A Finite Element-Inspired Hypergraph Neural Network: Application to Fluid Dynamics Simulations

TL;DR

Stabilized and accurate temporal roll-out predictions can be obtained using the $\phi$-GNN framework within the interpolation Reynolds number range and the network is also able to extrapolate moderately towards higher Reynolds number domain out of the training range.

Abstract

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($φ$)-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 $φ$-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.

Figures (20)

• Figure 1: Conversion of a representative computational mesh to graph: (a) schematic representing a portion of computational mesh, (b)-(c) two possible approaches to convert the mesh to a graph.
• Figure 1: Predicted versus ground truth lift and drag force curves between prediction time step 1 to 500 and 3501 to 4000, for the flow around airfoil test data sets with Reynolds number 2050, 2650 and 2950, when Reynolds number is not explicitly supplied as a input feature to the network.
• Figure 2: Conversion of a computational mesh to hypergraph: (a) schematic representing a portion of computational mesh, (b) converted hypergraph, (c) converted node-element hypergraph.
• Figure 2: Coefficient of determination of the predicted pressure field $\boldsymbol{p}^*$ at prediction time step 4000 when different amounts of training noise are used. Red line: Models trained with the setup described in Sec. \ref{['sec:exp']} with different amounts of training noise. Blue line: Changing random seed to 2 from the setup described in Sec. \ref{['sec:exp']}. Magenta line: Changing batch size to 8 from the setup described in Sec. \ref{['sec:exp']}.
• Figure 3: Schematic of the element and node update stages within each hypergraph message-passing layer: (a) element update stage described by Eq. \ref{['eq:ne+mpa']}, (b) node update stage described by Eq. \ref{['eq:ne+mpba']} and \ref{['eq:ne+mpbb']}.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4