In this talk, I will first give an elementary introduction to models and algorithms from two different fields: (1) machine learning, including logistic regression and deep neural networks, and (2) numerical PDEs, including finite element and multigrid methods.  I will then explore mathematical relationships between these different models and algorithms and demonstrate how such relationships can be used to understand, study and improve different aspects of deep neural networks, finite element and multigrid methods.  We will show that ReLU-DNN corresponds exactly to the traditional piecewise linear finite functions and [ReLU]^k-DNN leads to new finite element of piecewise polynomials of degree k with remarkable approximation properties.  We will demonstrate how a new convolutional neural network (CNN), known as MgNet, can be derived by making very minor modifications of a classic geometric multigrid method for the Poisson equation and then discuss the theoretical and practical potentials of MgNet.