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

Ismaël Zighed, Nicolas Thome, Patrick Gallinari, Taraneh Sayadi

TL;DR

This work tackles turbulence forecasting by decoupling scale dynamics into large-scale, stochastic evolution and small-scale closure. It introduces a two-task framework: (i) a probabilistic reduced-order model that learns large-scale dynamics in a latent space via a Variational Autoencoder and Transformer, producing ensembles that capture predictive uncertainty, and (ii) a Gaussian-process closure that maps reduced-space representations to full-resolution fields within a POD basis, yielding statistically consistent reconstructions. Applied to Kolmogorov flow, the approach achieves accurate first- and second-moment statistics, competitive uncertainty calibration (PICP/CRPS), and stable long-horizon PDFs, outperforming VAE and diffusion baselines while remaining computationally efficient. The modular, plug-and-play architecture enables real-time closure and broad applicability to multiscale turbulence problems, offering a principled data-driven path to robust, probabilistic turbulence prediction and control.

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. This is particularly the case in direct numerical simulation (DNS) applied to real-world turbulent applications. 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 as they accumulate errors through autoregression, severely degrading even short-term predictions. 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 (i.e. coherent motion) are learnt using an autoregressive model. This combines a VAE and Transformer architecture. The VAE projection is probabilistic, ensuring consistency between the model's stochasticity and the flow's statistical properties. To recover high-fidelity velocity fields from the filtered latent space, Gaussian Process (GP) regression is employed. This strategy has been tested in the context of a Kolmogorov flow exhibiting chaotic behaviour analogous to real-world turbulence. 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.

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

TL;DR

This work tackles turbulence forecasting by decoupling scale dynamics into large-scale, stochastic evolution and small-scale closure. It introduces a two-task framework: (i) a probabilistic reduced-order model that learns large-scale dynamics in a latent space via a Variational Autoencoder and Transformer, producing ensembles that capture predictive uncertainty, and (ii) a Gaussian-process closure that maps reduced-space representations to full-resolution fields within a POD basis, yielding statistically consistent reconstructions. Applied to Kolmogorov flow, the approach achieves accurate first- and second-moment statistics, competitive uncertainty calibration (PICP/CRPS), and stable long-horizon PDFs, outperforming VAE and diffusion baselines while remaining computationally efficient. The modular, plug-and-play architecture enables real-time closure and broad applicability to multiscale turbulence problems, offering a principled data-driven path to robust, probabilistic turbulence prediction and control.

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. This is particularly the case in direct numerical simulation (DNS) applied to real-world turbulent applications. 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 as they accumulate errors through autoregression, severely degrading even short-term predictions. 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 (i.e. coherent motion) are learnt using an autoregressive model. This combines a VAE and Transformer architecture. The VAE projection is probabilistic, ensuring consistency between the model's stochasticity and the flow's statistical properties. To recover high-fidelity velocity fields from the filtered latent space, Gaussian Process (GP) regression is employed. This strategy has been tested in the context of a Kolmogorov flow exhibiting chaotic behaviour analogous to real-world turbulence. 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.

Paper Structure

This paper contains 22 sections, 30 equations, 18 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Kolmogorov Flow
  4. Strategy for scale separation
  5. First task: Predicting Filtered dynamics
  6. Model architecture
  7. Model predictions
  8. Second Task: Gaussian process regression -- probabilistic small-scale closure
  9. Gaussian Process
  10. GPR applied for closure
  11. Full model prediction
  12. Evaluation on long term predictions
  13. Comparison to alternative ML architectures
  14. Conclusion
  15. Funding Statement
  16. ...and 7 more sections

Figures (18)

  • Figure 1: Velocity fields and corresponding kinetic energy signal.
  • Figure 2: Time-averaged energy spectrum displayed on a logarithmic scale.
  • Figure 3: Low-pass filter threshold keeping 90% of the energy
  • Figure 4: Low-pass filter on the energy spectrum and filtered energy spectrum
  • Figure 5: Effect of low-pass filter on velocity fields and kinetic energy.
  • ...and 13 more figures