Title: Nonlocal Mean Field Schr\"{o}dinger Bridge with Learned Interactions

Abstract: The Schr\"{o}dinger Bridge Problem aims to construct a stochastic process that links an initial distribution to a terminal distribution while minimizing energy expenditure. This study extends this framework to the Mean-Field Schr\"{o}dinger Bridge, specifically addressing interacting particle systems. When nonlocal interactions are involved, computing the resulting particle-dependent distributional terms can become computationally prohibitive, as the complexity scales quadratically with the number of particles, thereby rendering large-scale problems intractable. To overcome this limitation, we employ neural network surrogates to approximate these nonlocal interactions. Our proposed four-stage alternating algorithm significantly improves efficiency, lowering the per-step computational cost from quadratic to linear relative to the population size during inference. Furthermore, we establish Gr\"onwall-type stability bounds to characterize the propagation of surrogate errors into the generated trajectories. Numerical experiments conducted on tasks involving navigation and opinion dynamics demonstrate that the proposed approach accurately reproduces trajectories derived from analytical evaluations while substantially reducing training time.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC