Diffusion Bridge: Conditioned Stochastic Modeling

• Diffusion bridges are conditioned stochastic processes that connect two meaningful endpoint distributions instead of mapping data to a fixed Gaussian prior.
• They employ methodologies such as the Doob-h transform, time reversal, and guided-proposal constructions to achieve accurate endpoint matching in various applications.
• Recent advances—including denoising diffusion bridge models and bidirectional frameworks—enhance sampling efficiency and extend applicability to unpaired or partially observed scenarios.

A diffusion bridge is a diffusion process conditioned to connect prescribed endpoints or endpoint distributions rather than mapping data to a fixed Gaussian prior. In the classical setting, it is a diffusion process “conditioned to initialize and terminate at two given states”; in modern generative modeling, it is a stochastic process that connects two meaningful endpoint distributions, often paired or coupled, and may also be conditioned by sparse intermediate observations. Across these formulations, the central object is a path law on $C([0,T],\mathbb{R}^d)$ whose trajectories are constrained by endpoint information and, in some settings, by additional observation likelihoods along the path (Heng et al., 2021, Zhou et al., 2023, Tamir et al., 2023).

1. Conceptual scope and definitions

In the older simulation literature, a $(t_1,a,t_2,b)$-bridge is the process $X$ in $[t_1,t_2]$ conditional on $X_{t_1}=a$ and $X_{t_2}=b$. In contemporary machine-learning usage, the endpoint constraints are often random rather than fixed: one assumes paired endpoints $(x_0,x_T)\sim p(x,y)$, or, in image-to-image translation, paired samples $(y_A,y_B)\sim p(y_A,y_B)$, and learns a stochastic process whose endpoint law respects that coupling (Heng et al., 2021, Kieu et al., 12 Feb 2025).

This endpoint-centric view distinguishes diffusion bridges from standard diffusion models. Ordinary diffusion typically assumes one endpoint is a simple fixed prior, usually Gaussian, and learns to denoise toward a data distribution. Bridge models instead treat both endpoints as meaningful data distributions and construct a process between them. In paired translation, the process is required not merely to match marginal distributions, but to preserve correspondence between endpoint samples (Zhou et al., 2023, Kieu et al., 12 Feb 2025).

The same concept appears outside image translation. In computational anatomy, diffusion bridges are presented as the stochastic analogue of geodesics: deterministic landmark matching becomes conditioned stochastic landmark evolution, and bridge simulation becomes the probabilistic counterpart of inexact endpoint matching (Arnaudon et al., 2020). In time-series generation, a bridge can be interpreted as transport between the data distribution and a user-chosen prior distribution that better reflects temporal structure than a standard isotropic Gaussian endpoint (Park et al., 2024).

2. Foundational mathematical formulations

A common starting point is an Itô diffusion

$dX_t = b(t,X_t)\,dt + \sigma(t,X_t)\,dW_t,$

with $a(t,x)=\sigma(t,x)\sigma(t,x)^\top$. Conditioning is often expressed through an observation likelihood $(t_1,a,t_2,b)$0 at time $(t_1,a,t_2,b)$1, leading to

$(t_1,a,t_2,b)$2

and the conditioned process satisfies the Doob-$(t_1,a,t_2,b)$3-transform SDE

$(t_1,a,t_2,b)$4

This formulation directly accommodates exact endpoint constraints, partial terminal observation, and noisy terminal observation (Yang et al., 17 Feb 2025).

A second foundational route is time reversal. For a bridge conditioned on $(t_1,a,t_2,b)$5 and $(t_1,a,t_2,b)$6, the reverse-time dynamics can be written in terms of the score of the unconditioned transition density $(t_1,a,t_2,b)$7. “Simulating Diffusion Bridges with Score Matching” recasts bridge simulation as learning this reverse-time representation, derives a path-space KL objective, and then replaces the intractable score by a denoising-style score-matching loss based on short-time transition densities (Heng et al., 2021).

