Structure-Aware Spatial Kernels for Gaussian Process Distributed Port-Hamiltonian Systems

Tianyi Yang; Thomas Beckers

cs.LGOct 15, 2026

Structure-Aware Spatial Kernels for Gaussian Process Distributed Port-Hamiltonian Systems

Authors: Tianyi Yang, Thomas Beckers

Organizations: Department of Computer Science, Vanderbilt University, Nashville, TN 37212, USA · Institute for Measurement, Control and Microtechnology, Ulm University, 89081 Ulm, Germany

Abstract

Gaussian process distributed port-Hamiltonian systems (GP-dPHS) enable data-driven modeling of distributed dynamical systems while preserving their physical structure. However, standard stationary kernels can be inefficient for spatially heterogeneous systems because they neither exploit known input invariances nor reflect structured spatial dependencies. We address this limitation with a structure-aware kernel design that removes irrelevant state coordinates and decouples state and spatial dependencies through a tensor-product covariance. This modular construction allows qualitative knowledge of spatial heterogeneity to be incorporated by selecting an appropriate spatial kernel without requiring an analytical description of the underlying profile. Numerical experiments on a nonlinear string with periodic, smoothly varying, and semi-piecewise stiffness profiles demonstrate that structural decoupling consistently improves reconstruction accuracy, while profile-matched spatial kernels yield substantially larger gains in force-field and trajectory reconstruction.

Figures & tables

Fig. 1: Representative spatial stiffness profiles used in this study: periodic,

C^{2}

bumps, and semi-piecewise stiffness (top to bottom, respectively).

Model and profile	$\eta$	$P$	$\Delta_{\min}$
$\hat{h}_{2}$ , all profiles	$1\text{\times}{10}^{-1}$	$15$	$1\text{\times}{10}^{-3}$
$\hat{h}_{1}$ , periodic and constant	$1\text{\times}{10}^{-1}$	$15$	$1\text{\times}{10}^{-3}$
$\hat{h}_{1}$ , $C^{2}$ bumps	$2\text{\times}{10}^{-2}$	$50$	$1\text{\times}{10}^{-4}$
$\hat{h}_{1}$ , semi-piecewise	$1\text{\times}{10}^{-2}$	$50$	$1\text{\times}{10}^{-4}$

TABLE I: Optimization settings for the effort models, shared across kernel architectures.

Stiffness	Kernel	$\varepsilon_{q}$	$\varepsilon_{u}$	$\varepsilon_{E}$	Curl RMSE	Eff. RMSE	Force RMSE
Constant	RBF	$4.69\text{\times}{10}^{-2}$	$1.23\text{\times}{10}^{-2}$	$1.15\text{\times}{10}^{-3}$	$2.63\text{\times}{10}^{-3}$	$4.77\text{\times}{10}^{-2}$	$1.86\text{\times}{10}^{-1}$
Periodic	RBF	$5.60$	$1.18$	$1.11\text{\times}{10}^{4}$	$8.19\text{\times}{10}^{-1}$	$1.76$	$8.82$
Periodic	SD-RBF	$1.73$	$6.65\text{\times}{10}^{-1}$	$5.01\text{\times}{10}^{1}$	$3.56\text{\times}{10}^{-3}$	$1.50$	$6.30$
Periodic	SD-Per.	$8.14\text{\times}{10}^{-2}$	$2.06\text{\times}{10}^{-2}$	$3.52\text{\times}{10}^{-3}$	$3.56\text{\times}{10}^{-3}$	$6.03\text{\times}{10}^{-2}$	$3.08\text{\times}{10}^{-1}$
$C^{2}$ bumps	RBF	$5.47$	$1.18$	$1.64\text{\times}{10}^{3}$	$4.75\text{\times}{10}^{-1}$	$1.45$	$6.97$
$C^{2}$ bumps	SD-RBF	$1.27$	$7.13\text{\times}{10}^{-1}$	$5.10$	$3.72\text{\times}{10}^{-3}$	$1.23$	$5.97$

TABLE II: Average numerical metrics for the considered stiffness profiles and kernel configurations. SD-RBF, SD-Per., SD-Mat., and SD-CP denote structural decoupling with RBF, periodic, Matérn, and changepoint spatial kernels, respectively.

Fig. 2: Average reconstructed displacement trajectories for the periodic,

C^{2}

-bump, and semi-piecewise stiffness profiles. Columns correspond to rollout steps 0, 50, and 100. Curves compare the numerical reference, stationary RBF baseline, SD-RBF, and profile-matched structural-decoupling model. The matched models use periodic, Matérn, and changepoint spatial kernels for the three profiles, respectively.

Kernel Methods

Kernel methods use similarity functions and reproducing kernel spaces to model nonlinear relationships, quantify uncertainty, and compare data.

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Abstract

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