用小波变换提升动态系统学习的高频精度，改善长期预测稳定性。
Multi-Scale Wavelet Transformers for Operator Learning of Dynamical Systems
- 在小波域进行令牌化建模，分离高低频成分
- 多尺度小波注意力显著降低混沌系统误差
- 适合需要高保真频率响应的气象等复杂系统建模
近年来，数据驱动的动态系统代理模型比数值求解器快几个数量级。然而，许多基于机器学习的模型（如神经算子）存在谱偏差，会削弱常编码小尺度结构的高频分量。这一局限在天气预报等应用中尤为严重，高频信息失真会导致长期预测不稳定。为此，我们提出多尺度小波变压器（MSWT），在小波域中对系统动态进行建模。小波变换显式地在不同尺度上分离低频与高频内容。MSWT采用保持小波特性的下采样策略，保留高频特征，并使用基于小波的注意力机制捕捉跨尺度和频带的依赖关系。在混沌动态系统上的实验表明，模型显著降低误差并提升长期谱保真度。在ERA5气候再分析数据集上，MSWT进一步减少了气候偏差，证明其在真实世界预测场景中的有效性。
原文摘要 · Abstract (English)
Recent years have seen a surge in data-driven surrogates for dynamical systems that can be orders of magnitude faster than numerical solvers. However, many machine learning-based models such as neural operators exhibit spectral bias, attenuating high-frequency components that often encode small-scale structure. This limitation is particularly damaging in applications such as weather forecasting, where misrepresented high frequencies can induce long-horizon instability. To address this issue, we propose multi-scale wavelet transformers (MSWTs), which learn system dynamics in a tokenized wavelet domain. The wavelet transform explicitly separates low- and high-frequency content across scales. MSWTs leverage a wavelet-preserving downsampling scheme that retains high-frequency features and employ wavelet-based attention to capture dependencies across scales and frequency bands. Experiments on chaotic dynamical systems show substantial error reductions and improved long horizon spectral fidelity. On the ERA5 climate reanalysis, MSWTs further reduce climatological bias, demonstrating their effectiveness in a real-world forecasting setting.
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