Assistant professor in Mathematics at University of South Carolina.
Colloquia and RTG data science seminars.
Optimal transport and Mean field games workshop series
Draft "Variational conditional normalizing
flows for computing second-order mean field control problems" is online. Mean field control (MFC) problems have vast applications in artificial intelligence, engineering, and economics, while solving MFC problems accurately and efficiently in high-dimensional spaces remains challenging. This work introduces variational conditional normalizing flow (VCNF), a neural network-based variational algorithm for solving general MFC problems based on flow maps. Formulating MFC problems as optimal control of Fokker--Planck (FP) equations with suitable constraints and cost functionals, we use VCNF to model the Lagrangian formulation of the MFC problems. In particular, VCNF builds upon conditional normalizing flows and neural spline flows, allowing efficient calculations of the inverse push-forward maps and score functions in MFC problems. We demonstrate the effectiveness of VCNF through extensive numerical examples, including optimal transport, regularized Wasserstein proximal operators, and flow matching problems for FP equations. March 25, 2025.
Draft "Splitting Regularized Wasserstein Proximal Algorithms for Nonsmooth Sampling Problems" is online.
Sampling from nonsmooth target probability distributions is essential in various applications, including the Bayesian Lasso. We propose a splitting-based sampling algorithm for the time-implicit discretization of the probability flow for the Fokker-Planck equation, where the score function defined as the gradient logarithm of the current probability density function, is approximated by the regularized Wasserstein proximal. When the prior distribution is the Laplace prior, our algorithm is explicitly formulated as a deterministic interacting particle system, incorporating softmax operators and shrinkage operations to efficiently compute the gradient drift vector field and the score function. The proposed formulation introduces a particular class of attention layers in transformer structures, which can sample sparse target distributions. We verify the convergence towards target distributions regarding R\'enyi divergences under suitable conditions. Numerical experiments in high-dimensional nonsmooth sampling problems, such as sampling from mixed Gaussian and Laplace distributions, logistic regressions, image restoration with $L_1$-TV regularization, and Bayesian neural networks, demonstrate the efficiency and robust performance of the proposed method.
Feb 23, 2025.
Draft "Geometric calculations on density manifolds from reciprocal relations in hydrodynamics" is online.
Hydrodynamics are systems of equations describing the evolution of macroscopic states in non-equilibrium thermodynamics. From generalized Onsager reciprocal relationships, one can formulate a class of hydrodynamics as gradient flows of free energies. In recent years, Onsager gradient flows have been widely investigated in optimal transport-type metric spaces with nonlinear mobilities, namely hydrodynamical density manifolds. This paper studies geometric calculations in these hydrodynamical density manifolds. We first formulate Levi-Civita connections, gradient, Hessian, and parallel transport, and then derive Riemannian and sectional curvatures on density manifolds. We last present closed formulas for sectional curvatures of density manifolds in one dimensional spaces, in which the sign of curvatures is characterized by the convexities of mobilities. In examples, we present density manifolds and their sectional curvatures in zero range models, such as independent particles, simple exclusion processes, and Kipnis-Marchioro-Presutti models.
Jan 27, 2025.
Transport information geometry in Complex Dynamical systems, PDEs, Statistics, Optimization, Control and Games, Mathematical Data science, Graphs and Neural networks, and Scientific Computations.
