Accelerated evolution of Burgers' turbulence with coarse projective integration and deep learning

The evolution of a turbulent flow to a statistically steady state can be cast as a multiscale problem involving energy redistribution processes that take place on the long, large eddy turnover timescale and chaotic processes that take place on the much shorter timescale of the turbulence fluctuations. But the absence of a way to perform super-resolution reconstructions of the instantaneous velocity field from its lower-dimensional moments has prevented the use of standard multiscale computational approaches for accelerating turbulence simulations. We introduce an encoder-decoder recurrent neural network model, an architecture typically used in natural language processing, for translating between the instantaneous velocity field and energy spectrum in a one-dimensional turbulent flow. We use the model in a multiscale simulation scheme to evolve the stochastic Burgers' equation and recover the final, statistically stationary turbulent Burgers' velocity field up to 443 times faster in wall-clock time than using direct numerical simulation alone.