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.

翻译：在大规模网格上求解时间无关偏微分方程（PDE）对数据驱动的神经PDE求解器提出了重大挑战。我们引入了一种新颖的图重连技术，以应对其中的部分挑战，例如跨尺度信息聚合以及在非规则网格上的信息处理。所提出的方法通过连接远距离节点，增强了图神经网络（GNN）的全局交互能力。我们在三个数据集上的实验表明，基于GNN的方法在非规则网格上的时间无关PDE求解中树立了新的性能标准。最后，我们证明所提出的图重连策略能够提升基线方法的性能，并在其中一项任务中达到了最先进的结果。