RATL: Learning from Retrieved Residuals for Robust Multivariate Time-Series Forecasting

Retrieval-augmented generation (RAG) complements parametric models with retrieved external evidence. The same idea is attractive for continuous-output regression, but directly reusing retrieved target values is often not robust when samples differ in output level, numerical scale, or local dynamics. Moreover, conventional forecasting pipelines generally use residuals for model optimization and error diagnosis, but do not retain individual historical residual examples as memory that can be accessed at inference time.For multivariate time-series forecasting, we propose RATL, a plug-in residual-retrieval and feedback-correction method. RATL freezes a base forecaster to construct retrieval keys and turns its historical forecast residuals into a train-only memory specific to that base model. At inference time, RATL retrieves residual trajectories from similar historical contexts subject to causal availability constraints, then uses a set-aware router operating over forecast blocks and variables to select and combine these trajectories. Experiments show that historical residuals matched to the current context contain reusable forecasting information and that RATL improves frozen base forecasters in most experimental settings. Ablations further show that learned routing strengthens raw residual feedback, while validation-based correction-strength selection limits residual over-injection.On real-world benchmarks, we use iTransformer as the primary frozen base forecaster, compare against multiple strong forecasting baselines, and test transferability across backbones. The results show that RATL can further improve base-forecaster performance in most settings.Overall, RATL shifts the retrieved object from historical target values to base-model-specific historical forecast errors, providing a plug-in, residual-memory-based paradigm for learned feedback correction in continuous-output forecasting.

Compositional Spectral Prompts for LLM-based Online Time Series Forecasting

To address the sequential and evolving nature of time series, the Online Time Series Forecasting (OTSF) task has been extensively studied in multiple domains. Existing research focuses on adapting to non-stationary environments by employing memory buffer-based retrieval strategies. However, we observe that such frameworks struggle with long-term adaptation and fail to generalize to unseen patterns. To this end, we introduce CoSPOT, an LLM-based online time series forecasting framework that leverages a pre-trained LLM as the backbone online forecaster, motivated by its strong few-shot capabilities. For efficient online adaptation, CoSPOT keeps the LLM frozen and employs compositional spectral prompts grounded in frequency-domain bases to guide the model with the overall distribution of the input, thereby substantially reducing the number of parameters updated during the online phase. Specifically, CoSPOT decomposes time series into frequency bases and composes the corresponding spectral basis prompts according to their amplitudes, allowing unseen patterns to be represented as new combinations of learned basis prompts. Our extensive experiments on real-world datasets demonstrate the superiority and practicality of CoSPOT across challenging online scenarios, including extended online phases and cross-dataset settings with substantial distribution shifts. Our code is available at https://github.com/seungyoon-Choi/CoSPOT.

DynG-Diff: A State-Aware Dynamic Guidance Diffusion Framework for Probabilistic Time Series Forecasting

Probabilistic multivariate time series (MTS) forecasting is crucial for modeling complex dynamical systems. However, existing diffusion-based methods rely on task-specific conditional paradigms that lack flexibility and struggle with inherent "information heterogeneity"--the significantly varying noise levels and evolutionary patterns across variables. To address this, we propose DynG-Diff, a variable-sensitive dynamic guidance diffusion framework for probabilistic multivariate time-series forecasting: (1) DynG-Diff adopts a two-stage separated training strategy and uses an unconditional diffusion backbone to model the joint distribution of multivariate time series. (2) DynG-Diff introduces a lightweight state-aware policy network that adaptively infers variable reliability from real-time noisy states and one-step denoising estimates, outputting a dynamic guidance strength matrix. (3) DynG-Diff mathematically formulates this dynamic weight as the local precision of the observation distribution, enabling precise guidance for high-confidence variables during inference while filtering out interference from anomalous noise. Extensive experiments on real-world benchmarks demonstrate competitive probabilistic forecasting performance against state-of-the-art conditional diffusion models and improved robustness under severe observation corruption.The implementation code is available at: https://github.com/TT-20011031/DynG-Diff

