Artificial intelligence systems in critical fields like autonomous driving and medical imaging analysis often continually learn new tasks using a shared stream of input data. For instance, after learning to detect traffic signs, a model may later need to learn to classify traffic lights or different types of vehicles using the same camera feed. This scenario introduces a challenging setting we term Continual Multitask Learning (CMTL), where a model sequentially learns new tasks on an underlying data distribution without forgetting previously learned abilities. Existing continual learning methods often fail in this setting because they learn fragmented, task-specific features that interfere with one another. To address this, we introduce Learning with Preserving (LwP), a novel framework that shifts the focus from preserving task outputs to maintaining the geometric structure of the shared representation space. The core of LwP is a Dynamically Weighted Distance Preservation (DWDP) loss that prevents representation drift by regularizing the pairwise distances between latent data representations. This mechanism of preserving the underlying geometric structure allows the model to retain implicit knowledge and support diverse tasks without requiring a replay buffer, making it suitable for privacy-conscious applications. Extensive evaluations on time-series and image benchmarks show that LwP not only mitigates catastrophic forgetting but also consistently outperforms state-of-the-art baselines in CMTL tasks.
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Current unsupervised time series clustering methods often struggle to fully exploit the inherent characteristics of time series data and commonly adopt a two-stage training strategy that separates feature learning from the clustering process. To address these limitations, this paper proposes a novel deep clustering framework, Time-Frequency augmented Multi-level Contrastive Clustering (TFMCC). TFMCC employs a multi-scale time-frequency augmentation strategy, where each training iteration stochastically selects time and frequency scales to generate diverse augmented views, enhancing the model’s ability to learn robust and generalizable representations. In addition, a multi-level contrastive learning mechanism is introduced to jointly capture temporal dependencies, inter-sample similarities, and cluster structures. By jointly optimizing these components, TFMCC enables the learning of temporally-aware and clustering-friendly representations. Experimental results on 40 benchmark datasets demonstrate that TFMCC outperforms six existing methods in clustering accuracy.
Long-term series forecasting leverages historical observations to predict extended future sequences and plays a crucial role across various domains. However, conventional models relying on fixed-length lookback windows struggle with the inherent dynamic dependencies and multi-scale characteristics of time series data. These fixed windows either introduce noise through excessive length or omit critical patterns when too short, while optimal window sizes vary significantly with tasks and external conditions. To address this, we propose the Adaptive Lookback Window (ALW) framework, a wavelet transform-driven approach for multi-scale adaptive lookback window selection. ALW decomposes time series into distinct frequency components through wavelet transforms, quantifies the contribution of each historical time step via scale-specific attention mechanisms, and dynamically determines optimal window lengths through backward information accumulation and soft truncation techniques. Finally, refined input features are generated for downstream prediction models via a weighted reconstruction process. Extensive evaluations across multiple public benchmarks demonstrate that ALW, as an efficient plug-and-play technique, not only reduces the MSE of backbone models by an average of 3.2% but also alleviates hyperparameter tuning requirements and enables input feature dimensionality reduction, which curtails subsequent model computational costs.
Selecting an appropriate look-back horizon remains a fundamental challenge in time series forecasting (TSF), particularly in federated learning scenarios where data is decentralized, heterogeneous, and often non-independent. While recent work has explored horizon selection by preserving forecasting-relevant information in an intrinsic space, these approaches are primarily restricted to centralized and independently distributed settings. This paper presents a principled framework for adaptive horizon selection in federated time series forecasting through an intrinsic space formulation. We introduce a synthetic data generator that captures essential temporal structures in client data, including autoregressive dependencies, seasonality, and trend, while incorporating client-specific heterogeneity. Building on this model, we define a transformation that maps time series windows into an intrinsic representation space with well-defined geometric and statistical properties. We then derive a decomposition of the forecasting loss into a Bayesian term, which reflects irreducible uncertainty, and an approximation term, which accounts for finite-sample effects and limited model capacity. Our analysis shows that while increasing the look-back horizon improves the identifiability of deterministic patterns, it also increases approximation error due to higher model complexity and reduced sample efficiency. We prove that the total forecasting loss is minimized at the smallest horizon where the irreducible loss starts to saturate, while the approximation loss continues to rise. This work provides a rigorous theoretical foundation for adaptive horizon selection for time series forecasting in federated learning.
