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A Survey on Design-space Dimensionality Reduction Methods for Shape Optimization

Andrea Serani, Matteo Diez

TL;DR

This survey addresses the challenge of high-dimensional design spaces in external shape optimization and surveys methods for design-space dimensionality reduction. It traverses from linear approaches such as PCA/KLE/POD to nonlinear techniques including local KPCA, autoencoders, and GTM, and highlights physics-informed variants and Parametric Model Embedding that enable back-mapping to original parameters. The authors distinguish space reduction from direct dimensionality reduction, analyze a literature corpus to reveal trends (e.g., prevalence of direct and linear methods) and provide a framework for selecting methods based on geometry and simulation data. The work offers practical guidance for integrating reduced-dimensionality representations into optimization workflows, reducing computational cost, improving convergence, and enabling uncertainty quantification in complex shape-optimization problems.

Abstract

The rapidly evolving field of engineering design of functional surfaces necessitates sophisticated tools to manage the inherent complexity of high-dimensional design spaces. This survey paper offers a scoping review, i.e., a literature mapping synthesis borrowed from clinical medicine, delving into the field of design-space dimensionality reduction techniques tailored for shape optimization, bridging traditional methods and cutting-edge technologies. Dissecting the spectrum of these techniques, from classical linear approaches like principal component analysis to more nuanced nonlinear methods such as autoencoders, the discussion extends to innovative physics-informed methods that integrate physical data into the dimensionality reduction process, enhancing the physical relevance and effectiveness of reduced design spaces. By integrating these methods into optimization frameworks, it is shown how they significantly mitigate the curse of dimensionality, streamline computational processes, and refine the design exploration and optimization of complex functional surfaces. The survey provides a classification of methods and highlights the transformative impact of these techniques in simplifying design challenges, thereby fostering more efficient and effective engineering solutions.

A Survey on Design-space Dimensionality Reduction Methods for Shape Optimization

TL;DR

This survey addresses the challenge of high-dimensional design spaces in external shape optimization and surveys methods for design-space dimensionality reduction. It traverses from linear approaches such as PCA/KLE/POD to nonlinear techniques including local KPCA, autoencoders, and GTM, and highlights physics-informed variants and Parametric Model Embedding that enable back-mapping to original parameters. The authors distinguish space reduction from direct dimensionality reduction, analyze a literature corpus to reveal trends (e.g., prevalence of direct and linear methods) and provide a framework for selecting methods based on geometry and simulation data. The work offers practical guidance for integrating reduced-dimensionality representations into optimization workflows, reducing computational cost, improving convergence, and enabling uncertainty quantification in complex shape-optimization problems.

Abstract

The rapidly evolving field of engineering design of functional surfaces necessitates sophisticated tools to manage the inherent complexity of high-dimensional design spaces. This survey paper offers a scoping review, i.e., a literature mapping synthesis borrowed from clinical medicine, delving into the field of design-space dimensionality reduction techniques tailored for shape optimization, bridging traditional methods and cutting-edge technologies. Dissecting the spectrum of these techniques, from classical linear approaches like principal component analysis to more nuanced nonlinear methods such as autoencoders, the discussion extends to innovative physics-informed methods that integrate physical data into the dimensionality reduction process, enhancing the physical relevance and effectiveness of reduced design spaces. By integrating these methods into optimization frameworks, it is shown how they significantly mitigate the curse of dimensionality, streamline computational processes, and refine the design exploration and optimization of complex functional surfaces. The survey provides a classification of methods and highlights the transformative impact of these techniques in simplifying design challenges, thereby fostering more efficient and effective engineering solutions.
Paper Structure (25 sections, 43 equations, 27 figures)

This paper contains 25 sections, 43 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Shape Optimization Problem Formulation
  3. Tackling the Curse of Dimensionality
  4. Literature overview
  5. Research Questions
  6. Inclusion and Exclusion Criteria
  7. Databases and Keywords
  8. Search Procedure
  9. Scoping Review Outcomes
  10. Space Reduction Methods
  11. Bounds Narrowing
  12. Geometric filtering
  13. Indirect Dimensionality Reduction Methods
  14. Direct Dimensionality Reduction Methods: Representation Learning
  15. Linear dimensionality reduction methods
  16. ...and 10 more sections

Figures (27)

  • Figure 1: Shape modification example
  • Figure 2: Typical simulation-based design optimization block diagram for shape optimization
  • Figure 3: Example of a global optimizer convergence cost conditional to the design space characteristic dimension (data taken from serani2017random)
  • Figure 4: Classification tree for design-space dimensionality reduction in shape optimization
  • Figure 5: PRISMA flow chart
  • ...and 22 more figures