BézierFlow: Learning Bézier Stochastic Interpolant Schedulers for Few-Step Generation
Authors: Yunhong Min
Juil Koo
Seungwoo Yoo
Minhyuk Sung
Venue: ICLR 2026 Poster (2026)
Keywords: Stochastic interpolants
Bézier functions
Diffusion models
flow models
Primary area: generative models

Abstract: We introduce BézierFlow, a lightweight training approach for few-step generation with pretrained diffusion and flow models. BézierFlow achieves a 2–3× performance improvement for sampling with $\leq$ 10 NFEs while requiring only 15 minutes of training. Recent lightweight training approaches have shown promise by learning optimal timesteps, but their scope remains restricted to ODE discretizations. To broaden this scope, we propose learning the optimal transformation of the sampling trajectory by parameterizing stochastic interpolant (SI) schedulers. The main challenge lies in designing a parameterization that satisfies critical desiderata, including boundary conditions, differentiability, and monotonicity of the SNR. To effectively meet these requirements, we represent scheduler functions as Bézier functions, where control points naturally enforce these properties. This reduces the problem to learning an ordered set of points in the time range, while the interpretation of the points changes from ODE timesteps to Bézier control points. Across a range of pretrained diffusion and flow models, BézierFlow consistently outperforms prior timestep-learning methods, demonstrating the effectiveness of expanding the search space from discrete timesteps to Bézier-based trajectory transformations.

Published as a conference paper at ICLR 2026
BÉZIERFLOW: LEARNING BÉZIER STOCHASTIC IN- TERPOLANT SCHEDULERS FOR FEW-STEP GENERA-
TION
Yunhong Min"
Juil Koo"
Seungwoo Yoo
Minhyuk Sung
KAIST
(dbsghd363, 63days, dreamy1534, mhsuung acc.
ABSTRACT
We introduce BézierFlow, a lightweight training approach for few-step generation with pretrained diffusion and flow models. BézierFlow achieves 2-3x perfor- mance improvement for sampling with S NFEs while requiring only 15 min- utes of training. Recent lightweight training approaches have shown promise by learning optimal timesteps, but their scope remains restricted to ODE discretiza- tions. To broaden this scope, propose learning the optimal transformation of the sampling trajectory by parameterizing stochastic interpolant (SI) sched- ulers. The main challenge lies in designing parameterization that satisfies criti- cal desiderata, including boundary conditions, differentiability, and monotonicity of the SNR. To effectively meet these requirements, we represent scheduler func- tions as Bézier functions, where control points naturally enforce these properties. This reduces the problem learning an ordered set points in the time range, while the interpretation of the points changes from ODE timesteps to Bézier con- trol points. Across range of pretrained diffusion and flow models, BézierFlow consistently outperforms prior timestep-learnin fectiveness of expanding the search space from discrete timesteps to Bézicr- trajectory transformations. Project Page: http://bezierfloo.giibbb
methods, demonstrating the ef-
INTRODUCTION
Diffusion and flow models have achieved state-of-the-art performance, but at the cost of high com- putation due to their iterative generative processes. A large body of recent work (Lu et al.) 2022] Song et al. 2023 Liu et al. 2023] Tong et al. (025) has aimed to accelerate generation to just a few steps. For diffuston models, many dedicated solvers (Lu et al., 2022 2023 Zhang & Chen 2023
Zhao et al. 2023) tailored to their ODE formulations have been proposed. Although these methods substantially reduce the number of iterations from hundreds tens, they often insufficient to bring the steps down to only few. More recent distillation techniques, such as consistency mod- els (Song et al. 2023) and variants (Kim et al.. 2024 Berthelot et al., 2023 Zhou et al. 2025) for diffusion and ReFlow (Liu et al. 2023) for flow models, can reduce the number of steps to as few as one, but they require substantial fine-tuning, often hundreds thousands of GPU hours, even for small datasets.
notable line of recent work the lightweight training approach, which learns only a few param- eters with a pretrained model to improve output quality for small number of function evaluations (NFEs). Compared to the distillation techniques, such approaches require only tens of GPU minutes for training while achieving considerable improvements. The key questions for lightweight training are: (1) what to optimize, and (2) how to parameterize the variables. For the former, most previous work (Tong et al. 2025 Chen al. 2024; Xue et al. (024) has focused on learning the optimal sequence of timesteps for ODE solves, treating a nondecreasing sequence of timesteps as learnable variables. Most notably, recent teacher-foree forceing approach (Tong et al. 2025 that uses the outputs of a multistep adaptive solver as the teacher has demonstrated the effectiveness of learned ODE timesteps.
"Equal contribution.

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Published as a conference paper at ICLR 2026
Broadening our scope, explore variables beyond ODE timesteps that can be learned with lightweight training. As our first key contribution, we propose optimizing the sampling trajecto- ries themselves. The Stochastic Interpolant (SI) framework (Albergo et al., (023) provides a unified view of modern ODF-based generative models. In this framework, the state at any time is written as a linear interpolation between two endpoint samples: one drawn from the source (e.g., latent) distribution and the other from the target data distribution. The interpolation is governed by pair of dime-dependent coefficient functions, referred to as the SI scheduler. The scheduler fully specifies the geometry of the sampling trajectory. Different models adopt different schedulers, yet interchang- ing one scheduler for another at inference time does not change the endpoint marginal distributions. Inspired by this, we propose a lightweight training framework for learning an SI scheduler, which is equivalent to sampling path transformations that preserve the endpoints.
Our second key contribution lies in the parameterization of SI schedulers. The 1D continuous func- tions for SI schedulers must satisfy the following properties: (i) boundary conditions, ensuring that the endpoints of the coefficients are fixed, (ii) monotonicity, which guarantees strictly nondecreas- ing signal-to-noise ratio (SNR) along the sampling path, and (iii) differentiabilityy which ensures that a velocity field can be derived in the ODEs governed by the learned scheduler. To effectively parameterize the space of such functions, while restricting the scope to polynomials, we propose a Bézier-based parameterization, termed the Bézier SI Scheduler, which forms the core of our overall lightweight training framework, BézierFlow. A ID Bézier function naturally satisfies all of these properties: the boundary conditions can be enforced by simply setting the two end control points to the time range boundaries; the function is smooth and differentiable by the definition of polynomial Bézier curves; and monotonicity can be achieved by enforcing a nondecreasing order of control points, making the learning process identical to learning the ODE timesteps in previous work (Tong ct al. 2025).
In our experiments, we evaluate both diffusion and flow models across diverse datasets and ODE solvers. BézierFlow consistently outperforms existing acceleration techniques while requiring only lightweight training, taking around 15 minutes on single GPU. Extensive results further demon- strate the effectiveness of optimizing the sampling trajectories directly, rather than ODE timesteps, when coupled with our continuous Bézier-based parameterization.
2 RELATED WORK
There have been various attempts improve few-step generation in ODE-based generative models. One major line of work focuses on designing dedicated ODE solvers tailored to the dynamics of the models (Lu al. 2022 2023 Zhao et al., 2023) Zhang & Chen 2023). Although these methods require no additional training, they are unable to achieve high- fideliiyylity generation with only a few steps. Another line is distillation-based approaches (Song et al. 2023 Salimans & Ho 2022 Liu iet al., (202)), which have demonstrated impressive gains in the very low-NFE regime, but incur substantial computational cost in training. Despite these diverse strategies, our lightweight training approach is most closely related to the methods listed below, which we now discuss in more detail.
Learning ODE Solving Timesteps. Several methods aim to achieve high-fidelity generation with few NFEs by optimizing the ODE timesteps in a lightweight manner. Chen et al. (2024) frame ODE timestep learning as a selection problem under a fixed NFE budget. Based on statistics col- lected from multiple sampling trajectories. they allocate more steps to regions of high curvature and fewer to flatter regions. Xue et al. (024) optimize timesteps from the perspective of numeri- cal integration: given specific ODE solver, they minimize the accumulated local integration error along the trajectory. [Tong et al.] (2025) learn optimal timesteps through a data-driven distillation framework, where high-NFE sampler serves as the teacher and a low-NFE sampler the student. The timesteps are optimized by minimizing the discrepancy between their outputs starting from the same initial noise. Compared to these methods that learn optimal ODE timesteps, learn optimal stochastic interpolant schedulers and demonstrate superior performance over these approaches.
Learning Sampling Trajectories. Several works (Karras et al. 2022 Lipman et al. 2024 Pokle et al. 2024, Kim et al. 2025) have explored changing sampling trajectories at inference time to im- prove generation quality and diversity, while selecting from a few predefined stochastic interpolant schedulers (c.g., linear, VP, VE), rather than parameterizing the function space and finding the best scheduler through optimization. To our knowledge, Bespoke Solver (Shaul et al. 2024) is the only
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approach that leams an optimal sampling trajectory. Unlike method, however, it relies a dis- crete parameterization, which prevents direct derivation of first-order derivatives. These derivatives must instead be represented through auxiliary variables, introducing redundancy that can lead to inconsistencies between zeroth-order and first-order representations, and thus often fail to capture a truly differentiable function.
BACKGROUND: STOCHASTIC INTERPOLANT FRAMEWORK
Stochastic Interpolant (SI) AAlerrgo et al. 2023 is unified framework for generative modeling.
encompassing both ODE-based and SDE-based models (Song et al. 2021b; Ho et al. 2020
Song
al. 2021a Lipman et 2023). Given two marginal probability densities po, P1 : R
R 820.
stochastic interpolant x(t) is defined by the following stochastic process:
xx() a(()z) + (()2o + +()), L [0, 1].
