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

arXiv: https://arxiv.org/abs/2505.19038 · HTML: https://mlsanity.com/paper/2505.19038/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 83 references that are in the map.
- First posted on arXiv 25 May 2025.

## Builds on (most cited first, 30 of 32)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [CvT: Introducing Convolutions to Vision Transformers](https://mlsanity.com/paper/2103.15808/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [CoAtNet: Marrying Convolution and Attention for All Data Sizes](https://mlsanity.com/paper/2106.04803/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [CMT: Convolutional Neural Networks Meet Vision Transformers](https://mlsanity.com/paper/2107.06263/) (2021)
- [How Do Vision Transformers Work?](https://mlsanity.com/paper/2202.06709/) (2022)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Earthformer: Exploring Space-Time Transformers for Earth System Forecasting](https://mlsanity.com/paper/2207.05833/) (2022)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [PastNet: Introducing Physical Inductive Biases for Spatio-temporal Video Prediction](https://mlsanity.com/paper/2305.11421/) (2023)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [Spectral Bias in Practice: The Role of Function Frequency in Generalization](https://mlsanity.com/paper/2110.02424/) (2021)
- [On Measuring and Controlling the Spectral Bias of the Deep Image Prior](https://mlsanity.com/paper/2107.01125/) (2021)
- [OneForecast: A Universal Framework for Global and Regional Weather Forecasting](https://mlsanity.com/paper/2502.00338/) (2025)
- [Addressing Spectral Bias of Deep Neural Networks by Multi-Grade Deep Learning](https://mlsanity.com/paper/2410.16105/) (2024)
- [Modeling Spatio-temporal Dynamical Systems with Neural Discrete Learning and Levels-of-Experts](https://mlsanity.com/paper/2402.05970/) (2024)

## Built on by (most cited first, 9 of 9)

- [Frequency-Aligned Knowledge Distillation for Lightweight Spatiotemporal Forecasting](https://mlsanity.com/paper/2507.02939/) (2025)
- [NeuralOM: Neural Ocean Model for Subseasonal-to-Seasonal Simulation](https://mlsanity.com/paper/2505.21020/) (2025)
- [Breaking the Discretization Barrier of Continuous Physics Simulation Learning](https://mlsanity.com/paper/2509.17955/) (2025)
- [S^2-KD: Semantic-Spectral Knowledge Distillation Spatiotemporal Forecasting](https://mlsanity.com/paper/2512.00366/) (2025)
- [OMNIFLOW: A Physics-Grounded Multimodal Agent for Generalized Scientific Reasoning](https://mlsanity.com/paper/2603.15797/) (2026)
- [PnP-Corrector: A Universal Correction Framework for Coupled Spatiotemporal Forecasting](https://mlsanity.com/paper/2605.08935/) (2026)
- [Learning from History: A Retrieval-Augmented Framework for Spatiotemporal Prediction](https://mlsanity.com/paper/2510.24049/) (2025)
- [DIFFUMA: High-Fidelity Spatio-Temporal Video Prediction via Dual-Path Mamba and Diffusion Enhancement](https://mlsanity.com/paper/2507.06738/) (2025)
- [VISION: Prompting Ocean Vertical Velocity Reconstruction from Incomplete Observations](https://mlsanity.com/paper/2509.21477/) (2025)

## Abstract (arXiv)

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× 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.