Spectral Flow Learning Theory: Finite-Sample Guarantees for Vector-Field Identification

Chi Ho Leung; Philip E. Paré

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arXiv · 2509.25000

Spectral Flow Learning Theory: Finite-Sample Guarantees for Vector-Field Identification

Chi Ho Leung·Philip E. Paré

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Abstract

We study the identification of continuous-time vector fields from irregularly sampled trajectories. We introduce spectral flow learning, which learns in a windowed flow space using a lag-linear label operator that aggregates lagged Koopman actions. We provide finite-sample, high-probability (FS-HP) guarantees for the class of variable-step linear multistep methods (vLMM). The FS-HP rates are constructed using spectral regularization with qualification-controlled filters for flow predictors under standard source and filter assumptions. A multistep observability inequality links flow error to vector-field error and yields two-term bounds that combine a statistical rate with an explicit discretization bias from vLMM theory. Simulations on a controlled mass-spring system corroborate the theory and clarify conditioning, step-sample tradeoffs, and practical implications.

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BibTeX RIS

Chi Ho Leung, Philip E. Paré. 2025-09-29. Spectral Flow Learning Theory: Finite-Sample Guarantees for Vector-Field Identification. https://arxiv.org/abs/2509.25000

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