Probabilistic Time Series Forecasting

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Probabilistic time series forecasting studies not only what may happen next, but how uncertain that prediction is, often by producing quantiles, intervals, or complete future scenarios. Approaches range from Bayesian and statistical models, quantile regression, and conformal calibration to ensembles, diffusion and flow-based generators, and pretrained time-series foundation models. These methods are evaluated for accuracy, calibration, sharpness, and computational cost across settings such as energy, finance, health, and traffic, with increasing attention to multivariate dependence, irregular observations, and distribution shifts. A central tension is that better point accuracy does not guarantee reliable uncertainty: performance can vary across horizons and regimes, and foundation models do not consistently outperform simpler task-specific methods.

Latest papers

162 papers23 papers in the last four weeks, down 15% on the four weeks before. 0.1% of all new papers.

May 6, 2026

Forecasting Oncology Demand Trends with Boosting-Based Bayesian Conjugate Models

Accurate trend forecasting in healthcare time series is essential for planning and resource allocation. This paper proposes a Bayesian framework for predicting oncology demand trends, modeling weekly appointments as a Poisson process with a Gamma prior to the demand rate. To enhance adaptability and capture persistent directional patterns, we incorporate a residual-based boosting mechanism grounded in a Gamma-Log-Normal conjugate structure. This boosting approach allows the model to track both short- and long-term trend shifts while maintaining the analytical tractability of conjugate Bayesian updating. The methodology was evaluated on real oncology service data from Cariri, Ceara, Brazil, and compared against established baselines, including linear regression, ARIMA, naive forecasting, LSTM neural networks, and XGBoost. Results showed that the proposed model outperforms competing methods in trend detection accuracy, with gains in terms of percentage of correct direction of 38.25% in relation to the second best approach in some cases.

Ademir Batista dos Santos Neto, Tiago Alessandro Espinola Ferreira, Paulo Renato Alves Firmino

Bayesian Inference Time Series Forecasting Oncology Probabilistic Time Series Forecasting

May 5, 2026

Densification and forecasting of Sentinel-2 time series from multimodal SAR and Optical satellite data using deep generative models

Optical satellite image time series are extensively used in many Earth observation applications, including agriculture, climate monitoring, and land surface analysis. However, clouds and swath edges result in irregular sampling along the temporal dimension, limiting continuous monitoring. To address this issue, a growing body of work has focused on temporal densification and reconstruction of satellite image time series, with the objective of filling missing or cloud-contaminated observations within the temporal extent of the available data. While these approaches improve temporal continuity, they are inherently restricted to the reconstruction of the gaps within the observed time periods, and do not address the prediction of future observations. This work proposes a probabilistic deep learning framework for the densification and forecasting of Sentinel-2 time series by generating optical images at arbitrary past or future dates. The approach leverages multimodal satellite data by jointly exploiting Sentinel-2 optical and Sentinel-1 SAR observations. Unlike most existing works, we propose to focus on the uncertainty of the generated images. Experimental results demonstrate effective densification and forecasting, on sparse and temporally misaligned time series.

Véronique Defonte, Dawa Derksen, Alexandre Constantin +1

Thales Services Num´eriques 31670 Lab`ege, France · Centre National d’Etudes Spatiales 31400 Toulouse, France

Irregular Time-Series Modeling Remote Sensing Spatiotemporal Forecasting Probabilistic Time Series Forecasting

May 5, 2026

Training-Free Probabilistic Time-Series Forecasting with Conformal Seasonal Pools

We propose Conformal Seasonal Pools (CSP), a training-free probabilistic time-series forecaster that mixes same-season empirical draws with signed residual draws around a seasonal naive forecast. In an audited rolling-origin benchmark on the six time-series datasets where DeepNPTS was originally evaluated (electricity, exchange_rate, solar_energy, taxi, traffic, wikipedia), CSP-Adaptive significantly outperforms DeepNPTS on every metric we report -- CRPS (per-window paired Wilcoxon

p \approx 4 \times 10^{-10}

), normalized mean quantile loss (

p \approx 7 \times 10^{-10}

), and empirical 95% coverage (

p \approx 8 \times 10^{-45}

, mean 0.89 vs 0.66) -- while running over 500x faster on CPU. Coverage is the most decision-critical of these: a 0.95 nominal interval that contains the truth in only ~66% of cases fails the basic calibration desideratum and would not survive deployment in safety- or decision-critical settings. The failure mode is also more severe than aggregate coverage suggests: in the worst 10% of windows, DeepNPTS's prediction interval covers none of the H forecast horizons -- the entire multi-step trajectory misses the truth at every step simultaneously. This poses serious risk in safety- and decision-critical applications such as healthcare, finance, energy operations, and autonomous systems, where prediction intervals that systematically miss the truth across the entire planning horizon translate directly into misclassified patients, regulatory capital failures, grid imbalances, and safety-case violations. CSP achieves all of this with no learned parameters and no training. We argue training-free conformal samplers should be mandatory baselines when evaluating learned non-parametric forecasters.

Valery Manokhin

Independent researcher.

Time Series Forecasting Conformal Prediction Probabilistic Time Series Forecasting

May 5, 2026

Bi-Level Chaotic Fusion Based Graph Convolutional Network for Stock Market Prediction Interval

Financial market forecasting is inherently uncertain, yet most deep learning approaches rely on point predictions that provide only single-value estimates without quantifying uncertainty. Such predictions are insufficient for risk-aware decision-making, as they fail to capture the range of possible outcomes and the associated confidence of forecasts.The problem can be solved using prediction intervals, which allow obtaining an upper and lower bound for the prediction, thus enabling uncertainty representation in the model. Yet, the current methods tend to disregard relationships between assets or cannot simultaneously ensure good calibration and sharpness of the resulting intervals in dynamically changing market regimes. In our work, we propose a spatio-temporal graph-based approach with a bi-level chaotic fusion technique to solve this problem. Our model uses separate nonlinear transformation functions to estimate the interval center and width. Additionally, a volatility-aware gating mechanism is used to make predictions dependent on the regime in which the market operates. Temporal dependencies are considered by embedding graph structures and sequentially modeling them. Training is conducted according to a Lower-Upper Bound Estimation (LUBE) objective. Our experimental results show significant improvements compared to existing baselines (LSTM, GRU, GCN, HGNN) when applied to data from 2016 to 2026 with 43 leading companies in eight sectors of the NSE. It provides the lowest Winkler score (0.0778), tightest prediction intervals (PIAW = 0.1407), and highest coverage (PICP = 96.6%), with all differences statistically significant (p < 0.001) according to the Diebold-Mariano test.

