Probabilistic Forecasting

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Overview

Probabilistic forecasting estimates ranges, quantiles, event probabilities, or full predictive distributions rather than only a single future value. Methods include Bayesian and parametric models, ensemble post-processing and conformal calibration, and neural generative approaches such as diffusion and flow matching; language-model agents also forecast real-world events using evidence and probability aggregation. Researchers evaluate forecasts with proper scores such as CRPS and Brier score, alongside calibration, sharpness, and benchmarks in weather, energy, health, finance, and human mobility. A central challenge is balancing accuracy, reliable uncertainty, computational cost, and usefulness for downstream decisions, since strong average scores may hide failures in extremes or dependencies between outcomes.

Latest papers

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230 papers37 papers in the last four weeks, up 3% on the four weeks before. 0.2% of all new papers.

May 30, 2026

EnergyMamba: An Uncertainty-Aware Graph-Enhanced Selective State Space Model for Energy Consumption Prediction

Energy consumption prediction is essential for efficient grid management, demand-side optimization, and sustainable energy planning. Although advanced machine learning methods have been employed for better prediction performance, existing works have two key limitations: (1) they usually formulate this task as a purely time-series prediction problem without explicitly modeling the spatial dependencies among different regions, and (2) they fail to provide reliable predictions with uncertainty estimates under abnormal situations such as extreme weather events. To advance existing research, we propose EnergyMamba, an uncertainty-aware spatiotemporal learning framework for accurate and reliable energy consumption prediction, which comprises two key components: (i) a novel Graph-Enhanced Selective State Space Model (GE-Mamba) that injects spatial context learned from the grid topology into the temporal dynamics, enabling coupled spatiotemporal modeling, and (ii) an Adaptive Sequential Conformalized Quantile Regression (AS-CQR) module, which includes locally adaptive normalization and an online feedback mechanism to dynamically calibrate prediction intervals under potential distribution shifts. We evaluate EnergyMamba on four large-scale real-world datasets from Florida, New York, and California. Results show EnergyMamba achieves around 5% improvement in prediction accuracy and 6% improvement in uncertainty quantification over 15 state-of-the-art baselines.

Dahai Yu, Rongchao Xu, Lin Jiang +1

Florida State University · Tallahassee, Florida, USA

Selective SSMsTime Series ForecastingProbabilistic ForecastingConformal PredictionSpatiotemporal ForecastingElectricity Load Forecasting+1
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May 30, 2026

MUSCLE-NET: Predicted-Multiscale-Aware Network for Pedestrian Trajectory Forecasting

Accurate pedestrian trajectory prediction is essential for safe navigation in autonomous driving and intelligent transportation systems. Despite substantial progress made by recent methods, most existing approaches are limited in fully exploiting diverse observations and often overlook the scale dependency of future motion, treating multiscale features uniformly regardless of underlying motion dynamics. This limits their robustness across diverse pedestrian behaviors. To address these challenges, we propose a Predicted-MUltiSCale-Aware Network (MUSCLE-NET) for Pedestrian Trajectory Forecasting that integrates complementary multimodal cues with scale-adaptive prediction mechanisms. The proposed framework is built upon a Multiscale Multimodal Feature Extraction (MMFE) module, which combines multiscale representation, modality-aware recalibration, and directional cross-modal fusion to construct semantically aligned representations from bounding boxes, velocities, and pose information. Building on these features, a Multiscale Enhanced Hierarchical Prediction (MEHP) module performs prediction-aware future-motion refinement via a probabilistic coarse predictor, scale-aligned fusion, and progressive refinement, adaptively selecting scale-relevant cues to mitigate spatial drift. Extensive experiments on the JAAD and PIE benchmarks demonstrate that the proposed MUSCLE-Net achieves competitive performance and consistent gains compared with state-of-the-art trajectory prediction methods.

