Institutskolloquium des IPP 2024
Wissenschaftliches Kolloquium des IPP in Garching und Greifswald mit Videoübertragung
Many dynamical systems in physics and other fields possess some form of geometric structure, such as Lagrangian or Hamiltonian structure, symmetries, and conservation laws. Geometric numerical integration describes the research field that aims at developing numerical algorithms for solving such systems while preserving their geometric structure. Usually, such algorithms show greatly reduced errors and better long-time stability than non-structure-preserving algorithms. Scientific machine learning is a relatively new research field that aims at using machine learning methods for the solution of differential equations. In this presentation, we will explore how ideas of geometric numerical integration can be brought forward to the realms of scientific machine learning to make machine learning methods more robust. The talk starts with a brief review of numerical approximation and integration. Then, important mathematical structures of Hamiltonian systems will be reviewed, and the consequences of their non-preservation in numerical simulations highlighted. Basic structure-preserving algorithms for canonical Hamiltonian systems will be introduced and compared with their non-structure-preserving counterparts. In the second part of the talk, different approaches to using machine learning techniques for the solution of differential equations will be described. After exploring once more the consequences of disregarding the structure of the dynamical system, we will discuss how geometric structure can be incorporated into machine learning methods such as neural networks. The talk will close with an outlook on training-free algorithms.
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