Turb-L1: Achieving Long-term Turbulence Tracing By Tackling Spectral Bias

TL;DR

This work identifies spectral bias as the main bottleneck in long-term turbulence forecasting by deep learning models. It introduces Turb-L1, a multi-grid architecture with Hierarchical Dynamics Synthesis that explicitly learns full-spectrum dynamics across high, middle, and low frequencies, mitigating high-frequency errors during autoregressive rollout. Theoretical analysis links high-frequency error control to long-term stability, and empirical results on 2D isotropic turbulence demonstrate state-of-the-art long-term $L^2$ errors, $SSIM$, and accurate enstrophy spectra (including the $k^{-3}$ cascade). The combination of HDS and MG yields robust, physically realistic long-horizon predictions with strong cross-scale fidelity, offering a path toward reliable turbulence forecasting in scientific and engineering contexts.

Abstract

Accurately predicting the long-term evolution of turbulence is crucial for advancing scientific understanding and optimizing engineering applications. However, existing deep learning methods face significant bottlenecks in long-term autoregressive prediction, which exhibit excessive smoothing and fail to accurately track complex fluid dynamics. Our extensive experimental and spectral analysis of prevailing methods provides an interpretable explanation for this shortcoming, identifying Spectral Bias as the core obstacle. Concretely, spectral bias is the inherent tendency of models to favor low-frequency, smooth features while overlooking critical high-frequency details during training, thus reducing fidelity and causing physical distortions in long-term predictions. Building on this insight, we propose Turb-L1, an innovative turbulence prediction method, which utilizes a Hierarchical Dynamics Synthesis mechanism within a multi-grid architecture to explicitly overcome spectral bias. It accurately captures cross-scale interactions and preserves the fidelity of high-frequency dynamics, enabling reliable long-term tracking of turbulence evolution. Extensive experiments on the 2D turbulence benchmark show that Turb-L1 demonstrates excellent performance: (I) In long-term predictions, it reduces Mean Squared Error (MSE) by $80.3\%$ and increases Structural Similarity (SSIM) by over $9\times$ compared to the SOTA baseline, significantly improving prediction fidelity. (II) It effectively overcomes spectral bias, accurately reproducing the full enstrophy spectrum and maintaining physical realism in high-wavenumber regions, thus avoiding the spectral distortions or spurious energy accumulation seen in other methods.

Figures (9)

• Figure 1: Long-term turbulence prediction performance comparison.(a) Visual comparison of vorticity fields ($t$=20, $t$=99). Turb-L1 maintains high fidelity to the Ground-truth, while baselines exhibit excessive smoothing (e.g., FNO) or simulation collapse with artifacts (e.g., SimVP) driven by high-frequency error amplification. (b) Enstrophy spectrum density comparison ($t$=10, 30, 99). Baselines fail to capture high-frequency dynamics, increasingly deviating from the Ground Truth at high wavenumbers ($k$) over time, whereas Turb-L1 accurately preserves spectral characteristics.
• Figure 2: Overview of the Turb-L1 architecture. The model initially downscales the turbulence initial condition and embeds it into a latent space via a multi-grid encoder ($\mathcal{E}$). The core Hierarchical Dynamics Synthesis Mechanism ($\mathcal{HDS}$) then explicitly synthesizes dynamical features across different frequencies (high $\omega_H$, middle $\omega_M$, and low $\omega_L$) within this latent space. Subsequently, multi-scale aggregated features are upscaled by a multi-grid decoder ($\mathcal{D}$), ultimately generating high-fidelity turbulence predictions. The lower panel details the multi-grid operations---Restriction ($\mathcal{R}$), Prolongation ($\mathcal{P}$), and Iteration ($\circlearrowleft$)---illustrating how the model processes and transfers information across various grid scales to effectively capture cross-scale interactions and overcome spectral bias.
• Figure 3: Long-term prediction on 2D Decaying Isotropic Turbulence.(Left) Initial condition. (Center Columns 1-5) Vorticity fields at $t=10, 50, 99$ for Ground-truth, Turb-L1, FNO, SimVP, and U-Net. Turb-L1 maintains high fidelity, while FNO/U-Net show over-smoothing and SimVP (collapse icon) exhibits instability, underscoring Turb-L1's resilience to spectral bias. (Right) Quantitative evaluation: (Top Right.) SSIM Evolution, where Turb-L1 excels in structural similarity. (Bottom Right.) Relative $L^2$ Error Evolution, showing Turb-L1's superior accuracy and stability.
• Figure 4: Evolution of spectral error and visual prediction improvement for Turb-L1 during training. (a) Normalized spectral error for Turb-L1, ViT, and CNO at different training epochs. Turb-L1 shows significantly reduced error in high-wavenumber regions with training. (b) Visual comparison of Turb-L1's long-term predictions at early (Epoch: 50) and late (Epoch: 500) training stages, demonstrating enhanced capturing of high-frequency details and vortex structures.
• Figure 5: Comparison of enstrophy spectrum analysis. Left plot is Dataset I, Right plot is Dataset II.

Theorems & Definitions (2)

• Theorem 3.1: High-Frequency Error Control and Prediction Stability
• proof