# RESEARCH INTERESTS:

• **Numerical Analysis**: Minimum action methods for large deviation principle, Numerical methods for invariant probability measures, Machine learning methods and analysis, Numerical methods for stochastic differential equations and stochastic modeling, Uncertainty quantification.

• **Stochastic Analysis**: Stochastic Integration, White Noise Theory, Stochastic differential equations.

# WORK EXPERIENCE:

• **Visiting Assistant Professor: **

University of Massachusetts Amherst, Sep.2018–Present

• **Research Assistant: **

Louisiana State University, Feb.2016–May.2018

• __NSF Mathematical Sciences Graduate Internship Program:__

**Pacific Northwest National Laboratory, May.2017-Jul.2017**

• **Teaching Assistant:**

Louisiana State University, Aug.2012–Feb.2016

• **Teaching Assistant:**

Shanghai University, Sep.2011–Jan.2012

# FUNDS, AWARDS AND HONORS:

• __NSF Funding__, Co-PI.

Analysis and Data-Driven Computation for Nonequilibrium Thermodynamic Models

(Award Number: 2108628, PI: Dr. Yao Li, NSF Organization: DMS, Total Award Amount: $225,549.00), 2021-2024.

• **MSP Research Support Funds**

University of Massachusetts Amherst, 2018-2021

• **Pasquale Porcelli Graduate Student Academic Excellence Award**

Louisiana State University, 2013

• **Graduate School Enrichment**

Louisiana State University, 2012-2016

• **Innovation Fund of Academic Research for Graduate Students**

Shanghai University, 2011-2012

# PUBLICATIONS:

1. Jiayu Zhai, Matthew Dobson and Yao Li, *A deep learning method for solving Fokker–Planck partial differential equations*, Proceeding of Mathematical and Scientific Machine Learning (MSML21) (2021).

2. Xiaoliang Wan and Jiayu Zhai, *A minimum action method for dynamical systems with constant time delays*, SIAM Journal on Scientific Computing, (2021), 41(1), A541–A565.

3. Matthew Dobson, Yao Li and Jiayu Zhai, *Using coupling methods to estimate sample quality for stochastic differential equations*, SIAM/ASA Journal on Uncertainty Quantification (2021), 9(1), 135–162.

4. Matthew Dobson, Yao Li and Jiayu Zhai, *An efficient data-driven solver for Fokker-Planck equations: algorithm and analysis*, **Accepted**, Communications in Mathematical Sciences (2021), arXiv:1906.02600.

5. Xiaoliang Wan, Haijun Yu and Jiayu Zhai, *Convergence analysis of a finite element approximation of minimum action methods*, SIAM J. Numer. Anal., 56(3), 1597–1620.

6. Hui-Hsiung Kuo, Sudip Sinha and Jiayu Zhai, *Stochastic Differential Equations with Anticipating Initial Conditions*, Communications on Stochastic Analysis 12, no. 4 (2018) 473–485.

7. Chii-Ruey Hwang, Hui-Hsiung Kuo, Kimiaki Saitô and Jiayu Zhai, *Near-martingale property of anticipating stochastic integration*, Communications on Stochastic Analysis 11 (2017), no. 4, 491-504.

8. Chii-Ruey Hwang, Hui-Hsiung Kuo, Kimiaki Saitô and Jiayu Zhai, *A general Itô formula for adapted and instantly independent stochastic processes*, Communications on Stochastic Analysis 10, no.3 (2016) 341–362.

9. Zhongrui Shi and Jiayu Zhai, *λ-point and λ-property in generalized Orlicz spaces with Luxemburg norm*, Journal of East China Normal University (Natural Science), 1 (2012), 63–73.

# PREPRINTS AND WORKS IN PROGRESS:

1. Jiayu Zhai, *Quantify the dynamics from invariant distribution using GANs*, Work in progress (2021).

2. Jiayu Zhai, Xiaoliang Wan, Yao Li, *A flow-GANs for dynamics quantification via block-triangular mapping, Work in progress (2021).*

3. Yao Li, Jiayu Zhai, Matthew Dobson, Yaping Yuan, *A hybrid reinforcement method for injection measure in low density regions*, Work in progress (2021).

4. Yulong Lu, Jiayu Zhai, Yao Li, *Neural network representation of solutions to convection–diffusion equations in Barron Spaces*, Work in progress (2021).

5. Jiayu Zhai and Liwen Ouyang,, *A data-driven method for invariant probability measures of nonlinear dynamical systems driven by non-Gaussian Lévy processes*, Work in progress **(REU program as a supervisor at UMass Amherst)**.

6. Jiayu Zhai, Xiaoliang Wan and Bin Zheng, *Multigrid minimum action method*, In preparation.

7. Jiayu Zhai and Xiaoliang Wan, *A minimum action method by augmented Lagrange multipliers for delayed stochastic dynamical systems, In preparation.*

8. Jiayu Zhai, Huan Lei and Xiaoliang Wan, *MAM for systems with degenerated noises*, In preparation.

9. Hui-Hsiung Kuo, Sudip Sinha and Jiayu Zhai, *A Black-Scholes model with anticipating initial conditions*, **Preprint to be submitted** (2021).

