Finite-Dimensional Koopman Representation

• Finite-dimensional Koopman representation is a method that lifts nonlinear dynamics to a finite set of observables evolving under a linear operator, enabling precise state reconstruction.
• It uses Koopman eigenfunctions and invariant-subspace concepts to achieve exact closures in structured systems or generate approximate models in general nonlinear cases.
• The framework extends to controlled systems, delay-coordinate embeddings, and learning-based approximations, offering insights into error quantification and structural limitations.

Searching arXiv for recent and foundational papers on finite-dimensional Koopman representations. Searching arXiv for exact finite-dimensional Koopman embeddings and controlled/input-driven extensions. Searching arXiv for delay-coordinate, RKHS, and learning-based finite-dimensional Koopman approximations. A finite-dimensional Koopman representation is a lifted model of nonlinear dynamics in which a finite vector of observables evolves under a linear—or, for input-driven variants, structured linear parameter-varying or bilinear—state transition. For a discrete-time system $x^+=F(x)$, the Koopman operator acts on observables $g$ by $\mathcal K g=g\circ F$; for a continuous-time system $\dot x=f(x)$, the Koopman semigroup is $\mathcal K^t g=g\circ x(t;\cdot)$ with generator $\mathcal L g=\langle \nabla g,f\rangle$. The finite-dimensional problem is to find a lifting $z=\Phi(x)$ and matrices such that $z^+=Az$ or $\dot z=Az$, often together with a reconstruction $x=Cz$. Exact realizations require a finite Koopman-invariant subspace; otherwise one obtains projected, truncated, or learned approximations of an intrinsically infinite-dimensional operator (Mezić et al., 2 Jul 2026).

1. Operator-theoretic formulation

The classical finite-dimensional setting starts with a family of observables $g$0 spanning a subspace $g$1. If $g$2 is Koopman-invariant, $g$3, then the restriction of the Koopman operator to $g$4 is represented by a finite matrix. Writing $g$5, one has $g$6 in discrete time or $g$7 in continuous time. When the state itself is included among the observables, the original nonlinear dynamics can be recovered by a linear projection $g$8, so the lifted linear model is simultaneously predictive and state-reconstructive (Mezić et al., 2 Jul 2026).

Koopman eigenfunctions furnish the canonical coordinates for such representations. If $g$9, then a diagonal representation arises immediately; if generalized eigenfunctions are required, Jordan blocks appear instead. In the diagonalizable case, any finite-dimensional linear Koopman representation is conjugate to one generated by eigenfunctions. This connects the existence of a finite-dimensional representation directly to the spectral structure of the composition operator rather than to local tangent linearization (Mezic, 2020).

The geometry of the state space is encoded by level sets of eigenfunctions. If $\mathcal K g=g\circ F$0 is an eigenfunction, its level sets are mapped into one another by the flow, and joint level sets of several eigenfunctions define invariant partitions. A representation is faithful precisely when the joint level sets of the chosen observables separate points. This geometric viewpoint clarifies why finite-dimensional Koopman coordinates are more than a feature map: they organize invariant sets, stable and unstable structures, and equivalence classes under the dynamics (Mezic, 2020).

A broader usage of the term includes finite-dimensional nonlinear autonomous closure maps of the form $\mathcal K g=g\circ F$1, sometimes described as a representation eigenproblem. The linear invariant-subspace picture remains the central case in Koopman-based prediction and control, but this broader formulation is useful for learned latent dynamics in which closure is finite-dimensional without being strictly linear (Mezic, 2020).

2. Existence, obstructions, and structural conditions

Exact finite-dimensional Koopman representations are exceptional rather than generic. A state-inclusive invariant subspace is especially restrictive. One precise obstruction is topological: systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and therefore cannot admit a finite-dimensional linear Koopman subspace that includes the state globally. By contrast, systems with a single isolated fixed point may admit state-inclusive closures, particularly in special polynomial classes (Brunton et al., 2015).

The Koopman spectrum imposes a second obstruction. Point spectrum supports finite-dimensional linear closure, whereas continuous spectral components obstruct conjugacy to linear finite-dimensional dynamics. In the terminology of representation theory for Koopman systems, a finite-dimensional representation is not conjugate to a linear one when the chosen observables have nontrivial overlap with the continuous-spectrum component. This identifies a precise spectral boundary between exact finite linear closure and intrinsically non-finite or nonlinearly representable behavior (Mezic, 2020).

