This study presents a data-driven approach to reveal scale-invariant vortical structures in turbulent flows using Buckingham Pi-based sparse nonlinear scaling. The method identifies the influence of Pi groups on turbulent flow data, enabling the identification of interpolatory and extrapolatory vortical structures. The approach is applied to three-dimensional isotropic turbulence, where nonlinear scalings of velocity gradient tensor invariants are used to capture non-equilibrium effects of the energy cascade. The results show that machine-learning models reconstruct vortical structures well in interpolatory space but perform poorly in extrapolatory space. The study also demonstrates that the present approach allows for a comprehensive examination of flow fields, supporting training and testing of nonlinear machine-learning techniques. The findings suggest that nonlinear machine-learning models can capture scale-invariant characteristics of turbulent flows, even outside the range of training data. The study highlights the importance of considering both seen and unseen structures in turbulent flows, and provides guidance on the choice of machine-learning functions for robustness. The results also show that the present approach can be used for super-resolution analysis of turbulent flows, with the ability to reconstruct vortical structures even in untrained Reynolds number cases. The study concludes that the present approach provides a framework for developing robust machine-learning models and compiling necessary training data, enabling the use of complex models without naive training.This study presents a data-driven approach to reveal scale-invariant vortical structures in turbulent flows using Buckingham Pi-based sparse nonlinear scaling. The method identifies the influence of Pi groups on turbulent flow data, enabling the identification of interpolatory and extrapolatory vortical structures. The approach is applied to three-dimensional isotropic turbulence, where nonlinear scalings of velocity gradient tensor invariants are used to capture non-equilibrium effects of the energy cascade. The results show that machine-learning models reconstruct vortical structures well in interpolatory space but perform poorly in extrapolatory space. The study also demonstrates that the present approach allows for a comprehensive examination of flow fields, supporting training and testing of nonlinear machine-learning techniques. The findings suggest that nonlinear machine-learning models can capture scale-invariant characteristics of turbulent flows, even outside the range of training data. The study highlights the importance of considering both seen and unseen structures in turbulent flows, and provides guidance on the choice of machine-learning functions for robustness. The results also show that the present approach can be used for super-resolution analysis of turbulent flows, with the ability to reconstruct vortical structures even in untrained Reynolds number cases. The study concludes that the present approach provides a framework for developing robust machine-learning models and compiling necessary training data, enabling the use of complex models without naive training.