Time Series

Advanced deep learning-based approaches have been actively applied to forecast the spatiotemporal physical dynamics governed by partial differential equations (PDEs), which acts as a critical procedure in tackling many science and engineering problems. As real-world physical environments like PDE system parameters are always capricious, how to generalize across unseen out-of-distribution (OOD) forecasting scenarios using limited training data is of great importance. To bridge this barrier, existing methods focus on discovering domain-generalizable representations across various PDE dynamics trajectories. However, their zero-shot OOD generalization capability remains deficient, since extra test-time samples for domain-specific adaptation are still required. This is because the fundamental physical invariance in PDE dynamical systems are yet to be investigated or integrated. To this end, we first explicitly define a two-fold PDE invariance principle, which points out that ingredient operators and their composition relationships remain invariant across different domains and PDE system evolution. Next, to capture this two-fold PDE invariance, we propose a physics-guided invariant learning method termed iMOOE, featuring an Invariance-aligned Mixture Of Operator Expert architecture and a frequency-enriched invariant learning objective. Extensive experiments across simulated benchmarks and real-world applications validate iMOOE's superior in-distribution performance and zero-shot generalization capabilities on diverse OOD forecasting scenarios.

Computing worst-case robust strategies in pursuit-evasion games (PEGs) is time-consuming, especially when real-world factors like partial observability are considered. While important for general security purposes, real-time applicable pursuit strategies for graph-based PEGs are currently missing when the pursuers only have imperfect information about the evader's position. Although state-of-the-art reinforcement learning (RL) methods like Equilibrium Policy Generalization (EPG) and Grasper provide guidelines for learning graph neural network (GNN) policies robust to different game dynamics, they are restricted to the scenario of perfect information and do not take into account the possible case where the evader can predict the pursuers' actions. This paper introduces the first approach to worst-case robust real-time pursuit strategies (R2PS) under partial observability. We first prove that a traditional dynamic programming (DP) algorithm for solving Markov PEGs maintains optimality under the asynchronous moves by the evader. Then, we propose a belief preservation mechanism about the evader's possible positions, extending the DP pursuit strategies to a partially observable setting. Finally, we embed the belief preservation into the state-of-the-art EPG framework to finish our R2PS learning scheme, which leads to a real-time pursuer policy through cross-graph reinforcement learning against the asynchronous-move DP evasion strategies. After reinforcement learning, our policy achieves robust zero-shot generalization to unseen real-world graph structures and consistently outperforms the policy directly trained on the test graphs by the existing game RL approach.

We introduce WARP (**W**eight-space **A**daptive **R**ecurrent **P**rediction), a simple yet powerful model that unifies weight-space learning with linear recurrence to redefine sequence modeling. Unlike conventional recurrent neural networks (RNNs) which collapse temporal dynamics into fixed-dimensional hidden states, WARP explicitly parametrizes its hidden state as the weights and biases of a distinct auxiliary neural network, and uses input differences to drive its recurrence. This brain-inspired formulation enables efficient gradient-free adaptation of the auxiliary network at test-time, in-context learning abilities, and seamless integration of domain-specific physical priors. Empirical validation shows that WARP matches or surpasses state-of-the-art baselines on diverse classification tasks, featuring in the top three in 4 out of 6 real-world challenging datasets. Furthermore, extensive experiments across sequential image completion, multivariate time series forecasting, and dynamical system reconstruction demonstrate its expressiveness and generalization capabilities. Remarkably, a physics-informed variant of our model outperforms the next best model by more than 10x. Ablation studies confirm the architectural necessity of key components, solidifying weight-space linear RNNs as a transformative paradigm for adaptive machine intelligence.

