Dr. Hong’s theoretical and applied research addresses numerical analysis and mathematical foundation of machine learning and its applications to numerical partial differential equations (PDEs), data fitting, and image classification. This is based on building the connection between machine learning and numerical theory such as finite element analysis, numerical integration, iterative method and matrix analysis. These connections motivate us to apply the powerful approximation capability of neural networks solving high order PDEs, to design new training algorithms solving the optimization problem induced from the machine learning, and to develop new neural networks which are more efficient for training.