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

The Schr\"odinger Bridge Problem constructs a stochastic process that connects an initial distribution to a terminal distribution with minimum energy. This work considers its mean-field extension, the Mean-Field Schr\"odinger Bridge, for interacting particle systems. With nonlocal interactions, evaluating the resulting particle-dependent distributional terms can scale quadratically with the population size, which makes large-scale problems intractable. We address this bottleneck by approximating the nonlocal interactions with neural network surrogates. The resulting four-stage alternating algorithm reduces the per-step cost from quadratic to linear in the population size at inference. We also derive Gr\"onwall-type stability bounds that show how surrogate errors propagate to the generated trajectories. In numerical experiments on navigation and opinion-dynamics tasks, the proposed method reproduces trajectories obtained with analytical evaluation and reduces training time.