Yu Liu, Ming Huang, Xiao Ren +3

Guangdong Provincial Key Laboratory of Fully Actuated System Control Theory and Technology, School of Automation and Intelligent Manufacturing, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China · Department of Mechanical Engineering, City University of Hong Kong, Hong Kong SAR, China

Pedestrian Behavior ModelingPedestrian Trajectory PredictionProbabilistic ForecastingMultimodal Trajectory PredictionTrajectory PredictionMultiscale Time-Series Modeling
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May 29, 2026

Probabilistic Precipitation Nowcasting with Rectified Flow Transformers

Accurate weather forecasts are essential across various domains and are safety-critical in extreme weather conditions. Compared to simulation-based forecasting, data-driven approaches show greater efficiency, enabling short-term, high-resolution nowcasting. In particular, diffusion models proved effective in weather nowcasting due to their strong probabilistic foundation. However, existing methods rely on deterministic compression to reduce the complexity of high-dimensional weather data, limiting their ability to capture uncertainty in the decoding process. In this work, we introduce FREUD\textbf{FREUD}, a Fr\textbf{Fr}ame-wise E\textbf{E}ncoder and U\textbf{U}nited D\textbf{D}ecoder model based on rectified flow transformers for efficient compression of spatio-temporal weather data. Frame-wise encoding enables continuous forecast updates, while the unified video decoder ensures temporal consistency. Our uncertainty-preserving first stage allows us to capture aleatoric uncertainty via ensembling, which is particularly beneficial for extreme weather events with high decoding variability. We achieve state-of-the-art performance in precipitation nowcasting with a compact latent-space rectified flow transformer on the SEVIR benchmark and show further performance gains by model and test-time scaling. Code available here: https://github.com/CompVis/weather-rf

Johannes Schusterbauer, Jannik Wiese, Nick Stracke +2

Precipitation ForecastingRectified FlowProbabilistic ForecastingPrecipitation Nowcasting
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May 29, 2026

ForecastCompass: Guiding Agentic Forecasting with Adaptive Factor Memory

Agentic forecasting is important for decision-making in dynamic environments, but it remains challenging because agents must reason from incomplete, time-limited evidence and produce calibrated probabilities before outcomes are resolved. Memory provides a natural mechanism for transferring experience from resolved forecasts to future prediction tasks. However, existing agent-memory methods are not tailored to forecasting, as they typically store past interactions, reflections, or factual associations without explicitly representing reusable predictive factors or calibration knowledge. We propose ForecastCompass (FoCo), an adaptive factor-based memory framework for agentic forecasting. FoCo organizes forecasting experience with a hierarchical forecasting-task taxonomy, enabling retrieval task-relevant forecasting knowledge. It maintains two complementary memory components: factor memory, which captures reusable predictive dimensions, and reasoning memory, which encodes probability updating, uncertainty handling, and calibration principles. Using retrospective analyses as learning signals, FoCo iteratively revises memory through a verbalized memory-revision procedure, enabling the agent to accumulate transferable forecasting knowledge over time. Experiments on Prophet Arena and FutureX with GPT-5-mini and Gemini-2.5-Flash show that FoCo improves both probabilistic accuracy and calibration.

Yurui Chang, Yongkang Du, Yuanpu Cao +2

Pennsylvania State University

Probabilistic ForecastingMemory-Augmented Agents
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May 28, 2026

Spatio-temporal stochastic graph-based learning for infectious disease forecasting

Spatio-temporal graph-based models have typically been used to forecast new cases of infectious diseases such as COVID-19 and chickenpox outbreaks. However, the use of stochastic modelling into their learning process has been surprisingly under-investigated and rarely considered entire data sets of large countries. As a result, it is unknown whether these models would provide accurate forecasts in real-world disease spread scenarios. In this work, we propose a spatio-temporal stochastic graph-based architecture that integrates a stochastic formulation and uncertainty approximation process to forecast new infectious disease cases. We find that our approach can adapt to encode large and small population geographical networks within a single model architecture. Using two real-world data sets, COVID-19 in the US and chickenpox in Hungary, we report an enhanced effect of the proposed architecture across predictions of the 2022 first wave for COVID-19 in the US and comparative results of chickenpox waves during 2012-2014 in Hungary. By benchmarking with four spatio-temporal graph-based models, quantitative results show competitive overall weekly performance of the proposed approach on forecasting new cases for all 3,218 US counties and all 20 Hungary counties. The proposed approach can represent overall epidemic progression relative to baselines, though with a one-step delay; while exhibiting a reduced sensitivity to high-frequency and low-amplitude variability.

