Nonlocal Mean Field Schrödinger Bridge with Learned Interactions
2026-06-02 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a problem where many interacting particles are guided from a starting situation to an ending one with minimal effort. They focus on cases where particles affect each other in complex ways that can be very slow to compute when there are lots of particles. To speed things up, they replace these complex calculations with neural networks that approximate the interactions, making the computations much faster while keeping results accurate. They also provide math guarantees on how errors from these approximations affect the final outcomes. Their experiments show that this method works well for tasks like navigation and opinion dynamics, saving training time.
Schrödinger Bridge ProblemMean-Field TheoryInteracting Particle SystemsNonlocal InteractionsNeural Network SurrogatesGrönwall Stability BoundsTrajectory GenerationOpinion DynamicsNavigation TasksComputational Complexity
Authors
Daisuke Inoue, Mathieu Laurière, Dante Kalise
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.