Beschreibung:

Many examples of complex systems in physics, chemistry, and engineering can be modeled by deterministic or stochastic differential equations, which can be simulated on computers to make predictions about the system's future behaviour. However, it is not always an easy task to draw useful conclusions from the simulation data: in some cases, modelers are faced with a wealth of data which cannot simply be analyzed on a visual level. In other cases, we only have real-world measurement data, from which we would like to infer a valid differential equation model.

These and other questions can be addressed using an operator-theoretic framework to describe dynamical systems. This interdisciplinary approach draws on ideas from dynamical systems theory, some elements of linear operator theory, statistical learning, and of course insights from application areas.

In the course, we will first cover selected concepts from the disciplines mentioned above, then introduce the Koopman operator framework, which is at the heart of the class, and finally learn about its applications.    


Preliminary outline of the course:

- Matrix decompositions and linear regression problems (2 weeks).
- Review of Banach and Hilbert spaces, linear operators (1 week).
- Elements of deterministic and stochastic differential equations (2 weeks).
- The Koopman operator and its properties (2 weeks).
- Learning the Koopman operator from data: extended dynamic mode decomposition (1 week).
- System identification (SINDy) (1 week).
- Variational principles and model selection (2 weeks).
- Reproducing kernel Hilbert spaces (2 weeks).
- Kernel methods for dynamical systems (2 weeks).
- Selected applications (1 week).