RBF-Solver: A Multistep Sampler for Diffusion Probabilistic Models via Radial Basis Functions

Soochul Park; Yeon Ju Lee; SeongJin Yoon; Jiyub Shin; Juhee Lee; Seongwoon Jo

cs.LGMar 5, 2026

RBF-Solver: A Multistep Sampler for Diffusion Probabilistic Models via Radial Basis Functions

Authors: Soochul Park, Yeon Ju Lee, SeongJin Yoon, Jiyub Shin, Juhee Lee, Seongwoon Jo

Organizations: SteAI · MODULABS · Korea University · SoftAI · SBS · Bogonet

Abstract

Diffusion probabilistic models (DPMs) are widely adopted for their outstanding generative fidelity, yet their sampling is computationally demanding. Polynomial-based multistep samplers mitigate this cost by accelerating inference; however, despite their theoretical accuracy guarantees, they generate the sampling trajectory according to a predefined scheme, providing no flexibility for further optimization. To address this limitation, we propose RBF-Solver, a multistep diffusion sampler that interpolates model evaluations with Gaussian radial basis functions (RBFs). By leveraging learnable shape parameters in Gaussian RBFs, RBF-Solver explicitly follows optimal sampling trajectories. At first order, it reduces to the Euler method (DDIM). At second order or higher, as the shape parameters approach infinity, RBF-Solver converges to the Adams method, ensuring its compatibility with existing samplers. Owing to the locality of Gaussian RBFs, RBF-Solver maintains high image fidelity even at fourth order or higher, where previous samplers deteriorate. For unconditional generation, RBF-Solver consistently outperforms polynomial-based samplers in the high-NFE regime (NFE >= 15). On CIFAR-10 with the Score-SDE model, it achieves an FID of 2.87 with 15 function evaluations and further improves to 2.48 with 40 function evaluations. For conditional ImageNet 256 x 256 generation with the Guided Diffusion model at a guidance scale 8.0, substantial gains are achieved in the low-NFE range (5-10), yielding a 16.12-33.73% reduction in FID relative to polynomial-based samplers.

Figures & tables

Panel of Figure 1 — Figure 1: (a) During sampling, the three model evaluations at time points $t_{3}$ , $t_{2}$ , and $t_{1}$ are interpolated using both radial basis functions (RBFs) and a constant basis, each basis being multiplied by its weight. Only the values of the central green pixels are plotted; the points used for interpolation are shown in black, while unused future and past points are shown in gray. (b) For second- and third-order schemes, the curves show how the interpolation coefficients vary with the shape parameter $\gamma$ . As $\log\gamma\to-\infty$ , all coefficients converge to the same value, whereas as $\log\gamma\to\infty$ , they approach the coefficients of the Adams method.

Figure 2

Panel of Figure 3 — Figure 3: Conditional sampling results. ImageNet-128: FID $\downarrow$ on 50k samples; ImageNet-256 & Stable-Diffusion v1.4: FID $\downarrow$ /CLIP $\uparrow$ on 10k.

	NFE
	5	10	15	20	25	30	35	40
Adams Method	110.02	26.94	21.07	19.49	18.97	18.59	18.41	18.33
RBF-Solver w/o Constant	112.54	26.43	20.85	19.54	18.95	18.51	18.31	18.16
Adams Pred. + RBF Corr.	111.29	26.81	20.93	19.32	18.73	18.32	18.17	18.07
RBF Pred. + Adams Corr.	117.63	26.30	21.00	19.45	18.94	18.58	18.40	18.33
RBF-Solver	116.55	26.64	20.89	19.30	18.72	18.31	18.17	18.07

Table 1: From Adams method to RBF-Solver. Sample quality measured by FID

\downarrow

on ImageNet 64

\times

64 (Improved-Diffusion), evaluated on 50k samples.

