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：均值场薛定谔桥（MFSB）问题旨在设计一类最小耗能控制器，使得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演化至另一分布。与标准薛定谔桥不同，MFSB的动力约束来源于受控互作用粒子群的均值场极限，这一框架自然适用于描述大规模多智能体系统。由于非局部相互作用导致问题非凸，MFSB的计算极具挑战性。我们提出MFSB的Hopf-Cole变换推广形式，并基于此设计了一种Sinkhorn型递归算法，用于求解相应的积-偏微分方程组。在关于相互作用势的温和假设下，我们讨论了所提算法的收敛性保证。最后通过排斥与吸引相互作用的数值实例，验证了理论贡献的有效性。