Long-Term Time Series Forecasting

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Long-term time series forecasting uses past observations to predict many steps into the future, supporting planning when decisions must be made well ahead of time. Methods range from trend and seasonal decomposition, linear models, and patch-based mixers to Transformers, state-space models, and graph networks for linked locations or variables. Some approaches add physical dynamics, external signals, or retrieval of similar historical cases, while probabilistic models represent uncertainty as well as a single forecast. A central challenge is keeping forecasts stable under distribution shifts and accumulated errors; strong pointwise accuracy can still come at the cost of realistic uncertainty, structural fidelity, or efficiency.

Latest papers

115 papers20 papers in the last four weeks, up 82% on the four weeks before. 0.1% of all new papers.

Apr 1, 2026

DySCo: Dynamic Semantic Compression for Effective Long-term Time Series Forecasting

Time series forecasting (TSF) is critical across domains such as finance, meteorology, and energy. While extending the lookback window theoretically provides richer historical context, in practice, it often introduces irrelevant noise and computational redundancy, preventing models from effectively capturing complex long-term dependencies. To address these challenges, we propose a Dynamic Semantic Compression (DySCo) framework. Unlike traditional methods that rely on fixed heuristics, DySCo introduces an Entropy-Guided Dynamic Sampling (EGDS) mechanism to autonomously identify and retain high-entropy segments while compressing redundant trends. Furthermore, we incorporate a Hierarchical Frequency-Enhanced Decomposition (HFED) strategy to separate high-frequency anomalies from low-frequency patterns, ensuring that critical details are preserved during sparse sampling. Finally, a Cross-Scale Interaction Mixer(CSIM) is designed to dynamically fuse global contexts with local representations, replacing simple linear aggregation. Experimental results demonstrate that DySCo serves as a universal plug-and-play module, significantly enhancing the ability of mainstream models to capture long-term correlations with reduced computational cost.

Xiang Ao, Yinyu Tan, Mengru Chen

School of Software Engineering, Beijing Jiaotong University, Beijing, China

Long-Term Time Series Forecasting Multiscale Time-Series Modeling Time Series Decomposition

Mar 31, 2026

Meteorology-Driven GPT4AP: A Multi-Task Forecasting LLM for Atmospheric Air Pollution in Data-Scarce Settings

Accurate forecasting of air pollution is important for environmental monitoring and policy support, yet data-driven models often suffer from limited generalization in regions with sparse observations. This paper presents Meteorology-Driven GPT for Air Pollution (GPT4AP), a parameter-efficient multi-task forecasting framework based on a pre-trained GPT-2 backbone and Gaussian rank-stabilized low-rank adaptation (rsLoRA). The model freezes the self-attention and feed-forward layers and adapts lightweight positional and output modules, substantially reducing the number of trainable parameters. GPT4AP is evaluated on six real-world air quality monitoring datasets under few-shot, zero-shot, and long-term forecasting settings. In the few-shot regime using 10% of the training data, GPT4AP achieves an average MSE/MAE of 0.686/0.442, outperforming DLinear (0.728/0.530) and ETSformer (0.734/0.505). In zero-shot cross-station transfer, the proposed model attains an average MSE/MAE of 0.529/0.403, demonstrating improved generalization compared with existing baselines. In long-term forecasting with full training data, GPT4AP remains competitive, achieving an average MAE of 0.429, while specialized time-series models show slightly lower errors. These results indicate that GPT4AP provides a data-efficient forecasting approach that performs robustly under limited supervision and domain shift, while maintaining competitive accuracy in data-rich settings.

Prasanjit Dey, Soumyabrata Dev, Bianca Schoen-Phelan

ADAPT SFI Research Centre, School of Computer Science, Technological University Dublin, Ireland · ADAPT SFI Research Centre, School of Computer Science, University College Dublin · School of Computer Science, Technological University Dublin, Ireland

Air Quality Prediction LLM-Based Time Series Forecasting Multivariate Time Series Forecasting Long-Term Time Series Forecasting Zero-Shot Time Series Forecasting Parameter-Efficient Fine-Tuning

Mar 26, 2026

DSO: Dual-Scale Neural Operators for Stable Long-term Fluid Dynamics Forecasting

Long-term fluid dynamics forecasting is a critically important problem in science and engineering. While neural operators have emerged as a promising paradigm for modeling systems governed by partial differential equations (PDEs), they often struggle with long-term stability and precision. We identify two fundamental failure modes in existing architectures: (1) local detail blurring, where fine-scale structures such as vortex cores and sharp gradients are progressively smoothed, and (2) global trend deviation, where the overall motion trajectory drifts from the ground truth during extended rollouts. We argue that these failures arise because existing neural operators treat local and global information processing uniformly, despite their inherently different evolution characteristics in physical systems. To bridge this gap, we propose the Dual-Scale Neural Operator (DSO), which explicitly decouples information processing into two complementary modules: depthwise separable convolutions for fine-grained local feature extraction and an MLP-Mixer for long-range global aggregation. Through numerical experiments on vortex dynamics, we demonstrate that nearby perturbations primarily affect local vortex structure while distant perturbations influence global motion trends, providing empirical validation for our design choice. Extensive experiments on turbulent flow benchmarks show that DSO achieves state-of-the-art accuracy while maintaining robust long-term stability, reducing prediction error by over 88% compared to existing neural operators.

