Importance of equivariant and invariant symmetries for fluid flow modeling

TL;DR

This paper investigates how incorporating rotational symmetries through equivariant and invariant graph neural networks affects fluid flow modeling, finding that invariant representations lead to more accurate long-term predictions.

Contribution

It introduces a multi-scale equivariant GNN for fluid flow prediction and analyzes the impact of invariant versus equivariant representations on model accuracy.

Findings

Invariant quantities improve long-term prediction accuracy.

Invariant representations can be learned from velocity fields.

Equivariant architectures' effectiveness varies with flow type.

Abstract

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.