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arXiv · 2610.00785

VTV-FM: Flow Matching through Variational Terminal-Velocity Closure

Haoyang Jiang·Yuheng Li·Di Yang·Yanhai Xiong·Haipeng Chen·Yi He
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Abstract

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.

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Haoyang Jiang, Yuheng Li, Di Yang, Yanhai Xiong, Haipeng Chen, Yi He. 2026-09-30. VTV-FM: Flow Matching through Variational Terminal-Velocity Closure. https://arxiv.org/abs/2610.00785

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