# Numerical exploration of the range of shape functionals using neural networks

arXiv: https://arxiv.org/abs/2602.14881 · HTML: https://mlsanity.com/paper/2602.14881/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 58 references that are in the map.
- First posted on arXiv 16 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Maxout Networks](https://mlsanity.com/paper/1302.4389/) (2013)
- [CvxNet: Learnable Convex Decomposition](https://mlsanity.com/paper/1909.05736/) (2019)
- [Parametric Shape Optimization using the Support Function](https://mlsanity.com/paper/1809.00254/) (2018)
- [Volume-preserving geometric shape optimization of the Dirichlet energy using variational neural networks](https://mlsanity.com/paper/2407.19064/) (2024)
- [Numerical approximation of optimal convex shapes](https://mlsanity.com/paper/1810.10735/) (2018)
- [Numerical shape optimization among convex sets](https://mlsanity.com/paper/2203.06981/) (2022)
- [Optimization of Neumann Eigenvalues under convexity and geometric constraints](https://mlsanity.com/paper/2402.04117/) (2024)
- [Improved description of Blaschke--Santal'o diagrams via numerical shape optimization](https://mlsanity.com/paper/2501.01228/) (2025)
- [Meshless Shape Optimization using Neural Networks and Partial Differential Equations on Graphs](https://mlsanity.com/paper/2502.14821/) (2025)
- [Elastic energy of a convex body](https://mlsanity.com/paper/1406.7305/) (2014)
- [Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets](https://mlsanity.com/paper/1512.00993/) (2015)
- [Towards Optimal Gradient Bounds for the Torsion Function in the Plane](https://mlsanity.com/paper/1912.08376/) (2019)
- [The missing (A, D, r) diagram](https://mlsanity.com/paper/2005.05749/) (2020)
- [An application of the continuous Steiner symmetrization to Blaschke-Santal'o diagrams](https://mlsanity.com/paper/2011.08760/) (2020)
- [About the Blaschke-Santalo diagram of area, perimeter and moment of inertia](https://mlsanity.com/paper/2307.11658/) (2023)
- [Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domains](https://mlsanity.com/paper/2501.02281/) (2025)

## Built on by (most cited first, 1 of 1)

- [Parametrizing Convex Sets Using Sublinear Neural Networks](https://mlsanity.com/paper/2605.03520/) (2026)

## Abstract (arXiv)

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 ℝ2 and ℝ3, namely, the volume, the perimeter, the moment of inertia, the torsional rigidity, the Willmore energy, and the first two Neumann eigenvalues of the Laplacian.