Luz Stefani Sotomayor Valenzuela, Susanna Cramb, Darren Wraith

School of Public Health and Social Work, Queensland University of Technology, Brisbane, Australia · QUT Centre for Data Science, Queensland University of Technology, Brisbane, Australia

Epidemic ForecastingProbabilistic ForecastingSpatiotemporal ForecastingSpatiotemporal GNNsSpatial Epidemic Modeling
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May 28, 2026

Beyond MSE: Improving Precipitation Nowcasting with Multi-Quantile Regression

Deep-learning precipitation nowcasting models are often optimized using pointwise losses such as mean squared error or mean absolute error, which can lead to overly smooth forecasts and poor representation of heavy rainfall. This study investigates whether the predictive performance of an established deterministic nowcasting architecture can be improved by reformulating training as a multi-quantile regression problem. Using SmaAt-UNet as a core model, we compare MSE, MAE, and multi-quantile pinball-loss training on radar precipitation nowcasting over the Netherlands. The results show that multi-quantile training improves the central deterministic forecast, decreasing test-set MSE by 8.6% compared to a model trained using MSE, while also producing upper-quantile outputs that are useful for risk-sensitive prediction of heavy precipitation. These findings suggest that quantile regression provides a simple alternative to standard pointwise losses without requiring a new architecture or generative sampling procedure. The implementation of our models and training setup is available on GitHub.

Gijs van Nieuwkoop, Siamak Mehrkanoon

Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

Precipitation ForecastingQuantile RegressionProbabilistic ForecastingSpatiotemporal ForecastingPrecipitation Nowcasting
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May 27, 2026

Probabilistic bias adjustment of seasonal forecasts using generative machine learning: A case study of Arctic sea ice predictions

Seasonal climate predictions support planning and risk management by offering early information of the most likely-to-occur climate conditions in the coming months, and associated uncertainties. Ensemble forecasts enable this by simulating many plausible outcomes, allowing predictions to be expressed as usable probabilities. Large ensembles and high-resolution forecasts strengthen this guidance by better sampling uncertainty and capturing finer-scale processes but come with significant computational cost. Moreover, forecast ensembles drift and exhibit systematic biases and spatio-temporal errors that grow with lead time, requiring careful post-processing and calibration. A probabilistic post-processing framework based on conditional Variational Autoencoders (cVAEs) was developed at the Canadian Center for Climate Modeling and Analysis to generate large ensembles of bias adjusted seasonal predictions of Arctic sea ice. The generative model was designed to learn the observational distribution conditioned on the biased model prediction. This enables generation of arbitrarily large ensembles of well-calibrated, bias corrected forecasts with improved skill. Here, we extend this framework to address the loss of fine-scale energy and the characteristic blurriness in predictions, a known limitation of standard cVAEs. Specifically, we employ a generator in place of the Gaussian parametrized decoder in the cVAE and use Continuous Ranked Probability Score in the objective function instead of the Mean Square Error. We further use a higher resolution target dataset compared to the raw forecast. We show that the adjusted forecasts are better calibrated, more consistent with the observational distribution, and exhibit smaller errors than benchmark predictions, while also enhancing the resolution of the raw forecasts and improving sharpness and spectral power relative to the standard cVAE.

Parsa Gooya, Reinel Sospedra-Alfonso

Canadian Centre for Climate Modelling and Analysis, Environment and Climate Change Canada, Victoria, British Columbia, Canada

Time Series ForecastingProbabilistic ForecastingClimate ModelingModel CalibrationConditional Variational Autoencoder
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May 27, 2026

