schrödinger bridge matching

Existing Schrödinger bridge (SB) methods require sample access to both endpoint distributions, rendering them inapplicable when only unnormalized densities—e.g., energy functions—are available without samples. This work introduces the first sample-free, general-purpose SB modeling framework. We extend iterative proportional fitting (IPF) to the sample-free setting for the first time and integrate off-policy reinforcement learning to enable end-to-end learning of stochastic dynamics directly from energy functions. Further, we unify variational flow matching with diffusion process modeling to enhance expressivity and stability. The method significantly improves temporal discretization efficiency and enables exact probabilistic transport between multimodal distributions. We validate our approach on synthetic benchmarks and latent-space posterior sampling for generative models, demonstrating successful zero-training-data image-to-image translation.

This work addresses the problem of dynamic distributional transport. We propose a non-iterative neural network learning framework for efficiently solving entropy-regularized optimal transport. Methodologically, we introduce the first end-to-end, iteration-free learning approach for the Schrödinger bridge—without requiring iterative optimization—and enforce consistency between forward and backward bridge processes via a coupling-matching mechanism. Neural parameterization is grounded in analytically tractable diffusion processes, and we design coupled losses together with dynamical constraints to faithfully model the bridge dynamics. Theoretically, we provide convergence analysis. Experiments demonstrate that our method significantly outperforms baselines on distribution alignment tasks, achieving superior training efficiency, strong generalization across diverse distributions, and enhanced numerical stability.

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 lack of interpretability and resolution-invariant modeling capability of diffusion models in infinite-dimensional function spaces—such as images, time series, and probability density functions (PDFs). To this end, we extend stochastic optimal control theory to infinite dimensions. Our method establishes, for the first time, a rigorous equivalence between infinite-dimensional Doob h-transforms and stochastic optimal control; overcomes the fundamental challenge of undefined densities in infinite dimensions by constructing diffusion bridges without explicit density assumptions; and jointly leverages variational inference and functional optimization to directly compute optimal transport paths in function space. Experiments demonstrate that the proposed framework achieves resolution invariance in both distributional bridge learning and sampling tasks. It significantly improves fidelity and interpretability across diverse applications—including image generation, time-series interpolation, and PDF modeling—without dependence on spatial or temporal discretization.

Schrödinger Bridge (SB) methods in diffusion modeling require estimating intractable forward score functions and rely on costly implicit trajectory simulations, severely limiting scalability. To address this, we propose the Variational Schrödinger Diffusion Model (VSDM), the first SB framework integrating variational inference: it linearizes the forward score to enable simulation-free backward score training. We theoretically establish convergence of the variational score to the true backward score along sampling trajectories, eliminating the need for warm-up initialization and enhancing hyperparameter robustness. Experiments demonstrate that VSDM yields straighter, more anisotropic shape generation trajectories; achieves state-of-the-art performance on unconditional CIFAR-10 generation; effectively handles conditional time-series modeling; and significantly improves training efficiency and scalability in large-scale settings.

This work investigates how to efficiently transform a simple prior distribution into a complex target distribution that satisfies boundary constraints via stochastic trajectories in probability space, while ensuring path optimality. Building upon Schrödinger bridge theory, we develop a first-principles generative modeling framework that achieves distributional transformation by minimizing entropy deviation. Our approach establishes a unified mathematical foundation linking Schrödinger bridges with modern generative models—including diffusion models, score matching, and flow matching—and introduces a generalizable, task-oriented dynamic construction method. By integrating optimal transport, stochastic control, and path-space optimization, we devise an efficient computational toolkit for dynamic Schrödinger bridges, offering both theoretical grounding and practical improvement pathways for existing generative models.

This study addresses the failure of standard Schrödinger bridges to recover target distributions under initial distribution shifts by proposing a robust Schrödinger bridge framework. By integrating stochastic optimal control with distributionally robust optimization, we establish an exact variational formulation and design an alternating algorithm to learn a single controller that minimizes the worst-case objective under initial distributional uncertainty. Furthermore, Wasserstein and Sinkhorn gradient approximations are derived to enable efficient computation. Experimental results demonstrate that the proposed method significantly enhances robustness to input perturbations in tasks such as image translation, effectively reducing the sliced Wasserstein distance even under unseen noise levels.

This work investigates the construction of continuous martingales with prescribed marginal distributions in arbitrary dimensions and establishes a profound connection with the classical Schrödinger bridge problem. By introducing a weighted quadratic energy minimization framework, the authors extend the martingale Schrödinger bridge to the multidimensional setting and demonstrate its equivalence to the Föllmer martingale, a variational problem under convex order constraints, and a dual formulation of weak optimal transport. The main contribution lies in the first successful multidimensional generalization of this framework, proving that under irreducibility conditions, the continuous martingale Schrödinger bridge coincides with the Föllmer process. This result provides multiple equivalent characterizations, thereby unifying and significantly deepening the theoretical foundations of both classical and martingale Schrödinger bridges.

This work addresses the problem of constructing optimal transport trajectories between prescribed initial and terminal distributions under time-varying potential fields. Building upon the generalized Schrödinger bridge framework, it presents the first rigorous extension of Iterative Markov Fitting (IMF) to settings involving time-dependent potentials. By introducing a twisted Brownian motion as the reference process and leveraging the Feynman–Kac transformation, the authors formulate a bridge-matching loss that explicitly depends on the gradient of the potential. To enhance optimization stability, they further devise a trajectory variance reduction technique. Empirical evaluations on high-dimensional trajectory inference tasks—such as crowd navigation and single-cell dynamics modeling—demonstrate substantial improvements over existing methods, confirming the approach’s effectiveness and scalability in complex, dynamically evolving potential landscapes.

This study addresses the challenge that predicting interventions on stochastic system parameters relies on costly simulations, while existing models tend to degenerate into merely matching observational distributions by neglecting mechanistic sensitivity. To overcome this, we propose Tangent Schrödinger Bridge Matching, which pioneers the incorporation of mechanistic sensitivity supervision into the bridge matching framework. By jointly learning trajectories and parameter derivatives, combined with a dual-independent sampling objective to optimize mean responses, our approach eliminates estimation bias. We further establish theoretical bounds linking sensitivity precision to decision regret. Evaluated on PDE reaction-diffusion and Navier-Stokes fluid simulations, the proposed method significantly enhances parameter sensitivity estimation and counterfactual prediction accuracy. It rigorously preserves endpoint distribution consistency while effectively reducing tracking error in viscosity selection tasks.