Deep Generalized Schr\"odinger Bridges: From Image Generation to Solving Mean-Field Games

TL;DR

This paper introduces a neural network-based computational framework for generalized Schr"odinger Bridges, enabling efficient, mesh-free solutions applicable to generative modeling and mean-field games, bridging theory and practical algorithm design.

Contribution

It reinterprets GSBs as probabilistic models and develops a continuous, neural network-driven algorithm leveraging the nonlinear Feynman-Kac lemma for practical applications.

Findings

Effective in generative modeling tasks

Successful application to mean-field games

Outperforms traditional discretization methods

Abstract

Generalized Schr\"odinger Bridges (GSBs) are a fundamental mathematical framework used to analyze the most likely particle evolution based on the principle of least action including kinetic and potential energy. In parallel to their well-established presence in the theoretical realms of quantum mechanics and optimal transport, this paper focuses on an algorithmic perspective, aiming to enhance practical usage. Our motivated observation is that transportation problems with the optimality structures delineated by GSBs are pervasive across various scientific domains, such as generative modeling in machine learning, mean-field games in stochastic control, and more. Exploring the intrinsic connection between the mathematical modeling of GSBs and the modern algorithmic characterization therefore presents a crucial, yet untapped, avenue. In this paper, we reinterpret GSBs as probabilistic models and demonstrate that, with a delicate mathematical tool known as the nonlinear Feynman-Kac lemma, rich algorithmic concepts, such as likelihoods, variational gaps, and temporal differences, emerge naturally from the optimality structures of GSBs. The resulting computational framework, driven by deep learning and neural networks, operates in a fully continuous state space (i.e., mesh-free) and satisfies distribution constraints, setting it apart from prior numerical solvers relying on spatial discretization or constraint relaxation. We demonstrate the efficacy of our method in generative modeling and mean-field games, highlighting its transformative applications at the intersection of mathematical modeling, stochastic process, control, and machine learning.