Importance of equivariant and invariant symmetries for fluid flow modeling
This work addresses the problem of improving data-driven fluid flow modeling for computational physics applications, but it is incremental as it builds on existing geometric deep learning principles.
The authors investigated the effect of rotational equivariance in modeling fluid flows using graph neural networks, finding that modeling invariant quantities leads to more accurate long-term predictions, with results demonstrated on flow around a cylinder and buoyancy-driven shear flow.
Graph neural networks (GNNs) have shown promise in learning unstructured mesh-based simulations of physical systems, including fluid dynamics. In tandem, geometric deep learning principles have informed the development of equivariant architectures respecting underlying physical symmetries. However, the effect of rotational equivariance in modeling fluids remains unclear. We build a multi-scale equivariant GNN to forecast fluid flow and study the effect of modeling invariant and non-invariant representations of the flow state. We evaluate the model performance of several equivariant and non-equivariant architectures on predicting the evolution of two fluid flows, flow around a cylinder and buoyancy-driven shear flow, to understand the effect of equivariance and invariance on data-driven modeling approaches. Our results show that modeling invariant quantities produces more accurate long-term predictions and that these invariant quantities may be learned from the velocity field using a data-driven encoder.