提出新算法求解大规模多智能体系统的最优路径规划问题。
A Generalized Sinkhorn Algorithm for Mean-Field Schrödinger Bridge
- 基于广义霍普夫-科尔变换，设计递归型Sinkhorn算法求解积分-偏微分方程组。
- 在弱相互作用势下证明了算法的局部收敛性，数值实验验证了吸引力与排斥力场景。
- 适合研究多智能体系统、最优传输及随机控制的科研人员参考。
均值场薛定谔桥（MFSB）问题旨在设计一个最小能耗控制器，使具有非局部相互作用的扩散过程在固定截止时间前从初始分布演化到目标分布。与标准薛定谔桥不同，其动力学约束是大量受控交互智能体群体的均值场极限，是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，计算上极具挑战。本文提出针对MFSB的广义霍普夫-科尔变换，并在此基础上构建一种类Sinkhorn的递归算法以求解相关的积分-偏微分方程组。在对相互作用势的温和假设下，讨论了所提算法的收敛性保证。通过吸引与排斥相互作用的数值例子展示了理论贡献。
原文摘要 · Abstract (English)
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.
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