Time Series Forecasting

Multimodal time series forecasting is crucial in real-world applications, where decisions depend on both numerical data and contextual signals. The core challenge is to effectively combine temporal numerical patterns with the context embedded in other modalities, such as text. While most existing methods align textual features with time-series patterns one step at a time, they neglect the multiscale temporal influences of contextual information such as time-series cycles and dynamic shifts. This mismatch between local alignment and global textual context can be addressed by spectral decomposition, which separates time series into frequency components capturing both short-term changes and long-term trends. In this paper, we propose SpecTF, a simple yet effective framework that integrates the effect of textual data on time series in the frequency domain. Our method extracts textual embeddings, projects them into the frequency domain, and fuses them with the time series' spectral components using a lightweight cross-attention mechanism. This adaptively reweights frequency bands based on textual relevance before mapping the results back to the temporal domain for predictions. Experimental results demonstrate that SpecTF significantly outperforms state-of-the-art models across diverse multi-modal time series datasets while utilizing considerably fewer parameters. Code is available at https://github.com/hiepnh137/SpecTF.

Deep time-series forecasting can be formulated as a distribution balancing problem aimed at aligning the distribution of the forecasts and ground truths. According to Imbens' criterion, balancing two distributions requires that their first moments be identical with respect to any balancing function. We demonstrate that existing objectives enforce moment matching only for one or two predefined balancing functions, thus failing to achieve full distribution balance. To address this limitation, we propose direct forecasting with kernelized moment balancing (KMB-DF). Unlike existing objectives, KMB-DF adaptively selects the most informative balancing functions from a reproducing kernel Hilbert space (RKHS) to enforce sufficient distribution balancing. A tractable and differentiable formulation is derived to enable estimation from empirical samples and seamless integration into gradient-based training pipelines. Extensive experiments across multiple models and datasets show that KMB-DF consistently improves forecasting accuracy and achieves state-of-the-art performance. Code is available at https://anonymous.4open.science/r/KMB-DF-403C.

Real-world time series exhibit complex and evolving dynamics, making accurate forecasting extremely challenging. Recent multi-modal forecasting methods leverage textual information such as news reports to improve prediction, but most rely on token-level fusion that mixes temporal patches with language tokens in a shared embedding space. However, such fusion can be ill-suited when high-quality time-text pairs are scarce and when time series exhibit substantial variation in characteristics, thus complicating cross-modal alignment. In parallel, mixture-of-experts (MoE) architectures have proven effective for both time series modeling and multi-modal learning, yet many existing MoE-based modality integration methods still depend on token-level fusion. To address this, we propose Expert Modulation, a new mechanism for multi-modal time series prediction that conditions both routing and expert computation on textual signals, enabling direct and efficient cross-modal control over expert behavior. Through theoretical analysis and experiments, our proposed method demonstrates strong improvements in multi-modal time series prediction. The current code implementation is available at https://github.com/BruceZhangReve/MoME

We propose two methodologies for modelling aggregated supply and demand curves in the EPEX SPOT Day\char45 Ahead market, emphasizing generative models as a way to recover distributional variability. The first is a low\char45 dimensional parametric representation that yields deterministic point forecasts; the second is a high\char45 dimensional order\char45 level representation that samples from a conditional distribution of plausible curves. Both model the full curve structure, enabling the analysis of price sensitivity, volume sensitivity, and price impact. The parametric representation uses plateau levels, elastic\char45 region boundaries, and polynomial coefficients, forecast with eXtreme Gradient Boosting. The main contribution is the generative representation, which uses price arrivals and volume\char45 increment marks and is implemented with conditional Denoising Diffusion Probabilistic Models. Using French EPEX data from 2021 to 2024, we evaluate both approaches through curve reconstruction and a price\char45 maker storage optimization problem. The parametric implementation provides a deterministic reference, while the diffusion\char45 based implementation produces distributions of plausible curves and achieves higher realized profits and smaller gaps to an oracle benchmark in the storage application.

