Abstract Simulating turbulent fluid flows is a computationally prohibitive task, as it requires the resolution of fine-scale structures and the capture of complex nonlinear interactions across multiple scales. Consequently, extensive research has focused on analysing turbulent flows from a data-driven perspective. However, due to the complex and chaotic nature of these systems, traditional models often become unstable. To overcome these limitations, we propose a purely stochastic approach that separately addresses the evolution of large-scale coherent structures and the closure of high-fidelity statistical data. To this end, the dynamics of the filtered data are learnt using an autoregressive model. This combines a variational-autoencoder (VAE) and Transformer architecture. The VAE projection is probabilistic, ensuring consistency between the model’s stochasticity and the flow’s statistical properties. The mean realisation of stochastically sampled trajectories from our model shows relative L₁ and L₂ distances of 6% and 10%, respectively. Moreover, our framework enables the construction of meaningful confidence intervals, achieving a prediction interval coverage probability of 80% with minimal interval width. To recover high-fidelity velocity fields from the filtered space, Gaussian Process (GP) regression is employed. This strategy has been tested in the context of a Kolmogorov flow exhibiting chaotic behavior. We compare the performance of our model with state-of-the-art probabilistic baselines, including a VAE and a diffusion model. We demonstrate that our Gaussian process-based closure outperforms these baselines in capturing first and second moment statistics in this particular test bed, providing robust and adaptive confidence intervals.
Zighed et al. (Thu,) studied this question.