Uncertainty quantification and stability of neural operators for prediction of three-dimensional turbulence

Abstract: Turbulence poses challenges for numerical simulation due to its chaotic, multiscale nature and high computational cost. Traditional turbulence modeling often struggles with accuracy and long-term stability. Recent scientific machine learning (SciML) models, such as Fourier Neural Operators (FNO), show promise in solving PDEs, but are typically limited to one-step-ahead predictions and often fail over long time horizons, especially in 3D turbulence. This study proposes a framework to assess the reliability of neural operator models in turbulent flows. Using three-dimensional forced homogeneous isotropic turbulence (HIT) as a benchmark, we evaluate models in terms of uncertainty quantification (UQ), error propagation, and sensitivity to initial perturbations. Statistical tools such as error distribution analysis and autocorrelation functions (ACF) are used to assess predictive robustness and temporal coherence. Our proposed model, the factorized-implicit FNO (F-IFNO), improves long-term stability and accuracy by incorporating implicit factorization into the prediction process. It outperforms conventional LES and other FNO-based models in balancing accuracy, stability, and efficiency. The results highlight the importance of prediction constraints, time interval selection, and UQ in developing robust neural operator frameworks for turbulent systems.