Designing Poisson Integrators Through Machine Learning

Autor:	Vaquero, Miguel, de Diego, David Martín, Cortés, Jorge
Rok vydání:	2024
Předmět:	Mathematical Physics Computer Science - Machine Learning Mathematics - Differential Geometry Mathematics - Dynamical Systems Mathematics - Numerical Analysis 37J06 70H15 70H20 70G45 65L05 68T07 G.1.8 J.2
Druh dokumentu:	Working Paper
Popis:	This paper presents a general method to construct Poisson integrators, i.e., integrators that preserve the underlying Poisson geometry. We assume the Poisson manifold is integrable, meaning there is a known local symplectic groupoid for which the Poisson manifold serves as the set of units. Our constructions build upon the correspondence between Poisson diffeomorphisms and Lagrangian bisections, which allows us to reformulate the design of Poisson integrators as solutions to a certain PDE (Hamilton-Jacobi). The main novelty of this work is to understand the Hamilton-Jacobi PDE as an optimization problem, whose solution can be easily approximated using machine learning related techniques. This research direction aligns with the current trend in the PDE and machine learning communities, as initiated by Physics- Informed Neural Networks, advocating for designs that combine both physical modeling (the Hamilton-Jacobi PDE) and data. Comment: 5 pages, 5 figures
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2403.20139 Zobrazit plný text záznamu View this record from Arxiv