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Deep Learning · Sequential Models, Time series

Luca Masserano, Abdul Fatir Ansari, Boran Han, Xiyuan Zhang, Christos Faloutsos, Michael Mahoney, Andrew Wilson, Youngsuk Park, Syama Sundar Yadav Rangapuram, Danielle Maddix 等

How to best develop foundational models for time series forecasting remains an important open question. Tokenization is a crucial consideration in this effort: what is an effective discrete vocabulary for a real-valued sequential input? To address this question, we develop WaveToken, a wavelet-based tokenizer that allows models to learn complex representations directly in the space of time-localized frequencies. Our method first scales and decomposes the input time series, then thresholds and quantizes the wavelet coefficients, and finally pre-trains an autoregressive model to forecast coefficients for the forecast horizon. By decomposing coarse and fine structures in the inputs, wavelets provide an eloquent and compact language for time series forecasting that simplifies learning. Empirical results on a comprehensive benchmark, including 42 datasets for both in-domain and zero-shot settings, show that WaveToken: i) performs on par or better than recently proposed foundation models for forecasting while using a much smaller vocabulary (1024 tokens), and is competitive with modern deep learning models trained specifically on each dataset; ii) exhibits superior generalization capabilities, achieving the best average rank across all datasets for three complementary metrics; and iii) easily captures complex temporal patterns of practical relevance that are challenging for other recent pre-trained models, including trends, sparse spikes, and non-stationary time series with varying frequencies evolving over time.

Deep Learning · Sequential Models, Time series

Yuchen Wang, Hongjue Zhao, Haohong Lin, Enze Xu, Lifang He, Huajie Shao

This work aims to address the problem of long-term dynamic forecasting in complex environments where data are noisy and irregularly sampled. While recent studies have introduced some methods to improve prediction performance, these approaches still face a significant challenge in handling long-term extrapolation tasks under such complex scenarios. To overcome this challenge, we propose Phy-SSM, a general-purpose framework that integrates partial physics knowledge into state space models (SSMs) for long-term dynamics forecasting in complex environments. Our motivation is that SSMs can effectively capture long-range dependencies in sequential data and model continuous dynamical systems, while the incorporation of physics knowledge improves generalization ability. The key challenge lies in how to seamlessly incorporate partially known physics into SSMs. To achieve this, we decompose partially known system dynamics into known and unknown state matrices, which are integrated into a Phy-SSM unit. To further enhance long-term prediction performance, we introduce a physics state regularization term to make the estimated latent states align with system dynamics. Besides, we theoretically analyze the uniqueness of the solutions for our method. Extensive experiments on three real-world applications, including vehicle motion prediction, drone state prediction, and COVID-19 epidemiology forecasting, demonstrate the superior performance of Phy-SSM over the baselines in both long-term interpolation and extrapolation tasks. The source code will be publicly available upon publication.

Deep Learning · Sequential Models, Time series

Zhangyi Hu, Jiemin Wu, Hua XU, Mingqian Liao, Ninghui Feng, Bo Gao, Songning Lai, Yutao Yue

Irregular Multivariate Time Series (IMTS) forecasting is challenging due to the unaligned nature of multi-channel signals and the prevalence of extensive missing data. Existing methods struggle to capture reliable temporal patterns from such data due to significant missing values. While pre-trained foundation models show potential for addressing these challenges, they are typically designed for Regularly Sampled Time Series (RTS). Motivated by the visual Mask AutoEncoder's (MAE) powerful capability for modeling sparse multi-channel information and its success in RTS forecasting, we propose **VIMTS**, a framework adapting **V**isual MAE for **IMTS** forecasting. To mitigate the effect of missing values, VIMTS first processes IMTS along the timeline into feature patches at equal intervals. These patches are then complemented using learned cross-channel dependencies. Then it leverages visual MAE's capability in handling sparse multichannel data for patch reconstruction, followed by a coarse-to-fine technique to generate precise predictions from focused contexts. In addition, we integrate self-supervised learning for improved IMTS modeling by adapting the visual MAE to IMTS data. Extensive experiments demonstrate VIMTS's superior performance and few-shot capability, advancing the application of visual foundation models in more general time series tasks. Our code is available at https://github.com/WHU-HZY/VIMTS.

