as an operator-resolvent query, with implications for computation, decidability, and dynamical complexity (Caravelli et al., 7 Oct 2025).

6. Limitations and Challenges

Despite its strengths, the Koopman operator framework faces recognized challenges:

Model Expressiveness vs. Invariance: Achieving both a rich observable dictionary and closed invariance under the approximated Koopman operator is generally impossible. The trade-off is rigorously quantified in the oblique projection and principal vector subspace pruning frameworks (Uchida et al., 2023, Shah et al., 30 Mar 2026).
Basis/Observable Selection: The performance of finite-dimensional approximations depends critically on the chosen basis or feature space. Poor choices can lead to spurious modes, unphysical predictions, and loss of interpretability (Estornell et al., 25 Apr 2025, Hou et al., 27 Jan 2025).
Continuous Spectrum and Nonlinear Effects: Most data-driven methods recover only point spectrum, potentially missing subtle mixing and chaotic features. Generalized eigenfunctions and the design of dictionaries capturing continuous spectrum remain open problems (Budišić et al., 2012).
Scalability and Storage: High-dimensional models require substantial memory and computation; distributed/online algorithms and subspace primitives offer partial remedies (Shah et al., 30 Mar 2026, Azarbahram et al., 24 Sep 2025).
Interpretability and Guarantees: While hybrid and neural approaches offer expressive power, interpretability and formal error/control guarantees are still limited to simpler or specially structured models (Estornell et al., 25 Apr 2025, Uchida et al., 2023).

7. Frontiers and Outlook

Ongoing research directions in Koopman operator theory and applications include:

Adaptive and streaming dictionary selection, subspace pruning, and online learning with statistical guarantees (Shah et al., 30 Mar 2026, Hou et al., 27 Jan 2025).
Integration with optimal transport, quantum computing, and broader operator algebras for structure-preserving, high-expressivity modeling (Giannakis et al., 20 Mar 2026).
Unified frameworks for multi-agent, hierarchical, and real-time control incorporating game theory and time-scale separation (Bakker, 18 Jun 2025).
Deep learning architectures combining spectral, geometric, and operator-theoretic principles for robust prediction and control (Zhang et al., 17 Jun 2025, Liang et al., 2021).
Rigorously quantifying model uncertainty, expressive limitations, and data requirements in nonlinear, high-dimensional, and hybrid systems (Uchida et al., 2023).

The Koopman approach continues to provide a unifying, interpretable, and data-driven pathway for the analysis, identification, and control of complex nonlinear dynamical systems, with expanding influence across computational science, control, and machine learning (Budišić et al., 2012, Mezić, 2023).

Key references for further reading:

“Online Learning Koopman operator for closed-loop electrical neurostimulation in epilepsy” (Liang et al., 2021)
“A Unified Algebraic Framework for Subspace Pruning in Koopman Operator Approximation via Principal Vectors” (Shah et al., 30 Mar 2026)
“Operator is the Model” (Mezić, 2023)
“Applied Koopmanism” (Budišić et al., 2012)
“Koopman operator framework for spectral analysis and identification of infinite-dimensional systems” (Mauroy, 2021)
“Koopman Bilinearization of Nonlinear Control Systems” (Zhang et al., 2022)
“A Hybrid Framework for Efficient Koopman Operator Learning” (Estornell et al., 25 Apr 2025)
“SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting” (Zhang et al., 17 Jun 2025)
“Nonparametric Sparse Online Learning of the Koopman Operator” (Hou et al., 27 Jan 2025)
“Analog and Symbolic Computation through the Koopman Framework” (Caravelli et al., 7 Oct 2025)
“Distributed Koopman Operator Learning from Sequential Observations” (Azarbahram et al., 24 Sep 2025)
“Coopman Operator Framework for Modeling and Control of Off-Road Vehicle on Deformable Terrain” (Loya et al., 30 Mar 2026)
“Extracting Koopman Operators for Prediction and Control of Non-linear Dynamics Using Two-stage Learning and Oblique Projections” (Uchida et al., 2023)
“Koopman-based lifting techniques for nonlinear systems identification” (Mauroy et al., 2017)
“Koopman Analysis of the Singularly-Perturbed van der Pol Oscillator” (Katayama et al., 2024)
“Multi-Agent, Multi-Scale Systems with the Koopman Operator” (Bakker, 18 Jun 2025)
“Koopman and transfer operator techniques from the perspective of quantum theory” (Giannakis et al., 20 Mar 2026)
“A Koopman Operator Framework for Nonlinear Epidemic Dynamics: Application to an SIRSD Model” (Zinihi et al., 22 Aug 2025)
“On the Koopman operator of algorithms” (Dietrich et al., 2019)
“Koopman Operator, Geometry, and Learning” (Mezic, 2020)