几何算子学习与界面自编码器

中文摘要 AI 辅助

几何相关的偏微分方程定义了算子，其输入和输出可能各自由一个定向界面以及该界面上的函数组成，而大多数现有的神经算子是在固定域上构建的。本文首先建立了关于一般状态集之间连续算子的逼近定理。随后，我们通过界面自编码器（IAE）实例化该定理，该自编码器使用有限投影码在固定盒子中表示界面-函数状态，并通过零集提取和限制进行解码。对于张量余弦码，我们证明了均匀的$C^0$重建速率和管状局部梯度收敛性。这些估计产生了Hausdorff和图形-Hausdorff误差界，以及基于参考的拓扑证书。IAE既支持跨空间几何算子，将域几何映射到相关的偏微分方程解场，也支持空间内几何算子，在界面-函数状态之间进行映射。其有效性通过泊松方程、Hele-Shaw流中的界面合并以及含表面活性剂的两相Stokes流的数值实验得到验证。

英文摘要

Geometry-dependent PDEs define operators whose inputs and outputs may each consist of an oriented interface and a function on that interface, whereas most existing neural operators are formulated on fixed domains. In this paper, we first establish an approximation theorem for continuous operators between general state sets. We then instantiate the theorem with the Interface Autoencoder (IAE), which represents interface--function states in a fixed box using finite projection codes and decodes them by zero-set extraction and restriction. For tensor cosine codes, we prove a uniform $C^0$ reconstruction rate and tube-local gradient convergence. These estimates yield Hausdorff and graph-Hausdorff error bounds, together with a reference-based topology certificate. The IAE accommodates both cross-space geometric operators, mapping domain geometries to the associated PDE solution fields, and within-space geometric operators, mapping between interface--function states. Its effectiveness is demonstrated through numerical experiments on the Poisson equation, interface merging in Hele--Shaw flow, and two-phase Stokes flow with surfactant.

发表机构

• National University of Singapore（新加坡国立大学）
• Peking University（北京大学）

机构由 AI 辅助整理，请以论文原文为准。