Numerical exploration of the range of shape functionals using neural networks
This work addresses the challenge of characterizing inequalities between shape functionals for researchers in computational geometry and shape optimization, representing an incremental advancement with a novel method for a known bottleneck.
The authors tackled the problem of exploring Blaschke--Santaló diagrams for shape functionals by developing a numerical framework using neural networks and interacting particle systems, achieving effective exploration of diagrams for various geometric and PDE-type functionals in 2D and 3D convex bodies.
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.