Learning Non-Ideal Vortex Flows Using the Differentiable Vortex Particle Method

TL;DR

The paper tackles modeling non-ideal vortex flows by extending the differentiable vortex particle method (DVPM) to include viscous diffusion and non-conservative body forces. It adopts the Lamb-Oseen vortex, with an analytic solution from the Navier–Stokes equations, as a rigorous benchmark and demonstrates that the DVPM framework achieves higher accuracy than CNN and PINN baselines as well as the original DVPM across a range of conditions. The approach uses four neural networks to predict vortex attributes and a differentiable Biot-Savart kernel within an end-to-end training loop that couples Lagrangian vortex dynamics to Eulerian flow predictions. This work advances practical data-driven vortex modeling for non-ideal flows, including Coriolis effects, with potential impact on engineering and geophysical applications.

Abstract

Vortex flows are ubiquitous in both natural processes and engineering applications, including phenomena such as typhoons, water currents, and aerospace fluid dynamics. The vortex particle method, a computational approach grounded in vortex dynamics, has been extensively applied in aerodynamics, oceanography, turbulence, and aeroacoustics. With the recent introduction of machine learning into computational fluid dynamics, a hybrid framework known as the differentiable vortex particle method (DVPM) has been proposed, which integrates the vortex particle method with deep learning to enable efficient learning and prediction. However, the original formulation of DVPM is limited to ideal vortex flow conditions, such as inviscid flows without non-conservative body forces, which significantly restricts its practical applicability. In this study, we extend the differentiable vortex particle method beyond idealized flow scenarios to encompass more realistic, non-ideal conditions, including viscous flow and flow subjected to non-conservative body forces. We establish the Lamb-Oseen vortex as a benchmark case, representing a fundamental viscous vortex flow in fluid mechanics. This selection offers significant analytical advantages, as the Lamb-Oseen vortex possesses an exact analytical solution derived from the Navier-Stokes (NS) equations, thereby providing definitive ground truth data for training and validation purposes. Through rigorous evaluation across a spectrum of Reynolds numbers, we demonstrate that DVPM achieves superior accuracy in modeling the Lamb-Oseen vortex compared to conventional convolutional neural networks (CNNs) and physics-informed neural networks (PINNs). Our results substantiate DVPM's robust capabilities in modeling non-ideal vortex flows, establishing its distinct advantages over contemporary deep learning methodologies in fluid dynamics applications.

Figures (9)

• Figure 1: The physical model in our framework. $K\left(\left|\boldsymbol{\boldsymbol{x}}-\boldsymbol{x}\prime\right| / \delta\right)$ represents the kernel function, whose value is determined by multiple parameters: the dependent variable, the Euclidean distance between the vortex core and the point of interest in the flow field $\left|\boldsymbol{x}-\boldsymbol{x}_p\right|$, and the vortex radius $\delta$. Each vortex is characterized by a distinct vorticity $\omega$, which reduces to a scalar quantity in two-dimensional flow configurations.
• Figure 2: An overview of our framework.
• Figure 3: Comparative evaluation of Lamb-Oseen vortex prediction performance at $Re = 10$ among the proposed method and baseline models.
• Figure 4: Comparative evaluation of Lamb-Oseen vortex prediction performance at $Re = 100$ among the proposed method and baseline models.
• Figure 5: Comparative evaluation of Lamb-Oseen vortex prediction performance at $Re = 1000$ among the proposed method and baseline models.