# Enhancing Parametric Model Embedding with Physical Information for Effective Vehicle Design

Andrea Serani, Giorgio Palma, Matteo Diez, Jeroen Wackers, Domenico

Quagliarella, Stefano Gaggero

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Andrea Serani, Giorgio Palma, Matteo Diez, Jeroen Wackers, Domenico Quagliarella, et al. Enhancing Parametric Model Embedding with Physical Information for Effective Vehicle Design. Research Specialists' Meeting AVT-411 on Machine Learning and Artificial Intelligence for Military Vehicle Design, May 2025, Washington, United States. hal-05350445

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Distributed under a Creative Commons Attribution 4.0 International License
# Enhancing Parametric Model Embedding with Physical Information for Effective Vehicle Design

**Andrea Serani, Giorgio Palma, Matteo Diez**

National Research Council
Institute of Marine Engineering
ITALY

**Jeroen Wackers**

Ecole Centrale de Nantes/CNRS
FRANCE

**Domenico Quagliarella**

Italian Aerospace Research Centre
ITALY

**Stefano Gaggero**

University of Genoa
ITALY

## ABSTRACT

In this work, an extension of the parametric model embedding (PME) approach is presented, aiming to achieve more comprehensive and efficient design-space dimensionality reduction for vehicle design. PME, rooted in principal component analysis (PCA), not only identifies a smaller set of critical modes but also re-parameterizes the original variables—enabling direct and interpretable manipulations of shape modifications within the reduced space. Alongside the “physics-informed” version (PI-PME), which enriches geometry with low-fidelity distributed and lumped physical quantities, a “physics-driven” variant (PD-PME) is introduced that focuses exclusively on physical parameters. Both formulations employ PCA to capture the principal modes of variability yet differ in their balance between geometric and physical information. Through test cases involving the RAE-2822 airfoil, a bio-inspired underwater glider, a naval propeller, and the DTMB-5415 destroyer-type vessel, it is shown how the resulting frameworks provide a first-level assessment of design variability, offer interpretability regarding which original variables most strongly affect performance, and efficiently bridge geometric and physical considerations. Furthermore, lumped physical parameters can serve as a low-fidelity foundation for multi-fidelity optimization, directly leveraging the linear re-parameterization to drive the reduced design variables. Meanwhile, distributed physical parameters enable the construction of machine-learning-based reduced-order models to infer integral quantities of interest. By allowing the user to embed these insights early in the design process, PI-PME and PD-PME facilitate more robust, cost-effective exploration, paving the way for subsequent high-fidelity optimization and future goal-oriented formulations.

## 1 INTRODUCTION

Vehicle design in air, sea, and ground domains presents a multifaceted challenge, characterized by numerous geometric, functional, and performance constraints. Modern design processes must address a broad spectrum of requirements—ranging from energy efficiency and cost-effectiveness to safety and environmental
<figure><img src="image_2.png" /></figure>

# ENHANCING PARAMETRIC MODEL EMBEDDING WITH PHYSICAL INFORMATION

sustainability—while simultaneously accounting for aerodynamics, hydrodynamics, structural integrity, noise emissions, and regulatory standards. As the complexity of each vehicle concept increases, so does the dimensionality of the design space, often leading to an exponential growth in the number of parameters that define the shape and operating conditions. This phenomenon, commonly referred to as the *curse of dimensionality* [1], complicates the tasks of exploration, analysis, and optimization, since an extremely large solution space quickly becomes prohibitive to sample thoroughly.

To address these challenges, dimensionality reduction methods [2] have gained significant attention as tools to simplify high-dimensional design spaces without unduly compromising predictive accuracy. Among these methods, parametric model embedding (PME) [3] has emerged as a promising approach for shape parameterization and re-parametrization [4]. Rooted in principal component analysis (PCA), PME reduces the dimensionality of geometric data, capturing only the essential features while preserving key parametric relationships among design variables. In practical terms, this allows the original shape and its variations to be represented in a lower-dimensional latent space, which can be explored more efficiently and at lower computational cost. PME, however, focuses primarily on geometric variations, potentially overlooking critical physical phenomena—such as lift, drag, and viscous effects—that are integral to the performance of vehicles operating in fluid environments.

To overcome this limitation, an extended framework called physics-informed parametric model embedding (PI-PME) is here introduced. This approach enriches the data matrix with physical information, including lumped parameters (e.g., force coefficients or efficiencies) or distributed quantities (e.g., surface pressure fields), derived from low-fidelity simulations. Although this enrichment entails an upfront computational cost, the resulting latent space captures both geometric and physical variability, enabling a more informed and reliable assessment of performance trade-offs. By integrating physical insights at the earliest stages of design exploration, PI-PME provides a foundation that can guide subsequent high-fidelity simulations and optimizations.

Building on the same principle, a physics-driven parametric model embedding (PD-PME) is also proposed, which goes one step further by focusing solely on physical parameters and effectively excluding geometric variance. While this approach removes explicit shape information from the dimensionality reduction, it can be advantageous in scenarios where the primary objective is to characterize or optimize physical behaviors, and where the link between geometry and physics is well captured through simplified models. This exclusive emphasis on physics can also serve as a natural starting point for multi-fidelity optimization or for data-driven reduced-order modeling, where lumped or distributed physical data are used to predict key performance indicators at reduced computational expense.

The effectiveness of PI-PME and PD-PME becomes particularly apparent in real-world applications such as bio-inspired underwater gliders, naval propellers, or classical airfoils (e.g., the RAE-2822). In these cases, fluid-dynamic effects strongly impact performance, and omitting them from the dimensionality reduction step may lead to suboptimal designs and increased resource expenditure. By embedding physical information—whether partially, as in PI-PME, or exclusively, as in PD-PME—into the reduced space, it becomes possible to filter the vast design domain more intelligently, ensuring that only the most relevant configurations are explored. This synergy between geometric and physical factors not only improves the quality of early design-stage decisions but also streamlines the path to final optimization, cutting down both computational time and overall development costs. It may be noted that PME's flexible structure also makes it directly applicable to structural optimization. Instead of relying on aerodynamic or hydrodynamic simulations, one may incorporate structural properties (e.g., mass moments of inertia, stiffness distributions, or stress fields) into
the PME framework, enabling integrated assessments of mechanical performance [5, 6].