(1)
where (t) and (t) are interpolation coefficients between 2g Po and 21 P1, and I)) is a latent variable introducing stochasticityy. The process satisfies the boundary conditions z00) = 20 and (1) = 21 by enforcing (0)) = (1) 0, (11) = (0) and 70) = (1) 0.
While the general formulation written in Eq. is broad, many important generative models including diffusion (Song et al.] [2021a Ho al. 2020 Karras et al. [022), flow (Lipman al., 2023 [2024), and score-based (Song et al.) 2021b) models-can be expressed in a more specific
form, referred as one-sided interpolants:
zxt) = =(t))1+
where, as a common practice, Po = (0, 1) and pı = Paata thereby the latent variable is absorbed into the initial state To. By differentiating both sides of Eq. 2 it can be expressed in the following ODE form:
dx((t) a(l)ar + (1)xn, di
where we denote a time derivative by the dot. For these dynamics to be well defined, atl) and (() must be twice continuously differentiable (C²) to ensure that the divergence terms in the associated Pokker-Planck equation are well-defined.
Based on Eq. 2 within the SI framework, different ODE-based generative models, including diffu- sion, flow, score-based models, learn different but interchangeable quantities. Along the sampling path (z, t). flow models (1, t) approximate the velocity field (x) E[i; Iz = x... while dif- fusion models cs (z. approximate the expected initial random noise state mh (z(z) E[z0 2; = z²]. Finally, score-based models s (2, t) estimate the score function, which equivalent to the scaled version of the expected initial state: V logg (x(x) -o (z)) Thus, different types of gen- erative models are mathematically linked, and under the SI framework, a pretrained model of one type can be reinterpreted as another at inference. For convenience, we collectively refer to these ODE-based generative models as the SI model, denoted by SØ t), throughout the paper.
4 BÉZIERFLOW
4.1 PROBLEM DEFINITION
The objective of our work is to learn an optimal sampling trajectory that enables high-quality gen- eration with few NFEs (e.g.. < 10), while using pretrained diffusion or flow model.
We consider two sampling trajectories that share the same endpoints zi and 21 The source path refers to the trajectory used during model training, while the target path is newly optimized trajec- tory for inference. Although both trajectories share the same endpoints, we assume their intermedi- ate geometry matters: due to discretization error in ODE solving, output quality would depend on the path geometry. Given this assumption, we therefore aim to optimize the target path such that, even with only few of NFEs, its geometry produces sampling results comparable to those obtained along the source path with many steps.
Formally, given a pretrained SI model Sp. let E(ro, (4:) Sa) denote multistep ODE solver along the source path (the teacker), and (x M a few-step ODE solver along the target
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Teacher Trajectory (NFE-50)
Initial Trajectory (NFE=3)
BézieFFlow (NFE-3)
Bézier Functions with Varying Control Points
Target Prior
C
B(X)
C = (
C. (1-
Figure Illustration of Sampling Trajectories and Bézier Functions. the left, we visual- ize different sampling trajectories. While initial trajectories deviate from the target distribution, BézierFlow aligns them with those of the teacher using only NFE=3. On the right, we present ex- amples of 8-degree Bézier functions with different arrangements of control points.
path (the student), where {4 and {8 / are the respective timestep sets with M < N. Al- though both solvers start from the same initial state 2o ~ Po. they differ in the number NFEs and the sampling path. Let (xx1) denote the distribution induced by the teacher, and (21) the student's output distribution. Our objective is formulated as the following teacher-foreing KI. minimization:
min DxL (q(21) (x)o
0
In practice, we optimize Eq. 4 using the following tractable surrogate objective (Tong et al. 2025 which enforces the outputs of the two solvers to align with each other:
min ((0) = Exo-po [d({(Tro (t:). Sa), (o { Sb)]]
(5)
where d(-,-) is a distance metric such as LPIPS (Zhang et al. 2018). This lightweight optimization adjusts only the target scheduler coefficients while using the pretrained model, thereby improving few-step generation at minimal training cost. See the left of Fig. [ which compares sampling tra- jectories from the prior distribution the target distribution: (i) the teacher's trajectories with many steps (NFF=50), (ii) the student's ' initial trajectories, and (iii) trajectories optimized by BézierFlow. While the student's initial trajectories deviate from the target distribution at NFE=3, after training with BézierFlow, they closely follow those of the teacher despite the much smaller NFE.
4.2 SAMPLING PATH TRANSFORMATION
As discussed in Sec. in order to define two trajectories that share the same endpoints, we view this as transformation of the source path via the path reparameterization following prior works (Karras et al. 2022, Shaul et al. 2024 Pokle et al., 2024: Kim et al.. 2025).
In the SI framework, a sampling path is governed by a pair of interpolation coefficients in Eq. 2 which we refer to scheduler. We denote the coefficients of the source path as the source sched- uler (ake, ), and those of the target path as the target scheduler (ãk,, ã.). Specifically, to relate the two schedulers,
adopt scaling reparameterization trick (Karras et al. (022). where the target state Z, is defined from the source state x, as Z. = C. Here, C2 can be arbitrary scalar functions and 1, any invertible mapping from 8, but define them
a(t.) a(t))
a(e) a(e)
0<s<l, otherwise,
(6)
1,
p(t) a(l) o(t)' p(s) a(s) o(s)
t, = t(s) = p (p(s).
where both P and p are invertible the signal-to-noist ratio (SNR) increases monotonically time.
Applying the
to Eq. 3 the velocity in the target path (ā.) is expressed as
dī. (0, looglo c.) H, + C.
dt
a. (,(
ue
ds
ds
We note that replacing the source scheduler with the target scheduler based on Eq. 6 at inference valid for two reasons: (i) the endpoint marginal distributions are preserved, and (ii) the training
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objective of a SI model is invariant the choice of schedule long the SNR endpoints (minimum and maximum values) are the same. Thus, learning (āz, G.) only changes the geometry of the sampling path, and hence the few-step discretization behavior, without altering the underlying target distributions or requiring a different pretrained SI model. See App. forr more details.
4.3 BÉZIER STOCHASTIC INTERPOLANT SCHEDULER
In Sec. 4.2 we discussed what to optimize, namely the sampling path determined by the SI scheduler (ā.; T.). The next crucial question how to parameterize these ID continuous functions effectively. Since the space of arbitrary ID functions is prohibitively large, we employ ID Bézier parameteriza- tion, which offers strong expressiveness with a compact number of parameters- the control points. Moreover, Bézier functions naturally satisfy the three key requirements of the SI scheduler: (i) boundary conditions, as described in Sec. 3 (ii) monotonicity to ensure a strictly nondecreasing signal-to-noise ratio (SNR), and (iii) differentiability to compute the transformed velocity in Eq, g
n-degree Bézier curve defined as weighted linear combination of n + 1 control points {C,) where the weights are given by Bernstein basis polynomials b
(A) = ( (L- AE |0 B(A) = Σ be (A)C%, 1=0
(9)
As shown in Eq. 9 with only n control points, it can represent a wide range of trajectories. Unlike arbitrary ID polynomial functions, Bézier functions always pass through their control points in order, making it straightforward to enforce boundary conditions and monotonicity. See the right of Fig. [ which illustrates how ID Bézier functions B()), À E [0, 1] can represent diverse shapes under different control point arrangements while keeping the endpoints fixed.
Moreover, they are inherently smooth and infinitely differentiable (C*), with closed-form deriva- tive:
B(A) = n bi (A)(C()) -C),
(10)
which allows us to directly compute the transformed velocity in Eq. 8 at any time s. Specifically, parameterize (ss) and (s) as n-degree 1D Bézier functions, each defined by set of control
points:
n
(s) (a) 00)
(s) c(a) + 0op Σ
(²()) (a) - o0)
Σ
bin (s) c(o)
+00- (11)
1=0
For the boundary conditions, we fix the end control points (C/ c( = 0, cla¹ = cla² = 1) and treat only the interior control points parameters. Concretely, with learnable parameters g(a), (ia) E Rn-1 the control points are given by
((a)= = [0, (b(((a)) 1m-1, 1], ((a) [0, (b(()) 1n
(m) 1],
(12)
where o(0); Ei c is a softmax function, and (f); = Σ o00, is a cumulative soft- max function that ensures monotonicity. This monotonic parameterization ensures that p(s) = (s) is strictly nondecreasing on (0, 1), resulting in p exists.
4.4 CONNECTION TO PRIOR WORK
LD3 (Tong et al. 2025). From parameterization perspective, both LD3 and ours optimize the same type of parameter: a nondecreasing sequence of timesteps. However, their interpretations differ: LD3's parameters correspond directly to discrete ODE solver timesteps, whereas ours corre- spond to Bézier control points that form a continuous sampling path. Interpreting these parameters as an scheduler allows our approach to explore a much broader search space compared LD3. See App. B for the proof.
Bespoke Solver (Shaul et al. 2024). Bespoke solver similarly optimizes a new, target sampling path, but the key difference from our approach lies in the parameterization. Their parameterization
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is discrete: they learn ser- variables t, and C₂ in Eq. Such discrete parameterization requires separately modeling the derivatives tg. cz. thereby breaking the intrinsic connection between the values and their derivatives. This can yield mismatches between the predicted next value from numerical integration and the actual learned value, ultimately causing unstable optimization.
