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Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

Gustav Conradie, Nicolas Boullé, Jean-Christophe Loiseau, Steven L. Brunton, Matthew J. Colbrook

Abstract

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.

Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

Abstract

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.
Paper Structure (40 sections, 4 theorems, 68 equations, 14 figures, 1 table, 6 algorithms)

This paper contains 40 sections, 4 theorems, 68 equations, 14 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Data-driven dynamical systems
  3. The Koopman operator
  4. Square-integrable functions
  5. Reproducing kernel Hilbert spaces (RKHS)
  6. Controlling projection errors
  7. Galerkin methods
  8. Koopman mode decomposition
  9. Kernelized EDMD
  10. Computing spectra
  11. Related methods
  12. Koopman projection error bounds
  13. Estimation and numerical error
  14. Subspace invariance
  15. Other data-driven techniques
  16. ...and 25 more sections

Key Result

Theorem 3.1

Let $\theta_j(M,\epsilon_c;\mathcal{V},\mathcal{K}\mathcal{V})$ denote the output of alg:resDMD_angle for subspaces $\mathcal{V}$ and $\mathcal{K}\mathcal{V}$ using $M$ snapshots and cut-off parameter $\epsilon_c$, and let $\lambda_1$ be the smallest positive eigenvalue of $\lim_{M\rightarrow\infty}

Figures (14)

  • Figure 1: Workflow for computing error bounds on Koopman operators and learning a dictionary.
  • Figure 1: Left: The Duffing attractor. Middle: Principal angles on the Duffing oscillator for a finite subspace of Chebyshev polynomials. Right: One-step prediction errors for the corresponding principal observables. The solid line denotes a piecewise-linear best fit with a single breakpoint.
  • Figure 1: Sharpness of bounds. The exact errors and error bound formula to first-order (minimum of \ref{['KMD_bound_first_order', 'KMD_bound_first_order_2']}) and full-order (minimum of \ref{['bound_lemma', 'bound_lemma2']}) for a system with Lebesgue spectrum.
  • Figure 1: $L^2$ (top) and RKHS (bottom) norm forecast errors, error bounds and expected bounds for the Duffing oscillator (left) and Lorenz system (right), along with variance error bars including three standard deviations.
  • Figure 1: Left: Orbits of moons in the Pluto--Charon system with Pluto at the origin. Right: RKHS-norm errors, strict error bounds, expected errors and three standard deviation error bars for the Pluto--Charon system. While the strict bounds overestimate the true error, the expected errors track it closely.
  • ...and 9 more figures

Theorems & Definitions (8)

  • Theorem 3.1: Large data convergence
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Theorem 4.1: KMD projection error
  • Lemma 4.2
  • Proof 3
  • Proof 4: Proof of \ref{['kmd_norm_bound_thm']}