Neural networks with ReLU activations are a widely used model in machine learning. It is thus important to have a profound understanding of the properties of the functions computed by such networks. Recently, there has been increasing interest in the (parameterized) computational complexity of determining these properties. In this work, we close several gaps and resolve an open problem posted by Froese et al. [COLT '25] regarding the parameterized complexity of various problems related to network verification. In particular, we prove that deciding positivity (and thus surjectivity) of a function $f\colon\mathbb{R}^d\to\mathbb{R}$ computed by a 2-layer ReLU network is W[1]-hard when parameterized by $d$. This result also implies that zonotope (non-)containment is W[1]-hard with respect to $d$, a problem that is of independent interest in computational geometry, control theory, and robotics. Moreover, we show that (a) approximating the maximum within any multiplicative factor in 2-layer ReLU networks, (b) computing the $L_p$-Lipschitz constant for $p\in(0,\infty]$ in 2-layer networks, and (c) approximating the $L_p$-Lipschitz constant in 3-layer networks are all NP-hard and W[1]-hard with respect to $d$. Notably, our hardness results are the strongest known so far and imply that the naive enumeration-based methods for solving these fundamental problems are all essentially optimal under the Exponential Time Hypothesis.

Sequence modeling is currently dominated by causal transformer architectures that use softmax self-attention. Although widely adopted, transformers require scaling memory and compute linearly during inference. A recent stream of work linearized the softmax operation, resulting in powerful recurrent neural network (RNN) models with constant memory and compute costs such as DeltaNet, Mamba or xLSTM. These models can be unified by noting that their recurrent layer dynamics can all be derived from an in-context regression objective, approximately optimized through an online learning rule. Here, we join this line of work and introduce a numerically stable, chunkwise parallelizable version of the recently proposed Mesa layer (von Oswald et al., 2024), and study it in language modeling at the billion-parameter scale. This layer again stems from an in-context loss, but which is now minimized to optimality at every time point using a fast conjugate gradient solver. Through an extensive suite of experiments, we show that optimal test-time training enables reaching lower language modeling perplexity and higher downstream benchmark performance than previous RNNs, especially on tasks requiring long context understanding. This performance gain comes at the cost of additional flops spent during inference time. Our results are therefore intriguingly related to recent trends of increasing test-time compute to improve performance -- here by spending compute to solve sequential optimization problems within the neural network itself.

Time series foundation models (TSFMs) have recently gained significant attention due to their strong zero-shot capabilities and widespread real-world applications. Such models typically require a computationally costly pretraining on large-scale, carefully curated collections of real-world sequences. To allow for a sample-efficient pretraining of TSFMs, we propose CauKer, a novel algorithm designed to generate diverse, causally coherent synthetic time series with realistic trends, seasonality, and nonlinear interactions. CauKer combines Gaussian Process (GP) kernel composition with Structural Causal Models (SCM) to produce data for sample-efficient pretraining of state-of-the-art classification TSFMs having different architectures and following different pretraining approaches. Additionally, our experiments reveal that CauKer-generated datasets exhibit clear scaling laws for both dataset size (10K to 10M samples) and model capacity (1M to 783M parameters), unlike real-world datasets, which display irregular scaling behavior.

Motivated by the prevalence and success of machine learning, a line of recent work has studied learning-augmented algorithms in the streaming model. These results have shown that for natural and practical oracles implemented with machine learning models, we can obtain streaming algorithms with improved space efficiency that are otherwise provably impossible. On the other hand, our understanding is much more limited for the sliding window model, which captures applications where either recent data leads to better or older data must be expunged from the dataset, e.g., by privacy regulation laws. In this paper, we utilize an oracle for the heavy-hitters of datasets to give learning-augmented algorithms for a number of fundamental problems in the sliding window model, such as norm/moment estimation, frequency estimation, cascaded norms, and rectangular moment estimation. We complement our theoretical results with a number of empirical evaluations that demonstrate the practical efficiency of our algorithms on real and synthetic datasets.

Medical time series (MedTS) classification suffers from poor generalizability in real-world deployment due to inter- and intra-dataset heterogeneity, such as varying numbers of channels, signal lengths, task definitions, and patient characteristics. % implicit patient characteristics, variable channel configurations, time series lengths, and diagnostic tasks. To address this, we propose FORMED, a novel framework for repurposing a backbone foundation model, pre-trained on generic time series, to enable highly generalizable MedTS classification on unseen datasets. FORMED combines the backbone with a novel classifier comprising two components: (1) task-specific channel embeddings and label queries, dynamically sized to match any number of channels and target classes, and (2) a shared decoding attention layer, jointly trained across datasets to capture medical domain knowledge through task-agnostic feature-query interactions. After repurposing, FORMED achieves seamless adaptation to unseen MedTS datasets through lightweight label query training (0.1\% of parameters), eliminating the need for full fine-tuning or architectural redesign. We evaluate FORMED on 5 diverse MedTS datasets, benchmarking against 11 Task-Specific Models (TSM) and 4 Task-Specific Adaptation (TSA) methods. Our results demonstrate FORMED's dominant performance, achieving up to 35\% absolute improvement in F1-score (on ADFTD dataset) over specialized baselines. By decoupling domain-invariant representation learning from task-specific adaptation, FORMED establishes a scalable and resource-efficient paradigm for foundation model repurposing in healthcare. This approach prioritizes clinical adaptability over rigid task-centric design, offering a practical pathway for real-world implementation.