Huanshuo Dong, Hao Wu, Hong Wang +2

University of Science and Technology of China, Hefei, China · Tsinghua University, Beijing, China · Institute of Automation, Chinese Academy of Sciences, Beijing, China +1

PDE Operator Learning Neural Operators Computational Fluid Dynamics Long-Term Time Series Forecasting

Mar 18, 2026

TimeAPN: Adaptive Amplitude-Phase Non-Stationarity Normalization for Time Series Forecasting

Non-stationarity is a fundamental challenge in multivariate long-term time series forecasting, often manifested as rapid changes in amplitude and phase. These variations lead to severe distribution shifts and consequently degrade predictive performance. Existing normalization-based methods primarily rely on first- and second-order statistics, implicitly assuming that distributions evolve smoothly and overlooking fine-grained temporal dynamics. To address these limitations, we propose TimeAPN, an Adaptive Amplitude-Phase Non-Stationarity Normalization framework that explicitly models and predicts non-stationary factors from both the time and frequency domains. Specifically, TimeAPN first models the mean sequence jointly in the time and frequency domains, and then forecasts its evolution over future horizons. Meanwhile, phase information is extracted in the frequency domain, and the phase discrepancy between the predicted and ground-truth future sequences is explicitly modeled to capture temporal misalignment. Furthermore, TimeAPN incorporates amplitude information into an adaptive normalization mechanism, enabling the model to effectively account for abrupt fluctuations in signal energy. The predicted non-stationary factors are subsequently integrated with the backbone forecasting outputs through a collaborative de-normalization process to reconstruct the final non-stationary time series. The proposed framework is model-agnostic and can be seamlessly integrated with various forecasting backbones. Extensive experiments on seven real-world multivariate datasets demonstrate that TimeAPN consistently improves long-term forecasting accuracy across multiple prediction horizons and outperforms state-of-the-art reversible normalization methods.

Yue Hu, Jialiang Tang, Siwei Yu +3

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China · School of Computer Science and Engineering, Nanjing University of Science and Technology, China · Lee Kong Chian School of Medicine, Nanyang Technological University, Singapore +2

Non-Stationary Time Series Forecasting Multivariate Time Series Forecasting Long-Term Time Series Forecasting

Mar 12, 2026

VisiFold: Long-Term Traffic Forecasting via Temporal Folding Graph and Node Visibility

Traffic forecasting is a cornerstone of intelligent transportation systems. While existing research has made significant progress in short-term prediction, long-term forecasting remains a largely uncharted and challenging frontier. Extending the prediction horizon intensifies two critical issues: escalating computational resource consumption and increasingly complex spatial-temporal dependencies. Current approaches, which rely on spatial-temporal graphs and process temporal and spatial dimensions separately, suffer from snapshot-stacking inflation and cross-step fragmentation. To overcome these limitations, we propose VisiFold. Our framework introduces a novel temporal folding graph that consolidates a sequence of temporal snapshots into a single graph. Furthermore, we present a node visibility mechanism that incorporates node-level masking and subgraph sampling to overcome the computational bottleneck imposed by large node counts. Extensive experiments show that VisiFold not only drastically reduces resource consumption but also outperforms existing baselines in long-term forecasting tasks. Remarkably, even with a high mask ratio of 80%, VisiFold maintains its performance advantage. By effectively breaking the resource constraints in both temporal and spatial dimensions, our work paves the way for more realistic long-term traffic forecasting. The code is available at~ https://github.com/PlanckChang/VisiFold.

Zhiwei Zhang, Xinyi Du, Weihao Wang +2

Key Laboratory of Big Data & Artificial Intelligence in Transportation (Beijing Jiaotong University), Ministry of Education · School of Computer Science and Technology, Beijing Jiaotong University, Beijing 100044, China · Beijing Normal University, Beijing, China

Traffic Forecasting Long-Term Time Series Forecasting Spatiotemporal GNNs

Mar 11, 2026

Spatio-Temporal Forecasting of Retaining Wall Deformation: Mitigating Error Accumulation via Multi-Resolution ConvLSTM Stacking Ensemble

This study proposes a multi-resolution Convolutional Long Short-Term Memory (ConvLSTM) ensemble framework that leverages diverse temporal input resolutions to mitigate error accumulation and improve long-horizon forecasting of retaining-structure behavior during staged excavation. An extensive database of lateral wall displacement responses was generated through PLAXIS2D simulations incorporating five-layered soil stratigraphy, two excavation depths (14 and 20 m), and stochastically varied geotechnical and structural parameters, yielding 2,000 time-series deflection profiles. Three ConvLSTM models trained at different input resolutions were integrated using a fully connected neural network meta-learner to construct the ensemble model. Validation using both numerical results and field measurements demonstrated that the ensemble approach consistently outperformed the standalone ConvLSTM models, particularly in long-term multi-step prediction, exhibiting reduced error propagation and improved generalization. These findings underscore the potential of multi-resolution ensemble strategies that jointly exploit diverse temporal input scales to enhance predictive stability and accuracy in AI-driven geotechnical forecasting.

