Designing Poisson Integrators Through Machine Learning
This work addresses a specific challenge in mathematical physics and computational geometry for researchers in those fields, representing an incremental advancement by applying existing machine learning methods to a new context.
The paper tackled the problem of constructing Poisson integrators that preserve Poisson geometry by reformulating it as solving a Hamilton-Jacobi PDE, and they approximated the solution using machine learning techniques, achieving a method that combines physical modeling with data-driven approaches.
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.