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, graphs and graph networks are used to learn flow representations across multi-block-structured grids. 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, 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 in the order of 10−3. Furthermore, the proposed method reduces training costs by approximately 20% compared to a physics-informed neural network. To further verify the effectiveness of GC-FDM in multi-block processing, a 30P30N airfoil geometry is considered, and the predicted results are reasonably compared with those given by CFD. Finally, the applicability of GC-FDM to a three-dimensional (3D) case is tested using a 3D cavity geometry.