Hierarchical Modeling for Synthetic Turbulence Generation

We present a hierarchical framework for reconstructing one-dimensional turbulent time series from progressively downsampled representations. The original signal is decomposed dyadically—at each stage, every other data point is removed—until the deepest level approaches random Gaussian noise. The key challenge is to recover the full turbulence dynamics from these reduced representations while preserving their statistical and physical properties. Our hybrid surrogate modeling pipeline integrates classical autoregressive (ARIMA) and state-space models at the coarser levels with advanced temporal deep-learning architectures at deeper reconstruction stages. Specifically, recurrent models such as GRU-D and BRITS capture short- and medium-range dependencies, while attention-based models including SAITS and TimesNet handle long-range temporal correlations and multi-frequency dynamics. For uncertainty-aware and multi-scale reconstruction, we employ probabilistic and non-autoregressive approaches such as GP-VAE and NAOMI, ensuring robustness to irregular sampling. Each reconstruction stage is validated through statistical and physics-based diagnostics, including spectral scaling, intermittency, energy conservation, and long-range correlation metrics. This hierarchical organization yields an interpretable and physically consistent reconstruction of turbulent time series, demonstrating a principled integration of statistical, deep, and physics-informed modeling for high-fidelity surrogate generation.