Time series forecasting has witnessed significant progress with deep learning. While prevailing approaches enhance forecasting performance by modifying architectures or introducing novel enhancement strategies, they often fail to dynamically disentangle and leverage the complex, intertwined temporal patterns inherent in time series, thus resulting in the learning of static, averaged representations that lack context-aware capabilities. To address this, we propose the Dual-Prototype Adaptive Disentanglement framework (DPAD), a model-agnostic auxiliary method that equips forecasting models with the ability of pattern disentanglement and context-aware adaptation. Specifically, we construct a Dynamic Dual-Prototype bank (DDP), comprising a common pattern bank with strong temporal priors to capture prevailing trend or seasonal patterns, and a rare pattern bank dynamically memorizing critical yet infrequent events, and then an Dual-Path Context-aware routing (DPC) mechanism is proposed to enhance outputs with selectively retrieved context-specific pattern representations from the DDP. Additionally, we introduce a Disentanglement-Guided Loss (DGLoss) to ensure that each prototype bank specializes in its designated role while maintaining comprehensive coverage. Comprehensive experiments demonstrate that DPAD consistently improves forecasting performance and reliability of state-of-the-art models across diverse real-world benchmarks.

Accurate blood glucose forecasting using continuous glucose monitoring (CGM) data can support the early prediction of dysglycemic risk. However, current neural-network-based forecasting models treat CGM data as a purely numerical sequence without integrating the contextual information contained in CGM signal morphology. Recently, large language models (LLMs) have shown promise for time-series forecasting, yet their role as agentic context extractors in diabetes care remains largely unexplored. In this study, we bridge glucose forecasting and LLM-based contextualization by developing GlyRAG, a context-aware, retrieval-augmented forecasting framework that uses an LLM as a contextualization agent to summarize glucose morphology directly from a timed CGM window. The generated CGM-only narrative is embedded and fused with patch-based glucose representations, while a retrieval module incorporates similar historical training episodes through cross-attention. We evaluate GlyRAG on the OhioT1DM and AZT1D datasets for 5-, 30-, and 60-minute forecasting horizons. Compared with strong CGM-only baselines, GPT-4 GlyRAG significantly improves long-horizon root mean square error (RMSE) over PatchTST on both datasets. For example, RMSE decreases from 13.8 to 10.6 at 30 minutes and from 23.1 to 20.2 at 60 minutes on OhioT1DM. LLaMA 3.1 produces smaller but significant long-horizon gains, suggesting that the contextualization pipeline is not limited to GPT-4. Clinical error-grid analyses further show that approximately 85 percent of predictions fall within the clinically acceptable Clarke Error Grid Zones A and B. These results suggest that CGM-derived linguistic context and case-based retrieval can improve long-horizon glucose forecasting without requiring additional sensing modalities.

Predicting river flow in places without streamflow records is challenging because basins respond differently to climate, terrain, vegetation, and soils. Traditional basin attributes describe some of these differences, but they cannot fully represent the complexity of natural environments. This study examines whether AlphaEarth Foundation embeddings, which are learned from large collections of satellite images rather than designed by experts, offer a more informative way to describe basin characteristics. These embeddings summarize patterns in vegetation, land surface properties, and long-term environmental dynamics. We find that models using them achieve higher accuracy when predicting flows in basins not used for training, suggesting that they capture key physical differences more effectively than traditional attributes. We further investigate how selecting appropriate donor basins influences prediction in ungauged regions. Similarity based on the embeddings helps identify basins with comparable environmental and hydrological behavior, improving performance, whereas adding many dissimilar basins can reduce accuracy. The results show that satellite-informed environmental representations can strengthen hydrological forecasting and support the development of models that adapt more easily to different landscapes.

