A global structure-preserving kernel method for the learning of Poisson systems
Abstract
A structure-preserving kernel ridge regression method is presented that allows the recovery of globally defined, potentially high-dimensional, and nonlinear Hamiltonian functions on Poisson manifolds out of datasets made of noisy observations of Hamiltonian vector fields. The proposed method is based on finding the solution of a non-standard kernel ridge regression where the observed data is generated as the noisy image by a vector bundle map of the differential of the function that one is trying to estimate. Additionally, it is shown how a suitable regularization solves the intrinsic non-identifiability of the learning problem due to the degeneracy of the Poisson tensor and the presence of Casimir functions. A full error analysis is conducted that provides convergence rates using fixed and adaptive regularization parameters. The good performance of the proposed estimator is illustrated with several numerical experiments.
Cite
@article{arxiv.2504.13396,
title = {A global structure-preserving kernel method for the learning of Poisson systems},
author = {Jianyu Hu and Juan-Pablo Ortega and Daiying Yin},
journal= {arXiv preprint arXiv:2504.13396},
year = {2025}
}