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Shaping the Koopman dictionary by learning on the Grassmannian

Roland Schurig, Pieter van Goor, Karl Worthmann, Rolf Findeisen

TL;DR

This work addresses shaping the Koopman dictionary for nonlinear dynamics by learning on the Grassmannian to produce approximately invariant subspaces. It combines EDMD with a subspace design that fixes a prior observable set and optimizes the remaining components via a trust-region method on the Grassmann manifold, ensuring efficient, basis-invariant state predictions. The approach yields improved prediction accuracy and generalization, demonstrated on a Duffing oscillator, and offers a flexible framework for incorporating system structure and alternative optimization criteria. By leveraging differential-geometric concepts, the paper contributes a principled, scalable route to data-driven Koopman surrogates for complex dynamics.

Abstract

Extended dynamic mode decomposition (EDMD) is a powerful tool to construct linear predictors of nonlinear dynamical systems by approximating the action of the Koopman operator on a subspace spanned by finitely many observable functions. However, its accuracy heavily depends on the choice of the observables, which remains a challenge. We propose a systematic framework to identify and shape observable dictionaries, reduce projection errors, and achieve approximately invariant subspaces. To this end, we leverage optimisation on the Grassmann manifold and exploit inherent geometric properties for computational efficiency. Numerical results demonstrate improved prediction accuracy and efficiency. In conclusion, we propose a novel approach to efficiently shape the Koopman dictionary using differential-geometric concepts for optimisation on manifolds resulting in enhanced data-driven Koopman surrogates for nonlinear dynamical systems.

Shaping the Koopman dictionary by learning on the Grassmannian

TL;DR

This work addresses shaping the Koopman dictionary for nonlinear dynamics by learning on the Grassmannian to produce approximately invariant subspaces. It combines EDMD with a subspace design that fixes a prior observable set and optimizes the remaining components via a trust-region method on the Grassmann manifold, ensuring efficient, basis-invariant state predictions. The approach yields improved prediction accuracy and generalization, demonstrated on a Duffing oscillator, and offers a flexible framework for incorporating system structure and alternative optimization criteria. By leveraging differential-geometric concepts, the paper contributes a principled, scalable route to data-driven Koopman surrogates for complex dynamics.

Abstract

Extended dynamic mode decomposition (EDMD) is a powerful tool to construct linear predictors of nonlinear dynamical systems by approximating the action of the Koopman operator on a subspace spanned by finitely many observable functions. However, its accuracy heavily depends on the choice of the observables, which remains a challenge. We propose a systematic framework to identify and shape observable dictionaries, reduce projection errors, and achieve approximately invariant subspaces. To this end, we leverage optimisation on the Grassmann manifold and exploit inherent geometric properties for computational efficiency. Numerical results demonstrate improved prediction accuracy and efficiency. In conclusion, we propose a novel approach to efficiently shape the Koopman dictionary using differential-geometric concepts for optimisation on manifolds resulting in enhanced data-driven Koopman surrogates for nonlinear dynamical systems.

Paper Structure

This paper contains 18 sections, 17 theorems, 115 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Koopman Framework
  3. Koopman Linear Systems
  4. State Predictions in the Koopman Framework
  5. Invariant Subspaces for Prediction
  6. Computing Compressions from Data
  7. Extended Dynamic Mode Decomposition
  8. Subspace Extended Dynamic Mode Decomposition
  9. Finding an Optimal Subspace
  10. Optimisation over the Grassmann Manifold
  11. Riemannian structures of $\mathcal{ST}(r,d)$ and $\mathcal{GR}(r,d)$
  12. Simulation
  13. Conclusion
  14. Compression under a change-of-basis
  15. Coordinate Projection
  16. ...and 3 more sections

Key Result

Lemma 1

Let $V$ be a finite-dimensional subspace of $H$. Let $B = (\psi_1, \dots, \psi_M)$ and $E = (\varphi_1, \dots, \varphi_M)$ be two bases for $V$. Let $P \in \mathrm{GL}({M})$ be the change-of-basis matrix from $B$ to $E$. Then, for each $x_0 \in X$, the Koopman predictions $z_B(\cdot,x_0)$ relative t

Figures (8)

  • Figure 1: Estimation in lifted space for $n=1$, $M=2$ and $\Psi \colon x \mapsto (\cos(x), \sin(x))$.
  • Figure 2: Sketch of $V = T \oplus R$ and $W = T \oplus S \subseteq V$, with $R = S \oplus Q$.
  • Figure 3: Subspace $S$, together with the basis given by $U_2$ (green) and $U_3$ (orange).
  • Figure 4: Absolute difference between the mean errors of the state predictions over the horizon $N=20$ based on the reduced and the full dictionary on the domain $[-1,1] \times [-1,1]$.
  • Figure 5: Absolute difference between the mean errors of the state predictions over the horizon $N=20$ based on the reduced and the full dictionary on the domain $[-2,2] \times [-2,2]$.
  • ...and 3 more figures

Theorems & Definitions (40)

  • Lemma 1
  • Remark 1
  • Corollary 2
  • Corollary 3
  • proof
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • proof
  • ...and 30 more