Eshwar Sai Kandimalla, Sravan Chowdary Kankanala, Sumana Bhimineni +2

Department of Computer Science & Engineering, SRM University AP, India

Finance Financial Forecasting Financial Time Series Regression Prediction Intervals Probabilistic Time Series Forecasting Spatiotemporal GNNs+1

Apr 30, 2026

Differentiable latent structure discovery for interpretable forecasting in clinical time series

Background: Timely, uncertainty-aware forecasting from irregular electronic health records (EHR) can support critical-care decisions, yet most approaches either impute to a grid or sacrifice interpretability. We introduce StructGP, a continuous-time multi-task Gaussian process that couples process convolutions with differentiable structure learning to uncover a sparse, ordered directed acyclic graph (DAG) of inter-variable dependencies while preserving principled uncertainty. We further propose LP-StructGP, which augments StructGP with latent pathways-shared, temporally shifted trajectories inferred via subject-specific coupling filters and a softmax gating mechanism-to capture cross-patient progression patterns. Both models are trained under sparsity and acyclicity constraints (augmented Lagrangian, Adam) using scalable low-rank updates. Results: In simulations, the approach reliably recovers ground-truth graphs (Structural Hamming Distance approaching 0 as cohorts grow) and pathway assignments (high Adjusted Rand Index). On a MIMIC-IV septic shock cohort (n=1,008; norepinephrine, creatinine, mean arterial pressure), StructGP improves short-horizon (6 h) forecasting over independent-task baselines (average RMSE 0.68 [95%CI: 0.63--0.74] vs. 0.88 [0.83-0.94]) and, with 15 additional inputs, markedly outperforms unstructured kernels (0.63 [0.58-0.69] vs. 3.02 [2.85-3.18]) with superior calibration (coverage 0.96 vs. 0.84). On the PhysioNet Challenge (12k patients, 41 variables), StructGP attains competitive accuracy (MAE 3.72e-2) relative to a state-of-the-art graph neural model while maintaining calibrated uncertainty. Conclusion: These results show that structured process convolutions with latent pathways deliver interpretable, scalable, and well-calibrated forecasting for irregular clinical time series.

Ivan Lerner, Jean Feydy, Alexandre Kalimouttou +2

Universit´e Paris Cit´e and Universit´e Sorbonne Paris Nord, INRIA, Inserm, HEKA, Paris, 75015, France. · Department of Medical Informatics, Assistance Publique Hˆopitaux de Paris (AP-HP), Georges Pompidou European Hospital, Paris, France. · SIERRA, Inria Paris, Paris, 75015, France. +2

Irregular Time-Series Modeling Healthcare Causal Structure Learning Longitudinal EHR Modeling Probabilistic Time Series Forecasting Gaussian Process Regression

Apr 30, 2026

Probabilistic Circuits for Irregular Multivariate Time Series Forecasting

Joint probabilistic modeling is essential for forecasting irregular multivariate time series (IMTS) to accurately quantify uncertainty. Existing approaches often struggle to balance model expressivity with consistent marginalization, frequently leading to unreliable or contradictory forecasts. To address this, we propose CircuITS, a novel architecture for probabilistic IMTS forecasting based on sum-product networks (SPNs). Our model is flexible in capturing intricate dependencies between time series channels while structurally guaranteeing valid joint distributions. Experiments on five real-world datasets demonstrate that CircuITS achieves superior joint and marginal density estimation compared to state-of-the-art baselines.

Christian Klötergens, Lars Schmidt-Thieme, Vijaya Krishna Yalavarthi

ISMLL, University of Hildesheim, Germany · VWFS, Data Analytics Research Center

Irregular Time-Series Modeling Multivariate Time Series Forecasting Probabilistic Circuits Probabilistic Time Series Forecasting

Apr 27, 2026

Generative diffusion models for spatiotemporal influenza forecasting

Forecasting infectious disease incidence can provide important information to guide public health planning, yet is difficult because epidemic dynamics are complex. Current mechanistic and statistical approaches often struggle to capture multimodal uncertainty or emergent trends. Influpaint adapts denoising diffusion probabilistic models to epidemic forecasting. By encoding influenza seasons as spatiotemporal images in which pixel intensity represents incidence, Influpaint learns a rich distribution of disease dynamics from a hybrid dataset of surveillance and simulated trajectories. Forecasting is formulated as a conditional generation (inpainting) task from partial observations. We show that Influpaint generates realistic, diverse epidemic trajectories and achieves forecast accuracy that is competitive with leading ensemble methods in retrospective evaluation. In real-time evaluation during the 2023--2025 U.S. CDC FluSight challenges, performance improved substantially across seasons, with highly accurate but somewhat overconfident projections in 2024--2025. The best performance was achieved with a training dataset containing 30% surveillance and 70% simulated trajectories. These results show that diffusion models can capture important spatiotemporal structure in influenza dynamics and provide a flexible framework for probabilistic infectious disease forecasting.