For controlled nonlinear systems, recent work gives a sharp characterization of exact finite-dimensional Koopman linear embeddings. For the discrete-time system

$\mathcal K g=g\circ F$2

an exact embedding requires a lifting $\mathcal K g=g\circ F$3 and constant matrices $\mathcal K g=g\circ F$4 such that

$\mathcal K g=g\circ F$5

This exists if and only if the system can be transformed into a control-affine preserved (CAP) structure and the autonomous nonlinear subsystem admits a finite-dimensional Koopman closure that contains the state and all nonlinear terms needed for reconstruction. In this sense, the source of approximation error can be separated into an intrinsic structural limitation—absence of CAP—and a lifting limitation—failure to choose a sufficiently rich invariant subspace when CAP does exist (Shang et al., 16 Feb 2026).

Even when the autonomous part admits a finite-dimensional lift, systems with inputs generally do not yield an exact lifted LTI model. For continuous-time systems the lifted input term is typically state-dependent; for discrete-time systems it is generally state- and input-dependent. The resulting exact finite-dimensional form is therefore an LPV Koopman model,

$\mathcal K g=g\circ F$6

rather than $\mathcal K g=g\circ F$7 with constant $\mathcal K g=g\circ F$8. Constant-input-matrix LTI Koopman models remain approximations whose error depends on the deviation of the true input matrix from the chosen constant approximation (Iacob et al., 2022).

3. Constructive exact embeddings for special system classes

Although general existence is rare, several structured nonlinear classes admit constructive exact finite-dimensional embeddings. A basic example is the polynomial slow-manifold family studied through state-inclusive observables. For systems such as

$\mathcal K g=g\circ F$9

with polynomial $\dot x=f(x)$0, the observables formed by the state and the monomials appearing in $\dot x=f(x)$1 close exactly under differentiation. This produces a finite-dimensional linear lifted system whose eigenfunctions recover intrinsic coordinates on the invariant manifold (Brunton et al., 2015).

A more systematic constructive result is available for autonomous continuous-time lower-triangular polynomial systems,

$\dot x=f(x)$2

where each $\dot x=f(x)$3 is polynomial. For this class there exists an exact finite-dimensional lifting $\dot x=f(x)$4 containing the state such that

$\dot x=f(x)$5

The construction is recursive: one augments the state with all monomials generated under differentiation until closure is reached, and finite termination follows from the lower-triangular structure and finite polynomial degree. In the worked 4-state example, the resulting lifted state has dimension $\dot x=f(x)$6, and the trajectories of the nonlinear and lifted linear systems overlap exactly up to numerical error on the order of $\dot x=f(x)$7 (Iacob et al., 2023).

An even broader exact theory covers block-oriented polynomial systems assembled from series and parallel interconnections of LTI blocks and static polynomial nonlinearities. For these systems one can build a finite lifted state from Kronecker powers and obtain an exact polynomial input time-invariant Koopman form,

$\dot x=f(x)$8

with constant $\dot x=f(x)$9 and $\mathcal K^t g=g\circ x(t;\cdot)$0, $\mathcal K^t g=g\circ x(t;\cdot)$1 linear in $\mathcal K^t g=g\circ x(t;\cdot)$2, and $\mathcal K^t g=g\circ x(t;\cdot)$3 polynomial in $\mathcal K^t g=g\circ x(t;\cdot)$4. If the LTI blocks have no feedthrough, this simplifies to a bilinear time-invariant model. In the reported examples, duplicate monomials can be removed by projection, reducing a 17-dimensional raw lift to 12 dimensions in one case and a 931-dimensional raw lift to 103 dimensions in another, with output errors around $\mathcal K^t g=g\circ x(t;\cdot)$5, i.e. machine precision (Iacob et al., 20 Jul 2025).

These constructive results identify a common mechanism behind exact finite-dimensionality: closure is achieved when the nonlinearities belong to a finitely generated algebra that remains closed under the Koopman generator. Triangularity, polynomial degree bounds, and block-oriented acyclic interconnection are specific structural realizations of that mechanism.

