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

Since analytical solutions are usually not available, numerical solutions on discretized space and time domains are considered for predictive modeling. Leveraging state-of-the-art computational fluid dynamics (CFD) approaches based on finite volume [3] or finite element [4, 5] methods, one could obtain high-fidelity solutions that can be suitable for downstream design optimization and control purposes. However, the cost of performing such simulations is significant, and becomes prohibitively high for complex problems arising from real-world applications. This limitation of traditional CFD approaches has inspired the development of data-driven projectionbased reduced-order modeling techniques. Such models are usually used in an offline-online manner. In the offline stage, an approximation of the governing flow dynamics in a low-order linear subspace is constructed based on available fluid flow data collected. This approximation reduces the complexity of the problem in the online stage, making it possible to acquire fast, accurate predictions. Popular methods in this category include proper orthogonal decomposition (POD) [6, 7], dynamic mode decomposition (DMD) [8], along with many variants (e.g., [9, 10, 11]). However, these methods encounter difficulty when applied to scenarios with high Reynolds numbers and convection-dominated problems, whereas one needs a significantly large number of linear subspaces to achieve a satisfactory approximation.