Applying Neural Networks for Data Fitting and Numerical PDEs

主讲人：洪庆国 美国宾州州立大学

时间：2023年6月13日15：00

地点：三号楼332室

举办单位：数理学院

主讲人介绍：洪庆国，博士，美国宾州州立大学Assistant Research Professor。曾先后在奥地利科学院Radon研究所(RICAM)，德国Duisburg-Essen University, 美国宾州州立大学从事博士后研究。目前研究兴趣包括机器学习，迭代法，间断有限元方法及应用。在SIAM J. Numer. Anal., Math. Comp., Numer. Math., J. Comput. Phys., Comput. Methods Appl. Mech. Engrg.，Math. Models Methods Appl. Sci.和中国科学-数学等国内外期刊发表系列论文。

内容介绍：We develop new neural networks which are much easier to train for data fitting. These newly developed neural networks are motivated by finite element and spectral analysis. In addition, methods for solving PDEs using neural networks have recently become a very important topic. We provide an a priori error analysis for such methods. We first show that the generalization error arising from discretizing the energy integrals is bounded. Then we show that the resulting constrained optimization problem can be efficiently solved using a greedy algorithm, which replaces gradient descent. These importantly give a consistent analysis which incorporates the optimization, approximation, and generalization aspects of the problem. Some numerical results will be presented.