面向高斯过程分布式端口哈密顿系统的结构感知空间核函数

# Structure-Aware Spatial Kernels for Gaussian Process Distributed Port-Hamiltonian Systems
Source: [https://arxiv.org/html/2610.22575](https://arxiv.org/html/2610.22575)
Tianyi YangAffiliation:is with the Department of Computer Science, Vanderbilt University, Nashville, TN 37212, USAtianyi\.yang@Vanderbilt\.eduThomas BeckersAffiliation:is with the Institute for Measurement, Control and Microtechnology, Ulm University, 89081 Ulm, Germanythomas\.beckers@uni\-ulm\.de

###### 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\.

## IIntroduction

Data\-driven modeling has become increasingly important in control, robotics, and physical\-system identification, particularly when first\-principles models are incomplete or computationally expensive\[[1](https://arxiv.org/html/2610.22575#bib.bib1)\]\. In these applications, learned models should not only be able to predict trajectories but also remain consistent with the physical structure of the underlying system, as this consistency supports trustworthy predictions and reliable use in control of safety\-critical systems\. Physics\-informed machine learning addresses this requirement by incorporating prior knowledge of conservation laws, constitutive relations, and system interconnections into the learning process\[[2](https://arxiv.org/html/2610.22575#bib.bib3)\]\.

Recently, port\-Hamiltonian systems \(PHS\) have been increasingly used as an energy\-based modeling approach for nonlinear system identification\[[3](https://arxiv.org/html/2610.22575#bib.bib5)\]\. They provide a natural framework for structure\-preserving modeling in control by representing dynamics in terms of energy storage, interconnection, dissipation, and external ports\. This framework has motivated several data\-driven modeling approaches, including port\-Hamiltonian neural networks that learn energy functions, dissipation, and external forcing, and operator\-inference methods that identify structure\-preserving reduced\-order operators from data and a prescribed Hamiltonian form\[[4](https://arxiv.org/html/2610.22575#bib.bib6),[5](https://arxiv.org/html/2610.22575#bib.bib7)\]\. Within this broader class of approaches, Gaussian process distributed port\-Hamiltonian systems \(GP\-dPHS\) combine the port\-Hamiltonian representation for partial differential equations \(PDEs\) with nonparametric probabilistic learning, enabling unknown energy\-related quantities to be inferred from data while retaining an interpretable physical structure\[[6](https://arxiv.org/html/2610.22575#bib.bib8),[7](https://arxiv.org/html/2610.22575#bib.bib9)\]\. This Bayesian formulation provides predictive uncertainty estimates, while the distributed port\-Hamiltonian representation preserves the compositional structure of the physical system\.

Existing GP\-dPHS formulations typically employ squared\-exponential \(RBF\) kernels to learn the Hamiltonian or its associated dynamics\[[7](https://arxiv.org/html/2610.22575#bib.bib9)\]\. Although such kernels provide flexible general\-purpose priors, their stationary, distance\-based similarity structure can be restrictive in spatial modeling problems whose dependence patterns vary across the domain\[[8](https://arxiv.org/html/2610.22575#bib.bib12)\]\.

This limitation is particularly relevant for spatially heterogeneous continuum systems, which are ubiquitous in modern engineering and biological applications\. For instance, soft robotic manipulators may employ periodically spaced compliant sections to promote prescribed bending modes\. Additionally, piecewise stiffness profiles naturally emerge in multi\-material composite structures, whileC2C^\{2\}\-smooth profiles provide an idealized representation of smoothly graded materials\[[9](https://arxiv.org/html/2610.22575#bib.bib10)\]and heterogeneous biological tissues\[[10](https://arxiv.org/html/2610.22575#bib.bib11)\]\. In such systems, spatial locations can exhibit fundamentally different or recurring constitutive behavior despite belonging to the same physical domain\. Consequently, an effective learning model should capture these spatial relationships rather than relying solely on geometric proximity to determine similarity\.

The objective of this paper is to improve GP\-dPHS modeling of spatially heterogeneous systems by incorporating available structural knowledge into the covariance design\. In particular, we consider both known physical invariances and coarse information about spatial dependence\. To illustrate the limitations of standard kernel choices, we consider a one\-dimensional string with spatially varying stiffness, constant damping, and localized motor forcing\. While a stationary squared\-exponential kernel provides nearly exact reconstruction for spatially uniform stiffness, its performance deteriorates substantially for heterogeneous profiles\. In contrast, incorporating spatial priors aligned with the qualitative stiffness structure substantially improves reconstruction performance\.

Contribution:This paper proposes a systematic, modular framework for incorporating known physical invariances and qualitative spatial knowledge into GP\-dPHS kernel design\. The framework removes irrelevant state coordinates from individual effort models and decouples the remaining state and spatial dependencies through a tensor\-product covariance\. This separation provides a systematic interface for selecting spatial kernels according to coarse structural information, such as periodicity, finite smoothness, or transitions between distinct spatial regimes, without requiring an analytical description of the underlying heterogeneous profile\. In this way, prior knowledge is encoded directly in the covariance while preserving the underlying port\-Hamiltonian structure\. The approach is demonstrated on a nonlinear string with heterogeneous stiffness, where profile\-matched spatial kernels substantially improve force\-field and trajectory reconstruction over a stationary RBF baseline\.

The remainder of this paper is organized as follows\. Section[II](https://arxiv.org/html/2610.22575#S2)introduces the GP\-dPHS framework and learning problem\. Section[III](https://arxiv.org/html/2610.22575#S3)presents the proposed structure\-aware kernel design, and Section[IV](https://arxiv.org/html/2610.22575#S4)provides its numerical evaluation on a nonlinear string system\.

## IIBackground and Problem Formulation

In this section, we briefly introduce distributed Port\-Hamiltonian systems and Gaussian Process distributed Port\-Hamiltonian systems \(GP\-dPHS\)\. The latter provides a Bayesian framework for learning partially unknown PDE dynamics while preserving the underlying physical structure\.

### II\-ADistributed Port\-Hamiltonian Systems

Distributed Port\-Hamiltonian systems \(dPHS\) provide a systematic framework for representing physical systems governed by partial differential equations\. In contrast to finite\-dimensional Port\-Hamiltonian systems, the interconnection, dissipation, and input operators can contain spatial differential operators\. The dynamics are characterized by a Hamiltonian functional representing the total energy of the distributed system\.

Let111Vectors𝒂\{\\boldsymbol\{a\}\}and vector\-valued functions𝒇⁡\(⋅\)\{\\boldsymbol\{f\}\}\(\\cdot\)are denoted with bold characters\. Matrices are described with capital letters\.InI\_\{n\}is thenn\-dimensional identity matrix and0n0\_\{n\}the zero matrix\. The expressionA:,iA\_\{:,i\}denotes the i\-th column ofAA\. For a positive definite matrixΛ\\Lambda, the associated squared norm is‖𝒙−𝒚‖Λ2=\(𝒙−𝒚\)⊤​Λ​\(𝒙−𝒚\)\\\|\{\\boldsymbol\{x\}\}\-\{\\boldsymbol\{y\}\}\\\|\_\{\\Lambda\}^\{2\}=\(\{\\boldsymbol\{x\}\}\-\{\\boldsymbol\{y\}\}\)^\{\\top\}\\Lambda\(\{\\boldsymbol\{x\}\}\-\{\\boldsymbol\{y\}\}\)\.ℝ\>0\\mathbb\{R\}\_\{\>0\}denotes the set of positive real numbers, whileℝ≥0\\mathbb\{R\}\_\{\\geq 0\}is the set of non\-negative real numbers\.𝒞i\\mathcal\{C\}^\{i\}denotes the class ofiitimes continuously differentiable functions\.𝒵⊂ℝd\\mathcal\{Z\}\\subset\\mathbb\{R\}^\{d\}denote a compact spatial domain, and let𝒙⁡\(t,𝒛\)∈ℝn\{\\boldsymbol\{x\}\}\(t,\{\\boldsymbol\{z\}\}\)\\in\\mathbb\{R\}^\{n\}denote the state at timettand position𝒛\{\\boldsymbol\{z\}\}\. For each fixedtt, the distributed state𝒙⁡\(t,⋅\)\{\\boldsymbol\{x\}\}\(t,\\cdot\)belongs to a suitable space𝒳\\mathcal\{X\}of vector\-valued functions on𝒵\\mathcal\{Z\}\. The Hamiltonian functionalℋ:𝒳→ℝ\\mathcal\{H\}\\colon\\mathcal\{X\}\\to\\mathbb\{R\}is defined as

