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Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods

Diego Olguín, Axel Osses, Héctor Ramírez

TL;DR

The paper tackles the problem of provably bounding the error in stochastic Koopman operator approximations obtained via Kernel EDMD. It introduces a lifting-back operator to recover trajectories in the original state space and proves probabilistic $O(N^{-1/2})$ bounds for both the operator and the mean lifted trajectories, without requiring eigenvalue simplicity. The results rely on RKHS formalism with covariance operators and Hoeffding-type concentration, and are supported by numerical experiments on linear and nonlinear (SIR) systems that show the predicted convergence behavior. This work advances kernel-based operator estimation by providing explicit probabilistic guarantees and a mechanism to translate lifted representations back to the original dynamics, with practical implications for data-driven analysis of stochastic systems.

Abstract

In this article, we propose a new error bound for Koopman operator approximation using Kernel Extended Dynamic Mode Decomposition. The new estimate is $O(N^{-1/2})$, with a constant related to the probability of success of the bound, given by Hoeffding's inequality, similar to other methodologies, such as Philipp et al. Furthermore, we propose a \textit{lifting back} operator to obtain trajectories generated by embedding the initial state and iterating a linear system in a higher dimension. This naturally yields an $O(N^{-1/2})$ error bound for mean trajectories. Finally, we show numerical results including an example of nonlinear system, exhibiting successful approximation with exponential decay faster than $-1/2$, as suggested by the theoretical results.

Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods

TL;DR

The paper tackles the problem of provably bounding the error in stochastic Koopman operator approximations obtained via Kernel EDMD. It introduces a lifting-back operator to recover trajectories in the original state space and proves probabilistic bounds for both the operator and the mean lifted trajectories, without requiring eigenvalue simplicity. The results rely on RKHS formalism with covariance operators and Hoeffding-type concentration, and are supported by numerical experiments on linear and nonlinear (SIR) systems that show the predicted convergence behavior. This work advances kernel-based operator estimation by providing explicit probabilistic guarantees and a mechanism to translate lifted representations back to the original dynamics, with practical implications for data-driven analysis of stochastic systems.

Abstract

In this article, we propose a new error bound for Koopman operator approximation using Kernel Extended Dynamic Mode Decomposition. The new estimate is , with a constant related to the probability of success of the bound, given by Hoeffding's inequality, similar to other methodologies, such as Philipp et al. Furthermore, we propose a \textit{lifting back} operator to obtain trajectories generated by embedding the initial state and iterating a linear system in a higher dimension. This naturally yields an error bound for mean trajectories. Finally, we show numerical results including an example of nonlinear system, exhibiting successful approximation with exponential decay faster than , as suggested by the theoretical results.

Paper Structure

This paper contains 12 sections, 21 theorems, 118 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reproducing Kernel Hilbert Spaces
  4. Covariance and Koopman Operators
  5. Main results
  6. Proof of Theorem \ref{['theo:error_koop_sqrt_N_def']}
  7. Remarks of Theorem \ref{['theo:error_koop_sqrt_N_def']}
  8. Proof of Theorem \ref{['theo:lifted_traj']}
  9. Remarks of Theorem \ref{['theo:lifted_traj']}
  10. Numerical results
  11. Discussion
  12. Conclusion

Key Result

Theorem 1

Let $k$ be a positive semi-definite function on $\mathcal{X} \times \mathcal{X}$. Then, there exists a unique Hilbert space $\mathcal{H} \subset \mathbb{C}^{\mathcal{X}}$ with $k$ as its reproducing kernel. The subspace $\mathcal{H}_0$ of $\mathcal{H}$, generated by the functions $(k(\cdot, x))_{x \ where, for $(x_1, \dots, x_n) \in \mathcal{X}^n$ and $(y_1, \dots, y_m) \in \mathcal{X}^m$ we have

Figures (6)

  • Figure 1: Plot of the growth of the admissible probability of success $(1 - \delta_{\text{adm}, N})^2$ and the constant $\Tilde{C}_{\delta_{\text{adm}, N}}$.
  • Figure 2: Diagram explaning Extended Dynamic Mode Decomposition with this notation. Adapted from Williams et al. Williams2015ADecomposition.
  • Figure 3: Phase portraits of the linear system in three-dimensional space. The orange line represents the mean trajectory, while the blue lines show 30 trajectories sampled from the system distribution.
  • Figure 4: Errors for different values of the parameter $\alpha$ defining the system matrix $\mathbf{A}_\alpha$. The plots show the error committed by the kEDMD trajectories as a function of $N$. Dots and error bars represent 10 realizations, while the continuous lines correspond to fits of the form $A \cdot N^B$, with the fitted exponent $B$ indicated in the legend. The red line shows the theoretical upper bound $\Tilde{C}_\delta N^{-1/2}$ for $\delta = 10^{-15}$. The left panel presents the results in linear scale, and the right panel in log-log scale.
  • Figure 5: Phase portraits of the SIR system in three-dimensional space. The orange line represents the mean trajectory, while the blue lines correspond to 30 trajectories sampled from the system dynamics. The left panel shows the trajectories generated by the true system, and the right panel displays those generated using the kEDMD approximation.
  • ...and 1 more figures

Theorems & Definitions (47)

  • Definition 1: Transition measure
  • Definition 2: Invariant space
  • Definition 3: Reproducing Kernel Hilbert Space (RKHS) Mercer1909XVI.Equations
  • Definition 4: Positive Semi-Definite Function
  • Theorem 1: Moore–Aronszajn Aronszajn1950TheoryKernels
  • Theorem 2
  • Definition 5
  • Definition 6: Fractional Sobolev Space Adams2003SobolevSpaces
  • Definition 7: Interior Cone Condition Wendland2004ScatteredApproximation
  • Theorem 3: Tuo2016APropertiesWendland2004ScatteredApproximation
  • ...and 37 more