Wuchen Li

Assistant professor in Mathematics at University of South Carolina.

Google Scholar

Optimal transport and Mean field games seminar, Spring 2022.



Recent News

Draft "High order computation of optimal transport, meanfield planning, and mean field games" is online. We explore applying general high-order numericalschemes with finite element methods in the space-time domain for computingthe optimal transport (OT), mean-field planning (MFP), and MFG problems.We conduct several experiments to validate the convergence rate of the highorder method numerically. Those numerical experiments also demonstratethe efficiency and effectiveness of our approach. Feb 5, 2023.

Draft "Optimal Ricci curvature Markov chain Monte Carlo methods on finite states" is online. We construct a new Markov chain Monte Carlo method on finite states with optimal choices of acceptance-rejection ratio functions. We prove that the constructed continuous time Markov jumping process has a global in-time convergence rate in L^1 distance. The convergence rate is no less than one-half and is independent of the target distribution. For example, our method recovers the Metropolisā€“Hastings algorithm on a two-point state. And it forms a new algorithm for sampling general target distributions. Numerical examples are presented to demonstrate the effectiveness of the proposed algorithm. Feb 2, 2023.

Draft "A kernel formula for regularized Wasserstein proximal operators" is online. We study a class of regularized proximal operators in Wasserstein-2 space. We derive their solutions by kernel integration formulas. Numerical examples show the effectiveness of kernel formulas in approximating the Wasserstein proximal operator. Jan 24, 2023.

I am honored to receive the Air Force Office of Scientific Research YIP award, Transport Information Geometric Computations. Thanks for the consistent and strong support from AFOSR, Computational Mathematics program, Dec 14, 2022.

Draft "Master equations for finite state mean field games with nonlinear activations" is online. We formulate a class of mean field games on a finite state space with variational principles resembling continuous state mean field games. Several concrete examples of discrete mean field game dynamics on a two-point space are presented with closed formula solutions, including discrete Wasserstein distances, mean field planning, and potential mean field games. Dec 12, 2022.

Draft "Hypoelliptic entropy dissipation for stochastic differential equations (II)" is further renewed. We provide a short proof on the decomposition of non-gradient Fokker-Planck equations. More examples of Hessian matrices are given for one dimensional SDEs. They are inital steps for designing convergence guaranteed MCMC algorithms. Aug 11, 2022.

Draft "A primal-dual approach for solving conservation Laws with Implicit in Time Approximations" is online. We propose a framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods In particular, no nonlinear inversions are required! Specifically, we illustrate our approach using the finite difference scheme and discontinuous Galerkin method for the spatial scheme. July 15, 2022.

Draft "Computational mean field games on manifolds" is online. We formulate and compute the mean field games on discrete surfaces. Using triangular mesh representations, we design a proximal gradient method for variational mean field games. Numerical experiments on various manifolds are shown the effectiveness of methods. June 5, 2022.

Draft "Optimal neural network approximation of Wasserstein gradient direction via convex optimization" is online. We study the optimal neural network approximation of the Wasserstein gradient direction. Numerical experiments including PDE-constrained Bayesian inference and parameter estimation in COVID-19 modeling demonstrate the effectiveness of the proposed method. May 26, 2022.

Draft "Exponential Entropy dissipation for weakly self-consistent Vlasov-Fokker-Planck equations" is online. We study "Hessian matrices" for a kinectic Fokker-Planck equation with mean field interaction energy. Exponential convergence result in L1 distance is proved. Two examples of convergence rates are provided. April 27, 2022.

Draft "Mean field information Hessian matrices on graphs" is online. We study Hessian matrices of general energies in a graphic optimal transport metric space. A new mean field function namely transport information mean is introduced. A discrete Costa's entropy power inequality on a two point graph is derived. March 14, 2022.

Draft "Mean field Kuromoto models on graphs" is online. We study a mean field synnorization model on discrete domain using optimal transport on graphical models. A generalized Hopf-Cole transformation is studied. Analytical examples of synnorization models on two point graphs are discussed. March 3, 2022.

Draft "Entropy dissipation for degenerate stochastic differential equations via sub-Riemannian density manifold (I) " is renewed. We provide a new example of the exponential entropy convergence analysis for one dimensional degenerate SDEs. Feb 28, 2022.

Draft "Controlling conservation laws II: compressible Navier-Stokes equations" is online. We propose and compute solutions to a class of optimal control problems for barotropic compressible Navier-Stokes equations. Feb 23, 2022.

Draft "On a prior based on the Wasserstein information matrix" is online. We introduce a prior for the parameters of univariate continuous distributions, based on the Wasserstein information matrix. Several examples of Wasserstein priors, in particular skew Gaussian distributions, are presented. This is a natural second step for transport information statistics. Feb 8, 2022.

Draft "Controlling conservation laws I: entropy-entropy flux" is online. We propose to study a class of mean field control problems for conservation laws with entropy-entropy flux pairs. A variational structure, namely flux-gradient flows and their dual equations in entropy-entropy flux pair metric spaces, are introduced. The control of flux-gradient flows are useful in modeling complex dynamics and designing implicit time schemes for conservation laws. Nov 10, 2021.

Draft "Computational Mean-field information dynamics associated with Reaction diffusion equations" is online. We compute a general mean field control problems arised from nonlinear reaction diffusion equations. July 26, 2021.

Previous results

Research Interests

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







Recent Publications

Our paper "Wasserstein information matrix" is accepted in Information Geometry, Jan 2023.

Our paper "A primal-dual approach for solving conservation Laws with Implicit in Time Approximations" is accepted in Journal of Computational Physics, 2022.

Our paper "On a prior based on the Wasserstein information matrix" is accepted in Stati stics and Probability Letters, 2022.

Our paper "Mean field control problems for Vaccine distribution" is accepted in Research in the Mathematical Sciences, 2022.

Our paper "Computational Mean-field information dynamics associated with Reaction diffusion equations" is accepted in Journal of Computational Physics, 2022.

Our paper "Projected Wasserstein gradient descent for high-dimensional Bayesian inference" is accepted in SIAM/ASA Journal on Uncertainty Quantification, 2022.

Our paper "Controlling conservation laws II: compressible Navier-Stokes equations" is accepted in Journal of Computational Physics, 2022.

Our paper "A mean field game inverse problem" is accepted in Journal of Scientific Computing, 2022.

Our paper "Neural parametric Fokker-Planck equation" is accepted in SIAM journal on numerical analysis, 2022.

Our paper "Accelerated information gradient flow" is accepted in Journal of Scientific Computing, 2021.

Our paper "Transport information Bregman divergences" is accepted in Information Geometry, 2021.

Our paper "Tracial smooth functions of non-commuting variables and the free Wasserstein manifold" is accepted in Dissertationes Mathematicae, 2021.