When exact conditioning terms are intractable, guided-proposal constructions replace the true bridge drift by a tractable surrogate obtained from an auxiliary diffusion. In stochastic Hamiltonian systems for landmark-based shape evolution, this yields proposal bridges that remain usable in nonlinear, high-dimensional, partially observed, and hypoelliptic settings, with the mismatch between proposal and true bridge corrected by a Radon–Nikodym factor (Arnaudon et al., 2020). Classical one-dimensional work takes a different route: approximate bridges are built by splicing intersecting forward and time-reversed paths, and an exact bridge law is then recovered by a pseudo-marginal Metropolis–Hastings correction; the method applies to one-dimensional diffusion processes with finite speed-measure and has linear computational complexity in the length of the interval (Bladt et al., 2014).

The literature also distinguishes diffusion bridges from diffusion Schrödinger bridges. In the terminology used by “Neural Guided Diffusion Bridges,” a diffusion bridge conditions on fixed endpoints or terminal observations, whereas a Schrödinger bridge connects prescribed marginals (Yang et al., 17 Feb 2025).

3. Denoising diffusion bridge models

The modern generative formulation is crystallized by Denoising Diffusion Bridge Models (DDBMs). DDBM starts from a tractable bridge family with Gaussian conditional marginals

$(t_1,a,t_2,b)$8

where $(t_1,a,t_2,b)$9 is a schedule-dependent linear combination of $X$0 and $X$1. The model learns the bridge score $X$2 by denoising bridge score matching against the tractable conditional score $X$3. Reverse-time sampling is then performed with a bridge reverse SDE or bridge probability-flow ODE that combines the learned bridge score with the known Doob-$X$4-transform correction (Zhou et al., 2023).

This formulation makes diffusion bridges a direct model for paired endpoint transport rather than a conditional correction to a noise-to-data model. The same paper shows that ordinary diffusion is a special case: if the target endpoint is chosen as the standard diffusion noised version of the source, DDBM reduces to standard diffusion generation. It also relates bridges to OT-Flow-Matching and Rectified Flow by showing that a zero-noise VE bridge limit yields the straight-line transport path $X$5 (Zhou et al., 2023).

Empirically, DDBMs were introduced for paired image translation and restoration in both pixel and latent space. On Edges $X$6 Handbags, DDBM (VP) reports FID $X$7; on DIODE-Outdoor, DDBM (VP) reports FID $X$8; and when reduced back to unconditional generation, DDBM reports FID $X$9 on CIFAR-10 and $[t_1,t_2]$0 on FFHQ-64, comparable to EDM at the same NFE (Zhou et al., 2023).

4. Bidirectionality, endpoint behavior, and accelerated sampling

A major refinement is the claim that bridge models need not be intrinsically one-way. “Bidirectional Diffusion Bridge Models” begins from the Chapman–Kolmogorov equation for a conditional Markov process and shows that, once the bridge marginals conditioned on endpoints are chosen to be Gaussian, both forward and backward directional kernels admit analytical forms. This leads to the Bidirectional Diffusion Bridge Model (BDBM), which uses a single shared network to perform translation in both directions between two coupled distributions. In this view, forward and backward translation are two conditional transition kernels on the same bridge, not two unrelated processes (Kieu et al., 12 Feb 2025).

Another line of work argues that the standard score-matching-style bridge objective can be mis-specified near an endpoint. “Resolving Endpoint Underfitting in Diffusion Bridges via Noise Alignment” identifies endpoint underfitting near the target endpoint $[t_1,t_2]$1, characterized by variance drift and direction drift. Its diagnosis is that, in I2SB-style training, the network is forced to map a nearly deterministic input to a highly stochastic target. The proposed Noise-Aligned Diffusion Bridge (NADB) replaces that target by a noise-aligned mapping and shortens the bridge through a mean network,

$[t_1,t_2]$2

so that both input noise and target noise vanish at the endpoint (Gao et al., 27 May 2026).