In contrast, our Bézier parameterization ensures that the resulting scheduler (a", o.) is smooth and, in particular, satisfies the C2 condition required for the SI scheduler as discussed in Sec. 3 Consequently, the time-derivative terms the transformed velocity can be computed directly rather than learned separately, and the learned ODE trajectories are well-defined, thereby leading to more stable optimization.
Beyond the parameterization, Bespoke Solver also differs in its training objective: it relies on step- wise error minimization, whereas our method optimizes a global trajectory-Ievel loss. See App. C for the results comparing with Bespoke Solver trained under our loss, which isolate and highlight the benefit of our Bézier-based continuous parameterization.
EXPERIMENTS
Experiment Setup. We evaluate our BézierFlow (BF) on both diffusion and flow models across diverse datasets for image generation. For diffusion models, we adopt EDM (Karras et al. 2022)
with pretrained checkpoints on CIFAR-10 (32 x 32)
(Krizhevsky 2009), FFHQ (64 x 64) Karras 2020). For flow models, we use pretrained Krizhevsky 2009), FlowDCN (Wang et al.
2019). and AFHQv2 (64 x 64) (Choi et al.
ReFlow (Liu et al. 2023 on CIFAR-I0 (32 x 32)
2024 on ImageNet (256 x 256) (Deng et al. 2009).
and Stable Diffusion v3.5 (SD) (Esser al.,
2024 on MS-COCO (512 x 512) (Lin et al. 2014). All pretrained models are from their official
implementations.
Each model is paired with its dedicated ODE solver: UniPC (Zhao et al. 2023) and iPNDM (Zhang Chen 2023) for diffusion models, and Runge-Kutta integrators, RK1 (Buler) and RK2 (Mid- point), for flow models. We consider the following learning-based acceleration methods as baselines for comparisons with ours: DMN (Xue et al., al., (024), GITS (Chen et | [202]) and LD3 (Tong et al.) 2025). For flow models, additionally include Bespoke Solver (Shaul et al. (202)) which was specifically designed for RK1 and RK2 solvers The results of the base ODE solvers, without additional learning, are reported as reference. All baselines evaluated using their official implementations, except for Bespoke Solver, whose official code is not publicly available.
baseline,
Implementation Details. Methods based on the teacher-foreing framework, including LD3, Be- spoke Solver, and our BézierFlow, are trained and validated with the same number of samples for both training and validation: 200 each for CIFAR-10, 50 each for FFHQ, AFHQv2, and ImageNet, and 25 each for Stable Diffusion v3.5. following the experiment setup used in LD3 (Tong et al.) 2025). Training is performed for epochs on CIFAR-10. FFHQ AFHQv2 and 5 epochs on the others. For GITS (Chen et al., 2024), we precompute statistics using 256 sampling trajectories. For the training of BézierFlow, we use 32 control points for the Bézier parameterization unless stated otherwise. For all models, we initialize the target scheduler the linear SI scheduler, i.c., ā(s)
and ō (s) = 1 1 s. We set the timesteps uniformly in SNR p(s) for diffusion models and uniformly in time for flow models. Following LD3 (Tong al. 2025 also feed the learned decoupled timesteps (Li et al. 2024) to the neural network. See App. D for more details.
Quantitative Results. We report Fréchet Inception Distance (FID) (Heusel al. 2017 scores
across diverse datasets for diffusion models in Tab. Tan for flow models in Tab. 2 FID is computed between the reference set and 50K generated samples, where the test set for each dataset serves as the reference. For SD, both the reference and generated sets are constructed from disjoint subsets of 30K text prompts from MS-COCO, following the setup used in LD3 (Tong et al. 2025). Refer to App. for a more comprehensive comparison of SD, including text-image alignment metrics.
As shown Tab. 1 for few-step generation with pretrained diffusion models, BézierFlow consis- tently achieves the best FID on CIFAR-I0 10 across different NIEs, with especially large margins over the second-best at small NEEs (e.g., at NFE-A, BézierFlow: 9.55 vs. LD3: 12.04 with UniPC, and BézierFlow: 6.93 vs. LD3: 9.97 with iPNDM). On FFHQ and AFHQv2, BézierFlow outperforms or remains comparable to the baselines. The improvements are particularly strong at small NFEs, for example, BézierFlow: 17.05 vs. LD3 (the second-best): 22.48 at NFE=4 with UniPC.

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Table 1: FID comparison of few-step generation with difflusion models. Results of the base ODE solvers are reported on each top rows, highlighted in gray. Bold indicates the best results, and underline marks the second best.
Method
NFE
NFE=6
NFE=8
NFE=10
Method
NFE=4
NFE=66
NFE=S
NFE=10
CIFAR-I0 32
32
with EDM
(Karras
2022
(Teacher FID: 2.08)
UniPC
50,30
19.33
9.64
6.16
iPNDM
29.53
9.84
5.30
3.75
+ DMN
26.42
8.11
4.22
2.79
DMN
28.29
9.33
4.82
3.52
LD3
12.04
3,56
2.43
2.62
LD3
9.97
4.42
2.93
2.44
BezzerrFlow
9.55
3.13
2,40
2.09
BezierFlow
6.93
3,35
2.81
FFHQ
64 x 64 with EDM
Karras et
2022
(Teacher FID: 2.86)
UniPC
47.62
14.96
7.76
8.93
iPNDM
28.75
11.15
6.68
4,80
DMN
25.87
9,44
5,06
DMN
30.89
11.93
GITS
22,99
12.12
8.90
4.40
GITS
18.51
9.21
5,58
4.37
LD3
22.48
6.16
4,25
LD3
15.55
5.89
3.74
BézierFlow
17.05
7.43
3.82
BezierFlow
15.39
AFHQQ2
64 with EDM
Karras 2022
(Teacher 2.04)
UniPC:
23.59
10.15
7.76
6.38
iPNDM
15.14
6.12
3,80
3.01
DMN
30.39
14.40
3.98
3.69
DMN
33.21
15.95
5.99
5.29
GITS
13.20
7.50
3.89
3.94
GITS
14.31
5,81
3.88
3.57
18.17
BézierFlow
12.27
4.46
2.75
2.67
BézierFlow
14.44
4.69
2.63
2.16
Table 2: FID comparison of few-step generation with flow-based models. Results of the base ODE solvers are reported on each top rows, highlighted in gray. Bold indicates the best results, and underline marks the second best.
NFE=10
CIFAR 32 x 32 with ReFlow al.
(Tencher FID: 2.70)
RKI
52.78
26.30
17.40
13.30
RK2
25.36
12.12
9.17
7.89
Bespoke
45.31
18.08
11.88
9.25
Bespoke
39.45
64.87
16.67
13.34
GITS
47.42
26.11
GITS
11.84
-LD3
38.95
20.10
12.54
9.64
LD3
29.45
13.82
6.26
3.86
BézierFlow
20.64
9.67
7.30
5.51
BézierFlow
13.18
6.00
4.31
3.74
ImageNet 256
256 with FlowDON
Wang et al 2024
(Teacher FID: 15.89)
RKI
12.03
12.04
13.55
14.43
RK2
7.91
10.54
12.97
14.08
DMN
142.79
28.56
10.61
11.69
DMN
7.96
10.23
9.42
7,86
Bespoke
13.39
14.31
Bespoke
7.60
10.05
13.02
14,23
GITS
13.20
10.91
11.91
12.93
GITS
8.18
9.80
12,30
13.27
-LD3
LD3
BéxierFFoo
15.60
6,85
7.77
8.11
BéxierFlow
9.50
5.94
6.22
7.56
MS-COCO 512
512 with Stable Dillusion
Esser
2024
(Teacher FID: 12.13)
RKI
$7.93
21.50
RK2
DMN
113.24
46.02
31.58
24.41
DMN
36.33
16.45
27.09
17.36
Bespoke
134.21
52.51
23.70
20.69
Bespoke
20.18
13.26
GITTS
70.01
42.44
31.89
25.47
GITS
31.09
21.21
15.58
14.65
-LD3
55.31
36.85
20.37
19.76
LLD3
39.03
18.04
12.30
11.54
RézierFFlo
54.05
33.43
19.69
16.52
RézierFlow
33.94
16.41
12.20
11.02
When it comes to flow models, as shown in Tab. 2 BézierFlow achieves state-of-the-art results on CIFAR-10 10 with both RK1 and RK2, outperforming the others by clear margins. For example, we surpass the second-best LD3 by 18.31 at NFE=4 with RK1, and the second-best GITS by 9.66 at consistently obtain the best results across most NFEs, except
NFE=4 with RK2. On ImageNet, at NFE=4, again by large margins. On MS-COCÓ evaluated with Stable Diffusion v3.5, BézierFlow outperforms the baselines most NFEs, demonstrating generalizability to large-scale models.
Overall, these results demonstrate that BézierFlow attains the best or comparable performance to existing acceleration approaches across diverse experiment setups, including both diffusion and flow models, different NFEs, ODE solvers, and datasets.
Qualitative Results.
We present qualitative results for accelerated sampling of diffusion models in Fig. and flow models in Fig. Across both model classes, BézierFlow (BF) consistently produces

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Published as a conference paper at ICLR 2026
NPE]
UniPC
DMN
GITS
LD3
BF
NFE
UniPC
DMN
GITS
LD3
BF
FFHQ 64 64 with EDM
Karms
2022
AFHQv2 64
64 with FDM
Karras
2022
Figure 2: Qualitative comparisons samples generated using NFEs and on FFHQ and AFHQv2 datasets. We UniPC solver the base solver for both cases.