Process Model Forecasting (PMF) aims to predict how the control-flow structure of a process evolves over time by modeling the temporal dynamics of directly-follows (DF) relations, complementing predictive process monitoring that focuses on single-case prefixes. Prior benchmarks show that machine learning and deep learning models provide only modest gains over statistical baselines, mainly due to the sparsity and heterogeneity of the DF time series. We investigate Time Series Foundation Models (TSFMs), large pre-trained models for generic time series, as an alternative for PMF. Using DF time series derived from real-life event logs, we compare zero-shot use of TSFMs, without additional training, with fine-tuned variants adapted on PMF-specific data. TSFMs generally achieve lower forecasting errors (MAE and RMSE) than traditional and specialized models trained from scratch on the same logs, indicating effective transfer of temporal structure from non-process domains. While fine-tuning can further improve accuracy, the gains are often small and may disappear on smaller or more complex datasets, so zero-shot use remains a strong default. Our study highlights the generalization capability and data efficiency of TSFMs for process-related time series and, to the best of our knowledge, provides the first systematic evaluation of temporal foundation models for PMF.

Reliable forecasting of multivariate time series under anomalous conditions is crucial in applications such as ATM cash logistics, where sudden demand shifts can disrupt operations. Modern deep forecasters achieve high accuracy on normal data but often fail when distribution shifts occur. We propose Weighted Contrastive Adaptation (WECA), a Weighted contrastive objective that aligns normal and anomaly-augmented representations, preserving anomaly-relevant information while maintaining consistency under benign variations. Evaluations on a nationwide ATM transaction dataset with domain-informed anomaly injection show that WECA improves SMAPE on anomaly-affected data by 6.1 percentage points compared to a normally trained baseline, with negligible degradation on normal data. These results demonstrate that WECA enhances forecasting reliability under anomalies without sacrificing performance during regular operations.

Effective HFMD surveillance requires forecasts capturing both time-series patterns and contextual drivers such as school calendars, weather, and policy or surveillance reports. In clinical settings, forecasts must be trusted and actionable; thus, beyond point accuracy, decision-makers require concise, auditable explanations of why risk is expected to rise or fall. Classical models (e.g., ARIMA and Prophet) and foundation models (e.g., Chronos, Moirai, and TimesFM) treat external covariates as numerical inputs, lacking semantic reasoning to reflect epidemiological mechanisms or resolve conflicting signals. We propose a two-agent neuro-symbolic framework that decouples contextual interpretation from probabilistic forecasting. An LLM-based Event Interpreter ingests heterogeneous signals -- school schedules, weather summaries, government reports, and clinical guidelines -- and outputs a scalar transmission-impact signal. A Forecast Generator combines this signal with historical case counts to produce point forecasts that are mapped to probabilistic predictions through Poisson/negative-binomial moment matching. We focus on one-week-ahead rolling forecasts, aligning with weekly hospital-capacity planning and the rapid, context-driven inflections typical of HFMD. We evaluate on two datasets: Hong Kong surveillance (90 target weeks in 2023--2024) and Lishui hospital visits (33 target weeks in 2024). Against traditional and foundation-model baselines, our approach achieves competitive point accuracy while providing robust 90% intervals (coverage approximately 0.85--1.00) and concise rationales. This demonstrates that integrating domain knowledge through LLM-based agents can match strong numerical forecasters while yielding interpretable, context-aware forecasts aligned with public-health decision-making.

This study presents a controlled comparison of baseline Kolmogorov-Arnold Networks (KAN), implemented via PyKAN, and Long Short-Term Memory (LSTM) networks for the forecasting of stochastic, non-stationary financial time series. The two architectures are assessed in terms of predictive accuracy, computational efficiency, and interpretability, with accuracy measured by the Root Mean Square Error (RMSE) in normalised feature space. Under a direct multi-output forecasting protocol, LSTM attains clearly superior accuracy across all tested prediction horizons, consistent with its well-established effectiveness for sequential data modelling. Baseline KAN, although offering theoretical interpretability through the Kolmogorov-Arnold representation theorem, exhibits substantially higher error rates and limited practical applicability for time series forecasting in its standard form. Several specialised temporal variants -- including Temporal KAN and Time-Frequency KAN -- have since been proposed to address these sequential modelling limitations, but they lie outside the scope of the present study. KAN is observed to converge faster during training under the configurations tested, although direct runtime comparisons are constrained by methodological factors. These findings support the adoption of LSTM for accuracy-critical financial forecasting and establish an empirical baseline for standard KAN on stochastic sequential data, motivating further investigation of temporally-aware KAN architectures. The study benchmarks baseline KAN against baseline LSTM only; the results do not extend to specialised KAN variants designed for sequential data, nor to the broader family of temporal models.

