Generalized Koopman Embedding

• Generalized Koopman embedding is a framework that lifts nonlinear state and input dynamics into higher-dimensional spaces to achieve linear or bilinear representations.
• It employs structured designs, learned control encodings, and probabilistic methods to facilitate tractable prediction, spectral analysis, and control of complex systems.
• By reformulating control via LPV, coordinate-aware, and product-space models, the approach enables robust system identification and improved closed-loop performance.

Generalized Koopman embedding denotes a family of extensions of the classical Koopman framework in which nonlinear dynamics are represented through linear, bilinear, LPV, or spectrally regularized evolution in lifted coordinates that are not restricted to fixed state-only observables of autonomous systems. In the recent literature, this generalization appears through learned control encodings of the form $z_{k+1}=Az_k+B\,g_\phi(x_k,u_k)$, product-Hilbert-space operators $K:H_x\otimes H_u\to H_x$ with $z_{t+1}=K(z_t\otimes v_t)$, physics-structured coordinate changes such as momentum coordinates for Euler–Lagrangian systems, probabilistic families of Koopman-equivariant observables, and regularized generator eigenfunctions for systems with continuous spectrum (Shi et al., 2022, Lazar, 10 Aug 2025, Singh et al., 21 Sep 2025, Bevanda et al., 10 Feb 2025, Valva et al., 2024). The unifying objective is to obtain observables, latent coordinates, or functional representations in which nonlinear evolution becomes more tractable for prediction, control, spectral analysis, or identification.

1. From autonomous lifting to controlled and non-autonomous formulations

Classical Koopman theory treats autonomous dynamics by seeking observables whose evolution is linear under composition with the flow. For systems with inputs, that picture changes substantially. The paper “Koopman Form of Nonlinear Systems with Inputs” shows that, in both continuous and discrete time, the exact lifted representation of a nonlinear input-driven system generally has a linear state-transition part but a nonconstant input matrix. In continuous time, the lifted input term is generally state-dependent; in discrete time, it is generally state-and-input-dependent even for control-affine or input-linear systems. This yields an LPV Koopman form rather than an exact LTI model with constant $B$, and bilinear lifted models arise only under additional closure conditions on the lifted input vector fields (Iacob et al., 2022).

A different route is to generalize the control channel itself. “Deep Koopman Operator with Control for Nonlinear Systems” replaces the standard lifted control term $Bu_k$ by $B\,g_\phi(x_k,u_k)$, so that the learned dynamics remain linear in lifted state and encoded control even when the original system has state-dependent or more general nonlinear control dependence. In that framework, the lifted state is explicitly state-preserving, $z_k=[x_k;\,g_\theta(x_k)]$, the decoder is the fixed projection $x_k=Cz_k$, and the control channel can be instantiated as the linear-input baseline DKUC, the control-affine/state-dependent encoding DKAC, or the fully nonlinear encoding DKN. This broadens the class of controlled systems addressable by Koopman-inspired linear control, but it also introduces an inversion issue: DKN gives the best prediction performance, yet the paper does not use it for control because a unique inverse $g_\phi^{-1}$ is not available in practice (Shi et al., 2022).

2. Product spaces, hidden lifted behavior, and trajectory-space formulations

One line of generalization treats controlled Koopman models not as a modification of an autonomous state lift, but as an operator on a joint state-input function space. “From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma” constructs $H_x=L^2(X,\mu_x)$, $K:H_x\otimes H_u\to H_x$0, and the tensor-product space $K:H_x\otimes H_u\to H_x$1. In that setting, a nonlinear controlled system $K:H_x\otimes H_u\to H_x$2 admits an infinite-dimensional linear operator $K:H_x\otimes H_u\to H_x$3 such that

$K:H_x\otimes H_u\to H_x$4

where $K:H_x\otimes H_u\to H_x$5 and $K:H_x\otimes H_u\to H_x$6. The resulting lifted dynamics are linear as an operator equation and bilinear as a state-input evolution law, and the same construction underlies a nonlinear fundamental lemma derived from the bilinear structure of the lifted system (Lazar, 10 Aug 2025).

A complementary behavioral perspective appears in “Willems’ Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding”. There the starting assumption is an exact finite-dimensional controlled Koopman embedding,

$K:H_x\otimes H_u\to H_x$7

for a nonlinear discrete-time system $K:H_x\otimes H_u\to H_x$8, $K:H_x\otimes H_u\to H_x$9. The key result is that sufficiently rich nonlinear trajectories span the behavior of the hidden lifted linear system, so that prediction and control can be carried out directly from trajectory libraries without explicitly constructing the lifting functions. This reformulates generalized Koopman embedding as a trajectory-space characterization problem rather than an observable-design problem, but exactness is tied to a strong premise: the nonlinear system must admit an exact finite-dimensional Koopman linear embedding and the data must satisfy a lifted-excitation condition (Shang et al., 2024).