Sampling efficiency has become a separate research direction. Diffusion Bridge Implicit Models (DBIMs) generalize DDBMs through non-Markovian diffusion bridges on a chosen discrete sampling grid that preserve the same marginals and training objectives while interpolating between stochastic and deterministic samplers; the paper reports up to $[t_1,t_2]$3 faster sampling than vanilla DDBM and introduces a bridge-specific ODE together with “booting noise” at the initial step (Zheng et al., 2024). UniDB++ derives exact closed-form solutions for UniDB’s reverse-time SDEs, replaces noise prediction with a more stable data prediction model, adds an SDE-Corrector, and reports high-quality generation with up to $[t_1,t_2]$4 fewer sampling steps; it also shows that UniDB++ can recover DBIMs as special cases under some theoretical conditions (Pan et al., 23 May 2025).

A different acceleration strategy is amortized variational inference on path space. “Neural Guided Diffusion Bridges” starts from a guided proposal, adds a neural drift correction in Girsanov form, and minimizes

$[t_1,t_2]$5

through a tractable objective involving a quadratic control penalty and the guided-proposal correction term $[t_1,t_2]$6. Once trained, the method produces independent approximate bridge samples by a single forward SDE solve, without MCMC correction, SMC, or score estimation at sampling time; the paper also notes that the forward-KL objective is mode-seeking (Yang et al., 17 Feb 2025).

5. Beyond fully paired endpoint constraints

Several recent works relax the assumption that bridge learning is defined only by two fully paired endpoint marginals. “Transport with Support: Data-Conditional Diffusion Bridges” extends the dynamic Schrödinger bridge framework to sparse intermediate observations through the Iterative Smoothing Bridge (ISB). ISB alternates bridge fitting with Bayesian filtering-style corrections, uses particle filtering at observation times, and introduces differentiable resampling via entropy-regularized optimal transport, so that intermediate likelihood information changes both the sampled trajectories and the learned forward and backward drifts (Tamir et al., 2023).

“Structured Diffusion Bridges” addresses a different source of non-identifiability: if only the marginals $[t_1,t_2]$7 and $[t_1,t_2]$8 are known, there are infinitely many admissible couplings

$[t_1,t_2]$9

The paper therefore treats paired supervision as an optional heuristic rather than a prerequisite and restricts the bridge through marginal matching, endpoint cycle consistency, and trajectory-level cycle consistency. In this formulation, diffusion bridges become a constraint-composition framework that remains applicable in unpaired, semi-paired, and paired regimes (Kosman et al., 3 May 2026).

Bridge conditioning has also been widened in terms of observation models. The neural-guided framework explicitly supports Gaussian observation likelihoods $X_{t_1}=a$0, including exact endpoint conditioning, partial terminal observation, and noisy terminal observation. Its experiments focus on rare events, multimodality, partial observations, and state-dependent diffusion, where the paper argues that learning a variational bridge law can be more robust than score-learning from unconditional samples (Yang et al., 17 Feb 2025).

6. Time series, latent priors, and learned endpoints

In time-series generation, the bridge idea is used primarily to replace the standard Gaussian endpoint by a structure-aware prior. TimeBridge learns a bridge between the data distribution and a chosen prior distribution, with prior designs including a data-dependent Gaussian prior, a Gaussian-process prior with RBF kernel, and scale-preserving priors for conditional generation in which the condition itself is used as the endpoint. The paper emphasizes that the bridge mean is a linear combination of the target sample and the chosen endpoint, which preserves scale and temporal structure; empirically, TimeBridge-GP improves Context-FID and discriminative score relative to Diffusion-TS, and the VP bridge outperforms the VE bridge in its experiments (Park et al., 2024).

