Learning Robust Spectral Dynamics  for Temporal Domain Generalization
Authors: En Yu
Jie Lu
Xiaoyu Yang
Guangquan Zhang
Zhen Fang
Venue: NeurIPS 2025 poster ()
Keywords: Domain generalization
Temporal domain generalization
Concept drift
Primary area: general_machine_learning

Abstract: Modern machine learning models struggle to maintain performance in dynamic environments where temporal distribution shifts, \textit{i.e., concept drift}, are prevalent. Temporal Domain Generalization (TDG) seeks to enable model generalization across evolving domains, yet existing approaches typically assume smooth incremental changes, struggling with complex real-world drifts involving both long-term structure (incremental evolution/periodicity) and local uncertainties. To overcome these limitations, we introduce FreKoo, which tackles these challenges through a novel frequency-domain analysis of parameter trajectories. It leverages the Fourier transform to disentangle parameter evolution into distinct spectral bands. Specifically, the low-frequency components with dominant dynamics are learned and extrapolated using the Koopman operator, robustly capturing diverse drift patterns including both incremental and periodic drifts. Simultaneously, potentially disruptive high-frequency variations are smoothed via targeted temporal regularization, preventing overfitting to transient noise and domain uncertainties. In addition, this dual-spectral strategy is rigorously grounded through theoretical analysis, providing stability guarantees for the Koopman prediction, a principled Bayesian justification for the high-frequency regularization, and culminating in a multiscale generalization bound connecting spectral dynamics to improved generalization. Extensive experiments demonstrate FreKoo's significant superiority over state-of-the-art TDG methods, particularly excelling in real-world streaming scenarios with complex drifts and uncertainties.

Learning Robust Spectral Dynamics for Temporal Domain Generalization
En Yu, Jie Lu, Xiaoyu Yang, Guangquan Zhang, Zhen Fang
Australian Artificial Intelligence Institute (AAII),
University of Technology Sydney, Australia.
{en. yun,yu lu guangquun. zhang, zhen zhen. fang) @uts edu. au; Xiaoyu. Yang- mangg33ssudent .us. edu. au
Abstract
Modern machine learning models struggle to maintain performance in dynamic environments where temporal distribution shifts, i.e., concept drift, are prevalent. Temporal Domain Generalization (TDG) seeks to enable model generalization across evolving domains, yet existing approaches typically assume smooth incre- mental changes, struggling with complex real-world drifts involving both long-term structure (incremental evolution/periodicitt)) and local uncertainties. To overcome these limitations, we introduce FreKoo, which tackles these challenges through a novel frequency-domain analysis of parameter trajectories. It leverages the Fourier transform to disentangle parameter evolution into distinct spectral bands. Specif- ically, the low- low-frequency components with dominant dynamics learned and extrapolated using the Koopman operator, robustly capturing diverse drift patterns including both incremental and periodic drifts. Simultancously, potentially disrup- tive high-frequency variations are smoothed via targeted temporal regularization, preventing overfitting transient noise and domain uncertainties. In addition, this dual-spectral strategy is rigorously grounded through theoretical analysis, providing stability guarantees for the Koopman prediction, principled Bayesian justification for the high-frequency regularization, and culminating in a multiscale generaliza- tion bound connecting spectral dynamics to improved generalization. Extensive experiments demonstrate FreKoo's 's significant superiority over state-of ofthe-of-the-art TDG methods, particularly excelling in real-world streaming scenarios with complex drifts and uncertainties.
Introduction
Modern machine learning models face significant challenges dynamic environments where data distributions evolve over time [I] Unlike static settings that assume an IID relationship between training and test data, real-world scenarios often involve continuously generated data streams (c.g., user activity logs, sensor readings, financial transactions) exhibiting temporal dependencies and non non-stationarity due to factors like shifting user behavior, environmental variations, or system changes (2). These temporal dynamics lead to distributional shifts over time, known concept drift, which invalidates the alignment between historical and future out-of-distribution (OOD) data 3 Consequently, models trained past data frequently fail to generalize to future instances, leading to performance degradation and reduced reliability. This has spurred increasing interest in Temporal Domain Generalization (TDG), which seeks develop models capable of generalizing robustly across chronologically ordered source domains to unseen future target distributions 6
*Corresonnencee to Jie Lu and Zhen Fang
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Periodic drift
Incremontall drift
Deri
Arca 2
Model Space
Area
Smooth
O2
P
Area 4
Local overfitting
Time
Data distribution Dr
Figure 1: Illustaaion of challenges in TDG. Left: Complex drifting situations can involve both incremental shifts (e.g., Dj + D2 + D3) and long-term periodic returns (c.g.. D, resembling Di after a cycle). The underlying optimal parameters 0, evolve accordingly. Right: Within any domain Dr, uncertainties non-lID data concentrated in Areas 1, 2 and 4 compared to data in Area 3 can lead to local overfitting (solid blue line). Robust generalization requires converging to a smoother area (red dashed line) that is less sensitive such localized noise outliers.
Recent progress in TDG falls into two categories: data-driven and model-centric approaches. Data-driven methods enhance temporal robustness by simulating future distributions via OOD data generation [9] [0] or learning temporally invariant features [I.. Model-centric methods often leverage dynamic modeling, such as time-sensitive regularization parameter forecasting, to adapt models to evolving distributions 6 [2].
concept drift is primarily incremental However, existing methods often implicitly assume that focus on ensuring local smoothness between adjacent domains. This overlooks a crucial aspect of many real-world scenarios: 1) Difficulty in Modeling Long-term Periodic Dynamics: real-world concept drift frequently exhibits long-term periodicity (e.g., seasonality, weekly/daily user patterns, economic cycles), not just smoothly incremental changes [13
shown in the left of Figure T Current approaches struggle to capture these recurring patterns spanning longer time horizons [8] Their effectiveness often diminishes, particularly near phase transitions of periodic cycles where the inability to anticipate pattern recurrence hinders generalization. 2) Vulnerability to Overfitting typically segment continuous data streams into discrete temporal domains. However, the complex drift patterns inherent in real-world streams make it difficult to guarantee that data within each domain remains IID, as visualized in Appendix Fiigndi and non-IID characteristics within domains can complicate optimization and increase the risk of overfitting to comainn-secciff artifacts (illustrated in the right of Figure D ultimately undermining stable cross-temporal generalization [5].
Specife Uncertainties: current TDG methods
Consequently, localized uncertainties
These challenges underscore the necessity for principled frameworks capable of capturing long- term incremental/perrodiic dynamics while filtering local uncertainties. Intuitively, the trajectory of model parameters over time provides comprehensive reflection of the underlying concept drift, encapsulating its long-range evolution alongside its unpredictable uncertainties. Thus, we turn to analysis inherently
frequency-domain analysis of the parameter trajectories. Frequency-domen isolates dominant dynamics by representing them as distinct spectral peaks, facilitating robust modeling of long-term evolving patterns [16). Moreover, as analyzed in [17], spectral representations can reveal dynamic temporal structures often obscured by conventional statistical measures (e.g., mean, variance), thereby providing richer understanding of parameter evolution. Specifically, transforming trajectories into the frequency domain reveals that dominant evolving trends and periodic components typically manifest as energy concentrations low frequencies, whereas transient uncertainties typically dominate higher frequencies [18 19). This spectral separation naturally allows for targeted dynamic modeling: isolating and modeling the predictable long-term low-frequency dynamics while permitting robust management of disruptive high-frequency variations. Compared to purely time-domain approaches, this frequency-domen -domann perspective offers a more robust mechanism for modeling diverse temporal dynamics in real-wer world scenarios with complex drifts and uncertainties.
Motivated by this, we propose FreKoo, a Prequency-Koopnee Regularized framework that enhances temporal generalization by integrating spectral decomposition with Koopman operator theory. Specif- ically, as shown in Figure [ FreKoo decomposes model parameter trajectories into low-frequency and high-frequency components via the Fourier transform. Koopman operator theory (20) 21 is employed to learn stable linear model for the evolution of low-frequency dynamics, enabling robust prediction of incremental trends and periodicity. Simultaneously. it introduces a targeted temporal smooth regularization to the high-frequency components, promoting stable convergence and preventing overfitting to local uncertainties. Furthermore, we derive multiscale generalization

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bound that characterizes the interplay between Koopman stability, temporal regularization, and generalization robustness. This theoretical foundation provides rigorous insight into how FreKoo improves generalizability in dynamic environments, offering unified and principled solution to complex concept drifts in TDG. Our main contributions are summarized follows:
We pioneer a spectral analysis perspective on parameter trajectories for TDG. This enables principled disentanglement of complex and multiscaled temporal dynamics, offering novel pathway to address limitations of prior methods in handling long-term periodicity and domain-specific uncertainties.
We propose FreKoo, novel end-to-end framework that materializes this spectral insight. It synergistically combines Koopman operator extrapolation for stable low-frequency dynamies with targeted regularization of nigh-frequency uncertainties, thereby enhancing robustness against complex drifting situations.
We establish rigorous theoretical foundation for FreKoo via a novel multiscale gener- alization bound. connects FreKoo's ' spectral dominant dynamics and temporal smooth regularization to stable generalization. Extensive experiments further demonstrate FreKoo's significant superiority particularly in real-world scenarios with periodicity and uncertainties.
2 Related Works
Temporal Domain Generalization (TDG). TDG specifically tackles scenar- ios where data distributions evolve over time (i.e., concept drift), aiming to train models on historical data that generalize to near future domains [6]. This requires explicitly capturing and leveraging the dynamics of the distribution shifts over time. Current TDG research broadly follows two main directions. Data-driven approaches seek temporal robustness by either synthesizing future data via OOD generation 7 9 10), learning time-invariant representations [1]. Model-centric methods focus incorpo- rating dynamic adaptation mechanisms directly into the learning process, such as time-sensitive regularizations that encourage smooth decision boundaries over time [6] or forecasting future parameters based on past evolution (8] Furthermore, recognizing the continuous nature of time, Continuous TDG (CTDG) methods have been proposed, often treating time as a continuous index for adaptation [12 (22. However, despite progress, effectively modeling long-term periodic patterns and mitigating sensitiv- ity to intra-domain uncertainties leading to local overfitting remain key challenges across existing TDG methods.
D:
Dr+i
Space
Predicc
Model space
0
Purameter Trajectory 0= 10, 0,)
DFE: F() Filker: OB, F-()
Frequect Spocee
(O) (snige)
[9 Oraiet
Latena
dhow
Srow
neww
Phigh
Spuce
Koopman
K
K
Space
Broowe
Brazzow
Reconstructed Space
B
Figure FreKoo Framework. decomposes model parameter trajectories into low-frequency and high- frequency components via the Fourier transform. Then, the Koopman operator is employed to learn the evolution of low-frequency dynamics, enabling robust prediction of incremental trend and periodicity. Also, it introduces a targeted temporal difference regularization to the high- frequency components, promoting smooth convergence and preventing overfitting to local uncertainties.
Concept Drift. Concept drift signifies change in the underlying data distribution over time, ie.. P.+1 (X. y) * Pr(X, y). fundamentally challenges model reliability by requiring dynamic adaptation 3 Concept drift manifests in various forms, including sudden, gradual, incremental, and recurring (periodic) patterns. Much prior research treats as unpredictable and often employs detect-then-adapt strategies 25..2_ In contrast, TDG aims to leverage continuous predictable evolutionary dynamies for proactive generalization (8) However, existing TDG methods primarily model smooth incremental changes, consequently struggling to capture complex, long-range structures
3

