[2605.06680] On the Role of Strain and Vorticity in Numerical Integration Error for Flow Matching
[Submitted on 22 Apr 2026]
Title:On the Role of Strain and Vorticity in Numerical Integration Error for Flow Matching
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Abstract:Flow matching generates data by integrating a learned velocity field, where the number of integration steps (NFE) directly determines inference cost. We analyze which properties of the velocity field govern integration error by decomposing the velocity Jacobian into its symmetric part S (strain rate) and antisymmetric part Omega (vorticity). We prove that strain and vorticity play different roles: strain controls exponential error amplification through the logarithmic norm, while vorticity contributes only linearly to the local truncation error. We further show that the optimal transport velocity field is irrotational and has zero material derivative, implying second-order Euler accuracy; for exact displacement interpolation, the associated Lagrangian particle dynamics are integrated exactly by Euler. Motivated by this analysis, we study weighted Jacobian regularization with strain weight alpha and vorticity weight beta. Experiments on 2D synthetic data confirm the main theoretical predictions, showing up to 2.7x lower integration error at NFE=5. Preliminary CIFAR-10 experiments show consistent trends, with a lightweight fine-tuning procedure improving FID by 14 percent at NFE=10 while preserving high-NFE quality.
Comments: 16 pages, 7 figures. Preliminary version. Includes qualitative CIFAR-10 comparison and supporting synthetic experiments
Subjects:
Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV); Fluid Dynamics (physics.flu-dyn)
Cite as: arXiv:2605.06680 [cs.LG]
(or arXiv:2605.06680v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.06680
arXiv-issued DOI via DataCite
Submission history
From: Chenxi Tao [view email] [v1] Wed, 22 Apr 2026 21:34:37 UTC (15,760 KB)
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