NFE
RK2
DMN
GITS
Bespoke
LD3
BF
NFE
RK2
DMN
GITS
Bespoke
LD3
CIFAR- 10 32 32 with ReHlow Liu all
2023
ImageNet 256 x 256 with HowDCN (Wang et al.
2024
6
Figure 3: Qualitative comparisons of samples generated using NFEs 6 and on CIFAR-10 and ImageNet datasets. We use RK2 solver as the base solver for both cases.
sharper details and fewer artifacts at low NFEs. Notably, in the last row Fig. (right), the baselines fail to generate plausible animal face, whereas our method produces clear and realistic cat face. See App. F for more qualitative results.
Table 3: Generalizability of BézierFlow to unseen Table 4: Comparison with distillation NFFs. Each column corresponds to the inference methods on CIFAR-10. The middle column NFE. Baselines are trained with the same NFE used at reports FID and NFE. Our training time is inference, whereas ours is trained once with NFE=10 measured on an NVIDIA A6000, while dis- and directly applied to unseen NFEs. Reported results tillation times are directly from the original are FID scores on CIFAR-10 10 (lower better; best in papers on NVIDIA A100s. Best in bold.
bold).
(L)
Training Time
Method
RKI
RK2
CIFAR- 10 32x32
Diffusion
2.93 / NFE=2
days
GITS
26.11
19.89
15.34
11.84
8.77
6,58
BF UniPC
2.09/ NFE=10
15 minutes
Bespoke
18.08
11.88
9.25
64.87
16.67
13.34
CIFAR- 32x32 Flo
LD3
20.10
12.54
9.64
13.82
6.36
3,86
2-RF
3.85 NPE=2
days
BF (NFF==10)
18.50
9.02
6.01
9.57
5.32
3,71
BF w/ RK2
3.74 NFE==0
15 minutes
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Published as a conference paper at ICLR 2026
AFHQV2
CIFAR-10
4
n
n
15
8
8
n
n
10
n=16 16
10
n=16
15-
n=16
n= 16
n=
n=
10
n=
10
n=
5
4
NFE
NFE
NFE
NFE
(a) UniPC
(b) iPNDM
(c) RKI
(d) RK2
Figure 4: Effect of the degree of Bézier functions. Each line reports FID scores on CIFAR-10 across Bézier function degrees n from 4 to 32. Higher degrees yield lower FID, with convergence around n = 32.
Effect of the Degree of Bézier Functions. Fig. 4 shows the effect of the degree Bézier functions by varying the number of control points from 4 to 32. Across different datasets, NFEs, and base ODE solvers, increasing the number control points consistently improves FID, indicating that higher- degree Bézier functions provide greater expressiveness for learning optimal sampling trajectories. Empirically, we observe that the additional gains between n = 16 and n 32 become marginal, so adopt 32 as default. Note that the training time overhead from increasing is negligible, with only a few seconds difference between n = and n = 32.
Generalizabilityy to Unseen NFEs. As discussed in Sec. 4.4 unlike prior works (Xue et al,
2024 Chen et al. 20244 Tong al. 2025 Shaul et al. 2024) that learn discrete per-step vari- ables, BézierFlow learns the sampling trajectory with continuous functions, enabling generalization
NFEs unseen during training. As shown in Tab. 3 BézierFlow trained with NFE=10 also per- forms well at NFE=6 and NFE=8, achieving outperforming FID scores against the baselines trained directly at those NFEs.
Training Efficiency. In Tab. 4 we compare BézierFlow on CIFAR-10 (Krizhessyy 2009 against representative distillation-based approaches: Consistency Distillation (CD) (Song et al.) 2023) ) for diffusion models and 2-Rectified Flow (2-RF) (Liu et al. 2023) for flow models. While BézierFlow achieves better FID scores with larger inference NFE, its training cost is significantly lower, quiring only 15 minutes compared days for distillation, which corresponds to approximately 0.13% of the training time. A more comprehensive comparison of training time is provided in App. E.3
Combination with LD3. Although the search space LD3 is a subset of that of BézierFlow, LD3 optimizes discrete timesteps whereas BézierFlow learns continuous sampling paths, making it natural to ask whether jointly optimizing the two could offer complementary improvements. We therefore optimize both the target timesteps and the scheduler in unified framework. As shown in Tab. 5 of the Appendix, however, the combination does not offer clear advantages over using BézierFlow alone. This indicates that the advantages provided by LD3 are already captured within the learned scheduler, and thus do not translate into further gains when optimized jointly.
6 CONCLUSION
We introduce BézierFlow, lightweight training framework for few-step generation. By combining the optimization of sampling trajectories, rather than discrete ODE timesteps, with a Bézier-based continuous parameterization, BézierFlow achieves consistent improvements across diffusion and flow models with only minutes of training, surpassing existing lightweight training approaches. For future work, plan to explore alternative basis functions for Bézier functions, which may enable richer expressiveness with fewer control points.
Ethics Statement. We affirm adherence to the ICLR Code of Ethics. This work relies only on publicly available models and datasets and does not involve human subjects, user data, or personally identifiable information. We acknowledge the potential for misuse of generative Al and encourage responsible deployment and of our method.

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Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENTS
This work was supported by HITP grants NRS-2024-003999817, RS-2025-25441313, RS-2025- 25443318), funded by the Korean government (MSIT); the Industrial Technology Innovation Pro- gram (RS-2025-023113326), funded by the Korean government (MOTIE); the National Supercom- puting Center (KSC-2025-CRE--475)) and the DRB-KAIST SketchTheFuture Research Center.
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A VALIDITY OF SCHEDULER REPARAMETERIZATION
We provide a detailed exposition of two key properties discussed in Sec. 4.2 (i) the preservation of endpoint marginals, and (ii) the invaraince the SI training objective with respect to the choice of schedule, provided that the SNR endpoints are identical.
Proposition A.1. (Endpoint marginals equivalence) Let po and pı denote the endpoint marginals of It governed by scheduler (ave 0z), Po and pı denote those of another stochastic interpolant Ts. If I, Cg with c, and defined in Eq. 6 then the endpoint marginals preserved, i.e., Po Po and p1 = P1.
Proof. By the boundary conditions discussed in Sec. 3. any SI scheduler satisfies (0) 0, (1) 1, (0) 1, (11 =
(13)
From Eq. 6 C₂ 1 at both 0 and 1. By the definition of L. 1 --{(()))))) also have t==0 = and t==1 = 1. Hence, ش==0 = zt=0 and 1,=1 = It=1- Since the pair of endpoints (æo, Φ1) does not change under the sampling path transformation, the endpoint marginals also coincide: po Po and p1 = P1.
Proposition A.2. (Training objective invariance) For any two schedulers (ai, 0t) and (0t,, T.) with matching SNR endpoints, training SI model S. under either scheduler minimizes the same train- ing objective, hence yields equivalent optima.
Proof. As noted in Sec. different SI models (noise, velocity, or score predictors) learn different but interchangeable quantities, and can all be expressed in the denoiser form &g. For convenience, we therefore recall the continuous-tine time SI objective the denoiser form (Eq. 18 from Kingma al. 2023)): for schedule (01, 01) with p(t) = (tt)/((t) strictly increasing, define = p² and
Fy agy11 + TuZo:
Bty = dp (vV)), ou == of (vD)).
(14)
with Vmin p(0)3, Vmax p(1)2 Then,
Vmax
Lold; (o)) =
Ber zorupo [||=] Eel 1)1 dv.
(15)
Pmnn
Now consider another schedule (ā,, 7.) with the same SNR endpoints. Since p(t)
att)
and
o(1)
((s) a(s) are strictly increasing, the maps t + p(t)² and s + 2( are bijections onto
(s)
the common interval [Pmin Pmax]. Now, for each fixed 1/, we have
at)) a(s)
(16)
(1)= as))
which implies o, 2e and or Sy vp
Therefore, the interpolants satisfy
By = az #1 + 1 Jom))
Fy = āz x1 + -o
(17)
so that Z. T.. This shows that the two interpolants differ only by scalar rescaling factor, and since the integration limits are the same, we conclude
Luld; (x)) = Lol (ā., R.)].
(18)
B THEORETICAL ANALYSIS OF SAMPLING TRAJECTORY SPACES
In this section, we formally show that the family of sampling trajectories realizable by BézierFlow super set of that of L.D3 (Tong al. 2025). offering better optimization advantages.
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Published as a conference paper at ICLR 2026
Theorem B.1 (Inclusion of LD3 in the BézierFlow Trajectory Space). Let M and D denote the num- ber of sampling steps and the dimension of the state æ, respectively. Let XLDS. ABF C RMM++XD 1x be the sets of sampling trajectories over discrete timesteps realizable by LD3 and BézierFlow (parame- terized by Bézier curves of degree n. 2 M). Assuming the source sampling path defines a non-linear geometry, the trajectory space LDD3 is subset of that of BézierFlow:
XI.D3 & ABF.
(19)
Proof. Let [= and {sy]i denote the timesteps on the source scheduler and on the trajectory induced by the Bézier SI scheduler, respectively, with
0= 10 <4 <Im= =1, 0= 80 81 ....... < <MM =1
Define
ak = o+ ok = otx), Ste = 0g + T
O....,,M M. (20)
With this notation, any LD3 sampling trajectory over M timesteps can be written . E XLDS-
Since Bézier curve of degree 2 M can interpolate any M + 1 distinct values, there exists 0* such that
Ög- (8R) =
Okky
Tg (sk) = OR.
