Numerical exploration of the range of shape functionals using neural networks

TL;DR

This paper presents a neural network-based numerical framework for exploring shape functional inequalities via Blaschke--Santaló diagrams, enabling efficient shape optimization and sampling in convex geometry.

Contribution

It introduces a neural network parametrization of convex bodies and an interacting particle system for uniform sampling within shape diagrams, applied to geometric and PDE functionals.

Findings

Successfully demonstrated on diagrams involving volume, perimeter, and other functionals.

Achieved uniform sampling and effective exploration of shape inequalities.

Validated on convex bodies in 2D and 3D with various shape functionals.

Abstract

We introduce a novel numerical framework for the exploration of Blaschke--Santal\'o 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.