Generalized Extended Dynamic Mode Decomposition (gEDMD)

• gEDMD is a data-driven method that generalizes traditional EDMD by employing flexible, adaptive dictionaries to represent observables in nonlinear and stochastic systems.
• It uses a Galerkin projection framework to approximate transfer operators through empirical inner products, regularization, and learned or adaptive feature maps.
• Applications include multi-output regression, neural dictionary learning, and spectral analysis, addressing challenges in chaotic and high-dimensional dynamical systems.

Generalized Extended Dynamic Mode Decomposition (gEDMD) denotes a class of operator-theoretic, data-driven approximation methods that extend extended dynamic mode decomposition (EDMD) beyond a fixed, hand-chosen dictionary of observables. In the broader literature summarized here, gEDMD usually refers to EDMD with more flexible function spaces, regularization, multiple observables, learned or adaptive dictionaries, and, in an important specialization, approximation of the Koopman generator rather than only the Koopman operator (Boddupalli, 2021, Llamazares-Elias et al., 2024). The unifying viewpoint is Galerkin projection: nonlinear dynamics are lifted to a finite-dimensional space of observables, and the action of a transfer operator is approximated there by a matrix learned from data.

1. Operator-theoretic formulation

For a discrete-time system

$x^+ = F(x),$

the Koopman operator acts on observables $g$ by composition,

$Kg(x)=g(F(x)).$

For a continuous-time flow $\Phi^t$, the Koopman semigroup is

$K^t g(x)=g(\Phi^t(x)),$

and, when the semigroup is strongly continuous, the generator is

$$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$

In the stochastic case

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

the generator takes the Itô form

$$\mathcal{L}f(x)=b(x)\cdot \nabla f(x)+\frac12\operatorname{Tr}\big(\Sigma(x)\nabla^2 f(x)\big),$$

with $\Sigma=\sigma\sigma^\top$ (Alford-Lago et al., 2021, Llamazares-Elias et al., 2024).

EDMD begins by choosing a finite dictionary $\Psi=\{\psi_1,\dots,\psi_N\}$, spanning a trial space $g$0. In the classical Galerkin form, the Gram and structure matrices are

$g$1

where $g$2 is the operator of interest. The finite-dimensional approximation satisfies

$g$3

In the unified Monte Carlo framework of Colbrook, Kostic, and collaborators, EDMD is the case $g$4 or $g$5, whereas gEDMD is the case $g$6 or $g$7 (Llamazares-Elias et al., 2024).

With snapshot data $g$8, empirical inner products produce

$g$9

and the data-driven approximation is

$Kg(x)=g(F(x)).$0

This subsumes both operator EDMD and generator EDMD within the same empirical Galerkin construction (Llamazares-Elias et al., 2024).

2. From EDMD to generalized EDMD

In its classical form, EDMD assumes that the chosen trial space is approximately Koopman-invariant. For a dictionary $Kg(x)=g(F(x)).$1, one seeks a matrix $Kg(x)=g(F(x)).$2 such that

$Kg(x)=g(F(x)).$3

for observables $Kg(x)=g(F(x)).$4. With time-ordered data, the standard least-squares problem is

$Kg(x)=g(F(x)).$5

with solution obtained from the SVD or, in Galerkin notation, by $Kg(x)=g(F(x)).$6 (Alford-Lago et al., 2021, Boddupalli, 2021).

The central limitation is the dictionary. Classical EDMD requires the observables to be chosen a priori, and poorly chosen dictionaries can produce misleading or spurious spectral information when the span of the dictionary is not close to a Koopman-invariant subspace (Alford-Lago et al., 2021). This is the primary motivation for gEDMD: the generalization lies not in abandoning the Galerkin structure, but in enlarging the admissible observable spaces, changing the projection norm or operator, and introducing regularization or data-adaptive feature maps (Boddupalli, 2021, Llamazares-Elias et al., 2024).

