发表机构
William & Mary（威廉与玛丽学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
VTV-FM提出二阶流匹配框架，通过最小化加速度能量闭式推导终端速度，改进传输几何，在低维数据、物理场和CIFAR-10上超越一阶和高阶基线。
AI 中文摘要
流匹配（FM）通过拟合从简单源分布到数据分布的连续时间运动来学习生成性传输。大多数现有方法使用一阶桥：一旦源样本和目标样本配对，路径就是具有恒定速度的直线运动。具有最优传输（OT）的FM改进了配对，但桥本身仍然是线性的，限制了其建模曲线运动、加速度和方向变化的能力。一种自然的补救措施是使用二阶相空间动力学；然而，学习桥需要目标侧的终端速度信息，而静态数据集不提供这些信息。我们提出了变分终端速度流匹配（VTV-FM），这是一个二阶FM框架，通过最小化加速度能量来推导缺失的速度，为静态数据产生闭式闭合。相同的最小加速度变分构造也定义了用于训练的OT配对成本和加速度目标。在低维数据集、PDE控制的物理场和CIFAR-10上的实验表明，VTV-FM在传输几何和生成质量上优于一阶和高阶FM基线。
英文摘要
Flow matching (FM) learns generative transport by fitting continuous-time motion from a simple source distribution to the data distribution. Most existing methods use first-order bridges: once a source and a target sample are paired, the path is a straight motion with constant velocity. FM with optimal transport (OT) improves the pairing, but the bridge itself remains linear, limiting its ability to model curved motion, acceleration, and changing directions. A natural remedy is to use second-order phase-space dynamics; however, learning the bridge requires target-side terminal-velocity information that static datasets do not provide. We propose Variational Terminal-Velocity Flow Matching (VTV-FM), a second-order FM framework that derives the missing velocity by minimizing acceleration energy, yielding a closed-form closure for static data. The same minimum-acceleration variational construction also defines the OT pairing cost and the acceleration targets used for training. Experiments on low-dimensional datasets, PDE-governed physical fields, and CIFAR-10 show that VTV-FM improves transport geometry and generation quality over first-order and high-order FM baselines.
CommentsAccepted at NeurIPS 2026. Code: https://github.com/HaoyangJiang-WM/VTV-FM
