On the Role of Strain and Vorticity in Numerical Integration Error for Flow Matching

TL;DR

This paper analyzes how strain and vorticity affect numerical integration errors in flow matching, revealing their distinct roles and proposing regularization techniques to improve accuracy and efficiency.

Contribution

It provides a theoretical decomposition of velocity Jacobian into strain and vorticity, showing their different impacts on integration error and proposing regularization methods.

Findings

Strain controls exponential error amplification via the logarithmic norm.

Vorticity contributes only linearly to local truncation error.

Weighted Jacobian regularization improves integration accuracy and FID scores.

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.