Modeling long spatiotemporal sequences requires capturing both complex spatial correlations and temporal dependencies. Convolutional State Space Models (ConvSSMs) have been proposed to incorporate spatial modeling in State Space Models (SSMs) using the convolution of tensor-valued states and kernels. Yet, existing implementations remain limited to $1\times 1$ state kernels for computational feasibility, which limits the modeling capacity of ConvSSMs. We introduce a novel spatiotemporal model, ConvT3 (ConvSSM using Tridiagonal Toeplitz Tensors), designed to equivalently realize ConvSSMs with extended $3\times 3$ state kernels. ConvT3 structures a state kernel for its corresponding tensor to be composed as a structured SSM matrix on hidden state dimensions and a constrained tridiagonal Toeplitz tensor on spatial dimensions. We show that the structured tensor can be diagonalized, which enables efficient parallel training while leveraging $3\times 3$ state convolutions. We demonstrate that ConvT3 effectively embeds rich spatial and temporal information into the dynamics of tensor-valued states, achieving state-of-the-art performance on most metrics in long-range video generation and physical system modeling.

The design of training objective is central to training time-series forecasting models. Existing training objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which we found leading to the following two issues: (1) overlook the *label autocorrelation effect* among future steps, leading to biased training objective; (2) 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 training objective, addressing both of the 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 most preferable weights of the forecasting tasks with varying future steps. To achieve this, 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 performance of various forecast models, achieving state-of-the-art results. Code is available at https://anonymous.4open.science/r/QDF-8937.

Data augmentation is a fundamental technique in deep learning, widely applied in both representation learning and automated data augmentation (AutoDA). In representation learning, augmentations are used to construct contrastive views for learning task-agnostic embeddings, while in AutoDA the augmentations are directly optimized to improve downstream task performance. However, existing paradigms face critical limitations: representation learning relies on a two-stage scheme with limited adaptability, and current AutoDA frameworks are largely designed for image data, rendering them ineffective for capturing time series–specific features. To address these issues, we introduce **AutoDA-Timeseries**, the first general-purpose automated data augmentation framework tailored for time series. AutoDA-Timeseries incorporates time series features into augmentation policy design and adaptively optimizes both augmentation probability and intensity in a single-stage, end-to-end manner. We conduct extensive experiments on five mainstream tasks, including classification, long-term forecasting, short-term forecasting, regression, and anomaly detection, showing that AutoDA-Timeseries consistently outperforms strong baselines across diverse models and datasets.

Large time series models (LTMs) have recently demonstrated powerful capabilities for universal forecasting. However, these models still struggle to address the variety and nonstationarity of time series, resulting in an unsatisfying balance between forecasting performance and generalizability. Instead of breeding unceasingly new models for diverse domains, this paper proposes a novel framework, time-series adaptive transformation optimization (TATO), that enables a frozen pre-trained LTM to fit various downstream domains through an empirically optimal time-series transformation pipeline. Three representative types of time series transformations, including context slicing, scale normalization, and outlier correction, are constructed to help LTMs fit the target domain. A two-stage ranking is also designed to ensure the optimization's robustness by filtering out transformation pipelines with shortcomings in specific metrics. Extensive evaluations on state-of-the-art pre-trained LTMs and widely used datasets demonstrate that TATO performs universally well and significantly enhances domain-adaptive forecasting performance, achieving a maximum MSE reduction of 68.4% and an average decrease of 16.0%. In most cases, the time required by TATO to optimize time series transformation pipelines is under 2 minutes, making it practical for real-world applications. We have published the source code of TATO at https://anonymous.4open.science/r/TATO-D55C.

