报告人：蔡智强（普渡大学）

邀请人：数学与统计学院

报告时间：2023年12月12日（星期二）15：00-16:30

                 2023年12月13日（星期三）9:00-10:30

报告地点：科技楼南楼706室

报告题目：Neural Network Methods for Scalar Hyperbolic Conservation Laws（Ⅰ、Ⅱ）

报告摘要：Solutions of nonlinear scalar hyperbolic conservation laws (HCLs) are often discontinuous due to shock formation; moreover, locations of shocks are a priori unknown. This presents a great challenge for traditional numerical methods because most of them are based on continuous or discontinuous piecewise polynomials on fixed meshes. By employing neural network (NN), recently we proposed two NNbased methods for solving HCLs. One is a space-time approach (leastsquares neural network (LSNN) method), and the other is an explicit approach (evolving neural network (ENN) method) that emulates the underlying physics. Both the methods show a great potential to sharply capture shock without oscillation, overshooting, or smearing. The ENN method in one dimension is super accurate and efficient comparing with existing, well-developed mesh-based numerical methods. In this talk, I will give a brief introduction of NN as a class of approximating functions with “moving meshes” and use a simple example to show why the NN is superior to piecewise polynomials on fixed meshes when approximating discontinuous functions with unknown interface. I will then describe both approaches and discuss their pros and cons and related open problems.

报告人简介：Dr. Cai received his B.S. and M.S. from Huazhong University of Science and Technology, China in the respective Computer Science and Applied Mathematics, and his Ph.D from University of Colorado in Applied Mathematics in 1990. He went to Purdue as an associate professor in 1996 after serving as a postdoctoral fellow in the Brookhaven National Laboratory and the Courant Institute of New York University and as an assistant professor in the University of Southern California. He has been a summer visiting faculty at the Lawrence Livermore National Laboratory since 2003. His research is on numerical solution of partial differential equations with applications in fluid and solid mechanics. His primary interest was on accuracy control of computer simulations and self-adaptive numerical methods for complex systems before recently focusing on neural network for solving challenging partial differential equations.