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[2602.14881] Numerical exploration of the range of shape functionals using neural networks
Summary
This article presents a novel numerical framework for exploring shape functionals using neural networks, focusing on Blaschke–Santaló diagrams and convex bodies in various dimensions.
Why It Matters
The research provides insights into optimizing shape functionals, which can have significant implications in fields such as computer vision, robotics, and geometric optimization. By leveraging neural networks, the study enhances our understanding of geometric properties and their applications in artificial intelligence.
Key Takeaways
- Introduces a new framework for exploring shape functionals using neural networks.
- Utilizes Blaschke–Santaló diagrams to characterize inequalities in shape functionals.
- Demonstrates the method's effectiveness on geometric and PDE-type functionals.
- Employs a specific neural network architecture to maintain convexity during optimization.
- Highlights applications in convex bodies across multiple dimensions.
Mathematics > Optimization and Control arXiv:2602.14881 (math) [Submitted on 16 Feb 2026] Title:Numerical exploration of the range of shape functionals using neural networks Authors:Eloi Martinet, Ilias Ftouhi View a PDF of the paper titled Numerical exploration of the range of shape functionals using neural networks, by Eloi Martinet and 1 other authors View PDF HTML (experimental) 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. Comments: Subjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI) Cite as:...
