Toeplitz Based Spectral Methods for Data-driven Dynamical Systems
This work addresses the challenge of data-driven spectral estimation in dynamical systems for researchers and practitioners, representing an incremental improvement with a novel method for a known bottleneck.
The authors tackled the problem of estimating spectral properties of linear evolution operators in dynamical systems from equilibrium trajectory data without known equations, achieving recovery of spectral properties beyond standard data-driven methods.
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.