The following sections illustrate the mathematical underpinnings of PME, PI-PME, and PD-PME, including detailed discussions of how PCA is adapted to incorporate physical data, and how these frameworks can be applied to a range of high-dimensional vehicle design problems. Through examples, the benefits of early-stage integration of physical insights will be demonstrated, showing improved design robustness, a decrease in exhaustive computational campaigns, and the potential for more agile innovation cycles in modern vehicle development.

## 2 DESIGN-SPACE DIMENSIONALITY REDUCTION METHODS

In shape optimization problems aimed at minimizing a function $f(\mathbf{u})$, epistemic uncertainty manifests as limited prior knowledge about where the global optimum lies within the design space. While it is assumed that such an optimum exists, its position is not known a priori, thus requiring an extensive exploration of the domain. To formalize this, the design variable vector $\mathbf{u}$ may be considered as belonging to a stochastic domain $\mathcal{U}$, with an associated probability density function $p(\mathbf{u})$ that encodes the incomplete understanding of the feasible region. This uncertainty propagates through the modeling process, turning both geometric and physical parameters into random fields.

Dimensionality reduction methods—such as proper orthogonal decomposition, Karhunen-Loève expansion [7], singular Value Decomposition, and PCA—offer a systematic way to handle this high-dimensional, uncertain landscape. By isolating the principal modes of variation, these techniques distill large datasets and complex models into reduced representations that capture the key features most relevant to the optimization. In so doing, they mitigate the computational burden of exploring vast design spaces while preserving critical information about geometry and performance. As will be discussed in the following sections, integrating physical insights into these dimensionality reduction frameworks, for design-space assessment and subsequent shape optimization, provides a more robust path toward identifying optimal solutions under uncertainty.

### 2.1 Parametric Model Embedding

PME [3] is a design-space dimensionality reduction method that extends the standard PCA approach [7–9] by incorporating both shape deformations and design variables into a generalized feature space [2]. Specifically, PME applies PCA to an augmented matrix that includes the discretized shape deformation vector $\mathbf{d}$ and the original design variables $\mathbf{u}$. This extension allows PME to directly map the reduced design space back to the original design variables without the need for reparameterization, which is typically required in standard PCA approaches. As a result, PME offers a more robust and practical method for maintaining the integrity of the original design features while facilitating effective shape optimization.

Consider a manifold $\mathcal{G}$, which identifies the original shape/geometry. This manifold defines the geometric space in which the shape is parameterized by curvilinear coordinates $\xi \in \mathcal{G}$. The coordinates of the original shape are represented by $g(\xi) \in \mathbb{R}^n$ with $n = 1, 2$, or 3. Assume that, for the purpose of shape optimization, $g$ can be transformed to a deformed shape/geometry $g'(\xi, \mathbf{u})$ by

$$g'(\xi, \mathbf{u}) = g(\xi) + \delta(\xi, \mathbf{u}) \quad \forall \xi \in \mathcal{G} \qquad (14-1)$$
where $\delta(\xi, \mathbf{u}) \in \mathbb{R}^n$ is the resulting shape modification vector, defined by arbitrary shape parameterization or modification method (e.g., CAD parameterization, Bezier surfaces, free-form deformation, NURBS, etc.), and $\mathbf{u} \in \mathcal{U} \subset \mathbb{R}^M$ is the design-variable vector of dimension $M$.

<figure><img src="image_3.png" /><figcaption>Figure 14-1: Example of shape modification and discretization on a bi-dimensional geometry with notation, where $n = 2$ and $L = 4$.</figcaption></figure>

Discretizing $\mathcal{G}$ by $L$ elements (see, e.g., Fig. 14-1) of measure $\Delta\mathcal{G}_j$ (with $j = 1, \dots, L$), having $\mathbf{d}(\xi, \mathbf{u})$ as the discretization of $\delta(\xi, \mathbf{u})$, sampling $\mathcal{U}$ by a statistically convergent number of realizations $S$, so that $\{\mathbf{u}_k\}_{k=1}^S \sim p(\mathbf{u})$, and organizing $\hat{\mathbf{d}} = \mathbf{d} - \langle\mathbf{d}\rangle$ (with $\langle\cdot\rangle$ the mean value) in a data matrix $\mathbf{D}$ of dimensionality $[nL \times S]$

$$ \mathbf{D} = \begin{bmatrix} \hat{\mathbf{d}}_1^{(1)} & \dots & \hat{\mathbf{d}}_1^{(S)} \\ \vdots & \vdots & \vdots \\ \hat{\mathbf{d}}_n^{(1)} & \dots & \hat{\mathbf{d}}_n^{(S)} \end{bmatrix} = \begin{bmatrix} \hat{d}_{1,\xi_1}(\mathbf{u}_1) & \dots & \hat{d}_{1,\xi_1}(\mathbf{u}_S) \\ \vdots & \vdots & \vdots \\ \hat{d}_{L,\xi_1}(\mathbf{u}_1) & \dots & \hat{d}_{L,\xi_1}(\mathbf{u}_S) \\ \vdots & \vdots & \vdots \\ \hat{d}_{1,\xi_n}(\mathbf{u}_1) & \dots & \hat{d}_{1,\xi_n}(\mathbf{u}_S) \\ \vdots & \vdots & \vdots \\ \hat{d}_{L,\xi_n}(\mathbf{u}_1) & \dots & \hat{d}_{L,\xi_n}(\mathbf{u}_S) \end{bmatrix} \quad (14-2) $$

where $\hat{d}_{j,\xi_k}$ is the $k$-th component of the shape modification vector associated to the $j$-th element, defining $\hat{\mathbf{u}} = \mathbf{u} - \langle\mathbf{u}\rangle$, the embedding is achieved defining the matrix $\mathbf{P}$ of dimensionality $[(nL + M) \times S]$ as follows

