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Toeplitz Based Spectral Methods for Data-driven Dynamical Systems

Vladimir R. Kostic, Karim Lounici, Massimiliano Pontil

TL;DR

This work introduces a Toeplitz-based framework to estimate spectral objects of linear evolution operators (transfer/Koopman) from stationary trajectories without equation access. By representing analytic transforms of the generator as $F(L)=T(A_{ frac{}{ frac{ frac}}})$ and learning via RKHS-time-lag covariances, it unifies estimation of eigenstructure, resolvents, and frequency-selective filters in a scalable linear-algebraic scheme. The authors prove statistical consistency under $\beta$-mixing and derive practical estimators using primal and dual formulations with reduced-rank regularization, applicable to both stochastic and deterministic dynamics, including chaotic regimes where the spectrum is continuous. Empirical results on deterministic and chaotic Duffing systems illustrate improved spectral recovery and forecasting compared to standard data-driven methods, highlighting the framework’s potential for high-dimensional dynamical systems in physics and beyond.

Abstract

We introduce a Toeplitz-based framework for data-driven spectral estimation of linear evolution operators in dynamical systems. Focusing on transfer and Koopman operators from equilibrium trajectories without access to the underlying equations of motion, our method applies Toeplitz filters to the infinitesimal generator to extract eigenvalues, eigenfunctions, and spectral measures. Structural prior knowledge, such as self-adjointness or skew-symmetry, can be incorporated by design. The approach is statistically consistent and computationally efficient, leveraging both primal and dual algorithms commonly used in statistical learning. Numerical experiments on deterministic and chaotic systems demonstrate that the framework can recover spectral properties beyond the reach of standard data-driven methods.

Toeplitz Based Spectral Methods for Data-driven Dynamical Systems

TL;DR

This work introduces a Toeplitz-based framework to estimate spectral objects of linear evolution operators (transfer/Koopman) from stationary trajectories without equation access. By representing analytic transforms of the generator as and learning via RKHS-time-lag covariances, it unifies estimation of eigenstructure, resolvents, and frequency-selective filters in a scalable linear-algebraic scheme. The authors prove statistical consistency under -mixing and derive practical estimators using primal and dual formulations with reduced-rank regularization, applicable to both stochastic and deterministic dynamics, including chaotic regimes where the spectrum is continuous. Empirical results on deterministic and chaotic Duffing systems illustrate improved spectral recovery and forecasting compared to standard data-driven methods, highlighting the framework’s potential for high-dimensional dynamical systems in physics and beyond.

Abstract

We introduce a Toeplitz-based framework for data-driven spectral estimation of linear evolution operators in dynamical systems. Focusing on transfer and Koopman operators from equilibrium trajectories without access to the underlying equations of motion, our method applies Toeplitz filters to the infinitesimal generator to extract eigenvalues, eigenfunctions, and spectral measures. Structural prior knowledge, such as self-adjointness or skew-symmetry, can be incorporated by design. The approach is statistically consistent and computationally efficient, leveraging both primal and dual algorithms commonly used in statistical learning. Numerical experiments on deterministic and chaotic systems demonstrate that the framework can recover spectral properties beyond the reach of standard data-driven methods.
Paper Structure (8 sections, 5 theorems, 57 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 8 sections, 5 theorems, 57 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Linear operator perspective to stable dynamical systems
  3. Statistical learning of analytic transforms of the generator
  4. Toeplitz based empirical estimators
  5. Statistically consistent spectral Toeplitz estimators
  6. Implications.
  7. Experiments
  8. Discussion

Key Result

Proposition 3.1

Given $\Delta t>0$, let $C_{j}\colon\mathcal{H}\to\mathcal{H}$ be the $j\Delta t$-time-lagged cross-covariance operator Then the solution of the inverse problem eq:inv_prob is given by $G_{\mathcal{H}} = C_{0}^\dagger W_a$ , where represents the $a$-weighted time-lagged cross-covariance operator.

Figures (3)

  • Figure 1: Duffing oscilator on a simple attractor: training trajectory and samples (left), and estimated spectrum for three different models (right). True base frequency is $1/2\pi \approx 0.1592$.
  • Figure 2: Learning the Duffing oscillator on a simple attractor with different Toeplitz based transforms: preserving the spectral properties reduces the model's complexity and improves performance even with highly noisy observations. Bottom row: Toeplitz spectral filtering isolates the relevant frequencies (eigenvalues) in the range $(0.1, 10)$. Left: A low rank ($10$) is sufficient for accurate test-trajectory forecasting. Right: Increasing the rank to $100$ provides no significant improvement and instead seems to slightly increase noise-related variability.
  • Figure 3: Duffing Oscillator on strange attractor. Training trajectory and samples on the left, and resolvent response of velocity along the Nyquist region on the right.

Theorems & Definitions (13)

  • Example 2.1: Overdamped Langevin
  • Example 2.2: Ornstein-Uhlenbeck process
  • Example 2.3: Duffing oscillator
  • Proposition 3.1
  • proof
  • Theorem 3.2: Crouzeix's Theorem, see crouzeix2017numerical
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 3 more