Stabilizing distribution-free probabilistic forecasts

Multi-step-ahead forecasts are often updated as new observations become available, since shorter forecast horizons typically improve forecast quality. However, such improvements come at the cost of forecast instability, i.e., variability in forecasts for the same target period. This instability can trigger costly changes to plans formulated based on the forecasts and may erode trust in the forecasting system. In this work, we integrate forecast stability alongside forecast quality into the training of distribution-free probabilistic time-series forecasting models, allowing us to control this trade-off. We propose a method for generating stabilized forecasted conditional quantile functions using regression splines parameterized by a neural network. This approach enables joint optimization of quality and stability, as it allows us to directly penalize dissimilarities arising from forecast updates. Furthermore, it allows assigning varying importance to stabilizing different parts of the forecast distributions (e.g., central parts vs. tails) to focus on the parts most relevant for the intended downstream use (e.g., the upper tail for inventory management). We empirically evaluate the proposed method on two datasets with different statistical properties and show that it can effectively reduce forecast instability without a substantial loss in forecast quality, and that it can target stabilization effort toward specific parts of the forecast distributions.

Jente Van Belle, Honglin Wen, Wouter Verbeke +1

Faculty of Economics and Business, KU Leuven, Belgium · Department of Electrical Engineering, Shanghai Jiao Tong University, China · Dyson School of Design Engineering, Imperial College London, United Kingdom +3

Probabilistic ForecastingProbabilistic Time Series Forecasting
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May 26, 2026

Aligning LLMs with Human Uncertainty: A Beta-Bernoulli Calibrator for LLM Forecasting

Probabilistic forecasting estimates the likelihood of uncertain future events. To improve LLM forecasting, existing methods typically learn from binary outcomes to output verbalized forecasts. However, while aggregated human forecasts contain rich information in both the crowd probability estimate and the degree of agreement among forecasters, how to utilize these signals remains underexplored. To address this, we propose the Beta-Bernoulli Calibrator (BBC), which converts an initial point estimate forecast from any model into a distribution over event likelihood, using supervision from both binary outcomes and human forecasts. BBC models event likelihood p∼Beta(α,β)p \sim \text{Beta}(α, β) and outcome y∼Bernoulli(p)y \sim \text{Bernoulli}(p), with the mean as the calibrated point forecast and the variance as the epistemic uncertainty. Our results show that BBC generally provides better calibrated and more accurate forecasts than both traditional post-hoc calibration methods and models fine-tuned specifically for forecasting, while remaining lightweight and having good generalization. We also show that the epistemic uncertainty captured by BBC is a more reliable predictor of forecasting error than verbalized confidence.

Hui Dai, Ryan Teehan, Parsa Torabian +1

Agentic Learning AI Lab, New York University · Chronologies AI

Post-Hoc CalibrationProbabilistic ForecastingLanguage Model CalibrationLLM Uncertainty Estimation
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May 25, 2026

Agent-Centric Social Trajectory Prediction: A Free Energy Principle Perspective

Trajectory prediction methods have demonstrated remarkable capabilities in capturing complex motion patterns. However, existing methods rely on global state assumptions, suffer from insufficient belief inference under partial observability, and lack cognitive behavioral constraints in prediction. These limitations severely compromise both deployment feasibility and physical plausibility in real-world settings. In this work, we propose FEP-Diff, an agent-centric trajectory prediction framework grounded in the Free Energy Principle, aimed at achieving cognitively plausible predictions under realistic constraints. Specifically, a dual-branch spatiotemporal encoder extracts ego-motion dynamics and social interaction cues from local observations. Building upon this, a goal-conditioned belief learner infers multimodal latent belief distributions optimized via a free-energy objective, with a social consistency constraint on the local neighborhood graph to promote cognitive alignment among neighboring agents. Finally, a residual diffusion trajectory generator is conditioned on the learned belief representations with token-level proxy conditioning, producing precise and diverse future predictions. Extensive experiments on five public benchmarks demonstrate that FEP-Diff consistently outperforms state-of-the-art methods under restricted observability. Code: https://anonymous.4open.science/r/FEP-Diff-8876.

