Isokinetic Flow Matching for Pathwise Straightening of Generative Flows

Tauhid Khan

cs.LGApr 6, 2026

Isokinetic Flow Matching for Pathwise Straightening of Generative Flows

Authors: 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.

Figures & tables

Panel of Figure 1 — 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 $Dv/Dt$ .

Panel of Figure 2 — 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.

Item	Value
Backbone	DiT-S/2 Transformer (depth $=12$ , hidden dim $=384$ , heads $=6$ , patch size $=2$ , MLP ratio $=4.0$ ).
Trainable parameters	32,485,260 (conditional, $num\_classes=10$ );
	32,481,804 (unconditional, $num\_classes=1$ ).
Data / resolution	CIFAR-10 train split, RGB $32\times 32$ , random horizontal flip augmentation.
Batch size / epochs	256 / 2500.
Optimizer	AdamW ( $lr=5\times 10^{-4}$ , weight decay $=10^{-4}$ ).

Table 1: Training configuration for the main CIFAR-10 experiment.

Setup	Method	Best FID@1 (epoch)	Best FID@2 (epoch)	Best FID@4 (epoch)
Conditional	FM	245.3692 (1250)	78.8200 (2250)	27.2868 (1250)
Conditional	Iso-FM	83.8460 (1500)	27.1267 (750)	15.5443 (1250)
Conditional OT	Iso-FM	104.3007 (750)	17.7633 (750)	10.2254 (1000)
Unconditional	FM	327.1964 (500)	98.2889 (500)	47.7700 (500)
Unconditional	Iso-FM	143.6726 (2000)	36.9151 (1000)	25.0528 (1750)
Unconditional OT	FM	249.0945 (500)	79.6954 (500)	42.4719 (750)

Table 2: Best FID scores and epochs for the main CIFAR-10 experiment (DiT-S/2).

Panel of Figure 3 — 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.

Panel of Figure 4 — 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.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: Conditional samples: FM (left, epoch 1250) vs Iso-FM (right, epoch 1250).

Panel of Figure 6 — Figure 6: Unconditional OT samples: OT+FM (left, epoch 500) vs OT+Iso-FM (right, epoch 1250).

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Flow matching learns continuous-time vector fields that transport simple noise distributions into data for efficient generative modeling.

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Flow Matching has emerged as a leading framework for generative modeling, powering state-of-the-art systems such as FLUX and Stable Diffusion 3.5. However, the iterative nature of its ODE-based sampling process creates a fundamental efficiency bottleneck: the quality of generated samples is highly sensitive to the choice of step-sizes, and current models typically require 20 to 30 steps for good quality. In this work, we propose two lightweight, training-free algorithms, CAT-OV and CAT-OT that adapt step-sizes at inference time based on a novel connection between Flow Matching sampling and gradient flow. Our algorithms are computed efficiently by not requiring additional neural function evaluations. Specifically, CAT-OT estimates curvature over time via a finite-difference approximation of the time-derivative of the vector field, while CAT-OV approximates curvature over the state space via a gradient of the vector field. Under suitable conditions, both methods have truncation error bounds of constant order. Empirically, CAT-OV and CAT-OT outperform existing step-size heuristics in image quality metrics across four text- to-image Flow Matching models, reducing the number of generation steps required to reach comparable quality by up to 40%.

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