Leveraging Scale Separation and Stochastic Closure for Data-Driven Prediction of Chaotic Dynamics

Abstract: Simulating turbulent fluid flows is computationally very demanding, as it requires resolving fine-scale structures and capturing complex nonlinear interactions across multiple scales. This is especially true for direct numerical simulation applied to real-world turbulent problems. Consequently, much research has focused on analyzing turbulent flows from a data-driven perspective. However, because these systems are complex and chaotic, traditional models often become unstable as they accumulate errors over time, leading to significant degradation even in short-term predictions. To address these limitations, we propose a purely stochastic approach that separately models the evolution of large-scale coherent structures and the closure of high-fidelity statistical data. Specifically, the dynamics of the filtered data, representing coherent motion, are learned using an autoregressive model that combines a Variational Autoencoder with a Transformer architecture. The VAE projection is probabilistic, ensuring consistency between the model's stochasticity and the statistical properties of the flow. The mean realization of stochastically sampled trajectories from our model shows relative errors of 6 percent and 10 percent, respectively, compared to the test set. Furthermore, our framework allows the construction of meaningful confidence intervals, achieving a prediction interval coverage probability of 80 percent with minimal interval width. To recover high-fidelity velocity fields from the filtered latent space, we employ Gaussian Process regression. This strategy has been tested on a Kolmogorov flow exhibiting chaotic behavior similar to real-world turbulence.red latent space, we employ Gaussian Process regression. This strategy has been tested on a Kolmogorov flow exhibiting chaotic behavior similar to real-world turbulence.