DBMSolver: A Training-free Diffusion Bridge Sampler for High-Quality Image-to-Image Translation

Conference: CVPR 2026
arXiv: 2605.05889
Code: https://github.com/snumprlab/dbmsolver
Area: Diffusion Models / Image Generation / Image-to-Image Translation
Keywords: Diffusion Bridge Models, Training-free Sampler, Exponential Integrator, Semi-linear Structure, High-order ODE Solver

TL;DR

Addressing the slow sampling issue of Diffusion Bridge Models (DBM) in image-to-image translation (often requiring dozens or hundreds of network evaluations), DBMSolver requires no network modification or training. By revealing the "semi-linear structure" of Bridge SDEs/ODEs and deriving closed-form solutions using Exponential Integrators (EI), it surpasses previous SOTA results with only 6 steps (NFE). On DIODE, it reduces FID by 53% compared to second-order baselines at 20 NFE.

Background & Motivation

Background: Image-to-image translation (I2I, e.g., inpainting, colorization, stylization, semantic-to-image) has shifted from GANs to diffusion models. Diffusion Bridge Models (DBM, DDBM) utilize Doob's h-transform to establish a "diffusion bridge" between the source distribution \(p_T(\boldsymbol{x})\) and the target distribution \(p_0(\boldsymbol{x})\). They are currently the backbone for high-fidelity I2I due to their theoretical elegance.

Limitations of Prior Work: DBM sampling is excessively slow. The original Hybrid Heun sampler requires 119 network forward passes (NFE) for coherent results. Subsequent DBIM-2/3 reduced this to 20 NFE, but their high-order solvers rely on linear multi-step numerical approximations (non-closed-form), which suffer from significant approximation errors at low NFE.

Key Challenge: The prior \(p_T(\boldsymbol{x})\) for a diffusion bridge is an arbitrary image rather than pure Gaussian noise. This invalidates the theoretical foundation of fast solvers designed for "noise-to-image (N2I)" (e.g., DDIM, DPMSolver++), which assume \(p_T(\boldsymbol{x})\approx\mathcal{N}(\boldsymbol{0},\sigma_T^2\boldsymbol{I})\). Consequently, DBMs have been restricted to specialized but slow samplers.

Goal: To derive a fast and accurate plug-and-play sampler for DBMs without additional training or architectural changes, reducing NFE by an order of magnitude while maintaining or improving quality.

Key Insight: The authors re-examine the Bridge SDE (Eq. 2) and Bridge PF ODE (Eq. 5) followed by the reverse process of DBMs. They discover a previously overlooked semi-linear structure—linear with respect to \(\boldsymbol{x}_t\), with only the score term being non-linear. This structure is precisely what Exponential Integrators (EI) excel at solving accurately.

Core Idea: Utilize EI to obtain an exact closed-form solution for the linear term and apply Taylor expansion for the non-linear term containing the \(\boldsymbol{x}_0\)-prediction network. This derives a dedicated first-order SDE solver and a second-order ODE solver for DBMs, combined into a high-order sampling pipeline to replace crude numerical approximations.

Method

Overall Architecture

DBMSolver takes a source image (as a prior sample \(\tilde{\boldsymbol{x}}_T\sim p_T(\boldsymbol{x})\)) and a pre-trained \(\boldsymbol{x}_0\)-predicting DBM \(\textbf{D}_{\boldsymbol{\theta}}\) as input, producing the translated target image \(\tilde{\boldsymbol{x}}_0\) without altering network weights. The framework rewrites the DBM reverse SDE/ODE into a semi-linear form of "linear term + non-linear term," solves them separately, and organizes them into a three-stage sampling schedule:

• Initial Stochastic Step: Moves from \(s=T\) to \(t=T-\epsilon\) (\(\epsilon\approx10^{-4}\)) using the first-order Bridge SDE solver (Proposition 1, Eq. 8). This is handled separately because ODE solvers diverge as \(s\to T\) (coefficient \(\rho(\lambda_s,\lambda_T)\to0\)).
• Intermediate Deterministic Refinement: Iterates from \(t_{N-1}\) to \(t_1\) using the second-order Bridge PF ODE solver with \(k=2\) (Proposition 2, Eq. 9), gradually refining the noisy sample into a clean image.
• Final Euler Step: Moves from \(t_1\) to \(t_0=0\) by converting the \(\boldsymbol{x}_0\)-prediction to a score and applying a standard Euler update for high-fidelity output.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Source Prior<br/>x_T ~ p_T(x)"] --> B["Semi-linear Rewriting<br/>Bridge SDE/ODE = Linear + Non-linear"]
    B --> C["Initial Stochastic Step<br/>1st-order Bridge SDE (Eq. 8)"]
    C --> D["Intermediate Deterministic Refinement<br/>2nd-order Bridge ODE k=2 (Eq. 9)"]
    D -->|"Iterate t_{N-1}→t_1"| D
    D --> E["Final Euler Update<br/>x0-prediction to score"]
    E --> F["Translation Result x_0"]

