Generator Extended DMD (gEDMD)

• Generator Extended Dynamic Mode Decomposition (gEDMD) is a framework that directly approximates the infinitesimal Koopman generator to capture continuous-time dynamics.
• It employs Galerkin projection with smooth basis functions to compute spectral elements such as eigenvalues, eigenfunctions, and modes for system identification and control.
• gEDMD supports both deterministic and stochastic systems, offering convergence guarantees and finite-sample error bounds to ensure robust data-driven modeling.

Searching arXiv for recent and foundational papers on generator extended dynamic mode decomposition to support a comprehensive article. arxiv_search(query="generator extended dynamic mode decomposition Koopman generator gEDMD", max_results=10) Generator Extended Dynamic Mode Decomposition (gEDMD) is a data-driven framework for approximating the infinitesimal Koopman generator rather than a finite-time Koopman operator. In this respect it is a generator-level analogue of Extended Dynamic Mode Decomposition (EDMD): instead of learning how observables evolve over a prescribed lag time, it learns their infinitesimal time derivative from data. The method applies to deterministic and stochastic dynamical systems, supports computation of eigenvalues, eigenfunctions, and modes, and has been used for system identification, coarse-graining, and model predictive control (Klus et al., 2019). A complementary operator-theoretic view casts gEDMD as a Monte Carlo/Galerkin approximation of a linear operator on a finite-dimensional dictionary space, with convergence results and explicit finite-sample error bounds (Llamazares-Elias et al., 2024).

1. Operator-theoretic setting

For a deterministic ODE

$\dot{x} = b(x),$

the Koopman semigroup $\{\mathcal K^t\}$ acts on observables $f$ by

$(\mathcal K^t f)(x) = f(\Phi^t(x)),$

where $\Phi^t$ is the flow map. Its infinitesimal generator is

$$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$

and for differentiable observables,

$\mathcal L f = b\cdot \nabla_x f.$

For an Itô SDE

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$

the Koopman operator becomes a conditional-expectation propagator,

$(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$

and the generator is

$$\mathcal L f = b\cdot \nabla_x f + \frac12 a : \nabla_x^2 f,\qquad a=\sigma\sigma^\top.$$

These formulas are the basic objects gEDMD approximates (Klus et al., 2019).

This generator perspective is the defining distinction from EDMD. EDMD approximates the finite-time Koopman operator $\{\mathcal K^t\}$0 from snapshot pairs, whereas gEDMD approximates $\{\mathcal K^t\}$1 directly. In the unified framework of operator approximation, the algebraic estimator is similar in form, but the target operator differs:

$\{\mathcal K^t\}$2

This difference matters because the generator encodes continuous-time spectral information and avoids the matrix-logarithm step that would otherwise be required to infer a generator from a discrete-time Koopman approximation (Llamazares-Elias et al., 2024).

For interconnected systems, the generator viewpoint is particularly useful because it preserves linear dependence on couplings more cleanly than the discrete-time Koopman operator. In the modularized setting

$\{\mathcal K^t\}$3

with subsystem decomposition

$\{\mathcal K^t\}$4

the local generator admits an affine/bilinear decomposition with respect to neighbor-state components, which is the structural basis of modularized gEDMD (Guo et al., 2024).

2. Finite-dimensional approximation and regression form

gEDMD begins with a dictionary of sufficiently smooth basis functions

$\{\mathcal K^t\}$5

In Galerkin form, the projected generator is represented by matrices

$\{\mathcal K^t\}$6

so that

$\{\mathcal K^t\}$7

If $\{\mathcal K^t\}$8, then the projected generator acts as

$\{\mathcal K^t\}$9

This is the continuous-time analogue of the standard EDMD projection formula (Klus et al., 2019).

Empirically, with samples $f$0, one replaces the exact matrices by Monte Carlo estimates. In the notation of the unified approximation framework,

$f$1

and the data-driven approximation is

$f$2

Equivalently, gEDMD minimizes the empirical least-squares residual

$f$3

The paper on convergence rates interprets this estimator as the projection of the target operator onto the empirical subspace, not merely as a heuristic regression formula (Llamazares-Elias et al., 2024).

In the formulation of "Data-driven approximation of the Koopman generator: Model reduction, system identification, and control" (Klus et al., 2019), the same idea appears as

$f$4

for deterministic systems, or

$f$5

for stochastic systems. In the infinite-data limit, the empirical matrices converge to the Galerkin matrices, so gEDMD converges to the projection of the generator onto the chosen dictionary space (Klus et al., 2019).

Once $f$6 is available, approximate generator eigenpairs are obtained from

$f$7

The resulting $f$8 are approximate Koopman-generator eigenfunctions, and the associated time scales are written as

$f$9

This spectral interpretation is one of the main reasons gEDMD is used instead of a finite-lag EDMD approximation (Klus et al., 2019).