$$ \mathbf{P} = \begin{bmatrix} \mathbf{D} \\ \mathbf{U} \end{bmatrix} \quad \text{with} \quad \mathbf{U} = \begin{bmatrix} \hat{\mathbf{u}}^{(1)} & \dots & \hat{\mathbf{u}}^{(1)} \end{bmatrix} = \begin{bmatrix} \hat{u}_{1,1} & \dots & \hat{u}_{1,S} \\ \vdots & \vdots & \vdots \\ \hat{u}_{M,1} & \dots & \hat{u}_{M,S} \end{bmatrix} \quad (14-3) $$

where the matrix $\mathbf{U}$ is added to the data matrix $\mathbf{D}$ to which is associated a null weight $\mathbf{W}_u$ such that

$$ \mathbf{W}_u = \mathbf{0} \quad \text{and} \quad \mathbf{W} = \begin{bmatrix} \mathbf{W}_d & \mathbf{0} \\ \mathbf{0} & \mathbf{W}_u \end{bmatrix} \quad (14-4) $$
and so leading to a generalized PCA problem in the form

$$ \mathbf{AGWZ} = \mathbf{Z}\mathbf{\Lambda} \quad \text{with} \quad \mathbf{A} = \frac{1}{S}\mathbf{P}\mathbf{P}^{\top} \qquad (14-5) $$

where

$$ \mathbf{G} = \begin{bmatrix} \mathbf{G}_d & \mathbf{0} \\ \mathbf{0} & \mathbf{I} \end{bmatrix} \quad \text{and} \quad \mathbf{Z} = [\mathbf{z}_1 \dots \mathbf{z}_S] \quad \text{with} \quad \mathbf{z}_k = \begin{bmatrix} \mathbf{e}_k \\ \mathbf{v}_k \end{bmatrix} \qquad (14-6) $$

Here, $\mathbf{e}_k$ and $\mathbf{v}_k$ represent the eigenvector components associated to the shape modification d and design variable u vectors, respectively. The matrix $\mathbf{G}_d = \text{diag}(\mathbf{G}_1, \dots, \mathbf{G}_n)$ is block diagonal and has dimensionality $[nL \times nL]$, with each $[L \times L]$ k-th block being a diagonal matrix itself

$$ \mathbf{G}_k = \text{diag}(\Delta\mathcal{G}_1, \dots, \Delta\mathcal{G}_L) \qquad (14-7) $$

containing the measure $\Delta\mathcal{G}_j$ of the j-th element. Similarly, $\mathbf{W}_d = \text{diag}(\mathbf{W}_1, \dots, \mathbf{W}_n)$ is a block diagonal matrix of dimensionality $[nL \times nL]$, where each $[L \times L]$ k-th block $\mathbf{W}_k$ ($k = 1, \dots, n$) is itself a diagonal matrix defined as

$$ \mathbf{W}_k = \text{diag}(\rho_1, \dots, \rho_L) \qquad (14-8) $$

where $\rho_j$ (for $j = 1, \dots, L$) represents the arbitrary weight given to each element.

The solutions $\lambda_k$ and $\mathbf{z}_k$ of Equation (14-5) are finally used to construct the reduced dimensionality representation of the original parameterization; defining the desired confidence level $l$, with $0 < l \le 1$, the number of reduced design variables $N$ is chosen such that

$$ \sum_{k=1}^{N} \lambda_k \ge l \sum_{k=1}^{nL} \lambda_k = l\sigma^2 \quad \text{with} \quad \lambda_k \ge \lambda_{k+1} \qquad (14-9) $$

and the PME of the original design variables is finally achieved by using the eigenvector components $\mathbf{v}_k$ that embed (or contain) the reduced-order representation of the original design parameterization u as follows

$$ \mathbf{u} \approx \check{\mathbf{u}} = \langle \mathbf{u} \rangle + \sum_{k=1}^{N} x_k \mathbf{v}_k \qquad (14-10) $$

To reconstruct at least all the samples in D, the coefficients $\theta_j$, for $j = 1, \dots, S$, are evaluated projecting the matrix P on Z', that contains only the first N eigenvectors of Z, retaining the desired level of variance of the original design space, as follows

$$ \mathbf{\Theta} = \mathbf{P}^{\top}\mathbf{G}\mathbf{W}\mathbf{Z}' \qquad (14-11) $$

with $\mathbf{\Theta} = [\theta_1 \dots \theta_S]^T$. Consequently, the reduced design variables $\mathbf{x} = [x_1 \dots x_N]^T$ can be bounded such as

$$ \min_j \Theta_{jk} \le x_k \le \max_j \Theta_{jk} \quad k = 1, \dots, N. \qquad (14-12) $$

It may be noted that the overall methodology is independent of the specific shape modification method, which is seen as a black box by PME.
## 2.2 Physics-Informed Parametric Model Embedding

Introducing physical information within PME formulations translates into the definition of, similarly to Eq. 14-3, a new matrix $P_I$ as

$$ P_I = \begin{bmatrix} D \\ U \\ F \\ C \end{bmatrix} \quad \text{with} \quad F = \begin{bmatrix} \hat{f}_j^{(1)} & \cdots & \hat{f}_j^{(S)} \\ \vdots & \ddots & \vdots \\ \vdots & \ddots & \vdots \end{bmatrix} \quad \text{and} \quad C = \begin{bmatrix} \hat{c}_j^{(1)} & \cdots & \hat{c}_j^{(S)} \\ \vdots & \ddots & \vdots \\ \vdots & \ddots & \vdots \end{bmatrix} \quad (14-13) $$

where $F$ is a matrix containing distributed physical information (with $\hat{f}_j = f_j - \langle f_j \rangle$), such as, e.g., pressure distribution and/or velocity components, that cannot necessarily have to be defined on the geometry surface but can also belong to the field surrounding the object, like, e.g., wake and vortices, whereas $C$ contains lumped or scalar quantities of interest (with $\hat{c}_j = c_j - \langle c_j \rangle$), such as, e.g., lift and drag forces. It may be noted that the same physical information, can be collected for more than one operating condition.

