Kernel-Based EDMD (kEDMD)

• Kernel-based EDMD is a data-driven algorithm that approximates Koopman operators for nonlinear systems using reproducing kernel Hilbert space theory.
• It employs the kernel trick to automatically encode data geometry, bypass explicit basis construction, and scale to high-dimensional applications such as PDEs and fluid mechanics.
• Advanced techniques like Random Fourier Features and the Nystrom method enhance efficiency, while rigorous error bounds support robust control and reduced modeling.

Kernel-based Extended Dynamic Mode Decomposition (kEDMD) is a class of data-driven algorithms for approximating the action of the Koopman operator on observables of nonlinear dynamical systems. By leveraging reproducing kernel Hilbert space (RKHS) theory, kEDMD constructs linear finite-rank surrogates that enable prediction, analysis, control, and reduction of nonlinear systems without choosing an explicit (and typically large) dictionary of basis functions. Kernel-based lifting automatically encodes the geometry of the available data, resolves the curse of dimensionality, and admits rigorous analysis of approximation errors—including deterministic $L^\infty$-bounds under appropriate regularity assumptions.

1. Mathematical Foundation and Algorithmic Structure

The kernel-based EDMD algorithm begins by selecting a symmetric, positive-definite kernel $k: X \times X \to \mathbb{R}$ defined on the state space $X$. This kernel induces an RKHS $\mathcal{H}$ with canonical feature map $\Phi(x) = k(x,\cdot)$. Given a set of data pairs $(x_i, y_i)$ (with $y_i$ generated by the time-$t$ flow, $y_i = F^t(x_i)$, or from the transition kernel of a stochastic system), the kernel approach constructs empirical Gram matrices: $$K_X = [k(x_i, x_j)]_{i,j=1}^m,\quad K_Y = [k(y_i, x_j)]_{i,j=1}^m.$$

For the span $$V = \operatorname{span}\{ \Phi(x_1), \dots, \Phi(x_m)\}$$, the kEDMD estimator of the Koopman operator compression is given (with truncation or regularization indexed by $r$) by: $M_r^{m,t} = [K_X]_r^\dagger K_Y^\top,$ where $[K_X]_r^\dagger$ is the pseudoinverse of $K_X$ after rank truncation to $r$ (Philipp et al., 2023).

The operator analogue expresses this as a regression in the RKHS. Empirical covariance operators based on the data are: $$C^m = \frac{1}{m} \sum_{i=1}^m \Phi(x_i) \otimes \Phi(x_i), \quad C^{tm} = \frac{1}{m} \sum_{i=1}^m \Phi(x_i) \otimes \Phi(y_i).$$ The kEDMD Koopman estimator acting on $\mathcal{H}$ reads: $K_r^{m,t} = [C^m]_r^\dagger C^{tm}.$ For a function $f \in \mathcal{H}$, its propagation is approximated by

$$(K_r^{m,t} f)(x) = \sum_{i=1}^m \alpha_i k(x, x_i), \quad \text{with } \alpha = M_r^{m,t} f_X,$$

where $f_X = [f(x_1),\ldots,f(x_m)]^\top$.

This "kernel trick" bypasses explicit construction of a large basis and adapts to data geometry, enabling scaling to high-dimensional or complex state spaces where classical EDMD is infeasible.

2. Error Bounds, Uniform Approximation, and Invariance

A key advance in recent research is the derivation of deterministic $L^\infty$-error bounds for kEDMD (Köhne et al., 27 Mar 2024). For kernels such as Wendland functions—whose native spaces are Sobolev spaces $$H^{\sigma_{d,k}}(\Omega),\ \sigma_{d,k}=(d+1)/2 + k$$—the Koopman operator $\mathcal{K}_A$ composed with a $C^{\sigma_{d,k}}$-diffeomorphism $A$ preserves the native space. The regression problem in the kernel norm is equivalently solved by interpolation, and the error bounds follow from interpolation theory: $$\| \mathcal{K}_A - \hat{\mathcal{K}}_A \|_{(\mathcal{Y}) \to C_b(\mathcal{X})} \leq C h^{k+1/2},$$ where $h$ is the fill distance of the interpolation grid, $k$ is the smoothness parameter of the Wendland kernel, and $C$ depends only on kernel properties and system regularity.

The generalization to control systems involves showing that $kEDMD$ extends under less restrictive invariance conditions: it suffices that the RKHS is invariant under the Koopman operator, which is typically satisfied for widely used RBF kernels in practical systems (Philipp et al., 2023, Bold et al., 3 Dec 2024). For control-affine systems with constant or piecewise constant inputs, the affine structure is mirrored in the RKHS, and error bounds are established with explicit dependence on sample size, kernel regularity, fill distance, and in control applications, the time step $t$.

Regularization (by Tikhonov penalty $\lambda$ in the Gram matrix) ensures numerical stability and robustness, and the error bounds incorporate $\sqrt{\lambda}$ as a summand, reflecting regularization-induced bias.

3. Feature Generation, Computational Strategies, and Scalability

Feature selection for kEDMD includes two principal strategies (DeGennaro et al., 2017):

• Random Fourier Features: For translation-invariant kernels, features are drawn using Bochner's theorem; the kernel is approximated via Monte Carlo sample averages of complex exponentials. This approach is data-independent (beyond kernel parameter estimation) and supports adaptive feature addition by block updates of Gram matrices.
• Nystrom Method: For general symmetric positive-definite kernels, a subset of samples yields a finite-rank Gram matrix, whose leading eigenpairs approximate the expansion in Mercer's theorem. These eigenfunctions define the basis for the kernel approximation, and feature matrices can be interpolated efficiently (either on all data or only subsample points, trading accuracy for speed).

