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

En Yu; Xiaoyu Yang; Wei Duan; Guangquan Zhang; Jie Lu

cs.LGAug 23, 2026

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

Authors: En Yu, Xiaoyu Yang, Wei Duan, Guangquan Zhang, Jie Lu

Organizations: Australian Artificial Intelligence Institute (AAII), Faculty of Engineering and Information Technology, University of Technology Sydney (UTS), Australia

Abstract

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.

Figures & tables

Fig. 1 : Illustration of the three TDG challenges on a 2-Moons stream. The underlying distribution exhibits multi-scale periodicity, while empirical parameter estimates are corrupted by domain-specific uncertainties. Additionally, the irregularly sampled observations violate the time-invariant discrete transition assumed by standard sequence models.

Fig. 2 : FreKoo framework. It decomposes model parameter trajectories into low- and high-frequency components via the Fourier transform. Koopman dynamics extrapolate the low-frequency component, while temporal-difference regularization suppresses unstable high-frequency variations.

Fig. 3 : FreKoo++ framework. (1) Training Phase: Model parameters from irregular timestamps are encoded into a continuous latent trajectory and fitted by Koopman modal dynamics. Adaptive soft weighting disentangles persistent dominant dynamics from transient noise, yielding an optimized continuous spectral representation via joint optimization. (2) Inference Phase: For an arbitrary target timestamp

t_{s}>t_{T}

, FreKoo++ analytically extrapolates the persistent dominant component to

t_{s}

, freezes the transient residual at

t_{T}

, and decodes their superposition to obtain target parameters

\hat{\theta}(t_{s})

Methods	Classification				Regression
Methods	2-Moons-C $\downarrow$	Rot-MNIST-C $\downarrow$	Twitter $\uparrow$	Yearbook $\downarrow$	Cyclone $\downarrow$	House-C $\downarrow$
Offline	13.5 $\pm$ 0.3	6.6 $\pm$ 0.2	0.54 $\pm$ 0.09	8.6 $\pm$ 1.0	18.7 $\pm$ 1.4	19.9 $\pm$ 0.1
LastDomain	55.7 $\pm$ 0.5	74.2 $\pm$ 0.9	0.54 $\pm$ 0.12	11.3 $\pm$ 1.3	22.3 $\pm$ 0.7	20.6 $\pm$ 0.7
IncFinetune	51.9 $\pm$ 0.7	57.1 $\pm$ 1.4	0.52 $\pm$ 0.01	11.0 $\pm$ 0.8	19.9 $\pm$ 0.7	20.6 $\pm$ 0.2
IRM [ 1 ]	15.6 $\pm$ 0.2	8.6 $\pm$ 0.4	0.53 $\pm$ 0.11	8.3 $\pm$ 0.5	18.0 $\pm$ 0.8	19.8 $\pm$ 0.2
V-REx [ 21 ]	12.8 $\pm$ 0.2	8.6 $\pm$ 0.3	0.58 $\pm$ 0.05	8.9 $\pm$ 0.5	17.7 $\pm$ 0.5	20.2 $\pm$ 0.1

TABLE I : Performance comparison on CTDG datasets . The classification tasks report error rates (%) except for the AUC for the Twitter dataset. The regression tasks report MAE. ’-’ implies that the method does not support the task.

Methods	Classification					Regression
Methods	2-Moons-D $\downarrow$	Rot-MNIST-D $\downarrow$	ONP $\downarrow$	Shuttle $\downarrow$	Elec2 $\downarrow$	House-D $\downarrow$	Appliance $\downarrow$
Offline	22.4 $\pm$ 4.6	18.6 $\pm$ 4.0	33.8 $\pm$ 0.6	0.77 $\pm$ 0.10	23.0 $\pm$ 3.1	11.0 $\pm$ 0.36	10.2 $\pm$ 1.1
LastDomain	14.9 $\pm$ 0.9	17.2 $\pm$ 3.1	36.0 $\pm$ 0.2	0.91 $\pm$ 0.18	25.8 $\pm$ 0.6	10.3 $\pm$ 0.16	9.1 $\pm$ 0.7
IncFinetune	16.7 $\pm$ 3.4	10.1 $\pm$ 0.8	34.0 $\pm$ 0.3	0.83 $\pm$ 0.07	27.3 $\pm$ 4.2	9.7 $\pm$ 0.01	8.9 $\pm$ 0.5
CDOT [ 32 ]	9.3 $\pm$ 1.0	14.2 $\pm$ 1.0	34.1 $\pm$ 0.0	0.94 $\pm$ 0.17	17.8 $\pm$ 0.6	–	–
CIDA [ 38 ]	10.8 $\pm$ 1.6	9.3 $\pm$ 0.7	34.7 $\pm$ 0.6	–	14.1 $\pm$ 0.2	9.7 $\pm$ 0.06	8.7 $\pm$ 0.2

TABLE II : Performance comparison on DTDG datasets . Classification tasks report error rates (%), and regression tasks report MAE. “–” indicates that the method does not support the task.

