  A convolutional encoder-decoder-based transformer model is proposed for
autoregressively training on spatio-temporal data of turbulent flows. The
prediction of future fluid flow fields is based on the previously predicted
fluid flow field to ensure long-term predictions without diverging. A
combination of convolutional neural networks and transformer architecture is
utilized to handle both the spatial and temporal dimensions of the data. To
assess the performance of the model, a priori assessments are conducted, and
significant agreements are found with the ground truth data. The a posteriori
predictions, which are generated after a considerable number of simulation
steps, exhibit predicted variances. The autoregressive training and prediction
of a posteriori states are deemed crucial steps towards the development of more
complex data-driven turbulence models and simulations. The highly nonlinear and
chaotic dynamics of turbulent flows can be handled by the proposed model, and
accurate predictions over long time horizons can be generated. Overall, the
potential of using deep learning techniques to improve the accuracy and
efficiency of turbulence modeling and simulation is demonstrated by this
approach. The proposed model can be further optimized and extended to
incorporate additional physics and boundary conditions, paving the way for more
realistic simulations of complex fluid dynamics.
