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Learning Passive Continuous-Time Dynamics with Multistep Port-Hamiltonian Gaussian Processes

Chi Ho Leung, Philip E. Paré

TL;DR

This work tackles learning physically consistent continuous-time dynamics from noisy, irregular trajectories by introducing a multistep port-Hamiltonian Gaussian process (MS-PHS GP) that places a GP prior on the Hamiltonian surface $H(x)$ and encodes variable-step multistep integration constraints. By combining a port-Hamiltonian system kernel with multistep projections, MS-PHS yields closed-form posteriors for both the vector field $f(x)$ and $H(x)$, while preserving energy balance and passivity. The approach also provides a finite-sample, high-probability bound separating data-fit from discretization error, and anchors the Hamiltonian to fix the additive constant. Empirical results on Mass–Spring, Van der Pol, and Duffing oscillators show improved vector-field recovery and well-calibrated Hamiltonian uncertainty, even under substantial noise and timestamp jitter, making the method attractive for uncertainty-aware, physics-informed control tasks.

Abstract

We propose the multistep port-Hamiltonian Gaussian process (MS-PHS GP) to learn physically consistent continuous-time dynamics and a posterior over the Hamiltonian from noisy, irregularly-sampled trajectories. By placing a GP prior on the Hamiltonian surface $H$ and encoding variable-step multistep integrator constraints as finite linear functionals, MS-PHS GP enables closed-form conditioning of both the vector field and the Hamiltonian surface without latent states, while enforcing energy balance and passivity by design. We state a finite-sample vector-field bound that separates the estimation and variable-step discretization terms. Lastly, we demonstrate improved vector-field recovery and well-calibrated Hamiltonian uncertainty on mass-spring, Van der Pol, and Duffing benchmarks.

Learning Passive Continuous-Time Dynamics with Multistep Port-Hamiltonian Gaussian Processes

TL;DR

This work tackles learning physically consistent continuous-time dynamics from noisy, irregular trajectories by introducing a multistep port-Hamiltonian Gaussian process (MS-PHS GP) that places a GP prior on the Hamiltonian surface and encodes variable-step multistep integration constraints. By combining a port-Hamiltonian system kernel with multistep projections, MS-PHS yields closed-form posteriors for both the vector field and , while preserving energy balance and passivity. The approach also provides a finite-sample, high-probability bound separating data-fit from discretization error, and anchors the Hamiltonian to fix the additive constant. Empirical results on Mass–Spring, Van der Pol, and Duffing oscillators show improved vector-field recovery and well-calibrated Hamiltonian uncertainty, even under substantial noise and timestamp jitter, making the method attractive for uncertainty-aware, physics-informed control tasks.

Abstract

We propose the multistep port-Hamiltonian Gaussian process (MS-PHS GP) to learn physically consistent continuous-time dynamics and a posterior over the Hamiltonian from noisy, irregularly-sampled trajectories. By placing a GP prior on the Hamiltonian surface and encoding variable-step multistep integrator constraints as finite linear functionals, MS-PHS GP enables closed-form conditioning of both the vector field and the Hamiltonian surface without latent states, while enforcing energy balance and passivity by design. We state a finite-sample vector-field bound that separates the estimation and variable-step discretization terms. Lastly, we demonstrate improved vector-field recovery and well-calibrated Hamiltonian uncertainty on mass-spring, Van der Pol, and Duffing benchmarks.

Paper Structure

This paper contains 21 sections, 1 theorem, 35 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Notations
  4. Background
  5. Port-Hamiltonian Systems
  6. Gaussian Process Regression (GPR)
  7. Port–Hamiltonian System Kernel
  8. Method
  9. Problem Formulation
  10. Multistep Port-Hamiltonian System Kernel
  11. Multivariate Variable Step Size Multistep Projection
  12. Exact Inference via Multistep PHS Kernel
  13. Posterior Hamiltonian Surface
  14. Experiments
  15. Benchmarks
  16. ...and 6 more sections

Key Result

Proposition 1

Under the assumption that $H$ lies in the base RKHS induced by $k_{\rm base}$, for $0 < \eta \leq 1$, the following inequality holds with probability at least $1-\eta$: for some finite positive constant $c_{\rm obs}$, $C_{\rm fit}$, and $C_{\rm bias}$.

Figures (6)

  • Figure 1: Vector-field mean squared error (MSE) on three dynamical benchmarks—Mass-Spring, Van der Pol, and Duffing oscillators. Bars compare Gaussian-process regressors: multistep Port-Hamiltonian (MS-PHS-ab-1/2/3), multistep ODE (MS-ODE-ab-1/2/3), and a GP-PHS variant with Savitzky–Golay smoothing (GP-PHS-loess-2). Error bars indicate variability across runs. Lower is better.
  • Figure 2: Van der Pol vector-field comparison. Ground truth (gray streamlines); MS-PHS GP (blue); MS-ODE GP (orange). Black dots are observations; red shading indicates high posterior standard deviation (std) of $f_*$, computed as $\sqrt{\sum_{i=1}^n(\Sigma_{{f}})_{ii}}$, with $\Sigma_{{f}}$ defined in \ref{['eq:msphs-cov']}. MS-PHS preserves the correct flow direction even in uncertain regions, whereas MS-ODE does not.
  • Figure 3: 3D Hamiltonian landscape $H$ for the Duffing oscillator. The learned MS-PHS posterior mean $\mu_H$ surface (red) is overlaid with a reference/ground-truth surface $H_{\rm true}$ (green) and 95% posterior band $\mu_H \pm 2\sigma_H$ (translucent blue) for comparison. Black dots mark observed trajectory samples.
  • Figure 4: Duffing oscillator Hamiltonian error and uncertainty over the $(q,\dot q)$ plane. (Left) true absolute error between ground truth and learned mean, $|H_{\text{true}}-\mu_H|$. (Right) 95% posterior margin, $1.96\sigma_H$. The color bar shows magnitude; both error and uncertainty are small near the center and increase toward the domain boundary. Comparing the left panel and right panel shows that the $95\%$-posterior credible band ($\mu_H \pm 2\sigma_H$) tracks the true error: $|H_{\text{true}}-\mu_H|$.
  • Figure 5: Scaling of Hamiltonian posterior error and uncertainty with observation noise on the Duffing system. For each noise level $\sigma_x$, points show the median and bars the interquartile range (log scale). Panels share the y-axis for direct comparison. (Left) MS-PHS-ab-3: $\sigma_H^2$ rises in step with $H_{\text{mse}}$, indicating calibrated uncertainty across noise. (Right) GP-PHS-loess-2: $H_{\text{mse}}$ increases sharply while $\sigma_H^2$ lags, especially for $\sigma_x^2\ge 0.02$, suggesting underestimation of uncertainty.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Remark 2.2.1
  • Example 3.1: Explicit Forward Euler
  • Remark 3.2.1: Unmodeled GP Input Noise
  • Proposition 1: Vector-field finite-sample high-probability bound
  • proof
  • Remark 3.3.1