Optimization of probability flows with applications to sampling and solving PDEsMathematics of Data & Decisions
|Lingxiao Li, MIT
|Tue, Nov 21 2023, 1:10PM
In this talk, I will explore the theme of optimization of probability flows based on my two recent papers, for the application of sampling and solving a class of mass-conserving partial differential equations (PDEs). For the first half of the talk, I will show that by considering a carefully designed mollified interaction energy that converges to the chi-squared divergence as the mollifying effect vanishes, we can turn the problem of sampling into an optimization problem in the probability space. Proper discretization of this energy then yields a simple and efficient particle-based sampling algorithm suitable for both unconstrained and constrained domains. For the second half of the talk, I will introduce a discretization-free optimization framework for solving a wide class of mass-conserving PDEs, including the Fokker-Planck equations. The main observation is that the velocity field of the probability flow needs to be self-consistent, which gives rise to an efficient iterative optimization scheme that scales up to high dimensions and achieves strong performance.