- 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. PDF
- 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. PDF
- Draft "Mean field control problems for Vaccine distribution" is online. We design mean field control problems for optimal spatial distribution of vaccine. April 24, 2021. PDF
- Draft "A Fast Proximal Gradient Method and Convergence Analysis For Dynamic Mean Field Planning" is online. We design proximal gradient algorithms for a class of mean field planning problems and conduct related numerical analysis. Feb 26, 2021. PDF
- Draft "Projected Wasserstein gradient descent for high-dimensional Bayesian inference" is online. We apply first order transport optimization methods to design algorithms in inverse problems and Bayesian inferences. Feb 16, 2021. PDF
- Draft "Transport information Hessian distances" is online. We derive closed-form solutions for Hessian distances of information entropies in Wasserstein space. They are distances in term of Jacobi operators of pushforward mapping functions, namely "optimal Jacobi transport distances". We will apply the proposed distances in AI inference problems and MCMC algorithms. Feb 8, 2021. PDF
- Draft "Hypoelliptic Entropy dissipation for stochastic differential equations" is online. We derive a structure condition and an algebraic tensor to estimate the convergence rates of variable coefficients degenerate Langevin dynamics. Our method is based on a weighted Fisher information induced Gamma derivative method. We will apply the result to design degenerate MCMC algorithms with theoretical convergence guarantees. Feb 1, 2021. PDF
- Draft "Tracial smooth functions of non-commuting variables and the free Wasserstein manifold" is online. We study the optimal transport metric in free probabilties. It can be viewed as a natural first step to develop free transport information geometry. Jan 18, 2021. PDF
- Draft "Transport information Bregman divergences" is online. We study Bregman divergences in Wasserstein-2 space. In particular, we derive the transport Kullback-Leibler (KL) divergence, which is a Bregman divergence of negative Boltzmann-Shannon entropy in Wasserstein-2 space. Here the transport KL divergence is an Itakuraâ€“Saito type divergence in transport coordinates. Jan 4, 2021. PDF

- Draft "Entropy dissipation via information Gamma calculus: non-reversible stochastic differential equations" is online. We derive the other structure condition for non-reversible stochastic differential equations. We derive the Fisher information induced Gamma calculus to handle nongradient drift vector fields. From it, we obtain the explicit dissipation bound in terms of L1 distance and formulate the non-reversible Poincare inequality. An analytical example is provided for a non-reversible Langevin dynamic. Our result follows the Hessian operator of KL divergence in Wasserstein-2 space for non-symmetric vectors. Nov 16-25, 2020. PDF
- Draft "Quantum statistical learning via quantum Wasserstein natural gradient " is online. We formulate the quantum transport information matrix for quantum statistical learning and quantum computing. It is the first step for quantum transport information geometry. August 26, 2020. PDF
- Draft "Generalized Gamma z calculus via sub-Riemannian density manifold" is renewed. We further show the global in time convergence result for displacement group with a weighted volume on a compact region. August 10, 2020. PDF
- Draft "Information Newton's flow" is renewed. Towards the proposed Wasserstein Newton's flows for Bayesian sampling problems, we provide their particle implementations, in either affine models or RKHS, and derive the related convergence analysis. Several examples show the effectiveness of second-order sampling methods. August 4, 2020. PDF
- Draft "Generalized Gamma z calculus via sub-Riemannian density manifold" is renewed. We further show the global in time convergence result for displacement group with a weighted volume on a compact region. August 10, 2020. PDF
- Draft "Information Newton's flow" is renewed. Towards the proposed Wasserstein Newton's flows for Bayesian sampling problems, we provide their particle implementations, in either affine models or RKHS, and derive the related convergence analysis. Several examples show the effectiveness of second-order sampling methods. August 4, 2020. PDF
- Draft "A mean field game inverse problem" is online. We study the inverse problem in mean-field games, a.k.a. dynamics in transport density manifold. It is an initial numerical step to design optimization problems for learning observations of Hamiltonians in sample space. July 20, 2020 PDF
- Draft "Accelerate information gradient flow" is renewed. We introduce several accelerated gradient flows, based on the Kalman-Wasserstein metric and the Stein metric. Several ''accelerated'' interacting particle dynamics are designed for Bayesian sampling problems. Numerical examples of Bayesian regression problems demonstrate the effectiveness of their acceleration effects. June 2, 2020. PDF
- Draft "Controlling propagation of epidemics via mean-field games" is online. We introduce a mean-field SIR model for controlling the propagation of epidemics, such as COVID 19. We design the spatial SIR models with the population's velocity field as control variables. Numerical experiments demonstrate that the proposed model illustrates how to separate infected patients in a spatial domain effectively. June 1, 2020. PDF
- Draft "Computational methods for nonlocal mean field games" is online. We apply primal-dual algorithms to solve Mean field games with nonlocal interaction energies. Several applications of kernels, including robotic path planning problems, are demonstrated. PDF.
- Paper "Optimal transport natural gradient in statistical manifold with continuous sample space" has been accepted in Information geometry. April 14, 2020.