The same exact-embedding premise is used in “Online Tracking with Predictions for Nonlinear Systems with Koopman Linear Embedding”. There, the nonlinear online tracking problem is shown to be exactly equivalent to a lifted linear-quadratic tracking problem when a finite-dimensional embedding $z_{t+1}=K(z_t\otimes v_t)$0, $z_{t+1}=K(z_t\otimes v_t)$1, $z_{t+1}=K(z_t\otimes v_t)$2 exists. On that basis, the paper proves that the cumulative cost and dynamic regret of the nonlinear problem coincide with those of the lifted linear counterpart, and implements a model-free predictive controller via Willems’ fundamental lemma in the lifted setting (Pai et al., 8 Mar 2026).

3. Structure-aware and coordinate-aware exact embeddings

A major branch of generalized Koopman embedding replaces generic dictionary learning by constructive, physics- or structure-aware lifting. “Finite Dimensional Koopman Form of Polynomial Nonlinear Systems” proves that autonomous continuous-time lower-triangular polynomial systems of the form

$z_{t+1}=K(z_t\otimes v_t)$3

admit exact finite-dimensional Koopman liftings built recursively from monomials that close under differentiation. The lifted coordinates include the original state, the lifted dynamics are exactly linear, and the construction yields a finite invariant observable dictionary rather than a truncated Carleman expansion (Iacob et al., 2023).

The same constructive philosophy is extended to input-output systems in “Exact Finite Koopman Embedding of Block-Oriented Polynomial Systems”. For continuous-time block-oriented polynomial systems built from series and parallel interconnections of LTI blocks and static polynomial nonlinearities, the paper derives an exact finite-dimensional PITI Koopman representation,

$z_{t+1}=K(z_t\otimes v_t)$4

with constant state and output matrices and polynomial input dependence. When every LTI block has no feedthrough and the block chain does not begin with a static nonlinearity, this simplifies to an exact BLTI Koopman representation. The result is explicitly constructive and exact, but it is restricted to continuous-time, feedback-free, block-oriented polynomial systems (Iacob et al., 20 Jul 2025).

A more physically targeted coordinate redesign appears in “Generalized Momenta-Based Koopman Formalism for Robust Control of Euler-Lagrangian Systems”. That paper argues that, for Euler–Lagrangian systems, the apparent need for a bilinear Koopman model is often a consequence of using explicit coordinates $z_{t+1}=K(z_t\otimes v_t)$5. By switching to the implicit momentum state

$z_{t+1}=K(z_t\otimes v_t)$6

the control enters additively and only in the momentum channel, so the learned model can focus on passive nonlinear dynamics while using a fixed actuation matrix $z_{t+1}=K(z_t\otimes v_t)$7. The paper reports that only the passive operator $z_{t+1}=K(z_t\otimes v_t)$8 must be learned, the actuation matrix is not learned, and the number of learnable parameters is reduced by $z_{t+1}=K(z_t\otimes v_t)$9 relative to a conventional linear explicit-state Koopman model and by $B$0 relative to a bilinear Koopman formulation. This makes generalized Koopman embedding depend not only on observable choice but also on the physical state representation selected before lifting (Singh et al., 21 Sep 2025).

4. Learned, control-aware, and adaptive embeddings

Neural parameterizations have turned generalized Koopman embedding into a representation-learning problem. In “Control-aware Learning of Koopman Embedding Models”, the lifted state is

$B$1

with a fixed decoder $B$2. The paper shows that good state prediction does not imply good closed-loop control, because residual error in latent coordinates may be harmless for one-step state prediction yet accumulate under feedback. To address this, it proposes a second control-aware refinement stage that keeps the learned observable map fixed and updates $B$3 and $B$4 using closed-loop data, together with a deterministic continuous-function input-sampling strategy that better matches controller-generated actuation (Uchida et al., 2022).

“Adaptive Koopman Embedding for Robust Control of Complex Nonlinear Dynamical Systems” goes further by making the lifted model itself adaptive online. The framework first learns a nominal autoencoder-based Koopman architecture offline, then augments it with a feed-forward network that modifies the nominal lifted dynamics in response to mismatch between predicted and observed lifted states. The intended effect is improved generalization and robustness under measurement noise, disturbances, and parametric variations, while retaining the linear MPC workflow built around the nominal embedding (Singh et al., 2024).

A different response to model-selection difficulty is the two-stage hybrid strategy in “A Hybrid Framework for Efficient Koopman Operator Learning”. That work uses an SDP-based structural inference stage on a subset of data to estimate latent dimension, spectral composition, and memory/order, then trains a neural encoder-decoder and auxiliary eigenvalue networks using that inferred structure. The resulting embedding is neither purely hand-designed nor purely free-form neural; it is a learned latent linear representation whose architecture is shaped by convexly inferred Koopman structure, including delay-embedded state vectors $B$5 when memory is required (Estornell et al., 25 Apr 2025).

A continuous-time and uncertainty-aware variant appears in “Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability”. There the embedding is learned either in differential form, using the Koopman-generator condition $B$6, or in recurrent form, using $B$7. The architecture embeds an SVD-DMD branch into the autoencoder so that the neural network learns only the residual beyond the linear DMD embedding, and the continuous-time generator is constrained by a structural parameterization whose eigenvalues have non-positive real part. Mean-field variational inference is then used to quantify uncertainty in the learned embedding, operator, and predictions (Pan et al., 2019).

