A Closed-Form Diffusion Model for Learnring Dynamics from Marginal Observations

2025-11-11

AI Summary

Schrödinger Bridge (SB) problems suffer from the absence of closed-form solutions and reliance on unstable, computationally expensive iterative stochastic simulations. To address this, we propose the first unified closed-form diffusion dynamics modeling framework. Our method explicitly models the optimal transport dynamics from source to target distributions via score functions and analytically tractable diffusion processes—yielding non-iterative, deterministic solutions. It unifies and generalizes key special cases, including Gaussian SB and the Schrödinger–Föllmer process. Crucially, the framework entirely eliminates stochastic simulation. Empirically, it achieves substantial improvements in computational efficiency, numerical stability, and modeling accuracy on single-cell developmental trajectory inference and image restoration tasks—including inpainting and deblurring. By enabling scalable, interpretable, and deterministic generative modeling under sparse or marginal observations, our approach establishes a novel paradigm for learning dynamical processes from limited data.

Technology Category

Search and Optimization: Sampling/Simulation-based SearchReasoning under Uncertainty: Stochastic OptimizationComputer Vision: Diffusion Models for Vision

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingUser Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

Abstract

Score-based generative models learn transformations from a simple Gaussian to complex data distributions. To generalize these transformations between arbitrary distributions, recent work has focused on the Schr""odinger Bridge (SB) problem. However, SB solutions are rarely available in closed form, and existing methods rely on iterative stochastic simulations that are often unstable and costly. We introduce a closed-form framework for learning SB dynamics that unifies and extends previously known closed-form solutions, including the Schr""odinger F""ollmer process and the Gaussian SB. Notably, the classical Gaussian SB solution arises as an immediate corollary of our formulation. Based on this result, we develop a simulation-free algorithm that directly infers SB dynamics from samples of the source and target distributions. We demonstrate the approach in modeling single-cell developmental trajectories and in image restoration tasks such as inpainting and deblurring.

Problem

• Learning Schrödinger Bridge dynamics from marginal observations without iterative simulations
• Developing closed-form solutions for transforming arbitrary probability distributions
• Applying diffusion models to biological trajectory inference and image restoration tasks

Innovation

• Closed-form framework for learning Schrödinger Bridge dynamics
• Simulation-free algorithm infers dynamics from distribution samples
• Unifies and extends previously known closed-form SB solutions