解析流匹配中应变与涡量对积分误差的影响，指导更高效采样。
On the Role of Strain and Vorticity in Numerical Integration Error for Flow Matching
- 分解速度场雅可比矩阵，区分应变与涡量作用机制。
- 应变主导误差指数增长，涡量仅线性贡献局部误差。
- 加权正则化提升效率，5步内误差降低2.7倍，适配低算力场景。
流匹配通过积分学习到的速度场生成数据，积分步数（NFE）直接决定推理成本。我们分析速度场的哪些属性控制积分误差，将速度场的雅可比矩阵分解为对称部分S（应变率）和反对称部分Ω（涡量）。证明应变通过对数范数控制误差的指数放大，而涡量仅对局部截断误差产生线性影响。进一步表明最优传输速度场无旋且材料导数为零，意味着二阶欧拉精度；对于精确位移插值，对应的拉格朗日粒子动力学可通过欧拉法精确积分。基于此分析，研究了应变权重α和涡量权重β的加权雅可比正则化。在2D合成数据上的实验验证了主要理论预测，显示在NFE=5时积分误差最多降低2.7倍。初步的CIFAR-10实验显示一致趋势：轻量微调使FID提升14%（NFE=10），同时保持高步数下的高质量。
原文摘要 · Abstract (English)
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.
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