Deep Learning · Sequential Models, Time series

Wenjie Wu, Dexuan Huo, Hong Chen

Spiking Neural Networks (SNNs) have demonstrated remarkable potential across many domains, including computer vision and natural language processing, owing to their energy efficiency and biological plausibility. However, their application in long-term prediction tasks remains underexplored, which is primarily due to two critical challenges: (1) current SNN encoding methods are unable to effectively encode long temporal information, leading to increased computational complexity and energy consumption; (2) though Transformer-based models have achieved state-of-the-art accuracy in temporal prediction tasks, the absence of proper positional encoding for spiking self-attention restricts Spiking Transformer from effectively utilizing positional information, resulting in performance degradation. To address these challenges, we introduce an attention-free framework, **Spik**ing **F**ourier Network (**SpikF**), that encodes input sequences in patches and employs an innovative frequency domain selection mechanism to effectively utilize the sequential properties of time-series data. Extensive evaluations on eight well-established long-term prediction datasets demonstrate that SpikF achieves an averaged $1.9\\%$ reduction in Mean Absolute Error (MAE) compared to state-of-the-art models, while lowering total energy consumption by $3.16\times$. Our code is available at https://github.com/WWJ-creator/SpikF.

Theory · Probabilistic Methods

Lorenzo Lucchese, Mikko S. Pakkanen, Almut E. D. Veraart

The expected signature maps a collection of data streams to a lower dimensional representation, with a remarkable property: the resulting feature tensor can fully characterize the data generating distribution. This "model-free"' embedding has been successfully leveraged to build multiple domain-agnostic machine learning (ML) algorithms for time series and sequential data. The convergence results proved in this paper bridge the gap between the expected signature's empirical discrete-time estimator and its theoretical continuous-time value, allowing for a more complete probabilistic interpretation of expected signature-based ML methods. Moreover, when the data generating process is a martingale, we suggest a simple modification of the expected signature estimator with significantly lower mean squared error and empirically demonstrate how it can be effectively applied to improve predictive performance.

Deep Learning · Sequential Models, Time series

Grigory Bartosh, Dmitry Vetrov, Christian Andersson Naesseth

The Latent Stochastic Differential Equation (SDE) is a powerful tool for time series and sequence modeling. However, training Latent SDEs typically relies on adjoint sensitivity methods, which depend on simulation and backpropagation through approximate SDE solutions, which limit scalability. In this work, we propose SDE Matching, a new simulation-free method for training Latent SDEs. Inspired by modern Score- and Flow Matching algorithms for learning generative dynamics, we extend these ideas to the domain of stochastic dynamics for time series modeling, eliminating the need for costly numerical simulations. Our results demonstrate that SDE Matching achieves performance comparable to adjoint sensitivity methods while drastically reducing computational complexity.

Deep Learning · Sequential Models, Time series

Ricardo Buitrago Ruiz, Albert Gu

Recently, recurrent models such as state space models and linear attention have become popular due to their linear complexity in the sequence length. Thanks to their recurrent nature, in principle they can process arbitrarily long sequences, but their performance sometimes drops considerably beyond their training context lengths---i.e. they fail to length generalize.In this work, we provide comprehensive empirical and theoretical analysis to support the \textit{unexplored states hypothesis}, which posits that models fail to length generalize when during training they are only exposed to a limited subset of the distribution of all \textit{attainable} states (i.e. states that would be attained if the recurrence was applied to long sequences).Furthermore, we investigate simple training interventions that aim to increase the coverage of the states that the model is trained on, e.g. by initializing the state with Gaussian noise or with the final state of a different input sequence. With only 500 post-training steps ($\sim 0.1\%$ of the pre-training budget), these interventions enable length generalization for sequences that are orders of magnitude longer than the training context (e.g. $2k\longrightarrow 128k$) and show improved performance in long context tasks, thus presenting a simple and efficient way to enable robust length generalization in general recurrent models.

Deep Learning · Sequential Models, Time series

Qinglong Liu, Cong Xu, Wenhao Jiang, Kaixuan Wang, Lin Ma, Haifeng Li

Real-world time series inherently exhibit significant non-stationarity, posing substantial challenges for forecasting. To address this issue, this paper proposes a novel prediction framework, TimeStacker, designed to overcome the limitations of existing models in capturing the characteristics of non-stationary signals. By employing a unique stacking mechanism, TimeStacker effectively captures global signal features while thoroughly exploring local details. Furthermore, the framework integrates a frequency-based self-attention module, significantly enhancing its feature modeling capabilities. Experimental results demonstrate that TimeStacker achieves outstanding performance across multiple real-world datasets, including those from the energy, finance, and weather domains. It not only delivers superior predictive accuracy but also exhibits remarkable advantages with fewer parameters and higher computational efficiency.