3. Deterministic and stochastic system identification

In deterministic settings, gEDMD can be used to reconstruct the vector field. For the full-state observable $(\mathcal K^t f)(x) = f(\Phi^t(x)),$0, one has

$(\mathcal K^t f)(x) = f(\Phi^t(x)),$1

and if $(\mathcal K^t f)(x) = f(\Phi^t(x)),$2, then

$(\mathcal K^t f)(x) = f(\Phi^t(x)),$3

This is the basis for the paper’s statement that, when the full-state observable is included in the dictionary, SINDy is a special case of gEDMD (Klus et al., 2019).

In stochastic settings, gEDMD identifies both drift and diffusion. Using quadratic observables $(\mathcal K^t f)(x) = f(\Phi^t(x)),$4, the generator identity

$(\mathcal K^t f)(x) = f(\Phi^t(x)),$5

implies

$(\mathcal K^t f)(x) = f(\Phi^t(x)),$6

Accordingly, gEDMD identifies $(\mathcal K^t f)(x) = f(\Phi^t(x)),$7, rather than $(\mathcal K^t f)(x) = f(\Phi^t(x)),$8 itself, from the generator action on quadratic observables (Klus et al., 2019).

A central practical difficulty is that naive finite-difference estimates of drift and diffusion are highly sensitive to noise. For SDE data, the increment-based approximations

$(\mathcal K^t f)(x) = f(\Phi^t(x)),$9

inject large variance into the estimated generator matrix. The paper "Improvement of system identification of stochastic systems via Koopman generator and locally weighted expectation" therefore advocates conditional expectations of Kramers–Moyal type,

$\Phi^t$0

$\Phi^t$1

approximated in practice by Gaussian-kernel-weighted local averages, and shows that an additional clustering step is necessary to preserve nonlinear local structure accurately (Tahara et al., 2024).

A more recent stochastic application uses gEDMD to derive an effective SDE for coarse-grained observables of deterministic laser chaos. There the target process is

$\Phi^t$2

with $\Phi^t$3, where $\Phi^t$4 and $\Phi^t$5 are short-term cross-correlations encoding leader–laggard synchronization. The paper reports that direct gEDMD on raw data does not yield a meaningful model, whereas low-pass filtered data retain the leader-bias peak shift and the switching-time peak structure needed for the reinforcement-learning task (Fukushi et al., 6 Jun 2025).

4. Observable design, dictionaries, and structural variants

The quality of gEDMD depends strongly on the chosen observable space. This dependence is already explicit in the generator formulation: basis functions must be sufficiently smooth, and their span must capture the action of $\Phi^t$6 without causing severe conditioning problems (Klus et al., 2019).

Several strands of work address this issue by changing how observables are constructed.

Approach	Central idea	Representative paper
Classical gEDMD	Project $\Phi^t$7 onto a chosen smooth dictionary	(Klus et al., 2019)
Analytical Lie-derivative construction	Lift nonlinear terms by Lie derivatives to obtain an exact polynomialized observable set	(Netto et al., 2020)
Modularized gEDMD	Learn subsystem generators on local state spaces and couple them algebraically	(Guo et al., 2024)
Kernel gEDMD	Replace hand-crafted bases by kernel feature spaces with whitening/truncation	(Nateghi et al., 2024)

The analytical construction proposed in "On analytical construction of observable functions in extended dynamic mode decomposition for nonlinear estimation and prediction" starts from a nonlinear model

$\Phi^t$8

identifies elementary nonlinearities $\Phi^t$9, introduces lifted variables $$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$0, and differentiates them via Lie derivatives,

$$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$1

By repeating this lifting step, the nonlinear dynamics are transformed into a higher-dimensional system that is polynomial in the lifted variables. Although this work is framed in terms of EDMD, it is strongly aligned with gEDMD because the observable set is constructed explicitly from Lie derivatives, i.e. from generator dynamics (Netto et al., 2020).

The modularized formulation for interconnected systems uses subsystem-local dictionaries

$$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$2

with coordinate maps included so that

$$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$3

For each subsystem, gEDMD estimates matrix representations of local generators $$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$4 and $$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$5, which are then assembled into the parameterized local model

$$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$6

This reduces dimensionality because observables are defined on subsystem state spaces rather than on the full network state (Guo et al., 2024).

Kernel-based variants replace hand-designed dictionaries by a feature map built from kernel evaluations. In the coarse-graining paper, the kernel feature map is

$$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$7

with Gaussian and periodic Gaussian kernels used in practice. Since the kernel Gram matrix is typically ill-conditioned, the method introduces whitening and truncation,

$$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$8

to obtain a stable reduced generator on a whitened feature basis (Nateghi et al., 2024).