- Draft "Sub-Riemannian Ricci curvature via generalized Gamma z calculus" is online. We formulate generalized Ricci curvature tensors and Bochner's formula in a sub-Riemannian manifold. Several analytical examples, including the Hessenberg group, the Displacement group, and Martinet sub-Riemannian structure, have been given. April 6, 2020. PDF
- Our paper "Fisher information regularization schemes for Wasserstein gradient flows" has been accepted in Journal of Computational Physics. March 31, 2020.
- Draft "Optimal Transport of Nonlinear Control-Affine Systems" is online. We study the reachability and numerically compute optimal transport with sub--Riemannian or control affine structures. March 30, 2020. PDF
- Draft "Hessian metric via transport information geometry" is online. We extend and contain the classical optimal transport metric to the Hessian metrics. We observe that there are several connections with math physics equations. In particular, the transport Hessian Hamiltonian flow of negative Boltzmann--Shannon entropy satisfies the Shallow Water's equation; The transport Hessian metric is a particular mean field Stein metric. The transport Hessian metrics would be the key in AI and deep learning, following the study of transport information geometry. March 23, 2020. PDF
- Our paper "A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems" has been accepted in PNAS. March 2020.
- Draft
"Neural Fokker--Planck equation" is online. We propose Fokker-Planck equations within machine learning generative models. This approach allows us to approximate high dimensional Fokker-Planck equations. A neural kernelized mean-field stochastic differential equation is proposed. Numerical examples and analysis are provided. February 26, 2020. PDF - Draft
"Neural primal dual method" is online. We combine the generative adversary networks and primal-dual algorithms to solve general mean-field games. Here the primal (density) and dual (potential) variables are approximated by generators and discriminators, respectively. This method allows us to approximate the solution including 50 dimensions. Feb 24, 2020. PDF - Draft
"Information Newton flow: second-order optimization method in probability space" is online. We propose information Newton's flows for sampling optimization problems arisen in inverse problems and Bayesian statistics. Newton's Langevin dynamics, a.k.a Wasserstein Newton's flows of KL divergence, are derived. These can be viewed as a second-order algorithm for Bayesian sampling problems. Jan 13, 2020. PDF - Paper
"Ricci curvature for parametric statistics via optimal transport" has been accepted in Information Geometry. January 12, 2020.

- Paper
"Kernelized Wasserstein Natural Gradient" is accepted in ICLR 2020, Oral. Dec 20, 2019. - Draft
"A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems" is online. We provide a computational method to approximate solutions of high dimensional potential mean field games. Dec 4, 2019. PDF - Draft
"Tropical Optimal Transport in Phylogenetic Tree Space" is online. We study the dynamical optimal transport in Phylogenetic Tree Space. This is the first differential structure connection between optimal transport and Algebraic geometry with applications in biology, statistics, and data science. Nov 13, 2019. PDF - Draft
"Wasserstein information matrix" is online. We study the statistical and estimation properties of Wasserstein information matrices. The Wasserstein score function, the Wasserstein-Cramer-Rao bound, the Wasserstein efficiency, and the Poincare efficiency (interplay with Wasserstein and Fisher information matrices) are derived. Oct 24, 2019. PDF - Draft
"Generalized Gamma z calculus via sub-Riemannian density manifold" (104 pages) is online. A sysmetic calculus for studying degenerate drift diffusion process is provided. The further study is to estimate convergence rates of degenerate Langvein dynamics type MCMCs with variable drift coefficents, Oct 16, 2019.PDF - Paper
"Hessian transport gradient flows" , is accepted in Research in the Mathematical Sciences, Oct 11, 2019. - Paper
"Interacting Langevin Diffusions: Gradient Structure And Ensemble Kalman sampler" , is accepted SIAM journal on Applied dynamical system, Oct 10, 2019. - Paper
"Wasserstein Diffusion Tikhonov Regularization" is accepted in OTML Workshop NeurIPS, Oct 1, 2019. - Paper
"Equilibrium slection via optimal transport" is accepted in SIAM journal on Applied Math, Sep 23, 2019. - Paper
"Unnormalized Optimal transport" , known as GLOP, is accepted in Journal of Computational Physics, Sep 6, 2019. PDF - Paper
"Accelerated Information Gradient flow" is online. An accerlated gradient flow has been studied in probability space and Gaussian models for various information merics, including Fisher information metric and Wasserstein information metric, Sep 4, 2019. PDF - Paper
"Wasserstein Hamiltonian flow" is accepted in Journal of differential equations, August 24, 2019. PDF - Draft
"Fast algorithms for generalized unnormalized optimal transport" is online, August 12, 2019. PDF - Draft
"Diffusion hypercontractivity via generalized density manifold" is online, July 29, 2019. PDF - Draft
"Fisher information regularization schemes for Wasserstein gradient flows" is online, July 3, 2019. PDF - Draft
"Hessian transport gradient flows" is online, May 11, 2019.PDF - Paper
"Algorithm for Hamilton-Jacobi equation in density space" is accpeted in Journal of Scientific Computation, May 4, 2019. - Papers
"Parametric Fokker-Planck equation" ,"Affine Natural Proximal Learning" ,"Hopf-Cole transformation and Schrodinger problems" are accpeted in Geometry science of information, April 25, 2019. - Paper
"Wasserstein of Wasserstein Loss" is accepted in ICML, April 22, 2019.