RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains

TL;DR

RIGNO is a graph neural network framework designed to learn PDE solution operators on arbitrary domains, demonstrating high accuracy and robustness across diverse benchmarks and resolutions.

Contribution

The paper introduces RIGNO, a novel GNN-based neural operator that effectively handles PDEs on arbitrary domains with invariance to spatio-temporal resolution changes.

Findings

RIGNO outperforms existing neural operator baselines in accuracy.

RIGNO generalizes well to unseen spatial and temporal resolutions.

The model is validated on diverse PDE benchmarks.

Abstract

Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains. Our multi-scale model maps data between input/output point clouds by passing it through a downsampled regional mesh. The approach includes novel elements aimed at ensuring spatio-temporal resolution invariance. Our model, termed RIGNO, is tested on a challenging suite of benchmarks composed of various time-dependent and steady PDEs defined on a diverse set of domains. We demonstrate that RIGNO is significantly more accurate than neural operator baselines and robustly generalizes to unseen resolutions both in space and in time. Our code is publicly available at github.com/camlab-ethz/rigno.