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Adaptive Spectral-Koopman Dynamics Modeling for Temporal Domain Generalization

Tengxue Zhang, Yu Ke, Yang Shu, Chenchen Sun, Yisheng An, Chenjuan Guo, Bin Yang

arXiv:2610.02822v1cs.LGcs.AI

TL;DR

Temporal domain generalization addresses streaming data whose distributions shift over time, while existing approaches can overfit domain-specific noise or complicate parameter-space modeling. AdaSpecK combines spectral-Koopman dynamics with adaptive historical-context extraction. It is reported to achieve the best performance across eight benchmarks.

  • Problem

    TDG must generalize to future shifted domains, but data-space methods risk fitting domain-specific noise and parameter-space methods can be complex and unstable.

    Source: §1 · pp. 1–2

  • Method

    AdaSpecK filters latent trajectories spectrally, models their evolution with a Koopman operator, and uses target-conditioned attention and a learned router to select historical context.

    Source: §1 · p. 2

  • Results

    AdaSpecK achieves the best performance across all eight benchmarks and consistently outperforms continuous TDG baselines Koodos and Frekoo.

    Source: §1 · p. 3, §5.2 · p. 8

  • Takeaways & Limitations

    Its largest gains are reported on Twitter, YearBook, Cyclone, and Appliance, datasets characterized by multi-step future evolution.

    Source: §5.2 · p. 8

Abstract

from arXiv · show

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

1 INTRODUCTION

Temporal domain generalization must handle irregularly sampled domains, noise, and changing historical patterns. AdaSpecK combines spectral-Koopman dynamics with adaptive context extraction and is reported to achieve state-of-the-art performance across eight benchmarks.

  • Irregularly sampled domains make data-space learning prone to domain-specific noise, while parameter-space dynamics can be abstract, high-dimensional, and unstable.Source: §1 · pp. 1–2
  • Non-stationarity can obscure recurring temporal structure, and future domains may prioritize different historical cues.Source: §1 · p. 2
  • AdaSpecK filters latent representations to extract denoised low-frequency trajectories and learns a Koopman operator to model system dynamics.Source: §1 · pp. 2–3
  • Target-conditioned attention and adaptive expert routing selectively integrate historical contexts relevant to each future prediction step.Source: §1 · pp. 2–3
  • AdaSpecK achieves state-of-the-art performance on eight diverse classification and regression benchmarks.Source: §1 · p. 3

2 RELATED WORKS

Prior TDG work includes data-driven and parameter-adaptation strategies, while continuous-dynamics and spectral-learning methods offer complementary tools. The cited approaches leave gaps in modeling distribution shifts, explicit dynamics, and adaptive use of historical context.

  • TDG approaches include sequential meta-learning, future-distribution prediction, parameter-update extrapolation, probabilistic latent spaces, recurrent parameter generation, and continuous-time modeling.Source: §2 · p. 3
  • Prior TDG methods remain limited by data-space noise fitting, parameter-space complexity, and lack of explicit modeling of non-stationary historical contexts.Source: §2 · p. 3
  • Continuous dynamical systems model temporal evolution, while frequency-based methods expose periodic structures and multiscale trends through spectral representations.Source: §2 · p. 3
  • Continuous-dynamics approaches often overlook distribution shifts and structured context, whereas spectral methods typically lack explicit dynamic modeling.Source: §2 · p. 3

3 PROBLEM DEFINITION

TDG uses temporally evolving observed domains to train a predictor for future unseen out-of-distribution domains. The observed domains occur at arbitrary, ordered timestamps and undergo distribution drift.

  • The observed domains are ordered by arbitrary timestamps, with each domain containing input-label pairs.Source: §3 · p. 3
  • TDG seeks to generalize from observed domains to unseen future domains under evolving data distributions.Source: §3 · pp. 3–4

4 METHODOLOGY

AdaSpecK combines spectral-regularized Koopman dynamics with adaptive extraction of historical context for temporal domain generalization. Its method filters latent trajectories, models their evolution in Koopman space, and adapts historical information to each future prediction.

