Abstract: This talk is about using deep neural networks (DNNs) in solving PDE problems and will focus on two kinds of problems. In the first kind, the solutions are high-dimensional and the DNNs are used as an efficient way to represent these high dimensional solutions. The second kind of problems are concerned with the map between the coefficient and the solution operator of a PDE, where the DNNs are utilized to present these high-dimensional nonlinear maps. For each kind of problems, we show how to incorporate mathematics and physics into the design of network architectures. 

Bio: Lexing Ying is Professor of Mathematics at Stanford University since 2012. Prior to that, he was a professor at the University of Texas at Austin from 2006 to 2012. His research focuses on computational mathematics and scientific computing. He received his Ph.D. from New York University and was a postdoctoral scholar at California Institute of Technology from 2004 to 2006. He is a recipient of the Sloan Research Fellowship (2007), the National Science Foundation CAREER Award (2009), the Feng Kang Prize of Scientific Computing (2011), the James H. Wilkinson Prize in Numerical Analysis and Scientific Computing from SIAM (2013), and the Silver Morningside Medal in Applied Mathematics (2016).