Data-to-Energy Stochastic Dynamics

2025-09-30

AI Summary

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.

Technology Category

• Machine Learning: Imitation Learning & Inverse Reinforcement Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

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

Abstract

The Schrödinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms allow to infer such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schrödinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schrödinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method. Code: https://github.com/mmacosha/d2e-stochastic-dynamics

Problem

• Models Schrödinger bridges using unnormalized densities without data samples
• Generalizes iterative proportional fitting for data-free distribution transport
• Enables sampling from posterior distributions in generative model latent spaces

Innovation

• Generalizes IPF for data-free Schrödinger bridge modeling
• Learns diffusion coefficients to improve transport dynamics
• Enables sampling from unnormalized densities without data samples