Abstract: 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 or higher 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.  Finally, if time allows, we will report how to combine MgNet with meta-learning to develop a new Meta-MgNet for effective solution of parameterized partial differential equations.

Bio: Jinchao Xu is Verne M. Willaman Professor of Mathematics and Director of the Center for Computational Mathematics and Applications at Penn State. His main research interests are in the design, analysis, and application of numerical methods, especially multilevel and adaptive finite element methods, for systems of partial differential equations and problems with direct applications to science and engineering. His other research interests include the mathematical analysis, modeling and applications of deep neural networks. He was an invited speaker at the International Congress for Industrial and Applied Mathematics in 2007 as well as at the International Congress for Mathematicians in 2010. He is a Fellow of the Society for Industrial and Applied Mathematics (SIAM), the American Mathematical Society (AMS) and the American Association for the Advancement of Science (AAAS).