# Mean flow data assimilation using physics-constrained Graph Neural Networks

arXiv: https://arxiv.org/abs/2411.09476 · HTML: https://mlsanity.com/paper/2411.09476/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 61 references that are in the map.
- First posted on arXiv 14 Nov 2024.

## Builds on (most cited first, 13 of 13)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Scientific multi-agent reinforcement learning for wall-models of turbulent flows](https://mlsanity.com/paper/2106.11144/) (2021)
- [Expressivity of Deep Neural Networks](https://mlsanity.com/paper/2007.04759/) (2020)
- [A Perspective on Machine Learning Methods in Turbulence Modelling](https://mlsanity.com/paper/2010.12226/) (2020)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Learning Lagrangian Fluid Mechanics with E(3)-Equivariant Graph Neural Networks](https://mlsanity.com/paper/2305.15603/) (2023)
- [Background Error Covariance Iterative Updating with Invariant Observation Measures for Data Assimilation](https://mlsanity.com/paper/1910.09408/) (2019)

## Built on by (most cited first, 3 of 3)

- [Bridging Data and Physics: A Graph Neural Network-Based Hybrid Twin Framework](https://mlsanity.com/paper/2512.15767/) (2025)
- [Latent attention on masked patches for flow reconstruction](https://mlsanity.com/paper/2603.02028/) (2026)
- [An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study](https://mlsanity.com/paper/2605.28858/) (2026)

## Abstract (arXiv)

Despite their widespread use, purely data-driven methods often suffer from overfitting, lack of physical consistency, and high data dependency, particularly when physical constraints are not incorporated. This study introduces a novel data assimilation approach that integrates Graph Neural Networks (GNNs) with optimisation techniques to enhance the accuracy of mean flow reconstruction, using Reynolds-Averaged Navier-Stokes (RANS) equations as a baseline. The method leverages the adjoint approach, incorporating RANS-derived gradients as optimisation terms during GNN training, ensuring that the learned model adheres to physical laws and maintains consistency. Additionally, the GNN framework is well-suited for handling unstructured data, which is common in the complex geometries encountered in Computational Fluid Dynamics (CFD). The GNN is interfaced with the Finite Element Method (FEM) for numerical simulations, enabling accurate modelling in unstructured domains. We consider the reconstruction of mean flow past bluff bodies at low Reynolds numbers as a test case, addressing tasks such as sparse data recovery, denoising, and inpainting of missing flow data. The key strengths of the approach lie in its integration of physical constraints into the GNN training process, leading to accurate predictions with limited data, making it particularly valuable when data are scarce or corrupted. Results demonstrate significant improvements in the accuracy of mean flow reconstructions, even with limited training data, compared to analogous purely data-driven models.