Finite-difference-informed graph network for solving steady-state incompressible flows on block-structured grids

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.