	NFE
	5	10	15	20	25	30	35	40
Shared- $\gamma$	116.37	26.07	21.02	19.46	18.95	18.58	18.40	18.28
Split- $\gamma$ / Indep.	117.35	26.86	20.91	19.33	18.73	18.32	18.17	18.08
Split- $\gamma$ / Joint	116.55	26.64	20.89	19.30	18.72	18.31	18.17	18.07

Table 2: Shape parameter optimization. Sample quality measured by FID

\downarrow

on ImageNet 64

\times

64 (Improved-Diffusion), evaluated on 50k samples.

	NFE = 10		NFE = 20		NFE = 30		NFE = 40
	RBF-Solver	UniPC	RBF-Solver	UniPC	RBF-Solver	UniPC	RBF-Solver	UniPC
$p\!=\!3$	26.47	27.75	19.30	19.51	18.31	18.60	18.07	18.34
$p\!=\!4$	24.72	33.60	18.67	19.37	18.08	18.39	17.95	18.25
$p\!=\!5$	25.37	40.55	18.58	20.25	17.92	18.48	17.92	18.27
$p\!=\!6$	25.30	40.55	18.44	29.93	17.99	23.89	17.79	20.69
$p\!=\!7$	26.39	55.96	18.62	58.40	18.52	68.87	18.97	80.48

Table 3: High-order stability. Sample quality measured by FID

\downarrow

on the ImageNet 64

\times

64 (Improved-Diffusion), evaluated on 50k samples.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Evolution of key quantities across guidance scales 2.0, 4.0, 6.0, and 8.0. Variations in the predictor’s shape parameter modulate the magnitude of its zeroth-index coefficient, which in turn helps interpolate the rapidly changing trajectory of the model evaluations.

Panel of Figure 5 — Figure 5: Optimized predictor- and corrector-side shape parameters and their corresponding coefficient-magnitude ratios (CMRs). Rows represent guidance scales 2.0, 4.0, 6.0, and 8.0 (top → bottom); columns represent $\mathrm{NFE}=$ 5, 10, 15. Shape parameters are plotted on a logarithmic scale (blue = predictor, yellow = corrector). CMRs are color-coded as follows: green (0-index), red (1-index), purple (2-index), and brown (3-index).

Series	NFE
Series	5	6	8	10	12	15	20	25	30	35	40
Model Trajectory	27.37	11.39	5.97	4.68	4.76	3.54	2.91	2.73	2.65	2.61	2.61
Target Trajectory	26.65	13.05	5.32	4.13	3.84	3.02	2.66	2.56	2.51	2.50	2.49

Appendix

Table 4: Comparison of model and target trajectory. Sample quality is measured by FID

\downarrow

on the CIFAR-10 32×32 dataset (Score-SDE), evaluated on 50k samples.

Series	NFE
Series	5	6	8	10	12	15	20	25	30	35	40
$[-1,\,1]$	26.34	16.34	5.28	4.11	3.80	3.01	2.66	2.56	2.51	2.50	2.49
$[-2,\,2]$	26.65	13.05	5.32	4.13	3.84	3.02	2.66	2.56	2.51	2.50	2.49
$[-3,\,3]$	26.64	12.45	5.36	4.16	3.92	3.02	2.66	2.56	2.51	2.50	2.50

Appendix

Table 5: Shape parameter range ablation study. Sample quality is measured by FID

\downarrow

on CIFAR-10 32

\times

32 dataset (Score-SDE), evaluated on 50k samples.

Series	NFE
Series	5	6	8	10	12	15	20	25	30	35	40
$[-1,\,1]$	116.69	70.58	35.68	26.57	22.37	20.89	19.30	18.73	18.33	18.18	18.09
$[-2,\,2]$	116.55	70.68	34.15	26.47	22.72	20.89	19.30	18.72	18.31	18.17	18.07
$[-3,\,3]$	116.75	70.60	35.13	26.64	22.36	20.88	19.28	18.73	18.30	18.15	18.07

Appendix

Table 6: Shape parameter range ablation study. Sample quality is measured by FID

\downarrow

on ImageNet 64

\times

64 dataset (Improved-Diffusion), evaluated on 50k samples.

