Modeling Time Series Dynamics with Fourier Ordinary Differential Equations
This addresses the problem of capturing long-term dependencies and periodic structures in time series modeling for researchers and practitioners, representing an incremental improvement over Neural ODEs.
The paper tackled the limitations of Neural ODEs in modeling time series by proposing Fourier Ordinary Differential Equations (FODEs), which embed dynamics in the Fourier domain and include a learnable filtering mechanism, resulting in outperforming existing methods in accuracy and efficiency on various datasets.
Neural ODEs (NODEs) have emerged as powerful tools for modeling time series data, offering the flexibility to adapt to varying input scales and capture complex dynamics. However, they face significant challenges: first, their reliance on time-domain representations often limits their ability to capture long-term dependencies and periodic structures; second, the inherent mismatch between their continuous-time formulation and the discrete nature of real-world data can lead to loss of granularity and predictive accuracy. To address these limitations, we propose Fourier Ordinary Differential Equations (FODEs), an approach that embeds the dynamics in the Fourier domain. By transforming time-series data into the frequency domain using the Fast Fourier Transform (FFT), FODEs uncover global patterns and periodic behaviors that remain elusive in the time domain. Additionally, we introduce a learnable element-wise filtering mechanism that aligns continuous model outputs with discrete observations, preserving granularity and enhancing accuracy. Experiments on various time series datasets demonstrate that FODEs outperform existing methods in terms of both accuracy and efficiency. By effectively capturing both long- and short-term patterns, FODEs provide a robust framework for modeling time series dynamics.