ℋ⁡\[𝒙⁡\(t,⋅\)\]=∫𝒵H⁡\(𝒛,𝒙⁡\(t,𝒛\)\)​𝑑𝒛,\\mathcal\{H\}\[\{\\boldsymbol\{x\}\}\(t,\\cdot\)\]=\\int\_\{\\mathcal\{Z\}\}H\(\{\\boldsymbol\{z\}\},\{\\boldsymbol\{x\}\}\(t,\{\\boldsymbol\{z\}\}\)\)\\,\\mathrm\{d\}\{\\boldsymbol\{z\}\},\(1\)whereH:𝒵×ℝn→ℝH\\colon\\mathcal\{Z\}\\times\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}denotes the local Hamiltonian density\. The corresponding variational derivative is denoted byδx​ℋ\\delta\_\{x\}\\mathcal\{H\}\. Following the general dPHS formulation, the dynamics can be written as

∂𝒙∂t=\(𝒥−ℛ\)​δx​ℋ\+𝒢​𝒖,\\frac\{\\partial\{\\boldsymbol\{x\}\}\}\{\\partial t\}=\\left\(\\mathcal\{J\}\-\\mathcal\{R\}\\right\)\\delta\_\{x\}\\mathcal\{H\}\+\\mathcal\{G\}\{\\boldsymbol\{u\}\},\(2\)with output𝒚∈ℝm\{\\boldsymbol\{y\}\}\\in\\mathbb\{R\}^\{m\}defined by𝒚=𝒢∗​δx​ℋ\{\\boldsymbol\{y\}\}=\\mathcal\{G\}^\{\*\}\\delta\_\{x\}\\mathcal\{H\}\. Here,𝒥\\mathcal\{J\}is a skew\-adjoint differential operator describing the power\-conserving interconnection structure, with its boundary terms accounting for boundary power exchange\. The dissipation operatorℛ\\mathcal\{R\}is self\-adjoint and positive semidefinite on its specified domain, while𝒢\\mathcal\{G\}describes coupling to the external input𝒖\{\\boldsymbol\{u\}\}\. The operator domains and boundary conditions are chosen consistently with the associated energy balance\.

The Port\-Hamiltonian representation separates the physical interconnection structure from the Hamiltonian\. In particular,𝒥\\mathcal\{J\}andℛ\\mathcal\{R\}determine how energy is exchanged and dissipated, whereasℋ\\mathcal\{H\}characterizes the constitutive behavior of the system\. This separation is particularly useful for learning physical systems since complicated and potentially unknown nonlinearities can be represented through the Hamiltonian while retaining the known physical structure of the PDE\. A more detailed introduction can be found in\[[11](https://arxiv.org/html/2610.22575#bib.bib2)\]\.

### II\-BGaussian Process Distributed Port\-Hamiltonian Systems

For many physical PDE systems, the structure of the differential operators in \([2](https://arxiv.org/html/2610.22575#S2.E2)\) can be derived from first principles, whereas the Hamiltonian functional is only partially known\. This situation typically arises in highly nonlinear systems for which structural knowledge is available, but the exact governing equations are difficult to obtain\. An example is soft robotic systems, where complex and hard\-to\-characterize material behavior can give rise to unknown nonlinear dynamics\. Gaussian Process distributed Port\-Hamiltonian systems \(GP\-dPHS\) address this problem by learning the unknown Hamiltonian from data while preserving the dPHS structure\. We provide a brief introduction here, and more details can be found in\[[7](https://arxiv.org/html/2610.22575#bib.bib9)\]\.

Consider observations of the distributed system at temporal instantsti≥0t\_\{i\}\\geq 0and spatial locations𝒛j∈𝒵\{\\boldsymbol\{z\}\}\_\{j\}\\in\\mathcal\{Z\},

𝒟=\{ti,𝒛j,𝒙⁡\(ti,𝒛j\),𝒖⁡\(ti\)\}i=1,j=1Nt,Nz\.\\mathcal\{D\}=\\left\\\{t\_\{i\},\{\\boldsymbol\{z\}\}\_\{j\},\{\\boldsymbol\{x\}\}\(t\_\{i\},\{\\boldsymbol\{z\}\}\_\{j\}\),\{\\boldsymbol\{u\}\}\(t\_\{i\}\)\\right\\\}\_\{i=1,j=1\}^\{N\_\{t\},N\_\{z\}\}\.\(3\)The observations can first be used to estimate the state derivatives and, if necessary, to obtain a denser spatial representation of the state\. This results in an augmented training dataset containing states and their corresponding time derivatives,ℰ=\{X,X˙\}\\mathcal\{E\}=\\left\\\{X,\\dot\{X\}\\right\\\}, whereXXdenotes the spatially stacked states andX˙\\dot\{X\}their estimated temporal derivatives\.

The central idea of GP\-dPHS is to place a Gaussian Process prior on the unknown Hamiltonian\. Since Gaussian Processes are closed under linear transformations\[[12](https://arxiv.org/html/2610.22575#bib.bib4)\], the differential operator defining the dPHS dynamics can be applied directly to the GP\. Consequently, the dynamics in \([2](https://arxiv.org/html/2610.22575#S2.E2)\) admit the probabilistic representation

∂𝒙∂t∼𝒢​𝒫​\(𝒢​𝒖,kdPHS​\(𝒙,𝒙′\)\)\.\\frac\{\\partial\{\\boldsymbol\{x\}\}\}\{\\partial t\}\\sim\\mathcal\{GP\}\\\!\\big\(\\mathcal\{G\}\{\\boldsymbol\{u\}\},\\,k\_\{\\mathrm\{dPHS\}\}\(\{\\boldsymbol\{x\}\},\{\\boldsymbol\{x\}\}^\{\\prime\}\)\\big\)\.\(4\)Here,kdPHSk\_\{\\mathrm\{dPHS\}\}is a PHS\-constrained covariance function, and the operators may depend on a finite set of unknown physical parametersΘ\\Thetain the general GP\-dPHS formulation\. In the present study, the physical operators and their parameters are treated as known, as specified in Assumption[1](https://arxiv.org/html/2610.22575#Thmassum1)\.

Starting from a squared\-exponential covariance function for the Hamiltonian

kH​\(𝒙,𝒙′\)=σf2​exp⁡\(−‖𝒙−𝒙′‖22​ϕl2\),k\_\{H\}\(\{\\boldsymbol\{x\}\},\{\\boldsymbol\{x\}\}^\{\\prime\}\)=\\sigma\_\{f\}^\{2\}\\exp\\\!\\left\(\-\\frac\{\\\|\{\\boldsymbol\{x\}\}\-\{\\boldsymbol\{x\}\}^\{\\prime\}\\\|^\{2\}\}\{2\\phi\_\{l\}^\{2\}\}\\right\),\(5\)the PHS\-constrained covariance functionkdPHSk\_\{\\mathrm\{dPHS\}\}is obtained by applying the Port\-Hamiltonian operators to the covariance of the Hamiltonian \([5](https://arxiv.org/html/2610.22575#S2.E5)\), resulting in

kdPHS​\(𝒙,𝒙′\)=\(𝒥−ℛ\)​δx​kH​\(𝒙,𝒙′\)​δx′⊤​\(𝒥−ℛ\)⊤\.\\begin\{split\}k\_\{\\mathrm\{dPHS\}\}\(\{\\boldsymbol\{x\}\},\{\\boldsymbol\{x\}\}^\{\\prime\}\)&=\\left\(\\mathcal\{J\}\-\\mathcal\{R\}\\right\)\\delta\_\{x\}k\_\{H\}\(\{\\boldsymbol\{x\}\},\{\\boldsymbol\{x\}\}^\{\\prime\}\)\\delta\_\{x^\{\\prime\}\}^\{\\top\}\\left\(\\mathcal\{J\}\-\\mathcal\{R\}\\right\)^\{\\top\}\.\\end\{split\}\(6\)Thus, the physical structure is incorporated directly into the covariance function rather than imposed after learning\. The potentially unknown kernel hyperparametersϕl\\phi\_\{l\}andσf\\sigma\_\{f\}can be optimized by minimizing the negative log marginal likelihood\.