RecKAN: Kolmogorov-Arnold Networks with a Learnable Recursive Polynomial Basis

Kolmogorov--Arnold Networks (KANs) replace the fixed scalar weights of a standard network with learnable univariate functions on each edge, but existing variants still fix the \emph{basis} that those functions are built from: B-splines, Chebyshev polynomials, wavelets, or Jacobi polynomials, and learn only the combination weights over it. We introduce RecKAN, which instead defines the basis itself by a second order polynomial recurrence, $R_{n+1}(x) = (ax^2+bx+c)R_n(x) + (dx+e)R_{n-1}(x)$, whose five coefficients are learned jointly with the network. We show this recurrence recovers several classical polynomial families including both kinds of Chebyshev polynomials, Fibonacci, Pell, and Jacobsthal polynomials as special cases, and prove that its degree grows linearly in $n$ exactly on the sub-family containing all of them, giving a concrete sense in which the learned basis can move beyond any fixed classical choice. Across multiple benchmark datasets spanning image, text, biomedical time series classification, and time series forecasting, RecKAN outperforms three parameter-matched KAN baselines (Chebyshev, Jacobi, and spline based) on all classification tasks and achieves the lowest MSE on the ETTh1 forecasting benchmark. Additionally, when used as a classifier head with a convolutional backbone, RecKAN achieves higher accuracy than standard MLP heads on Fashion MNIST, CIFAR-10, and SVHN. On a synthetic function fitting benchmark it tracks a sharply oscillatory target that a parameter comparable MLP under fits. We further show that the learned recurrence coefficients are interpretable: on the task requiring the most local structure, training moves the basis away from the linear degree growth regime that contains every classical family we identify, consistent with our theoretical analysis of what that structural shift enables.

Learning to Difference: Adaptive Reversible Differencing (AdaRDiff) for Time Series Forecasting

Reliable long-horizon time series forecasting is an important yet difficult problem. Trends and seasonality introduce complex temporal structure that challenges learning-based forecasting models. Differencing, which subtracts nearby past values to remove such structure, is the classical remedy, but its reliance on hand-picked orders and periods has kept it largely absent from recent deep architectures. We propose \textbf{\underline{Ada}}ptive \textbf{\underline{R}}eversible \textbf{\underline{Diff}}erencing \textbf{(AdaRDiff)}, a generalized differencing approach that uses learnable weights to simplify the series through weighted differencing with previous time instants. This yields stabilized residuals on which forecasting is performed, after which the removed components are restored autoregressively to reconstruct the forecast, capturing trend and seasonality jointly through a single operator. This reconstruction admits a closed-form convolutional expression, which parallelizes on GPU and yields up to $33.7\times$ speedup over the naive recurrence. We furthermore rely on a two-phase training schedule that separates temporal structure discovery from reconstruction learning, as suggested by a theoretical analysis of the gradient when using a linear forecasting model. AdaRDiff attains state-of-the-art forecast accuracy across eight benchmarks spanning electricity, weather, traffic, and energy, at negligible parameter cost. Furthermore, it is designed as a plug-and-play module: integrating AdaRDiff improves eight diverse backbones, from linear models to Transformers, in the large majority of cases, by up to $25.9\%$ with a linear backbone and $18.3\%$ with iTransformer.

CEDAR: Controlled and Event-Driven Demand Forecasting via Residual Decomposition

Forecasting in large-scale e-commerce marketplaces is increasingly required to support planning: merchants need to evaluate sales outcomes under future action sequences such as budget schedules, rather than passively predicting what happens next. However, most existing time series forecasting (TSF) approaches remain inherently passive. Even when incorporating operational decisions as auxiliary covariates, they typically optimize for correlation-based extrapolation under historical policies. This design suffers from autoregressive inertia and conflates endogenous market evolution with decision-induced transitions, leading to policy-insensitive rollouts and unreliable counterfactual analysis. To bridge this gap, we propose CEDAR (Controlled and Event-Driven Demand forecasting via Action-aware Residual decomposition), a two-stage framework for robust decision-conditioned simulation. In Stage I, an Action-Interleaved Transformer learns controllable action-conditioned state transitions for rollout under planned interventions. In Stage II, a Residual Correction Module leverages external event signals and LLM-assisted text representations to align noisy event descriptions with product context and correct event-driven deviations. Our study is enabled by a large-scale real-world dataset from Alibaba 1688, comprising approximately 32 million product trajectories with paired state-action sequences and aligned event signals. Extensive offline experiments and online controlled experiments in production demonstrate that CEDAR consistently improves simulation accuracy over strong TSF baselines and delivers practical gains for real-world budget planning.

