用神经网络和图拉普拉斯实现无网格形状优化，提升精度与灵活性。
Meshless Shape Optimization using Neural Networks and Partial Differential Equations on Graphs
- 用神经网络参数化水平集函数，结合图拉普拉斯近似偏微分方程
- 可精确计算曲面法向与曲率，支持凸形状优化问题
- 摆脱网格限制，适合复杂几何形状的高效优化
形状优化旨在最小化定义在一组形状上的代价函数，通常由偏微分方程（PDE）约束。当缺乏解析解时，需依赖数值方法近似求解。水平集方法结合有限元法是目前最通用的形状优化方法之一，但仍受大多数基于网格方法的局限。本文提出一种完全无网格的水平集框架，利用神经网络参数化水平集函数，并采用图拉普拉斯近似底层PDE。该方法可精确计算几何量如表面法向与曲率，支持凸形状类优化问题的求解。
原文摘要 · Abstract (English)
Shape optimization involves the minimization of a cost function defined over a set of shapes, often governed by a partial differential equation (PDE). In the absence of closed-form solutions, one relies on numerical methods to approximate the solution. The level set method -- when coupled with the finite element method -- is one of the most versatile numerical shape optimization approaches but still suffers from the limitations of most mesh-based methods. In this work, we present a fully meshless level set framework that leverages neural networks to parameterize the level set function and employs the graph Laplacian to approximate the underlying PDE. Our approach enables precise computations of geometric quantities such as surface normals and curvature, and allows tackling optimization problems within the class of convex shapes.
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