After conditioning on the observationsℰ\\mathcal\{E\}, the learned dynamics can equivalently be written in distributed Port\-Hamiltonian form as

∂𝒙^∂t=\(𝒥−ℛ\)​δx​ℋ^\+𝒢​𝒖,\\frac\{\\partial\\hat\{\{\\boldsymbol\{x\}\}\}\}\{\\partial t\}=\\left\(\\mathcal\{J\}\-\\mathcal\{R\}\\right\)\\delta\_\{x\}\\widehat\{\\mathcal\{H\}\}\+\\mathcal\{G\}\{\\boldsymbol\{u\}\},\(7\)whereℋ^\\widehat\{\\mathcal\{H\}\}is endowed with the GP posterior distribution\. Hence, the GP\-dPHS does not represent only a single learned vector field, but a distribution over distributed Port\-Hamiltonian models\. An important consequence of this construction is that a sampleℋ^\\widehat\{\\mathcal\{H\}\}from the GP posterior is a deterministic Hamiltonian functional\. Substituting such a sample into \([7](https://arxiv.org/html/2610.22575#S2.E7)\) therefore yields a dPHS with the prescribed interconnection and dissipation structure\. As a result, the sampled models retain the physical energy structure of the underlying dPHS while the GP posterior provides a quantitative description of the remaining model uncertainty\.

### II\-CProblem Setting

We consider the problem of learning partially unknown constitutive relations of a*spatially heterogeneous*PDE system whose dynamics admit the distributed Port\-Hamiltonian representation \([2](https://arxiv.org/html/2610.22575#S2.E2)\) via GP\-dPHS\. Spatial heterogeneity may arise, for example, in soft robotic manipulators with geometric variations or composed of multiple materials with different stiffnesses\. Throughout the paper, we assume that the PDE system \([2](https://arxiv.org/html/2610.22575#S2.E2)\), with prescribed initial and boundary conditions, has a unique solution on the time interval considered, with sufficient temporal and spatial regularity for the derivatives used in the formulation\. The Hamiltonian functionalℋ\\mathcal\{H\}is assumed to be unknown due to, for example, nonlinear stress–strain relations\. We also assumeℋ\\mathcal\{H\}to be infinitely Fréchet differentiable with respect to the state on an appropriate state\-function space\. The objective is to obtain predictionsh^i:ℝn×𝒵→ℝ\\hat\{h\}\_\{i\}\\colon\\mathbb\{R\}^\{n\}\\times\\mathcal\{Z\}\\to\\mathbb\{R\}such that the learned model

∂𝒙^∂t=\(𝒥−ℛ\)​𝒉^​\(𝒙^,𝒛\)\+𝒢​𝒖,𝒉^=\[h^1⋯h^n\]⊤,\\displaystyle\\frac\{\\partial\\hat\{\{\\boldsymbol\{x\}\}\}\}\{\\partial t\}=\\left\(\\mathcal\{J\}\-\\mathcal\{R\}\\right\)\\hat\{\{\\boldsymbol\{h\}\}\}\(\\hat\{\{\\boldsymbol\{x\}\}\},\{\\boldsymbol\{z\}\}\)\+\\mathcal\{G\}\{\\boldsymbol\{u\}\},\\quad\\hat\{\{\\boldsymbol\{h\}\}\}=\\begin\{bmatrix\}\\hat\{h\}\_\{1\}&\\cdots&\\hat\{h\}\_\{n\}\\end\{bmatrix\}^\{\\top\},withh^i≈hi\(𝒙,𝒛\):=δ​ℋδ​xi\(𝒙,𝒛\),i=1,…,n\.\\displaystyle\\text\{with \}\\hat\{h\}\_\{i\}\\approx h\_\{i\}\(\{\\boldsymbol\{x\}\},\{\\boldsymbol\{z\}\}\):=\\frac\{\\delta\\mathcal\{H\}\}\{\\delta x\_\{i\}\}\(\{\\boldsymbol\{x\}\},\{\\boldsymbol\{z\}\}\),\\quad i=1,\\ldots,n\.\(8\)closely approximates the actual system dynamics in \([2](https://arxiv.org/html/2610.22575#S2.E2)\)\. Building on GP\-dPHS in\[[7](https://arxiv.org/html/2610.22575#bib.bib9)\], we especially focus on using coarse prior information about spatially varying constitutive properties to construct structure\-aware covariance functions\.

###### Remark 1

In particular, we do not assume that an analytical description of the spatially varying constitutive law with respect to𝐳\{\\boldsymbol\{z\}\}is available\. Instead, we have access to coarse structural information about its spatial dependence\. For example, this information may indicate that the constitutive properties vary periodically across the spatial domain, possess finite regularity, or transition between distinct spatial regimes as in a soft robot with varying geometry or materials\.

### II\-DAssumptions

To formalize the learning problem, we introduce the following assumptions\.

###### Assumption 1

The differential operators𝒥\\mathcal\{J\},ℛ\\mathcal\{R\}, and𝒢\\mathcal\{G\}defining the dPHS structure \([2](https://arxiv.org/html/2610.22575#S2.E2)\) are known\.

###### Assumption 2

The state of the PDE system \([2](https://arxiv.org/html/2610.22575#S2.E2)\) can be observed at temporal instantstit\_\{i\}and spatial locations𝐳j\{\\boldsymbol\{z\}\}\_\{j\}, yielding the set of observations\{𝐱⁡\(t1,𝐳1\),…,𝐱⁡\(t1,𝐳Nz\),…,𝐱⁡\(tNt,𝐳1\),…,𝐱⁡\(tNt,𝐳Nz\)\}\.\\left\\\{\{\\boldsymbol\{x\}\}\(t\_\{1\},\{\\boldsymbol\{z\}\}\_\{1\}\),\\\!\\ldots\\\!,\{\\boldsymbol\{x\}\}\(t\_\{1\},\{\\boldsymbol\{z\}\}\_\{N\_\{z\}\}\),\\\!\\ldots\\\!,\{\\boldsymbol\{x\}\}\(t\_\{N\_\{t\}\},\{\\boldsymbol\{z\}\}\_\{1\}\),\\ldots,\{\\boldsymbol\{x\}\}\(t\_\{N\_\{t\}\},\{\\boldsymbol\{z\}\}\_\{N\_\{z\}\}\)\\right\\\}\\\!\.

###### Assumption 3

The true Hamiltonian density is separable with respect to the scalar state components, taking the formH⁡\(𝐳,𝐱\)=∑i=1nHi​\(𝐳,xi\)H\(\{\\boldsymbol\{z\}\},\{\\boldsymbol\{x\}\}\)=\\sum\_\{i=1\}^\{n\}H\_\{i\}\(\{\\boldsymbol\{z\}\},x\_\{i\}\)\. Consequently, each effort componenthih\_\{i\}depends only onxix\_\{i\}and𝐳\{\\boldsymbol\{z\}\}and is independent ofxjx\_\{j\}forj≠ij\\neq i\.

###### Assumption 4

Coarse prior information about the spatial dependence of the unknown effort componentshi​\(xi,𝐳\)h\_\{i\}\(x\_\{i\},\{\\boldsymbol\{z\}\}\)is available\.

Assumption[1](https://arxiv.org/html/2610.22575#Thmassum1)is only mildly restrictive since𝒥\\mathcal\{J\}and𝒢\\mathcal\{G\}are typically matrix differential operators whose structure is determined by prior knowledge of the PDE system to be learned\. However, this assumption can be further relaxed by treating unknown physical parameters as hyperparameters during the learning process, see\[[7](https://arxiv.org/html/2610.22575#bib.bib9)\]for details\. Assumption[2](https://arxiv.org/html/2610.22575#Thmassum2)ensures that data can be collected from the PDE system and therefore requires observability of the state\. If the state cannot be directly observed, an appropriate observer must be employed\. Assumption[3](https://arxiv.org/html/2610.22575#Thmassum3)specifies the state invariances used in the covariance design, while Assumption[4](https://arxiv.org/html/2610.22575#Thmassum4)provides qualitative information for selecting the spatial covariance\.