Vk.
(21)
Hence
āg. (sk)
akk = p(tx):
po (sk) ==
(22)
og- (sk)
and therefore
Cak = σg. (sk)
te p- (sk)) = tk.
= 1.
(23)
(ter))
Using the sampling transformation in Eq.
Cop Csy Ftog
Ftas
Ak:
(24)
every x ALDA is also realizable by BézierFlow, and thus
Xr.a ABF.
(25)
For strictness, fix {tx
M {o and consider a target scheduler such that Tg(sx) * ox
po(ss) = p(tx),
for at least one k.
(26)
Then
(8(8k) o(tor)
(o(sk) # for least one k,
tsr = (po((s) tk
Cas
(27)
and the resulting trajectory satisfies
Cay Tta
Cas # 1 for some k.
(28)
Since LD3 is constrained to the fixed souree scheduler, which corresponds to sampling via Eq. with
C.. 1
(29)
so any trajectory with C., for some k cannot be realized by LD3. Thus
Xg E ABP and Xe $ XLDS.
(30)
which implics
Xi.Da Ç XBF.
(31)
n
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Table FID comparison of BézierFlow, LD3 and their combination, denoted as Both. The best results are highlighted in bold and the second best results are underlined. Gray cells indicate the base ODE solvers.
Method
NHEi
NFES6
NFES8
NFE=10
Method
NFEel
NFE=6
NFE=8
NF-:=10
UniPC
CIFAR-I0 with EDM (Teacher FID: 2,08)
UniPC
FFHQ with EDM (Teacher FID: 2.86)
LD3
12.04
3,56
2.43
2.62
LD3
22.48
6.16
4.25
2.92
BéxierFlow
9.55
3.13
2,40
2,09
BéxierFFlo
17.05
7.43
3.82
3.13
Booh
Both
RK2
CIFAR 10 with Refllow (Tencher FID: 2.70)
RK2
ImageNet with FlowDCN (Teacher FID: 15.89)
-LD3
29.45
13.82
6,26
3,86
LD3
7.59
10.17
12.75
14.04
BérierFFlo
BéxierFlow
Broh
12.23
5.50
3.74
3.17
Both
9.11
6.64
9.38
10.88
Table 6: FID comparison of BézierFlow, Bespoke Solver and Bespoke Solver trained with our training loss, denoted as Bespoke*. Results for the base solvers are reported on each top rows. The best results are highlighted in bold and the second best results are underlined. Gray cells indicate the base ODE solvers.
Method
NFE-4
NFE=6
NFE=8
NFE=10
Method
NEE=4
NPE=6
NFE=8
NFE=10
CTFAR
32 32 with ReFlow
Liu at.l
2023
(Tencher FTD: 2.70)
RKI
Bespoke
45.31
18.08
11.88
9.25
Bespoke
39.45
64.87
16.67
13.34
Bespoke*
17.28
7.65
Bespoke"
19.44
BézierFlow
20.64
9.67
7.30
5.51
BéziorFlow
13.18
6.00
4.31
3.74
Proposition B.2 (Better Optima under Larger Trajectory Spaces). Let XLDD,, XBr M++x be the trajectory spaces of LD3 and BézierFlow, respectively, and suppose
XLDS & XBF
(32)
as in Theorem B.1 Let L X R be any real-valued objective functional (e.g., distillation loss to teacher). Define the optimal objective values
LiDS = inf L(x), xeXida
Lis inf L(x). xeXBr
(33)
Then, the following inequality holds:
Lir
S Li
(34)
Proof. Recall that for any two sets A C B and an objective function f. the infimum over the superset less than equal to the infimum over the subset, i.e.,
inf zEB
fx) < inf f(x).
(35)
Since XLDS S XBF, applying this property directly yields:
Cer inf L(x) S inf XEXxF
[(x) Li.Da-
(36)
XEXIDA
Moreover, if there exists x' XBF \ XLDD such that L(x') < LiD3. then
SBF < Li
(37)
C COMPARISON OF SCHEDULER PARAMETERIZATTION: BÉZIERFLOW VS. BESPOKE SOLVER
As discussed in Sec 4.4 both Bespoke Solver (Shul et al. 2024 and BézierFlow aim to lear sam- pling trajectories, but differ in (i) parameterization and (ii) training objective. Bespoke Solver (Shaul et al. 2024) employs discrete per-step parameterization and minimizes step-wise & errors against
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teacher outputs, whereas BézierFlow adopts Bézier-based continuous parameterization and is trained with a global truncation loss, computed along the full trajectory from To to 21, with LPIPS (Zhang et al. 2018).
To ablate the effect of different training objectives and focus solely on paramcterization, we report additional quantitative results in Tab. where Bespoke* retains Bespoke Solver's parameterization but adopts the same training objective as ours. While Bespoke" improves over the original Be- spoke Solver, BézierFlow remains superior, with especially large gains at low NFEs, such +17.7 FID improvement at NFE=4 with RK1, underscoring the advantage of our Bézier-based continuous parameterization.
D IMPLEMENTATION DETAILS
We first describe the shared experimental setup for the methods based on teacher-forcing framework (Bespoke solver, LD3, and BézierFlow) and method -peecffcc configurations for BézierFlow and the baselines.
D.I SHARED SETUP
Teacher Data Generation. We generate teacher samples using the high-order adaptive solver RK45 (Butcher, 1996), except for Stable Diffusion v3.5 (Esser et al.] 2024), where we adopt RK2 with 30 NFEs. The same teacher samples are used for all baselines that rely on the teacher-forcing framework (e.g., Bespoke Solver, LD3).
Training. We train for 8 epochs on CIFAR-10 (Krizhevsky 2009), FFHQ Karras al. 2019), and AFHQv2 (Choi et al. (2020), and for 5 epochs on ImageNet (Deng et al. [209)] and Stable Diffusion v3.5 (Esser et al. 2024). the end each epoch, we perform validation and select the checkpoint with the best validation score for final evaluation. We use LPIPS (Zhang et al. (2088) as the distance
metric for LD3 Tong et al. 2025) and BézierFlow, and RMSE for Bespoke Solver (Shaul et al, 2024).
Evaluation. We report Fréchet Inception Distance (FID) (Heusel et al. 2017) scores computed against the reference set using 50K randomly generated samples. On ImageNet, generated sam- ples are drawn to match the class distribution of the reference set. For SD3.5, both reference and generated samples constructed from disjoint subsets of 30K text prompts from the MS-COCO validation set, following the setup of LD3 (Tong et al. 2025
D.2 BÉZIERFLOW TRAINING DETAILS
Target Timesteps. For diffusion models, the timesteps {s; ( NFE. are initialized to be uniformly spaced in terms of the hignal-to-noise ratio (SNR). For flow models, they are initialized to be uni- formly spaced in the time domain.
We initialize the Bézier scheduler with a linear SI scheduler, i.e., a(s) = 8 and a(s)= 1 Under the ID Bézier parameterization, this corresponds
Initialization.
g(a) 1, 8(0) 1, == 0.0.,...,,7
(38)
which places the 1 interior control points uniformly between the two endpoints. We use 32 control points in all experiments. For decoupled timesteps s. that are fed into the model, we set of 8; + o()) with o(9) initialized to zero, following L.D3 (Tong al. 2025).
Optimizer. We optimize the Bézier scheduler parameters 8(0), using RMSprop. and the de- coupled timesteps 0!() using SGD. For RMSprop. we set the momentum to 0.9 and weight decay to 0. The learning rate is 5 x 10-3 for CIFAR-10, FFHQ, and AFHQv2, and 10-1 for ImageNet and Stable Diffusion v3.5. For the decoupled timesteps, we use SGD with a learning rate of 1 x 10 for CIFAR-10 10 and ImageNet, 1 10 for FFHQ and AFHQy2, and 5 10-4 for Stable Diffusion v3.5, each further scaled by 1/NFE. We apply gradient clipping with a global norm threshold of 1.0 to all parameters.
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Table FID comparison of few-step generation with difflusion models at extremely low NFEs. Results of the base ODE solvers are reported on each top rows. Bold indicates the best results, and underline marks the second best. Gray cells indicate the base ODE solvers.
CIFAR 10 32 32 with EDM
FEEQ 64 x 64 with EDM
AFHQQ2 64 x 64 with EDM
Method
NFE=1
NFE=2
NFE=3
NFE=1
NFE=2
NFE=3
NFE=1
NFE=2
NFE=3
UniPC
377.15
168.35
57.45
280.61
104.57
59.54
312.37
64.62
44.52
DMN
160.65
66.03
142,57
64.99
141.99
70.01
53.21
25.13
L.D3
187.42
39.56
120.87
48,30
107.77
30.53
RézierFlow
125.03
55.07
33.72
26.31
iPNDM
377.15
153,31
47.68
280.61
102.50
45,70
312.37
79.32
38.16
61.54
76.28
GITS
153.33
43.71
105.78
32.33
95.47
26.40
32.19
23.85
BézierFlow
125.03
41.58
22.20
121.94
60.45
35.10
159.58
34,70
36.26
Table 8: FID comparison of few-step generation with flow models extremely low NFEs.
Results of the base ODE solvers are reported on each top rows. Bold indicates the best results, and underline marks the second best, Gray cells indicate the base ODE solvers.
