This study presents a novel hybrid approach that combines Graph Neural Networks (GNNs) with Reynolds-Averaged Navier Stokes (RANS) equations to enhance the accuracy of mean flow reconstruction across a range of fluid dynamics applications. Traditional purely data-driven Neural Networks (NNs) models, often struggle maintaining physical consistency. Moreover, they typically require large datasets to achieve reliable performances. The GNN framework, which naturally handles unstructured data such as complex geometries in Computational Fluid Dynamics (CFD), is here integrated with RANS equations as a physical baseline model. The methodology leverages the adjoint method, enabling the use of RANS-derived gradients as optimization terms in the GNN training process. This ensures that the learned model adheres to the governing physics, maintaining physical consistency while improving the prediction accuracy. We test our approach on multiple CFD scenarios, including cases involving generalization with respect to the Reynolds number, sparse measurements, denoising and inpainting of missing portions of the mean flow. The results demonstrate significant improvements in the accuracy of the reconstructed mean flow compared to purely data-driven models, using limited amounts of data in the training dataset. The key strengths of this study are the integration of physical laws into the training process of the GNN, and the ability to achieve high-accuracy predictions with a limited amount of data, making this approach particularly valuable for applications in fluid dynamics where data is often scarce.

翻译：本研究提出了一种新颖的混合方法，将图神经网络与雷诺平均纳维-斯托克斯方程相结合，以提高多种流体动力学应用中平均流场重建的精度。传统的纯数据驱动神经网络模型往往难以保持物理一致性，且通常需要大量数据集才能达到可靠性能。本文提出的框架将天然适用于处理非结构化数据（如计算流体力学中复杂几何形状）的图神经网络，与作为物理基准模型的RANS方程相融合。该方法利用伴随方法，使RANS导出的梯度能够作为优化项融入GNN训练过程，从而确保学习模型遵循控制物理规律，在保持物理一致性的同时提升预测精度。我们在多种CFD场景中测试了该方法，包括涉及雷诺数泛化、稀疏测量、平均流场去噪及缺损区域修复等案例。结果表明，与纯数据驱动模型相比，该方法在训练数据集有限的情况下，能显著提升重建平均流场的精度。本研究的核心优势在于将物理定律融入GNN训练过程，并能在数据有限的情况下实现高精度预测，这使得该方法对于数据通常稀缺的流体动力学应用具有重要价值。