用神经网络探索形状泛函的可能关系，高效绘制几何不等式图谱。
Numerical exploration of the range of shape functionals using neural networks
- 用可逆神经网络参数化凸体，保持凸性优化过程
- 通过粒子系统最小化瑞兹能量，实现图谱均匀采样
- 适用于二维三维凸体的几何与偏微分泛函分析
我们提出一种新的数值框架，用于探索布拉施克-桑塔洛图（Blaschke--Santaló diagrams），这类工具能有效刻画若干形状泛函之间的可能不等式关系。通过基于势函数的可逆神经网络架构，对任意维数的凸体进行参数化，确保形状优化过程中凸性得到内在保持。为实现图谱内部的均匀采样，从而获得充分描述，引入一个基于自动微分的交互粒子系统，以最小化瑞兹能量泛函。该方法在二维和三维空间中多个涉及体积、周长、转动惯量、抗扭刚度、威勒莫能、拉普拉斯算子前两个诺伊曼特征值等几何与偏微分型泛函的图谱上得到验证，展示了其有效性。
原文摘要 · Abstract (English)
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.
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