# Harnessing Equivariance: Modeling Turbulence with Graph Neural Networks

arXiv: https://arxiv.org/abs/2504.07741 · HTML: https://mlsanity.com/paper/2504.07741/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 45 references that are in the map.
- First posted on arXiv 10 Apr 2025.

## Builds on (most cited first, 12 of 12)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [A Perspective on Machine Learning Methods in Turbulence Modelling](https://mlsanity.com/paper/2010.12226/) (2020)
- [Generalizable data-driven turbulence closure modeling on unstructured grids with differentiable physics](https://mlsanity.com/paper/2307.13533/) (2023)
- [FLEXI: A high order discontinuous Galerkin framework for hyperbolic-parabolic conservation laws](https://mlsanity.com/paper/1910.02858/) (2019)
- [Deep Reinforcement Learning for Computational Fluid Dynamics on HPC Systems](https://mlsanity.com/paper/2205.06502/) (2022)
- [Importance of equivariant and invariant symmetries for fluid flow modeling](https://mlsanity.com/paper/2307.05486/) (2023)

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

- [Hierarchical-embedding autoencoder with a predictor (HEAP) as efficient architecture for learning long-term evolution of complex multi-scale physical systems](https://mlsanity.com/paper/2505.18857/) (2025)
- [FIGNN: Feature-Specific Interpretability for Graph Neural Network Surrogate Models](https://mlsanity.com/paper/2506.11398/) (2025)
- [Pixel-Resolved Long-Context Learning for Turbulence at Exascale: Resolving Small-scale Eddies Toward the Viscous Limit](https://mlsanity.com/paper/2507.16697/) (2025)
- [Multiscale Graph Neural Network for Turbulent Flow-Thermal Prediction Around a Complex-Shaped Pin-Fin](https://mlsanity.com/paper/2509.04463/) (2025)
- [Deep Wave Network for Modeling Multi-Scale Physical Dynamics](https://mlsanity.com/paper/2605.04198/) (2026)

## Abstract (arXiv)

This work proposes a novel methodology for turbulence modeling in Large Eddy Simulation (LES) based on Graph Neural Networks (GNNs), which embeds the discrete rotational, reflectional and translational symmetries of the Navier-Stokes equations into the model architecture. In addition, suitable invariant input and output spaces are derived that allow the GNN models to be embedded seamlessly into the LES framework to obtain a symmetry-preserving simulation setup. The suitability of the proposed approach is investigated for two canonical test cases: Homogeneous Isotropic Turbulence (HIT) and turbulent channel flow. For both cases, GNN models are trained successfully in actual simulations using Reinforcement Learning (RL) to ensure that the models are consistent with the underlying LES formulation and discretization. It is demonstrated for the HIT case that the resulting GNN-based LES scheme recovers rotational and reflectional equivariance up to machine precision in actual simulations. At the same time, the stability and accuracy remain on par with non-symmetry-preserving machine learning models that fail to obey these properties. The same modeling strategy translates well to turbulent channel flow, where the GNN model successfully learns the more complex flow physics and is able to recover the turbulent statistics and Reynolds stresses. It is shown that the GNN model learns a zonal modeling strategy with distinct behaviors in the near-wall and outer regions. The proposed approach thus demonstrates the potential of GNNs for turbulence modeling, especially in the context of LES and RL.