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like periodicity which are common in real-world streams and demand modeling approaches beyond simple monotonic progression assumptions.
Frequency Learning. Frequency learning has emerged as a promising paradigm for modeling non- stationary temporal dynamics in machine learning [26 27 By transforming time-domain signals into the spectral space, Irequency-awarg methods effectively capture periodic patterns and multi-scale trends that remain obscured in raw signals. For instance, prior works (28/30) leverage spectral decomposition to identify periodic structures in temporal data. FAN [9] further demonstrates that (requency-domane representations can reveal dynamic features including long-term trends and seasonal variations- that traditional statistical measures (e.g., mean, variance) fail to characterize. These insights provide foundational basis for our work, as we extend frequency learning to a novel perspective, modeling parameter trajectories under concept drift.
Methodology
3.1 Preliminary
Temporal Domain Generalization. Given sequence of II' temporal source domains (Dr)
each domain at timestamp is defined as D. = {xx]" .y4" 1i=1, ) sample-label pair, N, is the number of samples of Dr, and Nr and 3 denote the input and label spaces at timestamp t. We assume that each source domain D; is drawn from a time-dependent distribution Px(X,Y) and the sequence {Pr) exhibits concept drift, which may include incremental or long-term periodic trends. The objective of TDG is to build a model g(:; X + V/ using historical source domains (D,)T- that generalizes effectively to OOD future target domain DT+1 ~ PX, Y), without access to Dr+1 during training [6].
where (x(") yy ( X, x }4 is a
Challenges. Prevailing TDG methods, typically constrained by assumptions of incremental change or local smoothness, struggle to handle complex concept drifts. This manifests in two primary challenges. First, they fail to capture long-range periodic dynamics (Challenge 1), where the data distribution periodically recurs after L steps, i.c., 3, L E 2 s.t. P.(X, z Pi+KL (X, Y) for all k E Z+, despite undergoing short-term shifts Px(X, + Px+1(X,Y) Second, they ill-equipped filter out transient noise and domain-specific uncertainties (Challenge 2). a phenomenon we visualize in Appendix D.I Figure 5 Ultimately, these limitations prevent them from effectively balancing the stability and adaptability required in real-world scenarios.
3.2 Proposed Method: FreKoo
To address these challenges, we model the evolution of parameters 01,00,,..,00 0z and extrapolate predict fr+1, thereby enabling robust inference on future unseen domain Dr+1 with g(: 0r++)) Nr+1 + Vr+1. We establish dynamic system perspective for Temporal Domain Generalization (TDG) by modeling parameter evolution under complex drifting situations BT. Let 0, E denote the parameter of the model ge X, > 3% at time f, the evolution over time can be described by potentially nonlinear stochastic difference form:
0c+1 = 4 (0,) +t;
where c: RDO + D captures deterministic transitions induced by concept drift, and e, represents stochastic perturbations. This formulation explicitly addresses concept drift by treating TDG as parameter trajectory prediction problem governed by a stochastic nonlinear dynamical system.
Directly modeling the complex dynamics of the parameter trajectory is difficult, as it intertwines potentially difffrentt drifts and uncertainties in the real world. To overcome this, we introduce frequency-aware perspective as a powerful inductive bias. We hypothesize that different frequency components within the parameter trajectory θ correspond to distinct aspects of the concept drift: the low-frequency component captures the slowly- verrynng trends and the long- term periodic patterns (Challenge 1). The high-frequency component primarily reflects transient dynamics and domain- specific spurious noise (Challenge 2). By disentangling these frequency regimes, we can robustly model and extrapolate the stable structured low-frequency dynamics while suppressing the volatile, potentially misleading high-frequency variations. This insight motivates the core of our proposed FreKoo framework.

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3.2.1 Spectral Decomposition
Given a time-varying parameter trajectory è = [
E IR with T timesteps and D parameter dimensions, we utilize Fourier analysis to disentangle dominant dynamics from high- frequency fluctuations [16] [2]. Specifically, we first transfer it to frequency space via Discrete Fourier Transform (DFT) along the temporal axis, yielding a spectral representation O via:
@ =F()). Olf.d = Olt - f2x////
where F: FRR + Cree denotes the DFT operator and Vrre [T/2] represents the number of frequency components, serves as the frequency bin index while d indicates the parameter dimension index.
To identify dominant frequencies, we compute the average spectral magnitude across parameter dimensions as an energy proxy:
1 [(f,d]], Vf E {0, ... .Neegr 1}, My D
E
where My R_ represents the normalized energy contribution of frequency f. The magnitude vector M = [M, MI...., Mr E Nyee aggregates these values across all frequencies, serving as dimension-agnosse measure of frequency importance. This averaging operation inherently suppresses parameter- variations in oscillation patterns, ensuring robustness
localized noise in individual
dimensions. For specified energy preservation ratio E [0. 1J. we select the top-Q frequency indices Stop = {/1, .2,..., where Q = [TNg and M, Z M,, Z 2 Mfx ensuring Qtop contains the Q largest magnitudes. Then, we construct frequency-domain binary mask B E {0, N X that preserves selected frequencies:
Bif,d = {0
if f E 2tops for all d = I,.., D. otherwise,
(4)
Finally, can be decomposed into a component ttow (dominant frequencies) and a component Ohigh (high-frequency fluctuations) as follows:
0o e B,
Ohigh 0 2 (1-B),
too = F- F (0/oo) high F (0 (Ohsgh))
where @ denotes element-wise multiplication, represents a matrix of ones, and F is the corre- sponding inverse DFT. By the linearity of F-1, it holds that θ = O + This decomposition enables separate analysis of dominant dynamics Otow and high-frequency fluctuations Ohigh-
Therefore, the spectral decomposition enables targeted handling of distinct components, and Eq. be further expressed (0,) Low (0z, + Rhigh (0 kigh where 4 Piow and Rhigh govern distinct spectral regimes. This decomposition enables: 1) Explicit modeling of dominant evolving patterns through Plow. which exploits the long-term periodic and increment evolving patterns, i.c. targeting for Challenge 1; 2) Targeted regularization of uncertainties via Rhigh. i.c., targeting for Challenge 2. By jointly handling these regimes, the method enhances prediction stability against overfitting to domain-specte uncertainties while maintaining fidelity to long-term trends.
3.2.2 Robust Frequency Dynamics Learning
Koopman Operator for Dominant Dynamics. To model the dominant and temporally stable dynamics underlying the evolution of model parameters, we employ Koopman operator theory, which provides linear surrogate for nonlinear dynamical systems by mapping them into a higher- dimensional functional space 220 21 Specifically, it assumes that the state can be mapped into higher-dimensional Hilbert space via a measurement function ip, where the evolution governed ((x;,1),,
by infinite-dimensionall linear operator K, such that: K. (((x)) = tp ((t((.)))) where is the unknown nonlinear transition function that governs the temporal evolution of the system [16]. Formally, let {01,low.... ..., Or.,ow denote the low-free cow-frequency components of model
5

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parameters extracted via spectral decomposition. These components capture incremental and periodic variations that evolve gradually over time. We posit that this sequence admits near-linear evolution in latent space under a Koopman operator. Specifically, we seek an encoder & RRD - R. and decoder Φ
m DR such that the temporal evolution can expressed as:
Lovc ((0,,oow), 1, Kir,Jow 11 = o ((141,10m);
(6)
where K E I max is a learnable linear transformation that approximates the inite-dimensional Koopman operator. To enforce temporal consistency in the predicted low-frequency dynamics, we define a reconstruction loss:
Lkoop
111+1,low - 2 1.oww
which encourages accurate modeling of long-range parameter trends and mitigates error accumulation over time. This structure allows the model to forecast stable dynamics while preserving interpretability through linear temporal progression in the latent space.
Regularization for Freueerryy Components. While the low-frequene components capture dominant trends suit- able for extrapolation via Koopman dy- namics, the high-frequency components high bigh Or Aigh } often rep- resent transient noise or domain-specific uncertainties. Explicitly extrapolating these components could amplify noise and hinder generalization [15). There- fore, instead of modeling forward dynam- ies for the frggefreyunnyy part, we focus on regularizing its behavior and incorpo- rating its current state into the prediction.
Algorithm FreKoo End-to-End Learning Procedure
Require: Source domain datasets D (D..... Dr}: Hyperparameterss at 7, Koopman dimension m, Number of Epochs Nepoche.
Ensure: Trained
FreKoo model components (9, Vlow Phigh, 6- K) and target parameters 8r+1.
1: Initialize base model g()), Encoders Plows thigh De- coder Φ Koopman operator K E Rmxm.
2: for epoch = 1 to Nepocks do
3: for = to T do
4:
Build base model 9(:;0)) X, J/ - (D
We use an encoder thigh .R" > IR" to map the high-frequeney component into the latent space: high (0gh,
5: end for
6: Obtain parameters trajectory Q [0 ...., 6r|
7: Compute DFT: O = F(O) via Eq. 2
which captures the high-frequency infor-
8:
Compute spectral magnitudes M, via Eq.
mation at time L. To mitigate overfitting
9:
Determine Qtop based
to these potentially noisy high-frequency signals and encourage temporal smooth- ness in the underlying parameter trajec-
10:
Construct mask B via Eq. (4)
11: Compute IDFT Oowr Onigh via Eq.
12:
for = to 1
tory, we impose the regularization on the sequence hhigh
13:
2t.low Hlow (0r,low),
24+1.0o f- K K2r,low 15: Oct1.low x- ob (2
14:
T-1
Rhigh
Σ 2m1
1, Zt.high
16:
Re. f &higg (Ogh)sgh)
17:
0.41 + O ( (ow + &high (fe.
(8)
via Eq. 99)
This regularization encourages the high- frequency components to vary smoothly
18: end for
19: end for
over time, implicitly suppressing sharp,
20: Perform final spectral decomposition on learned θ to
transient changes that are less likely to generalize.
get OT 0 high.
21: 21 f- Koiow (OT,
22: high f- @high (iThhigh),
Parameter Prediction and Joint Opti- mization. To distinguish between high
23: 0rt1 t to (2T+,.0o + 2T Nigh)
and low frequencies, we use separate en- coders (plowr (high) ) for each frequency component, while sharing the same decoder (Φ-1) to ensure consistency the fusion of different components. To predict the parameters for the next timestamp. 0.,1, we combine the extrapolated frequencccc latent with the regularized high-frequency latent and get the reconstructed result through the shared decoder:
0 - (Károw (fr, ) + (highhOr,,ighh (0t,high)).
(9)

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This mechanism leverages the stability of the predicted wwwfrequency dynamics while incorporating the immediate context from the high-frequency component time t. To ensure the overall predicted parameters align with the actual sequence, we further introduce reconstruction error term:
Cree Σ 10+11 12+1112
(10)
Therefore, the final objective function integrates the task loss Lrack = E. [g((X;;; 0z), Yt), the Koopman dynamics loss for low frequencies, the high-frequency smoothness regularization, and the overall parameter reconstruction loss:
Liotal Lrask + aLCrea + [Looop + RRhighs
(11)
where a, B, and are hyperparameters controlling the relative contributions. This joint optimization framework encourages the model to learn parameter trajectory θ that not only performs well on historical tasks but also exhibits predictable low-frequency dynamics and smooth high-frequency variations, facilitating robust extrapolation to 07+1 for the unseen future domain. The whole learning process is detailed in Algorithm
3.3 Theoretical Insights
FreKoo's frequency-aware approach to TDG separates parameter dynamics into low-frequency (Qtow) for Koopman extrapolation and high-frequency (Origh) for regularized smoothing. This strategy is theoretically grounded to enhance generalization to unseen future domains. We outline generalization bound for the expected error on the target domain Dr+1- Let 07+1 be FreKoo's 's pre- dicted parameter for Dr+1 (derived from Eq. (9). and let 0++1 arg ming [ 0), Y)] be the optimal parameter under the target distribution Pr+1. Assume the loss é is Li-Lisschitz in its first argument, the base model g ()) is L.g-Lisschit w.r.t. 0, and our encoders/decoder satisfy Lipschitz conditions (Assumption iin Appendix B.I). Further, assume the hypothesis class
{X + g ø 6-- (=))) has Rademacher complexity bounded by C/vnn (Assumption 2 in Appendix B. where is the target domain sample size.
Theorem (Multiscale Generalization Bound) Under the stated assumptions, with probability at least 1 - 8 > O, the expected excess risk on the target domain satisfies;
Ert1 = Ep 1 (((g( (a::0(X;0440000 0r+1), Y)] Epr+i ( 07 +++Y)))
(1/0) Le + Eriggh) + OLLe Vn
C
(12)
)),
n
where Erom and Ehigh are errors in the predicted low and high-frequency latent components.
Discussion. Theorem I
decomposes the excess risk into two primary factors:
1) Parameter Prediction Error: For the first term Lç Ldee + Enigh), ), the low-frequency егтоr Elow is bounded by Koopman stability (Lemma D. while the gighffrequency error Enigh regulated via temporal smoothness (Lemma])
Lemma (Koopman Stability) For any initial low-frequency error eso and horizon h = T - tor eko S CxC(+hh) " where is the size of the largest Jordan block of K and Ck = k(V) for the Jordan busis IfK K''s 's spectral radius p(K) < 1, the bound sharpens to Ckp(K)>
Lemma (High-Frequency Smoothness Bias) Minimizing Reigh is equivalent to maximum-a- posteriori estimation the Gaussian random-walk prior h4thh 2t.high + Ees & ~ o2), with precision X 1/(20²).
2) Estimation Error: The (-)) term is a standard learning- complexity (C) and finite samples (n). Thus, FreKoo's distinct mechanisms, i.e., stable Koopman extrapolation for Elow and regularized smoothing for Enight directly address identifiable components of the generalization bound, promoting robust performance in evolving environments. more precise statement and proof are provided in Appendix B
-haming-theeortic theoretic bound related to hypothesis class
7