Jihoon Kim, Heejung Youn

Department of Civil and Environmental Engineering, Hongik University, Seoul, Republic of Korea · Department of Civil and Environmental Engineering, Hongik University, 94, Wausan-ro, Mapo-gu, Seoul, 04066, Republic of Korea

Spatiotemporal Forecasting Long-Term Time Series Forecasting Convolutional LSTM Deep Ensembles Structural Health Monitoring

Mar 10, 2026

Rethinking Adam for Time Series Forecasting: A Simple Heuristic to Improve Optimization under Distribution Shifts

Time-series forecasting often faces challenges from non-stationarity, particularly distributional drift, where the data distribution evolves over time. This dynamic behavior can undermine the effectiveness of adaptive optimizers, such as Adam, which are typically designed for stationary objectives. In this paper, we revisit Adam in the context of non-stationary forecasting and identify that its second-order bias correction limits responsiveness to shifting loss landscapes. To address this, we propose TS_Adam, a lightweight variant that removes the second-order correction from the learning rate computation. This simple modification improves adaptability to distributional drift while preserving the optimizer core structure and requiring no additional hyperparameters. TS_Adam integrates easily into existing models and consistently improves performance across long- and short-term forecasting tasks. On the ETT datasets with the MICN model, it achieves an average reduction of 12.8% in MSE and 5.7% in MAE compared to Adam. These results underscore the practicality and versatility of TS_Adam as an effective optimization strategy for real-world forecasting scenarios involving non-stationary data. Code is available at: https://github.com/DD-459-1/TS_Adam.

Yuze Dong, Jinsong Wu

School of Artificial Intelligence, Guilin University of Electronic Technology 1 Jinji Road, Guilin, 541004, Guangxi, China · Department of Electrical Engineering, University of Chile Av. Tupper 2007, Santiago, 8370451, Chile

Adaptive Gradient Methods Non-Stationary Time Series Forecasting Distribution Shift Robustness Long-Term Time Series Forecasting

Mar 9, 2026

Are We Winning the Wrong Game? Revisiting Evaluation Practices for Long-Term Time Series Forecasting

Long-term time series forecasting (LTSF) is widely recognized as a central challenge in data mining and machine learning. LTSF has increasingly evolved into a benchmark-driven ''GAME,'' where models are ranked, compared, and declared state-of-the-art based primarily on marginal reductions in aggregated pointwise error metrics such as MSE and MAE. Across a small set of canonical datasets and fixed forecasting horizons, progress is communicated through leaderboard-style tables in which lower numerical scores define success. In this GAME, what is measured becomes what is optimized, and incremental error reduction becomes the dominant currency of advancement. We argue that this metric-centric regime is not merely incomplete, but structurally misaligned with the broader objectives of forecasting. In real-world settings, forecasting often prioritizes preserving temporal structure, trend stability, seasonal coherence, robustness to regime shifts, and supporting downstream decision processes. Optimizing aggregate pointwise error does not necessarily imply modeling these structural properties. As a result, leaderboard improvement may increasingly reflect specialization in benchmark configurations rather than a deeper understanding of temporal dynamics. This paper revisits LTSF evaluation as a foundational question in data science: what does it mean to measure forecasting progress? We propose a multi-dimensional evaluation perspective that integrates statistical fidelity, structural coherence, and decision-level relevance. By challenging the current metric monoculture, we aim to redirect attention from winning benchmark tables toward advancing meaningful, context-aware forecasting.

Thanapol Phungtua-eng, Yoshitaka Yamamoto

Shizuoka University, Shizuoka, Japan

Time Series Forecasting Long-Term Time Series Forecasting Forecasting Benchmarks

Mar 7, 2026

N-Tree Diffusion for Long-Horizon Wildfire Risk Forecasting

Long-horizon wildfire risk forecasting requires generating probabilistic spatial fields under sparse event supervision while maintaining computational efficiency across multiple prediction horizons. Extending diffusion models to multi-step forecasting typically repeats the denoising process independently for each horizon, leading to redundant computation. We introduce N-Tree Diffusion (NT-Diffusion), a hierarchical diffusion model designed for long-horizon wildfire risk forecasting. Fire occurrences are represented as continuous Fire Risk Maps (FRMs), which provide a smoothed spatial risk field suitable for probabilistic modeling. Instead of running separate diffusion trajectories for each predicted timestamp, NT-Diffusion shares early denoising stages and branches at later levels, allowing horizon-specific refinement while reducing redundant sampling. We evaluate the proposed framework on a newly collected real-world wildfire dataset constructed for long-horizon probabilistic prediction. Results indicate that NT-Diffusion achieves consistent accuracy improvements and reduced inference cost compared to baseline forecasting approaches.

