# Learning Airfoil Flow Field Representation via Geometric Attention Neural Field
(2023) proposed INFINITY [23], which attempts to learn the signed distance field (SDF) of geometric shapes in flow fields through an auto-decoding process, aiming to achieve geometric generalization across various shapes. Recently, the Transformer architecture [24] has garnered widespread attention due to its remarkable success in natural language processing [25; 26; 27]. Its core component--the attention mechanism--has inspired efforts to apply it in computer vision [28; 29; 30]. [PERSON] et al. (2023) [31] utilized a Vision Transformer [28] to extract geometric features from images of geometric shapes. Additionally, in Transformer-based operator learning methods, such as GNOT [32] and Transolver [33], most approaches require improvements in algorithmic efficiency (transformer quadratic time complexity). While these methods are capable of handling point cloud data, most general operator learning techniques do not account for the specialization needed for geometric structures. These methods face challenges in accurately representing geometric shapes and minor changes. In particular, methods based on parametric representations of shapes lack a unified approach to shape parameterization, making it challenging for neural networks to learn mappings for parameterized shape representations across different schemes. Furthermore, image processing-based approaches generally only generalize to minor shape variations, and pixelization and voxelization lead to lossy representations of irregular geometries. These limitations render current methods particularly ineffective in capturing geometric changes, as shown in Figure 3, even though flow field structures can exhibit significant variations due to minor changes in geometry. To address this research gap, we introduce GeoAttention, which utilizes an attention mechanism to encode geometric information from point cloud representations. GeoAttention assigns different levels of importance to various parts of the geometry, as represented by the point cloud, based on different query latent features. This approach effectively encodes irregular geometric information into a unified latent vector. By capturing the mapping between flow fields and geometric relationships, GeoAttention significantly enhances the geometric generalization ability of INRs in solving physical problems. GeoAttention significantly enhances the geometric generalization ability of implicit neural representations (INRs) in solving physical problems and allows for more precise capture of subtle geometric variations, resulting in improved performance in predicting boundaries. The main contributions of this work can be summarized as follows:1. This paper introduces GeoAttention, a new method for geometric representation learning on irregular point clouds for flow field prediction. GeoAttention efficiently encodes geometric information without the need for voxelization or pixelization, enabling more accurate and flexible geometric representation. 2. By integrating GeoAttention with implicit neural field, we propose geometric attention neural field (GeoANF), which is capable of learning from completely random subsampling. It demonstrates robustness to different grid samplings and adaptability to various geometric shapes. 3. To verify the effectiveness of GeoANF, a series of experiments were conducted using the AirfRANS dataset [34]. Across four challenging tasks presented in the dataset, the GeoANF model consistently outperformed the baseline models. Additionally, we visualized and analyzed the normalized attention weight maps. These visualizations demonstrate the model's ability to assign different weights to geometric information based on query points.