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

2026-09-30

AI Summary

This study addresses the limitations of first-order flow matching in modeling curved trajectories and its reliance on terminal velocities unavailable in static data by proposing a second-order flow matching framework. Methodologically, we derive closed-form solutions via acceleration energy minimization to capture complex transport geometries. Furthermore, a variational terminal velocity closure mechanism is introduced to unify pairing costs with training objectives, thereby overcoming the constraints of linear interpolation bridges. By integrating second-order phase-space dynamics, variational inference, and optimal transport theory, this work significantly enhances generation quality and transport geometry accuracy across low-dimensional datasets, physical fields, and CIFAR-10 benchmarks.

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.

Problem

• Flow Matching
• Optimal Transport
• Second-order Dynamics
• Terminal Velocity
• Generative Model

Innovation

• Flow Matching
• Variational Terminal-Velocity
• Second-order Dynamics
• Optimal Transport
• Minimum Acceleration