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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Cite this article
APA 7
Jiang, H., Li, Y., Yang, D., Xiong, Y., Chen, H., & He, Y. (2026). VTV-FM: Flow Matching through Variational Terminal-Velocity Closure. https://omanscience.com/en/articles/vtv-fm-flow-matching-through-variational-terminal-velocity-closure
MLA 9
Jiang, Haoyang, et al. "VTV-FM: Flow Matching through Variational Terminal-Velocity Closure." https://omanscience.com/en/articles/vtv-fm-flow-matching-through-variational-terminal-velocity-closure.
Chicago (author–date)
Jiang, Haoyang, Yuheng Li, Di Yang, Yanhai Xiong, Haipeng Chen, and Yi He. 2026. "VTV-FM: Flow Matching through Variational Terminal-Velocity Closure." https://omanscience.com/en/articles/vtv-fm-flow-matching-through-variational-terminal-velocity-closure.
Harvard
Jiang, H., Li, Y., Yang, D., Xiong, Y., Chen, H. and He, Y. (2026) 'VTV-FM: Flow Matching through Variational Terminal-Velocity Closure', Available at: https://omanscience.com/en/articles/vtv-fm-flow-matching-through-variational-terminal-velocity-closure.
Vancouver
Jiang H, Li Y, Yang D, Xiong Y, Chen H, He Y. VTV-FM: Flow Matching through Variational Terminal-Velocity Closure. https://omanscience.com/en/articles/vtv-fm-flow-matching-through-variational-terminal-velocity-closure
IEEE
H. Jiang, Y. Li, D. Yang, Y. Xiong, H. Chen, and Y. He, "VTV-FM: Flow Matching through Variational Terminal-Velocity Closure," https://omanscience.com/en/articles/vtv-fm-flow-matching-through-variational-terminal-velocity-closure.