Spectral Modeling of Periodic Phenomena in Scientific Data via Fourier–Galerkin Representation

Periodic phenomena are ubiquitous across various scientific domains, such as climatic cycles. However, identifying and modeling these periodic behaviors in large-scale, heterogeneous scientific datasets pose significant challenges due to the inherent complexity and variability of phase shifts and signal misalignment. In this work, we propose a Fourier–Galerkin (FG) regression model that decomposes the time series into a linear trend and a periodic residual, using a FG basis for the residual. Our core model, FG-Linear, captures periodic behavior effectively by learning a linear combination of cosine and sine components, while addressing phase non-uniqueness with phase offsets for each frequency. We also introduce a deep spectral variant (FG-Deep), which incorporates a neural network to model non-linearities in the spectral features. Experimental results on the NOAA Global Summary of the Year climate dataset show that FG-Linear outperforms trend-only models and deep learning baselines, achieving the lowest test MSE. While FG-Deep provides flexibility, its performance is similar to FG-Linear on this small and regular dataset, demonstrating the efficiency and interpretability of the linear FG approach.