Modeling the dynamics of financial Limit Order Books (LOB) at the message level is challenging due to irregular event timing, rapid regime shifts, and the reactions of high-frequency traders to visible order flow. Previous LOB models require cumbersome data representations and lack adaptability outside their original tasks, leading us to introduce LOBERT, a general-purpose encoder-only foundation model for LOB data suitable for downstream fine-tuning. LOBERT adapts the original BERT architecture for LOB data by using a novel tokenization scheme that treats complete multi-dimensional messages as single tokens while retaining continuous representations of price, volume, and time. With these methods, LOBERT achieves leading performance in tasks such as predicting mid-price movements and next messages, while reducing the required context length compared to previous methods.

Time series forecasting is typically treated as a single-pass numerical prediction task based on historical observations. However, real-world forecasting often requires expert-like reasoning, where temporal patterns, contextual cues, case-based references, and domain knowledge are jointly considered and iteratively revised. Existing statistical, deep learning, foundation, and LLM-based forecasting models mainly rely on pattern fitting, direct prompting, or static context augmentation, leaving interactive and reflective forecasting insufficiently explored. In this work, we propose AlphaCast, a human wisdom-LLM intelligence co-reasoning framework for interactive time series forecasting. AlphaCast does not require real-time human intervention; instead, it operationalizes human wisdom as structured forecasting support, including time-series features, contextual cues, case-based references, and domain knowledge. Built upon an expert-like think-forecast-reflect workflow, AlphaCast comprises three agents: an Investigator that constructs forecasting context from available evidence, a Generator that produces context-grounded forecasts, and a Reflector that evaluates plausibility and provides feedback for refinement. Extensive experiments demonstrate that AlphaCast improves forecasting accuracy and produces interpretable forecast explanations, offering a promising step toward interactive and context-aware time series forecasting.

Training time-series forecasting models requires aligning the conditional distribution of model forecasts with that of the label sequence. The standard direct forecast (DF) approach resorts to minimizing the conditional negative log-likelihood, typically estimated by the mean squared error. However, this estimation proves biased when the label sequence exhibits autocorrelation. In this paper, we propose DistDF, which achieves alignment by minimizing a distributional discrepancy between the conditional distributions of forecast and label sequences. Since such conditional discrepancies are difficult to estimate from finite time-series observations, we introduce a joint-distribution Wasserstein discrepancy for time-series forecasting, which provably upper bounds the conditional discrepancy of interest. The proposed discrepancy is tractable, differentiable, and readily compatible with gradient-based optimization. Extensive experiments show that DistDF improves diverse forecasting models and achieves leading performance. Code is available at https://anonymous.4open.science/r/DistDF-F66B.

The design of learning objectives is central to training time-series forecasting models. Existing learning objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which leads to the following two challenges: (1) they overlook the label autocorrelation effect among future steps, leading to biased learning objectives; (2) they fail to set heterogeneous task weights for different forecasting tasks corresponding to varying future steps, limiting the forecasting performance. To fill this gap, we propose a novel quadratic-form weighted learning objective, addressing both issues simultaneously. Specifically, the off-diagonal elements of the weighting matrix account for the label autocorrelation effect, whereas the non-uniform diagonals are expected to match the preferred weights of the forecasting tasks with varying future steps. On this basis, we propose a Quadratic Direct Forecast (QDF) learning algorithm, which trains the forecast model using the adaptively updated quadratic-form weighting matrix. Experiments show that our QDF effectively improves the performance of various forecast models, achieving state-of-the-art results. Code is available at https://github.com/Master-PLC/QDF.

