Long-Term Time Series Forecasting

Follow this topic Ask about this topic

Momentum

8 papers in the last four weeks, against 2 the four weeks before. 0.1% of all new papers.

Jul 13Week of Sep 28

Latest papers 68

Oct 8, 2026cs.AI

AliO: Output Alignment Matters in Long-Term Time Series Forecasing

Long-term Time Series Forecasting (LTSF) tasks, which leverage the current data sequence as input to predict the future sequence, have become increasingly crucial in real-world applications such as weather forecasting and planning of electricity consumption. However, state-of-the-art LTSF models often fail to achieve prediction output alignment for the same timestamps across lagged input sequences. Instead, these models exhibit low output alignment, resulting in fluctuation in prediction outputs for the same timestamps, undermining the model's reliability. To address this, we propose AliO (Align Outputs), a novel approach designed to improve the output alignment of LTSF models by reducing the discrepancies between prediction outputs for the same timestamps in both the time and frequency domains. To measure output alignment, we introduce a new metric, TAM (Time Alignment Metric), which quantifies the alignment between prediction outputs, whereas existing metrics such as MSE only capture the distance between prediction outputs and ground truths. Experimental results show that AliO effectively improves the output alignment, i.e., up to 58.2% in TAM, while maintaining or enhancing the forecasting performance (up to 27.5%). This improved output alignment increases the reliability of the LTSF models, making them more applicable in real-world scenarios.

Kwangryeol Park, Jaeho Kim, Seulki Lee

Artificial Intelligence Graduate School Ulsan National Institute of Science & Technology (UNIST), South Korea · Artificial Intelligence Korea University, South Korea · Department of Computer Science and Engineering Ulsan National Institute of Science & Technology (UNIST), South Korea

Long-Term Time Series Forecasting Time Series Forecasting

Oct 6, 2026cs.LG

Structure-Aware Graph Abstention for Reliable Selective Forecasting

Selective forecasting abstains on high-risk test windows under a retained-coverage budget. Existing gates such as TEM (Brusokas et al., 2025) score each forecast as a whole; for multivariate outputs, trajectories can look plausible while violating dependencies among variables. We treat instance-level plausibility and relational consistency as distinct reliability axes and operationalize the latter via a learned sparse graph and a Dirichlet-style structural energy E_struct, trained with error-weighted graph regularization and score-error alignment. On seven long-horizon benchmarks and four backbones, structural gating often reduces selective MSE versus TEM at matched coverage, with the largest gains where cross-variable structure appears more informative in our benchmarks; gains are not universal, indicating a complementary abstention signal. Table 1 is a Protocol A ranking diagnostic (seed 2024); three-seed deployable Protocol B on an aligned subset is in Table 3 (full validation-to-test grids: Appendix A).

Jianxiang Xie, Belal Alsinglawi

University of New South Wales · Zayed University

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Selective Prediction

Sep 27, 2026cs.LG

From HL to H+L-1 Parameters: A Hankel-Toeplitz Forecaster for Long-Term Time Series Forecasting

Linear forecasters have shown competitive accuracy against Transformer-based models in long-term time series forecasting. We study how classical stationary prediction theory can guide parameter sharing for more compact linear forecasters. For centered second-order stationary processes with nonsingular history covariance, the minimum-MSE finite-window linear predictor factors into a Hankel cross-covariance matrix and an inverse Toeplitz covariance matrix. Shared lags and scale cancellation specify this predictor using

H+L-1

autocorrelations for lookback

L

and horizon

H

. Building on the innovations representation, our Hankel-Toeplitz Forecaster (HTF) learns one impulse response that defines both an inverse filter and a forecast map. We characterize the finite-history correction and, under summability assumptions, bound the excess risk of truncating the true filters. HTF uses

H+L-1

trainable coefficients while allowing a full-rank forecasting matrix. Across seven benchmarks at

L=336

, its horizon-averaged MSE is within 1.2% of Dense Linear on each dataset with 75-229 times fewer trainable parameters.

Chaoqi Zhang, Yu Wang, Haixu Tang

Indiana University Bloomington, IN, USA

Long-Term Time Series Forecasting Time Series Forecasting

Sep 27, 2026cs.LG

Diffusion-Based Rollouts as a Stabilization Mechanism for Long-Horizon Environmental Forecasting

Extending forecast lead times while maintaining predictive skill remains a major challenge in environmental forecasting. We investigate diffusion-based rollouts as a stabilization mechanism for recursive forecasting using low-dimensional water-level time series and high-dimensional precipitation fields. Across both modalities, diffusion suppresses recursive error growth, with the largest stabilization occurring where deterministic rollouts are most unstable. However, stabilization does not guarantee forecast fidelity. In the water-level experiments, forecasts progressively lose event-level fidelity as the rollout loses access to external predictive information, and trajectory-level comparisons show that diffusion can remain numerically stable while contracting toward central values and exhibiting reduced variability. In the precipitation experiments, which retain conditioning from numerical weather prediction throughout the rollout, diffusion better preserves spatial organization and event-detection skill. Together, these contrasting experiments indicate that diffusion can control recursive error amplification, while its practical benefit also depends on the predictive information available to constrain future evolution.

