brownian bridge diffusion modeling

This work addresses the problem of efficiently and accurately bridging arbitrary probability density functions within a bounded time horizon. Methodologically, it introduces a unified generative modeling paradigm based on stochastic interpolation processes, seamlessly integrating flow-based and diffusion-based models—supporting both deterministic ordinary differential equation (ODE) paths and stochastic differential equation (SDE) paths with tunable noise. A novel score-matching objective is derived for the first time; theoretical analysis proves that optimizing only a quadratic loss suffices for likelihood control, overcoming the traditional limitation of deterministic models requiring additional Fisher divergence regularization. By unifying Schrödinger bridge theory, the Fokker–Planck equation, and variational inference, the framework rigorously recovers the Schrödinger bridge solution under optimal interpolation and provides a unified estimator for both likelihood and cross-entropy.

This work addresses the challenge of efficiently and accurately performing conditional sampling π(x|y) in unconditional diffusion models, without introducing additional model approximation errors. The proposed method formulates conditional simulation as a partial stochastic differential equation (SDE) bridge inference problem on an augmented state space, and introduces the first unified framework integrating particle Gibbs samplers with pseudo-marginal sampling—both grounded in exact SDE bridges. By relying solely on Monte Carlo error and avoiding approximations such as posterior drift correction, the approach ensures strict Bayesian consistency and eliminates bias from heuristic corrections. Experiments on both synthetic and real-world datasets demonstrate that the method achieves superior sample fidelity and theoretical coherence compared to existing conditional diffusion techniques. It establishes a new paradigm for rigorous, principled conditional inference in diffusion models.

This work addresses weak interpolation controllability and training instability in conditional generation. We propose Conditional Stochastic Interpolation (CSI), a framework that enables differentiable and controllable transport from a reference distribution to a target conditional distribution by modeling conditional probability flows or stochastic differential equations (SDEs). Key contributions include: (i) the first explicit formulation of the conditional drift and score function as conditional expectations; (ii) an adaptive diffusion term that enhances training stability; (iii) a non-asymptotic error bound guaranteeing convergence and generalization; and (iv) support for parameter-free regression estimation, deterministic ODE sampling, and adaptive-diffusion sampling. Extensive experiments on standard image datasets demonstrate high-quality, high-fidelity conditional generation. The method combines theoretical rigor—grounded in conditional probability flow theory—with practical effectiveness, offering improved controllability, stability, and flexibility over existing approaches.

Standard Gaussian diffusion priors struggle to capture temporal structures, scale sensitivity, and fixed-point constraints inherent in time series. To address this, we propose TimeBridge—a novel framework that systematically introduces data- and time-dependent priors alongside scale-preserving constraint priors, enabling a learnable diffusion bridge mechanism for probabilistic transport from adaptive priors to the target data distribution. TimeBridge unifies unconditional and conditional generation, offering both flexibility and precise controllability. Evaluated on multiple benchmark time-series datasets, it achieves state-of-the-art performance in generation diversity, fidelity, and temporal consistency—significantly outperforming existing diffusion-based baselines.

Simulating diffusion bridges in infinite-dimensional spaces—arising from continuous representations of natural data—is hindered by intractable drift terms and the absence of closed-form solutions for conditional processes. Method: We propose the first end-to-end framework integrating score matching with neural operator learning (e.g., Fourier Neural Operator), directly learning discretization-equivariant infinite-dimensional bridge processes without explicitly solving stochastic differential equations. Contribution/Results: The method enables zero-shot generalization across arbitrary spatial resolutions. Evaluated on closed-form synthetic benchmarks and real-world biological morphological evolution tasks, it achieves significantly higher path fidelity compared to baseline methods, improves sampling efficiency by multiple-fold, and seamlessly adapts to multi-scale discretizations without retraining.

This work demonstrates that diffusion models, score-based generative models, and flow matching methods—despite their apparent formal differences—share a unified continuous-time generative mechanism. By constructing a measure-theoretic framework, the paper unifies these approaches as learning time-dependent vector fields that transport a reference distribution to the data distribution, with distributional evolution governed by the continuity equation and the Fokker–Planck equation. It establishes, for the first time under a common perspective, the equivalence and distinctions among the three paradigms, clarifies the relationship between probability flow ODEs and stochastic backward dynamics, and identifies flow matching as essentially a velocity field regression problem. The study further provides a systematic comparison of objective functions, sampling strategies, and discretization errors, links the framework to Schrödinger bridges and entropy-regularized optimal transport, and summarizes theoretical guarantees and open challenges regarding approximation capacity, stability, and scalability.

Existing stochastic generative models often conflate deterministic transport with diffusion-induced stochastic effects within a single velocity field, making it difficult to disentangle their respective contributions. This work proposes a natural transport–permeation decomposition that explicitly separates deterministic transport from diffusion-induced permeation in generative dynamics for the first time, and introduces a Bridge Matching framework to learn this decomposition. Built upon a flow-based architecture, the method integrates marginal and conditional modeling and leverages score function estimation to recover the permeation field, thereby enabling an interpretable decomposition of the velocity field. Experiments demonstrate that by modulating the weight of the permeation component, the sampling process can be flexibly controlled without compromising generation quality, significantly enhancing both interpretability and controllability of the model.

This work addresses the longstanding challenge in diffusion modeling of simultaneously enabling simulation-free training and finite-time generation. The authors propose a novel reference diffusion process whose marginal distributions exactly match the target distribution, and whose time-varying conditional distributions facilitate a well-defined reversal. This formulation reveals that score matching naturally arises as the consequence of reversing the reference process and further shows that conditional flow matching corresponds to its small-noise limiting case. The resulting framework is the first to jointly support training without requiring forward simulations and generation within a finite time horizon, thereby not only broadening the theoretical foundations of diffusion models but also enhancing their practical flexibility.

This work addresses the challenge of generating conditional stochastic processes in continuous spatiotemporal settings from arbitrary observation subsets—such as irregularly sampled data or future frames—by proposing an autoregressive generative framework based on non-Markovian diffusion bridges. The method unifies physical time and state evolution within a single continuous stochastic differential equation (SDE), innovatively initializing from neighboring states, injecting noise proportionally to temporal intervals, and explicitly embedding time into the SDE dynamics. The SDE is derived via path-space measure transformation, and training is performed using a path- and time-dependent denoising score matching algorithm. Empirical evaluations on video generation and weather forecasting demonstrate significant improvements over existing approaches, particularly under low-step sampling and irregular conditioning scenarios.

This work addresses the limitations of conventional machine learning approaches, which typically rely on deterministic forward prediction and thus struggle to model multiple plausible outcomes or support backward inference—capabilities essential for scientific workflows. To overcome this, we propose a novel system integrating a conditional diffusion model (DiffUNet²) with interactive visual analytics, enabling, for the first time, bidirectional probabilistic generation at arbitrary time points in scientific time-series data. Our approach facilitates exploration of branching timelines, state editing, and navigation within probability spaces, effectively transforming generative models into user-guided, hypothesis-driven discovery tools. We demonstrate the method’s efficacy through evaluations on five cross-domain scientific datasets, showing high predictive accuracy and high-quality probabilistic ensembles, and validate its practical utility in real-world scientific analysis through expert collaboration.