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

TL;DR

This paper introduces NLPME, a nonlinear extension of parametric model embedding, enabling more efficient shape parameterization with fewer latent variables while maintaining explicit backmapping.

Contribution

The paper develops NLPME, which replaces linear subspaces with nonlinear ones in PME, improving shape representation efficiency in parametric design.

Findings

NLPME achieves 5% reconstruction error with 5 latent variables, fewer than linear PME's 8.

NLPME reaches 1% error with 9 latent variables, compared to 15 for PME.

Most nonlinear compression gains are retained compared to deep autoencoders, with explicit backmapping.

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.