Marina Vicens-Miquel, Amy McGovern, Aaron J. Hill +2

University of Oklahoma, School of Meteorology, Norman, Oklahoma · University of Oklahoma, School of Computer Science, Norman, Oklahoma · NSF AI Institute for Research on Trustworthy AI in Weather, Climate, and Coastal Oceanography, Norman, Oklahoma +2

Precipitation Forecasting Long-Term Time Series Forecasting Time Series Forecasting Spatiotemporal Forecasting

Sep 23, 2026cs.LG

Learning Where to Look: A Shared Relative-Alignment Module for Time-Series Forecasting and PPG-to-Vital-Sign Reconstruction

PPG-to-vital-sign reconstruction turns a wrist-worn photoplethysmogram into clinical waveforms such as the ECG. Long-horizon multivariate time-series forecasting underpins planning in energy, weather, and traffic. Both generate a target sequence from a condition sequence, and current models hard-code where each target position reads it, as a same-position copy or seasonal recurrence, so neither transfers between tasks. We propose ROOSTER, one conditioning module that handles vital-sign reconstruction and time-series forecasting alike by learning this correspondence. Its core is a periodic-comb bias over the target-condition offset whose center, period, and sharpness are learned per head, so one module settles on the identity alignment or a seasonal lag and reports which it found. On vital-sign reconstruction from PPG, ROOSTER outperformed the published baselines on four heart-rate and respiratory-rate benchmarks. On multivariate time-series forecasting, it achieved the best horizon-averaged MSE on four benchmarks and outperformed the forecasting model it extends on 20 of 24 dataset-horizon settings under matched three-seed training. An ablation study indicated that the relative bias, not content matching, carried the alignment.

Ragamayi Puli, Shunya Nagashima

Neurogica Inc.

Multivariate Time Series Forecasting Long-Term Time Series Forecasting

Sep 20, 2026cs.LG

One Patch, Three Roles: What Is Actually Coupled in Autoregressive Time-Series Forecasting?

Patch-based autoregressive time-series forecasting often ties input representation, learned transitions, and recursive execution to one patch length. We ask which of these roles can be adjusted separately. A supporting atomic-encoding study finds greater sensitivity to model width than to atom grouping on the evaluated grid. Our main finding is that a frozen parent's recursive trajectory is easier to fit than the observed future with lightweight parallel exits. Autoregressive Trajectory Distillation (ATD) turns this into selectable ATD-1/2/4/8 execution, with ATD-1 exactly recovering the parent. On a paired four-data-set comparison, ATD-8 reaches

5.54\times

end-to-end speedup with stable quality across widths. Fewer calls do not automatically remove the parent's existing forecast error: ATD improves trajectory fidelity in all 21 seed runs but forecast accuracy in only 15 against matched clean-future supervision. We further find a correctable residual projection along a train-selected periodic history direction. Spectrum Tangent applies this correction without adding neural parameters or Transformer calls. At horizon 720, it reduces mean squared error (MSE) and mean absolute error (MAE) by 2.54% and 2.33% over seven data sets and two output widths, while remaining

3.24\times

faster than recursive inference. Level and shape projections sometimes disagree. Trajectory compressibility, the fidelity-accuracy mismatch, and the correction recur across three public AR parents. Together these results separate representation, transition, and execution as AR design axes. Code is available at https://github.com/RowanFFF/ATD-Spectrum-Tangent.

Ziang Li, Yue Huang, Guoxu Zhou +4

Guangdong University of Technology · Guangdong Polytechnic Normal University · Harbin Institute of Technology

Long-Term Time Series Forecasting Time Series Forecasting Autoregressive Decoding Knowledge Distillation Autoregressive Modeling

Sep 17, 2026cs.LG

SETTer: Sparse-Encoder Transformer for Long-term Multivariate Time Series Forecasting

Long-term multivariate time series plays a significant role in many application areas such as power systems, trading, etc. However, their accurate prediction is quite difficult for conventional forecasting methods as they often exhibit high dimensionality and complex relationships. Recent works show that transformer-based approaches are quite effective for long-term forecasting thanks to their attention mechanism. However, in the presence of complex high-dimensional inputs, they show evidence of oversmoothing, limited capacity, and opacity. To this end, this paper introduces SETTer, a transformer-based model that addresses these challenges by incorporating novel techniques for decoupled self-attention and hybrid masking. The proposed techniques enable SETTer to effectively capture the dominant short- and long-term patterns across the temporal and channel dimensions. In addition, we enrich the model layers with simple explainable structures that indicate the discriminative pattern of SETTer. We show that with a single-layer transformer architecture, SETTer can effectively model long-term dependencies in the presence of varying data complexities. Extensive experiments on real-word benchmark datasets for long-term multivariate time series forecasting demonstrate that SETTer outperforms state-of-the-art models in 88% of the scenarios.

Abraham Ezema, Chijioke Eze, Ferdinanda Ponci +1

Institute for Automation of Complex Power Systems, RWTH Aachen University, 52064 Aachen, Germany · RWTH Aachen University, 52064 Aachen, Germany · institute for Monitoring and Distributed Control for Power Systems Teaching and Research Area, RWTH Aachen University, 52064 Aachen, Germany +1

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Explainable Time Series Forecasting Time Series Forecasting Attention Mechanisms Time Series Transformers

Sep 17, 2026cs.LG

OceanMoE: Structured Conditional Sparse Computation for Long-Horizon Multivariate Ocean Forecasting

Multivariate ocean forecasting must exploit shared evolution in a coupled ocean system while adapting to the heterogeneous statistical and dynamical characteristics of different prediction variables and locations. Fully shared models may lack the flexibility to handle this heterogeneity, whereas fully independent models discard the common ocean context shared across variables. The key question is how to retain shared context in a unified model while allowing computation to specialize according to the prediction target and local state. We propose OceanMoE, a structured conditional sparse Mixture-of-Experts framework that combines sharing and specialization for multivariate ocean forecasting. OceanMoE fuses cross-variable information to construct target-specific local representations and uses them to perform content-conditioned sparse routing at each spatial location, with the number of active experts adapted to router confidence. In the decoder, routing is augmented with a learned geographic bias parameterized by spherical-harmonic spatial bases, while shared residual and seasonal pathways provide common cross-variable and month-dependent context. Experiments on long-horizon autoregressive ORAS5 forecasting show that OceanMoE lowers aggregate forecasting error in both evaluated settings and maintains lower geometric-mean normalized RMSE than the corresponding baselines over most later rollout months. Routing analyses further show that expert allocation varies with prediction targets and spatial locations. These results support structured conditional computation as a modeling strategy for balancing shared ocean context with adaptive specialization.

