Learning non-ideal vortex flows using the differentiable vortex particle method

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 (VPM), 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.