Georgia Institute of Technology College of Sciences

Finding a path between two probability distributions is a recurring problem in optimal transport, generative modeling, population dynamics, and control. A broad class of these problems can be formulated through an action functional defined over curves of probability distributions. The corresponding first-order optimality conditions lead to a coupled Hamilton–Jacobi and continuity system, which can be written as a boundary-value Hamiltonian flow in density space.

Computing these paths from samples becomes challenging in moderate and high dimensions. In this talk, I will discuss three approaches for learning and optimizing them: Parametric Density Path Optimization (PDPO), Hamiltonian Rectification (HR), and Neural Multiple Shooting (NMS) with a focus on NMS. 

The central idea of NMS is to reformulate the boundary-value problem in density space as a multiple-shooting problem in particle space. Since the particle dynamics are determined by the prescribed Hamiltonian system once the momentum is specified, the unknowns reduce to momentum maps that initialize a sequence of Hamiltonian initial-value problems. The resulting hybrid method combines neural approximation with classical multiple shooting: neural networks determine the momentum at the beginning of each shooting segment, while an ODE solver propagates the corresponding trajectories according to the prescribed dynamics.

I will illustrate the method through examples involving obstacle avoidance, interacting particle systems, double-integrator dynamics with smooth drag, and unicycle dynamics with obstacles and mean-field interactions.