# Modeling Time Series Dynamics with Fourier Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2510.04133 · HTML: https://mlsanity.com/paper/2510.04133/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 5 Oct 2025.

## Builds on (most cited first, 17 of 17)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [FNet: Mixing Tokens with Fourier Transforms](https://mlsanity.com/paper/2105.03824/) (2021)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Global Filter Networks for Image Classification](https://mlsanity.com/paper/2107.00645/) (2021)
- [Bias in Large Language Models: Origin, Evaluation, and Mitigation](https://mlsanity.com/paper/2411.10915/) (2024)
- [On Second Order Behaviour in Augmented Neural ODEs](https://mlsanity.com/paper/2006.07220/) (2020)
- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [ExARNN: An Environment-Driven Adaptive RNN for Learning Non-Stationary Power Dynamics](https://mlsanity.com/paper/2505.17488/) (2025)
- [Solar Photovoltaic Assessment with Large Language Model](https://mlsanity.com/paper/2507.19144/) (2025)

## Abstract (arXiv)

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.