Subgrid Closure for a Reduced Model of Turbulence using LSTM-augmented Runge-Kutta Integrator

In this work we investigate the capability of Artificial Neural networks to build Subgrid Closure for a Shell Model of turbulence. Shell Models are dynamical systems of Ordinary Differential Equations that have been shown to rather faithfully mimic the phenomenology of the energy cascade of the Naver Stokes Equations in Fourier space. Our method employs a novel custom-made Neural Network architecture comprising a classical integrator (Runge-Kutta 4^th order) for the large scales of turbulence, augmented with LSTM cells to obtain the values of the fluxes to the small scales. We are able to reproduce, within statistical error bars, the intermittent behavior found in the full model, obtaining the correct scaling laws for Eulerian and Lagrangian structure functions and outperforming classical physics based methods. This work demonstrates the capability of Machine Learning to capture complex multiscale dynamics and reproduce complex multi-scale and multi-time non-gaussian behaviors, opening up the possibility to tackle turbulence modelling in Navier-Stokes Equations.