Beijing Institute of Mathematical Sciences and Applications

Fan Sheng Xiong , Wu Yue Yang , Wen An Yong , Yi Zhu

Zhiqiang Cai

Thursday, November 3, 2022 10:00 AM - 11:30 AM

1129B

Zoom 537 192 5549 (BIMSA)

In this talk, I will present our recent works on neural networks (NNs) and its application in numerical PDEs. The first part of the talk is to use NNs to numerically solve scalar linear and nonlinear hyperbolic conservation laws whose solutions are discontinuous. I will show that the NN-based method for this type of problems has an advantage over the mesh-based methods in terms of the number of degrees of freedom. The second part of the talk is on our adaptive network enhancement (ANE) method. The ANE method is developed to address a fundamental, open question on how to automatically design an optimal NN architecture for approximating functions and solutions of PDEs within a prescribed accuracy. Moreover, to train the resulting nonconvex optimization problem, the ANE method provides a natural process of obtaining a good initialization.

Prof. 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. Before focusing on neural network for solving challenging partial differential equations, his second to last primary interest was on accuracy control of computer simulations and self-adaptive numerical methods for complex systems.