NFE	CIFAR-10	ImageNet 64 $\times$ 64	ImageNet 128 $\times$ 128	ImageNet 256 $\times$ 256
5	$1.916\pm 0.032$	$4.846\pm 0.006$	$4.356\pm 0.046$	$7.432\pm 0.024$
6	$2.361\pm 0.032$	$6.002\pm 0.026$	$5.346\pm 0.031$	$9.101\pm 0.025$
8	$3.333\pm 0.028$	$8.333\pm 0.019$	$7.384\pm 0.041$	$12.491\pm 0.024$
10	$4.207\pm 0.029$	$10.692\pm 0.012$	$9.461\pm 0.031$	$15.889\pm 0.027$
12	$5.205\pm 0.031$	$12.968\pm 0.009$	$11.465\pm 0.065$	$19.275\pm 0.033$
15	$6.524\pm 0.033$	$16.504\pm 0.009$	$14.566\pm 0.067$	$24.349\pm 0.041$

Appendix

Table 7: Runtime per NFE for shape-parameter optimization. Each per-NFE value is the time required to optimize one batch of size

M

. The Total row sums all per-NFE runtimes, multiplies the result by

K

, and then adds the target-set runtime. All values are reported as mean

\;\pm\;

std over 10 runs.

Dataset	Model	# Eval. samples	Section
Unconditional
CIFAR-10 32 $\times$ 32 [ 28 ]	Score-SDE [ 3 ]	50k	Appendix I.1.1
CIFAR-10 32 $\times$ 32 [ 28 ]	EDM [ 34 ]	50k	Appendix I.1.2
ImageNet 64 $\times$ 64 [ 40 ]	Improved-Diffusion [ 41 ]	50k	Appendix I.1.3
Conditional
ImageNet 128 $\times$ 128 [ 29 ]	Guided-Diffusion [ 4 ]	50k	Appendix I.2.1

Appendix

Table 9: Summary of experimental settings used in Additional Main Results

Model	NFE
Model	5	6	8	10	12	15	20	25	30	35	40
DPM-Solver++ (3)	28.50	13.46	5.32	4.01	4.04	3.32	2.89	2.75	2.68	2.65	2.62
UniPC– $B_{1}(h)$ (3)	23.69	10.38	5.15	3.96	3.93	3.05	2.73	2.64	2.60	2.59	2.57
RBF-Solver (3)	26.65	13.05	5.32	4.13	3.84	3.02	2.66	2.56	2.51	2.50	2.49
RBF-Solver (4)	20.17	12.45	5.71	4.26	3.90	2.87	2.60	2.54	2.49	2.49	2.48

Appendix

Table 10: Sample quality measured by FID

\downarrow

on CIFAR-10 32

\times

32 dataset (Score-SDE), evaluated on 50k samples. Numbers in parentheses indicate the solver order.

Panel of Table 11 — Table 11: Qualitative comparison of different samplers on CIFAR-10 (Score-SDE). Each image is randomly sampled and solver order 3. Columns correspond to samplers; rows indicate NFE values of 5, 10, and 15.

Model	NFE
Model	5	6	8	10	12	15	20	25	30	35	40
DPM-Solver++ (3)	24.54	11.87	4.36	2.91	2.45	2.17	2.05	2.02	2.01	2.00	2.00
UniPC- $B_{1}(h)$ (3)	23.75	12.74	4.22	3.01	2.41	2.08	2.01	2.00	2.00	2.00	2.00
UniPC- $B_{2}(h)$ (3)	23.52	11.11	3.86	2.85	2.37	2.07	2.01	2.00	1.99	1.99	1.99
RBF-Solver (3)	22.25	10.78	3.77	2.88	2.37	2.06	1.99	1.98	1.98	1.98	1.98
RBF-Solver (4)	14.91	7.67	4.90	3.71	2.50	2.05	2.00	1.99	1.99	1.99	1.98

Appendix

Table 12: Sample quality measured by FID

\downarrow

on CIFAR-10 32

\times

32 dataset (EDM), evaluated on 50k samples. Numbers in parentheses indicate the solver order.