# CONFERENCES, WORKSHOPS AND SUMMER SCHOOLS:

• **Mathematical and Scientific Machine Learning (MSML21)**

**Presented Talk:** A Deep Learning Method for Solving Fokker-Planck Equations

Zoom virtual conference, August 16–19, 2021.

• **SIAM Conference on Applications of Dynamical Systems (DS21)**

**Presented Talk:** A Neural Network Approximation for Invariant Measures in Stochastic Dynamical Systems

Zoom virtual conference, May 23–27, 2021.

• **The Second Northeast Conference on Dynamical Systems**

**Presented Talk:** A Data-Driven Solver for Steady State Distributions of Stochastic Dynamical Systems

University of Massachusetts Amherst, Amherst, MA, November 15–17, 2019.

• **SIAM Conference on Computational Science and Engineering (CSE18)**

**Presented Talk:**Transitions as Rare Events in Stochastic Delayed Systems

Spokane, WA, February 25– March 1 2019.

• **SIAM Conference on Uncertainty Quantification (UQ18)**

**Presented Talk:** Temporal Minimum Action Method for Delayed System

Orange County, CA, April 16-19, 2018.

• **The Finite Element Rodeo 2018**

Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, February 23-24, 2018.

• **Scientific Computing Around Louisiana (SCALA 2018)**

Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, February 2-3, 2018.

• **Joint Mathematics Meeting (JMM 2018)**

San Diego Convention Center, San Diego, CA, January 10-13, 2018.

• **The Finite Element Circus 2017**

**Presented Talk:** Convergence Analysis of Finite Element Approximation of Large Deviation Principle

College of Natural and Mathematical Sciences, UMBC, Baltimore, MD, October 20-21, 2017.

• **Scientific Computing Around Louisiana (SCALA2017)**

**Presented Talk:** Convergence Analysis of Finite Element Approximation of Large Deviation Principle

• **Scientific Computing Around Louisiana (SCALA2016)**

Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, February 12-13, 2016.

• **The 8th International Congresson Industrial and Applied Mathematics (ICIAM2015)**

Beijing, China, August 10-14, 2015.

• **The 3rd International Conference on Analytic Mathematics and its Application**

**Presented Talk:** λ-Point and λ-Property in Generalized Orlicz Spaces

Xinyang, China, August 8-12, 2011.

• **The 16th National Summer School of Mathematics for Graduate Students**

Chern Institute of Mathematics, Nankai University, Tianjin, China, August 1-19, 2011

• **International Conference on Convex Geometric Analysis, Integral Geometry and Related Topics**

Shanghai University, Shanghai, China, June 21-24, 2011.

# INVITED LOCAL TALKS:

• **Deep Density Estimation via Invertible Block-Triangular Mappings**

Seminar on Mathematics of Machine Learning

University of Massachusetts Amherst, Nov. 2021.

• **A Neural Network Approximation for Invariant Measures in Stochastic Dynamical Systems**

Nanjing University, May 2021.

• **Data-Driven Methods for Fokker–Planck Partial Differential Equations**

CCMA Seminar on Mathematics of Data and Computation

Pennsylvania State University and Peking University Jointly, Zoom presentation, Dec. 2021.

• **Minimum Action Methods for Transitions in Stochastic Dynamical Systems**

Applied Mathematics Seminar

Northeast Normal University, Zoom presentation, Oct. 2020.

• **Data-Driven Methods for Fokker–Planck Partial Differential Equations**

Scientific Computing and Numerical Analysis Seminar

Center for Computation and Technology, Louisiana State University, Cancelled due to COVID-19, Mar. 2020.

• **A Data-Driven Solver for Steady State Distributions of Stochastic Dynamical Systems**

Applied Mathematics Seminar

Nanjing, China, Dec. 2019.

• **A Data-Driven Solver for Steady State Distributions of Stochastic Dynamical Systems**

Applied Mathematics and Computation Seminar

University of Massachusetts Amherst, Sep. 2019.

• **Convergence Analysis of a Finite Element Approximation of Minimum Action Methods**

Numerical Methods Seminar

Worcester Polytechnic Institute, Nov. 2018.

• **Multilevel Convergence Analysis of a Finite Element Approximation of Minimum Action Methods**

Applied Mathematics and Computation Seminar

University of Massachusetts Amherst, Oct. 2018.

• **Markov Property (Series Talks)**

Weekly Group Seminar

University of Massachusetts Amherst, Fall 2018.

• **Multilevel Monte Carlo Methods for PDE**

Weekly Computational Group Seminar

Louisiana State University, Oct. 2017.

• **Banach Spaces, Orlicz Spaces and Related Theories (Series Talks)**

Weekly Graduate Student Seminar on Functional Analysis

Shanghai University, Sep. 2009-May. 2011.

• **Linear Topological Spaces**

Graduate Student Seminar

Shanghai University, Dec. 2009.

• **Convex Bodies: The Brunn-Minkowski Theory**

Graduate Student Seminar

Shanghai University, Dec. 2009.

# SUPERVISION EXPERIENCE:

• **Research Experiences for Undergraduates (REU)**, Summer 2020

Student: Liwen Ouyang

Topic: A data-driven method for invariant probability measures of nonlinear dynamical systems driven by non-Gaussian Lévy processes (paper in preparation)

University of Massachusetts Amherst