While Time Series Foundation Models (TSFMs) excel in zero-shot tasks, their behavior under continual fine tuning is poorly understood. We present the first systematic study of catastrophic forgetting in TSFMs (TimesFM-2.0, Chronos-2) versus a specialized SamFormer model across synthetic and real-world energy forecasting benchmarks. Our results show that while fine-tuning improves new task accuracy, it consistently triggers forgetting, though larger models exhibit greater inherent robustness. Notably, employing forgetting mitigation techniques such as DER, levels the playing field: it provides disproportionate gains to smaller models, allowing them to match TSFM performance by the end of the continual learning sequence. These findings suggest that in realistic, non-stationary scenarios, the high computational cost of large foundation models may not be justified over smaller models equipped with effective mitigation strategies.

Benchmark quality is critical for meaningful evaluation and sustained progress in time series forecasting, particularly with the rise of pretrained models. Existing benchmarks often have limited domain coverage or overlook real-world settings such as tasks with covariates. Their aggregation procedures frequently lack statistical rigor, making it unclear whether observed performance differences reflect true improvements or random variation. Many benchmarks lack consistent evaluation infrastructure or are too rigid for integration into existing pipelines. To address these gaps, we propose fev-bench, a benchmark of 100 forecasting tasks across seven domains, including 46 with covariates. Supporting the benchmark, we introduce fev, a lightweight Python library for forecasting evaluation emphasizing reproducibility and integration with existing workflows. Using fev, fev-bench employs principled aggregation with bootstrapped confidence intervals to report performance along two dimensions: win rates and skill scores. We report results on fev-bench for pretrained, statistical, and baseline models and identify promising future research directions.

Inherent temporal heterogeneity, such as varying sampling densities and periodic structures, has posed substantial challenges in zero-shot generalization for Time Series Foundation Models (TSFMs). Existing TSFMs predominantly rely on massive parameterization to absorb such heterogeneity, as their static tokenization and positional encoding schemes entangle diverse temporal patterns into a fixed representation space, encouraging memorization rather than adaptation. To address this limitation, we propose Kairos, a flexible and parameter-efficient TSFM dedicated to forecasting tasks, which decouples temporal heterogeneity from model capacity through a novel tokenization perspective. Kairos introduces a dynamic patching tokenizer and a mixture-of-size encoding that adapt observational granularity to local information density, enabling fine-grained temporal abstraction without increasing model width or depth. In addition, we design a multi-granularity positional embedding based on dynamic rotary encodings, which conditions on instance-level spectral features and temporal structure induced by dynamic patching tokenization, allowing robust modeling of diverse temporal dependencies. Trained on a novel Predictability-Stratified Time-Series (PreSTS) corpus, Kairos achieves superior zero-shot performance with substantially fewer parameters on two mainstream benchmarks, GIFT-Eval and Time-Series-Library. The project page is at https://foundation-model-research.github.io/Kairos .

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.

Time Series Forecasting (TSF) faces persistent challenges in modeling intricate temporal dependencies across different scales. Despite recent advances leveraging different decomposition operations and novel architectures based on CNN, MLP or Transformer, existing methods still struggle with static decomposition strategies, fragmented dependency modeling, and inflexible fusion mechanisms, limiting their ability to model intricate temporal dependencies. To explicitly solve the mentioned three problems respectively, we propose a novel Dynamic Multi-Scale Coordination Framework (DMSC) with Multi-Scale Patch Decomposition block (EMPD), Triad Interaction Block (TIB) and Adaptive Scale Routing MoE block (ASR-MoE). Specifically, EMPD is designed as a built-in component to dynamically segment sequences into hierarchical patches with exponentially scaled granularities, eliminating predefined scale constraints through input-adaptive patch adjustment. TIB then jointly models intra-patch, inter-patch, and cross-variable dependencies within each layer's decomposed representations. EMPD and TIB are jointly integrated into layers forming a multi-layer progressive cascade architecture, where coarse-grained representations from earlier layers adaptively guide fine-grained feature extraction in subsequent layers via gated pathways. And ASR-MoE dynamically fuses multi-scale predictions by leveraging specialized global and local experts with temporal-aware weighting. Comprehensive experiments on thirteen real-world benchmarks demonstrate that DMSC consistently maintains state-of-the-art (SOTA) performance and superior computational efficiency for TSF tasks. Code is available at https://github.com/1327679995/DMSC.

