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Workshop: The Symbiosis of Deep Learning and Differential Equations -- III

Multiscale Neural Operators for Solving Time-Independent PDEs

Winfried Ripken · Lisa Coiffard · Felix Pieper · Sebastian Dziadzio

Keywords: [ message passing ] [ GNN ] [ transformer ] [ Neural operator ] [ time-independent PDEs ]


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.

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