When Does Context Routing Help? A Systematic Study of Multi-Modal Fusion in Time Series Forecasting

Multi-modal time series forecasting methods integrate auxiliary context into temporal predictions through increasingly sophisticated fusion mechanisms. A growing body of work reports substantial gains, yet it is often unclear whether they reflect genuine use of the context or incidental architectural effects. We ask a narrower, checkable question: when can auxiliary context help a forecaster at all? We identify two dataset-level conditions that must both hold: (1) the target is not dominated by a last-value shortcut (low autocorrelation rho_h), and (2) the context carries information about the target beyond history (non-zero conditional mutual information delta; when delta=0 no predictor can benefit---a distribution-free result). Through controlled experiments on MoME (a 14.3B-parameter mixture-of-experts model, 6 datasets, 10 seeds) and four additional fusion mechanisms implemented within a single-backbone testbed (5 datasets), we find that when both conditions hold, text-conditioned expert modulation contributes a sizeable MSE reduction; when either fails, the contribution collapses to the capacity floor of the modulation pathway and carries no context-attributable signal. We establish causality through two interventions: adding a shortcut to MoME suppresses routing contribution by 77-93% across 3 datasets; progressively corrupting context quality drives the context-specific benefit from +44% to negative. We validate the autocorrelation component of our diagnostic on 27 Monash Archive datasets. We provide a calibrated pre-training diagnostic that, on the datasets we test, yields no false positives in well-powered settings. We are explicit about the asymmetry of our evidence: the negative arm is broadly reliable, while the large positive magnitudes come from a single model family (MoME) and are corroborated only in direction by the testbed.

Fuzzy-MoE: Interpretable Regime-Conditioned Expert Routing for Non-Stationary Multivariate Time Series Forecasting

In non-stationary multivariate time series, different variables and samples often exhibit heterogeneous latent dynamic states, while existing deep forecasting models usually compress them into a unified end-to-end mapping, leading to suboptimal modeling of time-varying dynamics and limited interpretability regarding which forecasting mechanism is activated under different latent states. To overcome these limitations, we reformulate time series forecasting as a unified framework of latent temporal state identification and interpretable expert routing, and propose Fuzzy-MoE, a fuzzy logic-based dynamic Mixture-of-Experts model. Fuzzy-MoE consists of multiple parallel expert mapping networks and a dual-view fuzzy router. By jointly exploiting local convolutional dynamics and global segmented statistics, the router infers latent temporal states and computes expert activation strengths through learnable Gaussian membership functions, enabling explicit IF-THEN rule-based expert selection. This fine-grained routing strategy allows different variables within the same sequence to activate different experts, effectively capturing heterogeneous temporal dynamics while improving model interpretability. Experimental results on multiple public time series benchmark datasets show that Fuzzy-MoE significantly outperforms mainstream forecasting methods in forecasting accuracy. Moreover, fuzzy memberships and rule activations provide interpretable routing diagnostics, demonstrating the effectiveness of the proposed framework in both forecasting performance and mechanism transparency. Unlike traditional MoE models that use black-box routing, Fuzzy-MoE`s routing is based on clear, interpretable fuzzy rules. This makes the expert selection transparent and traceable.

DecoVAE: a Lightweight Interpretable Trend-Seasonal VAE Framework for Efficient Probabilistic Time Series Forecasting

Probabilistic time series forecasting remains challenging, largely because modeling distinct trend and seasonal dynamics requires specialized approaches. Existing methods often fail to capture the unique inner properties of these components, lack interpretability, or suffer from heavy memory and runtime overhead. To address these limitations, we propose DecoVAE, a lightweight interpretable trend-seasonal VAE framework that explicitly decomposes time series into trend and seasonal components by applying domain-specific inductive biases. The trend stream enforces structural smoothness using a differential regularizer on the latent trajectory, analogous to the Hodrick-Prescott filter. Concurrently, the seasonal stream operates in the frequency domain via a complex Gaussian VAE, natively capturing the amplitude and phase of periodic patterns. Extensive evaluations across seven real-world benchmarks show that DecoVAE consistently outperforms strong baselines. It achieves reductions of up to 14.96\% in CRPS and 23.30\% in NMAE for short-term forecasting, and up to 52.68\% and 26.51\% for long-term horizons. Crucially, DecoVAE yields these accuracy gains while remaining highly efficient, reducing model weight by up to 93\% and accelerating speed by up to 74\% compared to the second-best method.

