Closure Modeling for Accelerated Multiscale Evolution of a 1-Dimensional Turbulence Model

Accelerating the simulation of turbulence to stationarity is a critical challenge in various engineering applications. This study presents an innovative equation-free multiscale approach combined with a machine learning technique to address this challenge in the context of the one-dimensional stochastic Burgers' equation, a widely used toy model for turbulence. We employ an encoder-decoder recurrent neural network to perform super-resolution reconstruction of the velocity field from lower-dimensional energy spectrum data, enabling seamless transitions between fine and coarse levels of description. The proposed multiscale-machine learning framework significantly accelerates the computation of the statistically stationary turbulent Burgers' velocity field, achieving up to 442 times faster wall clock time compared to direct numerical simulation, while maintaining three-digit accuracy in the velocity field. Our findings demonstrate the potential of integrating equation-free multiscale methods with machine learning methods to efficiently simulate stochastic partial differential equations and highlight the possibility of using this approach to simulate stochastic systems in other engineering domains.