Yucheng Xing, Xin Wang

Department of Electrical and Computer Engineering, Stony Brook University, Stony Brook, NY 11794, USA

Wildfire Forecasting Diffusion Models Spatiotemporal Forecasting Probabilistic Forecasting Long-Term Time Series Forecasting

Mar 6, 2026

UniMamba: A Unified Spatial-Temporal Modeling Framework with State-Space and Attention Integration

Multivariate time series forecasting is fundamental to numerous domains such as energy, finance, and environmental monitoring, where complex temporal dependencies and cross-variable interactions pose enduring challenges. Existing Transformer-based methods capture temporal correlations through attention mechanisms but suffer from quadratic computational cost, while state-space models like Mamba achieve efficient long-context modeling yet lack explicit temporal pattern recognition. Therefore we introduce UniMamba, a unified spatial-temporal forecasting framework that integrates efficient state-space dynamics with attention-based dependency learning. UniMamba employs a Mamba Variate-Channel Encoding Layer enhanced with FFT-Laplace Transform and TCN to capture global temporal dependencies, and a Spatial Temporal Attention Layer to jointly model inter-variate correlations and temporal evolution. A Feedforward Temporal Dynamics Layer further fuses continuous and discrete contexts for accurate forecasting. Comprehensive experiments on eight public benchmark datasets demonstrate that UniMamba consistently outperforms state-of-the-art forecasting models in both forecasting accuracy and computational efficiency, establishing a scalable and robust solution for long-sequence multivariate time-series prediction.

Xingsheng Chen, Xianpei Mu, Deyu Yi +6

School of Computing and Data Science, The University of Hong Kong, Hong Kong, China · School of Information Engineering, Beijing Institute of Graphic Communication, Beijing, China · Innovation Engineering College, Macau University of Science and Technology, Macau, China +1

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Selective SSMs Spatiotemporal Forecasting

Mar 5, 2026

Towards Efficient and Stable Ocean State Forecasting: A Continuous-Time Koopman Approach

We investigate the Continuous-Time Koopman Autoencoder (CT-KAE) as a lightweight surrogate model for long-horizon ocean state forecasting in a two-layer quasi-geostrophic (QG) system. By projecting nonlinear dynamics into a latent space governed by a linear ordinary differential equation, the model enforces structured and interpretable temporal evolution while enabling temporally resolution-invariant forecasting via a matrix exponential formulation. Across 2083-day rollouts, CT-KAE exhibits bounded error growth and stable large-scale statistics, in contrast to autoregressive Transformer baselines which exhibit gradual error amplification and energy drift over long rollouts. While fine-scale turbulent structures are partially dissipated, bulk energy spectra, enstrophy evolution, and autocorrelation structure remain consistent over long horizons. The model achieves orders-of-magnitude faster inference compared to the numerical solver, suggesting that continuous-time Koopman surrogates offer a promising backbone for efficient and stable physical-machine learning climate models.

Rares Grozavescu, Pengyu Zhang, Mark Girolami +1

Department of Engineering, University of Cambridge, Cambridge, UK · Alan Turing Institute, London, UK · INRIA, Paris, France

Spatiotemporal Forecasting PDE Surrogate Modeling Koopman Operator Learning Long-Term Time Series Forecasting Climate Modeling

Mar 5, 2026

Timer-S1: A Billion-Scale Time Series Foundation Model with Serial Scaling

We introduce Timer-S1, a strong Mixture-of-Experts (MoE) time series foundation model with 8.3B total parameters, 0.75B activated parameters for each token, and a context length of 11.5K. To overcome the scalability bottleneck in existing pre-trained time series foundation models, we perform Serial Scaling in three dimensions: model architecture, dataset, and training pipeline. Timer-S1 integrates sparse TimeMoE blocks and generic TimeSTP blocks for Serial-Token Prediction (STP), a generic training objective that adheres to the serial nature of forecasting. The proposed paradigm introduces serial computations to improve long-term predictions while avoiding costly rolling-style inference and pronounced error accumulation in the standard next-token prediction. Pursuing a high-quality and unbiased training dataset, we curate TimeBench, a corpus with one trillion time points, and apply meticulous data augmentation to mitigate predictive bias. We further pioneer a post-training stage, including continued pre-training and long-context extension, to enhance short-term and long-context performance. Evaluated on the large-scale GIFT-Eval leaderboard, Timer-S1 achieves state-of-the-art forecasting performance, attaining the best MASE and CRPS scores as a pre-trained model. Timer-S1 is released to facilitate further research.