4. Approximate finite-dimensional models from data and learning

Outside such special classes, finite-dimensional Koopman representations are typically approximate. The standard approximation is EDMD. Given snapshot pairs $\mathcal K^t g=g\circ x(t;\cdot)$6 with $\mathcal K^t g=g\circ x(t;\cdot)$7 and dictionary $\mathcal K^t g=g\circ x(t;\cdot)$8, EDMD computes

$\mathcal K^t g=g\circ x(t;\cdot)$9

as a least-squares approximation of the projected Koopman operator on the dictionary span. In continuous time, generator EDMD uses

$\mathcal L g=\langle \nabla g,f\rangle$0

The accuracy of such a model depends on both finite-sample estimation and the non-invariance of the chosen dictionary. This latter effect is quantified by the invariance proximity

$\mathcal L g=\langle \nabla g,f\rangle$1

which vanishes only for an exactly invariant subspace (Mezić et al., 2 Jul 2026).

Kernel and RKHS formulations replace a fixed dictionary by the finite span of kernel sections centered at the data. In this setting, DMD and EDMD become dual value-based and function-based approximations, and the learned finite-dimensional subspace is shaped by the sample locations. The regularized kernel Koopman estimator can be written entirely in terms of Gram matrices, for example

$\mathcal L g=\langle \nabla g,f\rangle$2

so the infinite-dimensional operator is approximated on a data-adaptive finite span without prescribing explicit basis functions a priori (Zanini et al., 2021).

A closely related but conceptually sharper result is the representer theorem for Koopman learning. In the discrete-time autonomous setting, regularized empirical risk minimization over the infinite-dimensional operator space $\mathcal L g=\langle \nabla g,f\rangle$3 admits an exact finite-rank optimizer of the form

$\mathcal L g=\langle \nabla g,f\rangle$4

under broad assumptions on the observables and regularizer. This yields a finite-dimensional optimization problem with no approximation or loss of precision at the optimization level, even though the learned operator still models nonlinear dynamics through finite data and finite observables (Khosravi, 2022).

Recent machine-learning approaches focus on learning nearly invariant subspaces rather than fixing them. One line learns locally supported, approximately orthogonal basis functions $\mathcal L g=\langle \nabla g,f\rangle$5 and a latent linear map $\mathcal L g=\langle \nabla g,f\rangle$6, producing

$\mathcal L g=\langle \nabla g,f\rangle$7

as a finite-dimensional approximation of the transfer or Koopman operator. Reconstruction of both $\mathcal L g=\langle \nabla g,f\rangle$8 and $\mathcal L g=\langle \nabla g,f\rangle$9, together with an $z=\Phi(x)$0-based local-support penalty, encourages the learned span to be nearly invariant and spectrally informative (Froyland et al., 8 May 2025). Another line, Koopmanizing Flows, assumes a nonlinear system is diffeomorphically related to a latent stable linear system $z=\Phi(x)$1, lifts $z=\Phi(x)$2 through a truncated monomial basis of order $z=\Phi(x)$3, and learns the diffeomorphism, linear dynamics, and decoder jointly; the finite feature dimension is

$z=\Phi(x)$4

The use of a Hurwitz parameterization guarantees asymptotic stability regardless of operator approximation accuracy, and on the LASA handwriting benchmark the method outperforms SKEL on DTWD, RMSE, and PCM (Bevanda et al., 2021).

Approximate closure can also be engineered through specialized dictionaries. State Inclusive Logistic Lifting augments the state with multivariate conjunctive logistic functions,

$z=\Phi(x)$5

to obtain finite approximate closure under differentiation. Its error is governed by the steepness parameter $z=\Phi(x)$6 and the mesh parameter $z=\Phi(x)$7, with improved closure as the logistic functions become sharper and the dictionary resolves the vector field more densely (Johnson et al., 2017). Information-theoretic latent models add a different perspective: they treat Koopman learning as an information bottleneck problem and use latent mutual information together with von Neumann entropy to avoid collapse onto a few dominant modes, thereby preserving a finite-dimensional latent Koopman subspace that remains predictive over long horizons (Cheng et al., 14 Oct 2025).

5. Delay coordinates, infinite-dimensional systems, and related operator variants

Time-delay observables provide a distinct route to finite-dimensional Koopman coordinates. Instead of selecting instantaneous observables, one embeds trajectory segments and projects them onto an orthonormal basis on a delay window. The resulting convolutional coordinates

$z=\Phi(x)$8

obey a universal generator representation

$z=\Phi(x)$9

where the matrix depends only on the chosen delay basis, not on the underlying dynamics. When the measured observable has a finite Koopman mode decomposition of order $z^+=Az$0, the first $z^+=Az$1 SVD delay coordinates span an invariant subspace and form an optimal finite-dimensional basis; the induced linear model coincides with DMD computed in those coordinates (Kamb et al., 2018).

