Title: ‘Learning and optimizing paths between probability distributions. From action minimization to boundary-valued Hamiltonian flows’.

Date:  November 4th 

Time: 9.00am (EST)

Room: 006

Zoom link: https://gatech.zoom.us/j/91925737499

Committee members:

Dr. Haomin Zhou (Advisor)

Dr. Peng Chen (Co-advisor)

Dr. Sung Ha Kang

Dr. Luca Dieci

Dr. Shu Liu

Abstract:

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.

We take three approaches for learning and optimizing them: Hamiltonian Rectification (HR), Neural Multiple Shooting (NMS), and Parametric Density Path Optimization (PDPO). In the talk, we are going to focus on Neural Multiple Shooting. 

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.