这篇论文用有限体积残差训练图网络，省了标注数据，效果还比有监督好，做热流体模拟的可以看看。
该研究提出用有限体积法残差训练注意力图神经网络，无需标注数据。在四个场景中验证，两个稳态基准上全场nRMSE为2.3-2.8%。与CFD参考吻合良好。在参数化瞬态案例中，该方法优于有监督基线且避免了数据生成成本。
Label-Free Finite-Volume-Residual Training of Attention Graph Neural Networks for Coupled Thermo-Fluid Fields
Neural surrogates are widely used in scientific machine learning for fast prediction of three-dimensional (3D) thermo-fluid fields. However, generating training data using conventional numerical solvers often incurs substantial computational and storage costs. We propose to train an attention graph neural network by minimizing the finite-volume method (FVM) residuals of the governing equations. These residuals are evaluated directly on the mesh, requiring no labeled data. We evaluate the trained surrogates against computational fluid dynamics (CFD) references and a data-supervised baseline across four scenarios. On the two steady-state benchmarks, the FVM-loss model achieves an all-field normalized root-mean-square error (nRMSE) of 2.3-2.8%. It demonstrates close agreement with the CFD references, including the buoyancy-energy coupling. On the two parametric transient cases, the FVM-loss model outperforms the supervised baseline in terms of accuracy, while avoiding the data-generation cost entirely. These results indicate that the FVM loss can provide a practical training signal for neural surrogates and reduce the model development cost.