Model	NFE
Model	5	6	8	10	12	15	20	25	30	35	40
DPM-Solver++ (3)	110.23	66.00	38.48	28.95	24.59	21.62	19.94	19.36	18.91	18.70	18.57
UniPC- $B_{1}(h)$ (3)	109.81	65.06	35.11	27.75	24.30	21.15	19.51	18.98	18.60	18.42	18.34
RBF-Solver (3)	116.55	70.68	34.15	26.47	22.72	20.89	19.30	18.72	18.31	18.17	18.07
RBF-Solver (4)	126.26	76.92	31.71	24.72	21.70	19.61	18.67	18.30	18.08	17.99	17.95

Appendix

Table 13: Sample quality measured by FID

\downarrow

on ImageNet 64

\times

64 dataset (Improved-Diffusion), evaluated on 50k samples. Numbers in parentheses indicate the solver order.

Panel of Table 14 — Table 14: Qualitative comparison of different samplers on ImageNet 64 $\times$ 64. Each image is randomly sampled and solver order $=$ 3. Columns correspond to samplers; rows indicate NFE values of 5, 10, and 15.

		NFE
		5	6	8	10	12	15	20	25
Guidance Scale = 2.0
	DPM-Solver++ (2)	15.32	10.60	7.16	6.01	5.41	4.93	4.50	4.30
	UniPC- $B_{2}(h)$ (2)	13.43	9.19	6.50	5.69	5.23	4.85	4.48	4.29
	RBF-Solver (2)	12.83	8.80	6.34	5.61	5.22	4.85	4.49	4.28
	RBF-Solver (3)	12.19	8.15	5.98	5.51	5.16	4.84	4.51	4.30

Appendix

Table 15: Sample quality measured by FID

\downarrow

on ImageNet 128

\times

128 dataset (Guided-Diffusion), evaluated on 50k samples. Numbers in parentheses indicate the solver order.

Panel of Table 16 — Table 16: Qualitative comparison of various samplers on ImageNet 128 $\times$ 128. Each image is randomly sampled with guidance scale 6.0 and solver order $=$ 2. Columns correspond to samplers; rows indicate NFE values of 5, 10, and 15.

		NFE
		5	6	8	10	12	15	20	25	30
Guidance Scale = 2.0
	DPM-Solver++ (2)	16.27	12.59	9.62	8.54	8.11	7.68	7.44	7.33	7.27
	UniPC- $B_{2}(h)$ (2)	15.00	11.53	9.05	8.20	7.83	7.62	7.36	7.23	7.23
	RBF-Solver (2)	15.05	11.45	9.11	8.28	7.87	7.71	7.43	7.31	7.28
	RBF-Solver (3)	14.77	11.11	9.00	8.40	8.05	7.78	7.45	7.31	7.28

Appendix

Table 17: Sample quality measured by FID

\downarrow

on ImageNet 256

\times

256 dataset (Guided-Diffusion), evaluated on 10k samples. Numbers in parentheses indicate the solver order.

Panel of Table 18 — Table 18: Qualitative comparison of different samplers on ImageNet 256 $\times$ 256. Each image is randomly sampled with guidance scale set to 8.0 and solver order $=$ 2. Columns correspond to samplers; rows indicate NFE values of 5, 10, and 15.

Panel of Table 20 — Table 20: Qualitative comparison of different samplers on four shared generation scripts extracted from MS-COCO 2014. Each cell displays a 512×512 image generated using Stable Diffusion v1.4, with classifier-free guidance scale set to 7.5 and NFE $=$ 5.