Yishun Zhu, Jian Wang

Hangzhou Institute for Advanced Study, University of the Chinese Academy of Sciences Zhejiang, China · Computer Network Information Center, Chinese Academy of Sciences Beijing, China

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Sparse Mixture-of-Experts Spatiotemporal Forecasting

Sep 11, 2026cs.LG

Large Distant Gradients Need Not Be Reliable: reliability-weighted credit assignment for long-horizon autoregressive forecasting

In autoregressive forecasting, long prediction rollouts provide distant supervision, but backpropagation through time (BPTT) carries gradients from those losses through many autoregressive steps. Repeated Jacobian products can make distant gradients dominate the update while amplifying predictable signal and unpredictable innovation together; a large distant gradient therefore need not carry reliable learning signal. Motivated by this, we introduce Internal Dual-Wiener routing (Internal-DW), a backward-only intervention that preserves the full forward rollout and all step losses while reliability-weighting internal gradient routes. At each residual block, we derive bounded Wiener gains for the identity and nonlinear routes that balance preserving predictable learning signal against suppressing unpredictable variation, and estimate them from route-level gradient statistics and an explicit noise model. In a controlled system with known gradient signal-to-noise ratio (SNR), we show that distant gradients can grow even as their SNR falls, and that Internal-DW reduces error in recovering predictable gradient signals and improves forecasting. On four history-dominated, weak-drive testbeds, Internal-DW reduces forecast error by 5.2%-13.8% relative to full BPTT, outperforms gradient clipping and Jacobian regularization on three testbeds, with similar performance on shear flow, and outperforms validation-selected truncated BPTT (TBPTT) on three. It also extends or preserves the fitted optimal training-horizon range across these four testbeds. Across benchmarks, the current Internal-DW estimator has a clear applicability boundary: its benefit diminishes or reverses when usable history is limited or when the selected sampler fails to represent dominant drive-dependent variation. The results show that retaining long-horizon supervision does not require trusting every backward contribution equally.

Junhao Zhao, David Michael Simberg, Jacob Kang +2

University of Maryland, College Park

Long-Term Time Series Forecasting Backpropagation Temporal Credit Assignment Autoregressive Modeling

Sep 8, 2026cs.LG

HypLTSF: A Hyperbolic Geometric View of Multi-Scale Hierarchies for Long-Term Time Series Forecasting

Multi-scale modeling has become an effective approach for long-term time series forecasting, capturing temporal patterns that range from fine-grained local dynamics to coarse global trends. Representations across these temporal scales are inherently hierarchical, with coarser scales abstracting and aggregating information from finer ones. While existing approaches readily exchange information across these scales, the hierarchy itself is typically left as an emergent byproduct of such interactions rather than captured as a geometric structure in its own right. In this paper, we introduce HypLTSF, a framework that endows the multi-scale hierarchy with a concrete geometric form by embedding scale-wise representations into the Poincaré ball, whose exponentially expanding volume naturally accommodates hierarchical structures. To align this geometry with the temporal hierarchy, HypLTSF imposes two constraints: (1) a radial constraint that orders embeddings by their level of abstraction, and (2) an angular constraint that groups fine-scale patterns sharing a common coarser-scale ancestor. Extensive experiments on long-term time series forecasting benchmarks show that HypLTSF achieves state-of-the-art performance, suggesting that explicitly modeling the multi-scale hierarchy as a geometric structure is effective for forecasting.

Namwoo Kim, Hyungryul Baik, Yoonjin Yoon

Urban AI Institute, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea · Department of Mathematical Science, KAIST, Daejeon, Republic of Korea · Department of Civil and Environmental Engineering, KAIST, Daejeon, Republic of Korea

Long-Term Time Series Forecasting Hierarchical Representation Learning Multiscale Time-Series Modeling Hyperbolic Representation Learning

Aug 11, 2026cs.LG

Long-Horizon Forecasting of Complete Financial Statements with Forma

Specialist training beats generalist scale when forecasting financial statements. To our knowledge, no prior work jointly forecasts complete financial statements beyond one year, yet in a discounted-cash-flow valuation most firm value sits past that window. We release ProForma-20Q, a reproducible benchmark for forecasting 78 statement line items 1-20 quarters ahead, for anonymized firms, from past statements and an industry code, scored by change-space

R^2

. On it, Forma, a transformer that reads statements as sets of (account, quarter, value) tuples and maximizes a masked-tuple Gaussian likelihood, beats every competitor we field: classical machine learning, chained gradient boosting, a zero-shot time-series foundation model, and frontier large language models. Its lead widens with horizon, where valuation needs accuracy most, and its Gaussian predictive intervals never under-cover. Forma's forecasts nearly satisfy accounting identities; exact coherence is recoverable at no statistically significant accuracy cost. Its tuple interface supports scenario analysis without retraining, and we show that pinning future revenue paths sharpens the rest of the statement.

Travis L. Johnson, Jiannan Jiang, Soumyabrata Chaudhuri +3

University of Texas at Austin, Austin, TX, USA

Long-Term Time Series Forecasting Financial Forecasting Forecasting Benchmarks Financial Time Series Probabilistic Time Series Forecasting

Aug 11, 2026cs.LG

Two-stage Odd Residual Flows for Mean-Preserving Probabilistic Time Series Forecasting

Probabilistic forecasting plays an essential role in risk-sensitive decision-making, particularly in long-horizon settings. However, existing approaches often face a fundamental trade-off between distributional flexibility and accurate mean prediction. Traditional parametric methods, such as Mean Variance Estimation (MVE), can suffer from degraded point accuracy when trained under joint Negative Log-Likelihood (NLL) objectives, while modern-flexible generative models, including Normalizing Flows and Diffusion Models, typically rely on costly Monte Carlo sampling and may yield suboptimal mean estimates. To address this limitation, we propose Two-stage Odd Residual Flows (TORF), a framework that decouples mean forecasting from uncertainty estimation. In the first stage, a pre-trained deterministic model is used to produce an accurate mean prediction. In the second stage, a Restricted Normalizing Flow, with strictly odd functions learns flexible residual distributions around the point forecast, guaranteeing mean preservation from the first stage without sampling. Experiments show that TORF achieves state-of-the-art deterministic accuracy (NMAE) while providing strong density estimation performance (CRPS) on short and long-horizon forecasting.

