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Long-term predictions of nonlinear dynamics of three-dimensional (3D)turbulence are very challenging for machine learning approaches. In this paper,we propose an implicit U-Net enhanced Fourier neural operator (IU-FNO) forstable and efficient predictions on the long-term large-scale dynamics ofturbulence. The IU-FNO model employs implicit recurrent Fourier layers fordeeper network extension and incorporates the U-net network for the accurateprediction on small-scale flow structures. The model is systematically testedin large-eddy simulations of three types of 3D turbulence, including forcedhomogeneous isotropic turbulence (HIT), temporally evolving turbulent mixinglayer, and decaying homogeneous isotropic turbulence. The numerical simulationsdemonstrate that the IU-FNO model is more accurate than other FNO-based modelsincluding vanilla FNO, implicit FNO (IFNO) and U-Net enhanced FNO (U-FNO), anddynamic Smagorinsky model (DSM) in predicting a variety of statistics includingthe velocity spectrum, probability density functions (PDFs) of vorticity andvelocity increments, and instantaneous spatial structures of flow field.Moreover, IU-FNO improves long-term stable predictions, which has not beenachieved by the previous versions of FNO. Besides, the proposed model is muchfaster than traditional LES with DSM model, and can be well generalized to thesituations of higher Taylor-Reynolds numbers and unseen flow regime of decayingturbulence.

Long-term predictions of turbulence by implicit U-Net enhanced Fourier neural operator