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A Structure-Preserving Kernel Method for Learning Hamiltonian Systems

Jianyu Hu, Juan-Pablo Ortega, Daiying Yin

TL;DR

This work tackles learning a nonlinear Hamiltonian $H$ from noisy Hamiltonian-vector-field observations while preserving the underlying symplectic structure. It introduces a structure-preserving kernel ridge regression approach that yields a closed-form estimator for $H$ via a differential Representer Theorem and connects it to Gaussian process regression under a specific regularization, enabling a rigorous error analysis. The authors derive a differential Gram matrix, prove the equivalence of GP posterior mean and the kernel estimator when $\lambda=\sigma^2/N$, and establish convergence rates for fixed and adaptive $\lambda$, with improvements under a coercivity condition. Numerical experiments on classic Hamiltonian systems (e.g., Double pendulum, Hénon–Heiles, Frenkel–Kontorova) demonstrate accurate recovery of Hamiltonians, robustness to non-convexities, and favorable comparisons with Hamiltonian neural networks. The results advance reliable, structure-preserving learning for autonomous Hamiltonian systems and lay groundwork for extensions to broader dynamical settings and online learning scenarios.

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.

A Structure-Preserving Kernel Method for Learning Hamiltonian Systems

TL;DR

This work tackles learning a nonlinear Hamiltonian from noisy Hamiltonian-vector-field observations while preserving the underlying symplectic structure. It introduces a structure-preserving kernel ridge regression approach that yields a closed-form estimator for via a differential Representer Theorem and connects it to Gaussian process regression under a specific regularization, enabling a rigorous error analysis. The authors derive a differential Gram matrix, prove the equivalence of GP posterior mean and the kernel estimator when , and establish convergence rates for fixed and adaptive , with improvements under a coercivity condition. Numerical experiments on classic Hamiltonian systems (e.g., Double pendulum, Hénon–Heiles, Frenkel–Kontorova) demonstrate accurate recovery of Hamiltonians, robustness to non-convexities, and favorable comparisons with Hamiltonian neural networks. The results advance reliable, structure-preserving learning for autonomous Hamiltonian systems and lay groundwork for extensions to broader dynamical settings and online learning scenarios.

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.
Paper Structure (41 sections, 23 theorems, 184 equations, 12 figures)

This paper contains 41 sections, 23 theorems, 184 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Conventions and summary of the main results.
  3. Comparison with some related results in the literature
  4. RKHS and Gaussian process regression
  5. RKHS: preliminaries
  6. Gaussian process regression
  7. Observation data regime.
  8. The GP regression for the Hamiltonian function.
  9. Reproducing properties of differentiable kernels on unbounded sets
  10. Structure-preserving kernel ridge regression
  11. Operator representations of the kernel ridge learning problem and its solution
  12. Equivalence of the Gaussian posterior mean estimator and the structure-preserving kernel estimator
  13. Estimation and approximation error analysis
  14. The approximation error.
  15. The noisy sampling error.
  16. ...and 26 more sections

Key Result

Theorem 2.6

Suppose that the observation noise term $\bm{\varepsilon}^{(n)}$ in observation regime is Gaussian and it is independent of $\mathbf{Z}_N$. Assume also that the the parameters $\theta, \sigma$ are known and that we are given the training dataset $(\mathbf{z}_N,\mathbf{x}_{\sigma^2,N})$ defined in tr where The symbols $K_{X_H,H}^{\theta}(\mathbf{z}_N, \mathbf{z}^*) = (K_{H,X_H}^{\theta})^{\top}(\m

Figures (12)

  • Figure 5.1: Double pendulum: (a) Ground truth potential (b) Potential of the learned Hamiltonian (c) Mismatch error after vertical shift
  • Figure 5.2: Hénon-Heiles system (a) Ground truth potential (b) Potential of the learned Hamiltonian (c) Mismatch error after vertical shift
  • Figure 5.3: Frenkel-Kontorova model (a) Ground truth potential (b) Potential of the learned Hamiltonian (c) Mismatch error after vertical shift
  • Figure 5.4: Ground truth potential
  • Figure 5.5: Learning with $N=500$ (a) Potential of the learned Hamiltonian (b) Mismatch error after vertical shift
  • ...and 7 more figures

Theorems & Definitions (58)

  • Example 2.1: Gaussian kernel
  • Example 2.2: Sobolev kernel
  • Definition 2.3: RKHS
  • Remark 2.4
  • Definition 2.5: Gaussian process
  • Theorem 2.6
  • proof
  • Theorem 2.7: Differential reproducing property
  • Corollary 2.8
  • proof
  • ...and 48 more