Dataset	Model	Compared Samplers	Section
Unconditional
CIFAR-10 32 $\times$ 32 [ 28 ]	Score-SDE [ 3 ]	DPM-Solver-v3 [ 39 ] , DC-Solver [ 38 ]	J.2.1
CIFAR-10 32 $\times$ 32 [ 28 ]	EDM [ 34 ]	DPM-Solver-v3 [ 39 ] , DC-Solver [ 38 ]	J.2.2
CIFAR-10 32 $\times$ 32 [ 28 ]	EDM [ 34 ]	AMED-Solver, AMED-Plugin [ 37 ]	J.2.3
FFHQ 64 $\times$ 64 [ 47 ]	EDM [ 34 ]	AMED-Solver, AMED-Plugin [ 37 ]	J.2.4
Conditional

Appendix

Table 21: Summary of experimental settings for auxiliary-tuning samplers.

Method	Parameter Type	Param. Count	Dataset Size	Offline Tuning Time *	Additional Network
DPM-Solver-v3 [ 39 ]	Empirical Model Stats.	1.1M–295M	1k–4k	$<$ 11 h / 8 GPUs	✗
AMED-Solver [ 37 ]	Neural-Net Weights	$\approx$ 9k	10k	$<$ 3 h / 1 GPU	✓
DC-Solver [ 38 ]	Compensation Ratios	$\mathrm{NFE}$	10	$<$ 5 min / 1 GPU	✗
RBF-Solver (Ours)	Shape Parameters	$2\,(\mathrm{NFE}\!-\!1)$	128	1 min - 1 h / 1 GPU **	✗

Appendix

Table 22: Comparison of solvers that employ auxiliary-parameter tuning. Param. Count is the total number of auxiliary-tuning parameters stored after the offline step; Dataset Size is the amount of data each sampler uses for tuning; Additional Network indicates whether a separate network is trained.

Model	NFE
Model	5	6	8	10	12	15	20	25	30	35	40
DPM-Solver-v3 (3)	12.76	7.39	3.93	3.39	3.23	2.89	2.69	2.62	2.58	2.58	2.57
DC-Solver (3)	48.27	22.65	6.46	4.64	5.51	4.39	3.86	2.68	2.59	2.52	2.48
RBF-Solver (3)	26.65	13.05	5.32	4.13	3.84	3.02	2.66	2.56	2.51	2.50	2.49
RBF-Solver (4)	20.17	12.45	5.71	4.26	3.90	2.87	2.60	2.54	2.49	2.49	2.48

Appendix

Table 23: Sample quality measured by FID

\downarrow

on CIFAR-10 32

\times

32 dataset (Score-SDE), evaluated on 50k samples. Numbers in parentheses indicate the solver order.

Model	NFE
Model	5	6	8	10	12	15	20	25	30	35	40
DPM-Solver-v3 (3)	12.19	8.56	3.50	2.51	2.23	2.10	2.02	2.00	2.00	1.99	1.99
DC-Solver (3)	29.69	12.52	5.21	2.88	3.15	2.37	2.07	2.16	2.15	2.11	2.17
RBF-Solver (3)	22.25	10.78	3.77	2.88	2.37	2.06	1.99	1.98	1.98	1.98	1.98
RBF-Solver (4)	14.91	7.67	4.88	3.71	2.50	2.05	2.00	1.99	1.99	1.98	1.98

Appendix

Table 24: Sample quality measured by FID

\downarrow

on CIFAR-10 32

\times

32 dataset (EDM), evaluated on 50k samples. Numbers in parentheses indicate the solver order.

Panel of Table 25 — Table 25: Qualitative comparison of different samplers on CIFAR-10 (EDM). Each image is randomly sampled and solver order $=$ 3. Columns correspond to samplers; rows indicate NFE values of 5, 10, and 15.

Model	NFE
Model	5	7	9	11	15	21	25	31	35	41
AMED-Solver (2)	7.28	4.20	3.78	4.00	3.40	3.35	3.26	3.06	2.95	2.62
AMED-Plugin (4)	6.77	3.80	2.66	2.31	2.00	1.97	1.99	2.07	2.01	1.99
RBF-Solver (3)	22.63	5.94	3.43	2.68	2.08	2.00	1.99	1.98	1.98	1.97
RBF-Solver (4)	15.01	5.29	5.06	3.01	2.09	2.01	1.99	1.98	1.98	1.97

Appendix

Table 26: Sample quality measured by FID

\downarrow

on CIFAR-10 32

\times

32 (EDM), evaluated on 50k samples. Numbers in parentheses indicate the solver order.