CLaST: Context-aware Contrastive VAE for Probabilistic Time Series Forecasting

Probabilistic forecasting models are widely used for time series forecasting in domains such as energy systems, finance, medicine, and transportation. In recent years, deep generative models have shown strong results on probabilistic forecasting, yet many conventional approaches struggle to capture internal temporal dependencies, leading to latent representations with limited expressive power. To address this limitation, we propose \textit{CLaST}, a VAE framework for probabilistic multivariate time series forecasting. Unlike existing generative models, CLaST learns embeddings that preserve contextual similarity between observations through our contrastive loss function. Experiments across nine widely adopted benchmarks demonstrate that CLaST consistently surpasses strong baseline methods. In short-term forecasting tasks, our approach achieves improvements of up to $16.4\%$ in CRPS and $14.4\%$ in NMAE over the second-best method. Furthermore, in long-term prediction CLaST attains superior overall performance, exceeding the second-best method by up to $48.6\%$ and $25.1\%$ in CRPS and NMAE, respectively.

Rethinking Patch Based Multivariate Time Series Forecasting with Semantic Structured Partitioning

Multivariate time series forecasting (MTSF) is a fundamental task in many real world applications. Existing patch based forecasting methods generally fall into three categories: fixed partitioning, multi-scale partitioning, and extendable partitioning. Fixed partitioning often breaks meaningful temporal boundaries, multi-scale partitioning may introduce redundant representations across scales, and extendable partitioning improves flexibility but still lacks an explicit mechanism for organizing semantic structure and modeling interactions among heterogeneous temporal patterns. To address these limitations, we propose SCPaT, a Transformer based framework built on semantic structured partitioning. SCPaT first decomposes input sequences into semantically consistent units through adaptive semantic unit generation, then constructs a dynamic semantic graph to model directed dependencies among these units and organize them into higher order semantic blocks. Based on these structured representations, an importance aware routing mechanism adaptively dispatches different semantic blocks to different experts for customized modeling. Extensive experiments on 12 real world datasets demonstrate the effectiveness of SCPaT.

AsyTO: Asymmetric Temporal Operator for Parameter-Efficient Multivariate Time Series Forecasting

Multivariate time-series forecasting faces a structural dilemma: sharing one temporal predictor across variables is parameter-efficient but forces heterogeneous variables through an identical history-to-future map, whereas learning an independent predictor per variable restores flexibility at a cost that grows with the product of variable count, context length, and horizon. We argue that this dilemma dissolves once the object being compressed is the forecasting operator rather than the observed series. Auditing per-variable linear history-to-future maps across standard benchmarks, we find that a phase-locked seasonal component paired with a compact residual operator outperforms a dense phase-blind reference in most audited settings. The residual transport is also directional: lag-invariant alternatives consistently underperform asymmetric history-to-future maps. Guided by this structure, we propose AsyTO, an Asymmetric Temporal Operator that factorizes the tensor of per-variable operators into shared but distinct history-reading and future-writing temporal modes with per-variable mode-wise gains, complemented by a low-rank periodic prototype and a cycle-separable factorization of the temporal modes. Each forecast reads only its own variable's history, so parameters and compute grow linearly in the number of variables. Across eleven benchmarks and multiple forecast horizons, AsyTO attains the best lightweight error in 30 of 44 dataset-horizon settings, locating at the accuracy-compute Pareto frontier.

Forecast Collapse in Time-Series Foundation Models

When forecasting hourly returns for 1,000 US equities, we observe an unexpected phenomenon: predictions become nearly flat and show poor stock ranking, as measured by cross-sectional correlation. We call this forecast collapse. Surprisingly, the phenomenon largely disappears when forecasting trading volume under the same setting. We investigate forecast collapse across time-series foundation models (TSFMs), twelve deep-learning forecasting models, and 97 public benchmark configurations, and find that it is closely tied to target predictability. We identify two distinct reasons behind it: low predictability limits the amplitude of calibrated point forecasts, while per-series objectives leave cross-series structure unidentified. These findings reveal a calibration-ranking tradeoff: optimizing squared error leads to flat predictions, whereas directly optimizing cross-sectional correlation improves ranking but can inflate forecast amplitude by more than an order of magnitude. To address this tradeoff, we introduce CalibRank, a simple objective that balances calibration and ranking. On Finance1K, CalibRank nearly triples cross-sectional correlation while keeping amplitude close to the target, and improves correlation on all tested models. Our results reveal a blind spot in conventional time-series evaluation: per-series metrics can hide failures in cross-series structure needed by downstream decisions.

