哈尔滨工业大学（深圳）学术讲座

演讲人Speaker：   西安交通大学 王飞  教授

题目Title：Randomized Neural Networks Methods for PDEs

时间Date：2023年 11 月 06 日       Time：09:00–10:00

地点Venue：腾讯会议481-660-457 

内容摘要Abstract： 

We propose a novel approach for solving partial differential equations (PDEs) based on randomized neural networks and the Petrov–Galerkin method, which we call the RNN-PG methods. This approach uses randomized neural networks to approximate unknown functions and enables a flexible choice of test functions, such as finite element basis functions, Legendre or Chebyshev polynomials, or neural networks. We apply the RNN-PG methods to various problems, including Poisson problems with primal or mixed formulations, and time-dependent problems with a space–time approach. This talk is based on the work on “Communications in Nonlinear Science and Numerical Simulation 127, (2023), 107518”, which is adapted from the work originally posted on arXiv.com by the same authors (arXiv:2201.12995, Jan 31, 2022).  The new ingredients include non-linear PDEs such as Burger’s equation and a numerical example of a high-dimensional heat equation. Numerical experiments show that the RNN-PG methods can achieve high accuracy with a small number of degrees of freedom. Furthermore, RNN-PG has several advantages, such as being mesh-free, handling different boundary conditions easily, solving time-dependent problems efficiently, and solving high-dimensional problems quickly. These results indicate the great potential of the RNN-PG methods in the field of numerical methods for PDEs. We will also discuss our recent work on local randomized neural networks with discontinuous Galerkin methods for PDEs (June 11, 2022, arXiv:2206.05577; May 25, 2023, arXiv:2305.16060).

个人简介(About the speaker):

王飞，西安交通大学教授，博士生导师。 2010年获浙江大学数学博士学位。2010年—2012年，在华中科技大学任教；2012年－2013年，为美国爱荷华大学客座助理教授；2013年－2016年，为美国宾州州立大学Research Associate；2015年入选西安交通大学青年拔尖人才B类，2017年入选陕西省青年百人，2022年入选西安交通大学青年拔尖人才A类。研究领域为数值分析与科学计算，主要研究兴趣包括：有限元分析及其应用，变分不等式的数值方法，求解偏微分方程的神经网络方法等。主持国家自然科学基金面上项目2项、青年基金1项，横向课题1项。已在国际 SCI 期刊发表论文四十多篇，其中包括计算数学方向的顶级期刊：SIAM J Numer. Anal.，IMA J Numer. Anal.，Numer. Math.，Comput. Methods Appl. Mech. Eng. 等。