###### Remark 2

Assumption[3](https://arxiv.org/html/2610.22575#Thmassum3)can be weakened by allowing separability between groups of state variables rather than between individual scalar components\. Let\{Ib\}b=1B\\\{I\_\{b\}\\\}\_\{b=1\}^\{B\}be a partition of the state indices and consider a Hamiltonian density of the formH⁡\(𝐳,𝐱\)=∑b=1BHb​\(𝐳,𝐱Ib\)H\(\{\\boldsymbol\{z\}\},\{\\boldsymbol\{x\}\}\)=\\sum\_\{b=1\}^\{B\}H\_\{b\}\(\{\\boldsymbol\{z\}\},\{\\boldsymbol\{x\}\}\_\{I\_\{b\}\}\)\. Fori∈Ibi\\in I\_\{b\}, the effort componenthih\_\{i\}may depend on all state variables in𝐱Ib\{\\boldsymbol\{x\}\}\_\{I\_\{b\}\}but remains independent of variables outside that group\. For example, a kinetic–potential decompositionH⁡\(𝐳,𝐩,𝐪\)=T⁡\(𝐳,𝐩\)\+V⁡\(𝐳,𝐪\)H\(\{\\boldsymbol\{z\}\},\{\\boldsymbol\{p\}\},\{\\boldsymbol\{q\}\}\)=T\(\{\\boldsymbol\{z\}\},\{\\boldsymbol\{p\}\}\)\+V\(\{\\boldsymbol\{z\}\},\{\\boldsymbol\{q\}\}\)preserves independence between momentum and configuration variables without requiring separability within either group\. However, independence between groups does not guarantee integrability among independently learned effort components within the same group and additional compatibility constraints are required to ensure that\.

## IIIStructured Kernel Design for GP\-dPHS

Before we present the proposed structure\-aware kernel in Subsection[III\-B](https://arxiv.org/html/2610.22575#S3.SS2), we first show the trivial extension of the GP\-dPHS model and the resulting deficits\.

### III\-ATrivial Extension

For a system with heterogeneous spatial structure, a straightforward modeling approach is to extend the input space of the GP\-dPHS kernel \([6](https://arxiv.org/html/2610.22575#S2.E6)\) to include the spatial variable𝒛\{\\boldsymbol\{z\}\}associated with that structure\. Thus, the input is extended to𝝃=\[𝒙⊤,𝒛⊤\]⊤\\boldsymbol\{\\xi\}=\[\{\\boldsymbol\{x\}\}^\{\\top\},\{\\boldsymbol\{z\}\}^\{\\top\}\]^\{\\top\}and used in the squared\-exponential Hamiltonian covariance in \([5](https://arxiv.org/html/2610.22575#S2.E5)\) as the augmented input space\. This covariance is assigned directly to each independently trained effort model\. By assigning a separate lengthscale to each coordinate, the trivial extended PHS kernel is given by

ktriv\(𝝃,𝝃′\)=σf2exp\[−12∑j=1n\+d\(ξj−ξj′\)2ℓj2\]\.\\displaystyle k\_\{\\mathrm\{triv\}\}\(\\boldsymbol\{\\xi\},\\boldsymbol\{\\xi\}^\{\\prime\}\)=\\sigma\_\{f\}^\{2\}\\exp\\left\[\-\\frac\{1\}\{2\}\\sum\_\{j=1\}^\{n\+d\}\\frac\{\(\\xi\_\{j\}\-\\xi\_\{j\}^\{\\prime\}\)^\{2\}\}\{\\ell\_\{j\}^\{2\}\}\\right\]\.\(9\)The coordinate\-specific lengthscalesℓj\\ell\_\{j\}control covariance decay along individual input dimensions and provide the automatic relevance determination \(ARD\) mechanism\[[13](https://arxiv.org/html/2610.22575#bib.bib13)\]\. Large lengthscales reduce sensitivity to variation in the corresponding coordinates, allowing the model to learn approximate input irrelevance\.

This trivial extension would learn each effort componenthih\_\{i\}independently using the complete input vector𝝃\\boldsymbol\{\\xi\}, without explicitly imposing known invariances or distinguishing the roles of state and spatial variables\. In addition, despite the ARD coordinate\-dependent scaling, the covariance remains stationary because it depends only on the input difference𝝃−𝝃′\\boldsymbol\{\\xi\}\-\\boldsymbol\{\\xi\}^\{\\prime\}, see\[[14](https://arxiv.org/html/2610.22575#bib.bib14), Sec\. 2\.1\]\. Consequently, the same distance\-based similarity rule is applied throughout the input domain\. Even though this kernel is universal on compact input domains, meaning that continuous effort functions can be approximated arbitrarily closely by functions from its reproducing kernel Hilbert space \(RKHS\)\[[15](https://arxiv.org/html/2610.22575#bib.bib15)\], the approximation property does not ensure efficient learning from limited data\. A generic ARD\-RBF covariance does not directly encode known invariances, recurring structure, or localized transitions between regimes\. For example, observations separated by one period along a spatial coordinate may represent equivalent physical conditions, while their RBF covariance decreases with separation\. In conclusion, the trivial extensionktrivk\_\{\\mathrm\{triv\}\}in \([9](https://arxiv.org/html/2610.22575#S3.E9)\) must therefore infer these relationships from observations, which can be inefficient, especially when training data are sparse or unevenly distributed\.

### III\-BStructure\-Aware Kernel

The trivial extension in the previous subsection treats all state and spatial coordinates uniformly, although prior knowledge may already specify which state variables are relevant to a particular effort component and provide qualitative information about its spatial dependence\. Under Assumptions[1](https://arxiv.org/html/2610.22575#Thmassum1)–[4](https://arxiv.org/html/2610.22575#Thmassum4), we exploit these two sources of prior information directly in the covariance design\. In particular, for theii\-th effort componenthih\_\{i\}, Assumption[3](https://arxiv.org/html/2610.22575#Thmassum3)implies thathih\_\{i\}depends only on the associated state componentxix\_\{i\}and the spatial coordinate𝒛\{\\boldsymbol\{z\}\}\. We therefore remove the irrelevant state coordinatesxjx\_\{j\},j≠ij\\neq i, from the GP input and separate the remaining state and spatial dependencies through the product kernel

ki​\(\(xi,𝒛\),\(xi′,𝒛′\)\)=kxi​\(xi,xi′\)​kz​\(𝒛,𝒛′\),k\_\{i\}\\big\(\(x\_\{i\},\{\\boldsymbol\{z\}\}\),\(x\_\{i\}^\{\\prime\},\{\\boldsymbol\{z\}\}^\{\\prime\}\)\\big\)=k\_\{x\_\{i\}\}\(x\_\{i\},x\_\{i\}^\{\\prime\}\)\\,k\_\{z\}\(\{\\boldsymbol\{z\}\},\{\\boldsymbol\{z\}\}^\{\\prime\}\),\(10\)wherekxik\_\{x\_\{i\}\}models similarity with respect to the relevant state component andkzk\_\{z\}models similarity with respect to the spatial coordinate\. The latter can be chosen according to the coarse structural information available from Assumption[4](https://arxiv.org/html/2610.22575#Thmassum4), without requiring an analytical description of the spatially varying constitutive law\.

###### Proposition 1

Letkxik\_\{x\_\{i\}\}andkzk\_\{z\}be positive semidefinite kernels on𝒳i⊆ℝ\\mathcal\{X\}\_\{i\}\\subseteq\\mathbb\{R\}and𝒵\\mathcal\{Z\}, respectively, with corresponding reproducing kernel Hilbert spacesℋxi\\mathcal\{H\}\_\{x\_\{i\}\}andℋz\\mathcal\{H\}\_\{z\}\. Then the covariance function in \([10](https://arxiv.org/html/2610.22575#S3.E10)\) is a positive semidefinite kernel on𝒳i×𝒵\\mathcal\{X\}\_\{i\}\\times\\mathcal\{Z\}and its associated RKHS is the Hilbert tensor product

ℋi=ℋxi⊗ℋz\.\\mathcal\{H\}\_\{i\}=\\mathcal\{H\}\_\{x\_\{i\}\}\\otimes\\mathcal\{H\}\_\{z\}\.\(11\)In particular,ℋi\\mathcal\{H\}\_\{i\}is the completion of finite sums of the form

f⁡\(xi,𝒛\)=∑r=1Rφr​\(xi\)​ψr​\(𝒛\),φr∈ℋxi,ψr∈ℋz\.f\(x\_\{i\},\{\\boldsymbol\{z\}\}\)=\\sum\_\{r=1\}^\{R\}\\varphi\_\{r\}\(x\_\{i\}\)\\psi\_\{r\}\(\{\\boldsymbol\{z\}\}\),\\quad\\varphi\_\{r\}\\in\\mathcal\{H\}\_\{x\_\{i\}\},\\psi\_\{r\}\\in\\mathcal\{H\}\_\{z\}\.\(12\)Consequently, the product covariance does not restrict the learned effort to a single separable function of the formf⁡\(xi\)​ψ​\(𝐳\)f\(x\_\{i\}\)\\psi\(\{\\boldsymbol\{z\}\}\)\.

