Frequency-Constrained Learning for Long-Term Forecasting

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.