A Geometry-Aware Message Passing Neural Network for Modeling Aerodynamics over Airfoils

TL;DR

This work introduces GeoMPNN, a geometry-aware message passing neural network for steady incompressible flows over airfoils. The method first builds a latent geometric representation of the airfoil via Surf2Vol on the surface and then propagates this information to the volume mesh through a directed bipartite graph, enabling efficient training on subsampled meshes while preserving full-mesh accuracy at test time. It enhances geometry expressiveness with augmented coordinate systems (trailing edge, polar-Cartesian, and basis embeddings) and canonicalizes inputs with respect to inlet velocity to boost generalization, achieving state-of-the-art performance in the NeurIPS ML4CFD competition on the AirfRANS dataset. The combination of Surf2Vol geometry encoding, advanced coordinate embeddings, and pressure transformations yields improved predictions for velocity, pressure, and turbulence, with practical implications for fast, geometry-robust airfoil surrogate modeling.

Abstract

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).

Figures (21)

• Figure 1: Full mesh (left) and subsampled mesh (right). To reduce training costs, we randomly sample 32K mesh points for each training example at each epoch. Evaluation is conducted at the full resolution.
• Figure 2: Scores for the MLP and GNN baselines across 8 runs.
• Figure 3: Difference in errors between subsampled and full resolution for MLP and GNN. We analyze the effect of increasing evaluation resolution by assessing the relative difference in the test error on the full resolution compared to the error on the subsampled resolution, defined in \ref{['eq:reldiff']}. As can be seen, increasing resolution substantially increases the error on each field for the GNN. Note that error on the turbulent viscosity field is omitted due to overfitting for both the MLP and GNN.
• Figure 4: Comparison of the Surf2Vol model to the GNN. Top: MSE errors on each field, where lower is better, and the targets are normalized as discussed in \ref{['sec:targets']}. Bottom: Scores in each category, where higher is better.
• Figure 5: Difference in errors between subsampled and full resolution for Surf2Vol model and GNN. The Surf2Vol model error does not change with increased resolution.