Numerical exploration of the range of shape functionals using neural networks

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.