Structure-preserving methods for simulation and learning of dynamical systems
Kathrin Flaßkamp (Saarland University)
The classical approach to predicting behavior of dynamical systems is to treat the mathematical modeling and the numerical integration as subsequent steps. Contrarily, structure-preserving methods, as they have been developed particularly for mechanical systems, intertwine them to generate discrete-time models, still based on first-principle modeling. Motivated by the growing interest in machine learning, we revisit these methods to study the potential of data-based enhancements.
This presentation gives insight into three aspects of this endeavor: Firstly, Hamiltonian neural networks which preserve and learn unknown system invariances, namely affine-linear Lie group symmetries, are introduced for system identification. Secondly, we present AI-enhanced symplectic numerical integrators which allow a structure-preserving efficiency increase. Lastly, we consider dynamical control systems possessing symmetry-induced relative equilibria, known as motion primitives. These primitives can be extracted from data as we demonstrate in an autonomous driving application.