Rainfall is an important environmental driver of water-quality variations through processes such as runoff, pollutant transport, dilution, and resuspension. Traditional mechanistic models can explicitly describe these processes but often require substantial process specification and site-specific calibration, limiting their flexibility under changing hydrological conditions. In this work, we explore a data-driven alternative by proposing RaiNet to jointly model multiscale water-quality dynamics and station-specific rainfall effects across relative lags and temporal scales. RaiNet employs LocTrend to capture irregular water-quality dynamics, constructs station-oriented rainfall events from gridded precipitation, and introduces XGateFusion for conditional lag-aware fusion across scales. We further release three real-world multimodal datasets comprising over 150,000 temporally aligned water quality observations and gridded precipitation raster images. Experiments show that RaiNet outperforms general time-series, water quality, diffusion-based, and spatiotemporal models by over 20%, while component-wise analyses confirm the distinct contribution of each module.

In the era of Industry 5.0, data-driven decision-making has become indispensable for planning in service systems such as tourism. This paper proposes a non-linear hybrid approach for forecasting tourist arrivals to Hong Kong from five major source markets before COVID-19 and to Sanya, China, through the COVID-19 shock. The method integrates multiple sources of Internet big data and employs a cooperative game-theoretic algorithm to identify coalitions of opinion leaders on social media, whose content then receives larger weights. The Empirical Wavelet Transform (EWT), applied causally, manages nonstationary attributes, and a Stacked Bi-directional Long Short-Term Memory (Stacked BiLSTM) network tuned by Bayesian optimization generates the forecasts. We prove that the Shapley values of the underlying trust-weighted coverage game have a closed form and that the coalition of maximal synergy can be found exactly in polynomial time. Relative to the same network fed only with past arrivals, Internet big data reduce the one-step-ahead root mean squared error by 4.1-41.7% across the Hong Kong markets and opinion-leader weighting by a further 10.1-49.4%, against 5.3% and 8.7% for Sanya. The approach compares favorably with published benchmarks and remains robust under volatile conditions, which suggests uses in logistics, supply chains and production planning.

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.

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.

Adapting pretrained Large Language Models (LLMs) for time series forecasting primarily relies on token-level linguistic-temporal alignment, leading to the stacking of logically disjointed tokens as input. While empirically effective, these methods overlook a fundamental capability of LLMs: modeling linguistic logic and structure, rather than merely processing token features. To address this limitation, we propose the Markovian-Guided Structure-Aware Alignment (MGSAA). Our core contribution is a framework that transcends pointwise feature matching to achieve global structural isomorphism between the linguistic and temporal domains. Specifically, MGSAA distills latent evolutionary patterns of language within LLMs into a Markovian state transition graph, which is transferred as a structural prior to the time series domain. Under this prior, time series patches are decoded into latent states and then aligned via state-constrained cross-attention. Ultimately, MGSAA generates a token sequence topologically isomorphic to the LLM's inherent mental structure, reactivating its reasoning capabilities for forecasting. Comprehensive evaluations across multiple benchmarks demonstrate that MGSAA achieves state-of-the-art performance, providing an innovative solution for cross-modal alignment in LLM for time series forecasting. Code is available at https://github.com/sunzju/MGSAA.

