An emerging trend in deep learning research focuses on the applications of graph neural networks (GNNs) for mesh-based continuum mechanics simulations. Most of these learning frameworks operate on graphs wherein each edge connects two nodes. Inspired by the data connectivity in the finite element method, we present a method to construct a hypergraph by connecting the nodes by elements rather than edges. A hypergraph message-passing network is defined on such a node-element hypergraph that mimics the calculation process of local stiffness matrices. We term this method a finite element-inspired hypergraph neural network, in short FEIH($\phi$)-GNN. We further equip the proposed network with rotation equivariance, and explore its capability for modeling unsteady fluid flow systems. The effectiveness of the network is demonstrated on two common benchmark problems, namely the fluid flow around a circular cylinder and airfoil configurations. Stabilized and accurate temporal roll-out predictions can be obtained using the $\phi$-GNN framework within the interpolation Reynolds number range. The network is also able to extrapolate moderately towards higher Reynolds number domain out of the training range.

翻译：深度学习研究的新趋势聚焦于图神经网络在基于网格的连续介质力学模拟中的应用。大多数此类学习框架基于图结构，其中每条边连接两个节点。受有限元方法中数据连通性的启发，我们提出了一种通过单元而非边连接节点来构建超图的方法。在该节点-单元超图上定义了一种超图消息传递网络，模拟局部刚度矩阵的计算过程。我们将该方法命名为受有限元启发的超图神经网络，简称FEIH($\phi$)-GNN。我们进一步为所提网络赋予旋转等变性，并探索其在非定常流体系统建模中的能力。该网络的有效性通过两个常见基准问题得以验证，即圆柱绕流和翼型构型。在插值雷诺数范围内，使用$\phi$-GNN框架可获得稳定且精确的时间演化预测。该网络还能适度外推至训练范围外的高雷诺数区域。