# Importance of equivariant and invariant symmetries for fluid flow modeling

arXiv: https://arxiv.org/abs/2307.05486 · HTML: https://mlsanity.com/paper/2307.05486/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 54 references that are in the map.
- First posted on arXiv 3 May 2023.

## Builds on (most cited first, 29 of 29)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Point-GNN: Graph Neural Network for 3D Object Detection in a Point Cloud](https://mlsanity.com/paper/2003.01251/) (2020)
- [Session-based Recommendation with Graph Neural Networks](https://mlsanity.com/paper/1811.00855/) (2018)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids](https://mlsanity.com/paper/1810.01566/) (2018)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Rethinking pooling in graph neural networks](https://mlsanity.com/paper/2010.11418/) (2020)
- [Learning continuous-time PDEs from sparse data with graph neural networks](https://mlsanity.com/paper/2006.08956/) (2020)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Mesh convolutional neural networks for wall shear stress estimation in 3D artery models](https://mlsanity.com/paper/2109.04797/) (2021)
- [Simulating Liquids with Graph Networks](https://mlsanity.com/paper/2203.07895/) (2022)
- [Diversified Multiscale Graph Learning with Graph Self-Correction](https://mlsanity.com/paper/2103.09754/) (2021)
- [RotEqNet: Rotation-Equivariant Network for Fluid Systems with Symmetric High-Order Tensors](https://mlsanity.com/paper/2005.04286/) (2020)

## Built on by (most cited first, 5 of 5)

- [Mesh-based Super-Resolution of Fluid Flows with Multiscale Graph Neural Networks](https://mlsanity.com/paper/2409.07769/) (2024)
- [Harnessing Equivariance: Modeling Turbulence with Graph Neural Networks](https://mlsanity.com/paper/2504.07741/) (2025)
- [Scalable and Consistent Graph Neural Networks for Distributed Mesh-based Data-driven Modeling](https://mlsanity.com/paper/2410.01657/) (2024)
- [Generalization vs. Memorization in Autoregressive Deep Learning: Or, Examining Temporal Decay of Gradient Coherence](https://mlsanity.com/paper/2509.00024/) (2025)
- [E3J: An Efficient and Open-Source Backend for Euclidean Equivariant Operations on GPU and TPU](https://mlsanity.com/paper/2609.35099/) (2026)

## Abstract (arXiv)

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.