Juncai He R.H. Bing Postdoctoral Fellow Department of Mathematics The University of Texas at Austin 2515 Speedway, PMA Austin, TX 78712 Phone: +1 616-227-5319 Email: jhe@utexas.edu

About me

I am currently a R.H. Bing postoctoral fellow in the Department of Mathematics at UT Austin.

I received the B.S. degree in Mathematics and Applied Mathematics from Sichuan University in 2014. In Summer of 2019, I received my Ph.D. degree in Computational Mathematics under the supervision of Prof. Jinchao Xu and Prof. Jun Hu at Peking University at Beijing, China. From 2019 to 2020, I worked as a Postdoctoral Scholar suprevised by Prof. Jinchao Xu in The Center for Computational Mathematics and Application (CCMA) in the Department of Mathematics at The Pennsylvania State University, University Park.

Research

• Deep Learning, Stochastic Optimization.

• Numerical Analysis, Finite Element Methods, Multigrid Methods.

My research interests are in algorithm development and theoretical analysis for machine learning and numerical methods for partial differential equations (PDEs). I have received broad and in-depth training in finite element, multigrid (MG) methods and machine learning. I have studied the finite element exterior calculus (FEEC) method both for its theoretical analysis and also the application in structure-preserving discretization for multi-physical systems. I have also applied techniques from numerical PDEs for understanding and improving deep learning models and algorithms in data science. In particular, I have worked on three different but related topics:

1. finite element methods and deep neural networks (DNNs);

2. multigrid methods and architectures of convolutional neural networks;

3. stochastic optimization methods.

Publications

• J. He, X. Jia, J. Xu, L. Zhang and L. Zhao. Make Regularization Effective in Training Sparse CNN. Computational Optimization and Applications. 2020. https:doi.org10.1007s10589-020-00202-1 ArXiv: 1807.04222v4.

• J. He, L. Li, J. Xu, and C. Zheng. ReLU Deep Neural Networks and linear Finite Elements. Journal of Computational Mathematics. 38(3): 502-527, 2020. doi:10.4208/jcm.1810-m2018-0096.

• J. He, K. Hu, J. Xu. Generalized Gaffney Inequality and Discrete Compactness for Discrete Differential Forms. Numerische Mathematik. 2019. https:doi.org10.1007s00211-019-01076-0.

• J. He and J. Xu. MgNet: A Unified Framework of Multigrid and Convolutional Neural Network. Science China Mathematics. 62(7): 1331–1354, 2019. https:doi.org10.1007s11425-019-9547-2.

• J. He, Y. Chen, L. Zhang and J. Xu. Constrained Linear Data-feature Mapping in Image Classification. ArXiv: 1911.10428. 2019.

Education

• Ph.D., Computational Mathematics, Peking University, 2014-2019
  Advisors: Prof. Jinchao Xu and Prof. Jun Hu
  Thesis: Finite Element Methods and Deep Neural Networks

• Visiting Ph.D. Research Scholar, Center for Computational Mathematics and Application (CCMA), Department of Mathematics, The Pennsylvania State University, Feb. 2016 - Jul. 2016, Oct. 2017 - Mar. 2018 and Mar. 2019 - May 2019.

• B.S., Mathematics and Applied Mathematics, Sichuan University, 2010-2014.

Teaching

Instructor at UT Austin for:

• M 408K: Differential Calculus, UT Austin, Aug. 2020 - Dec. 2020

Teaching assistant at Penn State and Peking University for:

• MATH 497: Deep Learning Algorithms and Analysis , Penn State University, Jul. 2019

• An Introduction for Applied Mathematics, Peking University, Feb. 2017 - Jun. 2017

• Advanced Linear Algebra I, Peking University, Sept. 2016 - Jan. 2017

• Calculus, Peking University, Sept. 2015 - Jan. 2016

The Workshops/ Minisymposium Organized

• Workshop on Mathematical Machine Learning and Application (Organizing Committee), December 14-16, 2020, Penn State University, USA.

• Minisymposium on “Multigrid and Machine Learning” (Co-organizer with Prof. Zuowei Shen and Prof. Jinchao Xu) in International Multigrid Conference, August 11-16, 2019, Kunming, China.

• 4th PKU Workshop on Numerical Methods for PDEs (Organizing Committee), October 30-31, 2018, Peking University, China.

• The First PKU Elite PHD Candidates Workshop on Computational Mathematics and 4th Beijing Graduate Students Workshop on Computational Mathematics (Chair of the Organizing Committee), September 9-12, 2018, Peking University, China.

Presentations

• Workshop on Mathematical Machine Learning and Application, Penn State University, University Park (Online), USA, Dec. 2020.

• Workshop on Computation and Applications of PDEs Based on Machine Learning, Jilin University, Changchun (Online), China, Jul. 2020.

• Data Science Seminar, Shanghai Jiao Tong University, Shanghai (Online), China, Mar. 2020.

• “Advances in Multilevel Methods: from PDEs to Data Intensive Studies” and “Multigrid and Machine Learning”, Minisymposiums in International Multigrid Conference, Kunming, China, Aug. 2019.

• Data Science Seminar, Tsinghua University, Beijing, China, May 2019.

• 16th Annual Meeting of the China Society for Industrial and Applied Mathematics, Chengdu, China, Sept. 2018.

• The First PKU Elite PHD Candidates Workshop on Computational Mathematics and 4th Beijing Graduate Students Workshop on Computational Mathematics, Peking University, Beijing, China, Sept. 2018.

• Workshop on Numerical Methods for PDEs, Peking University, Beijing, China, Jul. 2017.