schrödinger bridge theory

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.

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 Schrödinger Bridge (SB) problem under tree-structured costs and the entropy-regularized Wasserstein barycenter computation. We propose the first generalization of the Iterative Markov Fitting (IMF) framework to multi-marginal SB problems with arbitrary tree topologies. Methodologically, we unify the modeling of entropy-regularized optimal transport across arbitrary tree-structured marginals and design a streaming, dynamic fixed-point iteration scheme—overcoming scalability and adaptivity limitations of classical IPF and static barycenter algorithms. Key contributions include: (1) establishing the first IMF-based solution paradigm for tree-structured SB; (2) enabling online barycenter updates with rapid convergence; and (3) preserving IMF’s computational efficiency and numerical stability while supporting arbitrary tree structures. Experiments demonstrate significant improvements in convergence speed and numerical robustness, providing a novel tool for distribution alignment in streaming generative modeling.

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 aims to uncover the unifying mathematical principles underlying several prominent approaches in generative modeling. By constructing a high-level framework grounded in optimal transport theory, it systematically elucidates the intrinsic connections among diffusion models, flow matching, and Schrödinger bridges. The study demonstrates that these seemingly distinct methods can all be interpreted as special cases of optimal transport problems under varying constraints or approximations. This unified perspective not only clarifies the fundamental commonalities shared by diverse generative models but also establishes a theoretical foundation for their integration and further innovation.

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 work addresses the limitations of existing diffusion models in capturing discontinuous dynamics—such as abrupt changes, heavy-tailed distributions, and regime shifts—in real-world time series. The authors propose a novel jump-diffusion stochastic control framework grounded in the Schrödinger bridge formulation, which incorporates jump processes to model non-continuous behavior. By learning both the drift term and jump intensity directly from data, the method generates synthetic sequences on a fixed time grid that match the joint distribution of observed data. Integrating entropy-regularized optimal transport, stochastic control, and jump-diffusion processes, the approach significantly enhances the modeling of discontinuous temporal dynamics. Experiments on financial and energy datasets demonstrate that the proposed method outperforms conventional diffusion-based and state-of-the-art generative models in faithfully reproducing sudden shocks, heavy tails, and state transitions.

This work addresses the numerical instability and spurious correlations that arise in deterministic generative models under causal interventions, particularly when traversing low-density regions of the data distribution. To mitigate support mismatch induced by out-of-distribution interventions, we propose a Causal Schrödinger Bridge framework that formulates counterfactual reasoning as a structurally constrained entropy-regularized optimal transport problem. By constructing robust diffusion paths via stochastic differential equations, our approach ensures stable counterfactual generation. We establish a structural decomposition theorem that factorizes high-dimensional counterfactual bridging into locally robust transitions while rigorously preserving structural admissibility constraints. Experiments on high-dimensional interventions in Morpho-MNIST demonstrate that our method significantly outperforms deterministic baselines, maintaining superior structural consistency even under strong out-of-distribution shifts.

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.