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Analytic Extended Dynamic Mode Decomposition

Alexandre Mauroy, Igor Mezic

TL;DR

This work develops Analytic EDMD, an EDMD-type method that discovers the Koopman spectrum for analytic dynamical systems by projecting onto polynomial subspaces with a data-driven Taylor projection in an RKHS. By preserving the block-triangular structure of the Koopman operator, the method avoids spectral pollution and can recover lattice-structured eigenvalues and principal eigenfunctions with arbitrary accuracy using a fixed finite basis, given samples with positive measure. The authors provide convergence guarantees, error bounds, and a detailed numerical comparison showing superior performance relative to standard EDMD and kernel-based variants in many scenarios, including data far from equilibria and partial measurements. The approach has strong potential for analysis and control tasks, such as stability certification, isostable reduction, and spectrum-based network identification, with future directions including extensions to limit cycles and high-dimensional systems.

Abstract

We develop a novel EDMD-type algorithm that captures the spectrum of the Koopman operator defined on a reproducing kernel Hilbert space of analytic functions. This method, which we call analytic EDMD, relies on an orthogonal projection on polynomial subspaces, which is equivalent to a data-driven Taylor approximation. In the case of dynamics with a hyperbolic equilibrium, analytic EDMD demonstrates excellent performance to capture the lattice-structured Koopman spectrum based on the eigenvalues of the linearized system at the equilibrium. Moreover, it yields the Taylor approximation of associated principal eigenfunctions. Since the method preserves the triangular structure of the operator, it does not suffer from spectral pollution and, moreover, arbitrary accuracy on the spectrum can be reached with a fixed finite dimension of the approximation and with a (possibly non-uniform) sampling over an arbitrary set of nonzero measure. The performance of analytic EDMD is illustrated with numerical examples and is assessed through a comparative study with related methods. Finally, the method is complemented with theoretical results, proving strong convergence of the eigenfunctions and providing error bounds on the spectrum estimation.

Analytic Extended Dynamic Mode Decomposition

TL;DR

This work develops Analytic EDMD, an EDMD-type method that discovers the Koopman spectrum for analytic dynamical systems by projecting onto polynomial subspaces with a data-driven Taylor projection in an RKHS. By preserving the block-triangular structure of the Koopman operator, the method avoids spectral pollution and can recover lattice-structured eigenvalues and principal eigenfunctions with arbitrary accuracy using a fixed finite basis, given samples with positive measure. The authors provide convergence guarantees, error bounds, and a detailed numerical comparison showing superior performance relative to standard EDMD and kernel-based variants in many scenarios, including data far from equilibria and partial measurements. The approach has strong potential for analysis and control tasks, such as stability certification, isostable reduction, and spectrum-based network identification, with future directions including extensions to limit cycles and high-dimensional systems.

Abstract

We develop a novel EDMD-type algorithm that captures the spectrum of the Koopman operator defined on a reproducing kernel Hilbert space of analytic functions. This method, which we call analytic EDMD, relies on an orthogonal projection on polynomial subspaces, which is equivalent to a data-driven Taylor approximation. In the case of dynamics with a hyperbolic equilibrium, analytic EDMD demonstrates excellent performance to capture the lattice-structured Koopman spectrum based on the eigenvalues of the linearized system at the equilibrium. Moreover, it yields the Taylor approximation of associated principal eigenfunctions. Since the method preserves the triangular structure of the operator, it does not suffer from spectral pollution and, moreover, arbitrary accuracy on the spectrum can be reached with a fixed finite dimension of the approximation and with a (possibly non-uniform) sampling over an arbitrary set of nonzero measure. The performance of analytic EDMD is illustrated with numerical examples and is assessed through a comparative study with related methods. Finally, the method is complemented with theoretical results, proving strong convergence of the eigenfunctions and providing error bounds on the spectrum estimation.
Paper Structure (16 sections, 3 theorems, 36 equations, 8 figures, 4 tables, 1 algorithm)

This paper contains 16 sections, 3 theorems, 36 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Data-driven Taylor projection
  3. Analytic EDMD
  4. Finite-dimensional approximation
  5. Related works
  6. Spectral properties
  7. Algorithm
  8. Numerical simulations
  9. First examples
  10. Specific cases
  11. Numerical analysis
  12. Comparison with other methods
  13. Effect of measurement noise
  14. Choice of kernel
  15. Error bounds and convergence
  16. ...and 1 more sections

Key Result

Proposition 6.1

Let $\hat{\mathbf{K}}_{ij}= \mathbf{e}_i^T \mathbf{G}^{-1} \mathbf{e}_j'$ and $\mathbf{K}_{ij}=\langle e_i,K e_j \rangle_\mathcal{H}$. Then, MoreoverPart of this result was initially mentioned by Isao Ishikawa in a private communication., if the data points $\{\mathbf{x}_k\}_{k=1}^M$ are randomly drawn from a continuous distribution with support $\bar{X} \subseteq X$ of positive Lebesgue measure

Figures (8)

  • Figure 1: Computation of Koopman eigenvalues for a one-dimensional cubic dynamics.
  • Figure 1: Effect of the kernel on the estimated spectrum accuracy $\textrm{ESA}_1$ (first order) and the spectral pollution measure $\textrm{SPM}$. The results are averaged over $50$ simulations.
  • Figure 1: True error (estimated spectrum accuracy $\textrm{ESA}_\alpha$ with (a) $\alpha=1$ and (b) $\alpha=2$) and error bounds on the computation of Koopman eigenvalues for the map \ref{['eq:disc_map']}. Each dot corresponds to one simulation. The solid lines represent the errors averaged over $50$ simulations.
  • Figure 2: Computation of Koopman eigenvalues for the Van der Pol dynamics.
  • Figure 3: Taylor approximation (to the 8th order) of a principal Koopman eigenfunction for the stable Van der Pol dynamics. (The colormap shows the logarithm of the absolute value (left) and the value of the argument (right).)
  • ...and 3 more figures

Theorems & Definitions (16)

  • Remark 2.1: Interpretation of the inner product approximation
  • Example 2.2
  • Remark 3.1: Equilibrium
  • Remark 3.2: Regularization
  • Remark 3.3: Non-orthonormal basis
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Proposition 6.1
  • Proof 1
  • ...and 6 more