Mask the Redundancy: Evolving Masking Representation Learning for Multivariate Time-Series Clustering
PDF ↗Multivariate Time-Series (MTS) clustering discovers intrinsic grouping patterns of temporal data samples. Although time-series provide rich discriminative information, they also contain substantial redundancy, such as steady-state machine operation records and zero-output periods of solar power generation. Such redundancy diminishes the attention given to discriminative timestamps in representation learning, thus leading to performance bottlenecks in MTS clustering. Masking has been widely adopted to enhance the MTS representation, where temporal reconstruction tasks are designed to capture critical information from MTS. However, most existing masking strategies appear to be standalone preprocessing steps, isolated from the learning process, which hinders dynamic adaptation to the importance of clustering-critical timestamps. Accordingly, this paper proposes the Evolving-masked MTS Clustering (EMTC) method, whose model architecture comprises Importance-aware Variate-wise Masking (IVM) and Multi-Endogenous Views (MEV) generation modules. IVM adaptively guides the model in learning more discriminative representations for clustering, while the reconstruction and cluster-guided contrastive learning pathways enhance and connect the representation learning to clustering tasks. Extensive experiments on 15 benchmark datasets demonstrate the superiority of EMTC over eight SOTA methods, where the EMTC achieves an average improvement of 4.85% in F1-Score over the strongest baselines.
State-Derivative-Aware Neural Controlled Differential Equations for Multivariate Time Series Anomaly Detection and Diagnosis
PDF ↗Multivariate time series anomaly detection is a crucial factor in real-world applications but a challenging task due to the complex temporal dependencies and system dynamics. Reconstruction-based methods have made great improvements in recent years. However, we observe an issue these methods are suffering, that they primarily measure deviations in the time points themselves when performing anomaly detection but ignore changes in the dynamic properties of the system. In these cases, they are unable to produce sufficient reconstruction errors to detect anomalies, so some potential abnormal time points caused by the dynamic evolution of the system are missing. To address this problem, we propose a novel method, SDA2D, which models system dynamics by the derivative of the NCDE-derived state vector with respect to time, enabling the learning of reconstruction deviation and system evolution jointly. Our experimental results show that SDA2D achieves noticeable improvements in four benchmark datasets, and the visualization also provides further instructions for anomaly diagnosis, which helps locate the sources of these anomalies.
COGS: A Causal Representation Learning Framework for Out-of-Distribution Generalization in Time Series
PDF ↗Time series analysis is crucial in various fields such as healthcare and finance. However, environmental variations and the inherent non-stationarity of time series data often lead to out-of-distribution (OOD) scenarios, consequently causing model performance degradation. Most existing OOD generalization methods primarily focus on images or text, leaving time series analysis relatively underexplored. In this paper, we propose COGS, a novel framework that incorporates causal representation learning into the OOD generalization of time series. By imposing structural priors, our method identifies latent variables and learns a causal graph to disentangle causal variables from non-causal ones. These causal variables are then used to learn domain-invariant representations for stable prediction. Moreover, to tackle the challenge of the absence of domain labels, we further introduce a prototype-based domain discovery algorithm that infers domain labels in an unsupervised manner. The entire framework is optimized in a two-phase iterative manner, resulting in robust OOD performance. Extensive experiments on multiple real-world time series datasets demonstrate that our method achieves competitive performance compared to baseline methods.
Multivariable time series forecasting methods can integrate information from exogenous variables, leading to significant prediction accuracy gains. The transformer architecture has been widely applied in various time series forecasting models due to its ability to capture long-range sequential dependencies. However, a naïve application of transformers often struggles to effectively model complex relationships among variables over time. To mitigate against this, we propose a novel architecture, termed Spectral Operator Neural Network (Sonnet). Sonnet applies learnable wavelet transformations to the input and incorporates spectral analysis using the Koopman operator. Its predictive skill relies on the Multivariable Coherence Attention (MVCA), an operation that leverages spectral coherence to model variable dependencies. Our empirical analysis shows that Sonnet yields the best performance on 34 out of 47 forecasting tasks with an average mean absolute error (MAE) reduction of 2.2% against the most competitive baseline. We further show that MVCA can remedy the deficiencies of naïve attention in various deep learning models, reducing MAE by 10.7% on average in the most challenging forecasting tasks.
