Wuchen Li

Assistant professor in Mathematics at University of South Carolina.

Google Scholar

Optimal transport and Mean field games series, 2023.



Recent News

Draft "A time-fractional optimal transport and mean-field planning: Formulation and algorithm " is online. The time-fractional optimal transport (OT) and mean-field planning (MFP) models are developed to describe the anomalous transport of the agents in a heterogeneous environment such that their densities are transported from the initial density distribution to the terminal one with the minimal cost. We derive a strongly coupled nonlinear system of a time-fractional transport equation and a backward time-fractional Hamilton-Jacobi equation based on the first-order optimality condition. The general-proximal primal-dual hybrid gradient (G-prox PDHG) algorithm is applied to discretize the OT and MFP formulations, in which a preconditioner induced by the numerical approximation to the time-fractional PDE is derived to accelerate the convergence of the algorithm for both problems. Numerical experiments for OT and MFP problems between Gaussian distributions and between image densities are carried out to investigate the performance of the OT and MFP formulations. Those numerical experiments also demonstrate the effectiveness and flexibility of our proposed algorithm. Sep 2, 2023.

Draft "Noise-Free Sampling Algorithms via Regularized Wasserstein Proximals" is online. We consider the problem of sampling from a distribution governed by a potential function. This work proposes an explicit score-based MCMC method that is deterministic, resulting in a deterministic evolution for particles rather than a stochastic differential equation evolution. The score term is given in closed form by a regularized Wasserstein proximal, using a kernel convolution that is approximated by sampling. We demonstrate fast convergence on various problems and show improved dimensional dependence of mixing time bounds for the case of Gaussian distributions compared to the unadjusted Langevin algorithm (ULA) and the Metropolis-adjusted Langevin algorithm (MALA). We additionally derive closed form expressions for the distributions at each iterate for quadratic potential functions, characterizing the variance reduction. Empirical results demonstrate that the particles behave in an organized manner, lying on level set contours of the potential. Moreover, the posterior mean estimator of the proposed method is shown to be closer to the maximum a-posteriori estimator compared to ULA and MALA, in the context of Bayesian logistic regression. Aug 29, 2023.

Draft "Langevin dynamics for the probability of Markov jumping processes" is online. We study gradient drift-diffusion processes on a probability simplex set with finite state Wasserstein metrics, namely the Wasserstein common noise. A fact is that the Kolmogorov transition equation of finite reversible Markov jump processes forms the gradient flow of entropy in finite state Wasserstein space. This paper proposes to perturb finite state Markov jump processes with Wasserstein common noises and formulate stochastic reversible Markov jumping processes. We also define a Wasserstein Q-matrix for this stochastic Markov jumping process. We then derive the functional Fokker-Planck equation in probability simplex, whose stationary distribution is a Gibbs distribution of entropy functional in a simplex set. Finally, we present several examples of Wasserstein drift-diffusion processes on a two-point state space. July 2, 2023.

Draft "Minimal Wasserstein Surfaces" is online. In finite-dimensions, minimal surfaces that fill in the space delineated by closed curves and have minimal area arose naturally in classical physics in several contexts. No such concept seems readily available in infinite dimensions. The present work is motivated by the need for precisely such a concept that would allow natural coordinates for a surface with a boundary of a closed curve in the Wasserstein space of probability distributions (space of distributions with finite second moments). The need for such a concept arose recently in stochastic thermodynamics, where the Wasserstein length in the space of thermodynamic states quantifies dissipation while area integrals (albeit, presented in a special finite-dimensional parameter setting) relate to useful work being extracted. Our goal in this work is to introduce the concept of a minimal surface and explore options for a suitable construction. To this end, we introduce the basic mathematical problem and develop two alternative formulations. Specifically, we cast the problem as a two-parameter Benamou-Breiner type minimization problem and derive a minimal surface equation in the form of a set of two-parameter partial differential equations. Several explicit solutions for minimal surface equations in Wasserstein spaces are presented. These are cast in terms of covariance matrices in Gaussian distributions. June 25, 2023.

Draft "Generalized optimal transport and mean field control problems for reaction-diffusion systems with high-order finite element computation" is online. We design and compute a class of optimal control problems for reaction-diffusion systems. Numerical examples, including generalized optimal transport and mean field control problems between Gaussian distributions and a set of 12 image densities, demonstrate the effectiveness of the proposed modeling and computational methods. June 12, 2023.

Draft "A first-order computational algorithm for reaction-diffusion type equations via primal-dual hybrid gradient method " is online. We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy. May, 2023.

Draft "Primal-Dual Damping algorithms for optimization" is online. We propose an unconstrained optimization method based on the well-known primal-dual hybrid gradient (PDHG) algorithm. We first formulate the optimality condition of the unconstrained optimization problem as a saddle point problem. We then compute the minimizer by applying generalized primal-dual hybrid gradient algorithms. Theoretically, we demonstrate the continuous-time limit of the proposed algorithm forms a class of second-order differential equations, which contains and extends the heavy ball ODEs and Hessian-driven damping dynamics. Following the Lyapunov analysis of the ODE system, we prove the linear convergence of the algorithm for strongly convex functions. Experimentally, we showcase the advantage of algorithms on several convex and non-convex optimization problems by comparing the performance with other well-known algorithms, such as Nesterov's accelerated gradient methods. In particular, we demonstrate that our algorithm is efficient in training two-layer and convolution neural networks in supervised learning problems. April, 2023.

Draft "High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems" is online. We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. Numerical examples are presented to demonstrate the effectiveness of the methods in two-dimensional PDEs, including Wasserstein gradient flows, Fisher--Kolmogorov-Petrovskii-Piskunov equation, and two and four species reversible reaction-diffusion systems. March 15, 2023.

Draft "High order computation of optimal transport, meanfield planning, and mean field games" is online. We explore applying general high-order numerical schemes 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.

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 "A kernel formula for regularized Wasserstein proximal operators" is accepted in Research in Mathematics Sciences. Sep, 2023.

Our paper "Dynamical Optimal Transport of Nonlinear Control Affine Systems" is accepted in Journal of Computational Dynamics, Aug, 2023.

Our paper "High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems" is accepted in Journal of Computational Physics, July, 2023.

Our paper "High order computation of optimal transport, meanfield planning, and mean field games" is accepted in Journal of Computational Physics, July, 2023.

Our paper "Primal-Dual Damping algorithms for optimization" is accepted in Annals of Mathematical Sciences and Applications, June, 2023.

Our paper "Entropy dissipation for degenerate stochastic differential equations via sub-Riemannian density manifold" is accepted in Entropy, May, 2023.

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 Statistics 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.