The study of partial differential equations (PDE) through the framework of
deep learning emerged a few years ago leading to the impressive approximations
of simple dynamics. Graph neural networks (GNN) turned out to be very useful in
those tasks by allowing the treatment of unstructured data often encountered in
the field of numerical resolutions of PDE. However, the resolutions of harder
PDE such as Navier-Stokes equations are still a challenging task and most of
the work done on the latter concentrate either on simulating the flow around
simple geometries or on qualitative results that looks physical for design
purpose. In this study, we try to leverage the work done on deep learning for
PDE and GNN by proposing an adaptation of a known architecture in order to
tackle the task of approximating the solution of the two-dimensional
steady-state incompressible Navier-Stokes equations over different airfoil
geometries. In addition to that, we test our model not only on its performance
over the volume but also on its performance to approximate surface quantities
such as the wall shear stress or the isostatic pressure leading to the
inference of global coefficients such as the lift and the drag of our airfoil
in order to allow design exploration. This work takes place in a longer project
that aims to approximate three dimensional steady-state solutions over
industrial geometries.

本研究在深度学习的基础上，采用图神经网络对二维定常不可压纳维尔 - 斯托克斯方程在不同机翼几何形状下的解进行逼近，同时测试模型表现在体积和表面量如壁面剪切应力或等压力等的逼近性能，并推导出机翼的提升力和阻力等全局系数进行设计探索。