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Input data structure

PME-toolkit operates on datasets describing the variability of parametric geometries.

Data components

A dataset typically includes:

  • geometric data
  • design variables
  • optional physical quantities (PI-PME / PD-PME)

Global data matrix

PME-toolkit represents all information through a block-structured data matrix, where each column corresponds to one design realization.

PME data matrix structure

Figure: Structure of the PME data matrix. Geometry, design variables, optional distributed fields, and scalar quantities are stacked into a single matrix. This unified representation enables dimensionality reduction while preserving the link to the original parametric variables.

Geometry representation

Geometry is represented as a matrix:

\[ D \in \mathbb{R}^{n_{\text{features}} \times n_{\text{samples}}} \]

Where:

  • n_samples = number of design realizations
  • n_features = number of geometric degrees of freedom

Typical cases:

  • 2D geometry → stacked coordinates
  • 3D geometry → flattened mesh nodes

Example:

[x1, x2, ..., xN
 y1, y2, ..., yN
 z1, z2, ..., zN]

Design variables

Design variables are stored as:

\[ U \in \mathbb{R}^{n_{\text{variables}} \times n_{\text{samples}}} \]

These correspond to the original parametric model and are explicitly included in the data matrix, enabling analytical backmapping from the reduced space.

PME embedding matrix

PME internally constructs:

\[ \mathbf{P} = \begin{bmatrix} \mathbf{D} \\ \mathbf{U} \end{bmatrix} \]

This is the matrix used for dimensionality reduction.

Optional physical data

For PI-PME / PD-PME, additional matrices may be included:

  • pressure fields
  • performance indicators

These are appended as additional rows in P, extending the embedding with physical information.

Key requirement

All data must be:

  • aligned sample-wise
  • consistent in dimension
  • free of missing values (after filtering)

Summary

The entire PME workflow relies on a consistent data matrix representation, where each column represents one design realization and multiple data sources are combined into a unified embedding space.