The success of large-scale pre-training paradigm, exemplified by Large Language Models (LLMs), has inspired the development of Time Series Foundation Models (TSFMs). However, their application to financial candlestick (K-line) data remains limited, often underperforming non-pre-trained architectures. Moreover, existing TSFMs often overlook crucial downstream tasks such as volatility prediction and synthetic data generation. To address these limitations, we propose Kronos, a unified, scalable pre-training framework tailored to financial K-line modeling. Kronos introduces a specialized tokenizer that discretizes continuous market information into token sequences, preserving both price dynamics and trade activity patterns. We pre-train Kronos using an autoregressive objective on a massive, multi-market corpus of over 12 billion K-line records from 45 global exchanges, enabling it to learn nuanced temporal and cross-asset representations. Kronos excels in a zero-shot setting across a diverse set of financial tasks. On benchmark datasets, Kronos boosts price series forecasting RankIC by 93% over the leading TSFM and 87% over the best non-pre-trained baseline. It also achieves a 9% lower MAE in volatility forecasting and a 22% improvement in generative fidelity for synthetic K-line sequences. These results establish Kronos as a robust, versatile foundation model for end-to-end financial time series analysis.
Diffusion models have shown great promise in data generation, yet generating time series data remains challenging due to the need to capture complex temporal dependencies and structural patterns. In this paper, we present TSGDiff, a novel framework that rethinks time series generation from a graph-based perspective. Specifically, we represent time series as dynamic graphs, where edges are constructed based on Fourier spectrum characteristics and temporal dependencies. A graph neural network-based encoder-decoder architecture is employed to construct a latent space, enabling the diffusion process to model the structural representation distribution of time series effectively. Furthermore, we propose the Topological Structure Fidelity (Topo-FID) score, a graph-aware metric for assessing the structural similarity of time series graph representations. Topo-FID integrates two sub-metrics: Graph Edit Similarity, which quantifies differences in adjacency matrices, and Structural Entropy Similarity, which evaluates the entropy of node degree distributions. This comprehensive metric provides a more accurate assessment of structural fidelity in generated time series. Experiments on real-world datasets demonstrate that TSGDiff generates high-quality synthetic time series data generation, faithfully preserving temporal dependencies and structural integrity, thereby advancing the field of synthetic time series generation.
Modeling normal behavior in dynamic, nonlinear time series data is challenging for effective anomaly detection. Traditional methods, such as nearest neighbor and clustering approaches, often depend on rigid assumptions, such as a predefined number of reliable neighbors or clusters, which frequently break down in complex temporal scenarios. To address these limitations, we introduce the Granular-ball One-Class Network (GBOC), a novel approach based on a data-adaptive representation called Granular-ball Vector Data Description (GVDD). GVDD partitions the latent space into compact, high-density regions represented by granular-balls, which are generated through a density-guided hierarchical splitting process and refined by removing noisy structures. Each granular-ball serves as a prototype for local normal behavior, naturally positioning itself between individual instances and clusters while preserving the local topological structure of the sample set. During training, GBOC improves the compactness of representations by aligning samples with their nearest granular-ball centers. During inference, anomaly scores are computed based on the distance to the nearest granular-ball. By focusing on dense, high-quality regions and significantly reducing the number of prototypes, GBOC delivers both robustness and efficiency in anomaly detection. Extensive experiments validate the effectiveness and superiority of the proposed method, highlighting its ability to handle the challenges of time series anomaly detection.
Accurate forecasting of multivariate time series data remains a formidable challenge, particularly due to the growing complexity of temporal dependencies in real-world scenarios. While neural network-based models have achieved notable success in this domain, complex channel-dependent models often suffer from performance degradation compared to channel-independent models that do not consider the relationship between components but provide high robustness due to small capacity. In this work, we propose HN-MVTS, a novel architecture that integrates the hypernetwork-based generative prior with an arbitrary neural network forecasting model. The input of this hypernetwork is a learnable embedding matrix of time series components. To restrict the number of new parameters, the hypernetwork learns to generate the weights of the last layer of the target forecasting networks, serving as a data-adaptive regularizer that improves generalization and long-range predictive accuracy. The hypernetwork is only used during training, so it does not increase the inference time compared to the base forecasting model. Extensive experiments on eight benchmark datasets demonstrate that application of HN-MVTS to the state-of-the-art models (DLinear, PatchTST, TSMixer, etc.) typically improves their performance. Our findings suggest that hypernetwork-driven parameterization offers a promising direction for enhancing existing forecasting techniques in complex scenarios.
