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.

We introduce the notion of TDFOE studying the thermodynamic formalism on topological Markov shifts where the potential is random, depending on a Gibbs random walk on a compact metric space.

We introduce the associated Pressure Out of Equilibrium, and discuss its properties (e.g. variational principle), and applications (Azuma inequality for potentials expanding on average w.r.t. to a Gibbs measure).

We discuss the associated Semi-Ruelle operator which generates the POE, and its properties (e.g. existence of conformal measures, spectral radius). We provide an application by studying Effectively Expanding on Average diffeomorphisms on closed manifolds (this is a $C^1$-open condition, extends previously studied conditions).

We prove quasi-compactness (and later a spectral gap) for the Averaged Semi-Ruelle operator on a Sobolev space.

Notably, the diffeos are allowed to be highly dissipative. Finally, we mention a few applications such as: Dimension bounds on the stationary measure, and CLT and LLT for the (not necessarily invariant) volume measure.

On September 8, OpenAI announced a Lean-certified resolution of the Navier-Stokes Millennium Problem by furnishing a classical solution to the forced 3d Navier-Stokes equations which blows up at the origin in finite time. The purpose of this talk will be to provide (i) historical context and relevance for the Millennium Problem; and (ii) give an overall view of the techniques developed towards its resolution from the last ten years or so. I will avoid technical details, and endeavor to make the talk accessible to those outside PDE. While I will briefly address the ongoing priority dispute and allegations of misconduct by OpenAI, this will not be the focus of the talk.