Numerical exploration of the range of shape functionals using neural networks

• URL: http://arxiv.org/abs/2602.14881v1
• Date: Mon, 16 Feb 2026 16:10:58 GMT
• Title: Numerical exploration of the range of shape functionals using neural networks
• Authors: Eloi Martinet, Ilias Ftouhi,
• Abstract summary: We introduce a novel numerical framework for the exploration of Blaschke--Santal diagrams.<n>We introduce a parametrization of convex bodies in arbitrary dimensions using a specific invertible neural network architecture based on gauge functions.<n>To achieve a uniform sampling inside the diagram, and thus a satisfying description, we introduce an interacting particle system that minimizes a Riesz energy functional via automatic differentiation in PyTorch.<n>The effectiveness of the method is demonstrated on several diagrams involving both geometric and PDE-type functionals for convex bodies of $mathbbR2$ and $mathbbR
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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