Physics-constrained graph neural networks for solving adjoint equations

The adjoint method is widely used in gradient-based optimization with high-dimensional design variables. However, the cost of solving the adjoint equations in each iteration is comparable to that of solving the flow field, resulting in expensive computational costs. To improve the efficiency of solving adjoint equations, we propose a physics-constrained graph neural networks for solving adjoint equations, named ADJ-PCGN. ADJ-PCGN establishes a mapping relationship between flow characteristics and adjoint vector based on data, serving as a replacement for the computationally expensive numerical solution of adjoint equations. A physics-based graph structure and message-passing mechanism are designed to endow its strong fitting and generalization capabilities. Taking transonic drag reduction and maximum lift-drag ratio of the airfoil as examples, results indicate that ADJ-PCGN attains a similar optimal shape as the classical direct adjoint loop method. In addition, ADJ-PCGN demonstrates strong generalization capabilities across different mesh topologies, mesh densities, and out-of-distribution conditions. It holds the potential to become a universal model for aerodynamic shape optimization involving states, geometries, and meshes.