Graph neural network (GNN) is a promising approach to learning and predicting physical phenomena described in boundary value problems, such as partial differential equations (PDEs) with boundary conditions. However, existing models inadequately treat boundary conditions essential for the reliable prediction of such problems. In addition, because of the locally connected nature of GNNs, it is difficult to accurately predict the state after a long time, where interaction between vertices tends to be global. We present our approach termed physics-embedded neural networks that considers boundary conditions and predicts the state after a long time using an implicit method. It is built based on an E(n)-equivariant GNN, resulting in high generalization performance on various shapes. We demonstrate that our model learns flow phenomena in complex shapes and outperforms a well-optimized classical solver and a state-of-the-art machine learning model in speed-accuracy trade-off. Therefore, our model can be a useful standard for realizing reliable, fast, and accurate GNN-based PDE solvers. The code is available at https://github.com/yellowshippo/penn-neurips2022.

翻译：图神经网络（GNN）是学习和预测边界值问题（如带边界条件的偏微分方程）中物理现象的一种有前景的方法。然而，现有模型未能充分处理对此类问题可靠预测至关重要的边界条件。此外，由于GNN的局部连接特性，难以准确预测长时间后的系统状态，此时顶点间的相互作用倾向于全局化。我们提出了一种名为物理嵌入神经网络的方法，该方法考虑了边界条件，并采用隐式方法预测长时间后的状态。该模型基于E(n)-等变GNN构建，因此对不同形状具有高度的泛化性能。我们证明，该模型能学习复杂形状中的流动现象，并在速度-精度权衡上优于经过充分优化的经典求解器和最先进的机器学习模型。因此，我们的模型可作为实现可靠、快速且精确的基于GNN的偏微分方程求解器的实用基准。代码见：https://github.com/yellowshippo/penn-neurips2022。