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Modeling Time Series Dynamics with Fourier Ordinary Differential Equations

Muhao Guo, Yang Weng

TL;DR

The paper tackles the challenge that time-domain Neural ODEs struggle to model global periodic structures and discretely sampled data. It introduces Fourier Ordinary Differential Equations (FODEs), which learn dynamics in the frequency domain via FFT/IFFT and use a learnable Hadamard filter to remain aligned with discrete observations, formalized as $\dfrac{dx}{dt} = f_{FODE}(x,t;\theta_g) = \mathrm{IFFT}(\mathcal{M}(g(\mathcal{P}(\mathrm{FFT}(x)), t; \theta_g)))$. Key contributions include a concrete Fourier-domain construction with conjugate-symmetry enforcement, Lipschitz guarantees for well-posedness, adjoint-based training, and extensive experiments showing improvements in forecasting, classification, and physical dynamics tasks. The results demonstrate that combining spectral dynamics with a data-driven filter yields robust, efficient time-series models capable of capturing both long- and short-term patterns across diverse domains.

Abstract

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.

Modeling Time Series Dynamics with Fourier Ordinary Differential Equations

TL;DR

The paper tackles the challenge that time-domain Neural ODEs struggle to model global periodic structures and discretely sampled data. It introduces Fourier Ordinary Differential Equations (FODEs), which learn dynamics in the frequency domain via FFT/IFFT and use a learnable Hadamard filter to remain aligned with discrete observations, formalized as . Key contributions include a concrete Fourier-domain construction with conjugate-symmetry enforcement, Lipschitz guarantees for well-posedness, adjoint-based training, and extensive experiments showing improvements in forecasting, classification, and physical dynamics tasks. The results demonstrate that combining spectral dynamics with a data-driven filter yields robust, efficient time-series models capable of capturing both long- and short-term patterns across diverse domains.

Abstract

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.

Paper Structure

This paper contains 16 sections, 2 theorems, 17 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Discrete Fourier Transform
  4. Construct Dynamics in Fourier Domain
  5. Fourier Ordinary Differential Equations
  6. Element-Wised Filter
  7. Experiment
  8. Environment Setup
  9. Periodic Time Series Forecasting
  10. Physical Dynamics Forecasting
  11. Time Series Classification
  12. Time Series Forecasting
  13. Hidden State Analysis of FODE
  14. Ablation Study
  15. Evolution of filter K with different initialization
  16. ...and 1 more sections

Key Result

Lemma 1

Assume (1) $g(\cdot,t;\theta_g)$ is $L_g$‑Lipschitz in its first argument for every fixed $t$, and (2) $\mathcal{P}$ and $\mathcal{M}$ are $L_{\mathcal{P}}$‑ and $L_{\mathcal{M}}$‑Lipschitz, all with respect to the Euclidean norm. Because $\operatorname{FFT}$ and $\operatorname{IFFT}$ are bounded li

Figures (7)

  • Figure 1: Schematic of the proposed method. Blue region: The input time series $x(t_0)$ is first transformed to the frequency domain via FFT, where a neural operator learns the dynamics. An inverse FFT (IFFT) then maps the representation back to the time domain. Orange region: A learnable element-wise filter $K$ refines the final prediction $x(t_1)$. This design leverages both frequency and time-domain operations to capture complex patterns in the data.
  • Figure 2: Periodic-3D-A. Performance comparison of predictive models (RNN, NODE, and Ours) on the Periodic-3D-A dataset. The black lines represent the ground truth, while the green lines show predictions. The top row corresponds to a high-frequency amplitude of 0.05, and the bottom row corresponds to an amplitude of 0.1.
  • Figure 3: Periodic-3D-B. Performance comparison of predictive models (RNN, NODE, and Ours) on the Periodic-3D-B dataset. The black lines represent the ground truth, while the green lines show predictions. The top row corresponds to a high-frequency amplitude of 0.05, and the bottom row corresponds to an amplitude of 0.1.
  • Figure 4: Visual comparison of model predictions on four representative dynamical systems: Unstable Oscillator (far left), Forced Vibration (second from left), Lotka Volterra (third), and Glycolytic Oscillator (far right). The figure shows ground-truth training data (solid gray) and testing data (dashed gray), along with the initial state (black dot) and trajectories generated by RNN (green), NODE (blue), and FODE (red).
  • Figure 5: Predicted trajectories (colored) vs. ground truth (black) at 1, 3, and 200 epochs. RNN fails to capture the global pattern. NODE improves learning but shows convergence bias. FODE quickly recovers both local details and the global pattern, achieving stable, accurate predictions.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Definition 1: Fourier ordinary differential equation (FODE)
  • Lemma 1: Lipschitz continuity of $f_{\mathrm{FODE}}$
  • proof
  • Theorem 1: Existence and uniqueness
  • proof