# Learning Airfoil Flow Field Representation via Geometric Attention Neural Field
## 3 Experiment
### Result and Discussion
Figure 12 presents a comparison of the normalized mean squared error (MSE) of each model relative to the actual flow field values. The visualization results from the first to the fifth row sequentially represent the overall error, error of velocity-x, error of velocity-y, error of pressure, and error of turbulent viscosity. The GeoANF model has shown advantages over other baseline models across all listed evaluation metrics. Specifically, the majority of the errors in GeoANF's overall performance remain below 0.0139, while the baseline models exhibit large error regions exceeding 0.0373. Among the baseline models, the MLP demonstrates relatively better results in terms of overall error for the volume prediction. In the velocity field, GeoANF maintains errors below 0.00518 in most regions, whereas other models show large areas with errors exceeding 0.0139. Due to the relatively large absolute values of pressure, GeoANF's visualization also reveals error regions exceeding 0.013 in comparison to baseline models. However, GeoANF still outperforms all other models overall. In the prediction of viscosity, most errors are concentrated in the wake region, but compared to other models, GeoANF keeps errors below 0.0003 outside the wake, while the other models exhibit errors across non-wake regions. In the visualization of boundary pressure, we adopted the pressure coefficient approach to compare the baseline model with the results obtained from GeoANF and CFD solutions. The pressure coefficient \(C_{p}\) at the surface of the airfoil is defined as a dimensionless coefficient as follows:  
\[C_{p}:=\frac{\overline{p}-\overline{p}_{\infty}}{q_{\infty}},q_{\infty}:= \frac{1}{2}\mathcal{U}_{\infty}^{2}A \tag{9}\]  
Figure 12: Error comparison. Visualization of the error results with the real flow field, from top to bottom, the total error, the error of velocity-x, the error of velocity-y, the error of pressure and the error of \(\mu\)t.  
where \(\overline{p}\) is the mean-field reduced pressure, \(\overline{p}_{\infty}\) the far field pressure (set to 0 in the incompressible case), \(U_{\infty}\) is the magnitude of the inlet velocity and A is the characteristic area of the problem, we take here A = 1 \(m^{2}\). As illustrated in Figure 13, we present the visualization results of the boundary pressure coefficients for two cases. The inflow velocities for the first and second images are 80.247 and 49.099 m per second (m/s), respectively, with angles of attack of 1.078 and 14.068 degrees, respectively. GeoANF demonstrates the best alignment with CFD results in terms of \(C_{p}\) result. Nevertheless, GeoANF remains the best-performing model among them. However, it can be observed that Figure 13 first images, at the tail end of the airfoil, the \(C_{p}\) results from all deep learning methods fail to predict accurately. Figure 13: The pressure coefficient distribution on the airfoil surface for two CFD cases. We also compared the performance with relatively newer models. Due to the neural network's limitations in drag prediction, we focused on comparing the prediction performance of the lift coefficient. We evaluated the volume surface results in the full data regime, as well as lift predictions for two challenging extrapolation tasks, as shown in Table 5. GeoANF exhibits impressive performance, demonstrating advantages in predicting surface pressure and results in lift coefficients. Although its volume performance is inferior to that of other state-of-the-art (SOTA) models, these SOTA models typically possess a significantly larger number of network parameters, e.g., Transolver has approximately 2.5M parameters. As shown in Table 3, GeoANF has about 66,000 parameters, which is considerably fewer than the parameter requirements of models like Transolver, resulting in a lower training cost. As demonstrated in Table 6, the performance gap between the scarce data and full data tasks for the GeoANF model is minimal, highlighting its robust learning capability, particularly in handling scarce data.