Joseph Lemaitre, Justin Lessler

Department of Epidemiology, Gillings School of Global Public Health, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA · Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD 21205, USA · Carolina Population Center, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA

Epidemic Forecasting Spatiotemporal Forecasting Conditional Diffusion Models Probabilistic Time Series Forecasting Diffusion Models

Apr 22, 2026

Cold-Start Forecasting of New Product Life-Cycles via Conditional Diffusion Models

Forecasting the life-cycle trajectory of a newly launched product is important for launch planning, resource allocation, and early risk assessment. This task is especially difficult in the pre-launch and early post-launch phases, when product-specific outcome history is limited or unavailable, creating a cold-start problem. In these phases, firms must make decisions before demand patterns become reliably observable, while early signals are often sparse, noisy, and unstable We propose the Conditional Diffusion Life-cycle Forecaster (CDLF), a conditional generative framework for forecasting new-product life-cycle trajectories under cold start. CDLF combines three sources of information: static descriptors, reference trajectories from similar products, and newly arriving observations when available. Here, static descriptors refer to structured pre-launch characteristics of the product, such as category, price tier, brand or organization identity, scale, and access conditions. This structure allows the model to condition forecasts on relevant product context and to update them adaptively over time without retraining, yielding flexible multi-modal predictive distributions under extreme data scarcity. The method satisfies consistency with a horizon-uniform distributional error bound for recursive generation. Across studies on Intel microprocessor stock keeping unit (SKU) life cycles and the platform-mediated adoption of open large language model repositories, CDLF delivers more accurate point forecasts and higher-quality probabilistic forecasts than classical diffusion models, Bayesian updating approaches, and other state-of-the-art machine-learning baselines.

Ruihan Zhou, Zishi Zhang, Jinhui Han +2

Guanghua School of Management, Peking University, Beijing 100871, China · Department of Industrial Engineering and Decision Analytics, The Hong Kong University of Science and Technology, Hong Kong SAR, China

Time Series Forecasting Conditional Diffusion Models Probabilistic Time Series Forecasting

Apr 22, 2026

Trajectory-Aware Reliability Modeling of Democratic Systems

Failures in complex systems often emerge through gradual degradation and the propagation of stress across interacting components rather than through isolated shocks. Democratic systems exhibit similar dynamics, where weakening institutions can trigger cascading deterioration in related institutional structures. Traditional reliability and survival models typically estimate failure risk based on the current system state but do not explicitly capture how degradation propagates through institutional networks over time. This paper introduces a trajectory-aware reliability modeling framework based on Dynamic Causal Neural Autoregression (DCNAR). The framework first estimates a causal interaction structure among institutional indicators and then models their joint temporal evolution to generate forward trajectories of system states. Failure risk is defined as the probability that predicted trajectories cross predefined degradation thresholds within a fixed horizon. Using longitudinal institutional indicators, we compare DCNAR-based trajectory risk models with discrete-time hazard and Cox proportional hazards models. Results show that trajectory-aware modeling consistently outperforms Cox models and improves risk prediction for several propagation-driven institutional failures. These findings highlight the importance of modeling dynamic system interactions for reliability analysis and early detection of systemic degradation.

Dmitry Zaytsev, Valentina Kuskova, Michael Coppedge

Lucy Family Institute for Data & Society, University of Notre Dame, USA · Department of Political Science, University of Notre Dame, USA

Probabilistic Time Series Forecasting

Apr 21, 2026

Spatio-temporal modelling of electric vehicle charging demand

Accurate forecasting of electric vehicle (EV) charging demand is critical for grid management and infrastructure planning. Yet the field continues to rely on legacy benchmarks; such as the Palo Alto (2020) dataset; that fail to reflect the scale and behavioral diversity of modern charging networks. To address this, we introduce a novel large-scale longitudinal dataset collected across Scotland (2022 2025), which release it as an open benchmark for the community. Building on this dataset, we formulate EV charging demand as a spatio-temporal latent Gaussian field and perform approximate Bayesian inference via Integrated Nested Laplace Approximation (INLA). The resulting model jointly captures spatial dependence, temporal dynamics, and covariate effects within a unified proba bilistic framework. On station-level forecasting tasks, our approach achieves competitive predictive accuracy against machine learning baselines, while additionally providing principled uncertainty quan tification and interpretable spatial and temporal decompositions properties that are essential for risk-aware infrastructure planning.

Kaoutar Bouaachra, Yvenn Amara-Ouali, Yannig Goude +1

École Polytechnique de Paris · EDF Lab Paris-Saclay Palaiseau, France · INRIA Paris, DYOGENE/MATHNET Team

Electric Vehicle Charging Spatiotemporal Forecasting Probabilistic Time Series Forecasting

Apr 20, 2026

IDOBE: Infectious Disease Outbreak forecasting Benchmark Ecosystem

Epidemic forecasting has become an integral part of real-time infectious disease outbreak response. While collaborative ensembles composed of statistical and machine learning models have become the norm for real-time forecasting, standardized benchmark datasets for evaluating such methods are lacking. Further, there is limited understanding on performance of these methods for novel outbreaks with limited historical data. In this paper, we propose IDOBE, a curated collection of epidemiological time series focused on outbreak forecasting. IDOBE compiles from multiple data repositories spanning over a century of surveillance and across U.S. states and global locations. We perform derivative-based segmentation to generate over 10,000 outbreaks covering multiple outcomes such as cases and hospitalizations for 13 diseases. We consider a variety of information-theoretic and distributional measures to quantify the epidemiological diversity of the dataset. Finally, we perform multi-horizon short-term forecasting (1- to 4-week-ahead) through the progression of the outbreak using 11 baseline models and report on their performance. In addition to standard metrics such as NMSE and MAPE for point forecasts, we include probabilistic scoring rules such as Normalized Weighted Interval Score (NWIS) to quantify the performance. We find that MLP-based methods have the most robust performance, with statistical methods having a slight edge during the pre-peak phase. IDOBE dataset along with baselines are released publicly on https://github.com/NSSAC/IDOBE to enable standardized, reproducible benchmarking of outbreak forecasting methods.

