Long-term time series forecasting is a long-standing challenge in various applications. A central issue in time series forecasting is that methods should expressively capture long-term dependency. Furthermore, time series forecasting methods should be flexible when applied to different scenarios. Although Fourier analysis offers an alternative to effectively capture reusable and periodic patterns to achieve long-term forecasting in different scenarios, existing methods often assume high-frequency components represent noise and should be discarded in time series forecasting. However, we conduct a series of motivation experiments and discover that the role of certain frequencies varies depending on the scenarios. In some scenarios, removing high-frequency components from the original time series can improve the forecasting performance, while in others scenarios, removing them is harmful to forecasting performance. Therefore, it is necessary to treat the frequencies differently according to specific scenarios. To achieve this, we first reformulate the time series forecasting problem as learning a transfer function of each frequency in the Fourier domain. Further, we design Frequency Dynamic Fusion (FreDF), which individually predicts each Fourier component, and dynamically fuses the output of different frequencies. Moreover, we provide a novel insight into the generalization ability of time series forecasting and propose the generalization bound of time series forecasting. Then we prove FreDF has a lower bound, indicating that FreDF has better generalization ability. Extensive experiments conducted on multiple benchmark datasets and ablation studies demonstrate the effectiveness of FreDF.

翻译：长期时间序列预测是各类应用中长期存在的挑战。时间序列预测的核心问题在于方法应能有效捕捉长期依赖性。此外，时间序列预测方法在不同应用场景中需具备灵活性。尽管傅里叶分析为有效捕捉可复用周期模式以实现不同场景下的长期预测提供了替代方案，但现有方法常假设高频分量代表噪声，在时间序列预测中应予以剔除。然而，我们通过一系列动机实验发现，特定频率的作用随场景变化而异。在某些场景中，从原始时间序列中移除高频分量可提升预测性能，而在另一些场景中，移除高频分量反而损害预测性能。因此，有必要根据具体场景差异化处理不同频率。为此，我们首先将时间序列预测问题重新表述为在傅里叶域中学习各频率的传递函数。进一步，我们设计了频率动态融合（FreDF）方法，该方法独立预测各傅里叶分量，并动态融合不同频率的输出。此外，我们对时间序列预测的泛化能力提出了新颖见解，并推导了时间序列预测的泛化界。随后我们证明FreDF具有更低的泛化界，表明其具备更优的泛化能力。在多个基准数据集上开展的广泛实验及消融研究证实了FreDF的有效性。