Lehrstuhl für Mechatronik in Maschinenbau und Fahrzeugtechnik (MEC)

Dynamical Systems and Neural Networks

Motivation

In this course students shall learn the fundamental principles governing complex systems via the theory of dynamical systems. We shall mainly rely on PDEs and ODEs as prototypical models to capture the dynamics of various systems from vehicle motion to the spreading of diseases. In the first quarter of the course, we shall make a brief recapitulation of the basic topics such as wellposedness, phase space diagrams, stability theory and bifurcation theory, classes of PDEs and also look at standard numerical methods for solving ODEs and PDEs. With the emergence of neural networks as universal function approximators they offer compelling advantages for dealing with complex and high-dimensional PDEs and ODEs. The course shall focus on understanding how neural networks and the concepts of machine learning can be used to obtain solutions to different types of dynamical systems especially the ones specified using PDEs and ODEs. Going further, we shall also focus on the problem of estimating unobserved states and the problem of controlling the system states based on partial observations with the help of neural networks.  Finally, we shall see how the three core blocks of prediction, estimation and control can be integrated seamlessly in a network to obtain a generic unified algorithm for solving dynamical problems. By the course's end, students will have gained a solid foundational understanding of dynamical systems, ODEs, PDEs and neural networks, equipped with analytical and computational tools to address interdisciplinary problems effectively.

 

Course Content

Given below are the contents for the course:

  1. Recap on the basics of ODEs and PDEs from the viewpoint of functional theory
  2. Recap on numerical methods for ODEs (deterministic methods: Euler, RK, finite difference, finite elements)
  3. Neural networks (NNs) and approximation theorems
  4. NNs as a method for solving differential functional equations
  5. NNs as a method for solving functional state estimation problem
  6. NNs as a method for solving functional optimization problem
  7. Different types of NN design for PDEs and ODEs 
  8. Physics informed neural networks, neural differential equations and diffusion models
  9. Applications in process automation, automated robot and vehicle control

 

Literature

  • Strauss, W. A., Partial Differential Equations-An Introduction, John Wiley & Sons, Inc., New York, 1992
  • Evans, L. C., Partial differential equations, American Mathematical Society, 2010.
  • Birkhoff, G.D., Dynamical Systems, Volume 9. American Mathematical Soc., 1927.
  • Goodfellow, I., Bengio, Y., and Courville, A., Deep Learning, MIT press, 2016.
  • LeCun, Y., Bengio, Y., and Hinton, G., Deep learning, Nature, 521(7553):436–444, 2015
  • Bishop, C. M., and Others. Pattern Recognition and Machine Learning, Volume 4. Springer New York, 2006.

Dozent

Dr. Sandesh Hiremath
Gebäude 65, Raum 420
67663, Kaiserslautern
Phone: +49 631/205-3455
sandesh.hiremath(at)mv.uni-kl.de
 

Lecture

KIS entry
 

Exam

Written Exam: 120-150 Min.
Date: 17.03.2025
Time: 14:00 – 17:00
Location: Building. TBA
Credit Points: 4ECTS
KIS entry

 

Prerequisites

Multivariable calculus
Vector calculus
Linear algebra
Python programming

Script

OLAT