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Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces

Isao Ishikawa, Yuka Hashimoto, Masahiro Ikeda, Yoshinobu Kawahara

TL;DR

<3-5 sentence high-level summary> This work introduces JetEDMD, a jet-based extension of EDMD for estimating Koopman operators on RKHSs, enabling intrinsic observables that capture local dynamical structure and mitigate spectral pollution. By embedding the analysis in a rigged Hilbert space (Gelfand triple), the authors define an extended Koopman operator whose spectral decomposition is tractable and informative even when the original operator does not preserve the observable space. Theoretical results provide explicit error bounds and convergence rates for Gaussian and exponential kernels, along with generator-based analogues for continuous dynamics and a data-driven reconstruction procedure with performance guarantees. Empirically, JetEDMD demonstrates accurate eigenvalue/eigenfunction estimation and reliable reconstruction on classic nonlinear systems (Van der Pol, Duffing, Lorenz, Hénon).

Abstract

This paper presents a novel approach for estimating the Koopman operator defined on a reproducing kernel Hilbert space (RKHS) and its spectra. We propose an estimation method, what we call Jet Extended Dynamic Mode Decomposition (JetEDMD), leveraging the intrinsic structure of RKHS and the geometric notion known as jets to enhance the estimation of the Koopman operator. This method refines the traditional Extended Dynamic Mode Decomposition (EDMD) in accuracy, especially in the numerical estimation of eigenvalues. This paper proves JetEDMD's superiority through explicit error bounds and convergence rate for special positive definite kernels, offering a solid theoretical foundation for its performance. We also investigate the spectral analysis of the Koopman operator, proposing the notion of an extended Koopman operator within a framework of a rigged Hilbert space. This notion leads to a deeper understanding of estimated Koopman eigenfunctions and capturing them outside the original function space. Through the theory of rigged Hilbert space, our study provides a principled methodology to analyze the estimated spectrum and eigenfunctions of Koopman operators, and enables eigendecomposition within a rigged RKHS. We also propose a new effective method for reconstructing the dynamical system from temporally-sampled trajectory data of the dynamical system with solid theoretical guarantee. We conduct several numerical simulations using the van der Pol oscillator, the Duffing oscillator, the Hénon map, and the Lorenz attractor, and illustrate the performance of JetEDMD with clear numerical computations of eigenvalues and accurate predictions of the dynamical systems.

Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces

TL;DR

<3-5 sentence high-level summary> This work introduces JetEDMD, a jet-based extension of EDMD for estimating Koopman operators on RKHSs, enabling intrinsic observables that capture local dynamical structure and mitigate spectral pollution. By embedding the analysis in a rigged Hilbert space (Gelfand triple), the authors define an extended Koopman operator whose spectral decomposition is tractable and informative even when the original operator does not preserve the observable space. Theoretical results provide explicit error bounds and convergence rates for Gaussian and exponential kernels, along with generator-based analogues for continuous dynamics and a data-driven reconstruction procedure with performance guarantees. Empirically, JetEDMD demonstrates accurate eigenvalue/eigenfunction estimation and reliable reconstruction on classic nonlinear systems (Van der Pol, Duffing, Lorenz, Hénon).

Abstract

This paper presents a novel approach for estimating the Koopman operator defined on a reproducing kernel Hilbert space (RKHS) and its spectra. We propose an estimation method, what we call Jet Extended Dynamic Mode Decomposition (JetEDMD), leveraging the intrinsic structure of RKHS and the geometric notion known as jets to enhance the estimation of the Koopman operator. This method refines the traditional Extended Dynamic Mode Decomposition (EDMD) in accuracy, especially in the numerical estimation of eigenvalues. This paper proves JetEDMD's superiority through explicit error bounds and convergence rate for special positive definite kernels, offering a solid theoretical foundation for its performance. We also investigate the spectral analysis of the Koopman operator, proposing the notion of an extended Koopman operator within a framework of a rigged Hilbert space. This notion leads to a deeper understanding of estimated Koopman eigenfunctions and capturing them outside the original function space. Through the theory of rigged Hilbert space, our study provides a principled methodology to analyze the estimated spectrum and eigenfunctions of Koopman operators, and enables eigendecomposition within a rigged RKHS. We also propose a new effective method for reconstructing the dynamical system from temporally-sampled trajectory data of the dynamical system with solid theoretical guarantee. We conduct several numerical simulations using the van der Pol oscillator, the Duffing oscillator, the Hénon map, and the Lorenz attractor, and illustrate the performance of JetEDMD with clear numerical computations of eigenvalues and accurate predictions of the dynamical systems.
Paper Structure (37 sections, 57 theorems, 178 equations, 14 figures, 7 algorithms)

This paper contains 37 sections, 57 theorems, 178 equations, 14 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Summary of main results and their implications
  4. Intrinsic observables and jets
  5. Jets and derivatives of Dirac deltas
  6. Construction of the space of intrinsic observables in reproducing kernel Hilbert spaces
  7. Estimation of Perron--Frobenius operators
  8. Error analysis
  9. Several analysis with bounded Koopman operators
  10. Extended Koopman operators and rigged Hilbert spaces
  11. Rigged Hilbert spaces
  12. Extended Koopman operators
  13. Eigendecomposition of the extended Koopman operators
  14. Corresponding results to continuous dynamical systems
  15. The generator of Koopman operators on the canonical invariant subspaces
  16. ...and 22 more sections

