By learning the mappings between infinite function spaces using carefully designed neural networks, the operator learning methodology has exhibited significantly more efficiency than traditional methods in solving complex problems such as differential equations, but faces concerns about their accuracy and reliability. To overcomes these limitations, combined with the structures of the spectral numerical method, a general neural architecture named spectral operator learning (SOL) is introduced, and one variant called the orthogonal polynomial neural operator (OPNO), developed for PDEs with Dirichlet, Neumann and Robin boundary conditions (BCs), is proposed later. The strict BC satisfaction properties and the universal approximation capacity of the OPNO are theoretically proven. A variety of numerical experiments with physical backgrounds show that the OPNO outperforms other existing deep learning methodologies, as well as the traditional 2nd-order finite difference method (FDM) with a considerably fine mesh (with the relative errors reaching the order of 1e-6), and is up to almost 5 magnitudes faster than the traditional method.

翻译：通过精心设计的神经网络学习无限函数空间之间的映射，算子学习方法在求解微分方程等复杂问题时展现出比传统方法显著更高的效率，但其准确性和可靠性仍令人担忧。为克服这些局限性，结合谱数值方法的结构，本文引入了一种名为谱算子学习（SOL）的通用神经架构，并进一步提出其变体——正交多项式神经算子（OPNO），专门用于处理具有狄利克雷、诺伊曼和罗宾边界条件的偏微分方程。理论上证明了OPNO严格满足边界条件的性质及通用逼近能力。一系列具有物理背景的数值实验表明，OPNO在性能上优于现有的其他深度学习方法，甚至超越采用较细网格的传统二阶有限差分方法（相对误差达1e-6量级），且其计算速度比传统方法快近五个数量级。