Nonlinear filter and deep learning
As an important branch of control theory, nonlinear filtering refers to the estimation or filtering out of noise from a system where the underlying system dynamics are nonlinear. These problems arise in various fields, such as signal processing, robotics, and economics. The nonlinear property originating from the system makes it difficult to apply traditional linear filtering techniques such as the Kalman filter and its variants, particle filter, etc.
This course will focus on the mathematical foundation and algorithms to handle nonlinear filtering. In the 1983 International Congress of Mathematics, Brockett proposed the classification problem for finite-dimensional filters. We shall first introduce the so-called Brockett-Mitter program by applying geometric methods, e.g., Lie algebra. Later, we shall concentrate on the well-known Yau-Yau filter to introduce the theory and the corresponding developed numerical algorithms. Finally, combined with recent popular deep learning techniques, some novel filtering algorithms to deal with infinite-dimensional filters will be introduced.
After completing the course, participants will be familiar with important mathematical tools and their applications in nonlinear filtering. This course will also contain many cutting-edge academic results from the lecturer himself.
This course will focus on the mathematical foundation and algorithms to handle nonlinear filtering. In the 1983 International Congress of Mathematics, Brockett proposed the classification problem for finite-dimensional filters. We shall first introduce the so-called Brockett-Mitter program by applying geometric methods, e.g., Lie algebra. Later, we shall concentrate on the well-known Yau-Yau filter to introduce the theory and the corresponding developed numerical algorithms. Finally, combined with recent popular deep learning techniques, some novel filtering algorithms to deal with infinite-dimensional filters will be introduced.
After completing the course, participants will be familiar with important mathematical tools and their applications in nonlinear filtering. This course will also contain many cutting-edge academic results from the lecturer himself.
Lecturer
Date
19th March ~ 2nd July, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 14:20 - 16:55 | A3-1-101 | ZOOM 2 | 638 227 8222 | BIMSA |
Prerequisite
Basic knowledge of Calculus, linear algebra and differential equation.
Syllabus
1. Introduction to nonlinear filter problem, mathematical formulation, and popular filtering algorithms
Brockett-Mitter program of classification
2. Basic knowledge of Lie group and Lie algebra
3. Finite dimensional filter(I)
4. Finite dimensional filter(II)
5. Finite dimensional filter(III)
6. Finite dimensional filter(IV)
Optimal nonlinear filter
7. Basic knowledge of Partial differential equation
8. Yau-Yau filter (I)
9. Yau-Yau filter (II)
10. Yau-Yau filter (III)
11. Yau-Yau filter (IV)
Deep learning filter
12. Basic introduction to scientific machine learning
13. Deep learning-inspired filter algorithms (I)
14. Deep learning-inspired filter algorithms (II)
15. Deep learning-inspired filter algorithms (III)
16. Deep learning-inspired filter algorithms (IV)
Brockett-Mitter program of classification
2. Basic knowledge of Lie group and Lie algebra
3. Finite dimensional filter(I)
4. Finite dimensional filter(II)
5. Finite dimensional filter(III)
6. Finite dimensional filter(IV)
Optimal nonlinear filter
7. Basic knowledge of Partial differential equation
8. Yau-Yau filter (I)
9. Yau-Yau filter (II)
10. Yau-Yau filter (III)
11. Yau-Yau filter (IV)
Deep learning filter
12. Basic introduction to scientific machine learning
13. Deep learning-inspired filter algorithms (I)
14. Deep learning-inspired filter algorithms (II)
15. Deep learning-inspired filter algorithms (III)
16. Deep learning-inspired filter algorithms (IV)
Reference
Latest papers of the lecturer and the related literature.
Audience
Undergraduate
, Graduate
, Researcher
Video Public
Yes
Notes Public
Yes
Language
Chinese
, English
Lecturer Intro
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.