Multiscale Neural Operators for Solving Time-Independent PDEs
This work addresses a domain-specific problem for researchers and practitioners in computational physics and engineering by improving neural PDE solvers on irregular meshes, though it appears incremental as it builds on existing GNN methods.
The paper tackled the challenge of solving time-independent PDEs on large, irregular meshes by introducing a novel graph rewiring technique to enhance global interactions in GNNs, achieving state-of-the-art results in one task and setting new performance standards on three datasets.
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.