For forecasting, the Series-to-Series Diffusion Bridge Model ($X_{t_1}=a$1) replaces the usual data-to-noise process by a Brownian Bridge between a historical-data-derived prior $X_{t_1}=a$2 and the future target series $X_{t_1}=a$3,

$X_{t_1}=a$4

Because the reverse process is initialized at $X_{t_1}=a$5 rather than Gaussian noise, the model directly supports deterministic point forecasting through $X_{t_1}=a$6 and probabilistic forecasting through $X_{t_1}=a$7 (Yang et al., 2024).

Diffusion bridges have also been repurposed for representation learning. Diffusion Bridge AutoEncoders (DBAE) make the terminal state $X_{t_1}=a$8 depend on a low-dimensional latent code $X_{t_1}=a$9, then define the diffusion component as a Doob-$X_{t_2}=b$0-transform bridge between $X_{t_2}=b$1 and the inferred endpoint $X_{t_2}=b$2. This bridge view is used to address the “information split problem” of prior diffusion autoencoders: $X_{t_2}=b$3 becomes the information bottleneck, while $X_{t_2}=b$4 becomes a learnable endpoint rather than a static Gaussian prior (Kim et al., 2024).

A more explicit structural prior appears in Space-Time Diffusion Bridge, which introduces a linear Gaussian diffusion with matrix-valued drift, builds its endpoint-conditioned bridge, and then adds a nonlinear score-based refinement. The distinctive feature is “space-time mixing”: the drift may be non-diagonal, and one of the paper’s examples uses a graph Laplacian over image pixels, so the bridge is already spatially coupled before nonlinear correction (Behjoo et al., 2024).

7. Applications, comparative perspectives, and limitations

Applications span classical inference, generative modeling, and scientific simulation. In stochastic landmark dynamics, diffusion bridges are the stochastic analogue of geodesics and are used for landmark matching, parameter learning, and template estimation under partial observation and hypoelliptic noise structures (Arnaudon et al., 2020). In paired image translation and restoration, bridges model direct transport between degraded and clean, or source and target, distributions (Zhou et al., 2023). In time series, they support unconditional generation, trend-conditioned synthesis, imputation, and forecasting (Park et al., 2024, Yang et al., 2024).

The relation between diffusion bridges and flow matching has become a recurring point of comparison. DDBM already shows that a zero-noise VE bridge limit recovers the straight-line velocity $X_{t_2}=b$5 associated with OT-Flow-Matching and Rectified Flow (Zhou et al., 2023). A later comparative study recasts Diffusion Bridge and Flow Matching through stochastic optimal control, proves that under its UniDB-GOU assumptions the optimal control cost of Diffusion Bridge is lower than that of Flow Matching, and argues from an optimal-transport perspective that the interpolation coefficients $X_{t_2}=b$6 and $X_{t_2}=b$7 become increasingly ineffective when the training data size is reduced (Zhu et al., 29 Sep 2025).

The main limitations are formulation-dependent. DDBM and BDBM are built around paired data; BDBM explicitly notes that extending the framework beyond image domains, especially to multimodal settings such as image-text bridges, remains open (Zhou et al., 2023, Kieu et al., 12 Feb 2025). Neural Guided Diffusion Bridges minimize a forward KL on path space and therefore may underrepresent multiple bridge modes (Yang et al., 17 Feb 2025). NADB argues that many existing score-matching-style bridge objectives are asymptotically ill-conditioned near the target endpoint, and its theoretical claims are targeted rather than comprehensive (Gao et al., 27 May 2026). Structured Diffusion Bridges state that unpaired modality translation remains non-identifiable in general and that their constraints are heuristics rather than a full identifiability theory (Kosman et al., 3 May 2026). TimeBridge leaves class labels and time-independent conditions to future work (Park et al., 2024).

Taken together, the literature presents diffusion bridges as a family of conditioned stochastic transport models rather than a single algorithm. The common thread is endpoint-aware path construction. The differences lie in what is conditioned, how the bridge score or drift is learned, whether the formulation is exact or variational, how much structure is built into the reference process, and which compromise is preferred between exactness, speed, flexibility, and supervision.