###### Proof:

Consider arbitrary points\{\(xi\(j\),𝒛\(j\)\)\}j=1N\\\{\(x\_\{i\}^\{\(j\)\},\{\\boldsymbol\{z\}\}^\{\(j\)\}\)\\\}\_\{j=1\}^\{N\}\. LetKxK\_\{x\}andKzK\_\{z\}denote the Gram matrices generated bykxik\_\{x\_\{i\}\}andkzk\_\{z\}, respectively\. Since both kernels are positive semidefinite,Kx⪰0K\_\{x\}\\succeq 0andKz⪰0K\_\{z\}\\succeq 0\. The Gram matrix associated with \([10](https://arxiv.org/html/2610.22575#S3.E10)\) is given by

Ki=Kx∘Kz,K\_\{i\}=K\_\{x\}\\circ K\_\{z\},\(13\)where∘\\circdenotes the Hadamard product\. By the Schur product theorem,Ki⪰0K\_\{i\}\\succeq 0, and hencekik\_\{i\}is a valid positive semidefinite kernel\.

For product kernels, the corresponding RKHS is the Hilbert tensor product of the RKHSs associated with the individual factors,

ℋi=ℋxi⊗ℋz,\\mathcal\{H\}\_\{i\}=\\mathcal\{H\}\_\{x\_\{i\}\}\\otimes\\mathcal\{H\}\_\{z\},\(14\)see\[[16](https://arxiv.org/html/2610.22575#bib.bib16), Theorem 3\.4\]for details\. By definition, this space is obtained as the completion of finite linear combinations of elementary tensor\-product functionsφr​\(xi\)​ψr​\(𝒛\)\\varphi\_\{r\}\(x\_\{i\}\)\\psi\_\{r\}\(\{\\boldsymbol\{z\}\}\), which gives \([12](https://arxiv.org/html/2610.22575#S3.E12)\)\. Thus, although the covariance itself factorizes into a state and a spatial kernel, functions represented by its RKHS may contain general finite sums of interactions between state and spatial coordinates and are not restricted to a single separable product\. ∎

###### Remark 3\(Architectural modularity\)

An ARD\-RBF kernel on the reduced input\(xi,z\)\(x\_\{i\},z\)already admits an algebraic factorization into an RBF kernel onxix\_\{i\}and an RBF kernel on𝐳\{\\boldsymbol\{z\}\}, up to the overall output scale\. Hence, writing the covariance explicitly in the product form \([10](https://arxiv.org/html/2610.22575#S3.E10)\) does not by itself enlarge the model class when both factors are RBF kernels\. Its benefit is instead*architectural modularity*\. The state covariancekxik\_\{x\_\{i\}\}and spatial covariancekzk\_\{z\}become independent design components, so thatkzk\_\{z\}can be replaced by a kernel reflecting known heterogeneous structure without modifying the state\-dependent part of the model\. This separation therefore provides a systematic interface for incorporating coarse spatial prior knowledge while leaving the analytical form of the unknown constitutive relation to be learned from data\.

### III\-CSpatial Kernel Choices

A compatible spatial kernel can be selected using qualitative structural information inferred from preliminary observations, without requiring a detailed reconstruction of the underlying spatial profile\. For example, a periodic kernel can encode recurring spatial patterns, a Matérn\-5/25/2kernel can represent finitely smooth spatial variation, and a changepoint kernel can blend distinct covariance models across a transition between spatial regimes\. Parameters such as the fundamental period, changepoint location, and transition scale may be initialized from coarse estimates and subsequently optimized during GP training\. The string benchmark in Section[IV](https://arxiv.org/html/2610.22575#S4)applies these kernel choices to spatially varying stiffness\. Furthermore, we observe that the learned changepoint parameters and predictive performance in that benchmark remain insensitive to initialization errors\.

## IVNumerical Evaluation

We evaluate the proposed structure\-aware kernel design using a nonlinear string with heterogeneous stiffness\. We first describe the physical system and data\-generation procedure, followed by model training, trajectory reconstruction, and the numerical results\.

### IV\-AString Model

We consider a one\-dimensional string of lengthL=10L=10, with transverse displacementu⁡\(z,t\)u\(z,t\)at positionz∈\[0,L\]z\\in\[0,L\]and timet≥0t\\geq 0\. Under the unit\-density assumption, the momentum density coincides with the transverse velocity,p⁡\(z,t\)=∂u/∂tp\(z,t\)=\\partial u/\\partial t, while the strain isq⁡\(z,t\)=∂u/∂zq\(z,t\)=\\partial u/\\partial z\. With the state ordering𝒙=\[q,p\]⊤\{\\boldsymbol\{x\}\}=\[q,p\]^\{\\top\}, we consider the ground\-truth Hamiltonian functional given by

ℋ⁡\[𝒙\]=∫0LH⁡\(z,𝒙\)​𝑑z,\\displaystyle\\mathcal\{H\}\[\{\\boldsymbol\{x\}\}\]=\\int\_\{0\}^\{L\}H\(z,\{\\boldsymbol\{x\}\}\)\\,\\mathrm\{d\}z,withH⁡\(z,𝒙\)=12​p2\+12​k​\(z\)​q2\+12​q4,\\displaystyle\\text\{with\}\\quad H\(z,\{\\boldsymbol\{x\}\}\)=\\frac\{1\}\{2\}p^\{2\}\+\\frac\{1\}\{2\}k\(z\)q^\{2\}\+\\frac\{1\}\{2\}q^\{4\},\(15\)wherek⁡\(z\)k\(z\)is the spatially varying stiffness and the quartic term introduces nonlinear strain stiffening\. The analytical Hamiltonian generating reference data is*unknown*to the GP models, which learn its effort components independently\. The corresponding effort functions areh1:=δ​ℋ/δ​q=∂H/∂q=k⁡\(z\)​q\+2​q3h\_\{1\}:=\\delta\\mathcal\{H\}/\\delta q=\\partial H/\\partial q=k\(z\)q\+2q^\{3\}andh2:=δ​ℋ/δ​p=∂H/∂p=ph\_\{2\}:=\\delta\\mathcal\{H\}/\\delta p=\\partial H/\\partial p=p\. Thus, the dynamics take the distributed port\-Hamiltonian form

\[q˙p˙\]=\[0∂z∂z−γ\]​\[h1h2\]\+\[0Fext​\(z,t\)\],\\begin\{bmatrix\}\\dot\{q\}\\\\\[2\.84526pt\] \\dot\{p\}\\end\{bmatrix\}=\\begin\{bmatrix\}0&\\partial\_\{z\}\\\\\[2\.84526pt\] \\partial\_\{z\}&\-\\gamma\\end\{bmatrix\}\\begin\{bmatrix\}h\_\{1\}\\\\\[2\.84526pt\] h\_\{2\}\\end\{bmatrix\}\+\\begin\{bmatrix\}0\\\\\[2\.84526pt\] F\_\{\\mathrm\{ext\}\}\(z,t\)\\end\{bmatrix\},\(16\)whereγ=0\.05\\gamma=0\.05is the internal damping coefficient\. The external input is a localized periodic motor pulse applied near the center of the string withFext​\(z,t\)=exp⁡\[−10​\(z−L/2\)2\]​sin⁡\(2​π​t\)F\_\{\\mathrm\{ext\}\}\(z,t\)=\\exp\[\-10\(z\-L/2\)^\{2\}\]\\sin\(2\\pi t\)\. The initial conditions arep⁡\(z,0\)=0p\(z,0\)=0and

q⁡\(z,0\)=−4​\(z−zc\)​exp⁡\[−\(z−zc\)2\],zc=L/2\.\\displaystyle q\(z,0\)=\-4\(z\-z\_\{c\}\)\\exp\\\!\\left\[\-\(z\-z\_\{c\}\)^\{2\}\\right\],\\;z\_\{c\}=L/2\.\(17\)The boundary conditions are given by the homogeneous Dirichlet conditionsp⁡\(0,t\)=p⁡\(L,t\)=0p\(0,t\)=p\(L,t\)=0for allt≥0t\\geq 0, whileqqis left unconstrained at the boundary\.

