Diffusion Bridge Models Overview

• Diffusion bridge models are stochastic processes that condition a diffusion (SDE) on predetermined endpoints to interpolate between distributions.
• They enable applications such as high-fidelity image restoration, inverse problem solving, and time series forecasting by leveraging structured priors.
• Recent advances include bidirectional translation, constrained generation, and latent extensions that improve sampling speed and robustness.

A diffusion bridge model is a stochastic process constructed to interpolate between two specified endpoint distributions, typically representing a source and a target, by conditioning a diffusion process (often an SDE) on both initial and terminal states. This framework generalizes classical diffusion-based generative modeling by replacing the unconditional Gaussian initialization with arbitrary structured priors—enabling high-fidelity translation, conditional generation, and many specialized tasks across domains including image restoration, inverse problems, structured prediction, and time series modeling. This article provides a comprehensive review of diffusion bridge models, with an emphasis on recent advances such as system-embedded SDEs, latent- and fractional-bridge extensions, bidirectional translation, fast sampling, and constrained generation.

1. Mathematical Foundations of Diffusion Bridge Models

A diffusion bridge $\{X_t\}_{t\in[0,T]}$ is typically defined as a continuous-time Markov process conditioned on endpoints $X_0\sim p_0$ and $X_T\sim p_T$ or, equivalently, interpolating between deterministic $x_0$ and $x_T$. Such a process evolves according to a stochastic differential equation (SDE)

$dX_t = f(X_t, t)\,dt + g(t)\,dW_t$

with drift $f$, diffusion $g$, and $W_t$ standard Brownian motion. The unconditioned process drives $X_t$ from $p_0$ to $p_T$. The bridge version applies Doob’s $h$-transform, adding a time-inhomogeneous drift $g^2(t)\nabla_x \log p_T(x_T | X_t)$ so that $X_T = x_T$ almost surely: $$dX_t = f(X_t, t)\,dt + g^2(t) \nabla_x \log p_T(x_T | X_t)\,dt + g(t)\,dW_t, \quad X_0 \sim p_0,\, X_T = x_T.$$ The time-reversal (reverse SDE) for generative sampling from $p_0$ given $x_T$ is

$$dX_t = \left[f(X_t, t) - g^2(t)\{\nabla_x \log q(X_t | x_T) - \nabla_x \log p_T(x_T|X_t)\}\right]\,dt + g(t)\,d\bar W_t,$$

where $q(X_t | x_T)$ is the bridge law at $t$ given $x_T$, and $\bar W_t$ is backward-time Brownian motion. The deterministic sampling route leverages the associated probability-flow ODE.

The score $\nabla_x \log q(X_t | x_T)$ of the marginal bridge law is typically unknown, so bridge models train a neural network $s_\theta(x_t, t, x_T)$ (often via denoising score matching) to estimate it from data. This setup not only recovers the classical DDPM paradigm as a special case ($p_T=\mathcal N(0,I)$), but also unifies score-based diffusion, conditional (guided) diffusion, OT-Flow Matching, and Schrödinger bridges (Zhou et al., 2023, Sobieski et al., 30 Jun 2025, Zhang et al., 28 Oct 2024, Jiao et al., 20 Apr 2024).

2. Design Principles and Model Variants

Diffusion bridge models admit a broad design space:

• Stochastic Interpolant Family: Gaussian bridge kernels parameterized by mean/variance schedules $(\alpha_t, \beta_t, \gamma_t)$ (Zhang et al., 28 Oct 2024). For instance, with $\alpha_t=1-t$, $\beta_t=t$, $\gamma_t \propto \sqrt{t(1-t)}$, one recovers the Brownian bridge, while other schedules yield variance-preserving (VP) or variance-exploding (VE) processes (Zhou et al., 2023, Zhang et al., 28 Oct 2024).
• Score vs Denoising Parameterization: Many models predict either the conditional score or directly $x_0$ (Zhou et al., 2023, Zhang et al., 21 Apr 2025), with analytic conversion possible via the bridge’s Gaussian conditionals.
• Range–Nullspace Decomposition: For system-embedded bridges in inverse problems with known measurement operators $A$, decomposing the SDE drift and covariance into range and null-space components (e.g., $A^+A$ and $I-A^+A$) improves consistency and robustness under misspecification (Sobieski et al., 30 Jun 2025).
• Preconditioning and Base Noise: Introducing noise at $t=T$ by $X_T = y + b\cdot \xi$ (“boot noise”) enhances sample diversity at a slight cost in determinism (Zhang et al., 28 Oct 2024).
• Deterministic, Stochastic, and Fractional Bridges: Deterministic limits (zero noise) converge to flow-matching ODEs, while non-Markovian generalizations driven by fractional Brownian motion (fBM) capture long-range dependencies (Nobis et al., 3 Nov 2025).