When Denoising Hurts: Rethinking the Terminal Step of Diffusion Time Series Forecasters -- Extended Version

Diffusion models offer a natural way to model uncertainty in time series forecasting, yet their iterative sampling process is often treated as a uniformly beneficial refinement procedure. Our study challenges this view by examining how forecast quality evolves throughout reverse diffusion. We find that general temporal structure is often recovered at relatively high noise levels, whereas continued low-noise refinement can introduce statistical drift and degrade the final forecast. Our analysis further suggests that this behavior explains why prior methods often favor relatively narrow diffusion architecture and schedule design. Building on this observation, we propose a label-free global stopping criterion that detects the optimal termination point, eventually speeding up inference and improving predictive accuracy. Additionally, since early stopping terminates inference in high-noise regions, we propose a Bernoulli timestep sampler that concentrates training on this region while preserving coverage of the full diffusion process. Extensive experiments conducted across eight real-world datasets demonstrate the superior performance of our method compared to existing approaches.

Model-agnostic Retrieval-Augmented Extended Forecasting for time series

Time series forecasting with pretrained foundation models has demonstrated strong zero-shot capabilities. However, achieving optimal performance on time series with short or negligible historical data in domain-specific applications typically requires adaptation via either fine-tuning or RAG. While fine-tuning is effective, it incurs substantial computational costs. This work explores RAG within univariate time series (Retrieval Augmented Generation) as a more efficient alternative, in particular RAF (Retrieval Augmented Forecasting), and introduces RAEF (Retrieval-Augmented Extended Forecasting), a model-agnostic method built upon RAF. RAEF incorporates key refinements to the retrieval and aggregation mechanisms: (1) direct retrieval in input-space rather than embedding-space, reducing inference overhead, and (2) concatenation-based aggregation that preserves temporal structure instead of averaging. Empirical evaluation across multiple benchmark datasets demonstrates that RAEF outperforms RAF in both accuracy and inference overhead. Furthermore, comprehensive comparisons with zero-shot and fine-tuned foundation models show that RAEF achieves competitive or superior performance to fine-tuning while avoiding its computational burden, establishing it as a practical and scalable approach for domain adaptation in time series forecasting.

Into the ORBIT for Time Series: Training Regimes for Foundation Models

Time series foundation models (TSFMs) have advanced primarily through architectural innovation, while training regimes for large-scale heterogeneous corpora remain under-explored. As a result, pre-training distributions are often poorly controlled with respect to domain imbalance, context requirements, prediction horizons, and missingness. We introduce ORBIT (Omni-Range Bootstrap Incremental Training), a training paradigm that makes this distribution explicit and controllable. ORBIT combines Bootstrap Multi-Level Sampling, which controls dataset exposure and samples records, target variables, context windows, and prediction horizons, with Omni-Range Incremental Training, which varies context lengths and prediction horizons throughout a single training stage. Under ORBIT, we train Falcon-2.0, a simple univariate encoder-only Transformer with missingness-aware triple-channel patch tokenization and parallel patch prediction. We further introduce Rank-Guided Cross-Depth Alignment, a training objective that uses late-layer representations as stop-gradient teachers for shallow layers without additional inference cost. Evaluations on GIFT-Eval and fev-bench demonstrate strong zero-shot forecasting performance across diverse domains and frequencies.

FM-LLM: A frequency-enhanced mixture-of-experts framework for adapting LLMs to time series forecasting

Recent advances in Large Language Models (LLMs) have spurred cross-modal solutions for time-series forecasting. However, existing methods rely heavily on textual prompts for modality alignment-introducing nontrivial computational overhead and failing to leverage the rich spectral dynamics inherent in time-series data. To enable prompt-free, frequency-aware adaptation of frozen LLMs, we propose FM-LLM (Frequency-Enhanced Mixture-of-Experts for adapting LLMs to Time Series Forecasting), an autoregressive framework grounded in constrained asymmetric coupling. A Fourier Analysis Network (FAN)-based spectral token aligner injects structured harmonic representations directly into the frozen LLM with numerical compatibility. An asymmetric Mixture-of-Experts (MoE) decoder enforces role separation: shared experts with lightweight FAN layers reconstruct the global periodic backbone, while routed experts-restricted to standard FFNs-specialize in modeling non-periodic residual dynamics. A time-frequency hybrid loss function jointly optimizes temporal accuracy and spectral consistency, mitigating error accumulation during long-horizon autoregressive rollouts. Evaluated across eleven public benchmarks, FM-LLM achieves state-of-the-art performance on 59 out of 78 evaluation metrics. Compared to the strongest autoregressive LLM-based baseline, it delivers average improvements of 5.3% in MSE and 5.6% in MAE, with maximum gains reaching 8.0% for MSE and 8.4% for MAE. FM-LLM also demonstrates robust transferability, maintaining superior performance in 10% few-shot and zero-shot forecasting scenarios.

