A Structure-Preserving Kernel Method for Learning Hamiltonian Systems

TL;DR

This paper introduces a structure-preserving kernel ridge regression method for accurately recovering nonlinear Hamiltonian functions from noisy data, outperforming existing techniques and providing theoretical convergence guarantees.

Contribution

It extends kernel regression to Hamiltonian systems, proving a differential reproducing property, a Representer Theorem, and analyzing the relation to Gaussian estimators.

Findings

Method achieves superior numerical performance over existing techniques.

Provides convergence rates with fixed and adaptive regularization.

Numerical experiments validate the estimator's effectiveness.

Abstract

A structure-preserving kernel ridge regression method is presented that allows the recovery of nonlinear Hamiltonian functions out of datasets made of noisy observations of Hamiltonian vector fields. The method proposes a closed-form solution that yields excellent numerical performances that surpass other techniques proposed in the literature in this setup. From the methodological point of view, the paper extends kernel regression methods to problems in which loss functions involving linear functions of gradients are required and, in particular, a differential reproducing property and a Representer Theorem are proved in this context. The relation between the structure-preserving kernel estimator and the Gaussian posterior mean estimator is analyzed. A full error analysis is conducted that provides convergence rates using fixed and adaptive regularization parameters. The good performance of the proposed estimator together with the convergence rate is illustrated with various numerical experiments.