Yong Liu, Xingjian Su, Shiyu Wang +7

Tsinghua University · ByteDance

Time Series Forecasting Time-Series Foundation Models Long-Term Time Series Forecasting Sequence Modeling Mixture-of-Experts Models

Mar 4, 2026

XLinear: Frequency-Enhanced MLP with CrossFilter for Robust Long-Range Forecasting

Time series forecasters are widely used across various domains. Among them, MLP (multi-layer perceptron)-based forecasters have been proven to be more robust to noise compared to Transformer-based forecasters. However, MLP struggles to capture complex features, resulting in limitations on capturing long-range dependencies. To address this challenge, we propose XLinear, an MLP-based forecaster for long-range forecasting. Firstly, we decompose the time series into trend and seasonal components. For the trend component which contains long-range characteristics, we design Enhanced Frequency Attention (EFA) to capture long-term dependencies by leveraging frequency-domain operations. Additionally, a CrossFilter Block is proposed for the seasonal component to maintain the model's robustness to noise, avoiding the problems of low robustness often caused by attention mechanisms. Experimental results demonstrate that XLinear achieves state-of-the-art performance on test datasets. While keeping the lightweight architecture and high robustness of MLP-based models, our forecaster outperforms other MLP-based forecasters in capturing long-range dependencies.

Xiang Ao

School of Software Engineering, Beijing Jiaotong University, Beijing, China

Frequency-Domain Time Series Modeling Robust Time Series Forecasting Time Series Forecasting Long-Term Time Series Forecasting

Feb 20, 2026

Sub-City Real Estate Price Index Forecasting at Weekly Horizons Using Satellite Radar and News Sentiment

Reliable real estate price indicators are typically published at city level and low frequency, limiting their use for neighborhood-scale monitoring and long-horizon planning. We study whether sub-city price indices can be forecasted at weekly frequency by combining physical development signals from satellite radar with market narratives from news text. Using over 350,000 transactions from Dubai Land Department (2015-2025), we construct weekly price indices for 19 sub-city regions and evaluate forecasts from 2 to 34 weeks ahead. Our framework fuses regional transaction history with Sentinel-1 SAR backscatter, news sentiment combining lexical tone and semantic embeddings, and macroeconomic context. Results are strongly horizon dependent: at horizons up to 10 weeks, price history alone matches multimodal configurations, but beyond 14 weeks sentiment and SAR become critical. At long horizons (26-34 weeks), the full multimodal model reduces mean absolute error from 4.48 to 2.93 (35% reduction), with gains statistically significant across regions. Nonparametric learners consistently outperform deep architectures in this data regime. These findings establish benchmarks for weekly sub-city index forecasting and demonstrate that remote sensing and news sentiment materially improve predictability at strategically relevant horizons.

Baris Arat, Hasan Fehmi Ates, Emre Sefer

Ozyegin University, Istanbul, Turkey

Time Series Forecasting Long-Term Time Series Forecasting Multivariate Time Series Forecasting

Feb 20, 2026

Stable Long-Horizon Spatiotemporal Prediction on Meshes Using Latent Multiscale Recurrent Graph Neural Networks

Accurate long-horizon prediction of spatiotemporal fields on complex geometries is a fundamental challenge in scientific machine learning, with applications such as additive manufacturing where temperature histories govern defect formation and mechanical properties. High-fidelity simulations are accurate but computationally costly, and despite recent advances, machine learning methods remain challenged by long-horizon temperature and gradient prediction. We propose a deep learning framework for predicting full temperature histories directly on meshes, conditioned on geometry and process parameters, while maintaining stability over thousands of time steps and generalizing across heterogeneous geometries. The framework adopts a temporal multiscale architecture composed of two coupled models operating at complementary time scales. Both models rely on a latent recurrent graph neural network to capture spatiotemporal dynamics on meshes, while a variational graph autoencoder provides a compact latent representation that reduces memory usage and improves training stability. Experiments on simulated powder bed fusion data demonstrate accurate and temporally stable long-horizon predictions across diverse geometries, outperforming existing baseline. Although evaluated in two dimensions, the framework is general and extensible to physics-driven systems with multiscale dynamics and to three-dimensional geometries.

Lionel Salesses, Larbi Arbaoui, Tariq Benamara +2

Cenaero, Gosselies, Belgium

Spatiotemporal Forecasting Neural Surrogate Modeling Spatiotemporal GNNs Long-Term Time Series Forecasting Multiscale Time-Series Modeling

Feb 18, 2026

HPMixer: Hierarchical Patching for Multivariate Time Series Forecasting

In long-term multivariate time series forecasting, effectively capturing both periodic patterns and residual dynamics is essential. To address this within standard deep learning benchmark settings, we propose the Hierarchical Patching Mixer (HPMixer), which models periodicity and residuals in a decoupled yet complementary manner. The periodic component utilizes a learnable cycle module [7] enhanced with a nonlinear channel-wise MLP for greater expressiveness. The residual component is processed through a Learnable Stationary Wavelet Transform (LSWT) to extract stable, shift-invariant frequency-domain representations. Subsequently, a channel-mixing encoder models explicit inter-channel dependencies, while a two-level non-overlapping hierarchical patching mechanism captures coarse- and fine-scale residual variations. By integrating decoupled periodicity modeling with structured, multi-scale residual learning, HPMixer provides an effective framework. Extensive experiments on standard multivariate benchmarks demonstrate that HPMixer achieves competitive or state-of-the-art performance compared to recent baselines.