REATS: LLM Reasoning-based Ensemble Learning for Adaptive Time Series Forecasting

Due to the diversity of real-world time series, no single forecasting model consistently dominates across all samples. Ensemble learning addresses this by combining complementary model strengths, yet existing methods rely on fixed rules or black-box models based solely on numerical inputs, failing to leverage LLM reasoning for interpretable weighting decisions. We propose REATS, which leverages LLM reasoning capabilities as an intelligent ensemble router that jointly processes textual temporal pattern descriptions and numerical features to produce interpretable, sample-adaptive ensemble weights through chain-of-thought reasoning. To enable effective LLM-based ensembling, we study its key design choices and propose: (i) a structured input pipeline that transforms raw time series into hybrid textual--numerical representations with fixed token cost, enabling rule-based chain-of-thought construction without API dependency, augmented with retrieved similar-sample priors; (ii) a diverse multi-row weight supervision scheme coupled with a token-efficient percentage-table format that reduces numerical complexity and mitigates LLM hallucinations; and (iii) a two-stage fine-tuning framework combining SFT with GRPO, where a reciprocal reward mapping transforms the continuous unbounded MSE gap into bounded signals with amplified near-oracle sensitivity, addressing the uniform sensitivity and outlier-dominated advantage compression inherent in naive reward designs for regression-based GRPO. Experiments on eight benchmarks demonstrate that REATS outperforms competitive ensemble baselines while providing natural language explanations and demonstrating strong transfer learning and out-of-domain generalization to unseen candidate models.

Hybrid Neural-Classical Correction for Frozen Time Series Foundation Models: A Comprehensive Ablation Study on High-Frequency Stock Prediction

Foundation models for time series forecasting demonstrate impressive zero-shot generalization but often underperform on specialized domains such as high-frequency finance. We present a comprehensive study of hybrid neural-classical correction for adapting frozen TimesFM (200M parameters) to stock return prediction during the volatile opening trading hour. We compare two neural correction architectures - AttnCorrect (multi-head self-attention, approximately 471K parameters) and GatedLinear (low-rank bilinear projection with gating, approximately 49K parameters) - each augmented with Random Forest residual learning. Through systematic ablation across 10 major technology stocks (NVDA, MSFT, AAPL, GOOG, GOOGL, AMZN, META, AVGO, TSLA, NFLX) spanning 2 million data points, we reveal critical insights: (1) The hybrid neural-classical approach achieves 0.597 pooled correlation and 6.4x mean per-day correlation improvement over frozen TimesFM; (2) Classical residual learning (Random Forest) provides the largest single-component contribution, matching or exceeding the neural correction component; (3) Simpler neural architectures surprisingly outperform complex ones when classical residual learning is removed; (4) Self-attention provides the largest neural-only contribution. GatedLinear+RF achieves best overall performance with 9x fewer neural parameters than AttnCorrect+RF. We report three complementary correlation metrics - mean per-day, cross-day cumulative, and pooled - to provide a complete picture of predictive quality. Our results provide practical guidance: effective foundation model adaptation requires careful integration of neural and classical components, with classical methods playing a crucial complementary role.

End-to-End Neural Decomposition with Koopman Operators for Time-Series Forecasting

Koopman theory offers a linear-operator view of nonlinear sequence dynamics by lifting observations into a space where evolution is governed by a linear time-invariant Koopman operator. While the Koopman operator provides a linear representation of nonlinear dynamics, it is generally infinite dimensional and defined under time-invariant assumptions. To model non-stationary signals with frequency-dependent behavior, a frequency-varying extension is required. In recent years, deep learning has been increasingly employed to exploit its powerful function-approximation ability for learning the Koopman operator. In this study, we propose a novel approach called neural decomposition Koopman (NDKoop), an end-to-end architecture that integrates a learnable signal decomposition module with both frequency-independent and frequency-dependent Koopman based networks for sequence forecasting. To the best of our knowledge, this is the first work to jointly realize end-to end Koopman modeling and signal decomposition within a unified neural framework. We demonstrate that decomposing a signal into a frequency-independent trend component and a frequency-dependent periodic component, each governed by a corresponding Koopman operator, improves prediction accuracy when perfect linearization is unattainable. Numerical experiments across several forecasting benchmarks indicate that the proposed NDKoop provides strong performance.

