# FreKoo++: Learning Continuous Spectral Dynamics for Temporal Domain Generalization

arXiv: https://arxiv.org/abs/2608.22224 · HTML: https://mlsanity.com/paper/2608.22224/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 37 of its 59 references that are in the map.
- First posted on arXiv 23 Aug 2026.

## Builds on (most cited first, 30 of 37)

- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [Learning under Concept Drift: A Review](https://mlsanity.com/paper/2004.05785/) (2020)
- [Towards Stable Test-Time Adaptation in Dynamic Wild World](https://mlsanity.com/paper/2302.12400/) (2023)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors](https://mlsanity.com/paper/2305.18803/) (2023)
- [Modeling Irregular Time Series with Continuous Recurrent Units](https://mlsanity.com/paper/2111.11344/) (2021)
- [Wild-Time: A Benchmark of in-the-Wild Distribution Shift over Time](https://mlsanity.com/paper/2211.14238/) (2022)
- [Continuously Indexed Domain Adaptation](https://mlsanity.com/paper/2007.01807/) (2020)
- [Frequency Adaptive Normalization For Non-stationary Time Series Forecasting](https://mlsanity.com/paper/2409.20371/) (2024)
- [FilterNet: Harnessing Frequency Filters for Time Series Forecasting](https://mlsanity.com/paper/2411.01623/) (2024)
- [Koopman Neural Forecaster for Time Series with Temporal Distribution Shifts](https://mlsanity.com/paper/2210.03675/) (2022)
- [Temporal Domain Generalization with Drift-Aware Dynamic Neural Networks](https://mlsanity.com/paper/2205.10664/) (2022)
- [Training for the Future: A Simple Gradient Interpolation Loss to Generalize Along Time](https://mlsanity.com/paper/2108.06721/) (2021)
- [Generalizing to Evolving Domains with Latent Structure-Aware Sequential Autoencoder](https://mlsanity.com/paper/2205.07649/) (2022)
- [Adapting Multi-modal Large Language Model to Concept Drift From Pre-training Onwards](https://mlsanity.com/paper/2405.13459/) (2024)
- [Online Boosting Adaptive Learning under Concept Drift for Multistream Classification](https://mlsanity.com/paper/2312.10841/) (2023)
- [Continuous Temporal Domain Generalization](https://mlsanity.com/paper/2405.16075/) (2024)
- [Learning Robust Spectral Dynamics for Temporal Domain Generalization](https://mlsanity.com/paper/2505.12585/) (2025)
- [Drift-aware Collaborative Assistance Mixture of Experts for Heterogeneous Multistream Learning](https://mlsanity.com/paper/2508.01598/) (2025)
- [Generalizing across Temporal Domains with Koopman Operators](https://mlsanity.com/paper/2402.07834/) (2024)
- [Model Assessment and Selection under Temporal Distribution Shift](https://mlsanity.com/paper/2402.08672/) (2024)
- [Foresee What You Will Learn: Data Augmentation for Domain Generalization in Non-stationary Environment](https://mlsanity.com/paper/2301.07845/) (2023)
- [ATFNet: Adaptive Time-Frequency Ensembled Network for Long-term Time Series Forecasting](https://mlsanity.com/paper/2404.05192/) (2024)
- [Early Concept Drift Detection via Prediction Uncertainty](https://mlsanity.com/paper/2412.11158/) (2024)
- [SCoT: Unifying Consistency Models and Rectified Flows via Straight-Consistent Trajectories](https://mlsanity.com/paper/2502.16972/) (2025)
- [Generalized Incremental Learning under Concept Drift across Evolving Data Streams](https://mlsanity.com/paper/2506.05736/) (2025)
- [CODA: Temporal Domain Generalization via Concept Drift Simulator](https://mlsanity.com/paper/2310.01508/) (2023)
- [Distributionally Robust Policy Learning under Concept Drifts](https://mlsanity.com/paper/2412.14297/) (2024)

## Abstract (arXiv)

Temporal Domain Generalization (TDG) aims to learn from historical domains and generalize to unseen future distributions under concept drift. Nevertheless, prevailing TDG methods struggle with complex real-world streaming scenarios involving both multi-scale drift patterns (e.g., long-term periodicity intertwined with short-term incremental changes) and local uncertainties, especially in continuous settings where observations arrive irregularly. To address this limitation, we propose FreKoo++, a novel continuous spectral-dynamical framework that pioneers the unification of continuous Koopman modal dynamics with adaptive spectral disentanglement. Specifically, FreKoo++ maps source-domain parameters into a compact latent space, modeling their evolution as a superposition of learnable continuous modes where complex eigenvalues jointly encode oscillatory frequency and temporal growth or decay. This formulation naturally accommodates irregular timestamps and supports arbitrary horizon extrapolation without rigid discrete stepping. Furthermore, we propose a new adaptive soft spectral weighting mechanism backed by stability and spectral regularization, which automatically isolates persistent dominant dynamics from transient noise without relying on manual frequency thresholds. We derive modal approximation and generalization bounds that characterize how amplitude and eigenvalue estimation errors propagate with the prediction horizon. Extensive experiments on both discrete and continuous TDG benchmarks demonstrate that FreKoo++ achieves state-of-the-art performance under complex multi-scale drifts and irregular sampling.