Key Designs

1. Revealing the Semi-linear Structure: The Key to Exact Solutions Previous DBM samplers failed to recognize the internal structure of the Bridge SDE/ODE, resorting to global numerical integration which introduces large errors. The authors substitute the \(\boldsymbol{x}_0\)-predictor \(\textbf{D}_{\boldsymbol{\theta}}\) back into the Bridge PF ODE (Eq. 5), rewriting it into the semi-linear form \(\frac{\text{d}\boldsymbol{x}_t}{\text{d}t}=\underbrace{L(t)\,\boldsymbol{x}_t}_{\text{Linear}}+\underbrace{N(\textbf{D}_{\boldsymbol{\theta}}(\boldsymbol{x}_t),t,\boldsymbol{x}_T)}_{\text{Non-linear}}\). Once the linear term is isolated, an Exponential Integrator can yield an analytical exact solution for it, confining numerical approximation errors solely to the non-linear term.

2. First-order Bridge SDE Solver: Handling the Initial Stochastic Step Since ODE solutions diverge at \(t=T\), a specialized initial solver is required. The authors apply the EI method to the semi-linear Bridge SDE and use a first-order Taylor expansion (error \(O(\Delta t^2)\)) to derive Proposition 1: $\(\boldsymbol{x}_t=\frac{\text{SNR}_s}{\text{SNR}_t}\frac{\alpha_t}{\alpha_s}\boldsymbol{x}_s+\alpha_t\left(1-\frac{\text{SNR}_s}{\text{SNR}_t}\right)\textbf{D}_{\boldsymbol{\theta}}(\boldsymbol{x}_s)+\sigma_t\sqrt{1-\frac{\text{SNR}_s}{\text{SNR}_t}}\,\boldsymbol{z}_t\)$ where \(\boldsymbol{z}_t\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{I})\) and \(\text{SNR}_t:=\alpha_t^2/\sigma_t^2\). Using this only over the minimal interval \(s=T\to t=T-\epsilon\) ensures high accuracy even with a first-order approximation.

3. Second-order Bridge PF ODE Closed-form Solution: Precise Intermediate Refinement Intermediate steps are critical for quality. While DBIM-2/3 uses linear multi-step numerical methods, the authors derive an exact closed-form solution (Proposition 2, Eq. 9) for the Bridge PF ODE by using EI and change of variables. The core is an "exponentially weighted integral" \(\int_{\lambda_s}^{\lambda_t}\frac{e^{2\lambda}\textbf{D}_{\boldsymbol{\theta}}(\boldsymbol{x}_\lambda)}{\sqrt{\rho(\lambda,\lambda_T)}}\text{d}\lambda\), where \(\lambda_t:=\log(\alpha_t/\sigma_t)\). By performing a \((k-1)\)-order Taylor expansion on \(\textbf{D}_{\boldsymbol{\theta}}\) and solving the remaining integral analytically (Eq. 10) with \(k=2\), the solver achieves much tighter error bounds than numerical multi-step methods at low NFE.

4. Three-stage Schedule and Order Selection: Why k=2? The authors combine the propositions into a coherent algorithm (Algorithm 1). The choice of \(k=2\) is deliberate: for \(k\ge3\), the resulting integrals involve non-elementary antiderivatives that cannot be expressed in closed form using standard functions. This would force a return to numerical multi-step methods, re-introducing the errors found in DBIM-2/3. DBMSolver maintains the "analytical" boundary at \(k=2\) for the cleanest possible approximation.

Loss & Training

This is a training-free sampler. It directly reuses existing DBM checkpoints (e.g., DDBM weights for E2H/DIODE, DBIM weights for ImageNet inpainting). For datasets without pre-trained weights (Face2Comics, CelebAMask-HQ), a standard DBM is trained from scratch using an ADM U-Net, and then DBMSolver is applied for sampling.

Key Experimental Results

Main Results

Evaluations include sketch-to-image (E2H), surface-normal-to-image (DIODE), face caricaturization (Face2Comics), class-conditional center inpainting (ImageNet), and semantic-label-to-face (CelebAMask-HQ). Metrics used are FID/IS/LPIPS/MSE/CA, with NFE representing efficiency.