QFCQT: A Chaotically Gated Quantformer Framework for Volatile Time-Series Forecasting

Forecasting non-stationary time series remains difficult due to long-range dependencies, local volatility bursts, structural shifts, and nonlinear oscillatory behaviors. Although Transformer-based forecasters are effective for modeling long-term temporal dependencies, their feed-forward blocks typically rely on smooth static activations that are insufficiently sensitive to abrupt regime changes. Motivated by quantitative Transformer designs and oscillator-based nonlinear activations, we propose QFCQT, short for Quantum-Fractal-inspired Chaotically Gated Quantformer, for robust forecasting under complex volatile dynamics. Here, "quantum-fractal-inspired" denotes a computational analogy based on soft oscillator superposition and multi-scale nonlinear responses, rather than a formal quantum-mechanical or fractal-theoretic derivation. QFCQT consists of three main components: (1) a Quantformer-style numerical encoder that directly processes multivariate inputs via linear embedding; (2) a learnable Lee-oscillator activation module that maps scalar pre-activations to dynamic oscillatory responses and summarizes them through Max-over-Time pooling; and (3) a smooth-chaotic gated fusion mechanism that adaptively balances conventional smooth activations and chaos-sensitive responses. Furthermore, instead of using a single fixed oscillator, QFCQT employs a soft superposition of eight parameterized Lee oscillator families to adaptively capture different nonlinear response patterns across regimes. Experiments on ETTh1, ETTh2, and A-share Stock Index benchmarks show that QFCQT consistently outperforms strong baselines, including Informer, LogTrans, LSTMa, HAT, and COTN.

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.

TS-RAG: Retrieval Augmented Generation for Time Series Forecasting

While deep learning models, particularly transformer-based architectures, have shown impressive performance in time series forecasting, the application of retrieval-augmented generation (RAG) in this domain remains limited. Since RAG has proven effective in enhancing the capabilities of large language models by incorporating relevant external information, retrieving similar time series sequences as references might also improve accuracy in time series forecasting tasks. However, most time series models are constrained by limited training data, smaller parameter scales, and a lack of the extensive generative capabilities found in large language models. Simply concatenating reference sequences into the prompt, as done in language models, may not yield the expected results. To address these challenges, we propose a novel approach, TS-RAG, which leverages RAG to enhance forecasting performance. The framework introduces specially designed reference tokens to effectively fuse information from the input sequence with that from retrieved similar sequences, enabling a more robust capture of complex temporal dynamics. Experimental results demonstrate that TS-RAG achieves consistent state-of-the-art performance across several real-world forecasting benchmarks.

Multivariate Time Series Forecasting needs Cross Variable Loss

Multivariate time series forecasting presents unique challenges because future variables often co-evolve under shared system dynamics. While existing studies mainly focus on cross-variable dependencies in historical observations, dependencies among future values are much less explored. Specifically, modern forecasting models largely follow the Direct Forecasting (DF) paradigm, generating multi-step forecasts with point-wise objectives that do not explicitly constrain cross-variable structure. In this work, we show that the DF objective is mismatched in the presence of cross-variable and lagged dependencies, revealing an objective gap. To address this issue, we propose \textbf{C}ross-\textbf{V}ariable \textbf{Loss} (CvLoss), a plug-in structural regularizer that constrains forecast residuals on a cross-variable graph. CvLoss penalizes inconsistent edge-wise residual differences over forecast patches, encouraging consistency across both synchronous and asynchronous interactions. Our experiments show that CvLoss consistently improves competitive forecasting models, outperforms representative learning objectives, and is compatible with a variety of forecasting backbones.

