发表机构
ETH Zürich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出一种可独立于训练强制齐次狄利克雷边界条件的神经算子架构，适用于任意网格数据和一般几何结构，通过二维 PDE 算例验证了其有效性并统一了相关理论。
AI 中文摘要
科学机器学习中的算子学习关注无限维函数空间之间映射的近似，这类映射常作为偏微分方程（PDE）的解算子出现。神经算子已在从数据近似这类映射方面展现出广泛的经验成功，但大多数现有神经算子架构通过从数据训练间接强制边界条件，尽管边界条件往往是完全已知的；此外，现有明确强制边界条件的修改方法和方案存在不切实际的限制，包括边界光滑性要求、均匀网格要求以及可分离的盒状域要求。本研究提出一种架构，该架构独立于训练即可满足齐次狄利克雷边界条件，同时保留现有核积分神经算子架构的表达能力。这通过强制每个层的输出包含在输出域上拉普拉斯算子的齐次狄利克雷特征函数子集的张成空间中实现。该方法仅要求输出域具有 Lipschitz 边界且有界，对离散化选择无任何限制，使其适用于任意网格数据和一般几何结构。我们证明了所提架构的通用逼近性；此外，我们在分析中采用的方法证明了一大类核积分神经算子的通用逼近性，从而统一了多种算子学习方法的现有理论。我们在由二维 PDE 的系数到解映射定义的映射上验证了所提方法：正方形域上的 Darcy 流和圆形域上的 Helmholtz 方程，并与其他方法进行了比较。
英文摘要
Operator learning in scientific machine learning is concerned with approximation of maps between infinite-dimensional function spaces; such maps frequently arise as the solution operators of partial differential equations (PDEs). Neural operators have demonstrated broad empirical success at approximating such maps from data. However, most existing neural operator architectures enforce boundary conditions indirectly through training from data even though the boundary condition is often known exactly. Furthermore, existing modifications and approaches that do enforce boundary conditions explicitly suffer from impractical restrictions, including boundary smoothness, uniform grids, and separable, box-like domains. In this work, we propose an architecture which, independently of training, satisfies homogeneous Dirichlet boundary conditions, whilst simultaneously retaining the expressivity of existing kernel-integral neural operator architectures. This is achieved by enforcing the property that the output of each layer is contained in the span of a subset of the homogeneous Dirichlet eigenfunctions of the Laplacian on the output domain. The method requires only that the output domain be bounded with Lipschitz boundary and places no restriction on the choice of discretization, making it applicable to arbitrary mesh data and general geometries. We prove universal approximation for the resulting architecture; furthermore the approach we adopt in the analysis proves universality for a broad class of kernel-integral neural operators thereby uniting existing theory for a variety of operator learning methods. We validate the proposed method on maps defined by the coefficient to solution map in 2D PDEs: Darcy flow on a square domain and the Helmholtz equation on a circular domain. Comparisons are made with alternative methods.