A second axis of generalization is output structure. In multi-output EDMD, one distinguishes the operator approximation from the representation of observables themselves. If $Kg(x)=g(F(x)).$7 is a vector observable and $Kg(x)=g(F(x)).$8 collects its coefficients in the dictionary basis, then one approximates

$Kg(x)=g(F(x)).$9

and evolves all outputs by $\Phi^t$0. This leads naturally to multi-task regression and to regularized estimation of both $\Phi^t$1 and $\Phi^t$2 (Boddupalli, 2021).

Regularization is not peripheral in gEDMD. For the Koopman matrix, ridge regularization gives

$\Phi^t$3

For multiple outputs, Tikhonov regularization with prior structure yields

$\Phi^t$4

allowing known observables, such as state coordinates, to be regularized differently from unknown sensor outputs (Boddupalli, 2021). This makes gEDMD a framework for constrained inverse problems on observable spaces, not merely an eigen-analysis routine.

3. Principal methodological branches

The modern literature uses the label gEDMD for a heterogeneous but structurally coherent family of methods. Some papers use the term explicitly; others fall into the same class because they retain EDMD’s projection-and-regression core while generalizing the dictionary, function space, or operator.

Branch	Core generalization	Representative paper
Generator gEDMD	Approximate $\Phi^t$5 or $\Phi^t$6 instead of $\Phi^t$7	(Llamazares-Elias et al., 2024)
Multi-output EDMD	Jointly regress multiple observables and their iterates	(Boddupalli, 2021)
Dictionary learning	Learn $\Phi^t$8 from data with neural networks	(Li et al., 2017, Terao et al., 2021)
Autoencoder / latent EDMD	Learn an embedding and perform EDMD in latent coordinates	(Alford-Lago et al., 2021)
Eigenfunction-based lifting	Use learned Koopman eigenfunctions as lifting functions	(Folkestad et al., 2019)
Analytic RKHS EDMD	Project in analytic RKHSs via Taylor or polynomial subspaces	(Mauroy et al., 2024)
Symmetry-constrained EDMD	Impose equivariance and group-convolution structure	(Harder et al., 2024)
Rigged DMD	Approximate generalized eigenfunctions and continuous spectrum	(Colbrook et al., 2024)

Neural dictionary learning provides one of the most visible gEDMD instantiations. Li, Dietrich, Bollt, and Kevrekidis formulate a joint optimization

$\Phi^t$9

in which the dictionary $K^t g(x)=g(\Phi^t(x)),$0 is represented by a neural network and alternates with a regularized EDMD step (Li et al., 2017). Miyatake and collaborators replace the multilayer perceptron dictionary by a neural ODE, retaining the same EDMD-DL loss while reducing parameter count and memory footprint (Terao et al., 2021).

A closely related branch uses latent-coordinate learning. The DLDMD construction employs an encoder $K^t g(x)=g(\Phi^t(x)),$1 and decoder $K^t g(x)=g(\Phi^t(x)),$2, performs EDMD on encoded snapshots, and couples reconstruction, one-step EDMD residual, and multi-step prediction losses:

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

In the paper’s own interpretation, this searches for a learned feature map whose span is approximately Koopman-invariant and therefore sits squarely in the conceptual class of generalized EDMD (Alford-Lago et al., 2021).

KEEDMD generalizes EDMD in a different direction. Instead of arbitrary lifting functions, it learns a diffeomorphism $K^t g(x)=g(\Phi^t(x)),$4 that pulls back eigenfunctions of a nominal linear model to approximate Koopman eigenfunctions of the nonlinear system, then uses those eigenfunctions as the EDMD dictionary. The lifted state is

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

and the identified linear model has explicit structural blocks reflecting kinematics and eigenfunction evolution (Folkestad et al., 2019).

