Continuous-Time Koopman Operator

• Continuous-Time Koopman Operator is defined as a linear, infinite-dimensional operator acting on observables to capture the dynamics of continuous-time systems.
• Its methodology employs an infinitesimal generator and data-driven techniques such as EDMD to project and analyze nonlinear systems using linear spectral methods.
• The framework underpins rigorous analysis and control of deterministic, stochastic, and nonautonomous systems via modal decomposition and regularization strategies.

A continuous-time Koopman operator is a linear, typically infinite-dimensional, operator that describes the evolution of observables under the flow of a continuous-time dynamical system. This operator-theoretic formalism enables the spectral and modal analysis of nonlinear systems using linear techniques, by studying the action of the flow on function spaces rather than on the state space itself. The continuous-time setting is essential for theoretical rigor, spectral analysis, and computational schemes that directly encode the infinitesimal dynamics, providing a foundation for ergodic theory, data-driven identification of generators, and modern machine learning approaches to dynamical systems.

1. Definition and Functional-Analytic Structure

Let $(X,\mathcal{B},\nu)$ be a measurable phase space, and $\Phi^t:X\to X$ a measurable flow parameterized by $t\in\mathbb{R}$ or $[0,\infty)$, often assumed to preserve an invariant probability measure $\nu$. The continuous-time Koopman operator family $\{U^t\}_{t\geq 0}$ acts on an admissible space of observables (e.g., $L^2(X,\nu)$ or $C(X)$): $(U^t f)(x) = f(\Phi^t(x)).$ This family forms a strongly continuous one-parameter semigroup/unitary group in the ergodic, measure-preserving case, by linearity and the flow property. The infinitesimal generator $L$ (sometimes denoted $\Phi^t:X\to X$0 or $\Phi^t:X\to X$1) is defined, whenever the limit exists for $\Phi^t:X\to X$2 and $\Phi^t:X\to X$3,

$\Phi^t:X\to X$4

For smooth deterministic flows $\Phi^t:X\to X$5, this yields $\Phi^t:X\to X$6; for measure-preserving flows, $\Phi^t:X\to X$7 is skew-adjoint on $\Phi^t:X\to X$8. The Hille–Yosida and Stone's theorems guarantee that $\Phi^t:X\to X$9 forms the unique $t\in\mathbb{R}$0-semigroup generated by $t\in\mathbb{R}$1 under standard domain conditions (Servadio et al., 2021, Mauroy, 2021, Valva et al., 2024, Valva et al., 2023, Rosenfeld et al., 2019).

2. Spectral Theory and the Generator

The spectral theory of the continuous-time Koopman operator is central to decomposing nonlinear dynamics into coherent spatiotemporal modes. The spectral problem for $t\in\mathbb{R}$2,

$t\in\mathbb{R}$3

yields Koopman eigenfunctions $t\in\mathbb{R}$4 and eigenvalues $t\in\mathbb{R}$5 which determine the modal evolution: $t\in\mathbb{R}$6 In measure-preserving and ergodic cases, the spectral decomposition of $t\in\mathbb{R}$7 consists generally of pure point (discrete), continuous, and mixed components (via the spectral theorem and the Wold–Hahn decomposition) (Zhen et al., 2022, Valva et al., 2023). In the Hilbert space $t\in\mathbb{R}$8, this structure underpins expansion of observables and the temporal evolution of statistics, including autocorrelation dynamics and quasi-periodic vs mixing regimes.

For infinite-dimensional PDEs and random dynamical systems, the generator $t\in\mathbb{R}$9 may take the form of a Lie derivative or the Kolmogorov backward operator, e.g.,

$[0,\infty)$0

$[0,\infty)$1

Thus, generator-based spectral analysis encompasses deterministic, stochastic, and high-dimensional nonlinear systems (Mauroy, 2021, Črnjarić-Žic et al., 2017).

3. Data-Driven Identification and Galerkin Approximation

Finite-dimensional projections of the continuous-time Koopman operator and its generator are crucial for practical computation. The Extended Dynamic Mode Decomposition (EDMD) and Galerkin approaches construct a projected generator $[0,\infty)$2 on a chosen basis $[0,\infty)$3: $[0,\infty)$4 Least-squares and regression strategies, enabled by densely sampled trajectories (snapshot pairs or segments), allow for direct construction of $[0,\infty)$5 and thus finite-dimensional linear models for nonlinear dynamics (Mauroy, 2021, Meng et al., 2024, Servadio et al., 2021, Johnson et al., 2017, Mazouchi et al., 2021). Kernel-based and occupation-kernel approaches in RKHSs expand this to nonparametric, high-dimensional, and trajectory-based settings (Rosenfeld et al., 2019, Valva et al., 2023, Valva et al., 2024).

