Explainable deep learning for fluid dynamics using a Fourier-wavelet analysis framework*

We introduce a new framework that combines the spectral (Fourier) analyses of NNs and nonlinear physics, and leverages recent advances in theory and applications of deep learning, to move toward rigorous analysis of deep NNs for applications involving dynamical systems such as turbulent flows. We will use examples from subgrid-scale modeling of 2D turbulence and Rayleigh-Bernard turbulence, weather forecasting, and modeling gravity waves to show how this framework can be used to systematically address challenges about explainability, generalizability, and stability. For example, the framework shows that in many of such applications, millions of learned parameters in deep convolutional NNs reduce to a few classes of known spectral filters, such as low-pass and Gabor wavelets.