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Experiments
4.1 Experiment Settings
Datasets. Following []] we evaluate FreKoo on seven datasets with various drift types. In classifi- cation, the synthetic Rotated-Moon and Rot-MNIST benchmarks create incremental drift through steadily increasing rotation angles, whereas the real-world streams ONP, Shuttle, and Elec2 exhibit incremental, periodic unknown drifts, respectively. For regression, HousePrices and ApplianceEnn ergy also reflect real-world non-stationary. These diverse datasets, featuring various drifts and real-world uncertainties, form comprehensive test bed (details in Appendix
We compare our method against four groups of baselines. 1) Time-A Ainostinostic baselines: Offline, LastDomain and IncFinetune. 2) Continuous Domain Adaptation (CDA): CDOT
Baselines.
33
CIDA 22 3) TDG: GI [6, LSSAF 34 DDA 19, DRAIN { : 4) Continuous TDG: EvoS 35
Koodos 12 Details can be found in Appendix C.2 All results
averaged over 5 independent
runs, and we report the mean and standard deviation. The details of model implementations and hyper-
parameters are detailed in Appendix C.3
The code is available at https://iitub..oo//issyyyyyrrr
Table Performance comparisons terms of misclassification error (%) for classification and mean absolute error (MAE) for regression (both smaller the better).
Classification
Regression
Methods
2-Moons
Rot-MNIST
ONP
Shuttle
Flec2
House
Appliance
Offline
22.4 + 4.6
18.6 + 4.0
33.8 + 0.6
0.77 = 0.10
23.0 23.0=3 3.1
11.0 = 0.36
10.2
LastDomain
14.9 + 0.9
17.2 +
36.0 + 0.2
0.91 = 0.18
25.8 = 0.6
10.3 = 0.16
9.1
IncFinetune
16.7 3.4
10.1 0.8
34.0
0.83 = 0.07
27.3 4.2
9.7 + 0.01
8.9
CDOT 33
9.3 1.0
14.2+
+4.1+0.00
0.17
17.8 7.8+000 0.6
CIDA 22
10.8 1.6
9.7
34.7
+4.1
9.7 0.06
8.7 0.2
GI 6
3.5 1.4
1.3
0.84
0.09
1.7
9.6 0.02
8.2
LSSAE 34
9.911.1
9.6
38
0.02
16.1
DDA 9
9.7
7.7
0.21 0.3
12.8 118=111
9.5 + 0.12
6.1 +.1
DRAIN g
3.2 +
7.5
0.26 026+0.05
12.7
9.3 0.14
6,4 0.4
EvoS BS
3.0 0.4
7.3
0.2
0
11.8 0.5
9.8 + 0.10
7.2 0.1
Koodos
12
1.3 + 0.4
7.0 0.3
33.5 0.4
0.24 0.04
8.8 1 0.19
4.8 + 0.3
Fre (ours)
+ 0.3
6.9 0.7
32.3 0.3
0.20 0.02
9.2 0.7
9.0 0.11
4.0 +
4.2
Results and Analysis
As shown in Table [ our proposed FreKoo achieves state-of-the-an performance on six out of seven TDG benchmarks, significantly surpassing prior methods across both classification and regression tasks. FreKoo shows notable improvements on datasets characterized by synthetic incremental drifts, such as 2-Moons and Rot-MNIST, where it attains the lowest error rates. Crucially, excels particularly on benchmarks involving periodic drifts, like Elec2 and Appliance, achieving marked effectiveness with error rates of 9.2% and 4.0 MAE, respectively. This highlights the strength of our frequency-domain approach in capturing and leveraging underlying periodic dynamies, solv- ing Challenge 1. Furthermore, FreKoo's consistent leading performance across diverse real-world datasets (e.g., ONP, Elec2, Shuttle, Appliance) underscores its robustness against overfitting to domain-specific artifacts or local noise. This resilience stems from its dual mechanism: employing the Koopman operator to model stable low-frequency dynamies (trens/perriodccity) while simultane- ously regularizing tigh-frequency components to mitigate noise and prevent adaptation to transient, domain-specti -emain-specific characteristics, solving Challenge 2. Therefore, these results validate the efficacy of FreKoo's 's frequency-Koopman Koopman design for robustly handling complex concept drifts, while maintaining generalization capability in real-world dynamic environments. We also provide the visualization of the decision boundary and analysis compared with SOTA methods in Appendix [D2
4.3 Ablation Study
As shown in Table 2 our ablation study on diverse drift types (e.g., incremental drift Rotated Moons, periodic drift on Elec2 and Appliance) systematically evaluates FreKoo's core components. Removing the Koopman operator (w/o Koop.) severely degrades performance, underscoring its

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criticality for modeling parameter dynamics. Similarly, omitting frequency decomposition (w/o Freq.) significantly harms performance on periodic datasets (Elec2, Appliance), confirming spectral analysis is vital for disentangling stable wwwfrreuency trends from high-frequency noise, especially for complex periodic patterns.
In addition, we further analyze the train- ing objectives (Eq. (TT)) by removing in- dividual loss terms to highlight the impor- tance of each component. Removing the overall parameter reconstruction loss (w/o Crec: Eq. (TO)) decouples the learned latent dynamics from the final parameter predic- tion task and thus hurts fidelity. Excluding the Koopman latent consistency loss (w/o Lkoop. Eq. weakens the learned linear dynamics (K), impairing the reliable ex- trapolation of stable trends. Disabling the high-frequency smoothing regularization (w/o Rhigh. Eq. (8) increases sensitivity
Table 2: Ablation study. Comparison between FreKoo and its variants across datasets for classification and one dataset for regression.
Variants
2-Moons
Elec2
Appliance
w/o Koop. w/o Freq.
15.3 + 1.7
28.9 + 2.4
9.4 + 0.7
2.6 + 0.8
14.2 - 1.9
5.2 - 0.4
w/o free
9.1 11
12.2 = 1.0
4.3 + 0.4
w/o Croop
6.7 0.9
10.0 = 0.8
4.2 + 0.6
w/o Raigh
0.6
10.5 = 1.7
4.2 0.3
FreKoo
1.0 0.3
9.2 0.7
4.0 + 0.1
to domain-spectie noise and transient artifacts by allowing unfiltered high-frequency fluctuations (sgh))))) into the prediction (Eq. (9). These results collectively affirm that FreKoo's 's spectral decom-
position, Koopman modeling, high-frequency regularization, and associated losses all essential
for robust temporal domain generalization.
4.4 Periodicity Modeling Performance
To rigorously evaluate long-term periodicity handling, we extended the 10-domain rotating 2-Moons benchmark (0-9 domains, 18 increments) into a 37-domain sequence. This sequence embeds recurring cycles following the conceptual pattern (0=+9>>0~>9) 90) Training on the first domains and testing on the 37th forces models to exploit long-range pattern recurrence. We compare FreKoo against GI (6, DRAIN R) and Koodos [12]. As shown Figure 3a FreKoo significantly outperforms these methods on this periodic benchmark. This superior result demonstrates FreKoo's ' specific ability to capture and utilize the explicitly embedded long-term evolutions, validating robustness and enhanced generalization capabilities in real-world data streams with complex drifts.
In addition, Figure qualitatively illustrates FreKoo's parameter evo-
3b
0.04
Classiiiction Error (%)
Mean Value
0.02
lution: the low-frequency
component (Qww)) ef- fectively captures the underlying periodic trend of the raw parameters (θ), while the high-frequency component (Onigh) isolates transient variations. The reconstructed parameter (Ô) tracks these long-term evolutions smoothly. Such targeted spectral separation and dual modeling enable FreKoo to learn more stable parameter evolutions, mitigating overfitting to transient details and fostering improved generalization. A more extensive qualitative analysis is shown in Appendix D.3
0.02
Row iuremeter
Rocersoraztd foumder
0.00
61
DRAIN
Koodes
FreKoo lours)
Time Step Domain Index
(a) Quantitative performance
(b) Qualitative performance
Figure 3: Periodicity modeling performance on P-Moons dataset.
4.5 Sensitivity Analysis
We analyzed sensitivity to the spectral energy preservation ratio 7 and loss weights 8, 2- Moons with synthesized incremental drift and Appliance with real-world periodicity and uncertainties. As shown in Figure 4a] higher benefits 2-Moons, while an intermediate is optimal for Appliance. This underscores r's erucial role balancing the preservation essential low-frequency dynamics against filtering disruptive figh-frequenc noise, particularly for complex noisy real-world data. For or, G, 7, grid search over [0.01, 0.1, 1, 10, 100] (Figure 4b 4d) confirmed that optimal performance hinges on balancing reconstruction accuracy, Koopman stability, and high-frequest smoothing.

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2 Moons Appliance
10.0
2-Moons Applia
10.0
Moons Applance
Errors or MAE
Emors MAE
Emons or MAE
Errors MAE
7.5
5.0
2.5
2.5
Moons Appliangee
0.0
0.2
0.4
0.5 Values
0.8
1.0
0.01
0.1
1.0 Values
10.0
100.0
0.01
0.1
1.0
10.0
100.0
0.01
0.1
1.0
10.0
100.0
Values
Values
(a) Sensitivity 7
Sensitivity to
(c) Sensitivity 3
Sensitivity to
Figure 4: Parameters sensitivity on 2-Moons and Appliance datasets.
Conclusion and Limitations
This paper tackled key challenges of TDG, i.c., modeling long-term periodicity and achieving robustness against domain-specifiic uncertainties. Our proposed FreKoo introduces a novel frequency- aware perspective. It analyzes parameter trajectories in the frequency domain, disentangling low- frequency dynamics modeled via the Koopman operator from high-frequency noise smoothed via targeted regularization. Both theoretical analysis and extensive experiments validate FreKoo's effectiveness, demonstrating its significant generalization capabilities and robustness under complex concept drifts.
limitation of our work is the reliance on pre-defined heuristic for frequency separation, rather than adaptive mechanism that learns the optimal threshold from data. Furthermore, the current FreKoo framework is not designed for fully online or continuous settings. This restricts its applicability in real-world deployment scenarios that require continuous model adaptation dynamic data streams.
Acknowledgments and Disclosure of Funding
The work was supported by the Australian Research Council (ARC) under Laureate project FL190100149 and Discovery Project DP220102635.
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Appendix
A Additional Related Works
Domain Adaptation(Generaiiattio Domain Adaptation (DA) requires access to both source and target domain data during training, employing methods like domain-invarian learning, domain mapping and ensemble approaches to minimize domain discrepancy [36438]. Domain Generaliza- tion (DG) aims to train model on multiple source domains to generalize unseen target domains without access to target data, using techniques such as data augmentation, representation learning, and meta-learning 339 40 Both DG and DA frameworks, by treating domains as independent entities, often fail to account for the smooth evolutionary patterns present in time-ordered domain sequences.
Real-world Concept Drift. Concept drift signifies change in the underlying data distribution over time, formally occurring between time and t + 1 if the joint distribution Pt+1(X; X v) * Px(X, y). This non-stationarity poses fundamental challenge, requiring models to adapt dynamically to maintain predictive performance and reliability 1 23). While concept drift manifests in various forms, including sudden, gradual, incremental, and recurring (periodic) patterns, much prior research treats it as unpredictable, often employing detect-then-adaph strategies [4 25 41 In contrast, the Temporal Domain Generalization (TDG) setting frequently focuses on leveraging more predictable evolutionary dynamics for proactive generalization [8]. However, existing TDG methods often oversimplify these dynamics, primarily modeling smooth, incremental drifts while struggling to capture complex, long-range raage temporal structures like periodicity (a key limitation, Challenge I) which are prevalent in many real- world data streams yet demand different modeling approaches than simple monotonic progression.
Furthermore, a common methodology within TDG involves segmenting the continuous data stream into discrete temporal domains {Di, D2...., Dr}. assuming concept drift occurs primarily between these sequential domains (6). This discretization, while simplifying the problem, often struggles to perfectly align with real-world data streams, potentially violating the implicit assumption that data within each domain D, is Independent and Identically Distributed (IID). Consequently, individual domains may harbor internal non-IID structures or localized noise patterns [42]. The presence of intra-domain complexities leads sensitivity to domain-specific noise and artifacts, hindering robust generalization (a second key limitation, Challenge 2), particularly for TDG models designed
predominantly to address shifts occurring only the domain boundaries. Effectively handling both the diverse inter-domain evolutionary dynamics (including periodicity) and these intra-domain data characteristics is therefore crucial for robust temporal generalization.
Dynamics Learning via Koopman Theory. Koopman operator theory
20
offers a powerful
framework for analyzing nonlinear dynamical systems via linear operators acting on observables in an infinite-diment dimensioani function space. In practice, machine learning methods approximate this by learning transformations (often using autoencoders) into latent Koopman space where the system's dynamics can be linearly propagated [43 [6]. This learned linear structure facilitates efficient long-term prediction and control 44. While Koopman- bassed approaches have seen increasing use in time series forecasting [31], they typically focus on modeling the dynamics within given data stream (data-centric). Our work diverges by applying Koopman theory to model the evolution of model parameters, interpreting parameter changes under concept drift. Furthermore, we uniquely integrate this parameter-centric Koopman modeling with frequency- analysis for disentangling dynamics, which towards stable and robust generalization in complex drifting scenarios.
B Theoretical Analysis Details
This section provides supplementary details for the theoretical analysis presented in Section 3.3 including formal assumptions, detailed statements of the theorem and lemmas, and proofs.
B.1 Assumptions
Assumption (Encoder/Decoder Lipschitz) The encoders Vlow RR. D - R.m and thigh : R. - Rm are Lo Lipschit:::
v0,0 E R, [|0(0) o(0')l < Lo 110 - 0'11, for o E (Olows 3
(13)
20