Key Result

Proposition 1.2

The space $V_p$ is dense in $H$. Moreover, if $p$ satisfies $f(p) = p$, we have $C_f^*(V_{p,n}) \subset V_{p,n}$ for any $n \ge 0$. In addition, if $f(p)=p$ and the dual map $\iota'$ of $\iota$ is injective on the image of $\mathfrak{j}_{p,n}'$, the set of eigenvalues of $C_f^*|_{V_{p,n}}$ coincides

Figures (14)

  • Figure 1: The comparison of the computed eigenvalues of the dynamical system $f(x,y) := (x^2 - y^2 + x - y, 2xy + x + y)$ on $\mathbb{R}^2$ using data of $100$ pairs of sample from the uniform distribution on $[-1,1]^2$ and their images under $f$. The red $+$'s indicate the eigenvalues computed via EDMD using monomials of degree up to $10$. The blue $\times$'s indicate the estimated eigenvalues with JetEDMD. The green circles indicate set $\{\lambda_+^m\lambda_-^n\}_{m + n \le 5}$, where $\lambda_\pm = 1 \pm \mathrm{i}$ are the eigenvalues of the Jacobian matrix of $f$ at the origin.
  • Figure 2: Illustration of the data-driven reconstruction of the Lorenz system with Algorithm \ref{['algorithm: reconstruction, conti. with disc.']}, corresponding to the right images of Figure \ref{['fig: reconstruction_lorenz, discrete']}. Left: data used for estimation—300 input-output pairs, with inputs sampled uniformly from $[-10,10]^3$ and outputs given by the flow map at time $0.1$. Middle-left: estimated and theoretical eigenvalues of the generator of the Koopman operator. Middle-right: trajectory of the dynamical system reconstructed by our method. Right: true trajectory.
  • Figure 3: Eigenvalue estimation for the van der Pol oscillator ($\mu=1$) using Algorithm \ref{['algorithm: data-driven PF operator estimation, continuous']} (left) and Algorithm \ref{['algorithm: data-driven PF operator estimation, discrete']} with the matrix-log method in Remark \ref{['rmk: matrix log']} (middle, right). We use the exponential kernel $k^{\rm e}(x,y)=e^{x^\top y/4}$. Blue $\times$ marks show the eigenvalues of the estimated Perron--Frobenius operator $\widehat{\mathbf{A}}$; green circles show those of $A_F|{V{p,m}}$; plus signs show the eigenvalues of the Jacobian of $F(x,y)=(y,,\mu(1-x^2)y-x)$ at $p=(0,0)$. Left: continuous setting with $m=5$, $n=7$, $p=(0,0)$, $N=36$ samples drawn uniformly from $[-1,1]^2$, and their exact velocities. Middle: discrete setting with $m=5$, $n=15$, $T_s=0.5$, $p=(0,0)$, and $N=300$ input-output pairs from $[-1,1]^2$ and their images under $\phi^{T_s}$. Right: discrete setting with $m=5$, $n=20$, $T_s=1.0$, $p=(0,0)$, and $N=300$ input-output pairs as above.
  • Figure 4: Two domains of attraction of the Duffing oscillator \ref{['duffing']} (left). Left: two domains of attraction of the Duffing oscillator \ref{['duffing']}. Right: an odd estimated eigenfunction for eigenvalue $-1$ of the extended Koopman operator $A_F^\times$ for the Duffing oscillator \ref{['duffing']}, computed via Algorithm \ref{['algorithm: eigenfunction, continuous']} using the exponential kernel $e^{x^\top y}$ with input $m_1=m_2=10$, $n_1=n_2=16$, $p_1=(-1,0)$, $p_2=(1,0)$, and $N_1=N_2=7000$ samples drawn uniformly from $[-1.5,1.5]\times[-0.5,0.5]$, together with their exact velocities.
  • Figure 5: Left: the attractor (red scatter), the domain of attraction (blue region), and the invariant manifold (dark blue dashed line) of the fixed points of the H'enon map \ref{['henon']}. Middle: the heat map of the estimated eigenfunction with eigenvalue $0.3$ of the extended Koopman operator $C_f^\times$ for the Hénon map \ref{['henon']}, computed via Algorithm \ref{['algorithm: eigenfunction, discrete']} using the exponential kernel $e^{x^\top y/0.6}$ with input $m_1=6$, $n_1=30$, $p_1\approx(-1.131,-0.339)$, and $N_1=3000$ input--output pairs sampled uniformly from $p_1+[-0.25,0.25]^2$ and their images under $f$. Right: the contours of the eigenfunction overlaid on the left panel.
  • ...and 9 more figures

Theorems & Definitions (116)

  • Example 1.1
  • Proposition 1.2
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 106 more