Trajectory-Optimized Density Control with Flow Matching

Abstract: Optimal transport (OT) and Schr{\"o}dinger bridge (SB) problems have emerged as powerful frameworks for transferring probability distributions with minimal cost. However, existing approaches typically focus on endpoint matching while neglecting critical path-dependent properties -- particularly collision avoidance in multiagent systems -- which limits their practical applicability in robotics, economics, and other domains where inter-agent interactions are essential. Moreover, traditional density control methods often rely on independence assumptions that fail to capture swarm dynamics. We propose a novel framework that addresses these limitations by employing flow matching as the core modeling tool, where the flow model co-evolves with the control policy. Unlike prior methods that treat transport trajectories as mere interpolations between source and target distributions, our approach explicitly optimizes over the entire transport path, enabling the incorporation of trajectory-dependent costs and collision avoidance constraints. Our framework bridges optimal transport theory with mean field control, providing a principled approach to multiagent coordination problems where both endpoint alignment and path properties are critical. Experimental results demonstrate that our method successfully generates collision-free transport plans while maintaining computational efficiency comparable to standard flow matching approaches.