  Nonlinear machine learning for turbulent flows can exhibit robust performance
even outside the range of training data. This is achieved when machine-learning
models can accommodate scale-invariant characteristics of turbulent flow
structures. This study presents a data-driven approach to reveal
scale-invariant vortical structures across Reynolds numbers that provide
insights for supporting nonlinear machine-learning-based studies of turbulent
flows. To uncover conditions for which nonlinear models are likely to perform
well, we use a Buckingham Pi-based sparse nonlinear scaling to find the
influence of the Pi groups on the turbulent flow data. We consider nonlinear
scalings of the invariants of the velocity gradient tensor for an example of
three-dimensional decaying isotropic turbulence. The present scaling not only
enables the identification of vortical structures that are interpolatory and
extrapolatory for the given flow field data but also captures non-equilibrium
effects of the energy cascade. As a demonstration, the present findings are
applied to machine-learning-based super-resolution analysis of
three-dimensional isotropic turbulence. We show that machine-learning models
reconstruct vortical structures well in the interpolatory space with reduced
performance in the extrapolatory space revealed by the nonlinearly scaled
invariants. The present approach enables us to depart from labeling turbulent
flow data with a single parameter of Reynolds number and comprehensively
examine the flow field to support training and testing of nonlinear
machine-learning techniques.