For nonlinear infinite-dimensional systems such as PDEs, the Koopman semigroup can be defined on bounded continuous functionals, then projected onto a finite-dimensional space of functionals $z^+=Az$2. With basis $z^+=Az$3, one obtains a compressed semigroup $z^+=Az$4 and Lie generator $z^+=Az$5, and the data-driven approximation remains EDMD-like: $z^+=Az$6 The associated matrix logarithm

$z^+=Az$7

can be used both for spectral approximation and for identification of PDE coefficients through the first column of the generator estimate (Mauroy, 2021).

A further extension replaces observables or densities by wavefunctions. In the Koopman–von Neumann framework, the generator is

$z^+=Az$8

and Galerkin projection onto a finite basis yields

$z^+=Az$9

After whitening the basis, the finite-dimensional generator becomes skew-symmetric, so its exponential is unitary. This produces finite matrices suitable for spectral computation and, in principle, quantum-circuit realization, even for non-Hamiltonian dynamics (Klus et al., 9 Apr 2026).

The phrase “Koopman representation” also appears in operator-algebraic settings unrelated to state lifting. For self-similar groupoid actions on $\dot z=Az$0, the Koopman representation is generally infinite-dimensional, but it preserves a filtration by finite-dimensional cylinder-function spaces

$\dot z=Az$1

The resulting $\dot z=Az$2-algebra is residually finite-dimensional, meaning that finite-dimensional representations separate points even though the original Koopman representation is not itself finite-dimensional (Deaconu, 2021).

6. Exactness, approximation error, and unresolved issues

A recurrent misconception is that finite-dimensional Koopman modeling is principally a matter of enlarging the dictionary. The literature instead shows a structural dichotomy. If an exact invariant subspace exists, enlarging the observable set can help recover it. If the system lacks the requisite structure, no observable design can remove the intrinsic approximation error. In controlled systems this distinction is explicit: absence of CAP rules out exact finite-dimensional Koopman linear embeddings, whereas existence of CAP shifts the difficulty to finding a sufficiently rich lifting (Shang et al., 16 Feb 2026).

A second misconception is that a richer dictionary is automatically superior. In projection-based methods, larger spaces can reduce representation error while worsening non-invariance error and long-horizon drift. The core tradeoff is not only expressivity but also closure. This is why several recent methods optimize invariance explicitly, either by learning dynamically adapted bases, enforcing latent linearity with stable parameterizations, or regularizing against mode collapse in the latent covariance spectrum (Mezić et al., 2 Jul 2026).

The exact-versus-approximate distinction is especially important for systems with inputs. Much of the applied literature assumes a lifted LTI model $\dot z=Az$3, but the exact theory shows that even when the autonomous part is finitely representable, the input channel typically produces LPV or bilinear structure rather than a constant $\dot z=Az$4. A plausible implication is that model structure selection—LTI versus LPV versus bilinear—is not a secondary identification choice but a consequence of the nonlinear input mechanism (Iacob et al., 2022).

Finally, the literature contains claims of very strong general exactness that require careful scrutiny. The abstract of “Functional Dimensionality of Koopman Eigenfunction Space” states that the general form solution of the Koopman Partial Differential Equation is obtained, that the functional dimensionality of the Koopman eigenfunction space is finite and equal to the dimensionality of the dynamics, and that nonlinear dynamics therefore admit an error-free finite set of Koopman eigenfunctions together with a simple numerical method (Cohen et al., 2024). However, the supplied document text contains only template structure—title, metadata, theorem environments, and empty section headings—and does not include definitions, propositions, equations, derivations, flowbox statements, or algorithms. From the available text, the claimed theorem, its assumptions, and its numerical consequences cannot be assessed (Cohen et al., 2024).

In current usage, finite-dimensional Koopman representation therefore denotes a family of closely related but technically distinct constructions: exact invariant-subspace realizations for special structural classes, exact optimization-level reductions for certain learning problems, and approximate projected or learned latent models for generic nonlinear systems. The central question is not whether the Koopman operator is linear—it always is—but whether the nonlinear dynamics admit a finite observable algebra that closes under that linear action, and if not, how the residual non-closure should be quantified and controlled.