4.1 SPECTRAL-REGULARIZED KOOPMAN DYNAMICS MODELING

  • The spectral branch pools domains into a latent trajectory, filters it to retain low-frequency structure, then models its evolution in Koopman space.The filtering is intended to remove high-frequency components associated with sampling noise or domain-specific perturbations.Source: §4.1 · pp. 4–5
  • A three-part loss combines local Koopman transition fidelity, decoded-rollout supervision, and trajectory consistency to support accurate, stable latent dynamics.The consistency constraint aligns ground-truth trajectories with latent evolution, encouraging coherent, path-independent dynamics.Source: §4.1 · p. 5

4.2 CONTEXT-INFORMED HETEROGENEOUS PATTERN EXTRACTION

  • For each future step, target-conditioned attention uses a dynamic query to select semantically and temporally relevant segments from historical context.The resulting representation is a dynamic, target-specific historical summary rather than a fixed summary of the entire history.Source: §4.2 · p. 6
  • A router uses an environmental signature built from low-frequency amplitude and phase to select and aggregate expert patterns from the historical summary.The method also applies semantic orthogonality loss to encourage distinct expert patterns, then fuses the resulting pattern with target features for prediction.Source: §4.2 · pp. 6–7

5 EXPERIMENTS

AdaSpecK is evaluated against temporal and non-temporal baselines on eight classification and regression benchmarks, with the strongest overall results reported across all eight. Further analyses examine loss weighting, ablations, decision boundaries, spectral cutoff, and expert count.

5.1 EXPERIMENT SETTINGS

  • The evaluation spans eight classification and regression datasets and compares time-agnostic, discrete-temporal, and continuous-temporal methods.Classification uses error rate except for Twitter, which uses AUC; regression uses MAE, with five runs per method.Source: §5.1 · pp. 7–8

5.2 EXPERIMENTAL RESULTS

  • AdaSpecK achieves the best performance across all eight benchmarks and consistently surpasses continuous TDG baselines Koodos and Frekoo.The largest gains are reported on Twitter, YearBook, Cyclone, and Appliance, which involve multi-step future evolution.Source: §5.2 · p. 8
  • The full loss integration yields the best ablation results, supporting the value of adaptive historical modeling and environment-aware routing.Table 2 presents the ablation study, with best and second-best results distinguished in its caption.Source: Table 2 · p. 8, §5.2 · p. 9

5.3 FURTHER ANALYSIS

  • Moderate values of loss weights λK, λS, and λC give the best or near-best performance, while overly large coefficients slightly degrade accuracy, especially on Cyclone.The analysis covers Cyclone, House, and Appliance.Source: Figure 3 · p. 8, §5.3 · p. 9
  • On YearBook, AdaSpecK produces smoother decision boundaries and maintains consistent performance across test domains, unlike the reported distortions of Koodos and DeepODE.Figure 4 compares methods by row and test domains over time by column; blue and red dots denote the two classes.Source: §5.3 · p. 9, Figure 4 · p. 9
  • A moderate spectral cutoff ratio β performs best; excessively small values are suboptimal, while further increases noticeably reduce accuracy.The low-frequency branch is intended to retain denoised, slowly varying structure while excluding high-frequency noise.Source: Figure 5 · p. 9, §5.3 · p. 10
  • Using 4–8 experts offers the best balance, whereas too few may miss heterogeneous patterns and too many add complexity and may degrade results.The expert-count analysis considers K ∈ [2, 4, 6, 8, 10].Source: §5.3 · p. 10

6 CONCLUSION

AdaSpecK combines spectral-regularized Koopman dynamics with adaptive extraction of historical context and is reported to achieve state-of-the-art performance on eight benchmarks. The paper also describes its AI use, dataset ethics, and reproducibility practices.

6 CONCLUSION

  • AdaSpecK filters latent trajectories to retain denoised low-frequency patterns, models their evolution with a compact Koopman operator, and adaptively selects historical context.Target-conditioned attention summarizes relevant historical windows, while an environmental signature guides a learned router.Source: §6 · p. 10
  • Experiments across eight classification and regression benchmarks show state-of-the-art performance.Source: §6 · p. 10

AI USE STATEMENT

  • Generative AI tools were used to improve writing clarity and grammar, not to generate core ideas, research questions, or methodology.The authors state that they reviewed and edited AI-assisted work and take responsibility for the final content.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

  • The work uses publicly available benchmark datasets and involves no human subjects or release of personal or sensitive information.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

  • The authors report that experiments are reproducible and provide model, training, and evaluation details alongside released code and datasets.Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 DATASETS

The eight benchmarks cover classification and regression tasks with diverse temporal shifts, including synthetic changes and real-world non-stationarity.

