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

By learning the map between function spaces using carefully designed deep neural networks, the operator learning become a focused field in recent several years, and have shown considerable efficiency over traditional numerical methods on solving complicated problems such as differential equations, but the method is still disturbed with the concern of its accuracy and reliability. In this paper, combined with the structures and technologies of a popular numerical method, i.e. the spectral method, a general learning-based architecture named Spectral Operator Learning is introduced. One of its variants, Orthogonal Polynomials Neural Operator designed for partial differential equations with Dirichlet, Neumann and Robin boundary conditions is proposed later, of which the effectiveness, efficacy and accuracy of boundary conditions are illustrated by numerical experiments. The code will be available at https://github.com/liu-ziyuan-math/spectral_operator_learning after all the numerical results are summarised.