FID Comparison on DIODE (256×256):

Method	NFE	FID↓	IS↑	LPIPS↓	MSE↓
Hybrid Heun	119	4.43	6.21	0.244	0.084
DBIM-1	20	4.99	6.10	0.201	0.017
DBIM-2	20	4.40	6.11	0.200	0.017
DBIM-3	20	4.23	6.05	0.201	0.017
ODES3	28	2.29	5.92	0.203	0.018
DBMSolver (Ours)	6	3.38	6.00	0.196	0.015
DBMSolver (Ours)	20	2.06	6.00	0.198	0.018

At 20 NFE, FID is 2.06 vs. 4.40 for the second-order baseline DBIM-2, a 53% reduction. At only 6 NFE, the FID of 3.38 already outperforms Hybrid Heun (119 NFE) and all 20 NFE DBIM variants.

Results on Face2Comics / ImageNet Inpainting:

Task	Method	NFE	FID↓	Note
Face2Comics	DBIM-3	20	8.61	—
Face2Comics	DBMSolver	6	3.04	6 steps beats 20 steps DBIM
ImageNet Inpaint	DBIM-2	20	4.07	Latency 13.67s
ImageNet Inpaint	DBMSolver	6	4.98	Latency 3.66s (45.4x throughput)

Ablation Study

Configuration	Key Observation	Description
\(k=1\) (≈ DBIM-1)	Higher FID	1st-order Taylor expansion, large approximation error
\(k=2\) (DBMSolver)	Optimal FID	2nd-order closed-form solution, smaller error bounds
\(k\ge3\)	Non-analytical	Requires non-elementary integrals, reverts to numerical errors
1st-order SDE Start	Stable	Avoids ODE coefficient divergence at \(t=T\)

Key Findings

• Semi-linear structure is the source of quality: Exact solutions for linear terms combined with Taylor approximations for non-linear terms yield much smaller errors than the global numerical methods in DBIM-2/3.
• \(k=2\) is the optimal limit: Higher orders (\(k\ge3\)) introduce numerical errors due to loss of analytical tractability, making \(k=2\) the best trade-off between precision and analytical purity.
• Visual detail at low NFE: DBMSolver preserves fine structures (tree branches, textures, eye corners) at 6 NFE where DBIM remains blurry.

Highlights & Insights

• Dual Benefit of Training-free and Closed-form: As a plug-and-play replacement for DBM sampling, it reduces NFE by an order of magnitude (e.g., 6 vs. 119) while improving quality.
• Adapting EI to non-Gaussian priors: DPMSolver was assumed incompatible with DBMs due to the non-Gaussian prior. This work proves Bridge SDEs/ODEs are also semi-linear, bringing the N2I acceleration "dividend" to I2I tasks.
• Analytical Bound as an Error Bound: The insight that "elementary antiderivatives" define the maximum usable order is a valuable heuristic for designing other high-order diffusion solvers.

Limitations & Future Work

• Dependency on \(\boldsymbol{x}_0\)-prediction: The derivation is tied to DBMs trained with Bridge Score Matching. Other parameterizations would require re-derivation.
• Resolution limited to 256×256: Quality and speed gains at higher resolutions (512/1024) or in latent space DBMs haven't been fully explored.
• Trade-off at extreme low NFE: At 6 NFE on ImageNet, the FID is slightly higher than DBIM at 20 NFE, showing a remaining trade-off between extreme efficiency and absolute quality.

Related Work & Insights

• vs. Hybrid Heun (DDBM): HH alternates between 1st-order SDE and 2nd-order ODE solvers, requiring 119 NFE. DBMSolver uses EI for exact semi-linear solutions, achieving higher quality in only 6 NFE.
• vs. DBIM-1/2/3: DBIM is a non-Markovian sampler using numerical multi-step methods. DBMSolver provides analytical closed-form solutions at \(k=2\), avoiding the accumulation of numerical errors.
• vs. DPMSolver++: DPMSolver++ assumes a Gaussian prior. DBMSolver fills the gap for a closed-form high-order EI solver for non-Gaussian bridge models.

Rating

• Novelty: ⭐⭐⭐⭐⭐ First to reveal the semi-linear structure of Bridge SDEs/ODEs and derive dedicated closed-form EI solvers.
• Experimental Thoroughness: ⭐⭐⭐⭐ Covers 5 tasks and multiple baselines; however, resolution is capped at 256.
• Writing Quality: ⭐⭐⭐⭐ Clear hierarchical derivation (Semi-linear → SDE → ODE → Schedule).
• Value: ⭐⭐⭐⭐⭐ High practical value for real-time DBM deployment; code is open-source.

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