Assistant professor in Mathematics at University of South Carolina.

Colloquia and RTG data science seminars.

Optimal transport and Mean field games workshop series

Draft "Efficient Computation of Mean field Control based Barycenters from Reaction-Diffusion Systems" is online. We develop a class of barycenter problems based on mean field control problems in three dimensions with associated reactive-diffusion systems of unnormalized multi-species densities. This problem is the generalization of the Wasserstein barycenter problem for single probability density functions. The primary objective is to present a comprehensive framework for efficiently computing the proposed variational problem: generalized Benamou-Brenier formulas with multiple input density vectors as boundary conditions. Our approach involves the utilization of high-order finite element discretizations of the spacetime domain to achieve improved accuracy. The discrete optimization problem is then solved using the primal-dual hybrid gradient (PDHG) algorithm, a first-order optimization method for effectively addressing a wide range of constrained optimization problems. The efficacy and robustness of our proposed framework are illustrated through several numerical examples in three dimensions, such as the computation of the barycenter of multi-density systems consisting of Gaussian distributions and reactive-diffusive multi-density systems involving 3D voxel densities. Additional examples highlighting computations on 2D embedded surfaces are also provided. April 2, 2024.

Draft "A primal-dual hybrid gradient method for solving optimal control problems and the corresponding Hamilton-Jacobi PDEs" is online. March 6, 2024.

Draft "Numerical analysis on neural network projected schemes for approximating one dimensional Wasserstein gradient flows" is online. We provide a numerical analysis and computation of neural network pro- jected schemes for approximating one dimensional Wasserstein gradient flows. We approximate the Lagrangian mapping functions of gradient flows by the class of two-layer neural network functions with ReLU (rectified linear unit) activation functions. The numerical scheme is based on a projected gradient method, namely the Wasserstein natural gradient, where the projection is constructed from the L2 mapping spaces onto the neural network parameterized mapping space. We establish theoretical guarantees for the performance of the neural projected dynamics. We derive a closed-form update for the scheme with well-posedness and explicit consistency guarantee for a particular choice of network structure. General truncation error analysis is also established on the basis of the projective nature of the dynamics. Numerical examples, including gradient drift Fokker-Planck equations, porous medium equations, and Keller-Segel models, verify the accuracy and effectiveness of the proposed neural projected algorithm. Feb 27, 2024.

Draft "Mean field control of droplet dynamics with high order finite element computations" is online. Liquid droplet dynamics are widely used in biological and engineering applications, which contain complex interfacial instabilities and pattern formulation such as droplet merging, splitting, and transport. This paper studies a class of mean field control formulation towards these droplet dynamics. They are used to control and maintain the manipulation of droplets in applications. We first formulate the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. We then design an optimal control problem for these gradient flows. We lastly apply the primal-dual hybrid gradient algorithm with high-order finite element methods to simulate the proposed mean field control problems. Numerical examples, including droplet formation, bead-up/spreading, transport, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism. Feb 10, 2024.

Draft "Wasserstein proximal operators describe score-based generative models and resolve memorization" is online. Feb 9, 2023.

Draft "Fisher information dissipation for time inhomogeneous stochastic differential equations" is online. We provide a Lyapunov convergence analysis for time-inhomogeneous variable coefficient stochastic differential equations. Three typical examples include overdamped, irreversible drift, and underdamped Langevin dynamics. We first reformulate the probability transition equation of Langevin dynamics as a modified gradient flow of the Kullback–Leibler divergence in the probability space with respect to time-dependent optimal transport metrics. This formulation contains both gradient and non-gradient directions depending on a class of time-dependent target distributions. We then select a time-dependent relative Fisher information functional as a Lyapunov functional. We develop a time-dependent Hessian matrix condition, which guarantees the convergence of the probability density function of Langevin dynamics. We verify the proposed conditions for several time-inhomogenous Langevin dynamics. For overdamped Langevin dynamics, we prove the $O(t^{-\frac{1}{2}})$ convergence of $L^1$ distance for the simulated annealing dynamics with a strongly convex potential function. For the irreversible drift Langevin dynamics, we prove an improved convergence towards the target distribution in an asymptotic regime. We also verify the convergence condition for the underdamped dynamics. Numerical examples demonstrate the convergence results of time-dependent Langevin dynamics. Feb 5, 2024.

Draft "Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method" is online. A first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms are also discussed in the numerical section. Jan 30, 2024.

Draft "Tensor train based sampling algorithms for approximating regularized Wasserstein proximal operators" is online. We present a tensor train (TT) based algorithm designed for sampling from a target distribution and employ TT approximation to capture the high-dimensional probability density evolution of overdamped Langevin dynamics. This involves utilizing the regularized Wasserstein proximal operator, which exhibits a simple kernel integration formulation, i.e., the softmax formula of the traditional proximal operator. The integration poses a challenge in practical scenarios, making the algorithm practically implementable only with the aid of TT approximation. In the specific context of Gaussian distributions, we rigorously establish the unbiasedness and linear convergence of our sampling algorithm towards the target distribution. To assess the effectiveness of our proposed methods, we apply them to various scenarios, including Gaussian families, Gaussian mixtures, bimodal distributions, and Bayesian inverse problems in numerical examples. The sampling algorithm exhibits superior accuracy and faster convergence when compared to classical Langevin dynamics-type sampling algorithms. Jan 25, 2024.

Draft "A deep learning algorithm for computing mean field control problems via forward-backward score dynamics" is online. We propose a deep learning approach to compute mean field control problems with individual noises. The problem consists of the Fokker-Planck (FP) equation and the Hamilton-Jacobi-Bellman (HJB) equation. Using the differential of the entropy, namely the score function, we first formulate the deterministic forward-backward characteristics for the mean field control system, which is different from the classical forward-backward stochastic differential equations (FBSDEs). We further apply the neural network approximation to fit the proposed deterministic characteristic lines. Numerical examples, including the control problem with entropy potential energy, the linear quadratic regulator, and the systemic risks, demonstrate the effectiveness of the proposed method. Jan 19, 2024.

Applied mathematics; Transport information geometry in PDEs, Data science, Graphs and Neural networks, Complex Dynamical systems, Computation, Statistics, Control and Games.

Our paper "A first-order computational algorithm for reaction-diffusion type equations via primal-dual hybrid gradient method" is accepted in Jan, 2024.