Toeplitz Based Spectral Methods for Data-driven Dynamical Systems

2026-02-10

AI Summary

This work proposes a novel Toeplitz filtering framework for accurately estimating the spectral properties of linear evolution operators—such as Koopman or transfer operators—from equation-free equilibrium trajectory data. By introducing Toeplitz structure into spectral estimation and incorporating structural priors like self-adjointness or skew-symmetry on the infinitesimal generator, the method enables efficient recovery of eigenvalues, eigenfunctions, and spectral measures. Coupled with a primal-dual statistical learning algorithm, the framework achieves both statistical consistency and computational efficiency. Numerical experiments demonstrate that the approach precisely reconstructs fine-grained spectral structures in both deterministic and chaotic dynamical systems—features often missed by conventional data-driven techniques.

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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.

Problem

• spectral estimation
• dynamical systems
• Koopman operator
• data-driven
• Toeplitz

Innovation

• Toeplitz-based spectral methods
• data-driven dynamical systems
• Koopman operator
• infinitesimal generator
• spectral estimation