Model	NFE
Model	5	7	9	11	15	21	25	31	35	41
AMED-Solver (2)	14.80	8.82	6.30	5.29	4.40	3.85	3.53	3.33	3.21	2.93
AMED-Plugin (4)	12.48	6.64	4.24	3.70	2.93	2.61	2.57	2.53	2.50	2.55
RBF-Solver (3)	19.58	7.07	4.61	3.49	2.70	2.42	2.37	2.34	2.34	2.34
RBF-Solver (4)	15.58	7.93	8.71	4.19	2.60	2.34	2.30	2.29	2.30	2.31

Appendix

Table 27: Sample quality measured by FID

\downarrow

on FFHQ 64

\times

64 (EDM), evaluated on 50k samples. Numbers in parentheses indicate the solver order.

		NFE
		5	6	8	10	12	15	20	25
Guidance Scale = 2.0
	DPM-Solver-v3 (2)	14.75	10.99	8.79	8.05	7.74	7.61	7.35	7.23
	DC-Solver (3)	15.78	11.60	9.29	8.19	7.77	7.61	7.46	7.47
	RBF-Solver (2)	15.05	11.45	9.11	8.28	7.87	7.71	7.43	7.31
	RBF-Solver (3)	14.77	11.11	9.00	8.40	8.05	7.78	7.45	7.31

Appendix

Table 28: Sample quality measured by FID

\downarrow

on ImageNet 256

\times

256 (Guided-Diffusion), evaluated on 10k samples. Numbers in parentheses indicate the solver order.

Panel of Table 29 — Table 29: Qualitative comparison of different samplers on ImageNet 256 $\times$ 256. Each image is randomly sampled with guidance scale 8.0. DPM-Solver-v3 and RBF-Solver use order 2; DC-Solver uses order 3. Columns correspond to samplers; rows indicate NFE values of 5, 10, and 15.

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Formalizing the Sampling Design Space of Diffusion-Based Generative Models via Adaptive Solvers and Wasserstein-Bounded Timesteps

Sangwoo Jo, Sungjoon Choi

Department of Artificial Intelligence, Korea University, Seoul, South Korea.

Diffusion Model Sampling Diffusion Sampling Adaptive Sampling Diffusion Timestep Scheduling Diffusion Model Acceleration

Open paper AskPDF arXiv

Diffusion-based generative models have achieved remarkable performance across various domains, yet their practical deployment is often limited by high sampling costs. While prior work focuses on training objectives or individual solvers, the broader sampling design problem, specifically solver selection and scheduling, remains largely governed by static heuristics. We propose SDM, a principled, training-free sampling framework that adapts both the numerical solver and the timestep schedule to the intrinsic properties of the diffusion trajectory. By analyzing the PF-ODE dynamics, we show that velocity variation is small in high-noise stages and increases near the data manifold, identifying intervals where solver order is most consequential. In parallel, we introduce an offline-calibrated adaptive scheduling method that explicitly controls the local Wasserstein discretization error and projects the calibrated trajectory to a prescribed NFE budget. We further extend the formulation to a mixed-transition Wasserstein error bound, providing a unified error-propagation view of adaptive scheduling and solver selection within the overall SDM framework. Across standard benchmarks, with extensions to modern ODE samplers, high-resolution synthesis, and text-to-image generation, SDM achieves improved sample quality compared to baseline methods, attaining an FID of 1.93 on CIFAR-10, 2.41 on FFHQ, and 1.98 on AFHQv2, with a reduced number of function evaluations compared to existing samplers. Our code is available at https://github.com/aiimaginglab/sdm.

Abstract

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