Kiran Madhusudhanan, Christian Klötergens, Lars Schmidt-Thieme +1

Institute of Computer Science University of Hildesheim Hildesheim, Germany

Long-Term Time Series Forecasting Normalizing Flows Residual Learning Probabilistic Forecasting Probabilistic Time Series Forecasting

Aug 9, 2026cs.LG

Efficient Test-Time Scaling for LLM-based Time Series Forecasting

Long-term time series forecasting benefits from preserving global structure such as trends and seasonality. Recent LLM-based forecasters often improve accuracy through test-time scaling (e.g., iterative refinement), but these methods are computationally expensive and increasingly prone to global-shape mismatch as the prediction horizon extends. We propose SCALER, a coarse-to-fine forecasting framework that first employs a lightweight Transformer tailored to long-term shape modeling to predict a coarse representation of future dynamics. This predicted shape then serves as a compact guide for an LLM to perform test-time scaling via iterative coarse-to-fine residual token refinement, while processing substantially fewer tokens at each step. By guiding refinement with an explicit future-shape prediction, SCALER reduces reliance on long description prompts, and its fixed-step refinement avoids costly reward-model-based selection, further lowering computational overhead. Experimental results demonstrate that SCALER outperforms strong forecasting baselines in long-term, short-term and zero-shot forecasting while significantly reducing the inference cost associated with scaled LLM for time series forecasting. Code: https://github.com/xuanmay2701/SCALER.

Xuan-May Le, Minh-Tuan Tran, Ling Luo +3

The University of Melbourne Melbourne, Victoria, Australia · Monash University Melbourne, Victoria, Australia

Long-Term Time Series Forecasting Time Series Forecasting Test-Time Scaling LLM-Based Time Series Forecasting LLM Inference Acceleration

Aug 7, 2026cs.LG

KReF: Training-Free Retrieval for Long-Term Time-Series Forecasting and Predictive Uncertainty

Probabilistic long-term time-series forecasting commonly relies on trained models. Training-free conformal methods typically construct intervals around a pre-existing point forecaster and do not natively represent a complete predictive distribution; sequential variants additionally suffer from increasingly delayed feedback at long horizons. We propose KReF, a training-free retrieval framework that treats retrieved historical futures as a querylocal empirical predictive distribution. After robust preprocessing, KReF embeds each lookback using handcrafted statistics or frozen random Fourier features and retrieves similar historical lookback-future pairs. Their similarity weights directly define predictive masses, quantiles, CRPS, and a weighted-mean point forecast. KReF further uses the observed query lookback to construct a probability-integral-transform map and applies validation-selected expansion and shrinkage rates to adapt interval boundaries. Across six LTSF benchmarks and four horizons, KReF obtains the lowest CRPS in all 12 dataset-embedding settings and the lowest IS90 in 9 settings. Without gradient-based fitting, its point forecasts also match or surpass trained baselines on two of six datasets. An archive-oracle analysis further reveals substantial headroom under finer horizon- and channel-wise routing. These results establish retrieval as a useful and underexplored inductive bias for LTSF.

Yang Zhang, Rui Su

Long-Term Time Series Forecasting Retrieval-Augmented Time Series Forecasting Probabilistic Time Series Forecasting Regression Prediction Intervals

Aug 5, 2026cs.LG

Beyond Linear Dynamics: Neural Bilinear Dynamical Models for Time Series Forecasting

Time series in real-world applications are often generated by nonlinear dynamical systems, making accurate forecasting challenging. Existing approaches that explicitly model system dynamics typically rely on linear assumptions or Koopman-based linearizations, which may inadequately capture complex nonlinear behaviors and lead to error accumulation in long-horizon prediction. To address this limitation, we propose the Neural Bilinear Dynamical Model (NBDM), which models nonlinear system dynamics through a bilinear latent dynamical formulation. Specifically, NBDM leverages Koopman theory to lift the original nonlinear dynamics into a higher-dimensional latent space, where a bilinear dynamical model is constructed to characterize state evolution. To mitigate the approximation error introduced by bilinear representations, we further incorporate a parameterized error compensation term. Within this formulation, control inputs are explicitly integrated into the dynamics, using auxiliary variables when available and learned feedback signals otherwise. To handle scenarios with missing control inputs, we design a memory-enhanced controller that infers latent controls through multiplicative interactions between historical states and control signals. Experiments on five real-world datasets demonstrate that NBDM consistently outperforms competitive baselines in both given-control and missing-control settings, particularly for multi-step and long-horizon forecasting.

Mengzhou Gao, Huangqian Yu, Pengfei Jiao

Hangzhou Dianzi University Hangzhou, China

Dynamical Systems Long-Term Time Series Forecasting Latent Dynamics Modeling Time Series Forecasting

Aug 4, 2026cs.LG

CAMP: A Cycle-Aware Multi-Scale Patch Mixer for Time Series Forecasting

Real-world time series are often governed by recurring patterns, but their dominant periods may vary across datasets, forecasting settings, and individual input windows. Existing cycle-aware forecasters commonly rely on a single period selected at the dataset level, which can be restrictive when periodic behavior changes over time or when multiple cycles coexist. Moreover, patch-based models typically process all patch positions uni- formly, although patches farther from the forecast boundary may require broader contextual refinement, while recent patches contain information that should be preserved more directly. Af- ter cyclic behavior is removed, the remaining dynamics may also span multiple temporal resolutions and cannot be adequately de- scribed at a single scale. We introduce CAMP, a Cycle-Aware Multi-Scale Patch Mixer designed to address these challenges. The Adaptive Cycle Learning module identifies dominant fre- quencies separately for each input window and generates both historical and future cyclic components without requiring a pre- defined cycle length. The Horizon-Guided Patch Mixer intro- duces position-dependent refinement, allowing earlier patches to incorporate broader temporal context while preserving infor- mation close to the forecast boundary. CAMP further models the de-cycled residual through temporally aligned multi-resolution representations, enabling complementary dynamics at different scales to be captured within one forecasting framework. Across seven long-term forecasting benchmarks, CAMP achieves the best average MSE on six datasets and the best or tied-best MAE on six. It also obtains the highest MSE win count across sixteen settings on four PEMS traffic benchmarks.

