My primary research areas are applied probability and applied dynamical systems. I am interested in various rigorous and computational problems related to machine learning, statisticall mechanics, and neurosciences.
(Image from internet)
I, Stochastic dynamics, data-driven computing, and machine learning
My recent research focuses on understanding the interplay between stochastic dynamics, numerical analysis, and machine learning. On one hand, modern deep learning and statistical tools help us to understand complex stochastic dynamics in higher dimension. Several novel data-driving computational methods are developed to study invariant properties, ergodicity, and sensitivity analysis of stochastic dynamics. On the other hand, stochastic dynamics helps us to understand the mathematical foundation of machine learning.
Related papers:
- Data-driven computation methods for quasi-stationary distribution and sensitivity analysis, (with Yaping Yuan), Submitted (pdf)
- A deep learning method for solving Fokker-Planck equations, (with Matthew Dobson and Jiayu Zhai) MSML21: Mathematical and Scientific Machine Learning, Virtual Event, Aug 16-19, 2021 (pdf)
- Stationary distributions of persistent ecological systems, (with Alexandru Hening), Journal of Mathematical Biology 82(64), 2021 (pdf)
- Using coupling methods to estimate sample quality for stochastic differential equations, (with Matthew Dobson and Jiayu Zhai), SIAM/ASA J. Uncertainty Quantification, 9(1), 135-162, 2021 (pdf)
- An efficient data-driven solver for Fokker-Planck equations: algorithm and analysis, (with Matthew Dobson and Jiayu Zhai), Under Review (pdf)
- Numerical computations of geometric ergodicity for stochastic dynamics, (with Shirou Wang), Nonlinearity 33(12), 6935, 2020 (pdf)
- A data-driven method for the steady state of randomly purterbed dynamics, Communications on Mathematical Sciences, 17(4), 1045-1059,2019 , (pdf)
- Numerical Simulation of Polynomial-Speed Convergence Phenomenon (with H. Xu), Journal of Statistical Physics, 169(4), 2017 (pdf)
- Systematic measures of biological networks, part I: Invariant measures and entropy (with Yingfei Yi), Communications on Pure and Applied Mathematics, LXIX, 1777-1811. 2016 (pdf)
- A fast exact simulation algorithm for a class of Markov jump processes (with L. Hu), Journal of Chemical Physics, 143(18), 2015 (pdf)
- A limiting strategy for the back and forth error compensation and correction method for solving advection equations (with L. Hu, Y. Liu), Mathematics of Computation 85 (2016), 1263-1280 (pdf)
II: Nonequilibrium Statistical Mechanics
The derivation of macroscopic thermodynamic laws from microscopic Hamiltonian dynamics is a long standing challenge to mathematicians. Nonequilibrium thermodynamic models are also representative examples of "large" dynamical systems. I am interested in a variety of problems arising from nonequilibrium statistical mechanics, such as nonequilibrium kinetic particle systems, stochastic stability of nonequilibrium steady-states (NESS), and the microscopic derivation of macroscopic thermodynamic laws.
Related papers:
- From deterministic dynamics to thermodynamic laws II: Fourier's law and mesoscopic limit equa- tion, Under Review (pdf)
- Thermal conductivity and local thermodynamic equilibrium of stochastic energy exchange models (with Wenbo Xie), Journal of Statistical Mechanics: Theory and Experiment, 043205, 2019 (pdf)
- From billiards to thermodynamic laws: stochastic energy exchange model (with Lingchen Bu), Chaos: An Interdisci- plinary Journal of Nonlinear Science, 28, 093105, 2018, (pdf)
- On the polynomial convergence rate to nonequilibrium steady-states, Annals of Applied Probability, 28(6), 3765-3812, 2018 (pdf)
- Polynomial Convergence to Equilibrium for a System of Interacing Particles (with Lai-Sang Young), Annals of Applied Probability, 27(1), 2017, 65-90 (pdf)
- Local Thermodynamic Equilibrium for some Multidimensional Stochastic Models (with Peter Nandori and Lai-Sang Young) , Journal of Statistical Physics, 163(1), 61-91 (pdf)
- On the stochastic behaviors of locally confined particle systems, Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 073121(2015) (pdf)
- Nonequilibrium steady states for a class of particle systems (with Lai-Sang Young), Nonlinearity 27, page 607, 2014 (pdf)
- Existence of nonequilibrium steady state for a simple model of heat conduction (with Lai-Sang Young), Journal of Statistical Physics, pages 1 -- 24, 2013 (pdf)
(Image from internet)
III: Mathematical neuroscience
Understanding how cortex processes informations is an incredible complex task. From a mathematical point of view, neuronal networks are another representative examples of "large" dynamical systems that are both interesting and mathematically tractable. I am interested in nonlinear interaction of neurons in spiking neuronal networks, information-theoretic features of complex networks, simulations of neuronal network model, and connections between real and artificial neural networks.
Related papers:
- Unraveling the mechanisms of surround suppression in early visual processing, (with Lai-Sang Young) PLoS Computational Biology 17(4): e1008916 (pdf)
- Entropy, mutual information, and systematic measures of structured spiking neural networks, (with Wenjie Li) Journal of Theoretical Biology, 501(21), 110310, 2020 (pdf)
- Firing rate and spatial correlation in a stochastic neural field model (with Hui Xu), Journal of Mathematical Biology, 79, 1169–1204, 2019, (pdf)
- How well do reduced models capture the dynamics in models of interacting neurons? (with Logan Chariker and Lai-Sang Young), Journal of Mathematical Biology, 78(1-2), 85-115, 2019 (pdf)
- Systematic measures of biological networks, part II: Degeneracy, complexity and robustness. (with Yingfei Yi), Communications on Pure and Applied Mathematics, LXIX, 1952-1983, 2016 (pdf)
- Quantification of degeneracy in bio- logical systems for characterization of functional interactions between modules (with G. Dwivedi, W. Huang, M. Kemp and Y. Yi), Journal of theoretical biology, 302:2938, 2012 (pdf)
Other publications:
Fokker-Planck Equations on Discrete Spaces
- Fokker-Planck equations for a free energy functional or Markov process on a graph (with S-N. Chow, W. Huang and H-M. Zhou), Archive for Rational Mechanics and Analysis 203.3 (2012): 969-1008. (pdf)
- A free energy based mathematical study for molecular motors (with S-N. Chow, W. Huang and H-M. Zhou), Regular and Chaotic Dynamics 16.1-2 (2011): 117-127. (pdf)
- Convergence to global equilibrium for Fokker-Planck equations on a graph and talagrand-type inequalities (with R. Che, W. Huang and P. Tetali), Journal of Differential Equations 261, 2552-2583 (pdf)