Neural operators have been validated as promising deep surrogate models for solving partial differential equations (PDEs). Despite the critical role of boundary conditions in PDEs, however, only a limited number of neural operators robustly enforce these conditions. In this paper we introduce semi-periodic Fourier neural operator (SPFNO), a novel spectral operator learning method, to learn the target operators of PDEs with non-periodic BCs. This method extends our previous work (arXiv:2206.12698), which showed significant improvements by employing enhanced neural operators that precisely satisfy the boundary conditions. However, the previous work is associated with Gaussian grids, restricting comprehensive comparisons across most public datasets. Additionally, we present numerical results for various PDEs such as the viscous Burgers' equation, Darcy flow, incompressible pipe flow, and coupled reactiondiffusion equations. These results demonstrate the computational efficiency, resolution invariant property, and BC-satisfaction behavior of proposed model. An accuracy improvement of approximately 1.7X-4.7X over the non-BC-satisfying baselines is also achieved. Furthermore, our studies on SOL underscore the significance of satisfying BCs as a criterion for deep surrogate models of PDEs.

翻译：神经算子已被验证为求解偏微分方程的有前途的深度代理模型。然而，尽管边界条件在偏微分方程中具有关键作用，但仅有少数神经算子能稳健地强制实施这些条件。本文提出半周期傅里叶神经算子（SPFNO），一种新颖的谱算子学习方法，用于学习具有非周期性边界条件的偏微分方程的目标算子。该方法扩展了我们先前的工作（arXiv:2206.12698），该工作通过采用精确满足边界条件的增强型神经算子展现了显著改进。但先前工作依赖于高斯网格，限制了对大多数公共数据集的全面比较。此外，我们展示了针对多种偏微分方程（如黏性Burgers方程、达西流、不可压缩管流及耦合反应扩散方程）的数值结果。这些结果证明了所提模型的计算效率、分辨率不变性以及边界条件满足特性。相较未满足边界条件的基线模型，精度提升约1.7倍至4.7倍。进一步，我们对SOL的研究强调了将满足边界条件作为偏微分方程深度代理模型准则的重要性。