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

Abstract:The Schrödinger 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 Schrödinger 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 Grönwall-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.

Optimization and Control (math.OC); Machine Learning (cs.LG); Numerical Analysis (math.NA)

arXiv:2606.04265 [math.OC]

(or arXiv:2606.04265v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2606.04265 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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