Deep Learning · Sequential Models, Time series

Yihang Wang, Yuying Qiu, Peng Chen, Kai Zhao, Yang Shu, Zhongwen Rao, Lujia Pan, Bin Yang, Chenjuan Guo

With the growing availability of multi-domain time series data, there is an increasing demand for general forecasting models pre-trained on multi-source datasets to support diverse downstream prediction scenarios. Existing time series foundation models primarily focus on scaling up pre-training datasets and model sizes to enhance generalization performance. In this paper, we take a different approach by addressing two critical aspects of general forecasting models: (1) how to derive unified representations from heterogeneous multi-domain time series data, and (2) how to effectively capture domain-specific features to enable adaptive transfer across various downstream scenarios. To address the first aspect, we propose Decomposed Frequency Learning as the pre-training task, which leverages frequency-based masking and reconstruction to decompose coupled semantic information in time series, resulting in unified representations across domains. For the second aspect, we introduce the Time Series Register, which captures domain-specific representations during pre-training and enhances adaptive transferability to downstream tasks. Our model achieves the state-of-the-art forecasting performance on seven real-world benchmarks, demonstrating remarkable few-shot and zero-shot capabilities.

Deep Learning · Sequential Models, Time series

Xingjian Wu, Xiangfei Qiu, Hongfan Gao, Jilin Hu, Bin Yang, Chenjuan Guo

Probabilistic Time Series Forecasting (PTSF) plays a crucial role in decision-making across various fields, including economics, energy, and transportation. Most existing methods excell at short-term forecasting, while overlooking the hurdles of Long-term Probabilistic Time Series Forecasting (LPTSF). As the forecast horizon extends, the inherent nonlinear dynamics have a significant adverse effect on prediction accuracy, and make generative models inefficient by increasing the cost of each iteration. To overcome these limitations, we introduce $K^2$VAE, an efficient VAE-based generative model that leverages a KoopmanNet to transform nonlinear time series into a linear dynamical system, and devises a KalmanNet to refine predictions and model uncertainty in such linear system, which reduces error accumulation in long-term forecasting. Extensive experiments demonstrate that $K^2$VAE outperforms state-of-the-art methods in both short- and long-term PTSF, providing a more efficient and accurate solution.

Deep Learning · Sequential Models, Time series

Zhen Liu, Yicheng Luo, Boyuan Li, Emadeldeen Eldele, Min Wu, Qianli Ma

Shapelets are discriminative subsequences (or shapes) with high interpretability in time series classification. Due to the time-intensive nature of shapelet discovery, existing shapelet-based methods mainly focus on selecting discriminative shapes while discarding others to achieve candidate subsequence sparsification. However, this approach may exclude beneficial shapes and overlook the varying contributions of shapelets to classification performance. To this end, we propose a Soft sparse Shapes (SoftShape) model for efficient time series classification. Our approach mainly introduces soft shape sparsification and soft shape learning blocks. The former transforms shapes into soft representations based on classification contribution scores, merging lower-scored ones into a single shape to retain and differentiate all subsequence information. The latter facilitates intra- and inter-shape temporal pattern learning, improving model efficiency by using sparsified soft shapes as inputs. Specifically, we employ a learnable router to activate a subset of class-specific expert networks for intra-shape pattern learning. Meanwhile, a shared expert network learns inter-shape patterns by converting sparsified shapes into sequences. Extensive experiments show that SoftShape outperforms state-of-the-art methods and produces interpretable results.

General Machine Learning · Sequential, Network, and Time Series Modeling

Ruoxin Yuan, Guanhua Fang

This work introduces Residual TPP, a novel, unified, and lightweight approach for analyzing event stream data. It leverages the strengths of both simple statistical TPPs and expressive neural TPPs to achieve superior performance. Specifically, we propose the Residual Events Decomposition (RED) technique in temporal point processes, which defines a weight function to quantify how well the intensity function captures the event characteristics. The RED serves as a flexible, plug-and-play module that can be integrated with any TPP model in a wide range of tasks. It enables the identification of events for which the intensity function provides a poor fit, referred to as residual events. By combining RED with a Hawkes process, we capture the self-exciting nature of the data and identify residual events. Then an arbitrary neural TPP is employed to take care of residual events. Extensive experimental results demonstrate that Residual TPP consistently achieves state-of-the-art goodness-of-fit and prediction performance in multiple domains and offers significant computational advantages as well.

