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Numerical exploration of the range of shape functionals using neural networks

Eloi Martinet, Ilias Ftouhi

TL;DR

We address the numerical exploration of Blaschke--Santaló diagrams for convex bodies by introducing an invertible neural network representation based on gauge functions to parameterize convex sets, and a repulsive-particle scheme to uniformly sample the diagram in the space of shape functionals $F(\Omega)$ (e.g., $Vol$, $Per$, $W$, $T$, Willmore energy $E$, and Neumann eigenvalues). The method handles dimensions $d=2,3$ and functionals arising from geometry and PDEs via automatic differentiation, enabling computation of $Vol(\Omega)$, $Per(\Omega)$, $W(\Omega)$, $T(\Omega)$, $E(\Omega)$ and eigenvalues $\mu_k(\Omega)$. It yields dense, near-uniform coverage of the diagrams, reproduces known inequalities (e.g., Polya-type bounds) and characterizes boundary behavior through symmetric and non-symmetric classes, with results demonstrated in planar and space convex bodies. The implementation is open-source with notebooks, and the framework is extensible to additional functionals and possibly non-convex domains via extended diffeomorphic parametrizations.

Abstract

We introduce a novel numerical framework for the exploration of Blaschke--Santaló diagrams, which are efficient tools characterizing the possible inequalities relating some given shape functionals. We introduce a parametrization of convex bodies in arbitrary dimensions using a specific invertible neural network architecture based on gauge functions, allowing an intrinsic conservation of the convexity of the sets during the shape optimization process. To achieve a uniform sampling inside the diagram, and thus a satisfying description of it, we introduce an interacting particle system that minimizes a Riesz energy functional via automatic differentiation in PyTorch. The effectiveness of the method is demonstrated on several diagrams involving both geometric and PDE-type functionals for convex bodies of $\mathbb{R}^2$ and $\mathbb{R}^3$, namely, the volume, the perimeter, the moment of inertia, the torsional rigidity, the Willmore energy, and the first two Neumann eigenvalues of the Laplacian.

Numerical exploration of the range of shape functionals using neural networks

TL;DR

We address the numerical exploration of Blaschke--Santaló diagrams for convex bodies by introducing an invertible neural network representation based on gauge functions to parameterize convex sets, and a repulsive-particle scheme to uniformly sample the diagram in the space of shape functionals (e.g., , , , , Willmore energy , and Neumann eigenvalues). The method handles dimensions and functionals arising from geometry and PDEs via automatic differentiation, enabling computation of , , , , and eigenvalues . It yields dense, near-uniform coverage of the diagrams, reproduces known inequalities (e.g., Polya-type bounds) and characterizes boundary behavior through symmetric and non-symmetric classes, with results demonstrated in planar and space convex bodies. The implementation is open-source with notebooks, and the framework is extensible to additional functionals and possibly non-convex domains via extended diffeomorphic parametrizations.

Abstract

We introduce a novel numerical framework for the exploration of Blaschke--Santaló diagrams, which are efficient tools characterizing the possible inequalities relating some given shape functionals. We introduce a parametrization of convex bodies in arbitrary dimensions using a specific invertible neural network architecture based on gauge functions, allowing an intrinsic conservation of the convexity of the sets during the shape optimization process. To achieve a uniform sampling inside the diagram, and thus a satisfying description of it, we introduce an interacting particle system that minimizes a Riesz energy functional via automatic differentiation in PyTorch. The effectiveness of the method is demonstrated on several diagrams involving both geometric and PDE-type functionals for convex bodies of and , namely, the volume, the perimeter, the moment of inertia, the torsional rigidity, the Willmore energy, and the first two Neumann eigenvalues of the Laplacian.
Paper Structure (26 sections, 5 theorems, 75 equations, 8 figures)

This paper contains 26 sections, 5 theorems, 75 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Shape representation by an invertible neural network
  3. Previous works
  4. Proposed approach
  5. Encoding symmetries
  6. Sampling the diagram
  7. Numerical computation of the involved shape functionals
  8. Integral quantities
  9. Curvature-related quantities
  10. PDE-related quantities
  11. A note on the practical computation of shape quantities
  12. Code availability
  13. Obtained results
  14. The diagram Vol, Per, W
  15. The diagram Vol, Per, W for planar convex bodies
  16. ...and 11 more sections

Key Result

Proposition 2.1

Let $p$ be a positive, continuous, and sublinear function. The function $\phi:x\longmapsto \frac{\|x\|}{p(x)}x$ is a homeomorphism from $B$ to $\phi(B)$ and the image set $\phi(B)$ is convex.

Figures (8)

  • Figure 1: Numerical approximation of the diagram $\mathcal{D}^2_{VPW}$ for $d=2$.
  • Figure 2: Numerical approximation of the diagram $\mathcal{D}^2_{VPW}$ for planar and doubly symmetric convex bodies, along with the theoretically known lower boundary.
  • Figure 3: Numerical approximation of the diagram $\mathcal{D}^3_{VPW}$ for $d=3$
  • Figure 4: Numerical approximation of the diagram $\mathcal{D}^2_{VPT}$ for $d=2$ along with the smaller theoretically known superset.
  • Figure 5: Numerical approximation of the diagram $\mathcal{D}^2_{V\mu}$ for $d=2$ along with the theoretically known smaller superset.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 3.1: Th. 8.5.2 in borodachov_discrete_2019
  • Theorem 3.2: Th. 8.8.1 in borodachov_discrete_2019
  • Remark 4.1
  • Remark 5.1
  • Remark 5.2