Computational modeling of aerodynamics is a key problem in aerospace engineering, often involving flows interacting with solid objects such as airfoils. Deep surrogate models have emerged as purely data-driven approaches that learn direct mappings from simulation conditions to solutions based on either simulation or experimental data. Here, we consider modeling of incompressible flows over solid objects, wherein geometric structures are a key factor in determining aerodynamics. To effectively incorporate geometries, we propose a message passing scheme that efficiently and expressively integrates the airfoil shape with the mesh representation. Under this framework, we first obtain a representation of the geometry in the form of a latent graph on the airfoil surface. We subsequently propagate this representation to all collocation points through message passing on a directed, bipartite graph. We demonstrate that this framework supports efficient training by downsampling the solution mesh while avoiding distribution shifts at test time when evaluated on the full mesh. To enable our model to be able to distinguish between distinct spatial regimes of dynamics relative to the airfoil, we represent mesh points in both a leading edge and trailing edge coordinate system. We further enhance the expressiveness of our coordinate system representations by embedding our hybrid Polar-Cartesian coordinates using sinusoidal and spherical harmonics bases. We additionally find that a change of basis to canonicalize input representations with respect to inlet velocity substantially improves generalization. Altogether, these design choices lead to a purely data-driven machine learning framework known as GeoMPNN, which won the Best Student Submission award at the NeurIPS 2024 ML4CFD Competition, placing 4th overall. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

翻译：空气动力学计算建模是航空航天工程中的关键问题，通常涉及与翼型等固体物体相互作用的流动。深度代理模型作为一种纯数据驱动方法，基于仿真或实验数据学习从仿真条件到流场解的直接映射。本文研究固体物体表面不可压缩流动的建模问题，其中几何结构是决定空气动力学的关键因素。为有效融合几何信息，我们提出一种消息传递机制，能够高效且富有表现力地将翼型形状与网格表示相结合。在此框架下，我们首先通过翼型表面的潜在图获得几何表征，随后通过有向二分图上的消息传递将该表征传播至所有配点。我们证明该框架支持通过降采样求解网格实现高效训练，同时在完整网格测试时避免分布偏移。为使模型能够区分翼型相关的不同空间动态区域，我们采用前缘与后缘双坐标系表示网格点。通过将混合极坐标-笛卡尔坐标嵌入正弦与球谐基函数，进一步增强了坐标系表示的表现力。此外，我们发现将输入表示相对于来流速度进行规范化的基变换能显著提升泛化能力。这些设计共同构成了名为GeoMPNN的纯数据驱动机器学习框架，该框架在NeurIPS 2024 ML4CFD竞赛中获得最佳学生提交奖，总排名第四。我们的代码已作为AIRS库（https://github.com/divelab/AIRS）的一部分公开。