Abstract

We provide an a priori error analysis for methods solving PDEs using neural networks. We show that the resulting constrained optimization problem can be efficiently solved using greedy algorithms, which replaces stochastic gradient descent. Following this, we show that the error arising from discretizing the energy integrals is bounded both in the deterministic case, i.e. when using numerical quadrature, and also in the stochastic case, i.e. when sampling points to approximate the integrals. This innovative greedy algorithm is tested on several benchmark examples to confirm its efficiency and robustness. 

Short bio 

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., 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.