Jung Min Choi, Vijaya Krishna Yalavarthi, Lars Schmidt-Thieme

ISMLL, University of Hildesheim

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Multiscale Time-Series Modeling Residual Learning

Feb 18, 2026

SEMixer: Semantics Enhanced MLP-Mixer for Multiscale Mixing and Long-term Time Series Forecasting

Modeling multiscale patterns is crucial for long-term time series forecasting (TSF). However, redundancy and noise in time series, together with semantic gaps between non-adjacent scales, make the efficient alignment and integration of multi-scale temporal dependencies challenging. To address this, we propose SEMixer, a lightweight multiscale model designed for long-term TSF. SEMixer features two key components: a Random Attention Mechanism (RAM) and a Multiscale Progressive Mixing Chain (MPMC). RAM captures diverse time-patch interactions during training and aggregates them via dropout ensemble at inference, enhancing patch-level semantics and enabling MLP-Mixer to better model multi-scale dependencies. MPMC further stacks RAM and MLP-Mixer in a memory-efficient manner, achieving more effective temporal mixing. It addresses semantic gaps across scales and facilitates better multiscale modeling and forecasting performance. We not only validate the effectiveness of SEMixer on 10 public datasets, but also on the 2025 CCF AlOps Challenge based on 21GB real wireless network data, where SEMixer achieves third place. The code is available at the link https://github.com/Meteor-Stars/SEMixer.

Xu Zhang, Qitong Wang, Peng Wang +1

Shanghai Key Laboratory of Data Science, College of Computer Science and Artificial Intelligence, Fudan University, Shanghai, China · Harvard University, Cambridge, Massachusetts, United States

Long-Term Time Series Forecasting Multi-Scale Feature Fusion Multiscale Time-Series Modeling

Feb 18, 2026

Deep TPC: Temporal-Prior Conditioning for Time Series Forecasting

LLM-for-time series (TS) methods typically treat time shallowly, injecting positional or prompt-based cues once at the input of a largely frozen decoder, which limits temporal reasoning as this information degrades through the layers. We introduce Temporal-Prior Conditioning (TPC), which elevates time to a first-class modality that conditions the model at multiple depths. TPC attaches a small set of learnable time series tokens to the patch stream; at selected layers these tokens cross-attend to temporal embeddings derived from compact, human-readable temporal descriptors encoded by the same frozen LLM, then feed temporal context back via self-attention. This disentangles time series signal and temporal information while maintaining a low parameter budget. We show that by training only the cross-attention modules and explicitly disentangling time series signal and temporal information, TPC consistently outperforms both full fine-tuning and shallow conditioning strategies, achieving state-of-the-art performance in long-term forecasting across diverse datasets. Code available at: https://github.com/fil-mp/Deep_tpc

Filippos Bellos, NaveenJohn Premkumar, Yannis Avrithis +2

University of Michigan, Ann Arbor, MI, USA · Independent Scientist · Capital One, McLean, VA, USA

LLM-Based Time Series Forecasting Long-Term Time Series Forecasting

Dec 27, 2025

Robust Residual Correction via Selective Deployment for Time Series Forecasting

Post-hoc residual correction can reduce forecasting error, but always-on updates often fail to generalize: when residual signals are weak or noisy, expressive correctors can overfit and even increase the overall forecasting errors. We introduce CRC, a robust residual correction framework that couples a hybrid corrector with a validation-calibrated selective deployment rule. CRC proposes corrections via a conservative ridge "floor" plus conditional nonlinear refinement, but deploys them only when held-out evidence indicates net error reduction over a frozen baseline forecaster. CRC treats when to revise a forecast as a validation-calibrated deployment decision, not only how to fit residuals. Across standard long-sequence benchmarks and multiple backbones, CRC improves horizon-averaged MSE and MAE in most dataset-backbone pairs, with particularly large gains on complex datasets (e.g., over 20% relative MSE reduction on Traffic). We additionally report the non-degradation rate (NDR) as a stability diagnostic of harmful updates and ablate the validation-calibrated "firewall" that stabilizes MSE gains.

