  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.