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The shared decoder o R - R is Laec- Lips-cipsh
Vz, z' E IR" - S Lite || z'
(14)
This implies that small changes in the parameter space lead to bounded changes in the latent space, and vere-versa, ensuring stability of the transformations.
Assumption 2 (Model-class Capacity) The composite hypothesis class G = {x + g(z; の (2)) ]]]
E "R."", j. where g is the base model and do is the decoder, has an empirical Rademacher complexity (g) on any sequence of n. samples drawn frow target domain DY+1 (which is f-mixiig |45 46)) bounded as:
C
(9) S
(15)
Vn
For constant C > O, this assumption bounds the complexity of the function class our model can represent, crucial for generalization guarantees.
B.2 Detailed Lemma 1 and Proof
Lemma (Restatement of Lemma 1 Koopman Stability) For any initial low-frequency latent error etoloow 1oo low at initial time to and prediction horizon h (T 1) to
the propagated error tto+ = Ztothdoow Soothloow under the learned Koopman operator
K E _mxx is bounded. Specifically,
KK Now S [x(1++)00 110-1 |1eto II
(16)
where q is the size of the largest Jordan block of K, and CK =(V)= = I[[V] ||V-*|| number of the matrix V from the Jordan decomposition K VJV-1, If K is diagonalizable (i.e.. q 1) and its spectral radius p(K) = max; [K((K) < 1, the bound sharpens to:
iI is the condition
|| | eop S p()))(()) ||eto
(17)
Proof (Proof of Lemma 3 Jordan Canonical Form: Any square matrix K E <Rmx admits is block diagonal matrix. Each J; is a Jordan block corresponding to an eigenvalue Xa of K. is the matrix whose columns the eigenvectors and generalized eigenvectors of K. Let be the size of the largest Jordan block.
Jordan decomposition K VJV-1, where J = (ia J2, J2...., J.)
Power of Matrix via Jordan Form: The h-th power of K is Kh = (V Consequently, Jhh diag(J JJ Jh: J.......
VJV
Bound on the Norm of Jh: For a Jordan block J of size qi with eigenvalue Ad, and assuming [&d S for stability, standard result [47] states that there exists constant Oy such that for h we have,
|| S C₂ (1+
11 (Aa S C. (1+ (Al.))).
(18)
Thus, for the overall Jordan form matrix J. if max; 9a
| S C² + h1
(19)
for some constant C" that depends on the constants Cy and the number of blocks.
Error Propagation Bound: The propagated error K tow. Hss norm is:
K Now! J| eeto ||
(20)
< || || |e
(21)
I C(V)C" (1+ 13)-11 [[eto,
(22)
where x(V)= = =[[V]] || is the condition number of V. We define Ck = k(V)C" (or absorb C" into the definition presented in the lemma statement). This shows polynomial growth if > 1.
Diagonalizable Case: If K is diagonalizable, then J is a diagonal matrix of eigenvalues, A = diag (A1,..., ).) Then Jhh diag( (iag(>))-- (X),.... Xh)-
|| max X (max (2al)* p(K))*
(23)
21

---

In this case, the envor bound becomes:
1K S p()))(K)) [eettttool
(24)
If ((K 1, this implies exponential decay of the error, which is highly desirable for the stability of low-frequency dynamics. The loss term kkoop encourages learning such a stable K.
Discussion. We focus on the propagation of the low-frequency latent error Cr.low = kow ",low By recursively analyzing the single-step eπror recurrence Et+1,ow Kes,low + Ct.low> where Cillow (ooo 2iw is the single-step error, we express after an 1-step prediction as: Eto = K' etg + Σ1= eK'ttttt---i
Taking the norm, we get the multi-step error bound: ||eto S |||||| [|etg,tow|| + ||16:|1-11e66--11---oooolll This reveals that the final enror is not only dependent the initial error dow)) but also on the sum of all previous errors (ot,)om) accumulated along the entire prediction path. This provides a stronger justification full-trajee trajetrryyttry optimization approach. The stability of K bounds | and controls the propagation of intermediate errors. The objective function minimizes losses across all domains, directly works to reduce the magnitude of each single-step error ||e,towl]
our
B.3 Detailed Lemma 2 and Proof
Lemma 4 (Restatement of Lemma
Smoothness Bias) Minimizing the high-
frequency regularization term Rhigh -1 1 2t.high I2 is equivalent per- forming Maximum A Posteriori (MAP) estimation for the latent high-frequency sequence Zrigh hight ZT high under a Gaussian random-walk prior p(z++ hight (ighh (( o² I2. The precision this prior is Apriio 1/(20²).
Proof 2 (Proof of Lemma Gaussian Random- Walk Prior: Assume the high-frequency latent states htgh evolve according to a first-order Markov process:
highh Zt.high + Et. where & N(),
(25)
and & Gaussian noise vectors. The conditional probability (prior) is:
1 224 1,high
p(Zz41,hight 2r,high) (2mo2)m//2 exp
2
(26)
20t
Log-Likelihood of the Sequence under Prior: For a sequence Znigh (zi,tight 27, hight }. the joint probability under this prior model is p(Znigh) = p(2 IT 11 p(2t11,high legg)) The log-probability is:
log (hnggh)) = log ((11,tigh)) +
log 1ggg(,,,,
(27)
= loo (2
+
m 2
log log(2)00)
20 1 high Ze 12)
(28)
Maximum A Posteriori (MAP) Estimation: MAP estimation seeks to find Zrigh that maximizes
p(Zrigh). This is equivalent to maximizing log p( (Zighh), equivalently, minimizing log pl Znigh);
T
MAP high Zeiih
arg min
log pl p(21,)igh)
+
m log log 2TT²) +
20 11 - 21.
2
(29)
The terms - log p(z1. (ight) and m
2 2 are constant with respect to the sum of squared differ-
ences. Thus, the optimization simplifies lo:
MAP = arg min high
1l - 2e.high 113
(30)
Zough
20
22

---

Equivalence to Rhigh: The regularization term is Rhigh = Ei= || - high 12 Minimiz- ing Rhigh is equivalent to minimizing 2 Rhigh. since 26 is positive constant. This shows that minimizing Reigh equivalent to MAP estimation under the specified Gaussian random- walk prior with precision Xprior 1/(2""). This encourages smoothness the trajectory of high frequeryy components.
B.4 Detailed Proof of Theorem 1
Theorem (Restatement of Theorem - Multiscale Generalization Bound) Let 0r+1 be the predicted parameter for the target domain Dr+1 obtained via Eq. D. and let #T+1 arg ming ErT+1 (X 0), Y)) be the optimal parameter for the target distribution PT+1- The loss function ((u,z) y) is Li-Lisschizz in its first argument u. The base model g(x; 0) is Lgg Lipschitz with respect to its parameters 0. Under Assumptions [ (Lipschitz encoders/decoder) and (bounded Rademacher complexity C), the expected turget-domain excess risk Erisk == satisfies with probability at least - 8 O:
| (((g(X:0rr1)) Y) I ((((((;07+1))
C + Ensgh) 2L Vn L
1 log (1/0)
Erish I
(31)
2n
where Elow =12T+1Liow 1,o0m is the ervor in the predicted low-frequency latent component
(bounded by Lemma 3), Enigh [
1,igh is the error in the high-frequency latent
component (regularized by Rhigh as per Lemma 4 ve is the number of samples in the target domain, and B an upper bound on Lig ||2 2/1.
Proof 3 (Proof of Theorem 2 The proof involves combining bounds on parameter prediction error with standard generalization theory.
Bounding Excess Risk by Parameter Error: The excess risk can be related the difference in parameters using Lipschitz properties:
Erisk = Epr E 0TxK, - 01+1),
s Epr Le (((x00++)) - g(X: g(x::6/ 07 (by Li-Lisschitz of {)
(32)
- L2EPry1 [Lg (1rt1 - 07+1 (by Lipsccchi of g)
= Lebbo | 11r41-07.. 07+1
Since 0r+1 = o (2T+1) and 07-1 = o- (2 by Lice-hipschizz of -- (Assumption have:
7 we
|0r+1 - -7 Lider || 2--
(33)
The latent state z Plow Zhigh. So, 2r+1 1, 11
||TT+1 2
(| 1,tow
2,oo, + (2T 1,igh))
| 1,ow
27
(34)
Elow Enigh-
Therefore, the first term of the risk bound related to parameter prediction error is:
PredictionErrorTerm S Le Lag Lâco + Eligh)-
(35)
Evow is bounded by Lemma 3 and Enigh is controlled by the regularization from Lemma
Standard Generalization Error Bound (Estimation Error): The previous term relates the risk of our predictor θr+1 to the risk of the optimal 0 assuming 0 is within our hypothesis class. We also need to account for the generalization error from the empirical risk minimizer (over the unseen target domain DT+1) to the true risk. Let L(0) ((g((;0), Y). The composite function X + L())
23

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is LL Lipppscciizz where Lg = L.L.g Using a standard Rademacher complexity based generalization bound [45]: With probability at least 1 5 over the draw of samples for Dr
B3 log (1g(1
sup (EPr+1 [C(0)] - E. (c(O)) - 2I,I. + LuL.gg
(36)
OEHa
2n
where He is the space of parameters, Ge is the function class induced by x; 0) Rn (9a) is its
Rademacher complexity (here using Ra, (g) S C/ from Assumption where g the class g(X; Ø (x;() (z)),, and Be is a bound on 0) More directly, the generalization gap for our specific predictor 0r+1 (derived from 2 R is:
E Er ([l Y)] < (Xx((((X;00+)))
Tr. 2Le4g Jn + Le
C.
Ba (ogg(1/0) (37)
2n
where B bounds Laeo ||=|]- This term essentially bounds how much the true risk of 0T+1 1 can deviate from its (unobserved) empirical risk on DT+1- The excess risk desinition compares true risks. The parameter ervor term Eq. (35) captures the suboptimality of 0T+1 relative to 67 The estimation error term often appears when bounding (p Ep [(())))) The provided bound in Eq. (31) represents a structure where the first term is the approximation ervor (how far our best hypothesis 0r+1 is from 4 in terms of risk, scaled by Lipschitz constants) and the second and third terms represent the estimation error (how well one can estimate the risk of any hypothesis in the class G from n samples).
The bound structure in our method: Erisk S PredictionErrorTemm EstimationError. The Prediction Error IaIogItee + Enigh). The Estimation Er- ror, using Assumption [ and standard results for Lipschitz losses, can be written as Rr
Combining Terms:
C 2 R + ConfidenceTerm. Substituting Rn) S //: EstimationError as 2LeLg 7/
LeLg
Ba (log(1//)) Combining these yields Eq. 31
2n
C Experiment Settings
C.1 Datasets
Rotated 2 Moons. This benchmark adapts the 2-Moons dataset to model concept drift via rotation. It contains 1,800 2-dimensional samples across two classes, divided into 10 sequential domains. Fach domain is rotated 18 counter-clockwise relative the previous one. We train domains 0-8 and test on domain 9, where the drift is caused by the incremental rotation.
Rotated MNIST. We randomly sampled 1000 instances from the MNIST dataset and constructed a total of five domains by successively rotating them counter-clockwise by 15,, analogous to the Rotated 2 Moons setup. The first four rotated domains are used for training, while the fifth domain serves as the test set, creating incremental drift induced by progressive rotation transformations.
Online News Popularity E This dataset aggregates heterogeneous features of articles pub- lished by Mashable over two years, aiming to predict social media shares (popularity). comprises 39,797 samples with 58 features, where concept drift is characterized by temporal shifts in popularity patterns. We partition the data into time-ordered domains, using the first five for training and the last for testing. The dataset undergoes slight real-world concept drift over the observed time period.
Shuttle
The Shuttle dataset contains 58,000 instances of multi-class flight status classification under severe class imbalance. It is partitioned into 8 time-stamped domains using a chronological split: domains spanning timestamps 30-70 serve as training data, while the most recent period (70-80) is reserved for testing. The dataset also has real-world concept drifts over the observed time period.
Electrical D This dataset records electricity demand in province, addressing binary classification task to predict whether 30-minute demand exceeds or falls below the daily average for
24

