Vorticity–velocity physics-informed neural networks for spatio-temporal super-resolution of sub-sampled velocity measurements

Velocity measurement techniques, such as particle image velocimetry (PIV), face a trade-off between field of view, spatial resolution and sampling rate, so that small-scale vortices, shear layers and high-frequency turbulent motions are often under-resolved. Most physics-informed reconstructions use a velocity–pressure formulation, even though pressure is not measured in typical PIV experiments, so the Navier–Stokes constraints are only weakly enforced. We address this issue by formulating a vorticity–velocity physics-informed network (VVPINN), in which pressure is eliminated and incompressibility is enforced together with a vorticity transport equation, thereby directly constraining the velocity field and its derivatives. We then compare this formulation with a conventional velocity–pressure PINN (VPPINN) for spatio-temporal super-resolution of planar PIV data in three cases: a laminar multi-cylinder wake, a two-dimensional Taylor–Green vortex and an experimental two-cylinder wake. In the Taylor–Green vortex case, with identical architectures and training strategies, the VVPINN yields smaller velocity errors, reduces the $L_2$ errors in vorticity and shear by approximately $10\,\%$ , and the pressure gradient errors by up to approximately $30\,\%$ at moderate super-resolution factors, and produces instantaneous fields with more physically plausible vorticity, shear and fine-scale pressure gradient patterns. Spectral analysis shows that the temporal energy spectrum is recovered accurately, whereas the wavenumber spectra, particularly beyond the Nyquist wavenumber, remain more difficult to match because the training data strongly constrain the time histories at sampled locations, but only indirectly inform the smallest spatial scales. Overall, the results indicate that vorticity-based constraints provide a more effective route to physics-consistent super-resolution of sub-sampled PIV data than the conventional velocity–pressure formulation.