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.

1. Minshuo Chen, Hao Liu, Wenjing Liao, Tuo Zhao. [Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural Networks.](https://arxiv.org/abs/2011.01797) Submitted, 2021. 2. Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu. [Robust PDE Identification from Noisy Data](https://arxiv.org/abs/2006.06557). Submitted, 2021. 3. Yuchen He, Martin Huska, Sung Ha Kang, Hao Liu. [Fast Algorithms for Surface Reconstruction from Point Cloud](https://arxiv.org/abs/1907.01142). 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.](http://proceedings.mlr.press/v139/liu21e.html) 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](https://epubs.siam.org/doi/abs/10.1137/20M1354532). 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](https://arxiv.org/abs/2008.08103). 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](https://epubs.siam.org/doi/abs/10.1137/20M1314525). 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](https://www.math.hkust.edu.hk/~masyleung/Reprints/liuleu20.pdf). 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](https://ieeexplore.ieee.org/abstract/document/8890006). 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](https://ieeexplore.ieee.org/abstract/document/8968574). 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](https://arxiv.org/abs/2002.08461). 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](https://link.springer.com/article/10.1007/s10915-019-01080-4). 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](https://link.springer.com/article/10.1007/s10915-018-0839-y). 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](https://epubs.siam.org/doi/abs/10.1137/16M107236X). SIAM J. Sci. Comput., 39(4), A1616-A1646, 2017.