Render unto Numerics: Orthogonal Polynomial Neural Operator for PDEs with Nonperiodic Boundary Conditions

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C323-C348, August 2024.
Abstract. 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 differential equations, but faces concerns about their accuracy and reliability. To overcome these limitations through robustly enforcing boundary conditions (BCs), a general neural architecture named spectral operator learning is introduced by combining with the structures of the spectral numerical method. One variant called the orthogonal polynomial neural operator (OPNO) is proposed later, aiming at PDEs with Dirichlet, Neumann, and Robin BCs. The strict BC satisfaction properties and the universal approximation capacity of the OPNO are theoretically proven. A variety of numerical experiments with physical backgrounds demonstrate that the OPNO outperforms other existing deep learning methodologies, showcasing potential of comparable accuracy with the traditional second-order finite difference method that employs a considerably fine mesh (with relative errors on the order of [math]), and is up to almost 5 magnitudes faster over the traditional method. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/liu-ziyuan-math/spectral_operator_learning/tree/main/OPNO/Reproduce and in the supplementary materials (spectral_operator_learning-main.zip [669KB]).