Inference-time control of diffusion models aims to steer model outputs to satisfy new constraints without retraining. Previous approaches have mostly relied on heuristic guidance or have been coupled with Sequential Monte Carlo (SMC) for bias correction. In this paper, we propose a flexible alternative based on replica exchange, an algorithm designed initially for sampling problems. We refer to this method as the CREPE (Controlling with REPlica Exchange). Unlike SMC, CREPE: (i) generates particles sequentially, (ii) maintains high diversity in the generated samples after a burn-in period, and (iii) enables online refinement or early termination. We demonstrate its versatility across various tasks, including temperature annealing, reward tilting, model composition and classifier-free guidance debiasing, with competitive performance compared to prior SMC methods.

Time series forecasting has attracted significant attention in the field of AI. Previous works have revealed that the Channel-Independent (CI) strategy improves forecasting performance by modeling each channel individually, but it often suffers from poor generalization and overlooks meaningful inter-channel interactions. Conversely, Channel-Dependent (CD) strategies aggregate all channels, which may introduce irrelevant information and lead to oversmoothing. Despite recent progress, few existing methods offer the flexibility to adaptively balance CI and CD strategies in response to varying channel dependencies. To address this, we propose a generic plugin xCPD, that can adaptively model the channel-patch dependencies from the perspective of graph spectral decomposition. Specifically, xCPD first projects multivariate signals into the frequency domain using a shared graph Fourier basis, and groups patches into low-, mid-, and high-frequency bands based on their spectral energy responses. xCPD then applies a channel-adaptive routing mechanism that dynamically adjusts the degree of inter-channel interaction for each patch, enabling selective activation of frequency-specific experts. This facilitates fine-grained, input-aware modeling of smooth trends, local fluctuations, and abrupt transitions. xCPD can be seamlessly integrated on top of existing CI and CD forecasting models, consistently enhancing both accuracy and generalization across benchmarks.

We address multivariate time–series imputation by learning the velocity field of a data-conditioned ordinary differential equation (ODE) via flow matching. Our method, Time-Gated Multi-Scale Flow Matching (TG-MSFM), conditions the flow on a structured endpoint comprising observed values, a per-time visibility mask, and short left/right context, processed by a time-aware Transformer whose self-attention is masked to aggregate only from observed timestamps. To recon- cile global trends with local details along the trajectory, we introduce time-gated multi-scale velocity heads on a fixed 1D pyramid and blend them through a time- dependent gate; a mild anti-aliasing filter stabilizes the finest branch. At inference, we use a second-order Heun integrator with a per-step data-consistency projection that keeps observed coordinates exactly on the straight path from the initial noise to the endpoint, reducing boundary artifacts and drift. Training adopts gap-only supervision of the velocity on missing data coordinates, with small optional regu- larizers for numerical stability. Across standard benchmarks, Time-Gated Multi- Scale Flow Matching attains competitive or improved MSE/MAE with favorable speed–quality trade-offs, and ablations isolate the contributions of the time-gated multi-scale heads, masked attention, and the data-consistent ODE integration

Current methods for multivariate time series forecasting can be classified into channel-dependent and channel-independent models. Channel-dependent models learn cross-channel features but often overfit the channel ordering, which hampers adaptation when channels are added or reordered. Channel-independent models treat each channel in isolation to increase flexibility, yet this neglects inter-channel dependencies and limits performance. To address these limitations, we propose CPiRi, a channel permutation invariant (CPI) framework that infers cross-channel structure from data rather than memorizing a fixed ordering, enabling deployment in settings with structural and distributional co-drift without retraining. CPiRi couples spatio-temporal decoupling architecture with permutation-invariant regularization training strategy: a frozen pretrained temporal encoder extracts high-quality temporal features, a lightweight spatial module learns content-driven inter-channel relations, while a channel shuffling strategy enforces CPI during training. We further ground CPiRi in theory by analyzing permutation equivariance in multivariate time series forecasting. Experiments on multiple benchmarks show state-of-the-art results. CPiRi remains stable when channel orders are shuffled and exhibits strong inductive generalization to unseen channels even when trained on only half of the channels, while maintaining practical efficiency on large-scale datasets. The source code is released at https://github.com/JasonStraka/CPiRi.