Jianxiang Xie, Yuncheng Hua, Mingyue Cheng +2

School of Computer Science and Engineering, University of New South Wales, Sydney, Australia · State Key Laboratory of Cognitive Intelligence, School of Computer Science and Technology, University of Science and Technology of China, Hefei, China · The Hong Kong University of Science and Technology (Guangzhou)

Robust Time Series Forecasting Residual Learning Long-Term Time Series Forecasting Selective Prediction

Nov 17, 2025

Naga: Vedic Encoding for Deep State Space Models

This paper presents Naga, a deep State Space Model (SSM) encoding approach inspired by structural concepts from Vedic mathematics. The proposed method introduces a bidirectional representation for time series by jointly processing forward and time-reversed input sequences. These representations are then combined through an element-wise (Hadamard) interaction, resulting in a Vedic-inspired encoding that enhances the model's ability to capture temporal dependencies across distant time steps. We evaluate Naga on multiple long-term time series forecasting (LTSF) benchmarks, including ETTh1, ETTh2, ETTm1, ETTm2, Weather, Traffic, and ILI. The experimental results show that Naga outperforms 28 current state of the art models and demonstrates improved efficiency compared to existing deep SSM-based approaches. The findings suggest that incorporating structured, Vedic-inspired decomposition can provide an interpretable and computationally efficient alternative for long-range sequence modeling.

Melanie Schaller, Nick Janssen, Bodo Rosenhahn

Sequence Modeling State Space Models Long-Term Time Series Forecasting Multivariate Time Series Forecasting

Aug 24, 2025

GateTS: Versatile and Efficient Forecasting via Attention-Inspired routed Mixture-of-Experts

Accurate univariate forecasting remains a pressing need in real-world systems, such as energy markets, hydrology, retail demand, and IoT monitoring, where signals are often intermittent and horizons span both short- and long-term. While transformers and Mixture-of-Experts (MoE) architectures are increasingly favored for time-series forecasting, a key gap persists: MoE models typically require complicated training with both the main forecasting loss and auxiliary load-balancing losses, along with careful routing/temperature tuning, which hinders practical adoption. In this paper, we propose a model architecture that simplifies the training process for univariate time series forecasting and effectively addresses both long- and short-term horizons, including intermittent patterns. Our approach combines sparse MoE computation with a novel attention-inspired gating mechanism that replaces the traditional one-layer softmax router. Through extensive empirical evaluation, we demonstrate that our gating design naturally promotes balanced expert utilization and achieves superior predictive accuracy without requiring the auxiliary load-balancing losses typically used in classical MoE implementations. The model achieves better performance while utilizing only a fraction of the parameters required by state-of-the-art transformer models, such as PatchTST. Furthermore, experiments across diverse datasets confirm that our MoE architecture with the proposed gating mechanism is more computationally efficient than LSTM for both long- and short-term forecasting, enabling cost-effective inference. These results highlight the potential of our approach for practical time-series forecasting applications where both accuracy and computational efficiency are critical.

Kyrylo Yemets, Mykola Lukashchuk, Ivan Izonin

Department of Artificial Intelligence, Lviv Polytechnic National University, Lviv, 79005, Ukraine · Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, 5612 AE, Netherlands · The Bartlett School of Sustainable Construction, University College London, London, WC1E 7HB, United Kingdom

Time Series Forecasting Mixture-of-Experts Models Long-Term Time Series Forecasting Gated Attention

Jun 4, 2025

Temporal horizons in forecasting: a performance-learnability trade-off

When training autoregressive models to forecast dynamical systems, a critical question arises: how far into the future should the model be trained to predict for optimal performance? In this work, we address this question by analyzing the relationship between the geometry of the loss landscape and the training time horizon. Using dynamical systems theory, we prove that loss minima for long horizons generalize well to short-term forecasts, whereas minima found on short horizons result in worse long-term predictions. However, we also prove that the loss landscape becomes rougher as the training horizon grows, making long-horizon training inherently challenging. We validate our theory through numerical experiments and discuss practical implications for selecting training horizons. Our results provide a principled foundation for hyperparameter optimization in autoregressive forecasting models.

Pau Vilimelis Aceituno, Jack William Miller, Noah Marti +2

Institute of Neuroinformatics, ETH Zürich and University of Zürich, Winterhurerstrasse 190, Zürich 8057, Switzerland · School of Computing, National Australian University, 108 North Rd, Acton ACT 2601, Australia · Unit of Land Change Science, Swiss Federal Research Institute for Forest, Snow and Landscape Zürcherstrasse 111, Birmensdorf 8903, Switzerland

Long-Term Time Series Forecasting Time Series Forecasting Loss Landscape Geometry Hyperparameter Optimization Autoregressive Modeling

May 23, 2025

Time-o1: Time-Series Forecasting Needs Transformed Label Alignment

Training time-series forecasting models poses unique challenges in loss function design. Most existing approaches adopt temporal mean squared error, but this study reveals two critical limitations: (1) it ignores the presence of label autocorrelation, which biases it from the true label sequence likelihood; (2) it involves excessive number of tasks, which complicates optimization, especially for long-term forecasting. To address these issues, we introduce Time-o1, a transform-enhanced loss function for time-series forecasting. The central idea is to transform the label sequence into decorrelated components with discriminated significance. Models are then trained to align the most significant components, thereby effectively mitigating label autocorrelation and reducing task amount. Experiments demonstrate that Time-o1 achieves state-of-the-art performance and is compatible with various forecast models. Code is available at https://github.com/Master-PLC/Time-o1.

