Robust machine learning of turbulence through generalized Buckingham Pi-inspired pre-processing of training data*

Applications of machine learning to unseen situations are challenging for turbulent flows.  With turbulence resulting from strong nonlinear dynamics, categorizing seen and unseen events and structures are generally not clear-cut, especially for modern machine learning methods that can accommodate some stretching and rotational invariances. In such circumstances encountered in machine learning approaches, we develop a technique for pre-processing the training data by generalizing the Buckingham Pi theorem and combining it with sparse regression.  Identifying the appropriate non-dimensional scaling functions, we show that machine learning can be performed with enhanced accuracy and robustness.  We also note that the concept of interpolation and extrapolation becomes clearer through such scaling procedure, providing guidance on when users of machine learning techniques should be mindful of possible high-risk extrapolations.  Two and three-dimensional decaying isotropic turbulence are selected to demonstrate the robustness of the proposed approach and highlight its potential to enhance the training process for learning complex turbulent flow dynamics.