Aniruddha Adiga, Jingyuan Chou, Anshul Chiranth +7

Biocomplexity Institute, University of Virginia, Charlottesville, Virginia, USA · Department of Public and Ecosystem Health Cornell University College of Veterinary Medicine, Ithaca, New York, USA · Johns Hopkins University, Baltimore, Maryland, USA +2

Epidemic Forecasting Forecasting Benchmarks Spatiotemporal Forecasting Probabilistic Time Series Forecasting

Apr 20, 2026

Barrier-enforced multi-objective optimization for direct point and sharp interval forecasting

This paper proposes a multi-step probabilistic forecasting framework using a single neural-network based model to generate simultaneous point and interval forecasts. Our approach ensures non-crossing prediction intervals (PIs) through a model structure design that strictly satisfy a target coverage probability (PICP) while maximizing sharpness. Unlike existing methods that rely on manual weight tuning for scalarized loss functions, we treat point and PI forecasting as a multi-objective optimization problem, utilizing multi-gradient descent to adaptively select optimal weights. Key innovations include a new PI loss function based on an extended log-barrier with an adaptive hyperparameter to guarantee the coverage, a hybrid architecture featuring a shared temporal model with horizon-specific submodels, and a training strategy. The proposed loss is scale-independent and universally applicable; combined with our training algorithm, the framework eliminates trial-and-error hyperparameter tuning for balancing multiple objectives. Validated by an intra-day solar irradiance forecasting application, results demonstrate that our proposed loss consistently outperforms those in current literature by achieving target coverage with the narrowest PI widths. Furthermore, when compared against LSTM encoder-decoder and Transformer architectures--including those augmented with Chronos foundation models--our method remains highly competitive and can be seamlessly adapted to any deep learning structure.

Worachit Amnuaypongsa, Yotsapat Suparanonrat, Pana Wanitchollakit +1

Department of Electrical Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand · Department of Computer Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand

Multi-Objective Optimization Probabilistic Time Series Forecasting Regression Prediction Intervals

Apr 20, 2026

Causally-Constrained Probabilistic Forecasting for Time-Series Anomaly Detection

Anomaly detection in multivariate time series is a central challenge in industrial monitoring, as failures frequently arise from complex temporal dynamics and cross-sensor interactions. While recent deep learning models, including graph neural networks and Transformers, have demonstrated strong empirical performance, most approaches remain primarily correlational and offer limited support for causal interpretation and root-cause localization. This study introduces a causally-constrained probabilistic forecasting framework which is a Causally Guided Transformer (CGT) model for multivariate time-series anomaly detection, integrating an explicit time-lagged causal graph prior with deep sequence modeling. For each target variable, a dedicated forecasting block employs a hard parent mask derived from causal discovery to restrict the main prediction pathway to graph-supported causes, while a latent Gaussian head captures predictive uncertainty. To leverage residual correlational information without compromising the causal representation, a shadow auxiliary path with stop-gradient isolation and a safety-gated blending mechanism is incorporated to suppress non-causal contributions when reliability is low. Anomalies are identified using negative log-likelihood scores with adaptive streaming thresholding, and root-cause variables are determined through per-dimension probabilistic attribution and counterfactual clamping. Experiments on the ASD and SMD benchmarks indicate that the proposed method achieves state-of-the-art detection performance, with F1-scores of 96.19% on ASD and 95.32% on SMD, and enhances variable-level attribution quality. These findings suggest that causal structural priors can improve both robustness and interpretability in detecting deep anomalies in multivariate sensor systems.

Pooyan Khosravinia, João Gama, Bruno Veloso

Anomaly Localization Time-Series Anomaly Detection Time Series Transformers Probabilistic Time Series Forecasting Anomaly Detection

Apr 19, 2026

Prior-Fitted Functional Flow: In-Context Generative Models for Pharmacokinetics

We introduce Prior-Fitted Functional Flows, a generative foundation model for pharmacokinetics that enables zero-shot population synthesis and individual forecasting without manual parameter tuning. We learn functional vector fields, explicitly conditioned on the sparse, irregular data of an entire study population. This enables the generation of coherent virtual cohorts as well as forecasting of partially observed patient trajectories with calibrated uncertainty. We construct a new open-access literature corpus to inform our priors, and demonstrate state-of-the-art predictive accuracy on extensive real-world datasets.

César Ojeda, Niklas Hartung, Wilhelm Huisinga +6

University of Potsdam, Germany · Technical University Berlin, Germany · Freie Universität Berlin, Germany +2

Probabilistic Time Series Forecasting Healthcare Synthetic Data Generation Zero-Shot Time Series Forecasting Continuous Normalizing Flows Prior-Data Fitted Networks+2

Apr 17, 2026

Unveiling Stochasticity: Universal Multi-modal Probabilistic Modeling for Traffic Forecasting

Traffic forecasting is a challenging spatio-temporal modeling task and a critical component of urban transportation management. Current studies mainly focus on deterministic predictions, with limited considerations on the uncertainty and stochasticity in traffic dynamics. Therefore, this paper proposes an elegant yet universal approach that transforms existing models into probabilistic predictors by replacing only the final output layer with a novel Gaussian Mixture Model (GMM) layer. The modified model requires no changes to the training pipeline and can be trained using only the Negative Log-Likelihood (NLL) loss, without any auxiliary or regularization terms. Experiments on multiple traffic datasets show that our approach generalizes from classic to modern model architectures while preserving deterministic performance. Furthermore, we propose a systematic evaluation procedure based on cumulative distributions and confidence intervals, and demonstrate that our approach is considerably more accurate and informative than unimodal or deterministic baselines. Finally, a more detailed study on a real-world dense urban traffic network is presented to examine the impact of data quality on uncertainty quantification and to show the robustness of our approach under imperfect data conditions. Code available at https://github.com/Weijiang-Xiong/OpenSkyTraffic

Weijiang Xiong, Robert Fonod, Nikolas Geroliminis

Urban Transport Systems Laboratory (LUTS), EPFL, Lausanne, 1015, Switzerland

Traffic Forecasting Probabilistic Time Series Forecasting

Apr 16, 2026

Assessing the Performance-Efficiency Trade-off of Foundation Models in Probabilistic Electricity Price Forecasting

