Modern machine learning (ML) has achieved unprecedented empirical success in many application areas. However, much of this success involves trial-and-error and numerous tricks. These result in a lack of robustness and reliability in ML. Foundational research is needed for the development of robust and reliable ML. This talk consists of two parts. The first part will present the first mathematical theory of physics informed neural networks (PINNs) -one of the most popular deep learning frameworks for solving PDEs. Linear second-order elliptic and parabolic PDEs are considered. I will show the consistency of PINNs by adapting the Schauderapproach and the maximum principle. The second part will focus on some recent mathematical understanding and development of neural network training. Specifically, two ML phenomena are analyzed --"Plateau Phenomenon" and "Dying ReLU."New algorithms are developed based on the insights gained from the mathematical analysis to improve neural network training.

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