We present two novel generative geometric deep learning frameworks, termed Flow Matching PointNet and Diffusion PointNet, for predicting fluid flow variables on irregular geometries by incorporating PointNet into flow matching and diffusion models, respectively. In these frameworks, a reverse generative process reconstructs physical fields from standard Gaussian noise conditioned on unseen geometries. The proposed approaches operate directly on point-cloud representations of computational domains (e.g., grid vertices of finite-volume meshes) and therefore avoid the limitations of pixelation used to project geometries onto uniform lattices, as is common in U-Net-based flow matching and diffusion models. In contrast to graph neural network-based diffusion models, Flow Matching PointNet and Diffusion PointNet do not exhibit high-frequency noise artifacts in the predicted fields. Moreover, unlike such approaches, which require auxiliary intermediate networks to condition geometry, the proposed frameworks rely solely on PointNet, resulting in a simple and unified architecture. The performance of the proposed frameworks is evaluated on steady incompressible flow past a cylinder, using a geometric dataset constructed by varying the cylinder's cross-sectional shape and orientation across samples. The results demonstrate that Flow Matching PointNet and Diffusion PointNet achieve more accurate predictions of velocity and pressure fields, as well as lift and drag forces, and exhibit greater robustness to incomplete geometries compared to a vanilla PointNet with the same number of trainable parameters.

翻译：本文提出了两种新颖的生成式几何深度学习框架——流匹配PointNet与扩散PointNet，分别通过将PointNet融入流匹配模型与扩散模型，实现对不规则几何上流体流动变量的预测。在这些框架中，一个逆向生成过程以未见几何为条件，从标准高斯噪声中重建物理场。所提出的方法直接作用于计算域的点云表示（例如有限体积网格的网格顶点），从而避免了基于U-Net的流匹配与扩散模型中常见的、将几何投影至均匀网格所导致的像素化局限。与基于图神经网络的扩散模型相比，流匹配PointNet与扩散PointNet在预测场中不会出现高频噪声伪影。此外，不同于此类需要辅助中间网络进行几何条件化的方法，所提出的框架仅依赖PointNet，形成了简洁统一的架构。通过在圆柱绕流稳态不可压缩流动问题上进行评估——使用通过在不同样本间改变圆柱截面形状与方向构建的几何数据集——结果表明：相较于具有相同可训练参数量的基础PointNet，流匹配PointNet与扩散PointNet能够更准确地预测速度场、压力场以及升力与阻力，并对不完整几何表现出更强的鲁棒性。