Discretization and Statistical Consistency of Functional Flow Matching
Abstract
Functional flow matching learns to generate random functions, such as velocity fields or solutions of partial differential equations, by continuously transforming a simple probability distribution into a target distribution. In practice, however, training and sampling use only finitely many grid values or coefficients. Does refining these observations recover the intended function-space model, and how do observation, learning, and numerical errors affect the generated distribution? We address these questions by first proving that the velocity targets used for training converge in mean square when the finite reconstructions converge to the underlying functions and remain uniformly bounded. The observations need not be nested: each refinement may replace earlier sensor locations. We also give quantitative bounds for orthogonal projections and extend the result to point observations under suitable regularity and reconstruction assumptions. We then bound the discrepancy between the generated and target distributions in Wasserstein distance, separating observation, learning, numerical, and endpoint errors without requiring uniqueness of the population finite-dimensional dynamics. For a normalized quadrature neural operator representing the learned velocity field, we establish stability and learning bounds with constants independent of the number of sensors. Under fixed model dimension and boundedness assumptions, the statistical excess risk decreases at an inverse-sample-size rate, up to logarithmic factors. A Gaussian example combines these estimates into an explicit sampling-error rate.
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BibTeX
@misc{shikhman2026discretization,
title = {{Discretization and Statistical Consistency of Functional Flow Matching}},
author = {Lennon J. Shikhman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.04531},
eprint = {2608.04531},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04531},
}
RIS
TY - UNPB
TI - Discretization and Statistical Consistency of Functional Flow Matching
AU - Lennon J. Shikhman
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2608.04531
UR - https://arxiv.org/abs/2608.04531
AB - Functional flow matching learns to generate random functions, such as velocity fields or solutions of partial differential equations, by continuously transforming a simple probability distribution into a target distribution. In practice, however, training and sampling use only finitely many grid values or coefficients. Does refining these observations recover the intended function-space model, and how do observation, learning, and numerical errors affect the generated distribution? We address these questions by first proving that the velocity targets used for training converge in mean square when the finite reconstructions converge to the underlying functions and remain uniformly bounded. The observations need not be nested: each refinement may replace earlier sensor locations. We also give quantitative bounds for orthogonal projections and extend the result to point observations under suitable regularity and reconstruction assumptions. We then bound the discrepancy between the generated and target distributions in Wasserstein distance, separating observation, learning, numerical, and endpoint errors without requiring uniqueness of the population finite-dimensional dynamics. For a normalized quadrature neural operator representing the learned velocity field, we establish stability and learning bounds with constants independent of the number of sensors. Under fixed model dimension and boundedness assumptions, the statistical excess risk decreases at an inverse-sample-size rate, up to logarithmic factors. A Gaussian example combines these estimates into an explicit sampling-error rate.
ER -