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

Chi Ho Leung, Philip E. Paré

TL;DR

Spectral Flow Learning (SFL) addresses the problem of identifying continuous-time vector fields from irregular trajectories by learning in a windowed flow space with a Koopman-lag forcing operator that aggregates lagged actions. The authors develop a spectral-regularization approach with qualification-controlled filters to obtain finite-sample high-probability bounds for flow prediction and, via a multistep observability inequality and a uniform local truncation error bound, lift these guarantees to the vector field. The analysis encompasses variable-step linear multistep methods (AB/AM/BDF) with bounded step ratios and order-p consistency, and accommodates physics-informed vvRKHS designs to enforce dynamical constraints. The resulting two-term FS-HP bound separates a statistical rate, driven by the effective sample size $\\ell$, from a discretization bias governed by the expected anchor step size $   $ and $p$, enabling principled tradeoffs between data and discretization. This framework offers rigorous guarantees for sequentially learned flows and their lifted vector fields in identifying governing dynamics from irregular data.

Abstract

We study the identification of continuous-time vector fields from irregularly sampled trajectories. We introduce Spectral Flow Learning (SFL), 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 (vLLM). 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. This preliminary preprint states the results and sketches proofs, with full proofs and extensions deferred to a journal version.

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

TL;DR

Spectral Flow Learning (SFL) addresses the problem of identifying continuous-time vector fields from irregular trajectories by learning in a windowed flow space with a Koopman-lag forcing operator that aggregates lagged actions. The authors develop a spectral-regularization approach with qualification-controlled filters to obtain finite-sample high-probability bounds for flow prediction and, via a multistep observability inequality and a uniform local truncation error bound, lift these guarantees to the vector field. The analysis encompasses variable-step linear multistep methods (AB/AM/BDF) with bounded step ratios and order-p consistency, and accommodates physics-informed vvRKHS designs to enforce dynamical constraints. The resulting two-term FS-HP bound separates a statistical rate, driven by the effective sample size , from a discretization bias governed by the expected anchor step size and , enabling principled tradeoffs between data and discretization. This framework offers rigorous guarantees for sequentially learned flows and their lifted vector fields in identifying governing dynamics from irregular data.

Abstract

We study the identification of continuous-time vector fields from irregularly sampled trajectories. We introduce Spectral Flow Learning (SFL), 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 (vLLM). 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. This preliminary preprint states the results and sketches proofs, with full proofs and extensions deferred to a journal version.

Paper Structure

This paper contains 20 sections, 4 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup and Problem Formulation
  3. Notation
  4. Dynamical System and Flows
  5. Design Law (state visitation distribution)
  6. Step Law (time‑advance distribution)
  7. Problem Formulation (FS-HP bound)
  8. Variable Step Linear Multistep Method (vLMM) Theory
  9. General Form
  10. Assumptions for Deterministic Bias
  11. Spectral Regularization Learning Theory
  12. Population and Sampling Hilbert Spaces
  13. Inclusion Maps and Other Important Operators
  14. Filter Qualification and Lipschitz Condition
  15. Source Condition
  16. ...and 5 more sections

Key Result

Lemma 1

Let: then:

Figures (1)

  • Figure 1: Dependency graph linking assumptions, lemmas, theorems, corollaries, and propositions used in the SFL analysis.

Theorems & Definitions (14)

  • Remark 2.1
  • Definition 1: vLMM General Form
  • Definition 2: Order-$p$ consistency
  • Definition 3
  • Remark 4.1: Compatibility/transferring source between levels
  • Definition 4: Label Map and vLMM residual
  • Lemma 1: Observability via $\mathsf{B}$
  • proof
  • Lemma 2: Uniform LTE for vLMM residual
  • proof : Proof
  • ...and 4 more