Physics-Informed Neural Networks (PINNs) and the Deep Galerkin Method (DGM) have become widely used for physics-constrained learning and solving PDEs. In particular, DGM can solve high-dimensional (including infinite-dimensional) PDEs that are computationally intractable for traditional numerical methods. Due to the non-convexity of the PDE residual objective function used in these methods, the trained neural network may, in principle, converge to a local minimizer that does not correspond to a solution of the PDE. Consequently, there is a longstanding question regarding the mathematical foundations of these algorithms. For a class of semilinear PDEs, we prove that neural networks trained with gradient descent will converge to the PDE solution as the network width and training time become large. In the final part of the presentation, applications of machine learning to modeling turbulent flows in fluid mechanics will be discussed. A physics-based machine learning closure model is developed for the Reynolds-averaged Navier-Stokes (RANS) equations, which outperforms existing AI for Science methods (e.g., deep operator networks, PINNs, and TBNNs) on several canonical turbulent flow cases.