Juncai He
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Juncai He
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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:
finite element methods and deep neural networks (DNNs);
multigrid methods and architectures of convolutional neural networks;
stochastic optimization methods.
Publications
J. He, L. Li, J. Xu. ReLU Deep Neural Networks from the Hierarchical Basis Perspective. ArXiv: 2105.04156. 2021
Q. Chen, W. Hao, J. He. Power Series Expansion Neural Network. ArXiv: 2102.13221. 2021
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 NetworksVisiting 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
M 408N: Differential Calculus for Science, UT Austin, Jan. 2021 - May 2021
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.