Analogously, a corresponding block-diagonal weight matrix $W_I$ is introduced:

$$ W_I = \begin{bmatrix} W_d & 0 & 0 & 0 \\ 0 & W_u & 0 & 0 \\ 0 & 0 & W_f & 0 \\ 0 & 0 & 0 & W_c \end{bmatrix} = \mathrm{diag}(W_d, \underbrace{0}_{u}, W_f, W_c), \quad (14-14) $$

where $W_d$ is the block-diagonal matrix weighting the geometric entries (as in standard PME), $W_u$ the one for design variables, $W_f$ for distributed physical data $F$, and $W_c$ for lumped scalars $C$. Each block has to be set up to normalize its respective data by the inverse of the estimated variance. In practice, for a block associated with a data vector $y \in \{d, f, c\}$, one calculates

$$ \mathrm{Var}(y) = \sum_{j=1}^{S} (y_j - \langle y \rangle)^2 / S, \quad (14-15) $$

and then defines

$$ \rho_y = 1/\mathrm{Var}(y). \quad (14-16) $$

If the block contains multiple components (e.g., multi-dimensional fields or multiple operating conditions), each row or column of that block can be assigned its own weight $\rho_i$. Doing so ensures that geometric modifications and physical data all contribute comparably to the subsequent PCA procedure—i.e., no portion of the data dominates simply because it exhibits a larger raw variance.

Once the augmented data matrix $P_I$ and the block-diagonal weighting matrix $W_I$ are assembled, the eigenvalue problem remains analogous to the original PME formulation:

$$ A_I G_I W_I Z_I = Z_I \Lambda_I \quad (14-17) $$

where

$$ A_I = \frac{1}{S} P_I P_I^T, \quad G_I = \begin{bmatrix} G_d & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & 0 & G_f & 0 \\ 0 & 0 & 0 & I \end{bmatrix}, \quad Z_I = [z_{I,1} \dots z_{I,S}], \quad z_{I,k} = \begin{bmatrix} e_k \\ v_k \\ \phi_k \\ \pi_k \end{bmatrix}_I \quad (14-18) $$
<figure><img src="image_4.png" /></figure>

Here, $G_f$ accounts for the element size of the distributed physical vector, which does not necessarily correspond to $G_d$, while the identity block is applied also to the lumped scalars. The eigenvector solution $Z_I$ then provides the reduced representation for all data (where $\phi_k$ and $\pi_k$ are the eigenvector components that embed the distributed and lumped physical parameters), ensuring that the new PI-PME basis incorporates both shape and physics with consistent normalization. This yields a lower-dimensional yet physics-enriched space suitable for subsequent design-space exploration, optimization, or multi-fidelity modeling.

## 2.3 Physics-Driven Parametric Model Embedding

In some design scenarios, large geometric changes may not necessarily translate into significant variations in the associated physical phenomena. Consequently, the standard geometry-centric approach in dimensionality reduction can yield subspaces that capture a high geometric variance but do not correlate with improved physical performance. PD-PME addresses this issue by removing geometric deformations (D) altogether, focusing only on the physical data. Such a formulation aims to isolate and amplify the directions in the design space that have the strongest impact on physically relevant quantities, rather than those with merely large geometric variability.

By excluding D, the augmented data matrix reduces to

$$P_D = \begin{bmatrix} U \\ F \\ C \end{bmatrix}.$$

Because the objective is to let the physical variability guide the principal components, one assembles a block-diagonal matrix

$$W_D = \text{diag}(\underbrace{0}_{u}, W_f, W_c),$$

which assigns, as per standard PME and PI-PME, zero weight to the rows corresponding to U, while normalizing the distributed physical fields F and the lumped scalars C by the inverse of their respective variances, as per PI-PME (see, eqs. 14-15 and 14-16).

The solution of the generalized PCA problem

$$A_D G_D W_D Z_D = Z_D A_D, \quad \text{where} \quad A_D = \frac{1}{S} P_D P_D^T,$$

yields principal components strictly driven by physics, since the geometry has been removed and design variables are weighted to zero.

A relevant consideration arises when using only lumped (scalar) physical data. If the number of such scalars is smaller than the number of design variables, the rank of $P_D$ will be limited by the fewer of the two, resulting in a forced dimensionality reduction dictated by the size of the scalar data block rather than by a variance-based choice. Consequently, a minimum requirement for PD-PME to fully capture variability in the design vector is that the dimensionality of the physical information (e.g., the number of scalar data points) be at least equal to or larger than the number of design variables. Otherwise, the principal components will be constrained by the lower-dimensional block and may fail to represent all relevant modes of variation.
# 3 TEST CASES

To highlight the effectiveness and flexibility of the proposed approaches (PME, PI-PME, and PD-PME), four different test cases are considered. Each case targets a distinct geometry and set of operating conditions, thus allowing for a comprehensive assessment of how geometric and physical information can be combined (or, in the case of the physics-driven variant, used exclusively) to enable efficient design-space dimensionality reduction. The following subsections detail each test case in turn, describing both the parameterization strategies and the sources of physical data employed for PI-PME and PD-PME.

## 3.1 Test Case 1: RAE-2822 Airfoil

The first test case concerns the design optimization of the classical RAE-2822 airfoil. The design space, defined within the activities of the NATO-AVT-331 [10], includes $M = 20$ design variables; each variable is associated with a different shape function (3 polynomials, 6 Hicks-Henne bumps, and 1 Wagner function [11], see Fig. 14-2), acting either on the upper or lower surface of the airfoil [12]. An in-house code (WG2AER, developed at CIRA) parameterizes the airfoil as a linear combination of the parent geometry $g(\xi)$ and the modification functions $\delta$.