Koopman Invariants as Drivers of Emergent Time-Series Clustering in Joint-Embedding Predictive Architectures
PDF ↗Joint-Embedding Predictive Architectures (JEPAs), a powerful class of self-supervised models, exhibit an unexplained ability to cluster time-series data by their underlying dynamical regimes. We propose a novel theoretical explanation for this phenomenon, hypothesizing that JEPA's predictive objective implicitly drives it to learn the invariant subspace of the system's Koopman operator. We prove that an idealized JEPA loss is minimized when the encoder represents the system's regime indicator functions, which are Koopman eigenfunctions. This theory was validated on synthetic data with known dynamics, demonstrating that constraining the JEPA's linear predictor to be a near-identity operator is the key inductive bias that forces the encoder to learn these invariants. We further discuss that this constraint is critical for selecting this interpretable solution from a class of mathematically equivalent but entangled optima, revealing the predictor's role in representation disentanglement. This work demystifies a key behavior of JEPAs, provides a principled connection between modern self-supervised learning and dynamical systems theory, and informs the design of more robust and interpretable time-series models.
Amid recent advances for multivariate time series forecasting, self-supervised learning has emerged as a promising paradigm for deriving transferable knowledge from multi-domain data. Despite its effectiveness, existing approaches exhibit two critical limitations: (1) Underestimating the significance of multivariate dependencies in learning generalizable representations and (2) Failing to reconcile the complementary strengths of autoregressive and one-shot generative paradigms. In this work, we propose TimeCAP, a novel channel-aware pre-training framework that internalizes latent causal relationships among variables inherent in multi-domain data, and effectively transfers the acquired knowledge to downstream applications. Technically, we present a flexible channel-grouping learning approach, complemented by an adaptive meta-routing mechanism, enabling TimeCAP to parallel recognize intra-group local patterns while maintaining global coherence. Intra- and inter-group multivariate dependencies are captured through the self- and cross-attention with channel-aware mask, which strictly confine interactions among time-aligned, fine-grained multivariate tokens. To seamlessly unify two advanced generative paradigms, we propose a novel dynamic dual-head decoding and optimization strategy, empowering TimeCAP to leverage critical dependencies in the output series while avoiding cumulative errors over time. In the few-shot evaluation, TimeCAP achieves average MSE and MAE reductions of 11.8% and 6% over leading baselines, while also outperforming state-of-the-art models in full-shot and zero-shot settings by large margins.
Beyond Missing Data Imputation: Information-Theoretic Coupling of Missingness and Class Imbalance for Optimal Irregular Time Series Classification
PDF ↗Irregular time series (IRTS) are prevalent in real-world applications, where uneven sampling and missing data pose fundamental challenges to deep learning-based feature modeling. Although existing methods attempt to retain timestamp information, they often overlook the structured patterns embedded within the missingness itself, and tend to perform poorly when confronted with class imbalance exacerbated by data incompleteness. Specifically, temporal irregularity hinders the modeling of long-range dependencies and local patterns, while sparse observations limit representational capacity, disproportionately impairing minority classes and leading to severe classification bias. To address these deeply coupled challenges, we propose SPECTRA (Structured Pattern and Enriched Context-aware Temporal Representation Architecture), a unified framework for robust IRTS classification. SPECTRA introduces a frequency-guided observation encoder that reconstructs temporal dependencies in a stable manner, mitigating spectral distortion and information corruption. Complementarily, a missingness pattern encoder explicitly captures the dynamic evolution of missing data and leverages it as a discriminative signal. In addition, a prototype-constrained classification paradigm directly optimizes the geometric structure of the feature space, enhancing intra-class compactness and alleviating generalization bottlenecks caused by class imbalance. Extensive experiments on three public IRTS datasets—P12, P19, and PAM—demonstrate the superior performance of SPECTRA under both missing and imbalanced conditions.
FlowPath: Learning Data-Driven Manifolds with Invertible Flows for Robust Irregularly-sampled Time Series Classification
PDF ↗Modeling continuous-time dynamics from sparse and irregularly-sampled time series remains a fundamental challenge. Neural controlled differential equations provide a principled framework for such tasks, yet their performance is highly sensitive to the choice of control path constructed from discrete observations. Existing methods commonly employ fixed interpolation schemes, which impose simplistic geometric assumptions that often misrepresent the underlying data manifold, particularly under high missingness. We propose FlowPath, a novel approach that learns the geometry of the control path via an invertible neural flow. Rather than merely connecting observations, FlowPath constructs a continuous and data-adaptive manifold, guided by invertibility constraints that enforce information-preserving and well-behaved transformations. This inductive bias distinguishes FlowPath from prior unconstrained learnable path models. Empirical evaluations on 18 benchmark datasets and a real-world case study demonstrate that FlowPath consistently achieves statistically significant improvements in classification accuracy over baselines using fixed interpolants or non-invertible architectures. These results highlight the importance of modeling not only the dynamics along the path but also the geometry of the path itself, offering a robust and generalizable solution for learning from irregular time series.
