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A global structure-preserving kernel method for the learning of Poisson systems

Jianyu Hu, Juan-Pablo Ortega, Daiying Yin

TL;DR

The paper develops a global, structure-preserving kernel ridge regression approach to learn Hamiltonian functions on Poisson manifolds from noisy observations of Hamiltonian vector fields. It extends RKHS tools to Riemannian Poisson settings, using a differential reproducing property and a generalized differential Gram matrix to derive a closed-form estimator that remains globally defined across coordinate patches. The method addresses non-identifiability from Casimir functions via ridge regularization and proves convergence rates under a source condition, with error bounds that separate estimation and approximation errors. Through Lie-Poisson and non-Euclidean examples, the authors demonstrate accurate recovery of Hamiltonians (up to Casimirs when necessary) and preservation of symmetries and conserved quantities. The work advances structure-preserving learning on manifolds, offering exact or qualitative Hamiltonian recovery in diverse geometric settings and laying groundwork for extensions to more complex, possibly infinite-dimensional 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.

A global structure-preserving kernel method for the learning of Poisson systems

TL;DR

The paper develops a global, structure-preserving kernel ridge regression approach to learn Hamiltonian functions on Poisson manifolds from noisy observations of Hamiltonian vector fields. It extends RKHS tools to Riemannian Poisson settings, using a differential reproducing property and a generalized differential Gram matrix to derive a closed-form estimator that remains globally defined across coordinate patches. The method addresses non-identifiability from Casimir functions via ridge regularization and proves convergence rates under a source condition, with error bounds that separate estimation and approximation errors. Through Lie-Poisson and non-Euclidean examples, the authors demonstrate accurate recovery of Hamiltonians (up to Casimirs when necessary) and preservation of symmetries and conserved quantities. The work advances structure-preserving learning on manifolds, offering exact or qualitative Hamiltonian recovery in diverse geometric settings and laying groundwork for extensions to more complex, possibly infinite-dimensional 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.

Paper Structure

This paper contains 30 sections, 19 theorems, 196 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Results.
  3. Preliminaries on Poisson mechanics
  4. Hamiltonian systems on Poisson manifolds
  5. High-order differentials and random variables on Riemannian manifolds
  6. Compatible structures and local coordinate representations
  7. Structure-preserving kernel regressions on Poisson manifolds
  8. RKHS on Riemannian manifolds
  9. The structure-preserving learning problem
  10. Operator representations of the kernel estimator
  11. The global solution of the learning problem
  12. Symmetries and momentum conservation by kernel estimated Hamiltonians
  13. Estimation and approximation error analysis
  14. Test examples and numerical experiments
  15. Learning of Lie-Poisson systems
  16. ...and 15 more sections

Key Result

Proposition 2.18

Let $(P, \left\{\cdot , \cdot \right\})$ be a Poisson manifold equipped with a Riemannian manifold $g$. Then for any vector field $Y\in\mathfrak{X}(P)$ and any $h \in C^{\infty}(P)$: Furthermore, Casimir functions are constant along the flow of the vector field $-JY$, that is, $(-JY)[h] = 0$ for any vector field $Y\in\mathfrak{X}(P)$ and any $h\in\mathcal{C}(P)$.

Figures (8)

  • Figure 5.1: Rigid body dynamics: (a) Ground-truth Hamiltonian (b) Learned Hamiltonian with $N=500$ (c) Learned Hamiltonian adjusted by a Casimir function (d) Squared error of the predicted Hamiltonian vector field
  • Figure 5.2: Underwater Vehicle: (a) Ground-truth Hamiltonian (b) Learned Hamiltonian with $N=400$ (c) Squared error of the predicted Hamiltonian vector field
  • Figure 5.3: Gaussian kernel sections: (a) Ground-truth Hamiltonian (b) Learned Hamiltonian with $N=500$ (c) Absolute error of the predicted Hamiltonian function (d) Squared error of the predicted Hamiltonian vector field
  • Figure 5.4: Spherical 3-norm on $S^2\times S^2$: (a)(b) Ground-truth Hamiltonian (c)(d) Learned Hamiltonian with $N=1200$ (e)(f) Absolute Error of the predicted Hamiltonian function adjusted by constant.
  • Figure 5.5: Global heatmap of the spherical 3-norm on $S^2\times S^2$ with $N=1200$: (a) left: ground-truth Hamiltonian on the first sphere; right: learned Hamiltonian on the first sphere adjusted by constant (b) left: ground-truth Hamiltonian on the second sphere; right: learned Hamiltonian on the second sphere adjusted by constant.
  • ...and 3 more figures

Theorems & Definitions (54)

  • Definition 2.1
  • Example 2.2: Symplectic bracket
  • Example 2.3: Lie-Poisson bracket
  • Definition 2.4
  • Example 2.5: Rigid body Casimirs Marsden1994
  • Example 2.6: Underwater vehicle Casimirs leonard1997stability
  • Definition 2.7
  • Example 2.8: The linear momentum
  • Example 2.9: The angular momentum
  • Example 2.10: Lifted actions on cotangent bundles
  • ...and 44 more