We introduce mathematical methods for reducing complexity of deep neural networks 
in the context of computer vision for mobile and IoT applications such as sparsification and differentiable architecture search. We also describe applications in infectious disease prediction, and a deep learning and optimal 
transport (the deep particle) method in predicting invariant measures of 
stochastic dynamical systems arising in partial differential 
equation (PDE) modeling of transport in chaotic flows (e.g. rapid stirring of coffee and milk, raging forest fires in the wind).