Active research in time series classification and forecasting has led to the development of a wide range of machine learning models. For practitioners, the selection of a suitable model among these, along with their hyperparameters, remains a challenging task. While automated machine learning offers approaches for automatic selection of models for a given task, the practical efficacy of these methods is often limited, due to the computational complexity of searching over a large design space and the high dimensionality of time series datasets that poses additional challenges on generalisation quality. To fill this gap, we propose a meta-learning framework that transfers past knowledge from previous searches to recommend an architecture and its hyperparameters; specifically, this framework utilises a joint representation of deep neural architectures and time series datasets, and predicts the performance of neural architectures along with their hyperparameters on time series datasets. Our computational experiments reveal that the configurations proposed by our meta-learned surrogate achieve a performance gain of up to 34% on 4 out of the 8 forecasting datasets we considered and up to 60% on 36 out of 73 of our classification datasets, whilst reducing the computational cost to 10% of that required by the hyperparameter optimisation method HEBO to tune the architectures, showcasing the effectiveness of meta-learning in the time series domain.
Time series forecasting is crucial for applications in various domains. Conventional methods often rely on global decomposition into trend, seasonal, and residual components, which become ineffective for real-world series dominated by local, complex, and highly dynamic patterns. Moreover, the high model complexity of such approaches limits their applicability in real-time or resource-constrained environments. In this work, we propose a novel reliability-aware codebook-assisted time series forecasting framework (ReCast) that enables lightweight and robust prediction by exploiting recurring local shapes. ReCast encodes local patterns into discrete embeddings through patch-wise quantization using a learnable codebook, thereby compactly capturing stable regular structures. To compensate for residual variations not preserved by quantization, ReCast employs a dual-path architecture comprising a quantization path for efficient modeling of regular structures and a residual path for reconstructing irregular fluctuations. A central contribution of ReCast is a reliability-aware codebook update strategy, which incrementally refines the codebook via weighted corrections. These correction weights are derived by fusing multiple reliability factors from complementary perspectives by a distributionally robust optimization (DRO) scheme, ensuring adaptability to non-stationarity and robustness to distribution shifts. Extensive experiments demonstrate that ReCast outperforms state-of-the-art (SOTA) models in accuracy, efficiency, and adaptability to distribution shifts.
C3RL: Rethinking the Combination of Channel-independence and Channel-mixing from Representation Learning
PDF ↗Multivariate time series forecasting has drawn increasing attention due to its practical importance. Existing approaches typically adopt either channel-mixing (CM) or channel-independence (CI) strategies. CM strategy can capture inter-variable dependencies but fails to discern variable-specific temporal patterns. CI strategy improves this aspect but fails to fully exploit cross-variable dependencies like CM. Hybrid strategies based on feature fusion offer limited generalization and interpretability. To address these issues, we propose C3RL, a novel representation learning framework that jointly models both CM and CI strategies. Motivated by contrastive learning in computer vision, C3RL treats the inputs of the two strategies as transposed views and builds a siamese network architecture: one strategy serves as the backbone, while the other complements it. By jointly optimizing contrastive and prediction losses with adaptive weighting, C3RL balances representation and forecasting performance. Extensive experiments on seven models show that C3RL boosts the best-case performance rate to 81.4% for models based on CI strategy and to 76.3% for models based on CM strategy, demonstrating strong generalization and effectiveness.
Time series forecasting is fundamental to diverse applications, with recent approaches leverage large vision models (LVMs) to capture temporal patterns through visual representations. We reveal that while vision models enhance forecasting performance, 99% of their parameters are unnecessary for time series tasks. Through cross-modal analysis, we find that time series align with low-level textural features but not high-level semantics, which can impair forecasting accuracy. We propose OccamVTS, a knowledge distillation framework that extracts only the essential 1% of predictive information from LVMs into lightweight networks. Using pre-trained LVMs as privileged teachers, OccamVTS employs pyramid-style feature alignment combined with correlation and feature distillation to transfer beneficial patterns while filtering out semantic noise. Counterintuitively, this aggressive parameter reduction improves accuracy by eliminating overfitting to irrelevant visual features while preserving essential temporal patterns. Extensive experiments across multiple benchmark datasets demonstrate that OccamVTS consistently achieves state-of-the-art performance with only 1% of the original parameters, particularly excelling in few-shot and zero-shot scenarios.