The mean-field Schrödinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schrödinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.

翻译：平均场薛定谔桥（MFSB）问题旨在设计一种最小能耗控制器，引导具有非局部相互作用的扩散过程在规定时限内从给定分布到达另一分布。与标准薛定谔桥不同，MFSB的动态约束条件源自具有控制作用的相互作用智能体群体的平均场极限，这使其成为大规模多智能体系统的自然模型。由于非局部相互作用导致问题非凸，MFSB的计算极具挑战性。本文提出了MFSB的Hopf-Cole变换推广形式，并在此基础上设计了一种Sinkhorn型递归算法来求解相关的积分-偏微分方程组。在相互作用势能的温和假设下，我们讨论了所提算法的收敛性保证。通过具有排斥和吸引相互作用的数值示例，验证了理论贡献。