Anomaly detection is a critical task in computer vision with profound implications for medical imaging, where identifying pathologies early can directly impact patient outcomes. While recent unsupervised anomaly detection approaches show promise, they require substantial normal training data and struggle to generalize across anatomical contexts. We introduce D$^2$4FAD, a novel dual distillation framework for few-shot anomaly detection that identifies anomalies in previously unseen tasks using only a small number of normal reference images. Our approach leverages a pre-trained encoder as a teacher network to extract multi-scale features from both support and query images, while a student decoder learns to distill knowledge from the teacher on query images and self-distill on support images. We further propose a learn-to-weight mechanism that dynamically assesses the reference value of each support image conditioned on the query, optimizing anomaly detection performance. To evaluate our method, we curate a comprehensive benchmark dataset comprising 13,084 images across four organs, four imaging modalities, and five disease categories. Extensive experiments demonstrate that D$^2$4FAD significantly outperforms existing approaches, establishing a new state-of-the-art in few-shot medical anomaly detection.

Recent studies have demonstrated that test-time compute scaling effectively improves the performance of small language models (sLMs). However, prior research has mainly examined test-time compute scaling with an additional larger model as a verifier, leaving verification by sLMs underexplored. In this work, we investigate whether sLMs can reliably verify the output candidates under test-time scaling. We find that even with knowledge distillation from larger verifiers, sLMs struggle with verification tasks requiring memorization, such as numerical calculations and fact-checking. To address this limitation, we propose Tool-integrated verification (T1), a two-stage framework that first filters candidates with external tools and then uses an sLM for final verification, offloading memorization-heavy steps to tools such as a code interpreter. Within T1 we prove that offloading to external tools reduces the memorization burden on sLMs and improves test-time scaling performance. Experiments on the MATH benchmark demonstrate that, with T1, a Llama-3.2 1B model under test-time scaling outperforms the significantly larger Llama-3.1 8B model. Moreover, T1 improves the verification accuracy of both process reward models (PRMs) and critic models. Our findings highlight the potential of tool integration to substantially improve the verification abilities of sLMs.

Time series forecasting has long been dominated by advances in model architecture, with recent progress driven by deep learning and hybrid statistical techniques. However, as forecasting models approach diminishing returns in accuracy, a critical yet underexplored opportunity emerges: the strategic use of post-processing. In this paper, we address the last-mile gap in time-series forecasting, which is to improve accuracy and uncertainty without retraining or modifying a deployed backbone. We propose $\delta$-Adapter, a lightweight, architecture-agnostic way to boost deployed time series forecasters without retraining. $\delta$-Adapter learns tiny, bounded modules at two interfaces: input nudging (soft edits to covariates) and output residual correction. We provide local descent guarantees, $O(\delta)$ drift bounds, and compositional stability for combined adapters. Meanwhile, it can act as a feature selector by learning a sparse, horizon-aware mask over inputs to select important features, thereby improving interpretability. In addition, it can also be used as a distribution calibrator to measure uncertainty. Thus, we introduce a Quantile Calibrator and a Conformal Corrector that together deliver calibrated, personalized intervals with finite-sample coverage. Our experiments across diverse backbones and datasets show that $\delta$-Adapter improves accuracy and calibration with negligible compute and no interface changes.

Time series forecasting is of significant importance across various domains. However, it faces significant challenges due to distribution shift. This issue becomes particularly pronounced in online deployment scenarios where data arrives sequentially, requiring models to adapt continually to evolving patterns. Current time series online learning methods focus on two main aspects: selecting suitable parameters to update (e.g., final layer weights or adapter modules) and devising suitable update strategies (e.g., using recent batches, replay buffers, or averaged gradients). We challenge the conventional parameter selection approach, proposing that distribution shifts stem from changes in underlying latent factors influencing the data. Consequently, updating the feature representations of these latent factors may be more effective. To address the critical problem of delayed feedback in multi-step forecasting (where true values arrive much later than predictions), we introduce ADAPT-Z (Automatic Delta Adjustment via Persistent Tracking in Z-space). ADAPT-Z utilizes an adapter module that leverages current feature representations combined with historical gradient information to enable robust parameter updates despite the delay. Extensive experiments demonstrate that our method consistently outperforms standard base models without adaptation and surpasses state-of-the-art online learning approaches across multiple datasets.