Zhiqiang Cai, Professor, Great Bay University

Aihui Zhou, Professor

Neural Networks in Scientic Computing (SciML): Basics and Challenging Questions

10:00–11:00 December 29(Monday), N733

Neural networks (NNs) have demonstrated remarkable performance in computer vision, natural language processing, and many other tasks of arti cial intelligence. Recently, there has been a growing interest in leveraging NNs to solve partial di erential equations (PDEs). Despite the rapid proliferation of articles in recent years, research on NN-based numerical methods for solving PDEs in the context of science and en gineering is still in its early stages. Numerous critical open problems remain to be addressed before these methods can be broadly applied to solve computationally challenging problems.

In this talk, I will rst give a brief introduction of ReLU NNs from nu merical analysis perspective. I will then discuss our works on addressing some of critical questions such as

 • why use NNs instead of nite elements in scienti c computing? or for what applications, are NNs better than nite elements in ap proximation?

 • howtodevelop NNdiscretization methods that are not only physics informed but more importantly physics-preserved?

 • how to develop reliable and e cient training algorithms for NN discretization (non-convex optimization)?

 • for a given task, how to design a nearly optimal NN architecture within a prescribed accuracy?