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Research Interests

• 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.

Publication

Hao Liu, Shingyu Leung and Jianliang Qian. Operator Splitting/Finite Element Methods for the Minkowski Problem. Submitted, 2023.
Xue-Cheng Tai, Hao Liu, Raymond Chan. PottsMGNet: A mathematical explanation of encoder-decoder based neural networks. Submitted, 2023.
Xue-Cheng Tai, Hao Liu, Raymond Chan. Connections between Operator-splitting Methods and Deep Neural Networks with Applications in Image Segmentation. Submitted, 2023.
Hao Liu, Wenjing Liao. Learning Functions Varying along an Active Subspace. Submitted, 2022.
Hao Liu, Haizhao Yang, Minshuo Chen, Wenjing Liao, Tuo Zhao. Deep nonparametric estimation of operators between infinite dimensional spaces. Submitted, 2022.
Minshuo Chen, Hao Liu, Wenjing Liao, Tuo Zhao. Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural Networks. Submitted, 2021.

Hao Liu, Alex Havrilla, Rongjie Lai, Wenjing Liao. Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness.. Applied and Computational Harmonic Analysis, 68, 101602, 2024
Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu. Group projected subspace pursuit for identification of variable coefficient differential equations (GP-IDENT).. Journal of Computational Physics 494, 112526, 2023.
Jiahui Cheng, Minshuo Chen, Hao Liu, Tuo Zhao, Wenjing Liao. High dimensional binary classification under label shift: phase transition and regularization. Sampling Theory, Signal Processing, and Data Analysis 21 (2), 32, 2023.
Hao Liu, Xue-Cheng Tai, Ron Kimmel, Roland Glowinski. Elastica models for color image regularization. SIAM Journal on Imaging Sciences 16 (1), 461-500, 2023.
Hao Liu, Dong Wang. Fast operator splitting methods for obstacle problems.. Journal of Computational Physics 477, 111941, 2023
Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu. Numerical Identification of Nonlocal Potential in Aggregation.. Communications in Computational Physics, 32 (2022), pp. 638-670, 2022.
Hao Liu, Shingyu Leung and Jianliang Qian. An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Ampère Equation. Communications in Optimization Theory, 2022 (7), 2022.
Hao Liu, Minshuo Chen, Siawpeng Er, Wenjing Liao, Tong Zhang, Tuo Zhao. Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraint. International Conference on Machine Learning, PMLR 162:13669-13703, 2022.
Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu. Robust PDE Identification from Noisy Data.. SIAM Journal on Scientific Computing 44 (3), A1145-A1175, 2022.
Hao Liu, Xue-Cheng Tai, Roland Glowinski. An operator-splitting method for the Gaussian curvature regularization model with applications to surface smoothing and imaging.. SIAM Journal on Scientific Computing 44 (2), A935-A963, 2022.
Yuchen He, Martin Huska, Sung Ha Kang, Hao Liu. Fast Algorithms for Surface Reconstruction from Point Cloud. Mathematical Methods in Image Processing and Inverse Problems: IPIP 2018, Beijing, China, April 21–24, pp. 61-80, 2021.
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.
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.
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.
Yuchen He, Sung Ha Kang, Hao Liu. Curvature Regularized Surface Reconstruction from Point Cloud. SIAM Journal on Imaging Sciences, 13(4), 1834–1859, 2020.
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.
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.
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.
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.
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.
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.
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.

Academic Talk

1. Learning functions varying along an active subspace. SIAM Student Seminar, Georgia Institute of Technology, Feb. 2020
2. Approximate functions varying along an active subspace. Workshop on New Trends in Machine Learning and Numerical PDEs, Hong Kong Baptist University, Dec. 2019
3. A level set based variational principal flow method for nonparametric dimension reduction on Riemannian manifolds. Scientific Computing Seminars, University of Houston, Nov. 2018

Poster Presentation

1. Learning functions varying along an active subspace. 2020 Georgia Scientific Computing Symposium, Emory University, Feb. 2020.
2. Approximate functions varying along an active subspace. Workshop on Recent Developments on Mathematical/Statistical approaches in DAta Science (MSDAS), The University of Texas at Dallas, May. 2019.
3. A level set based variational principal flow method for nonparametric dimension reduction on Riemannian manifolds. 2019 Georgia Scientific Computing Symposium, Georgia Institute of Technology, Feb. 2019.
4. A level set based variational principal flow method for nonparametric dimension reduction on Riemannian manifolds. Meeting the Statistical Challenges in High Dimensional Data and Complex Networks, National University of Singapore, Feb. 2018.