Nonlocal Mean Field Schrödinger Bridge with Learned Interactions
For researchers working on large-scale interacting particle systems, this work provides a practical approximation method that reduces computational cost, though it is incremental as it combines existing techniques.
The paper tackles the computational bottleneck of nonlocal interactions in Mean-Field Schrödinger Bridge problems, which scale quadratically with population size. By approximating interactions with neural network surrogates, they achieve linear per-step cost at inference and demonstrate reduced training time on navigation and opinion-dynamics tasks.
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.