CIFAR 10 32 32 with Refloo
ImageNet 256 x 256 with FlowDCN
MS COCO 512 x 512 with SDv3.5
Method
NFE=1
NrF==2
NFE=3
NFE=1
NFE=2
NFE=3
NFE==
NFE=2
NFE==3
RKI
379.22
171.48
89.00
263.54
113.27
27.69
328.02
214.03
103,97
DMN
170.53
79.54
115.89
42.78
218.20
82.68
31.70
94.94
Respoke
471.18
405.94
265.77
264.81
1 114.66
28.27
324.94
212.16
98.91
LD3
182.40
81.35
126.29
85.16
150.21
85,16
BézierHoow
314.52
67.63
30.40
261.79
94.64
44.60
320.67
156.20
83.55
RK2
128,80
90.26
163.35
DMN
Bespoke
309.60
86.58
162.40
LD3
BéxierFlow
70.87
83,97
146.20
D.3 BASELINES
GITS (Chen et al. 2024). We adopt the official implementation code and follow the default num- ber of sampling trajectories, which is 256.
Bespoke Solver (Shaul al. 2024). Since official implementation code is publicly available, we re-implemented the method based on the descriptions in the original paper. We employ Adam optimizer with learning rate of 1 10-4, as we observed that the learning rate reported the paper (2 x 10 3) caused divergence and very high FID scores when training on relatively small datasets.
We adopt the official implementation code and follow the default training configurations. For timestep parameters, we use the same optimizer and match their learning rate to that of our scheduler. For the decoupled timesteps, follow the original parameterization and use SGD with a learning rate 0.1 following original paper.
LD3 Tong et al., 2025).
NPE except for ImageNet and Stable Diffusion v3.5, where we use 0.001
NFEE
E MORE QUANTITATIVE RESULTS
E. PROBING BÉZIERFLOW AT EXTREMELY LoW NFES
To stress-test BézierFlow the extreme low-NFE regime and identify where quality collapse begins, conduct additional experiments in the very low-NFE range (NFE < 3), which is even lower than the NFEs used in Sec. Except for the NFEs, all other experiment setups follow those used in Sec. 5
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Table Quantitative comparison of few-step generation on text-image alignment with Stable Diffusion (Esser et al., 2024). Results for the base solvers are reported each top rows. Bold indicates the best results, and underline marks the second best. Gray cells indicate the base ODE solvers.
Method
NFE-4
NFE=6
NFF=8
C
PickScore f CLIP
PickScore
CLIP
PickScore f
CLIP
PickScore
MS-COCO 512
512 with Stable Diffusion
Esser
2024
0.206
0.252
0.257
DMN
0.225
0.199
0.246
0.209
0.253
0.213
0.256
0.215
Bespoke
0.241
0.206
0.243
0.212
0.251
0.214
0.252
GITS
0.234
0.204
0.247
0.210
0.252
0.213
0.255
0,214
LD3
0.244
0.208
0.249
0.212
0.258
0.217
0.258
0.217
BézicrFlow
0.245
0.209
0.253
0.256
0.217
0.258
RK2
0.208
0.255
0.259
0.216
DMN
0.243
0.208
0.257
0,216
0.252
0.213
0,259
0,217
Bespoke
0.244
0.208
0.225
0.200
0,253
0,215
0.257
0.217
GITS
0.251
0.211
0.255
0.214
0.257
0.216
0.258
0,216
L.D
0.241
0.208
0.255
0.215
0.260
0.218
0.261
0.218
B/ziecFlow
0.210
0.258
0.260
0.217
0.263
0.219
Table 10: Quantitative comparison on training efficiency in few-step generation for diffusion and flow models C-FAR-10. All experiments conducted A6000 GPUs, except for the last row of distillation methods, which reports the performance of pretrained model from their official implementations (Song et al. 2023: Liu et al.) 2023). "Time" denotes wall-clock training time,
where s/m/d denote scconds/miinute/daay respectively.
Method
NFE=6
NFE=8
Method
NFE=6
NFE=8
FID +
Time
FID 4
Time 1
FID 4
Time J
FID
Time J
()) Non-distillation Methods
IPNDM
CIFAR- 1D with FDM (Tencher FID: 2.08)
RK2
CIFAR 10 with ReFlow (Teacher FID: 2.70)
DMN
9.33
5s
4,82
DMN
51.99
5s
21.43
GITS
6.80
30s
4.07
30s
GITS
11.84
30s
8.77
30s
Bespoke
Bespoke
64.87
30m
16.67
30m
L.3
4.42
10m
2,93
13m
LD3
13.82
10m
6.26
13m
BérierFlow
3.35
10m
2811
13m
BérierFlow
6.00
10m
4,31
13m
(2) Distillation Methods
CD
359.59
15m
343.59
15m
2-RE
12.12
15m
9.17
CD
4.24
6d
3.95
6d
+2RF
5.69
2d
5.45
24
2.82
(A0)
2.79
(A100)
2-RF
3.74
(A100)
3.68
(A
As summarized in Tab. and Tab. BézierFlow remains effective even at extremely low NFEs for both diffusion and flow models, improving over the base solvers by a substantial margin. Note that blank entries for RK2 simply reflect that RK2 only supports even numbers of function evaluations and has no timestep to learn in NFE=2. Furthermore, for NFE=1, timestep-learning methods cannot be applied, whereas scheduler-leamd learning approaches such as Bespoke Solver (Shaul al. 2024) and BézierFlow remain applicable.
E.2 TEXT-IMAGE ALIGNMENT FOR FOUNDATIONAL MODEL
To complement the zero-shot MS-COCO FID results of Stable Diffusion v3.5 (Fiser et al. 2024) in Tab. we provide additional evaluation results for more comprchensive assessment. We report CLIP score (Hessel et al. I) and PickScore (Kirstain et al.) (023), both which measure the alignment between the given text prompt and the generated image.
As shown in Tab. 9 BézierFlow achieves the best or second-bes performance across various NFEs, solvers, and evaluation metrics except for the CLIP Score at NFE=8 with the RKI solver. These additional results further corroborate the superiority pretrained stochastic interpolant model (Esser al. 2024
BézierFlow even with the large-scale 2.5B
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NPE]
UniPC
DMN
GITS
LD3
BF
NFE
UniPC
DMN
GITS
LD3
BF
FFHQ 64 64 with EDM
Karms
2022
AFHQv2 64
64 with FDM
Karmas
2022
Figure 5: Qualitative comparisons samples generated using NFEs and on FFHQ and AFHQv2 datasets. We UniPC solver the base solver for both cases.
NFE
RK2
DMN
GITS
Bespoke
LD3
BF
NFE]
RK2
DMN
GITS
Bespoke
LD3
BF
CIFAR 32 with ReFlow
[Liu et al.
2023
ImageNet 256
256 with FlowDCN
Wang
2024
Figure 6: Qualitative comparisons of samples generated using NFEs and on CIFAR-10 and ImageNet datasets. We use RK2 solver the base solver for both cases.
E.3 COMPARISON ON TRAINING EFFICIENCY WITH FEW-STEP GENERATION METHODS
For comprehensive and fair comparison of training efficiency beyond the Tab. report additional results at matched NFEs with varying training budgets in Tab. TO As shown, under the same NFEs, distillation-aan based approaches (Consistency Distillation (CD) (Song al. 2023) and 2-Rectified Flow (2-RF) (Liu et al. (023)) yield notably worse FID under the same lightweight training budget (15 minutes) and require substantially longer training time (2-6 days) to achieve FID comparable to BézierFlow, corresponding to roughly 200-600x x more training time. These results underscore BézierFlow 's highly training-effitn -ffficient acceleration, achieving in just a few minutes the performance that prior distillation-based approaches require several days of training to reach. Note that the 15-minute performance of 2-RF identical to that of the base pretrained model as this budget is fully spent on the data creation stage for ReFlow.
We also include training time comparisons against non-distillation baselines that accelerate genera- tion with lightweight training, including DMN, GITS, Bespoke Solver and LD3 (ue et al. 2024 Chen et al. 2024; Shaul et al., 2024: Tong et al. 2025). Among these lightweight acceleration methods, BézierFlow achieves the best FID, even outperforming LD3 under the same training bud- get. This demonstrates that BézierFlow offers a more favorable trade-off between training efficiency and sample quality.
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NFE
RKI
DMN
GITS
Bespoke
LD3
BézierFlow
MS-COCO 512 512 with Stable Diffusion Paser
2024
man standing up against a wall with his hands elasped together"
"A laptop computer sitting on top of a wooden table."
"Computer the derk nighttime in front of window." "
"A brown dog hanging it's head out of window."
NFF
RK2
DMN
GITS
Bespoke
LD3
BézierFlow
"A bus stopped in front of a tall red building"
"A fow pieces of luggage sitting on top of chair in an airport."
"A white bind walking shrough shallow of water.
"A young lody riding skis covered slope."
Figure 7: Qualitative comparisons of samples generated using NFEs 6 and 8 with Stable Diffu- sion v3.5 Esser et al. (2024). use RKI and RK2 as the base solver.
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Table 11: Quantitative comparison on unconditional 3D point cloud generation with Point Voxel Diffusion (PVD) (Zhou et al. 2021). Lower is better for CD-MMD (denoted as MMD) and ISD and higher better for CD-COV (denoted as COV). CD-MMD multiplied by 10 Results for the base solvers are reported each top rows. Bold indicates the best results, and underline marks the second best, Gray cells indicate base ODE solvers.
NFF
NFT=66
NFF=8
NFF=10
Mathod
MMD
COV 1
JSD 4.
