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By learning the mappings between infinite function spaces using carefullydesigned neural networks, the operator learning methodology has exhibitedsignificantly more efficiency than traditional methods in solving complexproblems such as differential equations, but faces concerns about theiraccuracy and reliability. To overcomes these limitations, combined with thestructures of the spectral numerical method, a general neural architecturenamed spectral operator learning (SOL) is introduced, and one variant calledthe orthogonal polynomial neural operator (OPNO), developed for PDEs withDirichlet, Neumann and Robin boundary conditions (BCs), is proposed later. Thestrict BC satisfaction properties and the universal approximation capacity ofthe OPNO are theoretically proven. A variety of numerical experiments withphysical backgrounds show that the OPNO outperforms other existing deeplearning methodologies, as well as the traditional 2nd-order finite differencemethod (FDM) with a considerably fine mesh (with the relative errors reachingthe order of 1e-6), and is up to almost 5 magnitudes faster than thetraditional method.

Render unto Numerics: Orthogonal Polynomial Neural Operator for PDEs with Non-periodic Boundary Conditions