<figure><img src="image_5.png" /><figcaption>Figure 14-2: Test case 1: RAE-2822 shape modification functions.</figcaption></figure>

Discretizing the airfoil with $L = 129$ grid points, the physical information required to train PI-PME and PD-PME is gathered from the XFOIL solver [13], under operating conditions of Mach = 0.4, Reynolds = $6.5 \times 10^6$, and a zero-degree angle of attack. Collected physical data include the pressure coefficient (as distributed information) and the lift, drag, and pitching moment coefficients (as lumped parameters).
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 14-3: Test case 2: AUG shape parameterization.</figcaption></figure>

## 3.2 Test Case 2: Autonomous Underwater Glider

The second test case addresses the design optimization of a bio-inspired autonomous underwater glider (AUG) with a manta-like shape, that has been selected as a test case for the activity of the NATO-AVT-404 on “Enhanced Design Processes of Military Vehicles through Machine Learning Methods”. The geometry is constructed as a continuous wing using a section-wise scheme and is by design spanwise symmetric; only half of the body is parameterized and then mirrored at the root section. As illustrated in Fig. 14-3, the model comprises three zones: the center body, the transition region, and the outer wing. Four sections define the half-span: (i) the root (section 1), (ii) the end of the main body (section 2), (iii) the end of the transition (section 3), and (iv) the tip (section 4). Each section is fully determined by ten variables: four parameters for the section’s NACA 4-digit airfoil (maximum camber *m*, maximum camber position *p*, thickness ratio *t*, chord *c*); three parameters for positioning the section’s leading edge (*x*₀, *z*₀, *s*₀); three parameters defining the rotation angles (pitch *ϑ*, roll *φ*, yaw *ψ*), applied sequentially about the leading edge. Section 1 retains only 2 degrees of freedom (*t* and *c*), while all other parameters are fixed. Consequently, the complete parametric model includes *M* = 32 design variables. The manta-like geometry is constructed via the OpenCASCADE CAD kernel [14] and meshed with Gmsh v4.12.1 [15].

The glider surface mesh consists of *L* = 784 elements, and the physical data (pressure coefficients, lift, and drag) are acquired via the PUFFIn solver [16], developed by ENSTA Bretagne, which combines an incompressible potential approach with viscous corrections. The operating conditions correspond to a uniform inflow velocity of 0.25 m/s in seawater at a depth of 1500 m, and an 8-degree angle of attack that yields near-optimal efficiency for the baseline configuration.

## 3.3 Test Case 3: Ship Propeller

Test case 3 considers the design optimization of the six-blade, right-handed, ship propeller of a cruise ship shown in Fig. 14-4. The reference propeller was designed for a nominal advance coefficient of about *J* = 0.87 and a cavitation index *σ*<sub>n</sub> (based on the rate of revolution) of 2.25. In this functioning condition, the delivered thrust corresponds to a thrust coefficient *K*<sub>T</sub> = 0.186. The parametric description of the propeller blade consists of B-Spline curves that describe the geometrical features, in radial and chordwise directions,
ENHANCING PARAMETRIC MODEL EMBEDDING WITH PHYSICAL INFORMATION

of the blade. This approach, extensively validated in several design-by-optimization cases [17, 18], provides a
robust and easily controllable representation of the blade geometry in terms of its fairness. Moreover, it allows for
the independent choice of quantities, like maximum sectional thickness, using criteria other than hydrodynamic
shape optimization (in this case, structural strength of the blade) which instead are hardly controllable when
using free-form deformations or B-Surfaces describing directly the suction and pressure side of the blade.

For this particular problem, the blade geometry is parametrized through B-Spline control polygons defining
radial distributions of non-dimensional chord (c/D, parameter name c), pitch (P/D, parameter name pd),
rake (rake/D, parameter name r), and maximum sectional camber (fmax/c, parameter name fh) of Fig. ??a.
The shape of sectional hydrofoils is included as well in the parametric model by describing, again by using
appropriate control polygons (see Fig. 14-5b), the non-dimensional chordwise sectional thickness (t/2tmax,
parameter name th) and camber (f/fmax, parameter name fh) distributions of the blade profile which, radial
section per radial section, are finally scaled with the corresponding maximum values. Given some constraints
(the radial position of control points at root and tip is fixed, leading- and trailing-edge points are always
given, and the chord at the tip never changes to comply with some limits of the flow solver adopted for the
hydrodynamic characterization of performances), control points, sequentially numbered from root to tip (or
from leading- to trailing-edge), are free to move within assigned ranges. Subscript “x” indicates the radial, or
chordwise, modification of the control point position, and subscript “y” refers instead to changes in the quantity
the control point describes. This description leads to a total of M = 38 design variables (5 for the chord, 8
for the maximum sectional camber and pitch each, 6 for the rake, 6 for the sectional camber line, and 5 for
the sectional thickness non-dimensional distribution) through which, at unchanged skew distribution (that of the
reference propeller), reconstruct any blade surface for the hydrodynamic analyses. To this aim, a low-fidelity
boundary element method (BEM) is employed. The BEM code [19] was developed at the University of Genoa
since early 2000 for analysis and design-by-optimization purposes. It is a Morino, Dirichlet-type boundary
condition implementation of panel methods for incompressible, potential flow solution, which makes use of the
“key blade” approach to deal with stationary or unsteady problems [20]. It includes a cavitation model (sheet
cavity model at leading edge and midchord, suction and pressure side [21]), wake alignment capabilities and the
iterative Kutta condition. For current analyses, the “key blade” is discretized with 1250 hyperboloidal panels
(Ł= 1326 nodes). Given the reference propeller design, which was developed to operate in the disturbed wake
of the ship, equivalent steady calculations were repeated (and data collected) for several functioning conditions
to have a broader view of the propeller behavior. Nominal operating conditions (i.e. propeller operating in the

<figure><img src="image_8.png" /><figcaption>Figure 14-4: Test case 3: geometry of the reference ship propeller; “Key-blade” with the surface panel discretization used for BEM calculations of performances.</figcaption></figure>
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>(a) Radial distributions (chord, pitch, rake, and maximum sectional camber, from left to right)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(b) Sectional distributions (camber line and chordwise thickness, left and right)</figcaption></figure>

<caption>**Figure 14-5: Test case 3: distributions of the geometrical features of the propeller.**</caption>

circumferentially averaged inflow axial wake) plus loaded and unloaded functioning under the opposite action of tangential velocities occurring at 90 and 270 deg. Blade positions into the ship wake were considered to collect distributed pressure data as well as lumped performance indicators such as thrust coefficient $K_T$, efficiency $\eta$, and tip-vortex intensity $\Gamma$.

