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

TL;DR

This paper introduces the orthogonal polynomial neural operator (OPNO), a neural network architecture designed for solving PDEs with various boundary conditions, demonstrating superior accuracy and speed over traditional methods and existing deep learning approaches.

Contribution

The paper develops the OPNO, a novel neural operator architecture with proven boundary condition satisfaction and universal approximation, tailored for PDEs with non-periodic boundaries.

Findings

OPNO outperforms existing deep learning methods in accuracy and speed.

OPNO achieves relative errors around 1e-6 in numerical experiments.

OPNO is up to nearly 5 times faster than traditional finite difference methods.

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 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.