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Isokinetic Flow Matching for Pathwise Straightening of Generative Flows

Tauhid Khan

Abstract

Flow Matching (FM) constructs linear conditional probability paths, but the learned marginal velocity field inevitably exhibits strong curvature due to trajectory superposition. This curvature severely inflates numerical truncation errors, bottlenecking few-step sampling. To overcome this, we introduce Isokinetic Flow Matching (Iso-FM), a lightweight, Jacobian-free dynamical regularizer that directly penalizes pathwise acceleration. By using a self-guided finite-difference approximation of the material derivative Dv/Dt, Iso-FM enforces local velocity consistency without requiring auxiliary encoders or expensive second-order autodifferentiation. Operating as a pure plug-and-play addition to single-stage FM training, Iso-FM dramatically improves few-step generation. On CIFAR-10 (DiT-S/2), Iso-FM slashes conditional non-OT FID at 2 steps from 78.82 to 27.13 - a 2.9x relative efficiency gain - and reaches a best-observed FID at 4 steps of 10.23. These results firmly establish acceleration regularization as a principled, compute-efficient mechanism for fast generative sampling.

Isokinetic Flow Matching for Pathwise Straightening of Generative Flows

Abstract

Flow Matching (FM) constructs linear conditional probability paths, but the learned marginal velocity field inevitably exhibits strong curvature due to trajectory superposition. This curvature severely inflates numerical truncation errors, bottlenecking few-step sampling. To overcome this, we introduce Isokinetic Flow Matching (Iso-FM), a lightweight, Jacobian-free dynamical regularizer that directly penalizes pathwise acceleration. By using a self-guided finite-difference approximation of the material derivative Dv/Dt, Iso-FM enforces local velocity consistency without requiring auxiliary encoders or expensive second-order autodifferentiation. Operating as a pure plug-and-play addition to single-stage FM training, Iso-FM dramatically improves few-step generation. On CIFAR-10 (DiT-S/2), Iso-FM slashes conditional non-OT FID at 2 steps from 78.82 to 27.13 - a 2.9x relative efficiency gain - and reaches a best-observed FID at 4 steps of 10.23. These results firmly establish acceleration regularization as a principled, compute-efficient mechanism for fast generative sampling.

Paper Structure

This paper contains 19 sections, 50 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Flow Matching and Eulerian Generative Models
  4. Flow Straightening and Acceleration Control
  5. Flow Map and Consistency Methods
  6. Background: Eulerian and Lagrangian Perspectives
  7. Isokinetic Flow Matching (Iso-FM)
  8. Dynamical Motivation: Acceleration and Sampling Error
  9. The MeanFlow Identity (Optional Connection)
  10. Practical Self-Guided Iso-FM
  11. Algorithmic Training Step.
  12. Optional OT Path Coupling
  13. Interpretation and Path Geometry
  14. Low-Dimensional Path Geometry Analysis.
  15. Experiments
  16. ...and 4 more sections

Figures (6)

  • Figure 1: Teaser: Iso-FM suppresses path bending in low-dimensional transport. Left: baseline FM trajectories are more curved than Iso-FM trajectories. Right: Iso-FM yields lower curvature, consistent with reducing the material acceleration $Dv/Dt$.
  • Figure 2: Low-dimensional geometric diagnostics of learned flows. Left (trajectory comparison): baseline FM paths exhibit stronger bending and non-uniform directional changes, while Iso-FM trajectories are more linear between source and target regions. Right (curvature comparison): the Iso-FM model maintains consistently lower curvature magnitude across integration time, empirically supporting the theoretical objective of acceleration suppression.
  • Figure 3: Conditional CIFAR-10 training dynamics (DiT-S/2): FID@2 and FID@4 versus epoch for FM, Iso-FM, and OT+Iso-FM.
  • Figure 4: Unconditional CIFAR-10 training dynamics (DiT-S/2): FID@2 and FID@4 versus epoch for FM, Iso-FM, OT+FM, and OT+Iso-FM.
  • Figure 5: Conditional samples: FM (left, epoch 1250) vs Iso-FM (right, epoch 1250).
  • ...and 1 more figures