## 3.4 Test Case 4: 5415 Destroyer-Type Vessel

The fourth test case targets the international benchmark 5415 hull [22], which is a geosymmetric replica of the DDG-51 (a US Navy destroyer-type vessel). Shape parameterization is based on a recursive set of $M = 27$ global modification functions applied within a hyper-rectangular region enclosing the demi-hull [23, 24], see, e.g., Fig. 14-6.

The surface mesh of the hull comprises $L = 2250$ grid nodes. Physical data include calm-water and seakeeping performance, extracted from the in-house linear potential-flow solver WARP [26] (developed by CNR-INM).
<figure><img src="image_12.png" /><figcaption>Figure 14-6: Test case 4: Example of shape modifications for the 5415 hull [25].</figcaption></figure>

with viscous corrections, alongside a linearized strip-theory code SMP [27]. Calm-water analyses provide the distribution of surface pressure and the wave-resistance coefficient; seakeeping analyses deliver estimates of the pitch motion (θ) RMS and the vertical acceleration (a<sub>B</sub>) RMS at the bridge. These integrated and distributed quantities collectively serve as physical inputs for PI-PME and PD-PME, allowing for a thorough exploration of how variations in hull geometry correlate with performance in real sea conditions.

4 RESULTS

To train the standard PME and its physics-informed extensions (PI-PME and PD-PME), an initial set of *S* = 16385 Sobol samples was generated for each test case. In addition to covering the geometric design space, physical simulations were performed at these samples whenever possible, providing distributed or lumped physical data. Any geometries that produced unfeasible outcomes (e.g., degenerate shapes, solver divergence, or *NaN* outputs) were discarded. This filtering process alone already serves as a preliminary assessment of the chosen parameterization, as it highlights whether certain geometric variations are physically inadmissible or numerically unstable.

An additional layer of screening was then applied to exclude extreme outliers. Specifically, all geometric-physical samples were retained only if the corresponding physical outputs fell within the ranges [*Q*1 − 3*IQR*, *Q*3 + 3*IQR*], where *Q*1 and *Q*3 are the first and third quartiles, respectively, and *IQR* = *Q*3 − *Q*1. This so-called extreme outlier rule was chosen to be sufficiently permissive, while remaining distribution-agnostic. As a result, only physically credible shapes and simulations populate the final data set used for dimensionality reduction.

Figure 14-7 presents how the fraction of variance (σ²) evolves as a function of the number of reduced components *N*, contrasting PME (purely geometric variance), PI-PME (geometry + physical data), and PD-PME (physical data only). These results are numerically summarized in Table 14-1 (e.g., how many modes are needed to reach 95% of the total variance). Meanwhile, Figures 14-8–14-11 provide deeper insights into how that variance is organized for each mode: on the left, each figure shows the normalized eigenvector components |**v**<sub>*k*</sub>|/max(*v*<sub>*ik*</sub>) along the vertical axis, plotted across the original design variables on the horizontal axis. Large peaks indicate
<figure><img src="image_13.png" /></figure>

<table><caption>Table 14-1: Design-space dimensionality reduction summary for the retaining the 95% of the problem variance.</caption><thead><tr><th>TC #</th><th>Geometry model</th><th>S Samples</th><th>M Original</th><th>N (dimensionality reduction%) PME</th><th>N (dimensionality reduction%) PI-PME</th><th>N (dimensionality reduction%) PD-PME</th></tr></thead><tbody><tr><td>1</td><td>RAE-2822</td><td>6,022</td><td>20</td><td>6 (70%)</td><td>7 (65%)</td><td>6 (70%)</td></tr><tr><td>2</td><td>AUG</td><td>5,330</td><td>32</td><td>8 (75%)</td><td>11 (66%)</td><td>5 (84%)</td></tr><tr><td>3</td><td>Propeller</td><td>13,615</td><td>38</td><td>8 (79%)</td><td>12 (68%)</td><td>8 (79%)</td></tr><tr><td>4</td><td>5415</td><td>5,842</td><td>27</td><td>14 (48%)</td><td>12 (66%)</td><td>7 (74%)</td></tr></tbody></table>

which design variables dominate that particular mode; on the right is shown the participation of each data source (e.g., geometry d, distributed pressure c<sub>p</sub>, or lumped coefficient like c<sub>D</sub>, c<sub>L</sub>, etc.) to variance retained by each mode. The latter plots reveal whether a mode primarily represents geometric variation, physical variation, or a mixture of both.

Taken together, these figures clarify whether geometry and physics are strongly correlated (leading to fewer modes or shared modes) or largely uncorrelated (leading to more modes, each distinctly owned by geometry or physics). Below is a case-by-case discussion.

PME, PI-PME, and PD-PME curves in the RAE airfoil problem lie close together (see Fig. 14-7a); each method

<figure><img src="image_14.png" /><figcaption>(a) TC#1: RAE-2822</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) TC#2: AUG</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(c) TC#3: Propeller</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(d) TC#4: 5415</figcaption></figure>

<caption>Figure 14-7: Variance resolved as a function of the reduced design variables.</caption>
<figure><img src="image_18.png" /></figure>

ENHANCING PARAMETRIC MODEL EMBEDDING WITH PHYSICAL INFORMATION

<figure><img src="image_19.png" /><figcaption>Figure 14-8: Design-space dimensionality reduction (left) eigenvectors $v_k$ that embed the original design variables and (right) participation to the variance retained by each eigenvector of geometrical and physical information for RAE-2822 (TC#1).</figcaption></figure>

needs a comparable number of modes ($N = 6$ or 7) to reach 95% or 99% variance. This suggests that geometric changes coincide strongly with variations in the pressure distribution and integrated aerodynamic coefficients. Figure 14-8 (left) shows a pretty similar embedding with eigenvectors almost coincident for PME, PI-PME, and PD-PME (see first and third mode), in particular, it can be seen how the first mode is mainly participated by $u_2$, $u_4$, and $u_6$, that correspond to the polynomial shape modification function applied to the lower side of the airfoil, highlighting also how the lower side modification is the major region affecting the aerodynamic performance. This is further confirmed by Figure 14-8 (right) where it is shown that geometry (d) and physical parameters ($c_p$, $c_L$, $c_D$, and $c_M$) share most of the first modes: a part from mode 2, that is mainly participated by the drag coefficient, no distinct pure physics or pure geometry mode dominates the decomposition. Because airfoil shape changes strongly affect the aerodynamic response under these conditions, the geometry and physics are highly correlated. Hence, adding physical data (PI-PME) does not inflate the necessary dimensionality, and removing geometry altogether (PD-PME) does not drastically reduce the dimension beyond PME, meaning that both geometry and physics point in similar directions in the design space.