Run-time scaling depends primarily on the number of selected features (basis functions) for random Fourier or Nystrom approaches. Efficient blockwise updates facilitate incremental enrichment of the feature space without full recomputation, yielding scalable algorithms even for large state dimension or number of snapshots.

4. Practical Applications and Numerical Results

kEDMD has been validated on high-dimensional dynamical systems where classical basis expansion is intractable:

• PDEs: For the Fitzhugh–Nagumo system ($d=100$) and experimental cylinder flow data ($d\approx 10,800$), both random Fourier and Nystrom features achieve rapid convergence of linear Koopman modes with as few as 500–600 basis functions (DeGennaro et al., 2017).
• Fluid Mechanics: In limited data regimes (e.g., fluid flow over cylinder), Laplacian kernels yielded more robust recovery of Koopman modes than Gaussian RBF kernels due to favorable operator-theoretic properties (closability, norm bounds) in the associated RKHS (Singh, 2023).
• Molecular Dynamics: kEDMD, formulated naturally in the RKHS of a chemical-similarity kernel, was used to identify metastable states in alanine dipeptide and folded/unfolded states in protein NTL9 via time-lagged Gram matrices followed by cluster analysis (Klus et al., 2018).
• Reduced Modeling and Coarse Graining: In kinetic modeling of molecular systems, kernel-based generator EDMD (gEDMD) is used to extract effective drift and diffusion in coarse-grained SDEs, with parameters determined via a force-matching-like minimization, and with comparison of dominant generator eigenvalues establishing kinetic fidelity (Nateghi et al., 24 Sep 2024).

5. Control, Stability Guarantees, and Closed-Loop Applications

Kernel-based EDMD underpins controller synthesis for nonlinear and controlled systems. For control-affine systems, kEDMD builds surrogate models

$\hat{F}(x,u) = \hat{g}_0(x) + \hat{G}(x) u$

using micro–macro grid decompositions (Bold et al., 3 Dec 2024), where local regression at each macro-grid center leverages nearest-neighbor "micro" data, followed by kernel interpolation to produce a global surrogate. Error bounds for the surrogate are explicit in the fill distance and cluster radius, and exhibit proportional dependence on the distance to the (controlled) equilibrium.

Crucially, the stability of the true system and the kEDMD surrogate is "bidirectionally" preserved: if a Lyapunov function verifies asymptotic stability for one system, it holds for the other up to practical asymptotic stability margins set by the error bounds. These principles have enabled the design of data-driven nonlinear Model Predictive Control (MPC) laws without terminal constraints, where constraint tightening offsets the approximation error and stability is guaranteed via uniform bounds (Bold et al., 15 Jan 2025, Bold et al., 3 Dec 2024). The error bounds, given in terms such as

$$\|F(x,u) - \hat{F}(x,u)\|_{\infty} \leq C(h_{\mathcal{X}} \operatorname{dist}(x, \mathcal{X}) + c_2 \varepsilon_c),$$

control the constraint violation probability and the practical region of attraction.

kEDMD also naturally yields bilinear or robust control surrogates with explicit error terms, making it suitable for LMI/SOS-based synthesis, robust MPC, and tube-based strategies. The boundedness and smoothness of the kernel and the data admissibly ensure the technical conditions required for controller design and feasibility.

6. Extensions, Limitations, and Future Research Directions

Recent developments generalize kEDMD to modularized frameworks for interconnected systems (Guo et al., 22 Aug 2024), where local Koopman approximations (constructed via kernel dictionaries) are combined using the system’s graph structure to alleviate the curse of dimensionality and support transfer learning. Group-convolutional kEDMD incorporates symmetry constraints to enforce equivariant surrogates, allowing fast predictions and eigenfunction computation for systems with spatial or rotational invariance (Harder et al., 1 Nov 2024).

Contemporary limitations include the conservatism of current error bounds (which can induce large data requirements), the sensitivity of performance to kernel choice and parameterization, and computational cost and memory constraints for dense kernel matrices on large datasets. Open research directions identified in recent surveys (Strässer et al., 2 Sep 2025) are:

• Automated dictionary (feature/kernel) learning for invariance and efficiency.
• Extending uniform error bounds to input–output or partial observation settings.
• Further refinement of error estimates and scalable regularization for high-dimensional or distributed control.
• Combining adaptive dictionary or kernel learning (e.g., neural networks with kernel constructions) to overcome static kernel limitations.

7. Summary Table: Key Properties and Recent Advances in kEDMD

Feature	kEDMD Approach	Practical Implication
Dictionary	Kernel-induced, implicit	Avoids explicit basis, adapts to data geometry
Error Bound	$L^\infty$ bounds via interpolation (Wendland)	Uniform, deterministic error control (rate $O(h^{k+1/2})$)
Control Extension	RKHS invariance sufficient for error bounds	Applies to control-affine and stochastic systems; less restrictive conditions
Numerical Stability	Built-in regularization (Tikhonov, cluster)	Robustness to ill-posed Gram matrices and noise
Scalability	Random Fourier, Nystrom, modular/localized grids	Efficient, adaptive, incremental feature addition; alleviates high-dimensionality
Stability Guarantees	Proportional error, Lyapunov transfer	Closed-loop practical stability for data-driven MPC and robust design

In conclusion, kernel-based EDMD is a mathematically rigorous and practically scalable paradigm for data-driven approximation, prediction, and control of nonlinear dynamical systems. Its foundation in RKHS theory enables error-quantified surrogates, robust controller synthesis, and applicability to a wide variety of domains, including high-dimensional PDEs, molecular kinetics, and modern distributed or modular systems. Active research continues to address scaling, adaptivity, and integration with deep learning for further performance gains and theoretical insight.