# Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors

arXiv: https://arxiv.org/abs/2305.18803 · HTML: https://mlsanity.com/paper/2305.18803/

## Key facts

- Cited by 113 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 54 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 30 of 35)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [Instance Normalization: The Missing Ingredient for Fast Stylization](https://mlsanity.com/paper/1607.08022/) (2016)
- [TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis](https://mlsanity.com/paper/2210.02186/) (2022)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Deeper, Broader and Artier Domain Generalization](https://mlsanity.com/paper/1710.03077/) (2017)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [N-BEATS: Neural basis expansion analysis for interpretable time series forecasting](https://mlsanity.com/paper/1905.10437/) (2019)
- [An Empirical Investigation of Catastrophic Forgetting in Gradient-Based Neural Networks](https://mlsanity.com/paper/1312.6211/) (2013)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization](https://mlsanity.com/paper/2106.06607/) (2021)
- [Less Is More: Fast Multivariate Time Series Forecasting with Light Sampling-oriented MLP Structures](https://mlsanity.com/paper/2207.01186/) (2022)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [N-HiTS: Neural Hierarchical Interpolation for Time Series Forecasting](https://mlsanity.com/paper/2201.12886/) (2022)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Domain Adaptation for Time Series Forecasting via Attention Sharing](https://mlsanity.com/paper/2102.06828/) (2021)

## Built on by (most cited first, 30 of 113)

- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Deep Time Series Models: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2407.13278/) (2024)
- [Sundial: A Family of Highly Capable Time Series Foundation Models](https://mlsanity.com/paper/2502.00816/) (2025)
- [TimeMixer++: A General Time Series Pattern Machine for Universal Predictive Analysis](https://mlsanity.com/paper/2410.16032/) (2024)
- [UniTS: A Unified Multi-Task Time Series Model](https://mlsanity.com/paper/2403.00131/) (2024)
- [UniST: A Prompt-Empowered Universal Model for Urban Spatio-Temporal Prediction](https://mlsanity.com/paper/2402.11838/) (2024)
- [Time-MMD: Multi-Domain Multimodal Dataset for Time Series Analysis](https://mlsanity.com/paper/2406.08627/) (2024)
- [AutoTimes: Autoregressive Time Series Forecasters via Large Language Models](https://mlsanity.com/paper/2402.02370/) (2024)
- [Frequency Adaptive Normalization For Non-stationary Time Series Forecasting](https://mlsanity.com/paper/2409.20371/) (2024)
- [Revitalizing Multivariate Time Series Forecasting: Learnable Decomposition with Inter-Series Dependencies and Intra-Series Variations Modeling](https://mlsanity.com/paper/2402.12694/) (2024)
- [TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting](https://mlsanity.com/paper/2410.04442/) (2024)
- [TimeXL: Explainable Multi-modal Time Series Prediction with LLM-in-the-Loop](https://mlsanity.com/paper/2503.01013/) (2025)
- [TSI-Bench: Benchmarking Time Series Imputation](https://mlsanity.com/paper/2406.12747/) (2024)
- [Non-stationary Diffusion For Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2505.04278/) (2025)
- [ST-MLP: A Cascaded Spatio-Temporal Linear Framework with Channel-Independence Strategy for Traffic Forecasting](https://mlsanity.com/paper/2308.07496/) (2023)
- [Ada-MSHyper: Adaptive Multi-Scale Hypergraph Transformer for Time Series Forecasting](https://mlsanity.com/paper/2410.23992/) (2024)
- [Attractor Memory for Long-Term Time Series Forecasting: A Chaos Perspective](https://mlsanity.com/paper/2402.11463/) (2024)
- [Learning Robust Spectral Dynamics for Temporal Domain Generalization](https://mlsanity.com/paper/2505.12585/) (2025)
- [VCformer: Variable Correlation Transformer with Inherent Lagged Correlation for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2405.11470/) (2024)
- [TimeEmb: A Lightweight Static-Dynamic Disentanglement Framework for Time Series Forecasting](https://mlsanity.com/paper/2510.00461/) (2025)
- [TimesBERT: A BERT-Style Foundation Model for Time Series Understanding](https://mlsanity.com/paper/2502.21245/) (2025)
- [FreCT: Frequency-augmented Convolutional Transformer for Robust Time Series Anomaly Detection](https://mlsanity.com/paper/2505.00941/) (2025)
- [Time-SSM: Simplifying and Unifying State Space Models for Time Series Forecasting](https://mlsanity.com/paper/2405.16312/) (2024)
- [Chimera: Effectively Modeling Multivariate Time Series with 2-Dimensional State Space Models](https://mlsanity.com/paper/2406.04320/) (2024)
- [K2VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2505.23017/) (2025)
- [FTS: A Framework to Find a Faithful TimeSieve](https://mlsanity.com/paper/2405.19647/) (2024)
- [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)
- [Phase-driven Domain Generalizable Learning for Nonstationary Time Series](https://mlsanity.com/paper/2402.05960/) (2024)
- [AdaWaveNet: Adaptive Wavelet Network for Time Series Analysis](https://mlsanity.com/paper/2405.11124/) (2024)

## Abstract (arXiv)

Real-world time series are characterized by intrinsic non-stationarity that poses a principal challenge for deep forecasting models. While previous models suffer from complicated series variations induced by changing temporal distribution, we tackle non-stationary time series with modern Koopman theory that fundamentally considers the underlying time-variant dynamics. Inspired by Koopman theory of portraying complex dynamical systems, we disentangle time-variant and time-invariant components from intricate non-stationary series by Fourier Filter and design Koopman Predictor to advance respective dynamics forward. Technically, we propose Koopa as a novel Koopman forecaster composed of stackable blocks that learn hierarchical dynamics. Koopa seeks measurement functions for Koopman embedding and utilizes Koopman operators as linear portraits of implicit transition. To cope with time-variant dynamics that exhibits strong locality, Koopa calculates context-aware operators in the temporal neighborhood and is able to utilize incoming ground truth to scale up forecast horizon. Besides, by integrating Koopman Predictors into deep residual structure, we ravel out the binding reconstruction loss in previous Koopman forecasters and achieve end-to-end forecasting objective optimization. Compared with the state-of-the-art model, Koopa achieves competitive performance while saving 77.3% training time and 76.0% memory.