Motivated by the applications discussed in Section[I](https://arxiv.org/html/2610.22575#S1), we consider three heterogeneous stiffness profiles: periodic,C2C^\{2\}bumps, and semi\-piecewise\. Figure[1](https://arxiv.org/html/2610.22575#S4.F1)illustrates the three heterogeneous profiles\. The transition in the semi\-piecewise profile isC2C^\{2\}smooth rather than strictly discontinuous, so that the spatial derivatives remain well\-defined across the interface for finite\-difference simulation and trajectory reconstruction\. A constant\-stiffness experiment provides a spatially homogeneous reference case\.

Fig\. 1:Representative spatial stiffness profiles used in this study: periodic,C2C^\{2\}bumps, and semi\-piecewise stiffness \(top to bottom, respectively\)\.
### IV\-BSimulation and Data Generation

For the generation of training data, the dynamics \([16](https://arxiv.org/html/2610.22575#S4.E16)\) are discretized on a uniform spatial grid withNz=100N\_\{z\}=100points\. LetDzD\_\{z\}denote the discrete spatial derivative operator on the uniform grid, withfi=f⁡\(zi\)f\_\{i\}=f\(z\_\{i\}\)\. At interior grid points, it uses the second\-order centered\-difference approximation

\[∂zf\]zi≈\[Dzf\]i:=fi\+1−fi−12​Δ​z,i=2,…,Nz−1\.\[\\partial\_\{z\}f\]\_\{z\_\{i\}\}\\\!\\approx\\\!\[D\_\{z\}f\]\_\{i\}:=\\frac\{f\_\{i\+1\}\-f\_\{i\-1\}\}\{2\\Delta z\},\\,i=2,\\ldots,N\_\{z\}\-1\.\(18\)At the endpoints,DzD\_\{z\}uses first\-order one\-sided differences\[Dz​f\]1:=f2−f1Δ​z\[D\_\{z\}f\]\_\{1\}:=\\frac\{f\_\{2\}\-f\_\{1\}\}\{\\Delta z\}and\[Dz​f\]Nz:=fNz−fNz−1Δ​z\.\[D\_\{z\}f\]\_\{N\_\{z\}\}:=\\frac\{f\_\{N\_\{z\}\}\-f\_\{N\_\{z\}\-1\}\}\{\\Delta z\}\.The endpoint momentum values are maintained at zero throughout integration, while no boundary values are prescribed for the strain\. The resulting system of ordinary differential equations is integrated using the adaptive Runge–Kutta method RK45, and the numerical solution is recorded at uniformly spaced output timesΔ​t=0\.05​s\\Delta t=0\.05\\,\\mathrm\{s\}for a simulation time ofT=5\.5​sT=5\.5\\,\\mathrm\{s\}\.

Each training sample consists of the augmented input𝝃i​j=\[p⁡\(zi,tj\),q⁡\(zi,tj\),zi\]⊤\\boldsymbol\{\\xi\}\_\{ij\}=\[p\(z\_\{i\},t\_\{j\}\),q\(z\_\{i\},t\_\{j\}\),z\_\{i\}\]^\{\\top\}paired with the corresponding values222This setup isolates the effect of kernel design from errors associated with reconstructing effort targets from trajectory observations\. In practice, the values ofh1h\_\{1\}andh2h\_\{2\}would be reconstructed from measurements ofq˙\\dot\{q\}andp˙\\dot\{p\}as described in\[[7](https://arxiv.org/html/2610.22575#bib.bib9)\]\.ofh1h\_\{1\}andh2h\_\{2\}\. From the complete space–time dataset, we select 2% of the samples for training\. Spatially stratified sampling allocates 25%, 50%, and 25% of the training budget to the left\-end regionz∈\[0,2\]z\\in\[0,2\], interior regionz∈\[2,8\]z\\in\[2,8\], and right\-end regionz∈\[8,10\]z\\in\[8,10\], respectively\. Within each stratum, samples are selected uniformly at random without replacement\. For each heterogeneous stiffness profile, the same fraction of the available space–time samples is selected\.

### IV\-CExperimental Design and Model Configurations

The experiments distinguish two contributions of the proposed methodology: imposing known state invariances and incorporating qualitative information about the spatial stiffness profile\. We use two independently trained scalar\-output GP models,h^1≈δ​ℋ/δ​q\\hat\{h\}\_\{1\}\\approx\\delta\\mathcal\{H\}/\\delta qandh^2≈δ​ℋ/δ​p\\hat\{h\}\_\{2\}\\approx\\delta\\mathcal\{H\}/\\delta p, for the constitutive and momentum efforts, respectively\. For each stiffness profile and sampling seed, all three configurations use the same training observations\.

The Hamiltonian in \([15](https://arxiv.org/html/2610.22575#S4.Ex2)\) implies thath1h\_\{1\}is independent of momentum andh2h\_\{2\}is independent of strain\. In the main comparisons, only the constitutive\-effort modelh^1\\hat\{h\}\_\{1\}is modified\. The momentum\-effort modelh^2\\hat\{h\}\_\{2\}retains the baseline formulation throughout, providing a controlled comparison and reflecting the comparatively simple relationh2=ph\_\{2\}=p\. We consider the following settings:

*Baseline:*Bothh^1\\hat\{h\}\_\{1\}andh^2\\hat\{h\}\_\{2\}use the complete input\(p,q,z\)\(p,q,z\)and an ARD\-RBF covariance\. This configuration does not explicitly impose either state invariance and follows the trivial extension as presented in Section[III\-A](https://arxiv.org/html/2610.22575#S3.SS1)\.

*SD\-RBF:*The constitutive\-effort modelh^1\\hat\{h\}\_\{1\}uses the reduced input\(q,z\)\(q,z\)and the product covariancekqRBF​\(q,q′\)​kzRBF​\(z,z′\)k\_\{q\}^\{\\mathrm\{RBF\}\}\(q,q^\{\\prime\}\)k\_\{z\}^\{\\mathrm\{RBF\}\}\(z,z^\{\\prime\}\)\. Removingppimposes the known momentum independence ofh1h\_\{1\}, while retaining its dependence on strain and spatial position\. As discussed in Section[III\-B](https://arxiv.org/html/2610.22575#S3.SS2), this product is equivalent to an ARD\-RBF covariance on\(q,z\)\(q,z\)\. The comparison with the baseline therefore isolates the effect of excluding the irrelevant momentum input\.

*SD\-matched \(proposed method\):*The effort modelh^1\\hat\{h\}\_\{1\}retains the input\(q,z\)\(q,z\)and the RBF strain factor, but uses the covariancekqRBF​\(q,q′\)​kzmatched​\(z,z′\)k\_\{q\}^\{\\mathrm\{RBF\}\}\(q,q^\{\\prime\}\)k\_\{z\}^\{\\mathrm\{matched\}\}\(z,z^\{\\prime\}\)\. Comparison with SD\-RBF therefore isolates the additional effect of selecting the spatial covariance according to the qualitative stiffness structure\. SD\-matched denotes SD\-Per\. with a periodic spatial kernel for periodic stiffness, SD\-Mat\. with a Matérn\-5/25/2spatial kernel for theC2C^\{2\}\-bump profile, and SD\-CP with a changepoint spatial kernel for the semi\-piecewise profile\.

The changepoint kernel blends two regional covariance functions using a smooth gatekzCP​\(z,z′\)=s¯​\(z\)​s¯​\(z′\)​kL​\(z,z′\)\+s⁡\(z\)​s​\(z′\)​kR​\(z,z′\),k\_\{z\}^\{\\mathrm\{CP\}\}\(z,z^\{\\prime\}\)=\\bar\{s\}\(z\)\\bar\{s\}\(z^\{\\prime\}\)k\_\{\\mathrm\{L\}\}\(z,z^\{\\prime\}\)\+s\(z\)s\(z^\{\\prime\}\)k\_\{\\mathrm\{R\}\}\(z,z^\{\\prime\}\),wheres\(z\)=\[1\+exp\(−\(z−z0\)/w\)\]−1s\(z\)=\[1\+\\exp\(\-\(z\-z\_\{0\}\)/w\)\]^\{\-1\}ands¯​\(z\)=1−s​\(z\)\\bar\{s\}\(z\)=1\-s\(z\)\. The kernelskLk\_\{\\mathrm\{L\}\}andkRk\_\{\\mathrm\{R\}\}are separately parameterized RBF covariances for the two spatial regimes\. The trainable parametersz0z\_\{0\}andw\>0w\>0specify the transition location and scale in the standardized spatial coordinate\. This construction represents a smooth transition between distinct covariance structures without prescribing the stiffness function itself\.