Large-scale renewable energy deployment introduces pronounced volatility into the electricity system, turning grid operation into a complex stochastic optimization problem. Accurate electricity price forecasting (EPF) is essential not only to support operational decisions, such as optimal bidding strategies and balancing power preparation, but also to reduce economic risk and improve market efficiency. Probabilistic forecasts are particularly valuable because they quantify uncertainty stemming from renewable intermittency, market coupling, and regulatory changes, enabling market participants to make informed decisions that minimize losses and optimize expected revenues. However, it remains an open question which models to employ to produce accurate forecasts. Should these be task-specific machine learning (ML) models or Time Series Foundation Models (TSFMs)? In this work, we compare four models for day-ahead probabilistic EPF (PEPF) in European bidding zones: a deterministic NHITS backbone with Quantile-Regression Averaging (NHITS+QRA) and a conditional Normalizing-Flow forecaster (NF) are compared with two TSFMs, namely Moirai and ChronosX. On the one hand, we find that TSFMs outperform task-specific deep learning models trained from scratch in terms of CRPS, Energy Score, and predictive interval calibration across market conditions. On the other hand, we find that well-configured task-specific models, particularly NHITS combined with QRA, achieve performance very close to TSFMs, and in some scenarios, such as when supplied with additional informative feature groups or adapted via few-shot learning from other European markets, they can even surpass TSFMs. Overall, our findings show that while TSFMs offer expressive modeling capabilities, conventional models remain highly competitive, emphasizing the need to weigh computational expense against marginal performance improvements in PEPF.

Jan Niklas Lettner, Hadeer El Ashhab, Veit Hagenmeyer +1

Karlsruhe Institute of Technology · Karlsruhe, Germany

Electricity Price Forecasting Time Series Forecasting Probabilistic Forecasting Time-Series Foundation Models Probabilistic Time Series Forecasting Power Systems

Apr 13, 2026

Probabilistic Prediction of Neural Dynamics via Autoregressive Flow Matching

Forecasting neural activity in response to naturalistic stimuli remains a key challenge for understanding brain dynamics and enabling downstream neurotechnological applications. Here, we introduce a generative forecasting framework for modeling neural dynamics based on autoregressive flow matching (AFM). Building on recent advances in transport-based generative modeling, our approach probabilistically predicts neural responses at scale from multimodal sensory input. Specifically, we learn the conditional distribution of future neural activity given past neural dynamics and concurrent sensory input, explicitly modeling neural activity as a temporally evolving process in which future states depend on recent neural history. We evaluate our framework on the Algonauts project 2025 challenge functional magnetic resonance imaging dataset using subject-specific models. AFM significantly outperforms both a non-autoregressive flow-matching baseline and the official challenge general linear model baseline in predicting short-term parcel-wise blood oxygenation level-dependent (BOLD) activity, demonstrating improved generalization and widespread cortical prediction performance. Ablation analyses show that access to past BOLD dynamics is a dominant driver of performance, while autoregressive factorization yields consistent, modest gains under short-horizon, context-rich conditions. Together, these findings position autoregressive flow-based generative modeling as an effective approach for short-term probabilistic forecasting of neural dynamics with promising applications in closed-loop neurotechnology.

Nicole Rogalla, Yuzhen Qin, Mario Senden +2

Department of Machine Learning and Neural Computing, Donders Institute for Brain, Cognition and Behaviour, Radboud University, Nijmegen, the Netherlands · Department of Cognitive Neuroscience, Maastricht University, Maastricht, the Netherlands · Maastricht Brain Imaging Center, Maastricht University, Maastricht, the Netherlands

Functional Magnetic Resonance Imaging Probabilistic Time Series Forecasting Conditional Generative Modeling Flow-Based Generative Modeling

Apr 10, 2026

PDE-regularized Dynamics-informed Diffusion with Uncertainty-aware Filtering for Long-Horizon Dynamics

Long-horizon spatiotemporal prediction remains a challenging problem due to cumulative errors, noise amplification, and the lack of physical consistency in existing models. While diffusion models provide a probabilistic framework for modeling uncertainty, conventional approaches often rely on mean squared error objectives and fail to capture the underlying dynamics governed by physical laws. In this work, we propose PDYffusion, a dynamics-informed diffusion framework that integrates PDE-based regularization and uncertainty-aware forecasting for stable long-term prediction. The proposed method consists of two key components: a PDE-regularized interpolator and a UKF-based forecaster. The interpolator incorporates a differential operator to enforce physically consistent intermediate states, while the forecaster leverages the Unscented Kalman Filter to explicitly model uncertainty and mitigate error accumulation during iterative prediction. We provide theoretical analyses showing that the proposed interpolator satisfies PDE-constrained smoothness properties, and that the forecaster converges under the proposed loss formulation. Extensive experiments on multiple dynamical datasets demonstrate that PDYffusion achieves superior performance in terms of CRPS and MSE, while maintaining stable uncertainty behavior measured by SSR. We further analyze the inherent trade-off between prediction accuracy and uncertainty, showing that our method provides a balanced and robust solution for long-horizon forecasting.

Min Young Baeg, Yoon-Yeong Kim

Spatiotemporal Forecasting Long-Term Time Series Forecasting Physics-Informed Diffusion Models Probabilistic Time Series Forecasting

Mar 26, 2026

A Distribution-to-Distribution Neural Probabilistic Forecasting Framework for Dynamical Systems

Probabilistic forecasts provide a principled representation of forecast uncertainty through predictive distributions. Directly propagating predictive distributions with conventional physical models requires solving distributional evolution equations, a task well-known to be computationally prohibitive for high-dimensional nonlinear systems. Consequently, multi-step predictive distributions are commonly approximated using ensembles or sampled trajectories. A novel distribution-to-distribution (D2D) neural probabilistic forecasting framework is proposed with compatible distributional encoding and decoding interfaces around a replaceable neural forecasting module. Kernel mean embeddings represent input distributions, while mixture density networks parameterise outputs, enabling recursive multi-step forecasting in distributional form. The framework is demonstrated on the Lorenz63 system. Results show that the D2D model captures nontrivial distributional evolution, produces skillful probabilistic forecasts without explicit ensemble simulation, and remains competitive with a naively initialised perfect-model benchmark. This provides proof-of-concept evidence that predictive distributions can be recursively propagated through a neural forecasting model and evaluated in terms of probabilistic forecast performance.