A.1 DATASETS

  • Rotated MNIST is a synthetic image-classification benchmark whose temporally ordered domains gradually rotate images, with later unseen domains used for prediction.Its main difficulty is steadily increasing geometric transformation rather than label noise.Source: §A.1 · p. 14
  • Twitter uses temporally separated tweet-collection windows across flu seasons to estimate influenza activity amid changing tweet volume, seasonal activity, and user behavior.The passage characterizes it as a real-world classification problem with stronger non-stationarity than synthetic rotation datasets.Source: §A.1 · p. 14
  • YearBook is a gender-classification benchmark of frontal portraits sampled by year, with long-term shifts in appearance, image quality, and social norms.Source: §A.1 · p. 14
  • Cyclone predicts tropical cyclone intensity from satellite imagery, with temporal shifts associated with seasonal and regional variation in storms.Source: §A.1 · p. 14
  • House predicts prices from tabular features collected across years, with drift reflecting changing market conditions and economic variation.Source: §A.1 · p. 14
  • Online News Popularity predicts article popularity from 58 features, using time-ordered domains over roughly two years and a final domain for testing.The passage describes its drift as relatively mild, reflecting gradual changes in social attention to content.Source: §A.1 · p. 14
  • Shuttle is a temporally ordered, imbalanced multiclass classification benchmark whose later periods test generalization to operational distribution changes.Source: §A.1 · p. 14
  • ApplianceEnergy predicts energy consumption from indoor sensor readings recorded every 10 minutes over several months and partitioned into chronological domains.Its temporal variation is associated with changing usage behavior and environmental conditions.Source: §A.1 · p. 14

A.2 BASELINE

The baseline set spans time-agnostic training, invariant learning, temporal adaptation, predictor forecasting, and continuous or frequency-aware dynamics.

  • Time-agnostic baselines include pooled Offline training, recent-domain-only LastDomain, sequential IncFinetune, and invariant-learning methods IRM and V-REx.IRM and V-REx favor predictors with stable risks across source domains.Source: §A.2 · p. 15
  • CIDA and TKNets represent temporal alignment or transfer, while DRAIN and DRAIN-∆t forecast future predictors from domain sequences, with DRAIN-∆t using time-gap information.Source: §A.2 · p. 15
  • DeepODE, Koodos, and FreKoo represent continuous temporal approaches, including latent dynamical systems, Koopman-space parameter evolution, and spectral-Koopman modeling.Source: §A.2 · p. 15

A.3 IMPLEMENTATION DETAILS

Implementation settings vary by dataset, while the experiments share an optimizer, learning-rate schedule, embedding size, low-frequency ratio, and integration method.

A.3 IMPLEMENTATION DETAILS

  • Across datasets, AdaSpecK uses Adam with cosine annealing, embedding dimension 32, low-frequency ratio 0.40, RK4 step size 0.2, learning rate 0.0005, and weight decay 0.005.Results are means and standard deviations over five independent runs; experiments use PyTorch on a 64-bit machine with an NVIDIA GeForce RTX 3090 GPU.Source: §A.3 · p. 15
  • Rotated MNIST uses a 3-layer CNN and Twitter an embedded categorical feature followed by a 3-layer MLP; both use 35 domains, segment length 20, and a 15-step horizon.Their latent dimensions are 64; spectral bins are 5 for Rotated MNIST and 4 for Twitter.Source: §A.3 · p. 15
  • Rotated MNIST and Twitter train for 200 epochs with batch size 1, using 8 and 4 experts and Koopman rollout lengths 15 and 10, respectively.Twitter's hidden dimension is 128; Rotated MNIST's is 256.Source: §A.3 · p. 15
  • YearBook uses a 3-stage CNN and Cyclone a 4-layer CNN; they use 28 and 50 domains, horizons of 12 and 22 steps, and segment length 20.Their latent dimensions are 128; YearBook uses 5 spectral bins and Cyclone 6.Source: §A.3 · pp. 15–16
  • YearBook trains 170 epochs and Cyclone 100, both with batch size 1, 8 experts, and rollout length 15; their hidden dimensions are 256 and 384, respectively.Both dataset backbones include dropout.Source: §A.3 · pp. 15–16
  • House, ONP, Appliance, and Shuttle use raw tabular inputs; their domain counts are 28, 5, 8, and 7, with segment lengths 20, 4, 4, and 4.All have latent dimension 64, hidden dimension 256, and 5 spectral bins; expert counts are 4, 4, 4, and 8, and rollout lengths are 6, 3, 3, and 5, respectively. Epochs are 100, 700, 500, and 350, with batch size 1 for House and 2 for the others.Source: §A.3 · p. 16
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