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Transition Flow Matching

Chenrui Ma

Abstract

Mainstream flow matching methods typically focus on learning the local velocity field, which inherently requires multiple integration steps during generation. In contrast, Mean Velocity Flow models establish a relationship between the local velocity field and the global mean velocity, enabling the latter to be learned through a mathematically grounded formulation and allowing generation to be transferred to arbitrary future time points. In this work, we propose a new paradigm that directly learns the transition flow. As a global quantity, the transition flow naturally supports generation in a single step or at arbitrary time points. Furthermore, we demonstrate the connection between our approach and Mean Velocity Flow, establishing a unified theoretical perspective. Extensive experiments validate the effectiveness of our method and support our theoretical claims.

Transition Flow Matching

Abstract

Mainstream flow matching methods typically focus on learning the local velocity field, which inherently requires multiple integration steps during generation. In contrast, Mean Velocity Flow models establish a relationship between the local velocity field and the global mean velocity, enabling the latter to be learned through a mathematically grounded formulation and allowing generation to be transferred to arbitrary future time points. In this work, we propose a new paradigm that directly learns the transition flow. As a global quantity, the transition flow naturally supports generation in a single step or at arbitrary time points. Furthermore, we demonstrate the connection between our approach and Mean Velocity Flow, establishing a unified theoretical perspective. Extensive experiments validate the effectiveness of our method and support our theoretical claims.
Paper Structure (57 sections, 4 theorems, 76 equations, 3 figures, 3 tables, 4 algorithms)

This paper contains 57 sections, 4 theorems, 76 equations, 3 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Flow Matching and Diffusion.
  4. Consistency Model and Mean Velocity Model.
  5. Transition Models and Flow Map.
  6. Preliminaries
  7. Notation and setup
  8. Random variables and realizations.
  9. Source/target distributions and coupling.
  10. Conditioning variable and conditional paths.
  11. General interpolant.
  12. Flow Matching
  13. Velocity fields.
  14. Generation ODE and continuity equation.
  15. Learning Flow Matching.
  16. ...and 42 more sections

Key Result

theorem 1

The gradients of the marginal Flow Matching loss and the conditional Flow Matching loss coincide: In particular, the minimizer of the conditional Flow Matching loss is the marginal velocity $v(x_t,t)$.

Figures (3)

  • Figure 1: 2D Generation Trajectory Visualization on Synthetic Data. Tested methods include: Flow Matchinglipman2023flow, Rectified Flowliu2023flow, Flow Map Matchingboffi2024flow, MeanFlowgeng2025mean.
  • Figure 2: Generation Results on ImageNet-256 under Varying NFE. As the number of function evaluations (NFE) increases from 1 to 10, the generated images exhibit progressively improved detail and fidelity. Notably, even single-step generation already produces reasonably good results.
  • Figure : TFM: Multi-step Sampling with Arbitrary Step Sizes

Theorems & Definitions (10)

  • theorem 1: Gradient equivalence of Flow Matching lipman2024flowmatchingguidecode
  • remark 1: Standard Flow Matching
  • theorem 2: Gradient equivalence of Transition Flow Matching
  • remark 2: Standard Transition Flow Matching
  • theorem 3: Gradient equivalence of Flow Matching lipman2024flowmatchingguidecode
  • remark 3: Standard Flow Matching
  • theorem 4: Gradient equivalence of Transition Flow Matching
  • proof : Proof of Theorem \ref{['proof:thm:MequivC']}
  • remark 4: Minimizer
  • remark 5: Standard Transition Flow Matching