Tianlin Yang, Hailiang Du, Louis Aslett

Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom · School of Mathematics, East China University of Science and Technology, Shanghai 200237, China · Data Science Institute, The London School of Economics and Political Science, London WC2A 2AE, United Kingdom

Mixture Density Networks Probabilistic Time Series Forecasting

Mar 25, 2026

Beyond Static Uncertainty: Modeling Temporal Uncertainty Dynamics for Probabilistic Time Series Forecasting

Real-world time series exhibit temporally structured uncertainty: volatility clusters in turbulent regimes, dissipates in stable periods, and shifts abruptly around structural breaks. Yet many probabilistic forecasting methods estimate predictive uncertainty as an independent per-step quantity, leaving the evolution and persistence of volatility regimes under-modeled. We formalize this missing dimension as temporal uncertainty dynamics and instantiate it in the Volatility Dynamics Variational Autoencoder (VolDy-VAE), a non-autoregressive generative forecaster with a location-scale decoder. VolDy-VAE combines a location path for mean prediction with a recurrent scale path that transfers and evolves a volatility hidden state from the look-back window to the forecasting horizon, enabling temporally coherent predictive variances. This design yields an adaptive attenuation mechanism: high-variance observations receive lower influence on the location estimate while their uncertainty is preserved through explicit scale predictions. We further provide a simplified regime-switching analysis showing that, when variances are known or consistently estimated, the volatility-aware objective reduces to inverse-variance weighting, whereas MSE-based estimators remain unbiased but statistically inefficient. Experiments on nine benchmarks show that VolDy-VAE improves forecasting accuracy and uncertainty calibration over competitive probabilistic and point-forecasting baselines while maintaining low inference latency; plug-in studies further indicate that the VolDy principle can benefit GAN, Koopman VAE, and Transformer backbones. The source code is publicly available at https://github.com/wangyijunlyy/VolDy-VAE.

Yijun Wang, Qiyuan Zhuang, Larysa Marchanka +1

Department of Computer Science, Southeast University, Nanjing, China · Francisk Skorina Gomel State University, Belarus

Probabilistic Time Series Forecasting Uncertainty Quantification Uncertainty Calibration Variational Autoencoders

Mar 23, 2026

Noise Titration: Exact Distributional Benchmarking for Probabilistic Time Series Forecasting

Modern time series forecasting is evaluated almost entirely through passive observation of single historical trajectories, rendering claims about a model's robustness to non-stationarity fundamentally unfalsifiable. We propose a paradigm shift toward interventionist, exact-statistical benchmarking. By systematically titrating calibrated Gaussian observation noise into known chaotic and stochastic dynamical systems, we transform forecasting from a black-box sequence matching game into an exact distributional inference task. Because the underlying data-generating process and noise variance are mathematically explicit, evaluation can rely on exact negative log-likelihoods and calibrated distributional tests rather than heuristic approximations. To fully leverage this framework, we extend the Fern architecture into a probabilistic generative model that natively parameterizes the Symmetric Positive Definite (SPD) cone, outputting calibrated joint covariance structures without the computational bottleneck of generic Jacobian modeling. Under this rigorous evaluation, we find that state-of-the-art zero-shot foundation models behave consistently with the context-parroting mechanism, failing systematically under non-stationary regime shifts and elevated noise. In contrast, Fern explicitly captures the invariant measure and multivariate geometry of the underlying dynamics, maintaining structural fidelity and statistically sharp calibration precisely where massive sequence-matching models collapse.

Qilin Wang

Independent Researcher

Probabilistic Time Series Forecasting Forecasting Benchmarks Non-Stationary Time Series Forecasting Chaotic Time-Series Prediction

Mar 19, 2026

Transformer-Based Predictive Maintenance for Risk-Aware Instrument Calibration

Accurate calibration is essential for instruments whose measurements must remain traceable, reliable, and compliant over long operating periods. Fixed-interval programs are easy to administer, but they ignore that instruments drift at different rates under different conditions. This paper studies calibration scheduling as a predictive maintenance problem: given recent sensor histories, estimate time-to-drift (TTD) and intervene before a violation occurs. We adapt the NASA C-MAPSS benchmark into a calibration setting by selecting drift-sensitive sensors, defining virtual calibration thresholds, and inserting synthetic reset events that emulate repeated recalibration. We then compare classical regressors, recurrent and convolutional sequence models, and a compact Transformer for TTD prediction. The Transformer provides the strongest point forecasts on the primary FD001 split and remains competitive on the harder FD002--FD004 splits, while a quantile-based uncertainty model supports conservative scheduling when drift behavior is noisier. Under a violation-aware cost model, predictive scheduling lowers cost relative to reactive and fixed policies, and uncertainty-aware triggers sharply reduce violations when point forecasts are less reliable. The results show that condition-based calibration can be framed as a joint forecasting and decision problem, and that combining sequence models with risk-aware policies is a practical route toward smarter calibration planning.

Adithya Parthasarathy, Aswathnarayan Muthukrishnan Kirubakaran, Akshay Deshpande +5

Predictive Maintenance Event Time Prediction Probabilistic Time Series Forecasting Time Series Transformers

Mar 16, 2026

Spatio-temporal probabilistic forecast using MMAF-guided learning

We present a theory-guided generalized Bayesian methodology for spatio-temporal raster data, which we use to train an ensemble of stochastic feed-forward neural networks with Gaussian-distributed weights. The methodology incorporates the dependence and causal structure of a spatio-temporal Ornstein-Uhlenbeck process into training and inference by enforcing constraints on the design of the data embedding and the related optimization routine. In inference mode, the networks are employed to generate causal ensemble forecasts by applying different initial conditions at different horizons. We call this workflow MMAF-guided learning. Experiments conducted on both synthetic and real data demonstrate that our forecasts remain calibrated across multiple time horizons. Moreover, we show that on such data, shallow feed-forward architectures can achieve performance comparable to, and in some cases better than, convolutional or diffusion deep learning architectures used in probabilistic forecasting tasks.

