Local neural operator for solving transient partial differential equations on varied domains

TL;DR

This paper introduces a local neural operator (LNO) that efficiently solves transient PDEs on varied domains, significantly reducing computational costs compared to traditional methods, with demonstrated success on fluid flow problems.

Contribution

The paper presents a novel local neural operator framework with boundary treatment strategies, enabling pre-trained models to generalize across different domains for transient PDEs.

Findings

LNO learns Navier-Stokes equations from data.

Pre-trained LNO predicts solutions on unseen domains.

LNO is about 1000× faster than finite element methods.

Abstract

Artificial intelligence (AI) shows great potential to reduce the huge cost of solving partial differential equations (PDEs). However, it is not fully realized in practice as neural networks are defined and trained on fixed domains and boundaries. Herein, we propose local neural operator (LNO) for solving transient PDEs on varied domains. It comes together with a handy strategy including boundary treatments, enabling one pre-trained LNO to predict solutions on different domains. For demonstration, LNO learns Navier-Stokes equations from randomly generated data samples, and then the pre-trained LNO is used as an explicit numerical time-marching scheme to solve the flow of fluid on unseen domains, e.g., the flow in a lid-driven cavity and the flow across the cascade of airfoils. It is about 1000$\times$ faster than the conventional finite element method to calculate the flow across the cascade of airfoils. The solving process with pre-trained LNO achieves great efficiency, with significant potential to accelerate numerical calculations in practice.