MMD
COV 1
JSD
MMD
COV
JSD
MMD
COV
JSD
DMN
1.10
16.79
0.27
0.68
26.42
0.13
1.50
15.06
0.32
0.67
19.51
0.27
GITS
5.32
9.52
0.56
9.14
0.74
0.63
1.20
20.25
0.51
0.90
16,30
0.23
21.23
20.74
21.48
BérierFlow
18:77
22.72
23.70
DMN
1,18
19:36
0.29
0.63
24.69
1.74
6.17
0.65
20,49
0.22
GITS
3.22
7.65
13.44
3.39
3.21
0.48
3.99
1.73
0.49
2.73
3.95
0.41
LD3
2,40
13.33
0.34
18.52
0.25
0.77
19.01
0.25
0.70
22.72
0.22
BéxeeFllow
18.52
24.52
F MORE QUALITATIVE RESULTS
We provide more qualitative results for accelerated sampling of diffusion models in Fig. 1 flow models in Fig. and Fig. 7 Across both model classes, BézierFlow (BF) consistently yields clearer structures and more faithful details compared to basclines under low NFEs.
G EXTENSION TO OTHER DOMAINS
BézierFlow is generic framework applicable not only to image synthesis but also to various gener- ative tasks within the stochastic interpolant framework. To demonstrate the versatility of our method and its robustness to different distance metrics beyond LPIPS, we conduct additional experiments on two distinct domains: 3D point cloud generation and layout generation.
G.1 UNCONDITIONAL 3D POINT CLOUD GENERATION
3D Point cloud generation involves creating 3D representations objects using discrete points, task essential for applications in robotics, autonomous driving, and 3D modeling. We evaluate BézierFlow using the Voxel Diffusion (PVD) model (Zhou et al. 2021), trained on the air- plane category of the ShapeNet dataset (Chang et al. 2015).
Experiment Setup. We adopt simple mean squared error (MSE) loss for both training and vali- dation. We generate 32 data pairs for both the training and validation sets using DPM-Solver (L et al.] 2022) with 64 NFEs. and train the model for 5 epochs. We compare method against the same set of baselines reported in Tab.
Evaluation Metrics. Following the evaluation protocol of PVD (Zhou 2021),
the quality of generated samples using three metrics based on the Chamfer Distance (CD): Mini- mum Matching Distance (CD-MMD). Coverage Score (CD-COV), and Jensen- Shannon Divergence (JSD).
Results. Tab. presents the quantitative results. BézierFlow consistently achieves the best or second-best performance on CD-MMD, CD-COV across all NFEs, substantially improving the base solvers and timestep-learning baselines (Xue et al.) 2024 CCen et al. 2024 Tong et al.) 2025]. Fig. g provides qualitative comparisons of generated 3D point clouds. where BézierFlow better preserves both the global shape and coverage of the target distribution.
G.2 UNCONDITIONAL. LAYOUT GENERATION
Layout generation aims to synthesize structural arrangements elements (e.g., UI components, document blocks), which is critical step in graphic design automation. We evaluate method
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NFE
iPNDM
DMN
GITS
LD3
BézierFlow
ShapeNet oirplane with PVD Thou
2021
Figure 8: Qualitative comparisons of 3D point cloud samples generated using NFEs and with PVD (Zhou et al., 2021). We use iPNDM as the base solver.
Table 12:
Quantitative comparison on unconditional layout generation with Layout- Flow (Guerreiro et al. 2024). Lower is better for FID, Alignment (denoted as Align.), Overlap. Results for the base solvers are reported on each top rows. Bold indicates the best results, and underline marks the second best. Gray cells indicate the base ODE solvers.
Method
NFEs4
NFEa6
NTE=8
NFE=10
Align
Overlap
Align.
Owerlap
FID
Align
Overlap
Align.
Overlip
RKI
55.88
0.40
0.60
22.75
0.35
0.56
11.56
0.30
0.51
7.93
0.27
0.52
DMN
10.96
GITS
41.08
0.37
0.57
1284
0.35
0.47
7.32
0.29
0.45
5,90
0.27
0.46
Respoke
213.61
0.92
1.01
201.20
0.88
0.67
168.49
0.63
01.99
171.11
B.63
0.56
0.28
3.20
IRK2
DMN
0.66
88.15
0.69
63.57
0.37
56.23
GUTS
102.11
0.49
0.42
$1.63
0.37
0.48
27.84
0.92
0150
8.25
01.22
0.47
Bernoke
126.80
0.61
0.47
187.62
0.86
0.37
32.99
0.38
0.54
21.54
0.36
0.50
RévierFlow
0.37
7.18
unconditional layout generation using LayoutFlow (Guerreiro et al. 2024), pretrained on the RICO dataset (Deka et al. 2017).
Experiment Setup. We adopt negative mean Intersection over Union (mloU) between the teacher and student layouts as the objective for both training and validation. We generate 50 noise-date pairs for both the training and validation sets using RK45 (Butcher 1996) solver, and train the model for 5 epochs. We compare our method against the same set of baselines reported in Tab.
Evaluation Metrics. Following the evaluation protocol of LayoutFlow (Guerreiro et al. 2024)
assess generation quality using Fréchet Inception Distance (FID) adapted for layouts, alongside
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NFE
RK1
DMN
GITS
Bespoke
LD3
BezierFlow
Teacher
RICO with LayonFlow
Guemeim al.
2024
6
Figure 9: Qualitative comparisons of layout samples generated using NFEs and 8 with Lay- outFlow (Guerreiro et al. 2024). We use RK1 as the base solver. The rightmost column shows teacher samples from RK45 solver.
Table 13: FID comparison of VDM, Multi-marginal SI (denoted as MMSI), and BézierFlow CIFAR-10. Results for the base solvers reported each top Bold indicates the best results, and underline marks the second best. Gray cells indicate the base ODE solvers.
Mathod
NFFe4
NFFa6
NFF#R
NFF=10
Mathod
NFFs4
NFF
NFF#8
NFF=10
CIFAR
al.
2.70)
RKI
52.78
26.30
17.40
13.30
RK2
25.36
12.12
9.17
7.89
VDM
54.72
22.06
19.10
19.00
VDM
36.24
25.74
16.37
12.39
MMSI
22.89
12.06
7.59
5.86
MMSI
20.82
9.03
7.57
7.79
BézierFlow
20.64
9.67
7.30
5.51
BézierFlow
13.18
6.00
4.31
3.74
Alignment and Overlap scores. For FID calculation, we employ the feature extractor from Layout- Diffusion (Zheng et al. 2023).
Results. As in Table 12
BézierFlow shows better performance than other baselines in terms
of FID and Alignment. Interestingly. it consistently outperforms the base solvers (RK1 and RK2) on all metrics at the same NEE settings. We provide qualitative comparison of the generated layouts in Fig. g BézierFlow produces layouts that most closely follow the teacher trajectory, preserving the spatial arrangement and aspect ratios of objects.
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H COMPARISON WITH OTHER SCHEDULER PARAMETERIZATIONS
In this section, we discuss prior work (Kingma et al. 2023 Albergo et al. 2024) that also learns op- timal SI schedulers and compare them against BézierFlow. We first clarify how these methods differ in their scheduler parameterizations, and then experimentally show that BézierFlow achieves supe- rior performance due to its compact parameterization that explicitly satisfies the core SI scheduler requirements: boundary conditions, monotonicity, and differentiability.
Varitional Diffusion Models (Kingma et al., 2023). Variational Diffusion Models (VDMs) model the signal-to-noise ratio function SNR(t) with a monotone neural network to satisfy mono- tonicity, aiming primarily to improve generative performance rather than sampling acceleration. However, this neural network contains more than 1024 parameters, and thus is not parameter- efficient. In contrast, BézierFlow uses much more compact parameterization with only n 32 control points in our experiments by leveraging 1-D Bézier functions, reducing the number of sched- uler parameters by roughly an order of magnitude.
Multi-Marginal Stochastic Interpolant (Albergo et al., 2024). Multi-Marginal Stochastic In- terpolant (Multi-Marginal SI) also learns a stochastic interpolant scheduler to improve generative performance, In the 2-marginal case, the (unnormalized) scheduler is parameterized as
2
(7))=
a(s) 1
ag sin
(s) -
Σ
by sin
(39)
with learnable coefficients d and by, which are then normalized via
(()) ā(s) (()) + (8))
o(s)
(()) a(s) +(())
(40)
While this parameterization enforces the boundary conditions a(0) - 0,) 0. a(1) 1, and (0) L, the induced SNR schedule p(s) (s))/((s) is not guaranteed to be monotonically increasing. In contrast, our Bézier-based parameterization explicitly satisfies the three core require- ments of an SI scheduler: (I) boundary conditions, (2) monotonicity, and (3) differentiability. This advantage is reflected in the quantitative comparison reported below.
Results. We report few-step generation FIDs on CIFAR-10 with ReFlow (Liu al.) 2023) for VDM (Kingma et al. 2023), Multi-Marginal SI (Albergo et al., 2024), and BézierFlow For VDM, we parameterize the SNR neural network as 3-layer ML. with hidden size 1024, following the original configuration in the paper, and set the trigonometric order of Multi-Marginal SI to K 32 so that its number of scheduler degrees of freedom matches our n = 32 Bézier parameterization. As shown in Tab. 13] BézierFlow consistently achieves the best FID across all NFEs and base solvers, outperforming VDM and Multi-Marginal SI under the same training setup. This highlights that our Bézier-basee parameterization. which satisfies the key requirements of an SI scheduler, provides more effective and stable way to learn SI schedulers for few-step generation than existing neural or trigonometric alternatives.