Leonardo Bardi, Imma Valentina Curato, Lorenzo Proietti

Department of Mathematics TU Chemnitz Reichenhainer Straße 41

Bayesian Neural Networks Spatiotemporal Forecasting Probabilistic Time Series Forecasting

Mar 14, 2026

JointFM-0.1: A Foundation Model for Multi-Target Joint Distributional Prediction

Despite the rapid advancements in Artificial Intelligence (AI), Stochastic Differential Equations (SDEs) remain the gold-standard formalism for modeling systems under uncertainty. However, applying SDEs in practice is fraught with challenges: modeling risk is high, calibration is often brittle, and high-fidelity simulations are computationally expensive. This technical report introduces JointFM, a foundation model that inverts this paradigm. Instead of fitting SDEs to data, we sample an infinite stream of synthetic SDEs to train a generic model to predict future joint probability distributions directly. This approach establishes JointFM as the first foundation model for distributional predictions of coupled time series - requiring no task-specific calibration or finetuning. Despite operating in a purely zero-shot setting, JointFM reduces the energy loss by 21.1% relative to the strongest baseline when recovering oracle joint distributions generated by unseen synthetic SDEs.

Stefan Hackmann

DataRobot Research

Time-Series Foundation Models Probabilistic Time Series Forecasting Zero-Shot Time Series Forecasting

Mar 12, 2026

Deep Generative Transformers for Probabilistic Time Series and Spatiotemporal Forecasting

Reliable uncertainty quantification is paramount for forecasting multivariate time series and spatiotemporal data. While Transformer architectures excel at sequence modeling, current probabilistic approaches typically rely on restrictive parametric likelihoods or quantile-based objectives, thereby limiting their ability to capture complex joint distributions in correlated time series. To overcome these limitations, we propose Enformer and its spatiotemporal extension, GEnformer. These models synthesize the expressive power of Transformers with engression, a stochastic learning paradigm for modeling conditional distributions. By injecting stochastic noise and optimizing a strictly proper scoring objective, our frameworks directly learn conditional predictive distributions without imposing parametric assumptions. This design ensures the generation of coherent multivariate trajectories while maintaining the Transformer's efficacy in modeling long-range dependencies and cross-series interactions. The probabilistic capability of Enformer is achieved with an asymptotic overhead of only a constant factor over a deterministic Transformer with an identical configuration. We extensively evaluate our frameworks on prominent multivariate benchmarks for temporal dynamics and real-world epidemic datasets for spatiotemporal dynamics. Empirical results demonstrate that both frameworks yield calibrated probabilistic forecasts and consistently outperform state-of-the-art baselines.

Rajdeep Pathak, Rahul Goswami, Madhurima Panja +2

SAFIR, Sorbonne University Abu Dhabi, United Arab Emirates. · SCAI, Sorbonne Universit´e, Paris, France. · Indian Institute of Technology Guwahati, Assam, India.

Probabilistic Time Series Forecasting Multivariate Time Series Forecasting Spatiotemporal Forecasting Time Series Transformers Spatiotemporal Transformers Conditional Generative Modeling

Mar 9, 2026

Bayesian Transformer for Probabilistic Load Forecasting in Smart Grids

The reliable operation of modern power grids requires probabilistic load forecasts with well-calibrated uncertainty estimates. However, existing deep learning models produce overconfident point predictions that fail catastrophically under extreme weather distributional shifts. This study proposes a Bayesian Transformer (BT) framework that integrates three complementary uncertainty mechanisms into a PatchTST backbone: Monte Carlo Dropout for epistemic parameter uncertainty, variational feed-forward layers with log-uniform weight priors, and stochastic attention with learnable Gaussian noise perturbations on pre-softmax logits, representing, to the best of our knowledge, the first application of Bayesian attention to probabilistic load forecasting. A seven-level multi-quantile pinball-loss prediction head and post-training isotonic regression calibration produce sharp, near-nominally covered prediction intervals. Evaluation of five grid datasets (PJM, ERCOT, ENTSO-E Germany, France, and Great Britain) augmented with NOAA covariates across 24, 48, and 168-hour horizons demonstrates state-of-the-art performance. On the primary benchmark (PJM, H=24h), BT achieves a CRPS of 0.0289, improving 7.4% over Deep Ensembles and 29.9% over the deterministic LSTM, with 90.4% PICP at the 90% nominal level and the narrowest prediction intervals (4,960 MW) among all probabilistic baselines. During heat-wave and cold snap events, BT maintained 89.6% and 90.1% PICP respectively, versus 64.7% and 67.2% for the deterministic LSTM, confirming that Bayesian epistemic uncertainty naturally widens intervals for out-of-distribution inputs. Calibration remained stable across all horizons (89.8-90.4% PICP), while ablation confirmed that each component contributed a distinct value. The calibrated outputs directly support risk-based reserve sizing, stochastic unit commitment, and demand response activation.

Sajib Debnath, Md. Uzzal Mia

O&M Analytics, AES Clean Energ; The AES Corporation; Louisville, CO 80027, USA · Department of Information and Communication Engineering; Pabna University of Science and Technology; Rajapur, Pabna, Bangladesh

Electricity Load Forecasting Probabilistic Time Series Forecasting Bayesian Neural Networks Uncertainty Calibration Time Series Transformers Attention Mechanisms

Mar 8, 2026

Uncertainty-Gated Generative Modeling

Financial time-series forecasting is a high-stakes problem where regime shifts and shocks make point-accurate yet overconfident models dangerous. We propose Uncertainty-Gated Generative Modeling (UGGM), which treats uncertainty as an internal control signal that gates (i) representation via gated reparameterization, (ii) propagation via similarity and confidence routing, and (iii) generation via uncertainty-controlled predictive distributions, together with uncertainty-driven regularization and calibration to curb miscalibration. Instantiated on Weak Innovation AutoEncoder (WIAE-GPF), our UG-WIAE-GPF significantly improves risk-sensitive forecasting, delivering a 63.5% MSE reduction on NYISO (0.3508

\rightarrow

0.1281), with improved robustness under shock intervals (mSE: 0.2739

\rightarrow

0.1748).

