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Group-Convolutional Extended Dynamic Mode Decomposition

Hans Harder, Feliks Nüske, Friedrich M. Philipp, Manuel Schaller, Karl Worthmann, Sebastian Peitz

TL;DR

This work addresses the challenge of learning finite-dimensional representations of the Koopman operator for high-dimensional, symmetry-rich dynamical systems. It introduces a group-convolutional EDMD framework, proving that the optimal EDMD matrix is equivariant under the system's symmetry group and can be represented implicitly by a convolution kernel learned in Fourier space via the generalized Fourier transform. The approach reduces data and computational requirements and enables fast eigenfunction estimation, demonstrated on two nonlinear PDEs: the Kuramoto–Sivashinsky equation and a high-dimensional spiraling-waves system. Empirical results show strong performance in low-data regimes and scalability to thousands of observables, underscoring the method's potential for efficient spectral analysis of equivariant dynamical systems. This work broadens the toolkit for Koopman analysis by harnessing group theory and convolutional structures to exploit symmetry in learning and prediction tasks.

Abstract

This paper explores the integration of symmetries into the Koopman-operator framework for the analysis and efficient learning of equivariant dynamical systems using a group-convolutional approach. Approximating the Koopman operator by finite-dimensional surrogates, e.g., via extended dynamic mode decomposition (EDMD), is challenging for high-dimensional systems due to computational constraints. To tackle this problem with a particular focus on EDMD, we demonstrate -- under suitable equivarance assumptions on the system and the observables -- that the optimal EDMD matrix is equivariant. That is, its action on states can be described by group convolutions and the generalized Fourier transform. We show that this structural property has many advantages for equivariant systems, in particular, that it allows for data-efficient learning, fast predictions and fast eigenfunction approximations. We conduct numerical experiments on the Kuramoto-Sivashinsky equation and a $λ$-$ω$ spiraling wave system, both nonlinear partial differential equations, providing evidence of the effectiveness of this approach, and highlighting its potential for broader applications in dynamical systems analysis.

Group-Convolutional Extended Dynamic Mode Decomposition

TL;DR

This work addresses the challenge of learning finite-dimensional representations of the Koopman operator for high-dimensional, symmetry-rich dynamical systems. It introduces a group-convolutional EDMD framework, proving that the optimal EDMD matrix is equivariant under the system's symmetry group and can be represented implicitly by a convolution kernel learned in Fourier space via the generalized Fourier transform. The approach reduces data and computational requirements and enables fast eigenfunction estimation, demonstrated on two nonlinear PDEs: the Kuramoto–Sivashinsky equation and a high-dimensional spiraling-waves system. Empirical results show strong performance in low-data regimes and scalability to thousands of observables, underscoring the method's potential for efficient spectral analysis of equivariant dynamical systems. This work broadens the toolkit for Koopman analysis by harnessing group theory and convolutional structures to exploit symmetry in learning and prediction tasks.

Abstract

This paper explores the integration of symmetries into the Koopman-operator framework for the analysis and efficient learning of equivariant dynamical systems using a group-convolutional approach. Approximating the Koopman operator by finite-dimensional surrogates, e.g., via extended dynamic mode decomposition (EDMD), is challenging for high-dimensional systems due to computational constraints. To tackle this problem with a particular focus on EDMD, we demonstrate -- under suitable equivarance assumptions on the system and the observables -- that the optimal EDMD matrix is equivariant. That is, its action on states can be described by group convolutions and the generalized Fourier transform. We show that this structural property has many advantages for equivariant systems, in particular, that it allows for data-efficient learning, fast predictions and fast eigenfunction approximations. We conduct numerical experiments on the Kuramoto-Sivashinsky equation and a - spiraling wave system, both nonlinear partial differential equations, providing evidence of the effectiveness of this approach, and highlighting its potential for broader applications in dynamical systems analysis.

Paper Structure

This paper contains 28 sections, 4 theorems, 55 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions and organization
  3. Notation
  4. Code and resources
  5. Preliminaries: The Koopman operator, EDMD, and symmetries
  6. Groups and group actions
  7. Equivariant matrices
  8. Convolution kernels
  9. The generalized Fourier transform
  10. Group representations
  11. The generalized Fourier transform
  12. Eigenvalues and eigenvectors
  13. Left eigenpairs
  14. Plancherel's identity
  15. Abelian groups
  16. ...and 13 more sections

Key Result

Lemma 1

Suppose $\mathcal{G}$ acts on $\mathcal{O}$ from the right, $K \in \mathbb{C}^{\mathcal{O} \times \mathcal{O}}$ is a matrix, and a left group action on $\mathbb{C}^\mathcal{O}$ is defined via $(g \cdot y)_o \coloneqq y_{o \cdot g^{-1}}$ for all $y \in \mathbb{C}^\mathcal{O}$ and $o \in \mathcal{O}$.

Figures (5)

  • Figure 1: A spring system of four nodes connected in a row with equal stiffness coefficients. This system is equivariant with respect to reflections along the center.
  • Figure 2: Left: Dataset used for the KS equation. The small dataset contains the system trajectory up until $t = 20$, the large dataset contains the system trajectory up until $t = 300$. Right: Comparison of the generated trajectories of the convolutional vs. full-scale EDMD approach on the test data.
  • Figure 3: Left: Computed eigenvalues of both approaches on both datasets. Each eigenvalue is colored by $\mathrm{SRR}_{\mathrm{eig}}$, allowing us to evaluate the quality of the computed eigenpair (smaller is better). Right: comparison of the leading eigenvalues on the large dataset.
  • Figure 4: Top two rows: datasets generated by solving the spiraling waves equation starting from different initial conditions. Bottom row: trajectory generated by convolutional EDMD on the test data.
  • Figure 5: Left: Density plot of all eigenvalues (ca. 69k) computed by convolutional EDMD. Right: randomly sampled eigenvalues colored by $\mathrm{SRR}_{\mathrm{eig}}$.

Theorems & Definitions (21)

  • Example 2.1
  • Example 2.2: Orbits
  • Example 2.3: Actions on PDEs
  • Remark 1: Orbits and free actions
  • Example 2.4
  • Lemma 1
  • proof
  • Example 2.5: Circulant matrices; heat equation
  • Example 2.6: Block-circulant matrices
  • Example 2.7: Symmetric spring system
  • ...and 11 more