Graph Transformers for inverse physics: reconstructing flows around arbitrary 2D airfoils

TL;DR

This work tackles the inverse physics problem of reconstructing full flow fields from sparse surface measurements around 2D airfoils by introducing the Flow Reconstruction Graph Transformer (FRGT), a hybrid architecture that combines local graph message-passing with global linear attention on meshes. By encoding geometric priors through GEN-based message passing and leveraging a Galerkin Transformer for long-range information exchange, FRGT achieves accurate predictions of pressure and velocity fields from boundary data with fast inference (~200 ms) on standard hardware. The authors generate a large, discretization-aware dataset of steady 2-D RANS simulations around diverse airfoils and demonstrate robustness to reduced sensor coverage and unseen geometries within the training envelope, providing insights into the relative importance of local versus global processing. The work suggests practical impact for real-time inverse-physics engines on meshes and outlines future directions, including physics-informed priors, uncertainty handling, and mesh-agnostic extensions to broaden applicability.

Abstract

We introduce a Graph Transformer framework that serves as a general inverse physics engine on meshes, demonstrated through the challenging task of reconstructing aerodynamic flow fields from sparse surface measurements. While deep learning has shown promising results in forward physics simulation, inverse problems remain particularly challenging due to their ill-posed nature and the difficulty of propagating information from limited boundary observations. Our approach addresses these challenges by combining the geometric expressiveness of message-passing neural networks with the global reasoning of Transformers, enabling efficient learning of inverse mappings from boundary conditions to complete states. We evaluate this framework on a comprehensive dataset of steady-state RANS simulations around diverse airfoil geometries, where the task is to reconstruct full pressure and velocity fields from surface pressure measurements alone. The architecture achieves high reconstruction accuracy while maintaining fast inference times. We conduct experiments and provide insights into the relative importance of local geometric processing and global attention mechanisms in mesh-based inverse problems. We also find that the framework is robust to reduced sensor coverage. These results suggest that Graph Transformers can serve as effective inverse physics engines across a broader range of applications where complete system states must be reconstructed from limited boundary observations.

Figures (12)

• Figure 1: Depiction of an inverse physics problem on a graph. Given a set of measurements on the sensed nodes in our input graph, we aim to inversely reconstruct the response solution at the remaining unmeasured graph nodes. Often, the number of sensed nodes is a fraction of the total nodes.
• Figure 2: Distribution of airfoil geometries visualized through their first two principal components. The training set (blue) and validation set (orange) span the primary shape variations, while the test set (red) holds an increased share of outlier shapes. We plot some examples of the test set to showcase some of the outlier shapes.
• Figure 3: Validation of the CFD pipeline against experimental measurements gregory1970low and numerical simulations krist1998cfl3d. The pressure coefficient distribution is shown for a NACA 0012 airfoil at $Re=6\cdot10^6$ for angles of attack of (a) 10$^\circ$ and (b) 15$^\circ$.
• Figure 4: Illustration of the simulation pipeline. Airfoil shapes are selected at random from subsets of the UIC database, then meshed and simulated in OpenFOAM. We use quasi-random Sobol sequences to draw the boundary conditions such that the entire parameter range is filled. We use a finite-volume scheme to convert the simulations into the input graphs.
• Figure 5: Geometrical features used as inputs to the model. Overall the signed distance field is plotted, with a zoom on the boundary length and relative edge direction between two nodes.