用神经网络演化曲面，自动优化形状以最小化弯曲能量。
Minimising Willmore Energy via Neural Flow
- 用神经网络参数化曲面，通过损失函数直接优化威尔莫尔能量。
- 成功生成了球面（亏格0）和克莱夫顿环面（亏格1）的最优形状。
- 首次尝试求解亏格2时的极小威尔莫尔曲面，推动开放问题进展。
本文引入闭合定向二维曲面在三维欧氏空间中的神经威尔莫尔流，作为一种自然演化过程，用于最小化威尔莫尔能量——即平均曲率的平方L²范数。采用神经网络架构建模从拓扑二维区域到三维空间的映射，学习过程通过类似物理信息神经网络（PINN）的损失函数，将威尔莫尔能量作为嵌入的泛函进行最小化。训练结果准确复现了亏格0情形下的理想球面与亏格1情形下的克莱夫顿环面。此外，亏格2情况的实验为解决该领域未解难题——寻找极小威尔莫尔曲面提供了新方法。
原文摘要 · Abstract (English)
The neural Willmore flow of a closed oriented $2$-surface in $\mathbb{R}^3$ is introduced as a natural evolution process to minimise the Willmore energy, which is the squared $L^2$-norm of mean curvature. Neural architectures are used to model maps from topological $2d$ domains to $3d$ Euclidean space, where the learning process minimises a PINN-style loss for the Willmore energy as a functional on the embedding. Training reproduces the expected round sphere for genus $0$ surfaces, and the Clifford torus for genus $1$ surfaces, respectively. Furthermore, the experiment in the genus $2$ case provides a novel approach to search for minimal Willmore surfaces in this open problem.
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