论文标题
RBF-MGN：使用物理信息图神经网络求解时空PDE
RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network
论文作者
论文摘要
物理知识的神经网络（PINN）最近作为解决方程（PDES）的代表性深度学习技术受到了极大的关注。最完全连接的基于网络的PINN使用自动差异来构建损失的损失功能，这些损失功能遭受缓慢的收敛性和困难的边界执法。此外，尽管基于卷积的神经网络（CNN）的PINN可以显着提高训练效率，但CNN在处理非结构化网格的不规则几何形状方面很难。因此，我们提出了一个基于图神经网络（GNN）和径向基函数有限差（RBF-FD）的新框架。我们将GNN引入物理知识的学习中，以更好地处理非结构化网格的不规则域。 RBF-FD用于构建微分方程的高精度差异格式，以指导模型训练。最后，我们对不规则域的泊松和波方程进行数值实验。我们说明了在不同的PDE参数，收集点数和几种类型的RBF上提出的算法的普遍性，准确性和效率。
Physics-informed neural networks (PINNs) have lately received significant attention as a representative deep learning-based technique for solving partial differential equations (PDEs). Most fully connected network-based PINNs use automatic differentiation to construct loss functions that suffer from slow convergence and difficult boundary enforcement. In addition, although convolutional neural network (CNN)-based PINNs can significantly improve training efficiency, CNNs have difficulty in dealing with irregular geometries with unstructured meshes. Therefore, we propose a novel framework based on graph neural networks (GNNs) and radial basis function finite difference (RBF-FD). We introduce GNNs into physics-informed learning to better handle irregular domains with unstructured meshes. RBF-FD is used to construct a high-precision difference format of the differential equations to guide model training. Finally, we perform numerical experiments on Poisson and wave equations on irregular domains. We illustrate the generalizability, accuracy, and efficiency of the proposed algorithms on different PDE parameters, numbers of collection points, and several types of RBFs.
