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$L^\infty$-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition

Frederik Köhne, Friedrich M. Philipp, Manuel Schaller, Anton Schiela, Karl Worthmann

TL;DR

The paper delivers the first deterministic, uniform-pointwise error bounds for kernel EDMD approximations of the Koopman operator on native RKHS spaces, by proving that Wendland RKHS are invariant under the Koopman flow and by recasting regression as interpolation in native spaces. It establishes a concrete error decomposition in terms of projection/interpolation operators and the operator norm of the Koopman map, with convergence rates linked to the kernel's smoothness and the fill distance $h$. The main theoretical contribution is a rigorous uniform bound of the form $igl\| ext{K}_A - ilde{ extK}_A igr\|_{ ext{N}(Y) o C_b(X)} \nleq igl\| I - S_ ext{X} igr\|_{ ext{N}(X) o C_b(X)} igl\| ext{K}_A igr\|_{ ext{N}(Y) o ext{N}(X)}$, together with analogous estimates for the variant $ ilde{ extK}_A^ ext{Y}$ and a decay $igl\|I - S_ ext{X} igr\|_{ ext{N}_{oldsymbol{ ext Phi}_{d,k}}(oldsymbol{ extOmega}) o C_b(oldsymbol{ extOmega})} \, o\, C h_{oldsymbol{ extX}}^{k+1/2}$. Empirical results on the Duffing and Lorenz systems corroborate the theory, showing that higher kernel smoothness and denser sampling yield faster convergence and improved multi-step prediction accuracy, with boundary effects mitigated by tailored node layouts.

Abstract

Extended dynamic mode decomposition (EDMD) is a well-established method to generate a data-driven approximation of the Koopman operator for analysis and prediction of nonlinear dynamical systems. Recently, kernel EDMD (kEDMD) has gained popularity due to its ability to resolve the challenging task of choosing a suitable dictionary by using the kernel's canonical features and, thus, data-informed observables. In this paper, we provide the first pointwise bounds on the approximation error of kEDMD. The main idea consists of two steps. First, we show that the reproducing kernel Hilbert spaces of Wendland functions are invariant under the Koopman operator. Second, exploiting that the learning problem given by regression in the native norm can be recast as an interpolation problem, we prove our novel error bounds by using interpolation estimates. Finally, we validate our findings with numerical experiments.

$L^\infty$-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition

TL;DR

The paper delivers the first deterministic, uniform-pointwise error bounds for kernel EDMD approximations of the Koopman operator on native RKHS spaces, by proving that Wendland RKHS are invariant under the Koopman flow and by recasting regression as interpolation in native spaces. It establishes a concrete error decomposition in terms of projection/interpolation operators and the operator norm of the Koopman map, with convergence rates linked to the kernel's smoothness and the fill distance . The main theoretical contribution is a rigorous uniform bound of the form , together with analogous estimates for the variant and a decay . Empirical results on the Duffing and Lorenz systems corroborate the theory, showing that higher kernel smoothness and denser sampling yield faster convergence and improved multi-step prediction accuracy, with boundary effects mitigated by tailored node layouts.

Abstract

Extended dynamic mode decomposition (EDMD) is a well-established method to generate a data-driven approximation of the Koopman operator for analysis and prediction of nonlinear dynamical systems. Recently, kernel EDMD (kEDMD) has gained popularity due to its ability to resolve the challenging task of choosing a suitable dictionary by using the kernel's canonical features and, thus, data-informed observables. In this paper, we provide the first pointwise bounds on the approximation error of kEDMD. The main idea consists of two steps. First, we show that the reproducing kernel Hilbert spaces of Wendland functions are invariant under the Koopman operator. Second, exploiting that the learning problem given by regression in the native norm can be recast as an interpolation problem, we prove our novel error bounds by using interpolation estimates. Finally, we validate our findings with numerical experiments.
Paper Structure (19 sections, 16 theorems, 98 equations, 3 figures, 5 tables)

This paper contains 19 sections, 16 theorems, 98 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Native spaces of kernels
  3. Interpolation in native spaces
  4. Regression in native spaces
  5. Koopman approximation by regression in the native norm
  6. The Koopman operator on native spaces
  7. Kernel EDMD
  8. Multi-step predictions
  9. Koopman operators on Native Spaces of Wendland functions
  10. Wendland functions and Sobolev spaces
  11. The Koopman operator preserves Sobolev regularity
  12. Uniform error bounds
  13. Numerical Examples
  14. Duffing oscillator
  15. Lorenz system
  16. ...and 4 more sections

Key Result

Proposition 2.1

The projection operator $S_\mathcal{X} : \mathcal{N}(X)\to V_\mathcal{X}$ admits a bounded extension $\widetilde{S}_\mathcal{X} : C_b(X)\to V_\mathcal{X}$, which is given by

Figures (3)

  • Figure 1: $L^\infty$-error for the predicted Duffing oscillator dynamics using the approximant $\widehat{\mathcal{K}}_A^\mathcal{Y} = S_\mathcal{X}\mathcal{K}_A S_\mathcal{Y}$ at smoothness degree $k=1$ with mesh sizes $\delta\in\{0.2,0.1,0.05\}$ from left to right.
  • Figure 2: Case $\mathcal{Y} = \mathcal{X}$: $L^\infty$-error for the predicted Duffing oscillator dynamics using the approximant $S_\mathcal{X}\mathcal{K}_A S_\mathcal{X}$ at smoothness degree $k=1$ with mesh sizes $\delta\in\{0.2,0.1,0.05\}$ from left to right.
  • Figure 3: $L^\infty$-error for the predicted Duffing oscillator dynamics using the approximant $S_\mathcal{X}\mathcal{K}_A S_\mathcal{Y}$ at smoothness degree $k=1$ with meshes of $N\in\{441,1681,6561\}$ Chebyshev nodes, from left to right.

Theorems & Definitions (37)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 27 more