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that time period. After removing instances with missing values, it contains 28,222 samples with 8 features. is partitioned into 30 two-week chronological domains, with the first 29 used for training and the 30th for testing. Seasonal variations in demand induce concept drift, making this real-world benchmark capturing both periodic and incremental drift patterns.
House Prices This dataset comprises housing price records from 2013 to 2019 for the regression task to predict property prices based on feature values. We treat each calendar year as a distinct domain, using 2013-2018 data for training and the final (2019) domain for testing. Concept drift emerges naturally from temporal economic shifts and market fluctuations over the years.
Appliances Energy Prediction P This dataset addresses regression modeling for predicting appliance energy consumption in a low-energy building. Comprising 10-minute sensor readings over 4.5 months in 2016, it is partitioned into 9 chronological domains. We train on the first eight domains and evaluate on the final (most recent) ninth domain, with concept drift arising from temporal shifts in energy usage patterns across the observation period.
C.2 Baselines
Time-agnostic methods. These methods do not consider the temporal drift, including Offline train on all source domains, train on the last source domain (LastDomain) and incrementally train on each source domain (IncFinetune).
Continuous Domain Adaptation (CDA). CDOT 33
predicts the future by transporting labelled samples of the last observed domain to an estimated target distribution via optimal transport, then retrains the classifier on those transported points. CIDA [22] leverages adversarial alignment with a probabilistic domain discriminator to model continuous domain shifts, while its probabilistic extension (PCIDA) enforces higher-order moment matching via Gaussian parameter prediction.
Temporal Domain Generalization (TDG). GI 6 regularizes temporal complexity by supervising the first-order Taylor expansion of time-sensitive model, enabling smooth adaptation to distribution shifts via adversarial selection of temporal perturbations. LSSAE (34) addresses evolving domain generalization by employing latent structure-aware sequential autoencoder to model dynamic shifts in both data sample space (covariate shift) and category space (concept shift). It leverages variational inference and temporal smoothness constraints capture continuous domain drift, enhancing gen- cralization to unseen target domains. DDA [9]
leverages an attention- based domain transformer to capture directional domain shifts and simulate future unseen domains through bi-level optimization with meta-learning. DRAIN S assumes that model parameters vary over time within a fixed network architecture and employs recurrent neural network autoregressively predict domain-optimal parameters for future timesteps through temporal dependency modeling.
Continuous Temporal Domain Generalization (CTDG). EvoS [35 proposes multi-scale atten- tion module (MSAM) to model evolving feature distribution patterns across sequential domains, dynamically standardizing features using predicted statistics to mitigate distribution shifts while employing adversarial training to maintain a shared feature space and prevent catastrophic forgetting. Koodos [12] models data and model dynamics as a continuous-time system via Koopman operator theory. It integrates prior knowledge and multi-objective optimization to synchronize model evolution with data drift.
All bascline results presented in Table D were directly sourced from their respective original publica- tions to ensure accurate and fair comparison.
C.3 Implementation Details
The architecture and implementation of backbones and prediction models for all datasets align with DRAIN g Specially, both the encoders and decoder employ a 4-layer MLP architecture with layer dimensions [1024, 512, 128, mi], where m 32 denotes the dimension of the Koopman operator. All experiments were conducted on a server with 187GB of memory, an Intel(R) Xcon(R) Gold 6226R CPU@2.90GH2, and two A100 GPUs.
25

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We adopt the Adam optimizer across all datasets, with distinct learning rates for the prediction module Ir pres encoder-decoder module (r and Koopman module krko For the 2-Moons dataset, the coder and Koopman learning rates are set Iroo
x 10 and prediction learning rate pre = 1 x 10-3, regulated by T = 0.9, a = 10, _ = = 1. The Rot-MNIST contiguration retains pre = Iroo = Irke = 1 x 10 and 5 = 0.9, a = 0.1, 5 = = 1. For ONP, we use creo = 10 for coder/Koopman and pre = 1 x 10-3 for prediction, combined with 7 = 0.8, = 0.1, 8 1, 7 = 0.01. The Shuttle dataset employs uniform learning rate 1 x 10-3 for all modules, and T = 0.9, = B = 7 = 1. For Elec2, Ir pre = 1 x 10-2, &rco 1 x 10-4 and Irko = 1 x 10-3 governed by = 0.1, a = 10, 3 = 0.1, 7 = 1. The House dataset shares the coder/Koopman learning rate 1 10-3 with prediction rate 1 x 10-2 with 0.3, 0.1, 3 10, = 1. Finally, Appliance maintains a uniform learning rate 1 x 10 across all modules with
T 0.8, a L. 1, 100.
D Supplementary Experiments
D.1 Visualization of Real-world Uncertainties
To illustrate the challenge of domain- specific uncertainties, Figure 5 vi- sualizes t-SNE embeddings of the first domain (Dz) from two real- world datasets, Elec2 and Appliance, with overlaid Kernel Density Esti- mates (KDE) highlighting distribu- tional structures. On both real-world datasets, the data are not uniformly distributed; instead, they form mul- tiple distinct high-density regions alongside sparser, peripheral points. These peripheral points, identified as Uncertainties/Noise, typically reside at the fringes of core data concentra- tions. Such observed heterogeneity within a single temporal domain un- derscores the frequent violation of the IID assumption. The co-existence of these localized concentrations and scattered uncertainties implies that models attempting to uniformly fit all data within a domain risk overfitting these domain-specific uncertainties, hereby impairing its ability to generalize to subsequent evolving domains. This vulnerability is precisely what Fre aims to mitigate through its targeted handling of different spectral components.
Core Data
20
Data UinoiaaildeesNNaaae
UncortairilesNaise
15
5
-5
10
-10
-15
-20
-10
10
-20
-10
10
20
(a) D1 on Elec2.
(b) D1 on Appliance.
Figure 5: Visualization of domain- uncertainties on real-world datasets.
D.2 Qualitative Analysis of Decision Boundary
To provide qualitative assessment of generalization capabilities, we visualize decision boundaries on the 2-Moons target domain, comparing FreKoo against representative TDG (DRAIN g)) and CTDG (Koodos [12) methods. As illustrated in Figure 6a] the DRAIN method exhibits decision boundary that, while attempting to separate the classes, appears somewhat convoluted and potentially overfitted to the specific distribution of the training domains. This can lead to suboptimal generalization on the unseen target domain. Figure 6b) shows the boundary Icarned by Koodos. While demonstrating a degree of adaptation, the boundary still shows some irregularities and does not perfectly capture the underlying structure of the target dlstribution. In contrast, FreKoo (Figure (6c) learns a notably smoother and more globally consistent boundary that effectively captures the target 2-Moons structure with less sensitivity to local variations. This smoother boundary is indicative of more robust generalization, suggesting that FreKoo's frequency-domain parameter analysis and dual modeling strategy successfully disentangle stable underlying dynamics from transient fluctuations, leading to more principled adaptation to the concept drift. This qualitative evidence aligns with the quantitative results, highlighting FreKoo's 's superior ability to generalize in temporally evolving environments.
26

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15
25
15
-55
-15-
-10
-10
-10
-35 -3.6 -13
-ts
-és do dl 1b
-55 -3o -ts
-i's
-Is
do dl
-5s -e -ls
-ts
-is do
1b
(a) DRAIN
(h) Koodos
(c) FreKoo
Figure 6: Visualization of decision boundaries on the 2-Moons dataset. Blue dots and red stars represent different data classes.
D.3 Qualitative Analysis of Frequency Analysis
Figure 7 visualizes the evolution of model parameters (averaged across dimensions) on P-2-Moons under varying spectral energy preservation ratios FreKoo decomposes the 'Raw Parameter' trajectory into a 'Low-Freq Component, which captures the smoother, dominant underlying trends, and 'High-Freq Component," which encapsulates more rapid, transient fluctuations often indicative of domain-specific noise or artifacts. As shown, the 'Low-Freq Component' (dashed magenta) isolates the core dynamic structure. The 'Reconstructed Parameter' (dashed green), derived from extrapolating the low-frequency dynamics and incorporating regularized high-frequency information, exhibits significantly enhanced smoothness compared the trajectory. This targeted spectral separation and dual modeling strategy allows Kre to learn a more stable parameter evolution, mitigating overfitting transient domain-specifie details and thereby fostering improved generalization to future temporal domains. The choice of r modulates the trade-off between fidelity to the core dynamics and robustness to high-frequency noise.
0.24
0.54
0.30
0.03
Weann Value
Maner Value
0.2
Raw Para ruter Roca intructdd Parsmeter
0:22
Rew Pertmnttrr Conperert Raca sifieceded
6,22
Raw Surirrder Corpxeert Raca Racindfieceed Porstete
0:3
Recandffecied Parsmete
002
602
10
10
Time Step
Step
Step
0.1
(b) 0.5
(c) 0.9
Figure 7: Visualization of parameter evolution for different components on the P-2-Moons dataset.
D.4 Running Time Analysis
We conducted running time tests on three classification datasets and one regression dataset, comparing against DRAIN, the most classic TDG method. We used the same batch size and number of training epochs for both methods to ensure a controlled comparison. As shown in Table 3 FreKoo's training time competitive and comparable to that of DRAIN. This demonstrates that the additional components in frame- work, such as the spectral decomposition and Koopman operator, do not introduce a significant computational overhead. Crucially, while maintaining similar efficiency, FreKoo consistently out- performs DRAIN in terms of generalization performance. This highlights that our method provides superior performance without sacrificing practical feasibility.
Table 3: Comparison of training time (seconds) between our method and two baselines across datasets for classifica- tion tasks and one datasets for regression tasks.
Datasets DRAIN FreKoo 14.26
2-Moons
ONP
Elec2
Appliance
17.28
34.45
23.30
20.23
34.83
17.19
20.42
27