Xingrui Gu, Haixi Zhang

Electrical Engineering and Computer Science, University of California, Berkeley, Berkeley, CA, USA · School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA

Time Series Forecasting Generative Modeling Probabilistic Time Series Forecasting Non-Stationary Time Series Forecasting Uncertainty Calibration

Mar 7, 2026

Deep Generative Spatiotemporal Engression for Probabilistic Forecasting of Epidemics

Accurate and reliable forecasting of epidemic incidences is critical for public health preparedness, yet it remains a challenging task due to complex nonlinear temporal dependencies and heterogeneous spatial interactions. Often, point forecasts generated by spatiotemporal models are unreliable in assigning uncertainty to future epidemic events. Probabilistic forecasting of epidemics is therefore crucial for providing the best or worst-case scenarios rather than a simple, often inaccurate, point estimate. We present deep spatiotemporal engression methods to generate accurate and reliable probabilistic forecasts on low-frequency epidemic datasets. The proposed methods act as distributional lenses, and out-of-sample probabilistic forecasts are generated by sampling from the trained models. Our frameworks encapsulate lightweight deep generative architectures, wherein uncertainty is quantified endogenously, driven by a pre-additive noise component during model construction. We establish geometric ergodicity and asymptotic stationarity of the spatiotemporal engression processes under mild assumptions on the network weights and pre-additive noise process. Comprehensive evaluations across six epidemiological datasets over three forecast horizons demonstrate that the proposal consistently outperforms several temporal and spatiotemporal benchmarks in both point and probabilistic forecasting. Additionally, we explore the explainability of the proposal to enhance the models' practical application for informed, timely public health interventions.

Rajdeep Pathak, Tanujit Chakraborty

SAFIR, Sorbonne University Abu Dhabi, United Arab Emirates · SCAI, Sorbonne Université, Paris, France

Epidemic Forecasting Spatiotemporal Forecasting Probabilistic Time Series Forecasting Spatial Epidemic Modeling

Mar 6, 2026

Ensemble Graph Neural Networks for Probabilistic Sea Surface Temperature Forecasting via Input Perturbations

Accurate regional ocean forecasting requires models that are both computationally efficient and capable of representing predictive uncertainty. This work investigates ensemble learning strategies for sea surface temperature (SST) forecasting using Graph Neural Networks (GNNs), with a focus on how input perturbation design affects forecast skill and uncertainty representation. We adapt a GNN architecture to the Canary Islands region in the North Atlantic and implement a homogeneous ensemble approach inspired by bagging, where diversity is introduced during inference by perturbing initial ocean states rather than retraining multiple models. Several noise-based ensemble generation strategies are evaluated, including Gaussian noise, Perlin noise, and fractal Perlin noise, with systematic variation of noise intensity and spatial structure. Ensemble forecasts are assessed over a 15-day horizon using deterministic metrics (RMSE and bias) and probabilistic metrics, including the Continuous Ranked Probability Score (CRPS) and the Spread-skill ratio. Results show that, while deterministic skill remains comparable to the single-model forecast, the type and structure of input perturbations strongly influence uncertainty representation, particularly at longer lead times. Ensembles generated with spatially coherent perturbations, such as low-resolution Perlin noise, achieve better calibration and lower CRPS than purely random Gaussian perturbations. These findings highlight the critical role of noise structure and scale in ensemble GNN design and demonstrate that carefully constructed input perturbations can yield well-calibrated probabilistic forecasts without additional training cost, supporting the feasibility of ensemble GNNs for operational regional ocean prediction.

Alejandro J. González-Santana, Giovanny A. Cuervo-Londoño, Javier Sánchez

Centro de Tecnologías de la Imagen Instituto Universitario de Cibernética, Empresas y Sociedad (IUCES) University of Las Palmas de Gran Canaria, Spain · Oceanografía Física y Geofísica Aplicada Instituto Universitario en Acuicultura Sostenible y Ecosistemas Marinos (ECOAQUA) University of Las Palmas de Gran Canaria, Spain

Probabilistic Time Series Forecasting Deep Ensembles Spatiotemporal Forecasting Spatiotemporal GNNs

Mar 4, 2026

Two-Stage Photovoltaic Forecasting: Separating Weather Prediction from Plant-Characteristics

Several energy management applications rely on accurate photovoltaic generation forecasts. Common metrics like mean absolute error or root-mean-square error, omit error-distribution details needed for stochastic optimization. In addition, several approaches use weather forecasts as inputs without analyzing the source of the prediction error. To overcome this gap, we decompose forecasting into a weather forecast model for environmental parameters such as solar irradiance and temperature and a plant characteristic model that captures site-specific parameters like panel orientation, temperature influence, or regular shading. Satellite-based weather observation serves as an intermediate layer. We analyze the error distribution of the high-resolution rapid-refresh numerical weather prediction model that covers the United States as a black-box model for weather forecasting and train an ensemble of neural networks on historical power output data for the plant characteristic model. Results show mean absolute error increases by 11% and 68% for two selected photovoltaic systems when using weather forecasts instead of satellite-based ground-truth weather observations as a perfect forecast. The generalized hyperbolic and Student's t distributions adequately fit the forecast errors across lead times.

Philipp Danner, Hermann de Meer

University of Passau, Innstraße 41, Passau, Germany

Solar Power Forecasting Solar Irradiance Forecasting Probabilistic Time Series Forecasting Weather Forecasting