Beijing Institute of Mathematical Sciences and Applications

Zhen Li , Xin Liang , Zhi Ting Ma , Hamid Mofidi , Li Wang , Fan Sheng Xiong , Shuo Yang , Wu Yue Yang

Xiao Pei Jiao

Thursday, March 6, 2025 3:00 PM - 4:00 PM

A3-4-312

Zoom 787 662 9899 (BIMSA)

Abstract: The nonlinear filtering problem, which dates back to the 1600s, aims to infer reliable state estimates from stochastic measurements. The introduction of the Kalman filter in the 1960s revolutionized fields such as aerospace engineering and navigation. Nevertheless, achieving optimal state estimation hinges on computing the conditional density, governed by the Duncan-Mortensen-Zakai (DMZ) equation introduced in the 1970s. In the 21st century, the Yau-Yau filter, was innovatively proposed to emerge as a groundbreaking tool for nonlinear filtering. The Yau-Yau filter remains a uniquely powerful method for effectively handling complex nonlinear systems, such as those involving cubic sensors. Building on the Yau-Yau framework, we introduced the Extended Direct Method (EDM) to address more general infinite-dimensional systems compared to the traditional Direct Method. EDM is supported by rigorous existence and uniqueness analyses, and numerical results demonstrate that this explicit algorithm can achieve near-optimal accuracy comparable to spectral methods. Additionally, we developed the Deep Generalized Galerkin Method based on Physics-Informed Neural Networks (PINNs), which accelerates the offline computations of the Yau-Yau filter while preserving its high accuracy. Numerical simulations validate the efficiency and precision of these advancements, highlighting their potential for broader applications in nonlinear filtering.

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.