Abstract: We present a novel deep learning method for computing eigenstates of the Schrödinger operator.  Our approach combines a novel loss function with an innovative neural network architecture that incorporates a-prior knowledge of the problem. These improvements enable the method to handle both high-dimensional problems and problems posed on irregular bounded domains. We successfully compute up to the first 30 eigenvalues for various Schrödinger operators. We analyze the generalization error in the framework of the sine-Barron space, in which a regularity result has been established with the aid of the operator theory. We also establish Barron regularity for the many-particle Schrödinger operator, which essentially improve Kato's classical regularity results in 1957.  This is a joint work with Yixiao Guo and Hao Yu.

Bio: Ming Pingbing is a research professor at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He currently serves as the Editor-in-Chief of the Journal on Numerical Methods and Computer Applications (Chinese), and is on the editorial boards of several journals including the Journal of Computational Physics and Science China Mathematics. His research primarily focuses on multiscale modeling and simulation of solids, as well as scientific machine learning. He is the author of over 70 publications in journals such as JAMS, CPAM, ARMA. He has made outstanding contributions to the mathematical theory of the Cauchy-Born rule and the theoretical prediction of the ideal strength of graphene. In 2023, he was awarded the 15th Feng Kang Prize in Scientific Computing.