5. Coarse-graining, reduced models, and control

A major application area for gEDMD is coarse-graining of high-dimensional stochastic dynamics. Given a coarse-graining map

$$\mathcal L f = \lim_{t\to 0}\frac{1}{t}\big(\mathcal K^t f - f\big),$$9

the coarse-grained generator is defined as

$\mathcal L f = b\cdot \nabla_x f.$0

where $\mathcal L f = b\cdot \nabla_x f.$1 denotes orthogonal projection onto functions of the reduced variable. In the reversible setting, the reduced dynamics again has the form of an SDE with effective drift and diffusion, and gEDMD applied to observables of the form $\mathcal L f = b\cdot \nabla_x f.$2 converges, in the infinite-data limit, to the Galerkin approximation of $\mathcal L f = b\cdot \nabla_x f.$3 (Klus et al., 2019).

The same paper proposes a separate identification strategy for reversible coarse-grained dynamics. First, force matching is used to estimate the effective potential $\mathcal L f = b\cdot \nabla_x f.$4 via the local mean force. Second, the diffusion is fitted from the generator matrix through an objective

$\mathcal L f = b\cdot \nabla_x f.$5

and the effective drift is reconstructed from

$\mathcal L f = b\cdot \nabla_x f.$6

This separation is significant because it allows potential and diffusion to be learned with different parameterizations and constraints (Klus et al., 2019).

Kernel-based coarse-graining develops this program further. For an SDE

$\mathcal L f = b\cdot \nabla_x f.$7

the slow kinetics are associated with the small eigenvalues of $\mathcal L f = b\cdot \nabla_x f.$8, so approximating the generator accurately yields implied time scales and metastable structure. The paper "Kinetically Consistent Coarse Graining using Kernel-based Extended Dynamic Mode Decomposition" learns a reduced generator, then fits an effective diffusion by minimizing

$\mathcal L f = b\cdot \nabla_x f.$9

and combines this with force matching to obtain a complete reversible coarse-grained SDE. It reports successful recovery of essential kinetic and thermodynamic properties in a two-dimensional model system and in alanine dipeptide (Nateghi et al., 2024).

Control is another canonical gEDMD application. In lifted coordinates

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$0

one obtains an approximately linear continuous-time surrogate

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$1

For control-affine settings, the generator framework is applied to a finite set of autonomous systems indexed by inputs $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$2, yielding a family of generator matrices $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$3. The resulting MPC problem is formulated directly in continuous time,

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$4

subject to

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$5

The same framework also admits switching-time optimization because the generator model is not tied to a fixed lag time (Klus et al., 2019).

6. Convergence theory, computational scaling, and conceptual boundaries

The strongest general theory currently available treats gEDMD as empirical operator projection. Under linear independence of the dictionary, continuity assumptions on $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$6 and $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$7, and i.i.d. sampling, the empirical matrices converge almost surely,

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$8

and consequently

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$9

The operator approximation also converges, and the framework provides explicit high-probability finite-sample error bounds, including noisy-observation bounds in which the effective variance term is enlarged by the noise contribution (Llamazares-Elias et al., 2024).

This theory also clarifies the practical role of conditioning. The Gram matrix condition number $(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$0 appears explicitly in the finite-sample bound, and numerical experiments show that monomial dictionaries can become extremely ill-conditioned. The same analysis notes that for second-order generators, piecewise linear FEM basis functions are not $(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$1, so the term $(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$2 required by the bound may be unavailable. The quality of gEDMD therefore depends not only on approximation power but also on regularity and numerical conditioning of the dictionary (Llamazares-Elias et al., 2024).

The curse of dimensionality remains a central limitation. Standard full-state EDMD on a network of dimension $(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$3 has complexity scaling like

$(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$4

whereas modularized learning for interconnected systems scales like

$(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$5

that is, exponentially only in subsystem dimension $(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$6, not in full network dimension. This is one motivation for modularization, local dictionaries, and transfer-learning strategies in networked systems (Guo et al., 2024).

A persistent source of confusion is the boundary between gEDMD and related Koopman approaches. Not every Koopman lifting method based on EDMD is generator-based. The paper "Extended Dynamic Mode Decomposition with Learned Koopman Eigenfunctions for Prediction and Control" uses the standard Koopman semigroup $(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$7 and an EDMD-style lifted regression model with learned Koopman eigenfunctions as observables, but it does not define or numerically approximate the Koopman generator $(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$8 in the gEDMD sense, does not solve a generator eigenproblem $(\mathcal K^t f)(x)=\mathbb E[f(\Phi^t(x))],$9 directly from data, and does not build the model from infinitesimal generator projection formulas. It is therefore more accurately described as Koopman eigenfunction learning plus EDMD-style lifted dynamics identification, not as gEDMD (Folkestad et al., 2019).

This distinction is substantive rather than terminological. gEDMD is organized around infinitesimal generator approximation, continuous-time spectral structure, and often direct access to drift, diffusion, or coarse-grained kinetics. Standard EDMD, by contrast, is organized around finite-lag semigroup approximation. The two methods are closely related, but they address different operator objects and inherit different numerical and modeling advantages.