Recent extensions include:

• Bidirectional Translation: Learning both $x_0 \to x_T$ and $x_T \to x_0$ in a single network by exploiting the symmetry in the Chapman-Kolmogorov equation for bridges, thus halving model resources (Kieu et al., 12 Feb 2025).
• Constrained Generation: Manually bridged models enforce hard geometric or logical constraints by augmenting the drift with gradients of differentiable distance penalties, supporting intersection of multiple constraints (Naderiparizi et al., 27 Feb 2025).
• Latent and Spherical Bridges: Latent Schrödinger bridge models leverage autoencoders to operate in compressed spaces, mitigating curse of dimensionality (Jiao et al., 20 Apr 2024), while domain-adapted bridges operate on spherical meshes or manifolds (Stoyanov et al., 10 Sep 2025).

3. Training Methodologies and Objectives

Bridge models are predominantly trained via supervised score matching over paired data $(x_0, x_T)$ (Zhou et al., 2023, Sobieski et al., 30 Jun 2025, Zhang et al., 21 Apr 2025). The canonical objective is

$$\mathcal{L}(\theta) = \mathbb E_{t, x_0, x_T, x_t} \left[\|s_\theta(x_t, t, x_T) - \nabla_{x_t} \log q(x_t | x_0, x_T)\|^2\right],$$

where $q(x_t | x_0, x_T)$ is the bridge conditional, often a tractable Gaussian.

Alternatives and enhancements include:

• Maximum Likelihood ELBO: For Gaussian bridges, latent variable EM or direct KL matching of forward and learned reverse processes can be applied (Ji et al., 29 Dec 2024).
• Consistency Training: Learning to solve the probability-flow ODE directly (i.e., mapping any $(x_t, t)$ to $x_0$), resulting in “consistency bridge models” that support aggressive sampler acceleration (He et al., 30 Oct 2024).
• Auxiliary and Structure-Preserving Losses: For dense prediction, adding image consistency or distribution alignment terms helps preserve task-relevant structure and capitalize on pretrained backbones (Ji et al., 29 Dec 2024).
• Constraint Annealing and Penalty Schedules: In hard constraint settings, control schedules $\gamma(t)$ for penalty strength ensure support concentrates on the desired set as $t\to0$ (Naderiparizi et al., 27 Feb 2025).
• Alternating Projections (Unpaired Learning): Extensions to Schrödinger bridge problems use iterative projection (reciprocal/Markovian) to fit unpaired marginals in transport tasks (Jiao et al., 20 Apr 2024, Nobis et al., 3 Nov 2025).

Empirically, the choice of parameterization, noise schedules, endpoint conditions, and auxiliary objectives can dramatically affect reconstruction quality, stability, and sampling efficiency.

4. Applications and Empirical Findings

Diffusion bridge models have achieved compelling results across a spectrum of tasks:

Domain	Application	Notable Models / Results
Inverse problems	Inpainting, superres	System-embedded bridges improve FID, PSNR, robustness (Sobieski et al., 30 Jun 2025)
Medical imaging	MRI $\leftrightarrow$ DTI translation	3D bridge models preserve clinical detail, benefit AD/sex classification (Zhang et al., 21 Apr 2025)
Image translation	Edges$\rightarrow$Handbags, DIODE	DDBM, DBIM, bidirectional bridges outpace pix2pix, rectified flow (Zhou et al., 2023, Kieu et al., 12 Feb 2025, Zheng et al., 24 May 2024)
Time series	Point and probabilistic forecasting	Series-to-Series Brownian bridges outperform CSDI, transformers (Yang et al., 7 Nov 2024)
Structured domains	Segmentation, point clouds, graphs	Grid and Ω-bridges match complex constraints, outperform multinomial diffusion (Liu et al., 2022, Naderiparizi et al., 27 Feb 2025)
Robotics / control	Trajectory generation / planning	Manual bridges yield constraint-satisfying plans in AVs (Naderiparizi et al., 27 Feb 2025)
Protein modeling	Future structure prediction	FDBM (fractional bridges) decrease RMSD vs Brownian baselines (Nobis et al., 3 Nov 2025)
Language-vision	Multilingual TTI, style transfer	Bridge mechanisms enable plugin-compatible, cross-language fusion (Liu et al., 2023)

A recurring observation is that embedding known system structure (measurement model, geometric constraints, informative priors) into the bridge SDE—either via drift design or endpoint specification—results in improved quality, robustness under test-time misspecification, and better utilization of limited data (Sobieski et al., 30 Jun 2025, Zhang et al., 21 Apr 2025, Ren et al., 14 Apr 2025, Naderiparizi et al., 27 Feb 2025).

