This study presents a data-driven approach to reveal scale-invariant vortical structures in turbulent flows across different Reynolds numbers. The authors use Buckingham Pi-based sparse nonlinear scaling to identify the influence of Pi groups on turbulent flow data, focusing on the invariants of the velocity gradient tensor in three-dimensional decaying isotropic turbulence. The scaling method not only identifies vortical structures that are interpolatory and extrapolatory but also captures non-equilibrium effects of energy cascade. The findings are applied to machine-learning-based super-resolution analysis, demonstrating that machine-learning models can reconstruct vortical structures well in the interpolatory space but perform less well in the extrapolatory space. The study provides insights into the robustness of nonlinear machine-learning models for turbulent flows and offers guidance on training and testing these models effectively.This study presents a data-driven approach to reveal scale-invariant vortical structures in turbulent flows across different Reynolds numbers. The authors use Buckingham Pi-based sparse nonlinear scaling to identify the influence of Pi groups on turbulent flow data, focusing on the invariants of the velocity gradient tensor in three-dimensional decaying isotropic turbulence. The scaling method not only identifies vortical structures that are interpolatory and extrapolatory but also captures non-equilibrium effects of energy cascade. The findings are applied to machine-learning-based super-resolution analysis, demonstrating that machine-learning models can reconstruct vortical structures well in the interpolatory space but perform less well in the extrapolatory space. The study provides insights into the robustness of nonlinear machine-learning models for turbulent flows and offers guidance on training and testing these models effectively.