Convergence theory shows that:

• As the dictionary and sample size increase, estimated eigenvalues and eigenfunctions converge pointwise or in operator norm under compactness/regularity assumptions (Mauroy, 2021, Rosenfeld et al., 2019, Zhen et al., 2022).
• Generator-based (continuous-time) DMD avoids the discretization artifacts and strong operator topology limitations inherent in classical, discrete-time DMD, yielding provably better norm-convergent approximations (Rosenfeld et al., 2019).

4. Spectral Approximation, Regularization, and Compactification

Analytical and computational spectral analysis of the generator $[0,\infty)$6 must address its typical unboundedness. Regularization and compactification techniques include:

• Bounded transforms: The Yosida resolvent, $[0,\infty)$7, or more generally, functional transforms (e.g. $[0,\infty)$8) produce bounded, compact, and skew-adjoint operators whose finite-dimensional spectra approximate that of $[0,\infty)$9 in a suitable limit (Valva et al., 2024, Valva et al., 2023).
• Kernel smoothing: Application of Markov semigroups or kernel integral operators $\nu$0 to the generator yields compact operators $\nu$1 admitting standard spectral decomposition, with convergence proven as $\nu$2 and $\nu$3 (Valva et al., 2024, Valva et al., 2023).
• Finite-rank compressions: Galerkin and SILL-type projections (e.g., via orthogonal polynomials or logistic dictionaries) enable practical system identification with explicit error bounds, and the capacity to drive dictionary and closure errors arbitrarily small (Zhen et al., 2022, Servadio et al., 2021, Johnson et al., 2017).

Theoretical results guarantee that these approximations preserve important operator-theoretic and group/semigroup invariances and have well-controlled convergence to the true spectrum and dynamics (Rosenfeld et al., 2019, Valva et al., 2023).

5. Extensions: Nonautonomous, Random, and Latent Linearized Koopman Frameworks

The continuous-time Koopman framework generalizes naturally to nonautonomous and stochastic settings:

• Nonautonomous linear systems: The Koopman operator family $\nu$4 is parameterized by $\nu$5, with spectral properties determined through instantaneous or local propagators (fundamental matrices), and computed by sliding-window Arnoldi/DMD methods. Correct assignment of observables or local transformation is required for spectral accuracy due to continuous-time biases (Maćešić et al., 2017).
• Random dynamical systems (RDS): The stochastic Koopman semigroup $\nu$6 is governed by the generator derived from the mean vector field or Kolmogorov operator, and its spectral properties admit explicit forms in linear cases, along with robust numerical approximation via sHankel-DMD (Črnjarić-Žic et al., 2017).
• Latent linear and diffeomorphic flows: Machine learning approaches construct invertible (diffeomorphic) coordinate lifts and ensure linear evolution in the latent space (e.g., $\nu$7), realized with neural networks, monomial bases, and loss functions ensuring stability and conjugacy to the nonlinear flow (Bevanda et al., 2021, Grozavescu et al., 2 Feb 2026).

6. Applications and Implications in Data-Driven Dynamical Systems

Continuous-time Koopman operator techniques justify and unify a range of data-driven spectral methods:

• Time series and SSA: Temporal autocorrelation functions, Hilbert-Schmidt operators, and singular spectrum analysis yield mode energies and frequencies consistent with pure-point Koopman spectra in the long-window limit, with rigorous convergence results (Zhen et al., 2022).
• System identification and control: Lifted linear generators inferred from data enable the identification of vector fields, derivation of Lyapunov functions, and prediction of nonlinear PDE dynamics, with explicit guarantees and robustness to noise (Mauroy, 2021, Meng et al., 2024, Mazouchi et al., 2021).
• Kernel and occupation kernel methods: Compact, spectral-resolvent-based approximations provide out-of-sample evaluation, spectral clustering of coherent observables, and consistency for both integrable and chaotic systems (Valva et al., 2024, Valva et al., 2023, Rosenfeld et al., 2019).

These frameworks enable operator-theoretic and statistical approaches to nonlinear dynamics from ergodic theory, spectral decomposition, to modern machine learning and control, in both theoretical and algorithmic dimensions.