Combining spectral analyses of turbulent flows and neural networks for explainable data-driven closure modeling

Recent studies have found promising results using machine learning (ML) techniques such as convolutional neural networks (CNNs) to develop data-driven subgrid-scape closures, e.g., for large-eddy simulations. However, the lack of interpretability of such deep NNs is a serious shortcoming, limiting the applications of such data-driven closures. Furthermore, NNs and similar techniques cannot be expected to work accurately outside their training manifold, i.e, they often do not extrapolate. Transfer learning (TL), which involves re-training some layers with a small amount of new data, offers a solution to this and a few recent studies have found promising results in simple test cases. Here, we present a framework, based on combining the spectral analysis of turbulent flows and spectral analysis of CNNs to 1) provide full explainability of what is learned during TL, and 2) provide insights into what the CNNs learn, from the input to the output. Using 2D turbulence as the test case, we show how this framework connects with the physics of the flow and some of the recent advances in the ML community on the training of NNs.