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

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 Schrödinger 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 Schrödinger Föllmer 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.