General Machine Learning · Sequential, Network, and Time Series Modeling

Annie Marsden, Evan Dogariu, Naman Agarwal, Xinyi Chen, Daniel Suo, Elad Hazan

We consider the problem of length generalization in sequence prediction. We define a new metric of performance in this setting – the Asymmetric-Regret– which measures regret against a benchmark predictor with longer context length than available to the learner. We continue by studying this concept through the lens of the spectral filter-ing algorithm. We present a gradient-based learn-ing algorithm that provably achieves length generalization for linear dynamical systems. We conclude with proof-of-concept experiments which are consistent with our theory.

General Machine Learning · Sequential, Network, and Time Series Modeling

Shuai Zhang, Chuan Zhou, Yang Liu, PENG ZHANG, Xixun Lin, Shirui Pan

Anomaly detection in continuous-time event sequences is a crucial task in safety-critical applications. While existing methods primarily focus on developing a superior test statistic, they fail to provide guarantees regarding the false positive rate (FPR), which undermines their reliability in practical deployments. In this paper, we propose CADES (Conformal Anomaly Detection in Event Sequences), a novel test procedure based on conformal inference for the studied task with finite-sample FPR control. Specifically, by using the time-rescaling theorem, we design two powerful non-conformity scores tailored to event sequences, which exhibit complementary sensitivities to different abnormal patterns. CADES combines these scores with Bonferroni correction to leverage their respective strengths and addresses non-identifiability issues of existing methods. Theoretically, we prove the validity of CADES and further provide strong guarantees on calibration-conditional FPR control. Experimental results on synthetic and real-world datasets, covering various types of anomalies, demonstrate that CADES outperforms state-of-the-art methods while maintaining FPR control.

General Machine Learning · Sequential, Network, and Time Series Modeling

Hanru Bai, Weiyang Ding

Neural ordinary differential equations (NODEs) have demonstrated strong capabilities in modeling time series. However, existing NODE- based methods often focus solely on the surface-level dynamics derived from observed states, which limits their ability to capture more complex underlying behaviors. To overcome this challenge, we propose KoNODE, a Koopman-driven NODE framework that explicitly models the evolution of ODE parameters over time to encode deep-level information. KoNODE captures the essential yet simple intrinsic linear dynamics that govern the surface dynamics by employing Koopman operators. Our framework operates at three hierarchical levels: the observed state dynamics, the parameter dynamics, and the Koopman linear dynamics, representing the fundamental driving rules of the state dynamics. The proposed approach offers significant improvements in two critical time series tasks: long-term prediction (enabled by the simple linear dynamics) and generalization to new data (driven by the evolving ODE parameters). We validate KoNODE through experiments on synthetic data from complex dynamic systems and real-world datasets, demonstrating its effectiveness in practical scenarios.

General Machine Learning · Sequential, Network, and Time Series Modeling

Xinyuan Fan, Bufan Li, Chenlei Leng, Weichi Wu

This paper introduces Graphon Attachment Network Models (GAN-M), a novel framework for modeling evolving networks with rich structural dependencies, grounded in graphon theory. GAN-M provides a flexible and interpretable foundation for studying network formation by leveraging graphon functions to define attachment probabilities, thereby combining the strengths of graphons with a temporal perspective. A key contribution of this work is a methodology for learning structural changes in these networks over time. Our approach uses graph counts—frequencies of substructures such as triangles and stars—to capture shifts in network topology. We propose a new statistic designed to learn changes in the resulting piecewise polynomial signals and develop an efficient method for change detection, supported by theoretical guarantees. Numerical experiments demonstrate the effectiveness of our approach across various network settings, highlighting its potential for dynamic network analysis.

