Advances in deep learning have enabled physics-informed neural networks to solve partial differential equations. Numerical differentiation using the finite-difference (FD) method is efficient in physics-constrained designs, even in parameterized settings. In traditional computational fluid dynamics(CFD), body-fitted block-structured grids are often employed for complex flow cases when obtaining FD solutions. However, convolution operators in convolutional neural networks for FD are typically limited to single-block grids. To address this issue, \blueText{graphs and graph networks are used} to learn flow representations across multi-block-structured grids. \blueText{A graph convolution-based FD method (GC-FDM) is proposed} to train graph networks in a label-free physics-constrained manner, enabling differentiable FD operations on unstructured graph outputs. To demonstrate model performance from single- to multi-block-structured grids, \blueText{the parameterized steady incompressible Navier-Stokes equations are solved} for a lid-driven cavity flow and the flows around single and double circular cylinder configurations. When compared to a CFD solver under various boundary conditions, the proposed method achieves a relative error in velocity field predictions on the order of $10^{-3}$. Furthermore, the proposed method reduces training costs by approximately 20\% compared to a physics-informed neural network. \blueText{To} further verify the effectiveness of GC-FDM in multi-block processing, \blueText{a 30P30N airfoil geometry is considered} and the \blueText{predicted} results are reasonable compared with those given by CFD. \blueText{Finally, the applicability of GC-FDM to three-dimensional (3D) case is tested using a 3D cavity geometry.

翻译：深度学习的发展使得物理信息神经网络能够求解偏微分方程。有限差分（FD）方法在物理约束设计中，甚至在参数化设置中，进行数值微分是高效的。在传统的计算流体动力学（CFD）中，获取有限差分解时，对于复杂的流动情况，通常采用贴体块结构化网格。然而，用于有限差分的卷积神经网络中的卷积算子通常局限于单块网格。为了解决这个问题，**我们使用图和图网络**来学习跨多块结构化网格的流动表示。**我们提出了一种基于图卷积的有限差分方法（GC-FDM）**，以无标签的物理约束方式训练图网络，从而能够在非结构化的图输出上进行可微分的有限差分操作。为了展示模型从单块到多块结构化网格的性能，**我们求解了参数化的稳态不可压缩Navier-Stokes方程**，包括顶盖驱动腔流以及单圆柱和双圆柱构型周围的流动。与CFD求解器在各种边界条件下的结果相比，所提方法在速度场预测上达到了$10^{-3}$量级的相对误差。此外，与物理信息神经网络相比，所提方法降低了约20%的训练成本。**为了**进一步验证GC-FDM在多块处理中的有效性，**我们考虑了一个30P30N翼型几何**，并且**预测**结果与CFD给出的结果相比是合理的。**最后，我们使用一个三维腔体几何测试了GC-FDM对三维（3D）情况的适用性。