# From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction

arXiv: https://arxiv.org/abs/2004.00574 · HTML: https://mlsanity.com/paper/2004.00574/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2020.

## Built on by (most cited first, 30 of 50)

- [Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors](https://mlsanity.com/paper/2305.18803/) (2023)
- [FreDF: Learning to Forecast in the Frequency Domain](https://mlsanity.com/paper/2402.02399/) (2024)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Bagging, optimized dynamic mode decomposition (BOP-DMD) for robust, stable forecasting with spatial and temporal uncertainty-quantification](https://mlsanity.com/paper/2107.10878/) (2021)
- [Time-o1: Time-Series Forecasting Needs Transformed Label Alignment](https://mlsanity.com/paper/2505.17847/) (2025)
- [Spectral Invariant Learning for Dynamic Graphs under Distribution Shifts](https://mlsanity.com/paper/2403.05026/) (2024)
- [Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems](https://mlsanity.com/paper/2403.12335/) (2024)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [DistDF: Time-Series Forecasting Needs Joint-Distribution Wasserstein Alignment](https://mlsanity.com/paper/2510.24574/) (2025)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Time Series Analysis in Frequency Domain: A Survey of Open Challenges, Opportunities and Benchmarks](https://mlsanity.com/paper/2504.07099/) (2025)
- [Frequency Spectrum is More Effective for Multimodal Representation and Fusion: A Multimodal Spectrum Rumor Detector](https://mlsanity.com/paper/2312.11023/) (2023)
- [Augmenting astrophysical scaling relations with machine learning: application to reducing the Sunyaev-Zeldovich flux-mass scatter](https://mlsanity.com/paper/2201.01305/) (2022)
- [Lite-Mind: Towards Efficient and Robust Brain Representation Network](https://mlsanity.com/paper/2312.03781/) (2023)
- [Not All Frequencies Are Created Equal:Towards a Dynamic Fusion of Frequencies in Time-Series Forecasting](https://mlsanity.com/paper/2407.12415/) (2024)
- [LETS Forecast: Learning Embedology for Time Series Forecasting](https://mlsanity.com/paper/2506.06454/) (2025)
- [Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning](https://mlsanity.com/paper/2202.05750/) (2022)
- [Deep Probabilistic Koopman: Long-term time-series forecasting under periodic uncertainties](https://mlsanity.com/paper/2106.06033/) (2021)
- [SDYN-GANs: Adversarial Learning Methods for Multistep Generative Models for General Order Stochastic Dynamics](https://mlsanity.com/paper/2302.03663/) (2023)
- [Extraction of instantaneous frequencies and amplitudes in nonstationary time-series data](https://mlsanity.com/paper/2104.01293/) (2021)
- [Active Learning of Quantum System Hamiltonians yields Query Advantage](https://mlsanity.com/paper/2112.14553/) (2021)
- [Lyapunov-Guided Representation of Recurrent Neural Network Performance](https://mlsanity.com/paper/2204.04876/) (2022)
- [When Graph Neural Networks Meet Dynamic Mode Decomposition](https://mlsanity.com/paper/2410.05593/) (2024)
- [FC2T2: The Fast Continuous Convolutional Taylor Transform with Applications in Vision and Graphics](https://mlsanity.com/paper/2111.00110/) (2021)
- [Physics-Informed Koopman Network](https://mlsanity.com/paper/2211.09419/) (2022)
- [Combining Slow and Fast: Complementary Filtering for Dynamics Learning](https://mlsanity.com/paper/2302.13754/) (2023)
- [Koopman AutoEncoder via Singular Value Decomposition for Data-Driven Long-Term Prediction](https://mlsanity.com/paper/2408.11303/) (2024)
- [Non-Stationary Time Series Forecasting Based on Fourier Analysis and Cross Attention Mechanism](https://mlsanity.com/paper/2505.06917/) (2025)
- [SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting](https://mlsanity.com/paper/2506.14113/) (2025)

## Abstract (arXiv)

We propose spectral methods for long-term forecasting of temporal signals stemming from linear and nonlinear quasi-periodic dynamical systems. For linear signals, we introduce an algorithm with similarities to the Fourier transform but which does not rely on periodicity assumptions, allowing for forecasting given potentially arbitrary sampling intervals. We then extend this algorithm to handle nonlinearities by leveraging Koopman theory. The resulting algorithm performs a spectral decomposition in a nonlinear, data-dependent basis. The optimization objective for both algorithms is highly non-convex. However, expressing the objective in the frequency domain allows us to compute global optima of the error surface in a scalable and efficient manner, partially by exploiting the computational properties of the Fast Fourier Transform. Because of their close relation to Bayesian Spectral Analysis, uncertainty quantification metrics are a natural byproduct of the spectral forecasting methods. We extensively benchmark these algorithms against other leading forecasting methods on a range of synthetic experiments as well as in the context of real-world power systems and fluid flows.