5. Probabilistic, spectral, and infinite-dimensional generalizations

Generalized Koopman embedding is not confined to deterministic finite-dimensional latent states. “Koopman-Equivariant Gaussian Processes” develops a probabilistic representation-learning framework in which trajectories are linear time-invariant in time but nonlinear in the initial condition. Instead of learning a deterministic encoder-decoder pair, the paper places Gaussian process priors over Koopman-style eigenfeatures and projects them into Koopman-equivariant subspaces using a trajectory-based symmetrization operator. The resulting kernel yields closed-form predictive distributions over future trajectories and an uncertainty-aware distribution over Koopman-like observables, so the embedding is probabilistic, nonparametric, and induced from past trajectory segments rather than only instantaneous states (Bevanda et al., 10 Feb 2025).

“Physics-informed spectral approximation of Koopman operators” generalizes spectral embeddings to continuous-time, measure-preserving ergodic systems with continuous spectrum. It applies a bounded transform to the Koopman generator, smooths it by a Markov semigroup of kernel integral operators, and obtains a skew-adjoint compact operator family whose eigendecomposition is computed from a variational generalized eigenvalue problem. The resulting eigenfunctions are not generally exact classical Koopman eigenfunctions of the original system, but they are regularized spectral coordinates that still evolve linearly under an approximate unitary dynamics and admit RKHS representatives for out-of-sample evaluation (Valva et al., 2024).

At the level of infinite-dimensional state spaces, “Koopman operator framework for spectral analysis and identification of infinite-dimensional systems” formulates Koopman theory directly for nonlinear dynamics on separable Hilbert spaces, such as PDEs. The Koopman semigroup acts on bounded continuous functionals, the Lie generator is defined pointwise by

$B$8

and finite-dimensional projections yield EDMD-like approximations built from functionals rather than only pointwise state coordinates. In that setting, generalized Koopman embedding becomes a lifting of field dynamics into a linear evolution of observables on function space, supporting both spectral analysis and identification of PDE coefficients (Mauroy, 2021).

A more geometric generalization appears in “Koopman Regularization”, which seeks a minimal functionally independent set of Koopman eigenfunctions, or equivalently unit velocity measurements, for continuous-time vector fields. The key idea is that a functionally independent set of Koopman coordinates provides an intrinsic embedding in which the dynamics are diagonal and from which the original vector field can be reconstructed through Jacobian inversion. The method turns the Koopman PDE into an optimization objective and functional independence into a feasibility condition, shifting attention from large redundant dictionaries to minimal coordinate systems (Cohen, 2024).

6. Control implications, common misconceptions, and limits

Several points in this literature are frequently misstated. First, exact Koopman models with inputs are not generically LTI models with constant $B$9. For broad classes of nonlinear systems, the exact lifted form is LPV, with state-dependent input matrices in continuous time and state-and-input-dependent input matrices in discrete time. Treating $Bu_k$0 as an exact universal target is therefore a modeling choice, not a theorem (Iacob et al., 2022).

Second, bilinear structure is not always intrinsic to the plant. The momentum-based Euler–Lagrangian construction shows that what appears bilinear in explicit $Bu_k$1 coordinates can become linear-in-control after a physically structured coordinate change to $Bu_k$2. This suggests that the distinction between linear and bilinear Koopman models may depend as much on state representation as on the underlying physics (Singh et al., 21 Sep 2025).

Third, predictive superiority does not automatically translate into a complete control pipeline. In the deep-control-encoding framework, DKN gives the best prediction performance, but the paper explicitly does not use it for control because the inverse map $Bu_k$3 is not uniquely available. A similar caution applies to control-aware learning more broadly: latent linearity that is adequate for one-step prediction may still generate poor feedback behavior if residual error propagates unfavorably in closed loop (Shi et al., 2022, Uchida et al., 2022).

Fourth, exact data-driven behavioral results rely on exact embedding assumptions. The nonlinear fundamental-lemma results and the lifted-regret equivalence in online tracking are rigorous only for systems that admit exact finite-dimensional Koopman linear embeddings and for data satisfying lifted-excitation conditions. The same papers are explicit that approximate embeddings, noise, imperfect predictions, and generic nonlinear systems remain outside their main theory (Shang et al., 2024, Pai et al., 8 Mar 2026).

Finally, exact finite-dimensional generalized Koopman embeddings are currently available only for structured classes. Lower-triangular polynomial systems, block-oriented polynomial systems, and Euler–Lagrangian systems with momentum coordinates are positive cases, but none of these results establishes a general theorem for arbitrary nonlinear controlled dynamics. A plausible implication is that “generalized Koopman embedding” should be understood less as a single canonical construction than as a design principle: choose observables, coordinates, operator domains, and regularizations so that the relevant nonlinear evolution becomes as linear, finite, and control-compatible as the system structure allows (Iacob et al., 2023, Iacob et al., 20 Jul 2025, Singh et al., 21 Sep 2025).