# Learning Long Term Dependencies via Fourier Recurrent Units

arXiv: https://arxiv.org/abs/1803.06585 · HTML: https://mlsanity.com/paper/1803.06585/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2018.

## Built on by (most cited first, 11 of 11)

- [FNet: Mixing Tokens with Fourier Transforms](https://mlsanity.com/paper/2105.03824/) (2021)
- [FiLM: Frequency improved Legendre Memory Model for Long-term Time Series Forecasting](https://mlsanity.com/paper/2205.08897/) (2022)
- [DCTdiff: Intriguing Properties of Image Generative Modeling in the DCT Space](https://mlsanity.com/paper/2412.15032/) (2024)
- [Vim-F: Visual State Space Model Benefiting from Learning in the Frequency Domain](https://mlsanity.com/paper/2405.18679/) (2024)
- [Fast-FNet: Accelerating Transformer Encoder Models via Efficient Fourier Layers](https://mlsanity.com/paper/2209.12816/) (2022)
- [Emergence of the Primacy Effect in Structured State-Space Models](https://mlsanity.com/paper/2502.13729/) (2025)
- [Oscillatory Fourier Neural Network: A Compact and Efficient Architecture for Sequential Processing](https://mlsanity.com/paper/2109.13090/) (2021)
- [Fourier-RNNs for Modelling Noisy Physics Data](https://mlsanity.com/paper/2302.06534/) (2023)
- [Blurry Window Attention](https://mlsanity.com/paper/2606.09862/) (2026)
- [Energy time series forecasting-Analytical and empirical assessment of conventional and machine learning models](https://mlsanity.com/paper/2108.10663/) (2021)
- [Generalizing to New Dynamical Systems via Frequency Domain Adaptation](https://mlsanity.com/paper/2507.00025/) (2025)

## Abstract (arXiv)

It is a known fact that training recurrent neural networks for tasks that have long term dependencies is challenging. One of the main reasons is the vanishing or exploding gradient problem, which prevents gradient information from propagating to early layers. In this paper we propose a simple recurrent architecture, the Fourier Recurrent Unit (FRU), that stabilizes the gradients that arise in its training while giving us stronger expressive power. Specifically, FRU summarizes the hidden states h(t) along the temporal dimension with Fourier basis functions. This allows gradients to easily reach any layer due to FRU's residual learning structure and the global support of trigonometric functions. We show that FRU has gradient lower and upper bounds independent of temporal dimension. We also show the strong expressivity of sparse Fourier basis, from which FRU obtains its strong expressive power. Our experimental study also demonstrates that with fewer parameters the proposed architecture outperforms other recurrent architectures on many tasks.