For the glider test case, the PME rapidly achieves a high fraction of geometric variance ($N = 8$ modes for 95%), whereas the PI-PME requires more modes ($N = 11$). In contrast, PD-PME dips below the PME line, reaching 95% in fewer modes ($N = 5$), as shown in Fig. 14-7b. Figure 14-9, shows, on the left, that while for PME the highest variance (retained by the first mode) is mainly due to the span ($s_{0,4}$) of the tip section, the variance of PI-PME and PD-PME are mainly affected by the maximum camber and the pitch angle of the end of the transition from the center body to the outer wing ($m_3$ and $\vartheta_3$). As shown in Fig. 14-9 (right) the first modes appear “decoupled” (i.e., geometry heavy or physics heavy): for both PI-PME and PD-PME the first mode is mainly participated by global forces ($L$ and $D$), the second mainly by the pressure distribution $c_p$, while the third mainly by the geometry (for PI-PME only). There is a partial uncorrelation between geometry and physics:
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>Figure 14-9: Design-space dimensionality reduction (left) eigenvectors $v_k$ that embed the original design variables and (right) participation to the variance retained by each eigenvector of geometrical and physical information for AUG (TC#2).</figcaption></figure>

large shape modifications do not necessarily produce proportional changes in lift or drag. Hence, PI-PME must retain more overall modes to capture both geometry-dominated and physics-dominated directions. Meanwhile, if the goal is purely performance-driven, PD-PME focuses on those physical variations alone, achieving a stronger reduction with fewer modes.

Concerning the propeller test case, PME captures geometric variability well in about $N = 8$ modes, but PI-PME demands significantly more ($N = 12$) to cover combined variance from geometry plus thrust, efficiency, vortex data, and pressure distributions. PD-PME, by contrast, can exceed 95% in only $N = 8$ modes if focusing purely on physical data. Figure 14-10 shows similar results as per the glider. The first mode of PME is participated by the variation of the intermediate control point of the chord ($c_{y,2}$), which obviously causes the most relevant modifications of the propeller shape, interpretable as variations of the expanded area of the blades. The second and third modes of the PME identify the rake (through $r_{y,2}$ and $r_{y,4}$) as the second most influential parameter on the geometry variance. This behavior is plausible too, since the “shift” of blade sections in the longitudinal plane, which is the modification induced by the rake, contributes more to differences in blade shapes than the localized modifications induced by changes in the camber and thickness (by radial or sectional perturbations of control points). First two modes of PI-PME and PD-PME are mainly participated by the physical information, while the third of PI-PME is mainly participated by the geometry, even if in this case a partial participation of the physical quantities is visible. In this respect, it is interesting to observe the very nice correlations between geometrical features and physical information revealed by both PI-PME and PD-PME. The first mode for both approaches, which is mainly participated by efficiency, identifies the pitch, and in particular its values at tip, as the most important responsible of the mode, which is a trend exactly in line with traditional design method outputs and guidelines. The second mode, for which again PI-PME and PD-PME are perfectly overlapped, establishes a correlation between the delivered thrust and the strength of the tip vortex circulation again with
ENHANCING PARAMETRIC MODEL EMBEDDING WITH PHYSICAL INFORMATION

<figure><img src="image_22.png" /><figcaption>Figure 14-10: Design-space dimensionality reduction (left) eigenvectors $v_k$ that embed the original design variables and (right) participation to the variance retained by each eigenvector of geometrical and physical information for propeller (TC#3).</figcaption></figure>

the pitch and maximum values of camber, which are, by analogy with 2D hydrofoils, responsible for sectional forces (lift) through the angle of attack (pitch) and lift coefficient at zero angle of attack (camber). When, instead, the focus is on physical information like the pressure coefficient, physics-informed models highlight the influence of pitch (responsible for hydrofoil suction peaks at the leading edge) and camber (responsible for pressure distributions on the central part of the hydrofoil) through modes 4 to 8. When purely geometrical analyses are employed, quantities like maximum sectional camber and hydrofoil sectional shape (camber and thickness) are completely discarded, being, from a geometrical point of view, responsible for minimal and localized shape variations. Embedding physical quantities activates these features which, indeed, regardless of the small geometrical variations they induce, have a critical influence on pressure distributions and the overall performance of the propeller. Geometry and physical performance are weakly correlated. This necessitates many extra modes in PI-PME to accommodate both sets of unaligned directions. PD-PME proves highly compressible because the pure physics space (i.e., lumps + distributions) is lower-dimensional in meaningful directions for performance.

Finally, looking at the 5415 results (see Fig. 14-11, PME needs $N = 14$ modes to exceed 95% variance, PI-PME, interestingly, needs fewer ($N = 12$) modes to reach the same coverage, while PD-PME reaches 95% variance in only $N = 7$ modes. The first mode in PI-PME is driven almost exclusively by lumped seakeeping parameters. The second mode largely captures the calm-water wave-resistance coefficient. Starting from the third mode onward, geometry and distributed pressure tend to move “in lockstep,” indicating a tight correlation between shape changes and local pressure distribution. At first glance, one might expect adding physics increases the total variance, as per the other three test cases. However, the crucial factor is that many purely geometric variations do not correlate with changes in both calm-water and seakeeping physical quantities. When geometry is combined with physics in PI-PME, the modes that represent “empty” geometric variance (irrelevant to performance) no
<figure><img src="image_23.png" /><figcaption>Figure 14-11: Design-space dimensionality reduction (left) eigenvectors $v_k$ that embed the original design variables and (right) participation to the variance retained by each eigenvector of geometrical and physical information for 5415 (TC#4).</figcaption></figure>

longer contribute significantly to the principal components. As a result, fewer total modes can end up describing all relevant variations.

