标题： 频率约束学习用于长期预测 显示英文标题

标题： Frequency-Constrained Learning for Long-Term Forecasting

摘要： 许多现实世界的时间序列表现出由物理定律、人类习惯或季节性周期引起的强周期结构。 然而，现代深度预测模型常常由于频谱偏差和缺乏频率感知的归纳先验而无法捕捉这些重复模式。 受这一差距的启发，我们提出了一种简单但有效的方法，通过频谱初始化和频率约束优化来显式建模周期性，从而增强长期预测。 具体来说，我们通过快速傅里叶变换（FFT）引导的坐标下降提取主要的低频成分，用这些成分初始化正弦嵌入，并采用双速学习计划在训练过程中保留有意义的频率结构。 我们的方法与模型无关，并可无缝集成到现有的基于Transformer的架构中。 在多种现实世界基准上的广泛实验表明，性能得到了一致的提升——尤其是在长时序上——突显了将频谱先验注入深度时间模型在稳健和可解释的长程预测中的优势。 此外，在合成数据上，我们的方法能够准确恢复真实频率，进一步验证了其在捕捉潜在周期模式方面的可解释性和有效性。 显示英文摘要

摘要： Many real-world time series exhibit strong periodic structures arising from physical laws, human routines, or seasonal cycles. However, modern deep forecasting models often fail to capture these recurring patterns due to spectral bias and a lack of frequency-aware inductive priors. Motivated by this gap, we propose a simple yet effective method that enhances long-term forecasting by explicitly modeling periodicity through spectral initialization and frequency-constrained optimization. Specifically, we extract dominant low-frequency components via Fast Fourier Transform (FFT)-guided coordinate descent, initialize sinusoidal embeddings with these components, and employ a two-speed learning schedule to preserve meaningful frequency structure during training. Our approach is model-agnostic and integrates seamlessly into existing Transformer-based architectures. Extensive experiments across diverse real-world benchmarks demonstrate consistent performance gains--particularly at long horizons--highlighting the benefits of injecting spectral priors into deep temporal models for robust and interpretable long-range forecasting. Moreover, on synthetic data, our method accurately recovers ground-truth frequencies, further validating its interpretability and effectiveness in capturing latent periodic patterns.