论文标题
物理包裹的神经网络：具有混合边界条件的图形神经PDE求解器
Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions
论文作者
论文摘要
图神经网络（GNN）是一种有前途的学习和预测边界价值问题中描述的物理现象的有前途的方法，例如具有边界条件的部分微分方程（PDE）。但是，现有模型对可靠预测此类问题所必需的边界条件不足。此外，由于GNN的局部联系性质，很难在长时间后准确预测状态，而顶点之间的相互作用往往是全球的。我们介绍了我们的方法称为物理包裹的神经网络，该神经网络考虑边界条件并使用隐式方法在长时间后预测状态。它是基于E（n） - 等级GNN构建的，从而在各种形状上产生了高概括性能。我们证明，我们的模型以复杂的形状学习流动现象，并优于速度精确的折衷方案中最先进的经典求解器和最先进的机器学习模型。因此，我们的模型可以成为实现可靠，快速和准确的基于GNN的PDE求解器的有用标准。该代码可在https://github.com/yellowshippo/penn-neurips2022上找到。
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.
