Designing Poisson Integrators Through Machine Learning
Miguel Vaquero, David Martín de Diego, Jorge Cortés
TL;DR
The paper addresses designing Poisson integrators that preserve Poisson geometry for integrable Poisson manifolds. It reframes Poisson diffeomorphisms as Lagrangian bisections in a local symplectic groupoid and reduces their construction to solving a Hamilton-Jacobi PDE, which is tackled via a machine-learning-inspired optimization. The key contributions include a general geometric framework, a practical ML-based approximation for the Hamilton-Jacobi equation, and a rigid-body demonstration that preserves the Hamiltonian and Casimir over long time horizons. This approach offers a flexible, geometry-respecting pathway for numerical integration with potential impact in physics and engineering where Poisson structures are central.
Abstract
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.