Research

My primary research areas are applied probability and applied dynamical systems. I am interested in various rigorous and computational problems related to machine learning, statistical mechanics, and neurosciences.

Stochastic Dynamics and Machine Learning

My recent research focuses on understanding the interplay between stochastic dynamics and machine learning. Modern deep learning and statistical tools help us to understand complex stochastic dynamics in higher dimension. Several data-driving computational methods are developed to study invariant properties, ergodicity, large deviation rate function, eigenfunction, and sensitivity analysis of stochastic dynamics. On the other hand, stochastic dynamics helps us to understand the mathematical foundation of machine learning, such as the landscape of a high dimensional loss surface.

Related Papers:

DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems (with Shirou Wang and Yicheng Liu), arXiv: 2508.09529
Landscape classification through coupling method (with Shirou Wang and Molei Tao), Stochastic Processes and their Applications, accepted
Sensitivity Analysis of Quasi-Stationary Distributions (QSDs) of Mass-Action Systems (with Yaping Yuan), SIAM/ASA Journal on Uncertainty Quantification 11, no. 4 (2023): 1164-1194
Artificial neural network solver for time-dependent Fokker–Planck equations (with Caleb Meredith) Applied Mathematics and Computation 457 (2023): 128185
Data-driven computation methods for quasi-stationary distribution and sensitivity analysis, (with Yaping Yuan), Journal of Dynamics and Differential Equations, 35, no. 3 (2023): 2069-2097
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
Stationary distributions of persistent ecological systems, (with Alexandru Hening), Journal of Mathematical Biology 82(64), 2021
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
An efficient data-driven solver for Fokker-Planck equations: algorithm and analysis, (with Matthew Dobson and Jiayu Zhai), Communications in Mathematical Sciences, 20(3), (2022)
Numerical computations of geometric ergodicity for stochastic dynamics, (with Shirou Wang), Nonlinearity 33(12), 6935, 2020
A data-driven method for the steady state of randomly purterbed dynamics, Communications on Mathematical Sciences, 17(4), 1045-1059,2019
Numerical Simulation of Polynomial-Speed Convergence Phenomenon (with H. Xu), Journal of Statistical Physics, 169(4), 2017
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)

Nonequilibrium Statistical Mechanics

I am interested in a variety of problems arising from nonequilibrium statistical mechanics. My recent study focuses on the nonequilibrium statistical properties of energy cascade models derived in wave turbulence theory.

Related Papers:

Non-equilibrium steady state for a three-mode energy cascade model (with Zaher Hani, Andrea Nahmod, and Gigliola Staffilani), arXiv:2505.16018
Thermal conductivity and local thermodynamic equilibrium of stochastic energy exchange models (with Wenbo Xie), Journal of Statistical Mechanics: Theory and Experiment, 043205, 2019
From billiards to thermodynamic laws: stochastic energy exchange model (with Lingchen Bu), Chaos: An Interdisci- plinary Journal of Nonlinear Science, 28, 093105, 2018
On the polynomial convergence rate to nonequilibrium steady-states, Annals of Applied Probability, 28(6), 3765-3812, 2018
Polynomial Convergence to Equilibrium for a System of Interacing Particles (with Lai-Sang Young), Annals of Applied Probability, 27(1), 2017, 65-90
Local Thermodynamic Equilibrium for some Multidimensional Stochastic Models (with Peter Nandori and Lai-Sang Young) , Journal of Statistical Physics, 163(1), 61-91
On the stochastic behaviors of locally confined particle systems, Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 073121(2015)
Nonequilibrium steady states for a class of particle systems (with Lai-Sang Young), Nonlinearity 27, page 607, 2014
Existence of nonequilibrium steady state for a simple model of heat conduction (with Lai-Sang Young), Journal of Statistical Physics, pages 1 -- 24, 2013

Mathematical Neurosciences

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:

Learning spiking neuronal networks with artificial neural networks: neural oscillations (with Ruilin Zhang, Zhongyi Wang, Tianyi Wu, Yuhang Cai, Louis Tao, and Zhuo-Cheng Xiao), Journal of Mathematical Biology 88, no. 6 (2024): 65
Unraveling the mechanisms of surround suppression in early visual processing, (with Lai-Sang Young) PLoS Computational Biology 17(4): e1008916
Entropy, mutual information, and systematic measures of structured spiking neural networks, (with Wenjie Li) Journal of Theoretical Biology, 501(21), 110310, 2020
Firing rate and spatial correlation in a stochastic neural field model (with Hui Xu), Journal of Mathematical Biology, 79, 1169-1204, 2019
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
Systematic measures of biological networks, part II: Degeneracy, complexity and robustness. (with Yingfei Yi), Communications on Pure and Applied Mathematics, LXIX, 1952-1983, 2016
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

Other/Earlier publications:

1, Epidemiology and Social Dynamics

Modeling, inference, and prediction in mobility-based compartmental models for epidemiology (with Ning Jiang and Weiqi Chu), SIAM Applied Mathematics, accepted
Artificial Neural Network Prediction of COVID-19 Daily Infection Count (with Ning Jiang, Charles Kolozsvary), Bulletin of Mathematical Biology 86, no. 5 (2024): 49

2, 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
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
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

3, Numerical Algorithms

A fast exact simulation algorithm for a class of Markov jump processes (with Lili Hu), Journal of Chemical Physics, 143(18), 2015
A limiting strategy for the back and forth error compensation and correction method for solving advection equations (with Lili Hu, Yingjie Liu), Mathematics of Computation 85, 2016, 1263 -- 1280

My research is supported by

UMass Amherst HEG Faculty Research Grant, $ 12,426, 2017-2018
Simons Collaboration Grant, $ 42,000, 2018-2018 (terminated early)
NSF DMS-1813246, $144,192, 2018-2021
NSF DMS-1900397, $23,600, 2019-2020
NSF DMS-2108628, $225,549, 2021-2024
NSF DMS-2510209, $225,000, 2025-2028