# Learning Airfoil Flow Field Representation via Geometric Attention Neural Field
## 3 Experiment
### Result and Discussion  
Figures 9 and 10 illustrate the visualization results for a AoA 14.068\({}^{\circ}\) and incoming flow velocity 49.099 m/s test case under the full data regime training configuration. Figure 8: GeoANF training process. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Running Time**} & \multirow{2}{*}{**\#Parameters**} \\  & **Training** & **Inference (\(\mu\)s)** & \\ \hline MLP & \(\sim\)2 h 20 min & \(13.3\pm 0.2\) & 19,988 \\ GraphSAGE & \(\sim\)4 h 20 min & \(20.9\pm 2.3\) & 29,204 \\ PointNet & \(\sim\)2 h 40 min & \(33.9\pm 3.5\) & 75,244 \\ Graph U-Net & \(\sim\)6 h 50 min & \(357.8\pm 36.6\) & 65,820 \\ GeoANF & \(\sim\)20 h 30 min & \(7.55\pm 2.0\) & 66,108 \\ \hline Simulation & & \(\sim\)25 min & \\ Dataset & & \(\sim\)20 days & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Training/inference time and number of parameters. For flow field visualization, we applied a spatial smoothing filter; as shown in Figure 9, we plotted contours for the magnitude of velocity, pressure, and turbulence viscosity. For velocity and pressure predictions, the flow field structure is well captured. For the streamline visualization, Figure 10 shows streamtraces of the velocity field across the entire space and the trailing edge vortex that appears. It can be seen that GeoANF accurately captures this detail. The GeoANF has comparable performance on this task. We compared the baseline model of airfrans, the baseline outcomes [34] are derived from AirfRANS, with our metrics being aligned with those from AirfRANS. The GeoANF model is trained five times to compute a mean and a standard deviation for full data  
Figure 10: Streamtrace visualization comparison. Figure 9: Flow field comparison between the prediction and CFD. regime, as illustrated in Table 4. Compared to the baseline model, GeoANF demonstrates improvements and robustness to random sampling. Specifically, the GeoANF model achieved notably reduced error rates in tasks related to volume flow field velocity and pressure fields. In terms of the predicted results for velocity and pressure, compared to the best results of the baseline model, there is an approximate 50% reduction in MSE. Furthermore, in the prediction of surface pressure, a more pronounced reduction in MSE was observed. Compared to the best results of the Graph U-Net model in predicting boundary pressure, our model reduces the error by a factor of 0.25. The calculated lift coefficient based on the predicted boundary pressure results also demonstrates improved accuracy, achieving a relative error of 0.08. Compared to the best results of the Graph U-Net model in the baseline, this represents a reduction to one-sixth of its error, the GeoANF model exhibited low standard deviation across various metrics over five instances of training, demonstrating robustness to random sampling. Additionally, a higher correlation was noted in the Spearman's rank correlation coefficient, indicating improved performance across various metrics. Figure 11 presents a visualization of the correlation between the lift and drag coefficients from CFD and the predicted surface pressure by various deep learning models, the GeoANF model achieves better outcomes in correlation evaluation metrics. With respect to the lift coefficient, the GeoANF model demonstrates a high degree of correlation with the CFD results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Mean Squared Error (MSE) on Normalized Fields**} & \\ \hline  & & **GeoANF** & **GraphSAGE** & **MLP** & **Graph U-Net** & **PointNet** \\ \hline \multirow{4}{*}{ \begin{tabular}{c} Volume \\ (\(\times 10^{-2}\)) \\ \end{tabular}  } & \(v_{x}\) & **0.43 \(\pm\) 0.07** & 0.83 \(\pm\) 0.01 & 0.95 \(\pm\) 0.06 & 1.52 \(\pm\) 0.34 & 3.50 \  
Since the values in most regions are relatively small, we applied a logarithmic scale to the color levels for visualization.