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.

翻译：本文提出了一种用于探索Blaschke--Santaló图的新数值框架，该图是描述给定形状泛函间可能不等式关系的有效工具。我们通过基于规范函数的特定可逆神经网络架构，实现了任意维度凸体的参数化表示，从而在形状优化过程中内在地保持了集合的凸性。为获得图内的均匀采样并实现对其的充分描述，我们引入了一种基于PyTorch自动微分机制、通过最小化Riesz能量泛函进行优化的相互作用粒子系统。该方法在涉及$\mathbb{R}^2$和$\mathbb{R}^3$凸体几何与PDE型泛函的多个图表中得到验证，包括体积、周长、转动惯量、扭转刚度、Willmore能量以及拉普拉斯算子的前两个Neumann特征值。