Designing Poisson Integrators Through Machine Learning
Miguel Vaquero David Martín de Diego Jorge Cortés
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.