| Abstract: |
| We are interested in the use of neural networks for shape optimization and their application in the exploration of shape functionals, also called Blaschke--Santal\`o diagrams. In a first part, we show how a certain neural network architecture allows one to represent convex bodies in any dimension without constraints. By leveraging the automatic differentiation capabilities of PyTorch, we demonstrate that this representation can be applied effortlessly to shape optimization problems. In a second part, in collaboration with Ilias Ftouhi, we show how to take advantage of this parametrization to provide a faithful numerical description of various Blaschke--Santal\`o diagrams, by representing each convex body as a repulsive electric charge in the diagram. |
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