Reviews:
[{"review_id": "VaDsiy61fi", "reviewer": "Reviewer_MuEU", "summary": "This paper introduces FreKoo for TDG (Temporal Domain Generalization). FreKoo first decomposes the model parameter into low and high frequency parts by DFT. FreKoo models the low frequency part by learnable Koopman operator, and suppresses high frequency part by adding regularizations. This paper gives a generalization bound for FreKoo, and demonstrates its practical advantage through experiments.", "questions": "1. Will the number of parameters in the model also affect the performance?\n2. Will overparameterization/overfitting of the model affect the selection of high and low frequency components?\n3. In appendix C3, the implementation details of the hyperparameters, different datasets and tasks have different learning rate for prediction module, encoder-decoder module, Koopman module. Also, the spectral pereservation ratio and loss weights are different as well. How these hyperparameters were selected? Is there any underlying intuition or guiding principle behind these choices? Furthermore, why do these parameter settings appear to generalize well to all datasets except the appliance dataset?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\no The spectral decomposition of the parameter space of the model is interesting and well-grounded.\no Using Koopman operator to model low-frequency components and regularizations for high-frequency components are technically solid.\no The paper provides theoretical results to validate the performance of FreKoo.\no The experiments are extensive, by evaluating across various well-known datasets and benchmarks.\no The paper is well-written and well-organized.\n\nWeaknesses:\no For experiment completeness, please also include the inference and training time. This information is critical for assessing the practical\nfeasibility of the method, especially in time-sensitive applications like online learning.\no The experimental comparison would be more robust with the inclusion of TKNets, another Koopman-based approach. Incorporating it would provide a clearer understanding of how the proposed method compares against existing alternatives in the same domain.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your valuable feedback and are grateful for the reviewer's recognition of our work's technical rigor, theoretical foundation, comprehensive experiments, and clear writing.\n\n### **W1：Runing time**\n\nWe thank the reviewer for this suggestion regarding experimental completeness. We agree that running time analysis is critical for assessing the practical feasibility of our method. \nAs shown in the following table, we conducted running time tests on three classification datasets and one regression dataset, comparing against DRAIN, the most classic TDG method. We used the same batch size and number of training epochs for both methods to ensure a controlled comparison. As the results show, FreKoo's training time is competitive and comparable to that of DRAIN. This demonstrates that the additional components in our framework, such as the spectral decomposition and Koopman operator, do not introduce a significant computational overhead. Crucially, while maintaining similar efficiency, FreKoo consistently outperforms DRAIN in terms of generalization accuracy (as shown in Table 1). This highlights the key advantage of our approach: it provides superior performance without sacrificing practical feasibility. \n\nRuning time (s)\n|Datasets|2-Moons|ONP|Elec2|Appliance| \n|-|-|-|-|-|\n|DRAIN|17.28|34.45|23.30|20.23|\n|FreKoo|14.26|34.83|17.19|20.42|\n\n### **W2：Comparison with TKNets.**\nThank you for your suggestion. Actually, when we initially designed the experiments, we considered the TKNets method. However, since the datasets used in that paper differ from ours, we had no optimal parameter settings to reference, so we did not include it in our initial comparison. Nevertheless, we have already cited that paper in our work, i.e., Ref[11]. The baseline methods currently compared in our paper were all obtained directly from their original publications, making them more reliable. To address your concern, we have now attempted to re-train TKNet on two classification and one regression datasets. The results are presented in the following table.\n||2-Moons|Rot-MNIST|Appliance| \n|-|-|-|-|\n|TKNets|3.9 ± 0.4|8.7 ± 0.1|6.6 ± 0.0|\n|FreKoo|1.0 ± 0.3|6.9 ± 0.7|4.0 ± 0.1|\n\nWhile the TKNets uses of Koopman theory in TDG by modeling the evolution of the data distribution, our FreKoo framework offers a distinct and robust perspective. Instead of analyzing the data space, FreKoo introduces a novel frequency-aware view by directly modeling the dynamics of the model parameters themselves. Our approach is specifically designed to tackle two critical challenges not addressed by prior works: the robust modeling of long-range periodicity and the filtering of domain-specific uncertainties. By decomposing the parameter trajectory into low- and high-frequency components and applying a dual-modeling strategy — Koopman extrapolation for structure and temporal smoothing noise. FreKoo provides a new principled solution that excels in scenarios with complex temporal dynamics. Thank you so much for your suggestion and we will further discuss this point in the revised version.\n\n\n### **Q1：Will the number of parameters affect the performance?**\nWe thank the reviewer for this insightful question. Yes, the number of parameters will affect performance. As with most deep learning models, changes to the architecture—such as modifying the backbone or altering the structure of the encoder-decoder—would lead to a different number of parameters and a corresponding change in performance, which could be either an improvement or a decline. This is a general characteristic of deep models and has also been analyzed in prior works like DRAIN. **To ensure a fair and controlled comparison, we maintained a consistent backbone architecture for all baselines, as detailed in our Appendix C.3**. This approach mitigates the risk of performance differences arising from architectural inconsistencies rather than the novelty of the methods themselves.\n\nNonetheless, we believe this is an important question. The interplay between model complexity, the number of parameters, and generalization performance in the context of TDG is a valuable area for deeper investigation. We plan to explore this potential relationship in our future work. Thank you again for raising this thoughtful point.\n\n### **Q2：Will overparameterization/overfitting of the model affect the selection of high and low frequency components?**\nWe thank the reviewer for this very insightful question that probes the core of our method. Yes, overparameterization and the resulting overfitting will affect the selection of high and low-frequency components. This is a crucial challenge that we explicitly considered in our design. If a model overfits to the transient noise within each domain $D_t$, the learned parameter trajectory $\\Theta$ will become erratic and dominated by high-frequency oscillations. This would \"pollute\" our spectral decomposition, causing energy to leak from the true low-frequency dynamics into the high-frequency bands, thus corrupting the signal we aim to model. To address this, our FreKoo framework is designed with built-in temporal regularization through its joint optimization objective (**Eq. 11**):\n\n1.  The **Koopman loss ($\\mathcal{L}_{{koop}}$)** penalizes unstructured, unpredictable low-frequency trajectories. This forces the model to find a more regular and stable underlying dynamic, implicitly resisting overfitting.\n2.  The **high-frequency regularization ($\\mathcal{R}_{{high}}$)** directly penalizes sharp, noisy fluctuations in the parameter trajectory.\n\nThese components act as a powerful regularizer against overfitting in the temporal dimension. They guide the model to learn a parameter trajectory that is not only accurate on the training data ($\\mathcal{L}_{{total}}$) but is also temporally smooth and predictable. This ensures a cleaner and more meaningful separation between the low-frequency signal and high-frequency noise, even in the presence of overparameterization.\n\n### **Q3：About hyper-parameter Analysis**\nThank you for your question regarding parameter tuning. For hyperparameters $\\alpha$, $\\beta$, and $\\gamma$, we used traditional grid search over $[0.01, 0.1, 1, 10, 100]$ while fixing other parameters, as described in **Section 4.5**. For the learning rate, we explored $[10^{-4}, 10^{-3}, 10^{-2}]$—a limited range that kept the tuning process manageable. These are standard ML tuning approaches, though admittedly not always adaptive. The spectral energy preservation ratio $\\tau$ remains our most important parameter. Different real-world datasets exhibit varying drift types, frequencies, and uncertainties, necessitating dataset-specific tuning. Since $\\tau$ is confined to [0,1], this adjustment remains relatively straightforward. Regarding the Appliance dataset, however, our method achieves SOTA (Table 1). Could you please let us know which part is your concern.\n\nThank you for raising this point. To be honest, parameter tuning remains a less-than-ideal aspect of our approach. We will focus on this area for further research in future work."}, {"review_id": "UgD5d39Xtj", "reviewer": "Reviewer_kzhb", "summary": "In time-series analysis, out-of-distribution generalization remains a significant challenge. The paper tackles the setting where the label-generating mechanism changes over-time. The goal is to be able to predict these changes ahead of time and adapt the model. \n\nThe proposed method treats the model parameters themselves as the signal. At every time step, an FFT vector is computed from the parameter trajectory estimated thus far. The learner separates the low-frequency components and the high frequency of the FFT trajectory and learns to predict the trajectory governed by the low frequency components and high frequency components. For the low-frequency part, the model uses the low frequency component $\\theta_t^{\\text{low}}$ and passes it via an encoder to output a latent. The latent evolution is modeled via a linear transformation (this part of the work is inspired from Koopman operators). For the temporal trajectory of the high-frequency components, a temporal difference regularization is imposed to smoothen out the changes. \n\nThe authors develop a multi-scale generalization bound to explain why this philosophy of frequency aware separation helps with improved generalization. Finally, the authors conduct experiments on seven benchmarks out of which five are classification based and two are regression based. In these experiments, the authors both establish that their approach turns out to be state-of-the-art but also provide an ablation analysis to show critical the role of different modeling components (Koopman operator, frequency decomposition etc.).", "questions": "Please refer to the weaknesses section to read the questions.", "limitations": "While the authors have a limitations section, the discussion of the limitations is rather sparse and can be expanded.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1. **Technical novelty** Treating the parameters as signal and separation of the signal into low and high-frequency component is a new method that to the best of my knowledge was not tried before. \n\n2. **Theoretical arguments** Low-frequency component's evolution depends on the Koopman operator. The authors show the sharpening of the generalization bound provided the spectral radius of the operator is less than one. Further, the authors propose a Bayesian justification for the smoothening regularizer used for the high-frequency components. \n\n3. **Solid empirical section** The results in the experiments section show that the approach is especially beneficial in modeling periodic drifts as shown by Elec 2 and Appliance dataset. The ablations and sensitivity plots are quite informative. \n\n**Weaknesses**\n\n1. **Writing** I found the paper poorly written. The method is hard to follow as it is broken down into many small subparts. The authors made the unfortunate choice of having the algorithm in the Appendix. The authors should take the pain of writing a simple step-by-step description of the key elements of the method. Especially since the method would first involve learning theta's for each domain, it makes it all the more confusing as that is not done in standard methods. \n\n2. **Vague claims:**  First let me quote, \"Prevailing TDG methods often fall short under complex concept drifts. Typically constrained by assumptions of incremental change or local smoothness, they struggle to simultaneously capture long-range dynamics like periodicity (Challenge 1) and filter transient noise or domain-specific uncertainties (Challenge 2), thereby failing to effectively balance stability with adaptability crucial for real-world scenarios\" \nFirstly, both these challenges are vaguely stated. Can the authors be more precise on what do each of these challenges mean in a mathematical sense? \nSecondly, when the authors state that the existing TDG methods fail to capture xyz, it would be best if authors can create a simple toy example. Through this toy task, the authors can highlight why the existing methods fail and why their method succeeds. Bear in mind, the toy task is to build intuition.  \nLet me quote \"This averaging operation inherently suppresses parameter-wise variations in oscillation patterns, ensuring robustness to localized noise in individual dimensions\". Now this is again a vague claim, which is not precise and has no evidence to back it up. Can the authors explain the choice of averaging better? or was it just the most natural heuristic? \n\n\n3. **On the role of Koopman operator** In the current form, the authors use the Koopman operators to motivate their loss function. I found the whole description via Koopman operators overloaded and unnecessary. The authors might have as well stated that they are modeling latent dynamics via a linear transformation. Here is a parallel I want to draw. Consider kernel SVMs, where one motivates featurization in infinite dimensional spaces and then uses representer theorems to show how Kernel matrix is all we need to characterize the solutions. In those descriptions, the infinite dimensional formulation is actually used. In this case, the authors are referring to an infinite dimensional object but using the straightforward linear usage of it. It is like referring to kernel SVMs, but invoking the linear kernel. Hence, I recommend that the authors tone down the jargon as it feels unncessary.\n\n\n4. **On the method** Overall I find that the method currently involves several heuristic choices and is not elegant. And one important lesson that stands test of time, is that simpler methods seem to outshine more complex counterparts in the long run. If in spirit of this, the authors could think of simplifications that they can do and suggest some simplifications for future work, that would be nice. \nDid the authors consider a simple transformer-based baseline? The transformer can take in as input the parameters theta and predict their changes? It will be nice to illustrate the failures of a transformer based baseline to model these long-term periodic changes in the parameters.", "was_revised": true, "final_justification": "The authors have succesfully addressed my key concerns regarding writing and usage of Koopman operator.", "rebuttal": "We sincerely thank the reviewer for your valuable feedbacks. We are also grateful for the reviewer's recognition of our work's strengths, including its technical novelty, theoretical arguments, and solid empirical results. We address your concerns and questions below.\n\n### **W1：Writing**\n\n**1. Writting logic:** We apologize for making confusion, and we briefly outline the logical flow to hopefully provide a clearer step-by-step picture:\n\n*   **Step 1:** A crucial point we wish to clarify is that our method is an **end-to-end joint optimization process**. We do not first discretely learn an independent optimal $\\theta_t$ for each domain $\\mathcal{D}_t$. Instead, the entire parameter trajectory $\\Theta = [\\theta_1, \\theta_2, ..., \\theta_T]$ is learned jointly by minimizing our overall objective function (**Eq. 11**). We will claim this in the revised verson.\n\n*   **Step 2: Spectral Decomposition (Section 3.2.1)**: Within this end-to-end learning loop, we apply the Discrete Fourier Transform (DFT) to decompose parameter trajectory into a **low-frequency component $\\Theta_{low}$** (capturing stable, long-term trends and periodicity) and a **high-frequency component $\\Theta_{high}$** (representing transient noise and uncertainties). This decomposition is the core of our frequency-aware perspective.