This paper introduces Fuzzy Attention Network-Based Transformer (FANTF), an approach for time series forecasting, classification, and anomaly detection. FANTF combines fuzzy logic concepts with transformer architectures through a novel fuzzy attention mechanism. The proposed mechanism is inspired by Type-2 fuzzy set theory. It uses a learnable Gaussian uncertainty perturbation to model uncertainty in attention relevance scores. This makes it possible for the model to accurately capture uncertainty in temporal relationships better than normal deterministic attention. Transformers capacity to record complex temporal patterns and multivariate connections is enhanced by FANTF. The approach is both architecture-agnostic and task-agnostic. The proposed fuzzy attention module can be used as a direct replacement for the standard attention layer. To evaluate its effectiveness, FANTF is integrated into five transformer backbones: Transformer, Informer, PatchTST, Crossformer, and iTransformer. The fuzzy attention mechanism produces softer and more diverse attention distributions. It provides uncertainty-aware interpretations through attention heatmap visualizations. Experimental evaluations on some real-life datasets reveal that FANTF enhances the forecasting performance, classification, and anomaly detection tasks over traditional transformer-based models. FANTF achieves statistically significant and competitive performance improvements over standard transformer-based baselines on most benchmark datasets (p < 0.05, Wilcoxon signed-rank test), observed under noisy and non-stationary time series conditions.

Time series forecasting (TSF) is important in real-world applications. Recently, Transformer-based methods have achieved strong performance by modeling token dependencies through attention mechanisms. However, existing methods are usually trained mainly with prediction error losses, which may cause models to exploit both critical and redundant token dependencies. Such redundant dependencies can introduce irrelevant information and weaken generalization. To address this issue, we propose a simple yet effective token dependency selection strategy. Specifically, by jointly introducing the attention entropy constraint and prediction error constraint, the model can identify fewer but more critical inter-token dependencies and perform forecasting based on them, thus avoiding the interference of redundant dependencies. The proposed method can be easily extended to various Transformer-based TSF models. Experiments on multiple TSF datasets demonstrate its effectiveness.

Time series forecasts are widely used to inform decisions under uncertainty. In many applications, these decisions involve classifying a time series based on outcomes that have yet to be observed. In this paper, we introduce Foreclassing, a new machine learning problem in which the class label of a time series is defined as a function of its future observations. We provide a mathematical formulation of Foreclassing and establish several theoretical results characterizing the problem. These results describe how additional future information affects Bayes classification risk, provide bounds on the classification error of learned Foreclassing models, and demonstrate how future observations used during training can yield finite-sample improvements over conventional classifiers. To provide a benchmark for future research, we propose ForeClassNet, a family of neural network models that jointly perform forecasting and classification while accounting for uncertainty in future outcomes. We investigate a variety of model architectures and evaluate their performance against conventional time series classifiers and forecasting-then-classify approaches. Experiments on datasets spanning weather, energy, and finance demonstrate the potential of Foreclassing to improve classification performance, probabilistic prediction, and downstream decision-making. Together, these theoretical and empirical contributions establish a foundation for future research on Foreclassing.

We introduce a local surrogate approach for explainable time-series forecasting. An initially non-interpretable predictive model to improve the forecast of a classical time-series 'base model' is used. 'Explainability' of the correction is provided by fitting the base model again to the data from which the error prediction is removed (subtracted), yielding a difference in the model parameters which can be interpreted. We provide illustrative examples to demonstrate the potential of the method to discover and explain underlying patterns in the data.

Recurrent neural networks are a successful neural architecture for many time-dependent problems, including time series analysis, forecasting, and modeling of dynamical systems. In the context of dynamical systems, training with backpropagation through time can lead to challenges arising from exploding or vanishing gradients. In this contribution, we introduce Koopman-informed recurrent neural networks, a computational approach to construct all weights and biases of a recurrent neural network without using gradient-based methods. The approach is based on a combination of random feature networks and Koopman operator theory for dynamical systems. The hidden parameters of a single recurrent block are sampled at random, while the outer weights are constructed using extended dynamic mode decomposition. This approach alleviates some problems with backpropagation commonly related to recurrent networks. The connection to Koopman operator theory also allows us to start using results in this area to analyze recurrent neural networks. In computational experiments on time series, forecasting for chaotic dynamical systems, control problems, and on real-world data, we observe that with comparable forecasting accuracy, the training time of the Koopman-informed recurrent neural networks is significantly improved when compared to models trained with commonly used gradient-based methods.