CROSS-DATASET TRANSFER OF BÉZIER SCHEDULER
We investigate whether a BézierFlow trained on one dataset can be reused on other datasets without retraining. Specifically, we train BézierFlow on CIFAR-10 with pretrained diffusion models (Karras) et al. 2022) and then evaluate two different datasets, FFHQ and AFHQv2. In Tab. [4 we report FIDs for the base ODE solvers, the datasct-specific scheduler (denoted as "BézierFlow")) and the scheduler reused the target datasets (denoted as "Transferred"".
As shown, although the scheduler trained only on 10, its performance on out-of-domain datasets still outperforms the base solvers and remains competitive with scheduler trained directly on the target dataset. This demonstrates that BézierFlow provides generally effective acceleration scheme even under domain shift.
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Table 14: Cross-dataset transfer of Bézier stochastic interpolant schedulers. Results for the base solvers are reported on each top rows. Bold indicates the best results, and underline marks the second best. Gray cells indicate the base ODE solvers.
Method
NHEa
NFE=6
NFESS
NFE=10
Method
NFEel
NFE=6
NFEE8
NFE=10
CIFAR-IO 32 x 32 with EDM
FFHQ 64 64 with EDM
UniPC
47.62
14.96
7.76
8.93
iPNDM
28.75
11.15
6.68
4,80
BesierFlow
17.05
7.43
3,82
3.13
BéxierFlow Transfered
15.39
7.84
5.56
3.75
Transferred
CIFAR 10 32 x 32 with EDM
AFTIQ22 64 x 64 with EDM
UniPC 23.59
10.15
7.76
6.38
iNDD BézierFlow Transfered 9.52
15.14
6.12
3.80
3,01
BéxierFlow 12:27
Transferedd
11.58
4.47
2.98
2.71
3.95
2.96
2.30
Table 15: Training time and peak GPU memory usage of BézierFlow for diffusion and flow models NFEs and 10 on single A6000 GPU.
NFE=4
NFE=10
Dataset/ Model
Training Time
VRAM
Training Time
VRAM
(1) Diffusion Models
CIFAR- 10 x2 with EDM (Kamaa et al 2022
8 minutes
GB
15 minutes
8 GB GB G
FFHQ 64x with EDM KKaras et A022
11 minutes
GB
minutes
AFHQv2 64 with EDMI Kanras ml 2022
11 minutes
G
minutes
(2) Flow Models
(Lin 2023
ImageNNr 256 x 256 with FlowDCN Waing or
2024
25 minutes 60 minutes
5
45 minutes
8 GB 22
MS-COCO x12 with Stable Diffussoa E350 at
2024
GB
minutes
J COMPUTATIONAL COSTS
We report wall-clock training time and peak GPU memory for BézierFlow across all datasets and both diffusion and flow models, evaluated NFE=4 and NFE=10 on single A6000 GPU. As shown in Tab. 15, BézierFlow trains in at most hour even for a 2.5B large-scale pretrained model (Esser al. 2024) at 512 512 resolution, while requiring only 22 GB of GPU memory. This makes the method practical even on a single commodity GPU commonly available in research labs. Despite this low training and memory cost, BézierFlow improves FID by large margins over the base model, e.g., from 50.30 to 9.55 (s=81% % relative improvement) at NFE=4 on CIFAR-10 and from 8.93 to 3.13 (4-6% % relative improvement) at NFE=10 on FFHQ, as shown in Tab.
The increase in training time from NFE=4 to NFE=10 is always less than 2x, and peak GPU mem- ory grows only mildly with NFE. This is because we apply gradient checkpointing over the student trajectory, activation memory scales only weakly with the number of steps, even for Stable Diffu- sion v3.5. These results indicate that BézierFlow scales to high-resolution, large models with modest computational overhead, making it practical hhug-and-play scheduler even for large-scale gen- crative models.
24


Reviews:
[{"review_id": "BRqBiAX5db", "reviewer": "Reviewer_wWWP", "summary": "The authors propose a new method for efficient sampling with generative models in the stochastic interpolants framework [Albergo et al. 2023], such as diffusion, score-based, and flow-based models. They optimize stochastic trajectories (i.e., the path from the latent space to the data distribution), parameterized by continuous functions $\\alpha(s)$ and $\\sigma(s)$, so that a scheduler requiring a small number of NFEs can be distilled from a more complex one in a teacher–student manner. The student scheduler parameters $\\bar{\\alpha}(s)$ and $\\bar{\\sigma}(s)$ are modeled using degree-$n$ 1-D Bézier curves with learnable control points. Bézier polynomials are chosen because they satisfy several constraints required by scheduler functions. The authors demonstrate effectiveness on multiple diffusion and flow-based models and datasets, achieving results that are better or comparable to the state of the art for low-NFE samplers.", "questions": "Beyond the additional details requested above, could the authors comment on potential alternatives to LPIPS when moving beyond vision tasks? Would their method retain the same advantages under different distance metrics?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper is well-written and easy to follow.\n* The use of Bézier polynomials to optimize schedulers for low-NFE samplers is interesting and well-motivated, and the authors conduct extensive experiments on multiple models within the stochastic interpolant framework.\n* In low-NFE regimes, the method achieves quality that is often superior to existing approaches while requiring minimal training cost.", "weaknesses": "* In the experiments section, both the “Generalizability to Unseen NFEs” and “Training Efficiency” subsections would benefit from more detailed explanations. In particular, I found the first one somewhat unclear, while the comparison in the latter subsection is a bit confusing. I suggest providing additional training-time comparisons at equal NFE settings, including against non-distillation-based methods.\n* All experiments are done on image-based models, so the choice of the LPIPS loss is justified, but it limits the scope of the evaluation.\n* Minor: typo at line 299.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6tQk75RbVb", "reviewer": "Reviewer_TYFB", "summary": "The paper proposes BézierFlow, a lightweight method to improve few-step generation for pretrained diffusion and flow models by learning the sampling trajectory—formulated as a stochastic interpolant (SI) scheduler—instead of only optimizing discrete ODE timesteps. The scheduler’s coefficient functions are parameterized as 1D Bézier curves whose control points enforce boundary conditions, differentiability, and (claimed) monotonic SNR. Training is a teacher-forcing alignment to a high-NFE teacher using a perceptual loss, and takes ~minutes.", "questions": "see Weakness", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper moves beyond learned timesteps to continuous path learning via SI schedulers, which conceptually unifies diffusion and flow settings and widens the search space versus LD3.  Besides, it also provides analyses for endpoint-marginal preservation and schedule-invariance of the SI training objective. \n\n2. This method only needs a few parameters and minutes of training, no finetuning of the base model, and it supports plug-and-play at inference.", "weaknesses": "1. Could the authors include more qualitative comparisons and broader evaluation metrics on SD3 models? Besides, for large-scale t2i model like SD3, except the FID values, other metrics, like CLIP are also important to evaluate the performance.\n\n2. Notice that training relies solely on LPIPS. Could this induce instability or mode collapse by using only this loss across most models training? \n\n3. The authors analyze briefly in Sec 4.4 the difference between BézierFlow and other schedulers, but this anlysis is not convincing to clarify its advantages over other schedulers. Could the authors provide more solid theoretical analysis?\n\n4. While “15 minutes” training time is appealing, results are only on CIFAR-10. Include runtime and memory analysis on larger datasets and models (e.g., ImageNet, SD3) to support the scalability claim.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UR1lpUHk3a", "reviewer": "Reviewer_2baL", "summary": "The paper proposes BézierFlow (BF), a lightweight, training-efficient way to learn a scheduler for Stochastic-Interpolant (SI) models so that few-step ODE sampling matches a strong many-step teacher. Schedulers $\\left(\\bar{\\alpha},\\bar{\\sigma}\\right)$ are parameterized as low-dimensional Bézier curves and optimized via learnable control points against the teacher trajectory. BézierFlow demonstrate high-quality few-step sampling ($\\approx$3–8 NFEs) with minimal tuning and no model retraining.", "questions": "– How far can BF push NFE down (e.g., $\\le2$) before quality collapses?\n\n– Does a scheduler learned on dataset A transfer to B (or to different guidance scales/resolutions) without re-tuning?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "– Writting is clear and easy to follow. \n\n– Using Bézier control points to model SI schedulers is simple and effective. \n\n– Competitive results vs. baselines at very low NFE.", "weaknesses": "– Although the target problem differs, VDM [1] and Multi-marginal SI [2] also optimize schedules to improve performance. A brief discussion contrasting these with BézierFlow would clarify the advantages of using Bézier curves versus alternative parameterizations (and the trade-offs without Bézier).\n\n– I’m curious about the method’s limits. Recent distillation works [3,4] reduce sampling to a few or even one step while maintaining quality. I understand BF is lightweight and not the same setting, but probing the few-to-one NFE regime would better reveal the method’s capability.\n\n[1] Kingma et al., “Variational Diffusion Models”, NeurIPS 2021\n\n[2] Albergo et al., “Multimarginal generative modeling with stochastic interpolants”, NeurIPS 2024\n\n[3] Zhou et al., “Inductive moment matching”, ICML 2025\n\n[4] Kim et al., “Consistency trajectory models: Learning probability flow ode trajectory of diffusion”, ICLR 2024", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)