\n\n*   **Step 3: Dual-Strategy Dynamics Modeling (Section 3.2.2)**: We then apply a distinct strategy to each component: For $\\Theta_{low}$, we learn a **Koopman operator $\\mathbf{K}$** to model its dominant dynamics in a latent space. This is guided by the $\\mathcal{L}\\_{{rec}}$ loss. For $\\Theta_{high}$, we apply a **temporal smoothing regularization ($\\mathcal{R}\\_{{high}}$)**. This prevents overfitting to domain-specific noise/uncertainties.\n\n*   **Step 4:** Finally, we give the prediction for the future parameter $\\hat{\\theta}_{t+1}$ and the joint optimzation objective, following by the Theoretical analtsis in Section 3.3\n\n**2. Position of Algorithm 1:** We are sorry to place Algorithm 1 in the Appendix due to the strict page limitations for the initial submission. Thank you for pointing out its importance. \n\nWe are grateful for your suggestions, and we will carefully revise the manuscript to improve its clarity and flow.\n\n### **W2：Vague claims**\n**1. Explanation of challenges and current limitations.** We apologize for the lack of mathematical formulations. To clarify:\n\n- **Challenge 1 (Periodicity)**: we refer **recurring drift** defined in [Learning under concept drift: A review. TKDE 2019] to formulate this challenge, i.e.,\n$\\exists L \\in \\mathbb{Z}^{+} : \nP_t(\\mathbf{X},Y) \\neq P_{t+1}(\\mathbf{X},Y)$ and\n$P_t(\\mathbf{X},Y) \\approx P_{t+kL}(\\mathbf{X},Y)\n\\forall k \\in \\mathbb{Z}^{+}$, where $L$ is Period length. Current TDG methods assume monotonic/smooth drift [6,8,12] failing to model such periodicity, and **Section 4.4 experiment** provide a experimental evidnece.\n\n- **Challenge 2 (Uncertainties):** Characterized by **transient noise and domain variations** that violate IID assumptions within temporal segments (Intro L45-48). While mathematically challenging to formalize, we provide intuitive visual evidence of these uncertainties in real-world scenarios through **Appendix Figure 5**. To our knowledge, no existing TDG method explicitly addresses this challenge.\n \nWe appreciate your suggestion again and will refine these points in the revision.\n\n**2:** We apologize for making confusion regarding the **averaging operation for computing spectral energy (Eq.3)**. If we were to select dominant frequencies based on each parameter dimension $d$ individually, we might select different sets of frequencies for different dimensions. This could lead to a fragmented and incoherent model of the overall dynamics. More critically, a single parameter dimension might exhibit a spurious, high-energy oscillation due to optimization noise or artifacts specific to that dimension. A model that latches onto this \"localized noise\" would not generalize well. By averaging the spectral magnitudes across all $D$ dimensions to get $M_f$, we are effectively performing a pooling operation. This computes a summary statistic that represents the overall importance of frequencyto the evolution of the entire parameter vector. We will revise the manuscript to incorporate these clarifications.\n\n### **W3: The role of Koopman operator.**\nWe sincerely thank the reviewer for this insightful comment and the thoughtful parallel drawn to Kernel SVMs. We would like to clarify that the Koopman framework is not just \"jargon,\" but a **fundamental concept that guides our entire model design**, distinguishing it from a generic linear transformation. The core of our data-driven Koopman approach is to simultaneously learn both the nonlinear feature map $\\phi$ (the encoder) and the linear transfermation $\\mathbf{K}$. We aim to discover a nonlinear coordinate system via our encoder $\\phi$ where the dynamics become globally linear. This is a significant departure from using a fixed feature map, which would be analogous to a \"linear kernel\". Our deep encoder learns a highly complex, data-driven feature map—a practice established in modern Koopman analysis [Deep learning for universal linear embeddings of nonlinear dynamics. *Nature communications 2018*]. Therefore, a more fitting analogy for our method would be learning the kernel function itself, rather than simply invoking a linear kernel. The \"infinite-dimensional\" nature of the Koopman operator is addressed by leveraging the universal approximation capabilities of a deep neural network $\\phi$ to find an optimal finite-dimensional basis where a linear operator $\\mathcal{K}$ suffices. In summary, the Koopman framework is not an unnecessary descriptor but the foundational blueprint for our model's architecture: it justifies the search for a nonlinear embedding $\\phi$, dictates the linear structure of the latent dynamics $\\mathcal{K}$, and provides the rationale for why this combination is effective for complex temporal dynamics.\n\nWe are grateful for this constructive feedback, as it will undoubtedly help us improve the accessibility and impact of our paper. We believe this revised presentation will strike a better balance, making the method easy to follow while still retaining the valuable theoretical context that Koopman theory provides.\n\n### **W4: On the method** \nWe thank the reviewer for this insightful comment on the elegance of our method and for raising the important point about the long-term value of simplicity. We appreciate the opportunity to discuss the design philosophy of FreKoo and the alternatives like Transformers.\n\n**On the method's principled design:**\nWhile FreKoo involves several components, we think that this is not an arbitrary integration but rather a **principled and structured design** derived from a new **frequency-aware perspective** on TDG. We decompose the parameter trajectory into a stable **low-frequency** component (long-range dynamics) and a transient **high-frequency** component (uncertainties). Each part of our method is a dedicated tool for this decomposition: the Fourier Transform for disentanglement, the Koopman operator for stable extrapolation, and targeted regularization for noise suppression. This structured approach is supported by a rigorous **theoretical analysis (Theorem 1)** and, importantly, provides a novel interpretable solution to TDG by allowing for the analysis of distinct temporal drift components.\n\n**On a Transformer-based Alternative:**\nThank you for the excellent suggestion of a Transformer-based baseline. A Transformer is indeed a powerful sequence model. We have not yet conducted this comparison, but we believe it is a very valuable direction for future work. Investigating the trade-offs between our structured-bias approach and a powerful general-purpose model like a Transformer would be highly insightful and we are grateful for the suggestion."}, {"review_id": "JQC61M8tSA", "reviewer": "Reviewer_2Mhj", "summary": "This paper introduces a novel method for Temporal Domain Generalization (TDG) named FreKoo that addresses performance degradation in machine learning models caused by complex concept drifts over time. By transforming model parameter trajectories into the frequency domain, FreKoo decomposes temporal dynamics into low-frequency components, which are extrapolated using the Koopman operator to capture long-term trends and periodicities, and high-frequency components, which are regularized to suppress transient noise and domain-specific uncertainties. The authors provide theoretical guarantees through a multiscale generalization bound and demonstrate FreKoo’s effectiveness in modeling stable parameter evolution. Experiments across classification and regression benchmarks show FreKoo outperforms existing TDG methods, particularly in scenarios with periodic or noisy distribution shifts.", "questions": "- Theorem 1 focus only on the relationship between domain $T$ and domain $T+1$, while the algorithm aggregates information from all previous domains to predict $T+1$. Can you extend the theoretical analysis to capture the influence of the full trajectory $\\{1, 2, ..., T\\}$ on the prediction of domain $T+1$? \n- The algorithm assumes that optimal model parameters evolve smoothly over time, which may not hold in practice, particularly in deep networks with high-dimensional non-convex loss landscapes. Can you provide empirical or theoretical justification for this assumption? For example, visualizing learned parameter trajectories on real-world datasets or quantifying smoothness across multiple runs could help.\n- The evaluation is limited to a static setting with fixed training/testing split (e.g., domains $1$ to $T$ for training, $T+1$ for testing). The authors should including more dynamic and realistic settings, such as: Varying the split index between training and test domains, evaluating on extended horizons (e.g., $T+2$, $T+3$) to test robustness under longer-range drift, using real-world high-dimensional datasets beyond the current benchmarks.\n- The method critically relies on the choice of spectral energy preservation ratio $\\tau$ and the decomposition into low- and high-frequency components, but these design choices are not deeply explored. Could you provide more systematic analysis on how sensitive the performance is to $\\tau$, and whether this threshold could be learned or adapted from data? Additionally, how does the model perform when trained with noisy or misaligned frequency partitions?\n- The use of Koopman operator assumes that low-frequency dynamics evolve approximately linearly in a latent space, but this may be violated in highly nonlinear or chaotic temporal processes. Can you discuss the expressivity limits of the Koopman formulation? Including empirical failure cases or comparisons to nonlinear forecasting alternatives (e.g., RNN-based parameter predictors) would be helpful.", "limitations": "- The theoretical analysis relies on Lipschitz continuity of neural networks and the linear Koopman operator to model low-frequency dynamics. These assumptions are difficult to guarantee in practice, particularly in high-dimensional and highly nonlinear deep learning settings.\n- The theory assumes that the future domain $T+1$ depends only on the immediately preceding domain $T$, neglecting long-range temporal dependencies that are common in real-world sequences. This simplification limits the theoretical alignment with the actual algorithm, which uses all past domains.\n- The method assumes that ground-truth model parameters evolve smoothly over time, which may not hold in complex real-world scenarios. In practice, multiple optimal models can exist with significantly different parameters, undermining the assumption of smooth parameter trajectories.\n- While the work focuses on academic benchmarks, the use of TDG methods in high-stakes domains (e.g., finance, healthcare, surveillance) could introduce risks if models generalize poorly across time or are misled by transient patterns. The authors should reflect on potential risks of over-reliance on extrapolation-based generalization and the consequences of failure under severe or abrupt distribution shifts.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n- The proposed idea makes sense. The key innovation lies in modeling model parameter dynamics in the frequency domain, which allows principled separation of stable (low-frequency) and noisy (high-frequency) components.\n- Code is provided for reproducibility purposes.\n\nWeaknesses:\n- The proposed theoretical results rest on strong assumptions, including the Lipschitz continuity of the neural networks and the dependence of the future domain $T+1$ solely on the immediately preceding domain $T$ via the Koopman operator. These assumptions are overly restrictive. For instance, enforcing Lipschitz continuity may not be practical for deep neural networks, which often exhibit highly non-smooth behavior. Additionally, assuming that domain $T+1$ depends only on domain $T$, and is conditionally independent of all earlier domains ($1, 2, \\dots, T-1$), oversimplifies the complexity of real-world temporal dynamics, which often exhibit long-range dependencies.\n- There is a notable gap between the theoretical analysis and the proposed algorithm. Specifically, Theorem 1 characterizes the relationship between domains $T$ and $T+1$ only, whereas the algorithm leverages information from all preceding domains to estimate domain $T+1$. Developing a theoretical bound that captures the influence of all prior domains [1] would provide a stronger alignment between theory and algorithm, leading to a more coherent and rigorous framework.\n- The proposed algorithm assumes the existence of ground-truth or optimal model parameters that evolve smoothly over time. However, this assumption is difficult to satisfy in practice, particularly in high-dimensional settings involving complex models and data. For instance, deep neural networks may admit multiple optimal solutions with significantly different parameter sets. This ambiguity challenges the practicality and applicability of the proposed approach in real-world scenarios.\n- The experimental design lacks comprehensiveness. The authors evaluate their method only in a static setting, where the model is trained on all but the last domain and tested solely on the final domain. To more thoroughly assess the effectiveness of the proposed algorithm, additional experimental settings should be considered. For example, a dynamic setting with varying split points between training and testing domains [2] would better reflect real-world scenarios. Evaluating model performance on further future timestamps (e.g., $T+2, T+3$) is also recommended to assess long-term generalization. Furthermore, experiments on real-world, high-dimensional datasets [3,4] are crucial to demonstrate the practical utility and scalability of the method.\n- The method is designed for batch TDG and doesn't support continual or online adaptation, limiting its applicability in real-time streaming environments.\n- The architecture involves multiple encoders and a Koopman operator, and work on parameter space instead on data space, which may introduce nontrivial computational and memory overhead compared to simpler TDG baselines.\n\nReferences:\n\n[1] Pham, Thai-Hoang, Xueru Zhang, and Ping Zhang. \"Non-stationary domain generalization: theory and algorithm.\" Uncertainty in artificial intelligence: proceedings of the... conference. Conference on Uncertainty in Artificial Intelligence. Vol. 2024. 2025.\n\n[2] Yao, Huaxiu, et al. \"Wild-time: A benchmark of in-the-wild distribution shift over time.\" Advances in Neural Information Processing Systems 35 (2022): 10309-10324.\n\n[3] Ginosar, Shiry, et al. \"A century of portraits: A visual historical record of american high school yearbooks.\" Proceedings of the IEEE International Conference on Computer Vision Workshops. 2015.\n\n[4] Lin, Zhiqiu, et al. \"The clear benchmark: Continual learning on real-world imagery.\" Thirty-fifth conference on neural information processing systems datasets and benchmarks track (round 2). 2021.", "was_revised": true, "final_justification": "The authors provide clarification as well as conduct additional experiment on model complexity. However, I still find the experimental design somewhat limited, particularly in terms of the datasets and evaluation settings considered. Additionally, the theoretical analysis does not offer substantial novel insights and is not strongly aligned with the proposed algorithm. That said, I believe this is a quite good paper overall, and the strengths outweigh the weaknesses. Accordingly, I have decided to increase my score.", "rebuttal": "We appreciate your insightful feedbacks. To improve clarity, we have reorganized the comments from weaknesses/questions/limitations into a unified structure (order may differ from the original). This helps streamline the review process for reviewers/ACs/SACs.\n\n### **1. Task and setting**\n**1.1. Static vs. continuous settings (W4/W5/Q3)**: Our experimental design ensures fair comparison with SOTA TDG baselines [6,8,12]—the established standard in this field, i.e., discrete domains. Your mentioned concerns are talked in **Limitations (Lines 376-377)**. Crucially, we considered the limitation arising from this discrete domain segmentation in **Introduction (Lines 45-51)**. This represents a significant advance over conventional TDG approaches. In addition, we acknowledge that we did not consider the continuous TD. However, we have included the continuous TD methods (*EvoS[nips2024]，Koodas[nips2024]*) as the comparision, demonstrating our method's competitive performance. Our **novel spectral analysis framework for parameter trajectory prediction** remains the primary contribution, rigorously validated through theoretical analysis and experiments.