  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.
