Multiscale Neural Operators for Solving Time-Independent PDEs

Winfried Ripken, Lisa Coiffard, Felix Pieper, Sebastian Dziadzio

Keywords: neural operator, GNN, transformer, message passing, time-independent PDEs

TL;DR: We propose and benchmark a novel rewiring technique for solving time-independent PDEs using message passing

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 benchmarks 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.

Submission Number: 5