\n\n**Abrupt distribution shifts**: As discussed in **Related Works (Lines 119-129)**, TDG fundamentally aims to leverage predictable evolutionary dynamics for proactive generalization. Abrupt shifts usually means unpredictable and thus violate the core assumption of TDG methods including ours, which is beyond this paper's scope.\n\n**1.2. (W4)**: While we did not use your suggested datasets [\"A century of portraits: A visual historical record of american high school yearbooks\"; \"The clear benchmark: Continual learning on real-world imagery\"], our evaluation was conducted on **5 real-world benchmarks** which are popular in TDG and concept drift. They feature varying numbers of features, complex drift patterns and real-world uncertainties. The consistent SOTA performance across these diverse tasks demonstrates the practical utility and robustness of FreKoo in real-world scenarios. \n\nWe sincerely appreciate your valuable suggestions and references. These will be incorporated in the revised manuscript.\n\n### **2. Assumptions (W1/W3/Q2)**\n**2.1. Lipschitz assumption (W1)**: The Lipschitz assumption is a widely-used tool in the theoretical analysis of domain adaptation and generalization [*\"Continuous temporal domain generalization, nips2024\"; \"Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks, ICML2019\"*]. **The primary purpose of this assumption is to provide a formal framework for analyzing how errors in the latent parameter space propagate to the final prediction risk**. On the one hand, the high-frequency component is not forward-predicted but regularized via temporal smoothness, which naturally suppresses sharp transitions in the parameter space. This acts as an implicit regularizer that effectively controls local Lipschitz behavior in practice. On the other hand, while the Lipschitz constant of deep neural networks is theoretically unbounded, our implementation employs weight decay. This standard regularization technique penalizes large weights, which in turn empirically constrains the model's complexity and helps control its Lipschitz constant in practice.\n\nThank you so much for raising this point to help enhance the theoretical soundness of our work. We shall provide further clarification regarding these assumptions in the revised version\n\n**2.2 Smooth parameter evolution (W3/Q2:)**: We acknowledge that multiple optimal solutions may exist if we treat each domain's optimization independently. However, our method's core strength is its joint optimization (Eq.(11)) in an end-to-end way, which selects a solution sequence $[\\theta_1, \\theta_2, \\dots, \\theta_T]$ that explicitly enforces a coherent temporal structure. This objective function explicitly regularizes the learned parameter trajectory to exhibit desirable temporal properties. By minimizing this, our framework navigates the complex solution space to select a sequence of parameters that not only performs well on historical tasks but is also structurally predictable and smooth.\n\n### **3. Long-range dependencies (W2/Q1)**\n**3.1. Our method captures long-range dependencies from $0 \\rightarrow T$:**\n * Spectral decomposition is applied to the entire historical parameter trajectory $\\Theta = [\\theta\\_1, \\theta\\_2, ..., \\theta_T]$. This transformation inherently encodes the full history of parameter evolution into the frequency domain. Long-range dependencies naturally manifest as concentrated energy in the low-frequency components ($\\Theta\\_{low}$).\n* The Koopman operator learns the dynamics on the whole low-frequency latent representations via Eq.(6). Since $\\Theta\\_{low}$ already encapsulates information from the full history, the learned evolution $\\hat{\\theta}_{t+1} = \\phi^{-1}(\\mathbf{K} \\phi\\_{low}(\\theta\\_{t, low}) + \\phi\\_{high}(\\theta\\_{t, {high}}))$ is effectively propagating a state that is imbued with historical context. \n\n**3.2 Theorem 1:** The primary goal of this theorem is to offer a qualitative decomposition of the generalization error. It provides a principled justification for our algorithm's dual-component design by showing how the total risk can be broken down into: the low-frequency dynamics ($\\mathcal{E}\\_{low}$) controlled by the stability of the Koopman operator (Lemma 1), and the high-frequency noise ($\\mathcal{E}\\_{high}$) managed by temporal smoothing regularization (Lemma 2). While the bound is presented for a single step, the logic applies inductively, and the state used for prediction already contains the information from all prior $T$ domains. \n\n**We also extended it**.    We focus on the propagation of the low-frequency latent error $e_{t,low}= \\hat{z}\\_{t,low}- z^{\\*}\\_{t,low}$. By recursively analyzing the single-step error recurrence $e_{t+1,low} = K e_{t,low} + e_{t,low}$, where $e_{t,low} = K z^{\\*}\\_{t,low} - z^{\\*}\\_{t+1,low}$ is the single-step model mismatch, we can express the error after an $h$-step predictions as:\n\n$\ne_{t_0+h,low} = K^h e_{t_0,low} + \\sum_{i=0}^{h-1} K^i e_{t_0+h-1-i, low}\n$\n\nTaking the norm, we get the multi-step error bound:\n\n$\n\\|e_{t_0+h,low}\\| \\le \\|K^h\\| \\cdot \\|e_{t_0,low}\\| + \\sum_{i=0}^{h-1} \\|K^i\\| \\cdot \\|e_{t_0+h-1-i, low}\\|\n$\n\nThis formulation reveals that the final error is not only dependent on the **initial error** ($e_{t_0,low}$), but also on the **sum of all previous errors** ($e_{t,low}$) accumulated along the entire prediction path.\n\nThis provides a stronger justification for our full-trajectory optimization approach: The stability of $K$ **Lemma 1** bounds $\\|K^i\\|$ and controlls the propagation of intermediate errors; The objective function minimizes losses across all domains, directly works to reduce the magnitude of each single-step mismatch error $\\|e_{t,low}\\|$, thereby minimizing the second term in the bound. A more detailed discussion will be included in the revised appendix.\n\n### **4. Computational and memory overhead (W6)**\nThe computational cost of the additional components is modest. The **Fourier Transform** is efficient (O(T log T)) and applied only to the short sequence of $T$ domains. The encoders and Koopman operator are lightweight MLPs and a linear layer adding minimal overhead. To prove this, we compared the running time with the most popular DRAIN method.\n|Datasets|2-Moons|ONP|Elec2|Appliance| \n|-|-|-|-|-|\n|DRAIN|17.28|34.45|23.30|20.23|\n|FreKoo|14.26|34.83|17.19|20.42|\n\nAs the results show, FreKoo's training time is competitive and comparable to that of DRAIN. We think the introduced overhead is manageable and highly justified by its superior performance. The added components are vaulable for modeling the complex dynamics that simpler methods fail to capture.\n\n### **5. Analysis of $\\tau$ (Q4)**\nWe analyzed the sensitivity to the energy preservation ratio $\\tau$ in **Section 4.5**. The optimal $\\tau$ reflects a principled trade-off: higher $\\tau$ is better for clean structured drifts (2-Moons), while a more moderate $\\tau$ excels on noisy real-world data (Appliance) by filtering out disruptive high-frequency components. We explicitly highlight adaptive threshold $\\tau$ as a future work in our **Limitation (Lines 374-376)**. Our current work establishes the foundational effectiveness of using a fixed partition, paving the way for these more advanced adaptive methods.\n\n### **6. Discuss about Koopman operator (Q5)**\nWe thank the reviewer for this excellent and insightful question.\n\nRather than assuming linearity in the original system, Koopman theory finds a latent space (via a powerful nonlinear mapping $\\phi$ learned by the encoder) that can be linearized [*A data–driven approximation of the koopman operator: Extending dynamic mode decomposition, Journal of Nonlinear Science 2015*]. Theoretically, $\\phi$ can approximate many nonlinear systems (per the universal approximation theorem) given sufficient capacity.  In this sense, Koopman theory provides a framework for modeling nonlinear dynamics. However, for fully chaotic systems with irregular or unpredictable dynamics, we believe modeling becomes inherently difficult, and the Koopman framework may also fail. This may go beyond the task assumptions of TDG, but it remains an important and thought-provoking direction for future exploration. \n\n**FreKoo vs. RNN-based methods**: Actually, DRAIN is an RNN-based method. However, DRAIN attempts to fit noisy parameter trajectories directly with a single nonlinear model. In contrast, FreKoo simplifies the learning process by explicitly decomposing the trajectories into low-frequency (signal) and high-frequency (noise) components using the Fourier transform, before modeling the dynamics. Our experimental results (see Table 1) also show that FreKoo consistently outperforms DRAIN."}, {"review_id": "OtGBBKGuar", "reviewer": "Reviewer_mg4n", "summary": "This paper introduces FreKoo, a novel framework for Temporal Domain Generalization (TDG) designed to address concept drift, particularly complex drifts involving long-term periodicity and domain-specific uncertainties. The core idea is to analyze the trajectories of model parameters in the frequency domain. The authors provide theoretical grounding for their approach, including stability guarantees for Koopman prediction, a Bayesian justification for high-frequency regularization, and a multiscale generalization bound. Extensive experiments on synthetic and real-world datasets demonstrate FreKoo's superiority over existing SOTA TDG methods.", "questions": "See the Strengths and Weaknesses section.", "limitations": "Yes", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. Addresses Critical TDG Limitations: This paper tackles two critical limitations of current TDG methods: (i) the difficulty of modeling long-term periodic dynamics, and (ii) vulnerability to overfitting domain-specific uncertainties. This is particularly relevant for the complex drift scenarios encountered in real-world applications.\n\n2. Novel Methodology: The proposed method, FreKoo, leverages a novel and well-motivated approach by applying spectral analysis to model parameter trajectories for TDG.\n\n3. Rigorous Theoretical Foundation: The paper provides a rigorous theoretical foundation for the proposed method.\n\n4. Comprehensive Experiments: The thorough experimental setup, encompassing a diverse set of synthetic and real-world benchmarks with various drift types, convincingly validates the paper's claims.\n\n5. Clear and Compelling Presentation: The paper is generally well-written and clearly structured, featuring a compelling problem statement and motivation.\n\nWeaknesses and Suggestions for Improvement:\n\n1. The paper mentions using encoders/decoders $\\left(\\varphi, \\varphi^{-1}\\right)$ for the Koopman operator. While Appendix C. 3 specifies MLP architectures, a little more detail in the main text about how K is learned (e.g., is it simply a linear layer optimized via L_koop, or are there specific constraints to encourage stability beyond the loss term?) would be helpful.\n\n2. Lemma 1 provides a tighter bound if K's spectral radius ρ(K) < 1. Is this condition actively enforced during training, or is it an assumption for the tighter bound?\n\n3. FreKoo operates on model parameter trajectories. While this is an effective proxy for concept drift, a brief discussion on the assumed relationship between parameter evolution in frequency bands and the actual shifts in P(X,Y) could add further depth. For instance, does a strong periodic component in θ_low necessarily imply a periodic shift in the data distribution itself?\n\n4. Throughout the paper, there appears to be an interchangeable use of \"periodical\" and \"periodic\" when referring to time- based cyclical patterns (e.g., \"long-term periodical dynamics\" vs. \"periodic drifts\"). For clarity and consistency, it is recommended to uniformly use \"periodic\" as it is the more standard adjective in this context.\n\n5. Please check and complete the missing author information for reference [11] (currently listed as \"--\").", "was_revised": true, "final_justification": "This paper presents a novel and methodologically sound contribution with strong theoretical foundations. The authors have successfully addressed all previously raised concerns, and the comprehensive experimental evaluation on real-world datasets convincingly demonstrates the effectiveness of the proposed approach.", "rebuttal": "We sincerely thank the reviewer for your valuable comments, which significantly strengthen our work. We are particularly grateful for your recognition of our contributions' key strengths: technical novelty, theoretical foundation, comprehensive experiments, and presentation quality. Below we address all concerns point-by-point.\n\n### **W1: Explanation about $\\mathbf{K}$**\n\n**Learning $\\mathbf{K}$:** In our implementation, $\\mathbf{K}$ is learned via a standard linear layer. It is not learned in isolation but is jointly optimized with the encoders and decoder via backpropagation as defined in our overall objective **Eq. (11)**. Thank you your suggestion, and we will refine this part in the revised version.\n\n### **W2: Clarification of K's spectral radius ρ(K) < 1.**\nWe appreciate your suggestion on the learning mechanism for the transformation $\\mathbf{K}$. As the reviewer correctly infers, ensuring the stability of $\\mathbf{K}$ is crucial for preventing error explosion and achieving a tighter theoretical bound. To address this, we actively enforce a stability constraint during training in our code (please refer our offered code of the model implementation). Therefore, this condition is not merely a theoretical assumption for the tighter bound in Lemma 1; it is a practical constraint actively enforced in our implementation. This ensures that the learned operator is stable, which in turn helps to minimize error accumulation and leads to the more robust long-term forecasting performance demonstrated in our experiments.\n\n### **W3: Explanation about the relationship between parameter evolution and data distribution**\n\nWe thank the reviewer for this excellent question regarding the connection between parameter and data dynamics.\n\nOur framework's end-to-end learning process directly establishes this relationship. As detailed in **Algorithm 1**, the parameter trajectory $\\Theta$ is learned jointly by minimizing the task loss across all data domains. This ensures that the learned parameter evolution is an intrinsic reflection of the shifts in the data distribution $P(X,Y)$.\n\nTherefore, a strong periodical component in $\\Theta_{low}$ is indeed a direct consequence of a periodical shift in the data itself, as this is the optimal trajectory for minimizing loss on such data. We empirically validate this relationship in our **Periodicity Modeling experiment (Section 4.4)**. The visualizations in **Figure 3(b)** provide clear evidence that our model successfully captures the induced periodical structure in the parameter space.\n\nThank you for the insightful question. We will emphasize this connection more explicitly in the revised manuscript.\n\n### **(W4/W5) \"Periodical\" vs. \"Periodic\" and Missing reference information [11]:** \nThank you for your suggestion. We will revise the manuscript to consistently use \"periodical\" for improved clarity and uniformity, and will thoroughly verify all content—including references—to ensure accuracy."}]

Author rebuttal:

Meta-review: This paper introduces a novel method for dealing with temporal distribution shifts, by leveraging Fourier analysis and Koopman operator theory. It addresses a tricky problem in time series analysis by separating low and high frequency components of the signal and using the Koopman operator to extrapolate the former. The paper further comes with a solid experimental evaluation and a theoretical basis. 

Some initial concerns about formal clarity and the specific role of the Koopman operator, the model assumptions, training \& inference times, and missing baselines were comprehensively addressed by the authors in their rebuttal through runtime evaluations, additional comparisons, and a number of specific clarifications. Although a few concerns remained about limitations of the experimental setups, all referees uniformly agreed on acceptance, and so this was an easy decision.

A few notes from my side: The focus of this work is on classification and regression tasks, less so on forecasting, which the authors might want to make clearer from the outset. Also, the paper currently lacks a Limitations section, where remaining shortcomings discussed with the referees, like the model’s inability to deal with abrupt distribution shifts could be discussed (the authors argue the latter are essentially unpredictable, but I do not concur with this, see lit. on predicting tipping points).

Overall this is a very good paper and I go with the referees on recommending acceptance.
Decision: Accept (poster)