In the field of fluid numerical analysis, there has been a long-standing problem: lacking of a rigorous mathematical tool to map from a continuous flow field to discrete vortex particles, hurdling the Lagrangian particles from inheriting the high resolution of a large-scale Eulerian solver. To tackle this challenge, we propose a novel learning-based framework, the Neural Vortex Method (NVM), which builds a neural-network description of the Lagrangian vortex structures and their interaction dynamics to reconstruct the high-resolution Eulerian flow field in a physically-precise manner. The key components of our infrastructure consist of two networks: a vortex representation network to identify the Lagrangian vortices from a grid-based velocity field and a vortex interaction network to learn the underlying governing dynamics of these finite structures. By embedding these two networks with a vorticity-to-velocity Poisson solver and training its parameters using the high-fidelity data obtained from high-resolution direct numerical simulation, we can predict the accurate fluid dynamics on a precision level that was infeasible for all the previous conventional vortex methods (CVMs). To the best of our knowledge, our method is the first approach that can utilize motions of finite particles to learn infinite dimensional dynamic systems. We demonstrate the efficacy of our method in generating highly accurate prediction results, with low computational cost, of the leapfrogging vortex rings system, the turbulence system, and the systems governed by Euler equations with different external forces.

翻译：在流体数值分析领域，长期存在一个难题：缺乏严格的数学工具将连续流场映射到离散涡旋粒子，这阻碍了拉格朗日粒子继承大规模欧拉求解器的高分辨率特性。为应对这一挑战，我们提出了一种基于学习的新型框架——神经涡旋法（NVM），该方法通过构建拉格朗日涡旋结构及其相互作用动力学的神经网络描述，以物理精确的方式重建高分辨率欧拉流场。该基础设施的关键组件包含两个网络：涡旋表征网络，用于从基于网格的速度场中识别拉格朗日涡旋；涡旋相互作用网络，用于学习这些有限结构的潜在控制动力学。通过将这两个网络嵌入涡度-速度泊松求解器，并利用高分辨率直接数值模拟获得的高保真数据训练其参数，我们能够在传统涡旋方法（CVMs）无法企及的精度水平上预测精确的流体动力学。据我们所知，本方法是首个能够利用有限粒子运动学习无限维动力系统的方法。我们通过实验证明了该方法在生成蛙跳涡环系统、湍流系统及受不同外力作用的欧拉方程系统的高精度预测结果时的有效性，且计算成本低廉。