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

arXiv: https://arxiv.org/abs/2509.25000 · HTML: https://mlsanity.com/paper/2509.25000/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2025.

## Built on by (most cited first, 1 of 1)

- [Learning Passive Continuous-Time Dynamics with Multistep Port-Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2510.00384/) (2025)

## Abstract (arXiv)

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.