Align-RAG: Alignment Is All You Need for TSFM In-Context Learning

Retrieval-augmented forecasting promises to adapt frozen Time Series Foundation Models (TSFMs) to new domains without fine-tuning, but recent methods typically rely on learned fusion modules, i.e., trained adapters that merge retrieved examples into the backbone's forecast, based on the assumption that frozen backbones cannot dynamically incorporate retrieved context on their own. We show this assumption is unnecessary. We introduce Align-RAG, a training-free method that applies a closed-form per-pair amplitude rescaling and integer-lag phase shift to retrieved past-future windows before they enter a frozen backbone's context. With no learned parameters, Align-RAG outperforms the state-of-the-art trained retrieval adapter on a frozen Chronos-Bolt on all seven datasets of the standard benchmark (avg -3.75% MSE), showing that the gains previously attributed to learned fusion are recoverable without any training. Align-RAG further improves zero-shot MSE on four additional frozen TSFMs with various architectures by 2.5% to 13.7% per backbone with no per-backbone tuning. To probe why alignment helps, we compare the frozen backbone's prediction shift under aligned demonstrations to the closed-form ridge prediction shift on the same pairs. We find that aligned demonstrations induce prediction shifts that track a closed-form ridge predictor on the same pairs, with a future-shuffle control ruling out a futures-averaging account. Together, these results indicate that frozen TSFMs already support dynamic in-context use of retrievals, and that closed-form alignment should be the default baseline for retrieval-augmented forecasting before any fusion module is trained. Code available at: https://github.com/masadi-99/align-rag

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.

RouteTS: Frequency-Time Routing for Time Series Forecasting

Real-world time series inherently intertwine global periodic structures with localized non-stationary variations. Existing approaches process these heterogeneous dynamics within a single computational domain, incurring fundamental limitations: time-domain models suffer from periodic misalignment over long horizons, while frequency-domain models over-smooth transient spikes. We argue that the optimal computational domain is not a property of the model, but of the data itself. Based on this principle, we propose RouteTS, a unified forecasting framework that partitions the frequency spectrum via amplitude routing and delegates components to their mathematically optimal domains. Dominant frequencies are processed by a complex-valued linear predictor in the frequency domain to preserve periodic structure, while residual spectral energy is reverted to the time domain and modeled by a lightweight MLP for local variations. Extensive experiments demonstrate that RouteTS achieves competitive prediction accuracy across diverse real-world datasets, with routing decisions guided by the underlying spectral signature. Furthermore, the lightweight design of RouteTS provides significant computational efficiency advantages, offering a principled solution to the longstanding dilemma between global periodicity and local transience.

POEM: Phase-Aware $\mathrm{SO}(2)$ Feature Rotation for Time Series Forecasting Under Periodicity Drift

Deep learning has advanced time series forecasting, but periodicity drift, in which cycle timing and phase vary over time, remains a challenging problem. Existing methods predominantly model these sequences on fixed time grids, suffering from a limited ability to accommodate phase-related variation. To address this limitation, we propose \textbf{POEM}, a phase-aware forecasting framework based on latent feature rotation using the special orthogonal group in two dimensions, denoted by $\mathrm{SO}(2)$. POEM aims to reduce the phase-related variability by learning a phase-correction coordinate and applying an invertible $\mathrm{SO}(2)$-based rotation to paired latent features. To extrapolate this correction coordinate, Directional Phase Increment Attention (DPIA) retrieves historical phase increments from similar temporal contexts and integrates them into future phase corrections. Experiments demonstrate that POEM achieves competitive performance, while qualitative visualizations suggest that the learned phase-aware transformation makes latent trajectories more regular.

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.

Beyond Magnitude and Shape: A Direction-Aware Loss for Time Series Forecasting

The direction of change --- whether a series will move up or down --- is often as important as its exact value in decisiondriven applications such as risk management and financial forecasting. However, most forecasting losses optimize either point magnitude or shape and frequency structure, and none explicitly targets the direction of change. In this paper, we find that MSE-trained forecasters fail on the direction of small moves. To address this, we propose CosDir, a simple yet effective direction-aware loss that aligns the difference vectors of the prediction and the target via cosine similarity. Being scale-invariant, CosDir keeps a directional gradient on small moves, re-injecting learning signal exactly where MSE neglects it. CosDir is a lightweight, plug-in term that attaches to any backbone without architectural modification. Since the best ratio for mixing the directional and magnitude terms differs across datasets, we further propose CosDir-UW, an extension that makes this ratio adaptive by learning it during training, matching a per-dataset tuned weight with no hyperparameter. We conduct over 100K experiments, demonstrating that our method consistently and significantly improves directional accuracy while preserving magnitude accuracy, and that it outperforms various loss functions. Code is available at: https://github.com/seunghan96/cosdir.