General Machine Learning · Sequential, Network, and Time Series Modeling

Quan Zhou, Changhua Pei, Fei Sun, HanJing, Zhengwei Gao, haiming zhang, Gaogang Xie, Dan Pei, Jianhui LI

Time series anomaly detection (TSAD) underpins real-time monitoring in cloud services and web systems, allowing rapid identification of anomalies to prevent costly failures. Most TSAD methods driven by forecasting models tend to overfit by emphasizing minor fluctuations. Our analysis reveals that effective TSAD should focus on modeling "normal" behavior through smooth local patterns. To achieve this, we reformulate time series modeling as approximating the series with smooth univariate functions. The local smoothness of each univariate function ensures that the fitted time series remains resilient against local disturbances. However, a direct KAN implementation proves susceptible to these disturbances due to the inherently localized characteristics of B-spline functions. We thus propose KAN-AD, replacing B-splines with truncated Fourier expansions and introducing a novel lightweight learning mechanism that emphasizes global patterns while staying robust to local disturbances. On four popular TSAD benchmarks, KAN-AD achieves an average 15% improvement in detection accuracy (with peaks exceeding 27%) over state-of-the-art baselines. Remarkably, it requires fewer than 1,000 trainable parameters, resulting in a 50% faster inference speed compared to the original KAN, demonstrating the approach's efficiency and practical viability.

Deep Learning · Sequential Models, Time series

Wenzhe Niu, Zongxia Xie, Yanru Sun, Wei He, Man Xu, Chao Hao

Recent research has shown an increasing interest in utilizing pre-trained large language models (LLMs) for a variety of time series applications. However, there are three main challenges when using LLMs as foundational models for time series forecasting: (1) Cross-domain generalization. (2) Cross-modality alignment. (3) Error accumulation in autoregressive frameworks. To address these challenges, we proposed **LangTime**, a **lan**guage-**g**uided unified model for **time** series forecasting that incorporates cross-domain pre-training with reinforcement learning-based fine-tuning. Specifically, LangTime constructs Temporal Comprehension Prompts (TCPs), which include dataset-wise and channel-wise instructions, to facilitate domain adaptation and condense time series into a single token, enabling LLMs to understand better and align temporal data. To improve autoregressive forecasting, we introduce TimePPO, a reinforcement learning-based fine-tuning algorithm. TimePPO mitigates error accumulation by leveraging a multidimensional rewards function tailored for time series and a repeat-based value estimation strategy. Extensive experiments demonstrate that LangTime achieves state-of-the-art cross-domain forecasting performance, while TimePPO fine-tuning effectively enhances the stability and accuracy of autoregressive forecasting.

Deep Learning · Sequential Models, Time series

Yihang Wang, Yuying Qiu, Peng Chen, Yang Shu, Zhongwen Rao, Lujia Pan, Bin Yang, Chenjuan Guo

Existing works on general time series forecasting build foundation models with heavy model parameters through large-scale multi-source pretraining. These models achieve superior generalization ability across various datasets at the cost of significant computational burdens and limitations in resource-constrained scenarios. This paper introduces LightGTS, a lightweight general time series forecasting model designed from the perspective of consistent periodical modeling. To handle diverse scales and intrinsic periods in multi-source pre-training, we introduce Periodical Tokenization, which extracts consistent periodic patterns across different datasets with varying scales. To better utilize the periodicity in the decoding process, we further introduce Periodical Parallel Decoding, which leverage historical tokens to improve forecasting. Based on the two techniques above which fully leverage the inductive bias of periods inherent in time series, LightGTS uses a lightweight model to achieve outstanding performance on general time series forecasting. It achieves state-of-the-art forecasting performance on 9 real-world benchmarks in both zero-shot and full-shot setting with much better efficiency compared with existing time series foundation models

Social Aspects · Privacy

Jan Schuchardt, Mina Dalirrooyfard, Jed Guzelkabaagac, Anderson Schneider, Yuriy Nevmyvaka, Stephan Günnemann

Many forms of sensitive data, such as web traffic, mobility data, or hospital occupancy, are inherently sequential. The standard method for training machine learning models while ensuring privacy for units of sensitive information, such as individual hospital visits, is differentially private stochastic gradient descent (DP-SGD). However, we observe in this work that the formal guarantees of DP-SGD are incompatible with time series specific tasks like forecasting, since they rely on the *privacy amplification* attained by training on small, unstructured batches sampled from an unstructured dataset. In contrast, batches for forecasting are generated by (1) sampling sequentially structured time series from a dataset, (2) sampling contiguous subsequences from these series, and (3) partitioning them into context and ground-truth forecast windows. We theoretically analyze the privacy amplification attained by this *structured subsampling* to enable the training of forecasting models with sound and tight event- and user-level privacy guarantees. Towards more private models, we additionally prove how data augmentation amplifies privacy in self-supervised training of sequence models. Our empirical evaluation demonstrates that amplification by structured subsampling enables the training of forecasting models with strong formal privacy guarantees.