Two structurally specialized branches are especially notable. Analytic EDMD works in an RKHS of analytic functions, projects orthogonally onto polynomial subspaces, and exploits the triangular structure of the Koopman operator near a hyperbolic equilibrium. Because the projection is a Taylor projection in an analytic RKHS, the method does not suffer from spectral pollution and can achieve arbitrary spectral accuracy with fixed finite dimension by increasing data (Mauroy et al., 2024). Group-convolutional EDMD, by contrast, generalizes EDMD under finite-group equivariance assumptions. Under those assumptions, the optimal EDMD matrix is equivariant, can be represented by a group convolution kernel, and becomes block-diagonal under the generalized Fourier transform, which yields data-efficient learning and fast eigenfunction approximation (Harder et al., 2024).

4. Spectral interpretation and function-space dependence

A defining issue in gEDMD is that the meaning of the computed spectrum depends on the function space. For analytic expanding circle maps, Slipantschuk, Wormell, and collaborators show that EDMD with Fourier-type dictionaries approximates compact Perron–Frobenius and Koopman operators on Hardy–Hilbert spaces of analytic functions, not merely a naive $K^t g(x)=g(\Phi^t(x)),$6 Koopman operator. In that setting, the finite-dimensional matrices are genuine Galerkin approximations of compact operators, and eigenvalues converge exponentially with dictionary size (Slipantschuk et al., 2019).

A parallel result for chaotic circle maps clarifies the least-squares side. Wormell proves that, in the infinite-data limit and for trigonometric polynomial dictionaries, the least-squares projection error is exponentially small even under non-uniform analytic sampling measures, by means of a new approximation theorem in orthogonal polynomials on the unit circle. Forecasts and Koopman spectral data therefore converge exponentially fast with dictionary size to physically meaningful limits in this analytic setting (Wormell, 2023).

These results sharpen a recurring lesson: gEDMD is not meaningful independently of the chosen observable space. If the function space is too large or poorly matched to the dynamics, continuous spectrum and spectral pollution can dominate. If it is chosen to reflect analyticity, hyperbolicity, or symmetry, the finite-dimensional approximation can inherit compactness, triangularity, or equivariance and thus acquire a legitimate spectral interpretation (Slipantschuk et al., 2019, Mauroy et al., 2024, Harder et al., 2024).

Continuous spectrum exposes the limitation of ordinary eigenpair extraction most starkly. Rigged DMD addresses this by combining measure-preserving EDMD with high-order kernels and a rigged Hilbert space $K^t g(x)=g(\Phi^t(x)),$7. Instead of approximating only $K^t g(x)=g(\Phi^t(x)),$8 eigenfunctions, it computes wave-packet approximations of generalized eigenfunctions through the resolvent and smoothed spectral projections, thereby covering both discrete and continuous spectral components (Colbrook et al., 2024). A plausible implication is that gEDMD, in its most general form, includes not only richer dictionaries but also richer spectral objects.

5. Convergence, error, and robustness

The 2024 convergence theory of Colbrook, Kostic, and collaborators places EDMD and gEDMD inside a unified Monte Carlo Galerkin framework and proves convergence of the approximating operator and its spectrum under non-restrictive conditions (Llamazares-Elias et al., 2024). For fixed dictionary size $K^t g(x)=g(\Phi^t(x)),$9, the empirical matrix $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$0 converges almost surely to the Galerkin matrix $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$1 as the number of samples $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$2. Under approximation assumptions on the trial spaces $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$3 and, for unbounded operators, on the graph-norm spaces $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$4, the Galerkin approximation converges strongly to the target operator as $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$5 (Llamazares-Elias et al., 2024).

The same framework yields spectral convergence. For empirical eigenpairs $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$6 of $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$7, subsequences converge to eigenpairs of the Galerkin operator; in the dictionary limit, subsequences of Galerkin eigenpairs converge to eigenpairs of the full operator. In the joint limit of growing data and growing dictionary, subsequences of empirical eigenpairs converge to eigenpairs of the target operator $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$8, including the generator in the gEDMD case (Llamazares-Elias et al., 2024).