Hao Wang, Licheng Pan, Zhichao Chen +5

Xiaohongshu Inc. · State Key Lab of General AI, School of Intelligence Science and Technology, Peking University · Gaoling School of Artificial Intelligence, Renmin University of China +4

Long-Term Time Series Forecasting Time Series Forecasting

May 21, 2025

Human Supervisor Workload Prediction: Lag Horizon Selection

Teleoperation systems must be aware of the human's workload during missions to maintain operator performance. Prior work employed wearable physiological sensor response metrics to estimate current human workload; however, these estimates only enable robots to respond to under- or overload conditions reactively. Current human workload prediction approaches are limited to very short prediction horizons and fail to investigate variable lag horizons' impact on those predictions. This manuscript investigates physiological sensor driven human workload prediction focusing on the impact of lag horizons on both univariate and multivariate time series forecasting models, with longer prediction horizons than the workload prediction state-of-the-art (i.e., > 30 seconds using Long Short-Term Memory networks). Models were trained using data from a 64 participant non-sedentary supervisory environment NASA Multi-Attribute Task Battery-II human subjects evaluation. A key finding is that univariate workload predictions required 240 second lag horizons, whereas multivariate workload predictions sufficed with 120 second lag horizons. This finding indicates additional workload components reduce lag horizon requirements, enabling more efficient models with longer prediction horizons.

Mark-Robin Giolando, Julie A. Adams

Collaborative Robotics and Intelligent Systems Institute at Oregon State University, Corvallis, OR, USA

Multivariate Time Series Forecasting Long-Term Time Series Forecasting

Jan 31, 2025

BEAT: Balanced Frequency Adaptive Tuning for Long-Term Time-Series Forecasting

Long-term time-series forecasting supports a wide range of applications, including weather prediction and electricity demand planning. Frequency-domain methods address this task by decomposing observations into components that describe temporal variations at different scales. However, separate representations do not by themselves provide an explicit mechanism for adjusting the training emphasis across components. Under a shared forecasting objective, the frequency-specific networks can retain different levels of coefficient prediction error, motivating an error-dependent adjustment to their gradients. To this end, we propose BEAT (Balanced frEquency Adaptive Tuning), a framework that combines frequency-specific error monitoring with adaptive gradient modulation. We design a Frequency-Specific Monitor that compares predicted and target wavelet coefficients in a common normalized space and expresses each discrepancy relative to a reference error computed from the detail components. We further introduce a Dynamical Gradient Balancer that converts these ratios into positive, bounded coefficients. Components with higher relative errors receive larger gradient weights, whereas those with lower relative errors receive smaller weights. A shared modulation-strength parameter controls the departure from unmodulated training, and the monitoring and balancing operations are used only during training. Experiments on seven real-world datasets show that BEAT achieves competitive performance against state-of-the-art forecasting methods.

Zhixuan Li, Naipeng Chen, Seonghwa Choi +2

College of Computing and Data Science, Nanyang Technological University, Singapore · Department of Electrical and Electronic Engineering, Yonsei University, Korea

Long-Term Time Series Forecasting Frequency-Domain Time Series Modeling Adaptive Gradient Methods Frequency-Domain Feature Learning

Aug 2, 2024

Spatio-Temporal Partial Sensing Forecast for Long-term Traffic

Traffic forecasting uses recent measurements by sensors installed at chosen locations to forecast the future road traffic. Existing work either assumes all locations are equipped with sensors or focuses on short-term forecast. This paper studies partial sensing forecast of long-term traffic, assuming sensors are available only at some locations. The problem is challenging due to the unknown data distribution at unsensed locations, the intricate spatio-temporal correlation in long-term forecasting, as well as noise to traffic patterns. We propose a Spatio-temporal Long-term Partial sensing Forecast model (SLPF) for traffic prediction, with several novel contributions, including a rank-based embedding technique to reduce the impact of noise in data, a spatial transfer matrix to overcome the spatial distribution shift from sensed locations to unsensed locations, and a multi-step training process that utilizes all available data to successively refine the model parameters for better accuracy. Extensive experiments on several real-world traffic datasets demonstrate its superior performance. Our source code is at https://github.com/zbliu98/SLPF

Zibo Liu, Zhe Jiang, Zelin Xu +5

Department of Computer & Information Science & Engineering, University of Florida · Department of Computer Science, University of Kentucky

Long-Term Time Series Forecasting Spatiotemporal Forecasting Traffic Forecasting