Numerical exploration of the range of shape functionals using neural networks
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| Format: | Preprint |
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2026
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| _version_ | 1866913125215764480 |
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| author | Martinet, Eloi Ftouhi, Ilias |
| author_facet | Martinet, Eloi Ftouhi, Ilias |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_14881 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Numerical exploration of the range of shape functionals using neural networks Martinet, Eloi Ftouhi, Ilias Optimization and Control Artificial Intelligence 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. |
| title | Numerical exploration of the range of shape functionals using neural networks |
| topic | Optimization and Control Artificial Intelligence |
| url | https://arxiv.org/abs/2602.14881 |