Beijing Institute of Mathematical Sciences and Applications

Shing Toung Yau

Xiao Pei Jiao

Monday, May 8, 2023 4:30 PM - 5:00 PM

Online

Schrodinger equation plays an important role in quantum physics and many other fields in applied mathematics such as control theory. In nonlinear filtering, under the framework of Yau-Yau algorithm, Schrodinger equation at a short time interval plays a crucial role for state estimation problem. In this paper, Schrodinger equation with polynomial potential has been investigated and explicit fundamental solution has been written down. By applying Euler operator theory developed in estimation algebra theory, uniqueness and existence of fundamental solution are verified. We propose two numerical algorithms for Schrodinger equation including the first order approximation and PINN based solver. We compare our algorithms with classical spectrum method. Numerical results demonstrate that our two new algorithms are superior to the traditional filtering algorithms in effectiveness and accuracy.

Xiaopei Jiao received his bachelor's degree from the Zhiyuan College of Shanghai Jiao Tong University and his Ph.D. from the Department of Mathematical Sciences at Tsinghua University. He subsequently worked as a postdoctoral researcher at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) and at the University of Twente in the Netherlands. His current research interests include finite-dimensional filtering theory, Yau-Yau filtering methods, physics-informed neural networks, and bioinformatics. His research focuses primarily on: (1) using geometric tools such as Lie algebras for solving partial differential equations and classifying nonlinear systems; (2) designing novel numerical algorithms based on physics-informed neural networks.