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

ISMLL, University of Hildesheim, VWFS Data Analytics Research Center (VWFS-DARC), Hildesheim, Germany

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

Jul 31, 2026cs.LG

TFGformer: Multivariate Time Series Forecasting via Time-Frequency Graph Learning and Covariate Fusion

Large-scale multivariate time series from heterogeneous IoT sensors demand accurate long-term forecasting for resource scheduling and predictive maintenance. While recent time series foundation models exhibit strong generalization, they rely on static parametric knowledge and lack dynamic access to external historical patterns during inference. Retrieval-Augmented Generation (RAG) offers a potential remedy, yet its application to time series forecasting is challenged by magnitude variations across heterogeneous sources and the mismatch between historical similarity and future consistency. We propose CrossRAG, a retrieval-augmented forecasting framework that integrates Shape-Aware Memory (SAM) with RevIN normalization for magnitude-robust shape-level retrieval, Future-Consistent Contrastive (FCC) learning to distinguish informative references from hard negatives with similar history but divergent futures, and Cross-Attention Temporal Fusion (CATF) to fuse retrieved historical--future reference pairs into the backbone's representations at the representation level. Experiments on seven public benchmarks show that CrossRAG consistently outperforms both parametric-only baselines and existing retrieval-augmented forecasting methods.

Yu Sun, Yuan Chang, Xiaohou Shi +1

Beijing University of Posts and Telecommunications Beijing, China · China Telecom Research Institute Beijing, China

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Retrieval-Augmented Time Series Forecasting Time-Series Foundation Models

Jul 23, 2026cs.LG

Nipping the Butterfly Effect in the Bud: Self-Output Fine-Tuning for Autoregressive Weather Prediction

Long-horizon weather forecasting is a fundamental challenge in atmospheric science, for which autoregressive Deep Learning Weather Prediction (DLWP) has emerged as the primary paradigm. Although the autoregressive pipeline is highly scalable and flexible, its prediction errors grow rapidly over long forecasting horizons. In this work, we study this error growth phenomenon from both theoretical and empirical perspectives. Our analysis reveals that the growth is driven by a feedback loop between output errors and input distribution shifts. Specifically, the autoregressive process amplifies small initial output errors, which progressively corrupt subsequent input distributions, echoing the butterfly effect in atmospheric science and ultimately deteriorating forecasting accuracy over longer horizons. Furthermore, we show that this distributional shift originates at the earliest stage of inference, with out-of-distribution signatures detectable as early as the first autoregressive step. To mitigate this issue, we propose \textbf{Self-Output Fine-Tuning (SOFT)}, a plug-and-play strategy that leverages the model's own one-step predictions to calibrate the biased input distribution encountered at the first step. Extensive experiments demonstrate that, despite its simplicity, SOFT achieves state-of-the-art performance on long-horizon forecasting tasks and substantially reduces both prediction errors and distributional discrepancy. The success of SOFT highlights the importance of reexamining the fundamental pipeline of deep learning weather prediction, representing a critical pipeline advance for atmospheric science.

Yun-Ye Cai, Hsuan-Tien Lin

National Taiwan University · Taipei, Taiwan

Long-Term Time Series Forecasting Distribution Shift Weather Forecasting Autoregressive Modeling

Jul 16, 2026cs.LG

Asymmetric Peak-Aware Loss for Peak-Critical Time Series Forecasting

In many operational time-series forecasting applications, such as crowd demand forecasting, the risk related to under-prediction is substantially higher than that of over-prediction. Accurate prediction of rare demand spikes plays a critical role in downstream tasks. Yet most time-series forecasters are trained with symmetric objectives (e.g., MSE, MAE) and evaluated primarily on aggregate error, which can mask failures in extreme-values and peak-timing predictions. We introduce Asymmetric Peak-Aware Loss (APAL), a simple, model-agnostic objective that (i) penalizes under-predictions more heavily and (ii) increases the training weight of peak regions within each forecast window. We further propose a peak-critical evaluation protocol that complements MAE/MSE with channel-wise tail error (Top-10% and Top-1%) and peak metrics (precision, recall, F1 under timing tolerance, and peak timing error). We evaluate APAL on long-horizon multivariate forecasting across five state-of-the-art backbones, with a focus on pedestrian demand forecasting using (i) a production-ready subset of the City of Melbourne pedestrian hourly count dataset and (ii) a beach visitor count dataset. The generality of the loss function for time-series forecasting is tested on additional benchmarks. Across peak-critical datasets and settings, APAL improves tail accuracy and peak-prediction quality while exposing a controllable trade-off with aggregate error, making it a practical solution when peak-prediction failures are the dominant operational concern.

Theivaprakasham Hari, Yanan Xin, Winnie Daamen +2

Department of Transport and Planning, Faculty of Civil Engineering and Geosciences Delft University of Technology, Delft, The Netherlands · Mobility Innovation Centre Delft University of Technology, Delft, The Netherlands

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Cost-Sensitive Learning Time Series Forecasting

Jul 15, 2026cs.AI

A Comparative Analysis of Machine Learning Models for Long and Short-Term Forecasting of the Egyptian Stock Market: A Focus on EGX30

This study concentrates on predicting stock prices in the Egyptian market, focusing on the EGX30, an influential financial hub in the Middle East. While most research focuses on global stocks, there's a growing need to understand stock trends in developing countries like Egypt. The study compares different machine learning models for forecasting EGX30 trends, covering short and long-term predictions. Using historical EGX30 data, including metrics like root mean squared error, Mean Absolute Percentage Error, and coefficient of determination, models like K-Nearest Neighbours, random forest, extreme gradient boosting, long short-term memory networks, and gated recurrent unit networks were evaluated. The goal is to determine the most effective models for EGX30 prediction, considering Egypt's unique market dynamics. Insights from this study aid investors in making informed decisions. Results show that the Gated Recurrent Unit (GRU) outperformed the other models in the one-week, one-month, and two-months while the eXtreme Gradient Boosting (XGBoost) model outperformed others in the one-day predictions, highlighting their usefulness in predictive analysis for financial markets. The study also showed the importance of using the ensemble techniques, especially in the long-term predictions which proved better results reaching 5 times the GRU in the two-month predictions. Additionally, the study notes the surprisingly good performance of K-Nearest Neighbours (KNN) on long-term predictions, suggesting its enduring relevance and potential for future applications in the fintech domains.