## 5 DISCUSSION

One of the clear benefits of using extended versions of PME is the ability to incorporate lumped physical metrics and distributed fields, enabling a tighter link between geometry and performance. All four test cases define lumped metrics, each representing integral performance indicators at specific operating conditions. By embedding these scalars, low-fidelity predictions of these lumped metrics can be tightly coupled with the reduced design variables from PME-based methods, simplifying multi-fidelity optimization. Furthermore, early sensitivity analyses are made easier looking at, e.g., the 2D contour plots (see Fig. 14-12) reveal how varying just one or two reduced coordinates shifts integral performance (e.g., drag, thrust, seakeeping motion) away from the baseline configuration. This “lumped-level” lens is invaluable for quick screening, where high-fidelity CFD simulations are too expensive to run exhaustively. By building on these low-fidelity, designers can prune unpromising shape variations early on. Furthermore, this low-fidelity model can be already used to illustrate the trade-offs between multiple objectives (e.g., lowering drag vs. improving lift, increasing thrust vs. reducing vortex intensity, or minimizing calm-water resistance while controlling seakeeping motions), as well as providing a first Pareto optimal solutions (see Fig. 14-13).

This approach allows to quickly visualize constraints, identify high-potential regions, and serve as a stepping stone, guiding more elaborate, high-dimensional or high-fidelity optimization, focusing computational resources on the most promising design clusters.
ENHANCING PARAMETRIC MODEL EMBEDDING WITH PHYSICAL INFORMATION

<figure><img src="image_24.png" /><figcaption>(a) TC#1: RAE-2822</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(b) TC#2: AUG</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(c) TC#3: Propeller</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>(d) TC#4: 5415</figcaption></figure>

**Figure 14-12: Contour maps of lumped physical information as a function of the first two reduced design variables x.**
Alongside lumped metrics, the embedding often includes distributed fields—pressure over the surface for the airfoil and glider, or near-blade flows for the propeller. These data prove critical if the final goal is to train machine-learning reduced-order models (ROMs) capable of predicting integral quantities of interest. For instance, for the RAE-2822 the pressure distributions can be aggregated into a small set of modes that reconstruct aerodynamic loads, while for the propeller it provides richer mode shapes and ensures variations that affect vortex shedding, cavitation risk, or tip flow details. Since the embeddings reduce the dimension of these distributed fields, surrogate ROMs can be trained more efficiently, predicting crucial integral outputs without re-running the solver for every shape perturbation.

<figure><img src="image_28.png" /><figcaption>**Figure 14-13:** Scatter plot of lumped physical parameters usable as objective functions in a multi-objective problem.</figcaption></figure>
# 6 CONCLUSIONS

This work has demonstrated how the parametric model embedding (PME) framework can be extended to include physical information—either partially (PI-PME) or exclusively (PD-PME)—in a manner that seamlessly combines geometry and performance data within a single reduced space. Across a range of test cases, including the RAE-2822 airfoil, a bio-inspired underwater glider, a ship propeller, and the DTMB-5415 destroyer-type hull, the results showed that adding physics to PME does not always raise the number of modes needed for a target variance level, especially when physical behavior strongly correlates with geometric modifications. In cases where geometry and physics are poorly aligned, however, PI-PME incorporates both purely geometric and purely physical directions, inevitably increasing the dimensionality. This difference reveals how important it is to identify whether the chosen geometric parameterization meaningfully covers the performance-critical aspects of the design.

PI-PME preserves the comprehensive interplay between shape and physics and is particularly useful when geometric feasibility, manufacturing requirements, and interlinked performance metrics must remain in the same design space. PD-PME, by removing geometry, directly focuses on physics-driven changes and can yield a reduced representation when many shape variations do not meaningfully affect integral objectives such as resistance, lift, or thrust. In this role, PD-PME also serves as a diagnostic to highlight situations where geometric degrees of freedom provide little to no performance advantage. Lumped physical quantities—such as lift, drag, thrust, or calm-water resistance—can be incorporated at lower fidelity for early screening or coarse optimization, while distributed fields like surface pressure or vortex data enrich the understanding of local flow phenomena and facilitate training machine-learning-based reduced-order models for predicting global performance measures.

A clear benefit of these methods lies in their interpretability. By revealing which original design variables contribute most strongly to each embedding mode, the extended PME approaches guide engineers to focus on shape changes that matter for performance, rather than for mere geometric variety. Early-stage screening becomes more efficient, as shown by low-dimensional contour plots and Pareto fronts generated from PI-PME or PD-PME analyses, allowing for fast identification of promising design regions without incurring the costs of exhaustive high-fidelity simulation. Ultimately, whether one relies on PME alone (when geometry largely dictates performance), integrates physical data using PI-PME (when shape-physics coupling is important), or centers on PD-PME (when physics alone can drive meaningful decisions), these variations collectively form a flexible toolkit for modeling, analyzing, and optimizing complex design spaces.

# ACKNOWLEDGMENTS

This work has been conducted within the NATO-AVT-404 Research Task Group on “Enhanced Design Processes of Military Vehicles through Machine Learning Methods”. CNR-INM and the University of Genoa acknowledge the support of the Italian Ministry of University and Research (MUR) through the National Recovery and Resilience Plan (PNRR), Sustainable Mobility Center (CNMS), Spoke 3 Waterways, CN00000023 - CUP B43C22000440001. CNR-INM is also grateful to MUR through PRIN 2022 program, project BIODRONES, 20227JNM52 - CUP B53D23005560006. CNR-INM finally acknowledges the CINECA award under the ISCRA initiative, for the availability of high-performance computing resources and support. CIRA work was supported by the CIRA internal project OPTIWING (OPTImization for WINg Generation).
<figure><img src="image_29.png" /></figure>

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