Multiscale Neural Operators for Solving Time-Independent PDEs

TL;DR

This paper introduces a graph rewiring technique for neural operators that improves solving time-independent PDEs on large, irregular meshes by enhancing global information aggregation and outperforming existing methods.

Contribution

A novel graph rewiring approach that enhances neural PDE solvers' ability to handle irregular meshes and improves performance on multiple datasets.

Findings

GNN methods set new performance standards for PDEs on irregular meshes.

Graph rewiring boosts baseline methods, achieving state-of-the-art results.

The approach effectively aggregates information across scales and irregular structures.

Abstract

Time-independent Partial Differential Equations (PDEs) on large meshes pose significant challenges for data-driven neural PDE solvers. We introduce a novel graph rewiring technique to tackle some of these challenges, such as aggregating information across scales and on irregular meshes. Our proposed approach bridges distant nodes, enhancing the global interaction capabilities of GNNs. Our experiments on three datasets reveal that GNN-based methods set new performance standards for time-independent PDEs on irregular meshes. Finally, we show that our graph rewiring strategy boosts the performance of baseline methods, achieving state-of-the-art results in one of the tasks.