Yanping Wu, Ji Zhang, Hao Chen +2

University of Glasgow Glasgow, UK · Southwest Jiaotong University Chengdu, China

Multi-Agent Trajectory PredictionProbabilistic ForecastingFree Energy PrincipleTrajectory PredictionProbabilistic Trajectory Prediction
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May 23, 2026

Deep Learning-Enabled Prediction of Geoeffective CMEs Using SOHO and SDO Observations

Understanding and forecasting the geoeffectiveness of a coronal mass ejection (CME) is crucial for protecting infrastructure in the near-Earth space environment and on Earth. In this study, we present a novel fusion model to forecast the geoeffectiveness of CME events. Our model combines convolutional neural networks for feature learning and a prediction network for feature fusion and event classification. The model is trained by observations from instruments including the Large Angle Spectroscopic Coronagraph (LASCO) on board the Solar and Heliospheric Observatory (SOHO) and the Atmospheric Imaging Assembly (AIA) and Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO). The trained model is then used to predict whether an Earth-reaching CME will cause a geomagnetic storm and/or the probability that the CME will cause such a storm. Experimental results based on a five-fold cross validation scheme demonstrate the good performance of our fusion model, achieving a mean true skill statistic (TSS) score of 0.703 when the model is used as a deterministic prediction tool, and a mean Brier score of 0.095 when the model is used as a probabilistic forecasting tool, where a TSS score of 1 or a Brier score of 0 indicates perfect performance. This work contributes to forecasting the causal relationship between Earth-directed CMEs and geomagnetic storms in solar-terrestrial interactions.

Zhaoxin Yan, Jason T. L. Wang, Haimin Wang +5

Institute for Space Weather Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA · Department of Computer Science, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA · Center for Solar-Terrestrial Research, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA +1

Probabilistic Forecasting
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May 22, 2026

Graph-based Complexity Forecasts in UK En Route Airspace Using Relevant Aircraft Interactions

Effectively managing Air Traffic Control Officer (ATCO) workload is crucial in maintaining operational safety. Group supervisors use tools that estimate upcoming traffic load to aid decision-making. However, industry-standard models can fail to capture the nuances of upcoming air traffic complexity. This study presents a probabilistic approach to forecast the complexity of an airspace sector using the number of relevant aircraft pairs, i.e., those that require monitoring or deconfliction by a controller, as a proxy measure for ATCO workload. We adapted an existing filter algorithm to make it suitable for use in London Middle Sector (LMS), a complex airspace sector with multiple flows of traffic above some of the busiest airports in Europe. Through iterative feedback with ATCOs, the algorithm was refined and extended to handle specific geometric and operational considerations. The updated algorithm outperformed the original, with an F1-score of 0.84 compared to 0.69 on a labelled set of 50 traffic scenarios. To produce forecasts of future numbers of relevant aircraft pairs in the sector, a graph representation of the LMS route network was constructed, standardising the spatial fidelity of route legs. The forecasting method accounts for uncertainty in aircraft arrival times by modelling the probability of each aircraft occupying route segments at future query times. When combined with historic distributions of relevant interactions and a live operational data stream, predictions of upcoming ATCO workload could be made up to 45 minutes in advance. The proposed method to forecast upcoming workload showed a significantly stronger correlation with actual relevant interactions (Spearman's ρ=0.68ρ= 0.68) than a standard traffic volume prediction (ρ=0.55ρ= 0.55). The resulting data-driven tool shows promise for use by group supervisors to inform sector configuration and ATCO rostering decisions.

Edward Henderson, George De Ath, Nick Pepper

The Alan Turing Institute · University of Exeter

Air Traffic ManagementIntelligent Transportation SystemsProbabilistic ForecastingTraffic Forecasting
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May 21, 2026

Is Capability a Liability? More Capable Language Models Make Worse Forecasts When It Matters Most

We document inverse scaling in LLMs on forecasting problems whose underlying time series exhibit superlinear growth and tail risk of regime change, a structure common in finance and epidemiology. On these tasks, more capable models produce worse distributional forecasts. The pattern appears on ForecastBench-Sim (FBSim), a contamination-free, simulated-world benchmark we release, in forecasting synthetic SIR epidemics with a matched linear control, and replicates in real-world datasets on COVID-19, measles, housing markets, and hyperinflation. A per-quantile decomposition shows the failure concentrates at the upper tail, which more capable models shift upward to track aggressive extrapolations of growth, while the lower tail stays put. A within-family study of Llama-3.1 shows that both model scale and post-training independently contribute to this effect. Domain knowledge does not reliably rescue calibration. This inverse scaling does not appear on single-threshold metrics common in LLM forecasting benchmarks, reversing the sign of the capability--accuracy relationship on identical outputs. Single-threshold scoring at conventional cutoffs misses the upper-tail cost; tail-inclusive scoring reverses the sign of the capability--accuracy relationship on the same outputs. We recommend that LLM forecasting evaluations use continuous (and unbounded) measures of accuracy alongside bounded binary threshold metrics.