Muhammed Walid, Ahmed El-Naeimy, Hosam Moubarak +1

Cyber Physical Lab, Department of Computer Science & Engineering · Egypt-Japan University Of Science and Technology, Alexandria, Egypt · Accounting Department +1

Long-Term Time Series Forecasting Recurrent Neural Networks Financial Time Series

Jul 14, 2026cs.LG

TSSM: Triaxial State Space Model for Global Station Weather Forecasting with Temporal-Variable-Historical Modeling

Global Station Weather Forecasting (GSWF) is pivotal for localized and extreme weather prediction over key regions. Despite efforts to exploit look-back windows, existing methods show limited accuracy gains and struggle with extreme events and error accumulation. These limitations stem from overreliance on short-term patterns, which are insufficient to capture chaotic weather dynamics, especially under partial observations. To address this problem, we propose a novel Triaxial State Space Model (TSSM) with a history-enhanced Temporal-VariableHistorical paradigm, which incorporates period-aligned historical weather data to compensate for long-term, large-scale periodic, and full-window weather patterns beyond the temporal lookback window. Specifically, TSSM stacks historical samples into period-aligned batches, where forecasting is causally supported by historical and current observations. Temporal, variable, and historical scanning are designed to capture axial temporal dependencies, variable correlations, and historical evolution. This structure is hierarchically shared to model seasonal to extreme events while alleviating misalignment across historical patterns. TSSM achieves SOTA performance on Weather-5K, the largest station weather dataset to date, with 10% and 61% gains in accuracy and extreme event metrics, and obtains 95% best or second-best results on human-involved datasets. Its advantages are more pronounced in long-horizon and iterative forecasting, reaching a 37.5% gain at 240h and up to 103.5% under a 48h times 5 iterative setting. Moreover, TSSM retains > 90% performance under up to 80% missing observations, compared with < 43% for baselines, demonstrating robustness and practical potential for reliable GSWF in global in-situ observation networks.

Songru Yang, Zili Liu, Tao Han +7

Department of Aerospace Intelligent Science and Technology, School of Astronautics, Beihang University, and with the Key Laboratory of Spacecraft Design Optimization and Dynamic Simulation Technologies, Ministry of Education, Beihang University, Beijing 100191, China · Shanghai Artificial Intelligence Laboratory, Shanghai 200232, China · Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong 999077, China +1

Long-Term Time Series Forecasting State Space Models Weather Forecasting

Jul 9, 2026cs.LG

RhyMix: A Lightweight Adaptive Multi-Rhythm Network for Long-Term Time Series Forecasting

Real-world time series exhibit complex dynamics characterized by multiple simultaneous temporal patterns: short-term fluctuations, periodic seasonal cycles, long-term trends, and irregular abrupt changes. However, many existing forecasting architectures rely on single-path temporal modeling--transformers capture long-range dependencies but smooth local variations, convolutions capture local patterns but have limited receptive fields, and linear models are efficient but cannot capture nonlinear dynamics. To address this, we introduce RhyMix (RHYthm MIXture), a hybrid neural architecture designed around a parallel dual-path modeling paradigm with adaptive gating mechanisms. RhyMix integrates two complementary encoding branches: (i) a Cyclic Path that incorporates explicit seasonal inductive bias through learnable cyclic embeddings, capturing predictable rhythmic patterns; and (ii) a lightweight Multi-Scale Temporal Convolutional Network with Channel Attention Path that employs multi-scale depthwise dilated convolutions to capture temporal dependencies across different receptive fields. A key innovation is the use of adaptive gating at multiple levels: a path gate dynamically combines four specialized forecasting heads (Direct, Trend-Seasonal Decomposition, Local Convolution, and Periodic Fusion) per sample and channel, while a hybrid gate adaptively balances the Cyclic and MSTCN-CA Paths based on input characteristics. This design ensures the model adapts to specific temporal patterns while maintaining linear complexity in sequence length, channels, and prediction horizon. Across extensive benchmarks on 12 real-world datasets for long-term forecasting, RhyMix achieves state-of-the-art performance on 10 of 12 datasets. The model remains lightweight (~40K params) with linear complexity and low-latency inference (<5ms),suitable for resource-constrained edge devices and real-time deployment.

Sumit Satishrao Shevtekar, Chandresh Kumar Maurya

Long-Term Time Series Forecasting Time Series Forecasting Temporal Convolutional Networks Multiscale Time-Series Modeling

Jul 7, 2026cs.LG

NEST: Tackling Dataset-Level Distribution Shifts via Regime-Oriented Mixture-of-Experts

Accurate long-term forecasting in complex systems is frequently compromised by dataset-level distribution shifts, where diverse underlying behavioral modes and evolving system states drive the dynamic multivariate time-series. While existing methods predominantly focus on local temporal shifts, they fail to explicitly model the global structural challenge where datasets are composites of distinct operational regimes. In this paper, we propose NEST, a specialized framework designed to model and recompose these evolving structures through a two-phase dense MoE architecture. NEST first facilitates structural specialization by partitioning the dataset into distinct operational regimes through unsupervised clustering in a principled moment-entropy space. We introduce a regime-oriented router mechanism that generates initial expert weights based on temporal content, subsequently refined through geometric modulation to regime centroids. Crucially, rather than acting as monolithic predictors, individual experts function as specialized kernels that capture regime-specific dynamics by evolving unique variate-attention patterns. Extensive evaluations on diverse benchmarks, including heterogeneous network traffic and physical phenomena, demonstrate that NEST consistently achieves state-of-the-art performance. Our code and datasets are available at https://github.com/Aaralshin/NEST

