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

arXiv: https://arxiv.org/abs/2610.00785 · HTML: https://mlsanity.com/paper/2610.00785/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 19 of its 30 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 19 of 19)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Self-Supervised Visual Planning with Temporal Skip Connections](https://mlsanity.com/paper/1710.05268/) (2017)
- [Bespoke Solvers for Generative Flow Models](https://mlsanity.com/paper/2310.19075/) (2023)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Towards Hierarchical Rectified Flow](https://mlsanity.com/paper/2502.17436/) (2025)
- [Terminal Velocity Matching](https://mlsanity.com/paper/2511.19797/) (2025)
- [High-Order Matching for One-Step Shortcut Diffusion Models](https://mlsanity.com/paper/2502.00688/) (2025)
- [Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows](https://mlsanity.com/paper/2503.09069/) (2025)
- [CurveFlow: Curvature-Guided Flow Matching for Image Generation](https://mlsanity.com/paper/2508.15093/) (2025)
- [Improving Rectified Flow with Boundary Conditions](https://mlsanity.com/paper/2506.15864/) (2025)
- [Improving Flow Matching by Aligning Flow Divergence](https://mlsanity.com/paper/2602.00869/) (2026)
- [OAT-FM: Optimal Acceleration Transport for Improved Flow Matching](https://mlsanity.com/paper/2509.24936/) (2025)

## Abstract (arXiv)

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.