# Learning Airfoil Flow Field Representation via Geometric Attention Neural Field
## 2 Methodology
### GeoANF  
GeoANF is capable of implicitly representing flow field conditions under various free-stream velocities, airfoil shapes, and angles of attack. The core of this approach lies in utilizing the latent space representation of geometric information generated by the GeoAttention, concatenating it with the features of the query point cloud, and outputting key physical quantities of the flow field such as velocity, pressure, and turbulent viscosity. Figure 5a provides an overview for GeoANF, including both the inference and training phases. We utilize the NN block to project the feature space into the latent space or to decode the latent space back into the observed values. As shown in Figure 5b, the NN block consists of a combination of three linear layers and two ReLU activation functions, which enables the model to capture nonlinear features effectively. Inference: GeoANF takes the airfoil boundary points (assumed to consist of \(m\) points) and \(n\) feature space query points as inputs, each represented in a \(d_{p}\)-dimensional space, Figure 5b displays a basic module, the NN block, which maps them into the neural representation space \(H\). The query points, in conjunction with the airfoil boundary points, are fed into the GeoAttention module to obtain a geometric information representation \(g\). Subsequently, we concatenatd these representations and pass them into the neural field for decoding into the flow field. Training: the computation of GeoANF is largely consistent with that of inference. For the dataset, we utilize CFD results, which are typically stored in spatially discretized grid points or elements. The training process of GeoANF reduces these data to point clouds and observed values as inputs. We input the airfoil boundary points from the mesh as geometric information and perform random sampling across the entire mesh (including the airfoil boundary points) as feature query points. The corresponding flow field physical quantities serve as labels, and the entire system is trained in an end-to-end manner. As shown in Figure 5d, the neural field module employs an NN block, followed by processing through a ReLU MLP with batch normalization [35]. This step is aimed at further optimizing the training process and enhancing the model's generalization capability. Batch normalization, a widely used technique in deep learning, accelerates the training process and reduces the model's sensitivity to the initial weight selection by normalizing the distribution of each batch of data. In processing flow field data, the physical quantities of the flow field, such as velocity and pressure, may exhibit dramatic spatial variations, making it challenging for the model to converge stably and quickly. By applying batch normalization, the impact of these variations can be mitigated, resulting in a more stable training process and aiding the model in converging to a more optimal solution. The final stage of the model involves decoding the high-dimensional space features, such as velocity and pressure, back to the target flow field through another NN block. Figure 5: GeoAttention neural field overview. (**a**) provides an overview of GeoANF, including both the inference and training phases. Airfoil boundary points (assumed to consist of \(m\) points) and \(n\) feature space query points, are each represented in a \(d_{p}\)-dimensional space. These inputs are processed through distinct NN block to the neural representation space \(\mathcal{H}\). GeoAttention processes the query points, in conjunction with the airfoil boundary points, obtain a geometric representation \(g\). Subsequently, we concatenate \(z\) and \(g\), which passing the neural field to decode into the flow field. (**b**) shows that the basic module used by the neural network is three-layer ReLU MLP, which has \(c_{i}\) input and \(c_{o}\) output channels. (**c**) Shows the process of GeoAttention. A point representation \(z_{i}\), calculates attention weights with geometric representations of the airfoil boundary points, obtaining \(g_{i}\).