Lanhao Li, Bingshu Xie, Lijun Sun +3

1Beihang University

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Time Series Forecasting OOD Generalization Mixture-of-Experts Models Time Series Clustering

Jun 30, 2026cs.LG

StateFlow: Dual-State Recurrent Modeling for Long-Horizon Time Series Forecasting

Long-horizon multivariate time series forecasting (LTSF) remains challenging due to non-stationarity, regime shifts, and error accumulation. The Variability-Aware Recursive Neural Network (VARNN) is designed to track such variability by maintaining a residual-memory state driven by one-step prediction errors. However, its original formulation is limited to one-step sequence regression and does not directly support multi-step forecasting. In this work, we extend VARNN to long-horizon forecasting and introduce StateFlow, a recurrent forecasting framework that uses VARNN as a dual-state recurrent backbone to capture two complementary signals from the lookback sequence: a hidden-state trajectory representing primary temporal dynamics, including trend, seasonality, level changes, and recurring patterns, and a residual-memory trajectory representing structured local prediction deviations, driven from a nonlinear recurrent transformation of errors between one-step base predictions and observed values. A chunk-based decoder separately summarizes these trajectories and maps them to the future horizon for direct multi-step forecasting. We further employ a two-stage optimization strategy that first trains the VARNN encoder through a one-step base prediction objective to optimize the internal representations over the lookback sequence, and then trains a horizon-specific decoder for direct multi-step forecasting. Experiments on standard LTSF benchmarks show that StateFlow achieves competitive performance against strong linear, recurrent, convolutional, and Transformer-based baselines while preserving linear recurrent encoding and a compact model design.

Haroon Gharwi, Yue Dai, Kai Shu

Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA · Department of Computer Science, Emory University, Atlanta, GA, USA

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Time Series Forecasting Recurrent Neural Networks Non-Stationary Time Series Forecasting

Jun 26, 2026cs.LG

TA-SparseMG: Trend-Aware Sparse Forecasting via Multi-Scale Gating for Long-Term Time Series

Long-term time series forecasting finds extensive applications in domains such as power demand, traffic flow, meteorological observation, and renewable energy dispatch. Forecasting dynamically varying long-term time series poses inherent challenges, including statistical nonstationarity, local high-frequency disturbances, and coupled cross-period dependencies, which make it difficult for lightweight models to balance parameter efficiency and forecasting performance. To address this issue, this study presents TA-SparseMG, a lightweight cross-period forecasting model built on SparseTSF's sparse cross-period modeling framework. It incorporates three key modules: a trend-aware reversible instance normalization module, a scale-adaptive gated denoising module, and a multiscale gated-attention MLP forecasting module. The trend-aware normalization module captures input-window statistics and calibrates forecast-window distributions, effectively mitigating distribution shift. The scale-adaptive gated denoising module performs feature smoothing and residual suppression before period rearrangement, thereby reducing interference from high-frequency perturbations. The multiscale gated attention prediction module strengthens the prediction head's adaptive representational capacity via conditional gating and feature modulation. Extensive experiments across multiple LTSF benchmarks demonstrate that the proposed TA-SparseMG consistently achieves superior, stable performance. Ablation studies confirm that each module independently improves distribution adaptation, input robustness, and cross-period feature mapping capability.

Wenchao Liu, Hongbing Wang, Youji Zhu +2

School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China. · School of Big Data and Computer Science, Guizhou Normal University, Guiyang 550025, China.

Long-Term Time Series Forecasting Non-Stationary Time Series Forecasting Multiscale Time-Series Modeling

Jun 26, 2026cs.LG

GNBAN: Graph Neural Basis Attention Networks for Long-Horizon Forecasting over Large Entity Sets

Demand forecasting at the bottom of a retail hierarchy requires predicting tens of thousands of correlated long-horizon series across products, stores, and regions. Modern systems must scale across massive catalogs, capture shared demand dynamics, and remain interpretable enough to be trusted. Classical statistical methods need a separate model per series and are hard to manage at scale; deep autoregressive models struggle as the joint state grows to tens of thousands of dimensions; and recent graph-based forecasters, while capturing cross-entity dependencies, often produce opaque long-horizon forecasts. We propose GNBAN (Graph Neural Basis Attention Network), an end-to-end architecture combining heterogeneous graph representation learning with an interpretable basis-decomposition head. Retail data are represented directly as a heterogeneous graph derived from the relational schema, so a single model serves the entire catalog. Rather than predicting the horizon directly, GNBAN decomposes each forecast into trend, seasonal, and generic components. Its key innovation is a per-basis attention mechanism: each basis function keeps its own learnable query and retrieves information independently from the entity's historical neighborhood, letting different bases specialize to distinct temporal patterns while preserving interpretability. On two large-scale benchmarks, M5 Walmart and Favorita Grocery Sales, evaluated under matched protocols, GNBAN improves volume-weighted WRMSSE by roughly 4-5% over a matched graph baseline. Qualitative analysis shows the learned decomposition exposes trend, seasonal, and residual demand drivers without post-hoc explanation methods. These results demonstrate that scalable relational forecasting and interpretable forecast decomposition can be achieved together in a unified graph-based framework.

