Current Research Interests

My research focuses on deep learning theory, PDE learning, numerical PDEs and image processing.

• Deep learning theory: I develope approximation theories and statistical learning theories of deep neural networks on various problems, especially when data have some low-dimensional structures.
• PDE learning: I design efficient and robust algorithms for PDE learning from noisy data sets.
• Numerical PDEs: I focus on using the level set method and operator-splitting method to solve various problems and nonlinear PDEs. My recent works proposed operator-splitting method based numerical solvers for the Monge-Ampère type equations.
• Image processing: I design image regularization models and efficient algorithms by operator-splitting methods.

 

 

Selected Publications

1. Minshuo Chen, Hao Liu, Wenjing Liao, Tuo Zhao. Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural Networks. Submitted, 2021.
2. Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu. Robust PDE Identification from Noisy Data. Submitted, 2021.
3. Yuchen He, Martin Huska, Sung Ha Kang, Hao Liu. Fast Algorithms for Surface Reconstruction from Point Cloud. Accepted by Proceeding of International Workshop On Image Processing and Inverse Problems, 2021.
4. Hao Liu, Minshuo Chen, Tuo Zhao, Wenjing Liao. Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks. International Conference on Machine Learning, 6770-6780, 2021.
5. Hao Liu, Xue-Cheng Tai, Ron Kimmel, Roland Glowinski. A Color Elastica Model for Vector-Valued Image Regularization. SIAM Journal on Imaging Sciences 14 (2), 717-748, 2021.
6. Roland Glowinski, Shingyu Leung, Hao Liu, Jianliang Qian. On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Ampère Operator. ESAIM: Control, Optimisation and Calculus of Variations, 26, 118, 2020.
7. Yuchen He, Sung Ha Kang, Hao Liu. Curvature Regularized Surface Reconstruction from Point Cloud. SIAM Journal on Imaging Sciences, 13(4), 1834–1859, 2020.
8. Hao Liu, Shingyu Leung. A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints. Numer. Math. Theor. Meth. Appl., 13, pp. 620-643, 2020.
9. Yazhou Hu, Wenxue Wang, Hao Liu, Lianqing Liu. Reinforcement Learning Tracking Control for Robotic Manipulator with Kernel-Based Dynamic Model. IEEE Transactions on Neural Networks and Learning Systems, 31(9), 3570 - 3578, 2020.
10. Yazhou Hu, Wenxue Wang, Hao Liu, Lianqing Liu. Robotic Tracking Control with Kernel Trick-based Reinforcement Learning. 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 997-1002, 2019.
11. Hao Liu, Shingyu Leung. An Alternating Direction Explicit Method for Time-Dependent Evolution Equations with Applications to Fractional Differential Equations. Methods and Applications of Analysis, Special Issue in Honor of Roland Glowinski, 26(3), 249-268, 2019.
12. Hao Liu, Roland Glowinski, Shingyu Leung and Jianliang Qian. A Finite Element/Operator-Splitting Method for the Numerical Solution of the Three Dimensional Monge-Ampère Equation. Journal of Scientific Computing, 81(3), 2271-2302, 2019.
13. Roland Glowinski, Hao Liu, Shingyu Leung and Jianliang Qian. A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic Monge-Ampère Equation. Journal of Scientific Computing, 79(1), 1-47, 2019.
14. Hao Liu, Zhigang Yao, Shingyu Leung and Tony F. Chan. A Level Set Based Variational Principal Flow Method for Nonparametric Dimension Reduction on Riemannian Manifolds. SIAM J. Sci. Comput., 39(4), A1616-A1646, 2017.