Method-level variants
Variant	2-Moons-C	Twitter	Cyclone
	Err. $\downarrow$	AUC $\uparrow$	MAE $\downarrow$
Fixed spectrum	9.7 $\pm$ 1.5	0.64 $\pm$ 0.11	17.2 $\pm$ 0.7
Hard gating	6.7 $\pm$ 2.8	0.67 $\pm$ 0.09	18.7 $\pm$ 0.3
Frozen gating	2.8 $\pm$ 0.3	0.70 $\pm$ 0.01	16.2 $\pm$ 0.2
Objective-level variants

TABLE III : Ablation studies on representative CTDG benchmarks. We report error rate (%,

\downarrow

) for 2-Moons-C, AUC (

\uparrow

) for Twitter, and MAE (

\downarrow

) for Cyclone. The best performance in each column is in bold .

Panel of Figure 7 — Fig. 4 : Qualitative spectral analysis on 2-Moons-C. (a) The parameter trajectory is decomposed into a smooth dominant component and a short-lived transient residual. (b) Learned modes are organized in the frequency–decay plane, where adaptive gates separate persistent modes from transient ones without fixed DFT bins or manual frequency thresholds. (c) Continuous-time inference extrapolates the dominant dynamics while freezing the transient component, producing a coherent future trajectory.

Panel of Figure 8 — Fig. 5 : Qualitative comparison of decision boundaries learned on 2-Moons-C across six unseen future domains. Red curves denote learned decision boundaries alongside the corresponding classification error rates.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Predictive model ( $\theta\in\mathbb{R}^{D}$ )	Latent encoder-decoder ( $\phi,\phi^{-1}$ )
2-Moons-C	MLP: [2, 50, 50, 1] + ReLU + Sigmoid	$\phi$ : [ $D\to 1024\to 512\to 128\to m$ ] $\phi^{-1}$ : [ $m\to 128\to 512\to 1024\to D$ ]
Rot-MNIST-C	$3\times$ [Conv(32/32/64, $k{=}3$ ) + ReLU + MaxPool( $2$ )] + MLP: [576, 128, 10] + LogSoftmax
Twitter	Embedding Network + MLP [32, 128, 32, 1] + ReLU + Sigmoid
Yearbook	$3\times$ [Conv(32/32/64, $k{=}3$ ) + ReLU + MaxPool( $2$ )] + MLP: [1024, 128, 32, 1] + ReLU + Sigmoid
Cyclone	$4\times$ [Conv(32/32/64/64, $k{=}3$ ) + ReLU + MaxPool( $2$ )] + MLP: [1024, 128, 32, 1] + ReLU
House	MLP: [30, 400, 400, 1] + ReLU

Appendix

TABLE IV : Detailed network architectures for the task predictive model and the uniform latent autoencoder (

\phi,\phi^{-1}

) across all benchmark datasets. All autoencoder hidden layers employ ReLU activations.

Panel of Figure 10 — Fig. 6 : t-SNE visualization on all 15 future domains of Rot-MNIST-C (colored by class). Each subfigure is one future domain with its step index and classification error.

Panel of Figure 11 — Fig. 7 : t-SNE visualization on all 12 future domains of Yearbook (colored by class). Each subfigure is one future domain with its step index and classification error.

Panel of Figure 12 — Fig. 8 : Hyperparameter sensitivity analysis of FreKoo++ with respect to objective loss weights $\alpha$ ( $\mathcal{L}_{\mathrm{rec}}^{(C)}$ ), $\beta$ ( $\mathcal{L}_{\mathrm{fit}}^{(C)}$ ), $\gamma$ ( $\mathcal{R}_{\mathrm{spec}}$ ), and $\delta$ ( $\mathcal{R}_{\mathrm{stab}}$ ) swept over $\{0.01,0.1,1.0,10,100\}$ across six benchmark datasets.

Panel of Figure 13 — Fig. 9 : Sensitivity of the number of Koopman modes $K$ . The red star marks the adopted $K=32$ .

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Open paper AskPDF arXiv

Temporal Domain Generalization (TDG) has emerged to address real-world streaming data with distribution shifts over time. However, existing methods are either prone to overfitting to domain-specific noise in the data space or become overly complex and less interpretable in the parameter space. To bridge these gaps, we propose AdaSpecK, a spectral-Koopman framework with adaptive context extraction for TDG. To mitigate noise fitting to irregularly sampled domains, we introduce spectral-regularized Koopman dynamics modeling, which applies spectral-aware filtering in the latent space to extract denoised low-frequency trajectories and learn a Koopman operator to model the system dynamics in a linearized space. To model complex historical environments under non-stationarity, we design a context-informed heterogeneous pattern extraction mechanism. Specifically, we employ a target-conditioned attention module to attend to distinct past windows, producing a dynamic, target-specific historical summary. By constructing an environmental signature from the current evolutionary pattern, our model adaptively perceives which aspects of the past context are most informative for future prediction via a learned router. Extensive experiments on eight diverse classification and regression benchmarks demonstrate that AdaSpecK achieves state-of-the-art performance. The code and datasets are available at \href{}{https://anonymous.4open.science/r/Ada-Spec-K}.

Abstract

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