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.

Time series modeling aims to encode historical sequence to predict future data, which is crucial in diverse applications: long-term forecast in weather prediction [3, 40], short-term prediction in industrial maintenance [24, 7, 35], and missing data imputation in healthcare [30]. A key challenge in time series modeling, distinguishing it from canonical regression tasks, is the presence of autocorrelation. It refers to the dependence between time steps, which exists in both the input and label sequences. To accommodate autocorrelation in input sequences, diverse forecast models have been developed [28, 5, 8], exemplified by recurrent [29], convolution [37] and graph neural networks [25, 4, 11]. Recently, Transformer-based models, utilizing self-attention mechanisms to dynamically assess autocorrelation, have gained prominence in this line of work [20, 26, 13, 38]. Concurrently, there is a growing trend of incorporating frequency analysis into forecast models [41, 21].