###### Remark 4

To satisfy Assumption[3](https://arxiv.org/html/2610.22575#Thmassum3), the momentum\-effort modelh^2\\hat\{h\}\_\{2\}should depend only on\(p,z\)\(p,z\), excludingqq\. However, our numerical experiments indicate that removingqqfromh^2\\hat\{h\}\_\{2\}has no noticeable effect on performance\.

### IV\-DTraining and Reconstruction

Both effort models are implemented as exact GPs and trained in full\-batch mode by minimizing the negative exact marginal log likelihood with Adam\. Kernel hyperparameters use the default GPyTorch initialization unless otherwise specified\. Training uses double precision and a maximum of10001000iterations\. Early stopping is applied, and the checkpoint attaining the lowest training loss is restored\. Table[I](https://arxiv.org/html/2610.22575#S4.T1)lists the learning rateη\\eta, early\-stopping patiencePP, and minimum\-improvement thresholdΔmin\\Delta\_\{\\min\}\. The settings are shared across kernel architectures within each stiffness profile\. The constitutive\-effort models for theC2C^\{2\}\-bump and semi\-piecewise profiles use smaller learning rates, longer patience, and smaller improvement thresholds to accommodate their convergence behavior\. The momentum\-effort model uses the same settings across all profiles\.

TABLE I:Optimization settings for the effort models, shared across kernel architectures\.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\.During evaluation, the trained GPs predict the effort functions at the current integrated state\. The baseline models useh^1​\(p,q,z\)\\hat\{h\}\_\{1\}\(p,q,z\)andh^2​\(p,q,z\)\\hat\{h\}\_\{2\}\(p,q,z\)\. Structural decoupling usesh^1​\(q,z\)\\hat\{h\}\_\{1\}\(q,z\)andh^2​\(p,q,z\)\\hat\{h\}\_\{2\}\(p,q,z\)\. After transformation back into physical units, the predicted efforts are substituted into the discretized port\-Hamiltonian dynamics at interior grid points

q˙i=\[Dz​h^2\]i,p˙i=\[Dz​h^1\]i−γ​h^2,i\+Fext​\(zi,t\)\.\\dot\{q\}\_\{i\}=\[D\_\{z\}\\hat\{h\}\_\{2\}\]\_\{i\},\\qquad\\dot\{p\}\_\{i\}=\[D\_\{z\}\\hat\{h\}\_\{1\}\]\_\{i\}\-\\gamma\\hat\{h\}\_\{2,i\}\+F\_\{\\mathrm\{ext\}\}\(z\_\{i\},t\)\.\(19\)Here,DzD\_\{z\}is the discrete spatial derivative operator defined in Section[IV\-B](https://arxiv.org/html/2610.22575#S4.SS2), using the same interior and endpoint stencils as the reference simulation\. The reconstruction enforcespGP​\(0,t\)=pGP​\(L,t\)=0p\_\{\\mathrm\{GP\}\}\(0,t\)=p\_\{\\mathrm\{GP\}\}\(L,t\)=0\. The resulting system of ODEs is integrated using the same RK45 method as the reference simulation, producingpGP​\(z,t\)p\_\{\\mathrm\{GP\}\}\(z,t\)andqGP​\(z,t\)q\_\{\\mathrm\{GP\}\}\(z,t\)\. Each simulation uses the same initial conditions, boundary conditions, external forcing, and evaluation grid as the reference trajectory from which training observations are sampled\. This evaluates trajectory reconstruction from sparse observations of that trajectory rather than generalization to new initial conditions or forcing\. For both reference and predicted trajectories, displacement is reconstructed from strain by numerical integration over the spatial domain𝒵\\mathcal\{Z\}\.

### IV\-EResults

Table[II](https://arxiv.org/html/2610.22575#S4.T2)summarizes the numerical results\. The relative errorsεq\\varepsilon\_\{q\},εu\\varepsilon\_\{u\}, andεE\\varepsilon\_\{E\}quantify the overall reconstruction accuracy for strain, displacement, and physical energy, respectively, over the complete trajectory, while the curl RMSE measures violations of the mixed\-partial integrability condition between the learned effort functions\. The effort RMSE evaluates the local accuracy of the learned constitutive efforth^1\\hat\{h\}\_\{1\}, whereas the force RMSE measures the error after applying the spatial derivativeDz​h^1D\_\{z\}\\hat\{h\}\_\{1\}, which directly affects the reconstructed dynamics\. As an additional baseline, we consider a constant\-stiffness case representing a spatially homogeneous system\. As expected, the GP\-dPHS model with standard RBF kernel performs well in this setting; however, its performance deteriorates substantially for heterogeneous profiles\.

Comparing RBF with SD\-RBF examines the effect of removingppfrom the input toh^1\\hat\{h\}\_\{1\}while retaining the RBF kernel family\. SD\-RBF reduces strain, displacement, and energy errors for all three heterogeneous profiles, while also improving effort and force RMSE\. The displacement error decreases by approximately43\.6%43\.6\\%,39\.6%39\.6\\%, and17\.1%17\.1\\%for the periodic,C2C^\{2\}\-bump, and semi\-piecewise profiles, respectively\. The trajectory reconstruction improvements are therefore more pronounced for the periodic andC2C^\{2\}\-bump cases, with smaller gains for the semi\-piecewise profile\. Despite these improvements, substantial reconstruction errors remain, indicating that removing the irrelevant input alone does not adequately capture the heterogeneous structure\.

Structural decoupling also reduces the curl RMSE by one to two orders of magnitude for the stiffness profiles considered in this work\. Sinceh^2\\hat\{h\}\_\{2\}is independent ofpp, the cross\-derivative∂h^1/∂p\\partial\\hat\{h\}\_\{1\}/\\partial pvanishes identically\. The remaining term∂h^2/∂q\\partial\\hat\{h\}\_\{2\}/\\partial qis not constrained to vanish, so these configurations reduce the integrability residual without enforcing exact integrability\. Within each profile, SD\-RBF and SD\-matched have the same reported curl RMSE because they share the momentum\-effort model and both excludeppfrom the constitutive\-effort model\.

Using prior information on the stiffness profile as we propose with the structure\-aware kernel \([10](https://arxiv.org/html/2610.22575#S3.E10)\) produces substantially greater improvements\. SD\-Per\., SD\-Mat\., and SD\-CP achieve displacement errors of2\.06×10−22\.06\\times 10^\{\-2\},1\.03×10−11\.03\\times 10^\{\-1\}, and6\.90×10−26\.90\\times 10^\{\-2\}, respectively\. These values correspond to reductions of approximately98\.3%98\.3\\%,91\.3%91\.3\\%, and93\.2%93\.2\\%relative to the baseline, and96\.9%96\.9\\%,85\.6%85\.6\\%, and91\.8%91\.8\\%relative to SD\-RBF\. The matched models also attain the lowest strain and energy errors and the lowest effort and force RMSE among the tested configurations for each heterogeneous profile\. Their additional reconstruction improvements occur without a further reduction in curl RMSE, highlighting the contribution of the spatial covariance beyond the imposed momentum invariance\. Examples of the predicted trajectories are shown in Figure[2](https://arxiv.org/html/2610.22575#S4.F2), where our structure\-aware design \(SD\-matched\) most closely reproduces the actual trajectory\.

Fig\. 2:Average reconstructed displacement trajectories for the periodic,C2C^\{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\.These experiments demonstrate the benefit of our approach: exploiting known input invariances provides consistent improvements over the baseline, while incorporating profile\-matched spatial priors yields further substantial gains\. In particular, the best performance is achieved by combining structural decoupling with a spatial covariance adapted to the stiffness profile\.