Accelerated and implicit samplers (e.g., ODE-based, Heun solvers, consistency distillation) achieve substantial speedups ($4\times$–$50\times$ reduction in NFEs, with equal FID) compared to legacy SDE-based sampling, with deterministic or almost-deterministic flows for high-throughput applications (Zheng et al., 24 May 2024, Wang et al., 28 Dec 2024, He et al., 30 Oct 2024, Zhang et al., 21 Apr 2025).

5. Theoretical Analysis and Consistency Guarantees

Recent work formalizes statistical and computational guarantees for diffusion bridge models:

• Consistency of Marginals: Embedding the measurement operator (e.g., $A$ in SDB) ensures that the time-marginals of the SDE exactly match those induced by the measurement and recovery process, which is critical for ill-posed inverse problems (Sobieski et al., 30 Jun 2025).
• Error Bounds and Rate Analysis: Under regularity conditions, supervised bridge learning enjoys finite-sample/statistical efficiency, with end-to-end Wasserstein (or KL) bounds that balance discretization, score-learning, and encoding errors. In latent-SB models, reducing ambient dimension via autoencoding mitigates the curse of dimensionality (Jiao et al., 20 Apr 2024, Liu et al., 2022).
• KL-Optimality of Residual and Fractional Bridges: Analytical models such as the residual diffusion bridge (RDBM) and fractional (fBM-driven) bridges are shown to be KL-optimizing with respect to their path-space distributions. They provably outperform memoryless (Markovian) counterparts for data exhibiting temporal correlation and spatial adaptivity (Wang et al., 27 Oct 2025, Nobis et al., 3 Nov 2025).
• Closure of Manual Bridges under Constraint Addition: Sum of distance-to-constraint gradients yields a valid bridge drift for the intersection of constraints, supporting systematic constraint composition for safe generative modeling (Naderiparizi et al., 27 Feb 2025).
• Sample Diversity and Determinism: Purely deterministic bridge flows tend to collapse sample diversity; introducing controlled base noise at the endpoint restores diversity, which is quantitatively measured by average feature distance (AFD) (Zhang et al., 28 Oct 2024).

6. Extensions, Limitations, and Future Directions

• Nonlinear and Non-Gaussian Forward Models: Extending the system embedding method to nonlinear (possibly learned) forward operators is an open direction, with local linearization or mixture models as plausible routes (Sobieski et al., 30 Jun 2025).
• Unpaired and Multimodal Bridging: Iterative Markov fitting enables unpaired data translation and Schrödinger bridge problems (Jiao et al., 20 Apr 2024, Nobis et al., 3 Nov 2025). Multi-domain and cross-modal bridges (text-image, spherical surfaces) are active research areas (Liu et al., 2023, Stoyanov et al., 10 Sep 2025).
• Scalability and Transfer: Latent-bridge constructions, plugin compatibility, and distribution-aligned normalization enable integration with foundation models and high-resolution dense prediction backbones (Ji et al., 29 Dec 2024).
• Sampling Speed and Compression: ODE solvers, consistency distillation, and non-Markovian bridge approximations reduce computation and support invertibility for encoding/interpolation (Wang et al., 28 Dec 2024, He et al., 30 Oct 2024, Zheng et al., 24 May 2024).
• Theoretical Generalization: Tighter sample complexity rates, discretization error control, and extension to constrained or manifold domains (e.g., geometry, graph-structured data) remain open theoretical fronts (Jiao et al., 20 Apr 2024, Stoyanov et al., 10 Sep 2025).
• Practical Guidelines: Modelers are advised to (i) embed known system structure when available, (ii) choose schedule and noise schemes appropriate to task and data, (iii) balance speed, accuracy, and diversity via the design of endpoint conditions and noise levels, and (iv) precondition scores for numerically stable and efficient inference (Sobieski et al., 30 Jun 2025, Zhang et al., 28 Oct 2024).

7. Impact and Broader Implications

Diffusion bridge models represent a unifying and extensible framework for conditional, constrained, and structure-aware generative modeling. By encoding domain structure, leveraging informative priors, and fitting stochastic paths between arbitrary endpoint distributions, these models deliver state-of-the-art performance in atomic-resolution structure prediction, high-fidelity image restoration/conversion, robust system inversion, and beyond (Sobieski et al., 30 Jun 2025, Nobis et al., 3 Nov 2025, Zhang et al., 21 Apr 2025). The principle of explicitly embedding available measurement, constraint, or prior knowledge within the bridge emerges as a critical methodological advantage for real-world and safety-critical applications.

Anticipated research includes advances in nonlinear/implicit bridge design, scalable consistency distillation, generalization to complex geometric and multimodal spaces, and principled adaptive noise scheduling. The diffusion bridge paradigm continues to inspire foundational progress in both theory and large-scale practical deployment.