Understanding Turbulence via Machine Learning

Turbulence remains one of the most challenging and fundamental open problems in classical physics. Despite the apparent simplicity of the governing Navier–Stokes equations, their nonlinear and multiscale nature gives rise to complex behaviors making analytical treatment extremely difficult. The goal of this work is to assess the capability of machine learning to identify and interpret the interaction structure underlying the energy cascade in turbulence. Shell models are systems of coupled ordinary differential equations designed to mimic the energy cascade of turbulent system across discretized scales, and thus with a reduced number of degrees of freedom, while successfully reproducing many statistical features of fully developed turbulence. First, we employ sparse nonlinear regression techniques to extract relevant interactions from data generated by a shell model. We apply Lasso regression using three complementary strategies: growing-window analysis, batch-wise model aggregation, and greedy interaction selection. Results demonstrate that expected interactions consistently emerge as the most relevant terms, with coefficients dominating by several orders of magnitude. The model generally succeeds in identifying the expected interactions from the SABRA shell model, though quality of learning degrades for higher shell indices. The batch approach significantly reduces unexpected interactions while the greedy method confirms that three interactions per shell capture essential dynamics, aligning with theoretical expectations.