## VConclusion

This work introduced a structure\-aware kernel design framework for GP\-dPHS of spatially heterogeneous systems, combining known input invariances with a modular separation of state and spatial dependencies\. This enables coarse knowledge of spatial heterogeneity to be incorporated directly into the covariance without requiring an analytical description of the underlying profile\. Numerical experiments demonstrate that structural decoupling consistently improves upon the standard baseline, while substantially larger gains are achieved when the spatial covariance is matched to the qualitative stiffness structure\. These results highlight structure\-aware kernel design as an effective mechanism for exploiting spatial prior knowledge in GP\-dPHS, with extensions to higher\-dimensional and experimental systems providing promising directions for future work\.

## References

- \[1\]\(2022\)Data\-driven control: overview and perspectives\.In2022 American Control Conference \(ACC\),Vol\.,pp\. 1048–1064\.External Links:[Document](https://dx.doi.org/10.23919/ACC53348.2022.9867266)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p1.1)\.
- \[2\]M\. Raissi, P\. Perdikaris, and G\.E\. Karniadakis\(2019\)Physics\-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations\.Journal of Computational Physics378,pp\. 686–707\.External Links:ISSN 0021\-9991,[Document](https://dx.doi.org/https%3A//doi.org/10.1016/j.jcp.2018.10.045),[Link](https://www.sciencedirect.com/science/article/pii/S0021999118307125)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p1.1)\.
- \[3\]A\. van der Schaft and D\. Jeltsema\(2014\)Port\-hamiltonian systems theory: an introductory overview\.Foundations and Trends in Systems and Control1\(2\-3\),pp\. 173–378\.External Links:ISSN 2325\-6818,[Document](https://dx.doi.org/10.1561/2600000002),[Link](https://doi.org/10.1561/2600000002),https://www\.emerald\.com/ftsys/article\-pdf/1/2\-3/173/11146089/2600000002en\.pdfCited by:[§I](https://arxiv.org/html/2610.22575#S1.p2.1)\.
- \[4\]S\. A\. Desai, M\. Mattheakis, D\. Sondak, P\. Protopapas, and S\. J\. Roberts\(2021\)Port\-hamiltonian neural networks for learning explicit time\-dependent dynamical systems\.Phys\. Rev\. E104,pp\. 034312\.External Links:[Document](https://dx.doi.org/10.1103/PhysRevE.104.034312),[Link](https://link.aps.org/doi/10.1103/PhysRevE.104.034312)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p2.1)\.
- \[5\]Y\. Geng, L\. Ju, B\. Kramer, and Z\. Wang\(2025\)Data\-driven reduced\-order models for port\-hamiltonian systems with operator inference\.Computer Methods in Applied Mechanics and Engineering442,pp\. 118042\.External Links:ISSN 0045\-7825,[Document](https://dx.doi.org/https%3A//doi.org/10.1016/j.cma.2025.118042),[Link](https://www.sciencedirect.com/science/article/pii/S0045782525003147)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p2.1)\.
- \[6\]T\. Beckers, J\. Seidman, P\. Perdikaris, and G\. J\. Pappas\(2022\)Gaussian process port\-hamiltonian systems: bayesian learning with physics prior\.In2022 IEEE 61st Conference on Decision and Control \(CDC\),Vol\.,pp\. 1447–1453\.External Links:[Document](https://dx.doi.org/10.1109/CDC51059.2022.9992733)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p2.1)\.
- \[7\]K\. Tan, P\. Li, and T\. Beckers\(2024\)Physics\-constrained learning of PDE systems with uncertainty quantified port\-Hamiltonian models\.InProceedings of the 6th Annual Learning for Dynamics &; Control Conference,A\. Abate, M\. Cannon, K\. Margellos, and A\. Papachristodoulou \(Eds\.\),Proceedings of Machine Learning Research, Vol\.242,pp\. 1753–1764\.External Links:[Link](https://proceedings.mlr.press/v242/tan24a.html)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p2.1),[§I](https://arxiv.org/html/2610.22575#S1.p3.1),[§II\-B](https://arxiv.org/html/2610.22575#S2.SS2.p1.1),[§II\-C](https://arxiv.org/html/2610.22575#S2.SS3.p1.2),[§II\-D](https://arxiv.org/html/2610.22575#S2.SS4.p2.1),[footnote 2](https://arxiv.org/html/2610.22575#footnote2)\.
- \[8\]J\. Ton, S\. Flaxman, D\. Sejdinovic, and S\. Bhatt\(2018\)Spatial mapping with gaussian processes and nonstationary fourier features\.Spatial Statistics28,pp\. 59–78\.Note:One world, one healthExternal Links:ISSN 2211\-6753,[Document](https://dx.doi.org/https%3A//doi.org/10.1016/j.spasta.2018.02.002),[Link](https://www.sciencedirect.com/science/article/pii/S2211675317302890)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p3.1)\.
- \[9\]S\. Chi and Y\. Chung\(2006\)Mechanical behavior of functionally graded material plates under transverse load—part i: analysis\.International Journal of Solids and Structures43\(13\),pp\. 3657–3674\.External Links:ISSN 0020\-7683,[Document](https://dx.doi.org/https%3A//doi.org/10.1016/j.ijsolstr.2005.04.011),[Link](https://www.sciencedirect.com/science/article/pii/S0020768305002052)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p4.1)\.
- \[10\]F\. Serwane, A\. Mongera, P\. Rowghanian, D\. A\. Kealhofer, A\. A\. Lucio, Z\. M\. Hockenbery, and O\. Campàs\(2017\)In vivo quantification of spatially varying mechanical properties in developing tissues\.Nature Methods14\(2\),pp\. 181–186\.External Links:ISSN 1548\-7105,[Document](https://dx.doi.org/10.1038/nmeth.4101),[Link](https://doi.org/10.1038/nmeth.4101)Cited by:[§I](https://arxiv.org/html/2610.22575#S1.p4.1)\.
- \[11\]R\. Rashad, F\. Califano, A\. J\. van der Schaft, and S\. Stramigioli\(2020\)Twenty years of distributed port\-hamiltonian systems: a literature review\.IMA Journal of Mathematical Control and Information37\(4\),pp\. 1400–1422\.Cited by:[§II\-A](https://arxiv.org/html/2610.22575#S2.SS1.p3.1)\.
- \[12\]C\. Jidling, N\. Wahlström, A\. Wills, and T\. B\. Schön\(2017\)Linearly constrained Gaussian processes\.Advances in neural information processing systems30\.Cited by:[§II\-B](https://arxiv.org/html/2610.22575#S2.SS2.p3.1)\.
- \[13\]C\. Williams and C\. Rasmussen\(1995\)Gaussian processes for regression\.InAdvances in Neural Information Processing Systems,D\. Touretzky, M\.C\. Mozer, and M\. Hasselmo \(Eds\.\),Vol\.8,pp\.\.External Links:[Link](https://proceedings.neurips.cc/paper/_files/paper/1995/file/7cce53cf90577442771720a370c3c723-Paper.pdf)Cited by:[§III\-A](https://arxiv.org/html/2610.22575#S3.SS1.p1.2)\.
- \[14\]J\. Snoek, K\. Swersky, R\. Zemel, and R\. Adams\(2014\)Input warping for bayesian optimization of non\-stationary functions\.InProceedings of the 31st International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.32,Beijing, China,pp\. 1674–1682\.External Links:[Link](https://proceedings.mlr.press/v32/snoek14.html)Cited by:[§III\-A](https://arxiv.org/html/2610.22575#S3.SS1.p2.1)\.
- \[15\]C\. A\. Micchelli, Y\. Xu, and H\. Zhang\(2006\)Universal kernels\.Journal of Machine Learning Research7\(95\),pp\. 2651–2667\.External Links:[Link](http://jmlr.org/papers/v7/micchelli06a.html)Cited by:[§III\-A](https://arxiv.org/html/2610.22575#S3.SS1.p2.1)\.
- \[16\]K\. Albrecht, J\. Entzian, and A\. Iske\(2025\)Product kernels are efficient and flexible tools for high\-dimensional scattered data interpolation\.Adv\. Comput\. Math\.51\(2\)\.External Links:ISSN 1019\-7168,[Link](https://doi.org/10.1007/s10444-025-10226-y),[Document](https://dx.doi.org/10.1007/s10444-025-10226-y)Cited by:[§III\-B](https://arxiv.org/html/2610.22575#S3.SS2.p3.2.1)\.