A nonlinear extension of parametric model embedding for dimensionality reduction in parametric shape design

2026-05-12

AI Summary

In high-dimensional parametric shape design, linear dimensionality reduction struggles to capture nonlinear geometric variations effectively, limiting optimization and exploration efficiency. This work proposes a nonlinear latent variable representation that generalizes linear subspaces to an interpretable nonlinear latent space while preserving an explicit inverse mapping back to the original design parameters. The approach integrates nonlinear latent modeling, a parameter decoder, and a CAD-driven forward geometry generation pipeline, achieving both strong compression performance and engineering interpretability. Demonstrated on an underwater glider case study, the method reconstructs shapes with only 5% error using just five latent variables—outperforming linear PME, which requires eight—and achieves 1% reconstruction error with nine latent variables compared to fifteen for linear methods, significantly surpassing both conventional linear techniques and black-box deep autoencoders.

Abstract

Dimensionality reduction is essential in simulation-based shape design, where high-dimensional parameterizations hinder optimization, surrogate modeling, and systematic design-space exploration. Parametric Model Embedding (PME) addresses this issue by constructing reduced variables from geometric information while preserving an explicit backmapping to the original design parameters. However, PME is intrinsically linear and may become inefficient when the sampled design space is governed by nonlinear geometric variability. This paper introduces a nonlinear extension of PME, denoted NLPME. The proposed framework preserves the defining principle of PME -- geometry-driven latent variables and parameter-mediated reconstruction -- while replacing the linear reduced subspace with a nonlinear latent representation. Geometry is not reconstructed directly from the latent variables; instead, the latent representation is decoded into admissible design parameters, and the corresponding geometry is recovered through a forward parametric map. The method is assessed on a bio-inspired autonomous underwater glider with a 32-dimensional parametric shape description and a CAD-based geometry-generation process. NLPME reaches a 5\% reconstruction-error threshold with \(N=5\) latent variables, compared with \(N=8\) for linear PME, and a 1\% threshold with \(N=9\), compared with \(N=15\) for PME. Comparison with a deep autoencoder shows that most of the nonlinear compression gain can be retained while preserving an explicit backmapping to the original design variables. The results establish NLPME as a compact, admissible, and engineering-compatible nonlinear reduced representation for parametric shape design spaces.

Problem

• dimensionality reduction
• parametric shape design
• nonlinear geometric variability
• design-space exploration
• simulation-based design

Innovation

• nonlinear dimensionality reduction
• parametric shape design
• geometry-driven latent representation
• explicit backmapping
• NLPME