Janak M. Patel, Anirudh Deodhar, Dagnachew Birru

Phi Labs, Quantiphi Marlborough, MA 01752, USA

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Heterogeneous GNNs Time Series Forecasting Time Series Decomposition

Jun 25, 2026cs.AI

PMDformer: Patch-Mean Decoupling Information Transformer for Long-term Forecasting

Long-term time series forecasting (LTSF) plays a crucial role in fields such as energy management, finance, and traffic prediction. Transformer-based models have adopted patch-based strategies to capture long-range dependencies, but accurately modeling shape similarities across patches and variables remains challenging due to scale differences. To address this, we introduce patch-mean decoupling (PMD), which separates the trend and residual shape information by subtracting the mean of each patch, preserving the original structure and ensuring that the attention mechanism captures true shape similarities. Futhermore, to more effectively model long-range dependencies and capture cross-variable relationships, we propose Trend Restoration Attention (TRA) and Proximal Variable Attention (PVA). The former module reintegrates the decoupled trend from PMD while calculating attention output. And the latter focuses cross-variable attention on the most relevant, recent time segments to avoid overfitting on outdated correlations. Combining these components, we propose PMDformer, a model designed to effectively capture shape similarity in long-term forecasting scenarios. Extensive experiments indicate that PMDformer outperforms existing state-of-the-art methods in stability and accuracy across multiple LTSF benchmarks. The code is available at https://github.com/aohu1105/PMDformer.

Ao Hu, Liangjian Wen, Jiang Duan +7

Southwestern University of Finance and Economics · Artificial Intelligence and Digital Finance Key Laboratory of Sichuan Province · Chengdu Everimaging Science and Technology Co., Ltd. +4

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Cross-Attention Time Series Forecasting Time Series Transformers

Jun 22, 2026cs.LG

Learning the Koopman Operator using Attention Free Transformers

Learning Koopman operators with autoencoders enables linear prediction in a latent space, but long-horizon rollouts often drift off the learned manifold, leading to phase and amplitude errors on systems with switching, continuous spectra, or strong transients. We introduce two complementary components that make Koopman predictors more robust. First, we add an attention-free latent memory (AFT) block that aggregates a short window of past latents to produce a corrected latent before each Koopman update. Unlike multi-head attention, AFT operates in linear time and adds only

\approx

30k parameters (

3d^2 + T^2

, fewer than matched multi-head attention), yet captures the local temporal context needed to suppress error divergence. Second, we propose dynamic re-encoding: lightweight, online change-point triggers (EWMA, CUSUM, and sequential two-sample tests) that detect latent drift and project predictions back onto the autoencoder manifold. Across three benchmark systems -- Duffing oscillator, Repressilator, IRMA -- our model consistently reduces error accumulation compared to a Koopman autoencoder and matched-capacity multi-head attention. We also compare against GRU and Transformer autoencoders, evaluated both from initial conditions and with a 50-step context, and find that Koopman+AFT (with optional re-encoding) attains markedly lower long-horizon error while maintaining lower inference latency. We report improvements over horizons up to 1000 steps, together with ablations over trigger policies. The result is a fast, compact predictor that stays on the learned manifold over long horizons.

Mohammed Nagdi, Evangelos-Marios Nikolados, Alexey Yermakov +3

Institute for Bioengineering, School of Engineering and Centre for Engineering Biology, University of Edinburgh, Edinburgh, United Kingdom · Electrical and Computer Engineering and Applied Mathematics, University of Washington, Seattle, WA, USA · Electrical and Computer Engineering and Computer Science & Engineering, University of Washington, Seattle, WA, USA

Long-Term Time Series Forecasting Koopman Operator Learning Latent Dynamics Modeling Transformer

Jun 22, 2026cs.LG

Distribution-Aware Diffusion-LLM for Robust Ultra-Long-Term Time Series Forecasting

Time series forecasting is a fundamental machine learning task. Recent work has explored Large Language Models (LLMs) for this purpose due to their strong generalization, pattern recognition, and zero-shot or few-shot capabilities. Despite their suitability for long-context learning, LLMs face challenges in multimodal settings: they lack calibrated probabilistic modeling for non-text data and struggle to align heterogeneous representations. To address these issues, we propose a new framework Diffusion-LLM that integrates a conditional diffusion model into an LLM-based forecasting pipeline. This joint design enables learning the conditional distribution of future data while improving semantic alignment in a shared latent space. We evaluate Diffusion-LLM on six long-term forecasting benchmarks, including ETT, Weather, and ECL. Our method consistently outperforms existing LLM-based baseline, achieving notable gains in ultra-long-term and few-shot forecasting and demonstrating the value of distribution-aware regularization for enhancing robustness and generalization in time series LLMs.

Falguni Ghosh, Vahid Hashemi, Bernhard Kainz

Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, 91052, Germany · AUDI AG, Auto-Union-Straße 1, 85057 Ingolstadt, Germany · Imperial College London, London SW7 2RH, United Kingdom

Long-Term Time Series Forecasting Probabilistic Forecasting LLM-Based Time Series Forecasting Conditional Diffusion Models

Jun 16, 2026cs.LG

Multiple cyclicity and Wavelet Decomposition with Channel Correlation for Long-term Time Series Forecasting

Cyclicity and trend are important components of time series data and many studies based on cyclicity and trend have achieved good results in long-term time series forecasting. However, we believe that current work neglects the influence of real-world inter-channel correlations in time series data which leads to suboptimal predictions. Furthermore, these models rely on complex designs to capture diverse information so that resulting in low computational efficiency. To address this challenge, we propose McWC, a long-term time series forecasting model that separately models the cyclicity, trend, and inter-channel correlations. Specifically, McWC first decouples cyclical information from data using a multi-layer cyclicity construction module. Then, it extracts inter-channel correlations using multi-layer perceptron. Next, it models and fuses the multi-layer high-frequency and low-frequency information from data using a multi-level wavelet decomposition module. Finally, it aggregates the results of different components to obtain the output. Simultaneously, we decouple intra-channel autocorrelations by calculating a loss function in the frequency domain. Experiments on six real-world datasets demonstrate that McWC achieves state-of-the-art performance, exhibiting excellent computational efficiency and historical information extraction capabilities.

Bin Wang, Heming Yang, Jinfang Sheng

School of Computer Science and Engineering, Central South University, HuNan, China.

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