报告题目：Deep Petrov-Galerkin Method for Solving Partial Differential Equations
报告人：王飞，西安交通大学教授
报告时间：2023.05.26上午9:00-10:00
腾讯会议：828-523-453
会议密码：123456
会议链接：//meeting.tencent.com/dm/JWXJMdB908Ph
摘要：Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this talk, we propose a neural network-based numerical method to solve partial differential equations. In this new framework, the method is designed on weak formulations, and the unknown functions are approximated by deep neural networks and test functions can be chosen by different approaches, for instance, basis functions of finite element methods, neural networks, and so on. Because the spaces of trial function and test function are different, we name this new approach by Deep Petrov-Galerkin Method (DPGM). The resulted linear system is not necessarily to be symmetric and square, so the discretized problem is solved by a least-squares method. Take the Poisson problem as an example, mixed DPGMs based on several mixed formulations are proposed and studied as well. In addition, we apply the DPGM to solve two classical time-dependent problems based on the space-time approach, that is, the unknown function is approximated by a neural network, in which temporal variable and spatial variables are treated equally, and the initial conditions are regarded as boundary conditions for the space-time domain. Finally, several numerical examples are presented to show the performance of the DPGMs, and we observe that this new method outperforms traditional numerical methods in several aspects: compared to the finite element method and finite difference method, DPGM is much more accurate with respect to degrees of freedom; this method is mesh-free, and can be implemented easily; mixed DPGM has good flexibility to handle different boundary conditions; DPGM can solve the time-dependent problems by the space-time approach naturally and efficiently. The proposed deep Petrov-Galerkin method shows strong potential in the field of numerical methods for partial differential equations.
报告人简介：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项目3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。
邀请人：杜锐