Quantitatively, the sampling error is governed by matrix concentration estimates for the empirical Gram and structure matrices. Up to logarithmic factors and basis-dependent quantities such as $$\mathcal{L}f(x)=\lim_{t\downarrow 0}\frac{1}{t}\big(K^t f(x)-f(x)\big).$$9, the data complexity scales like $dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t,$0, and the empirical operator error decays like $dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t,$1 (Llamazares-Elias et al., 2024). Noise changes constants but not the convergence order: the noisy-data estimates retain the same $dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t,$2 scaling, provided the noise is mean-zero and bounded in probability (Llamazares-Elias et al., 2024).

Conditioning remains decisive. The error bounds depend explicitly on $dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t,$3 and $dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t,$4, and numerical experiments show that Gaussian and finite-element dictionaries are markedly more robust to noise than monomials, largely because of Gram-matrix conditioning (Llamazares-Elias et al., 2024). This is consistent with the warning, already emphasized in the EDMD literature, that rich dictionaries can be numerically harmful when they are redundant or nearly linearly dependent (Boddupalli, 2021, Li et al., 2017).

6. Applications, limitations, and recurrent misconceptions

The application range of gEDMD is broad. Learned-dictionary variants have been tested on the Duffing oscillator, Van der Pol oscillator, Lorenz-63, and the Kuramoto–Sivashinsky equation, where they outperform standard DMD or reduce parameter count substantially relative to fixed-dictionary baselines (Alford-Lago et al., 2021, Terao et al., 2021). KEEDMD improves state prediction and closed-loop trajectory tracking for a simulated cart-pole system by using learned Koopman eigenfunctions as lifting functions in an EDMD-style model (Folkestad et al., 2019). Multi-output EDMD is aimed at settings where observables are sensor outputs rather than full states, including cases where some outputs are known and others are only observed through data (Boddupalli, 2021). Symmetry-aware group-convolutional EDMD has been demonstrated on the Kuramoto–Sivashinsky equation and a $dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t,$5-$dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t,$6 spiraling wave system, both nonlinear PDEs (Harder et al., 2024). Rigged DMD has been applied to systems with Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds-number lid-driven flow (Colbrook et al., 2024).

Several misconceptions recur. First, gEDMD is not synonymous with deep learning. Neural dictionaries and autoencoder embeddings are prominent, but analytic RKHS projections, regularized multi-output formulations, symmetry-constrained EDMD, generator approximations, and rigged-Hilbert-space methods are equally part of the generalization (Mauroy et al., 2024, Boddupalli, 2021, Colbrook et al., 2024). Second, generalization does not remove the problem of function-space design. It redistributes it across basis choice, kernel choice, latent dimension, regularization, symmetry assumptions, and operator choice (Alford-Lago et al., 2021, Llamazares-Elias et al., 2024). Third, finite-dimensional Koopman models remain limited for strongly chaotic systems. In DLDMD, for example, Lorenz-63 attractor geometry is recovered reasonably well, but long-horizon pointwise prediction remains poor (Alford-Lago et al., 2021). Rigged DMD can be read as a response to this limitation: when continuous spectrum dominates, generalized eigenfunction decompositions become more appropriate than ordinary eigendecompositions (Colbrook et al., 2024).

Open problems in the cited literature are correspondingly structural. They include automated selection or regularization of latent dimension in learned EDMD (Alford-Lago et al., 2021), extension of rigorous rates from i.i.d. sampling to time-correlated trajectory data (Llamazares-Elias et al., 2024), treatment of non-analytic or singular sampling measures in chaotic dynamics (Wormell, 2023), extension of symmetry-constrained EDMD beyond finite groups or exact equivariance (Harder et al., 2024), and generalization of rigged spectral methods to non-unitary Koopman operators (Colbrook et al., 2024). Across these branches, the central theme remains unchanged: gEDMD seeks finite, data-driven representations of transfer operators whose spectral content is both computationally accessible and dynamically meaningful.