Difficulty-Calibrated Interpolation Paths for Conditional Flow Matching
Authors: Airin Akter Tania, Md Raihan Khan
Organizations: Dept. of Electrical and Electronic Engineering Khulna University of Engineering & Technology Khulna, Bangladesh · Dept. of Electrical and Electronic Engineering North Western University Khulna, Bangladesh
Conditional Flow Matching trains generative models by regressing a network onto the velocity of a prescribed noise-to-data interpolation path. The interpolation schedule that shapes this path is known to affect convergence and sample quality, yet it is invariably fixed in advance, independent of both the data and the model. We show that the regression difficulty of Conditional Flow Matching varies systematically along the path, and we propose Difficulty-Calibrated Flow Matching, which derives the schedule from the model itself: a short pilot run with the linear path records the per-time loss, and the schedule is set to the quantile function of this difficulty profile, so the trajectory lingers where the velocity is hardest to learn. The method has a single hyperparameter, leaves the training objective and its gradient equivalence intact, composes with classifier-free guidance, and adds about two percent training overhead. In controlled experiments on CIFAR-10, MNIST, and Fashion-MNIST with an identical compact U-Net, the calibrated path attains the best FID on CIFAR-10 at full sampling budget and clearly outperforms all fixed schedules in the large-batch, few-update regime, precisely the setting where compute is scarcest.
Figures & tables
Fig. 1: Conceptual illustration of DC-FM. A sample xt travels from noise x0 to data x1 along the interpolation path while the network vθ regresses the target velocity ut under an MSE loss. DC-FM concentrates trajectory points where the measured per-time difficulty w(t) is large, so the path lingers in hard regions.
Fig. 2: Overview of Difficulty-Calibrated Flow Matching (DC-FM). (a) Standard CFM pipeline: an interpolant mixes a data sample x1 and noise x0 , a U-Net regresses the target velocity under the flow-matching loss, and samples are drawn by integrating the ODE at inference. (b) The interpolation schedule α(t) for linear, cosine, sigmoid, and our calibrated path; only α(t) changes across the controlled comparison. (c) DC-FM measures per-time difficulty w(t) in a linear pilot and sets the schedule to the quantile function of wγ , so the trajectory slows through hard regions of the path and moves quickly through easy ones.
Fig. 3: Interpolation . schedule α(t) (left), its speed α′(t) (middle), and the pilot difficulty w(t) (right) on CIFAR-10 (top), Fashion-MNIST (middle), and MNIST (bottom). DC-FM slows down ( α′ small) exactly where w(t) is large.
Method
FID@100
FID@20
CIFAR-10
Rectified Flow (linear) [ 9 ]
5.44
5.85
GVP (cosine) [ 6 ]
5.90
5.71
Logistic (sigmoid)
5.30
6.53
DC-FM (ours)
5.13
5.15
MNIST
TABLE I: Main results: FID ( ↓ ) at NFE 100 / 20 on CIFAR-10, MNIST, and Fashion-MNIST. Best per column in bold . Schedules are named by their proposal.
γ
0.1
0.5
1
1.5
2.0
FID@100
7.43
7.90
5.81
8.85
9.52
FID@20
7.74
8.44
6.42
11.04
16.21
TABLE II: Effect of the emphasis exponent γ on DC-FM (MNIST). Best per row in bold . Linear baseline shown for reference ( ≈ 7.4 FID@100 at this budget).
B
10
20
40
80
FID@100
8.63
5.81
7.45
7.71
FID@20
9.11
6.42
8.06
8.37
TABLE III: Effect of the number of difficulty bins B on DC-FM (MNIST). Best per row in bold .
Method (batch =512 )
FID@100
FID@20
Rectified Flow (linear) [ 9 ]
10.16
10.61
GVP (cosine) [ 6 ]
10.71
22.44
Logistic (sigmoid)
11.05
13.57
DC-FM (ours)
8.13
8.39
TABLE IV: Effect of batch size on MNIST (FID ↓ , NFE 100/20). Best per column in bold . DC-FM’s advantage grows in the large-batch, few-update regime.
Fig. 4: Uncurated DC-FM samples (NFE =100 ) on CIFAR-10 (top) and Fashion-MNIST (bottom).
Timestep sampling p(t) is a central design choice in Flow Matching models, yet common practice increasingly favors static middle-biased distributions (e.g., Logit-Normal). We show that this choice induces a speed--quality trade-off: middle-biased sampling accelerates early convergence but yields worse asymptotic fidelity than Uniform sampling. By analyzing per-timestep training losses, we identify a U-shaped difficulty profile with persistent errors near the boundary regimes, implying that under-sampling the endpoints leaves fine details unresolved. Guided by this insight, we propose Curriculum Sampling, a two-phase schedule that begins with middle-biased sampling for rapid structure learning and then switches to Uniform sampling for boundary refinement. On CIFAR-10, Curriculum Sampling improves the best FID from 3.85 (Uniform) to 3.22 while reaching peak performance at 100k rather than 150k training steps. Our results highlight that timestep sampling should be treated as an evolving curriculum rather than a fixed hyperparameter.
Flow matching trains a neural network to regress the conditional velocity along a linear interpolant between noise and data, and the number of network evaluations~(NFE) sets the cost of sampling. The straight-line interpolant carries an implicit choice: the sample moves at constant speed throughout the trajectory. We relax this choice and introduce Velocity Scheduled Flow Matching~(VSFM), which replaces the conditional target x1−x0 with v(t)(x1−x0) for any nonnegative profile v:[0,1]→R≥0 satisfying ∫01vdt=1. We study six polynomial profiles drawn from motion planning. The first use of VSFM is at inference time: a pretrained linear flow-matching model can be sampled under any admissible profile by integrating its ODE on a non-uniform τ-schedule, with no retraining and no additional computation; on CIFAR-10 this lowers FID by up to 19.8%. Training from scratch under a braking profile gives a further reduction of 17.4% at 4~NFE. Both gains follow from the local truncation error of the Euler integrator on the induced grid.
Vitalii Bondar
Cherkasy State Technological University Cherkasy, Ukraine
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.