Nick Merrill, Jaeho Lee, Ezra Karger

Forecasting Research Institute · UC Berkeley

Language Model Scaling LawsForecasting BenchmarksProbabilistic ForecastingLLM-Based Time Series ForecastingLanguage Model CalibrationProbabilistic Time Series Forecasting
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May 21, 2026

Aurora Hunter: A Two-Stage Framework for Probabilistic Visibility Forecasting

Aurora visibility at a given location requires two physically distinct conditions to hold at once: aurora occurring overhead, governed by solar wind-magnetosphere coupling, and observing conditions that permit detection, governed by cloud cover and moonlight. Approaches that conflate the two weaken the space-weather signal and limit cross-site generalizability. We develop Aurora Hunter, a two-stage cascade that separates occurrence prediction from observing-condition assessment. Stage 1 uses 43 physics-driven features to predict P(aurora identifiable in all-sky images) via gradient-boosted trees trained on joint Tromso+Kiruna data (about 16,600 hours, 2015-2023). Stage 2 models P(unobscured image class | identifiable) with logistic regression on 15 observing-condition features, trained on hours with identifiable aurora. The cascade P(visible) = P(identifiable) x P(unobscured | identifiable) achieves retrospective ROC-AUC of 0.958 (Tromso test, 2019-2020) and 0.933 (independent Kiruna, 2024), improving on the occurrence stage used alone by +0.095 and +0.097. Transfer to Skibotn, the one station withheld entirely (2022-2025), is limited. SHAP analysis identifies magnetic local time, the Kp x nightside interaction and the three-hour mean Kp as the dominant features (44% of attribution), consistent with auroral oval physics. Hemisphere-wide occurrence maps combine the Stage 1 amplitude, on a clear-sky basis, with the Feldstein oval parameterization. By providing location-specific visibility probabilities with measured reliability rather than coarse geomagnetic indices, the framework links space weather research to practical observation planning. An operational proof of concept is available at https://aurora-hunter.onrender.com

Zongyuan Ge, Chenwaner Zhang, Haoyang Li +4

College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China · School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650504, People’s Republic of China · Engineering Research Center of Advanced Marine Physical Instruments and Equipment, Ministry of Education, Qingdao 266100, China +1

Probabilistic ForecastingSpatiotemporal Forecasting
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May 21, 2026

A Posterior-Predictive Variance Decomposition for Epistemic and Aleatoric Uncertainty in Wind Power Forecasting

Accurate wind power forecasting requires reliable uncertainty quantification, yet most existing methods report a single predictive uncertainty that conflates epistemic and aleatoric sources. This paper applies the law of total variance to the joint setting of heteroscedastic neural network regression and Bayesian posterior approximation, deriving an explicit decomposition of total uncertainty (TU) into aleatoric (AU) and epistemic (EU) components. The resulting estimators are compatible with standard posterior-approximation methods and with ββ-NLL training to regulate the mean--variance learning trade-off. A wind power--specific evaluation framework is proposed to validate disentanglement without access to ground-truth uncertainty labels, comprising three modules: controlled synthetic experiments to verify responses to heteroscedastic noise and distribution shift; data-property--driven validation on a real-world wind turbine SCADA dataset; and dataset-size scaling experiments to examine the predicted asymptotic behavior of EU. Across synthetic and real-world experiments, the decomposed AU and EU components respond in theoretically consistent directions to noise structure, distributional shift, and training-scale variation, supporting the theoretical consistency and operational utility of the proposed decomposition and evaluation protocol.

Yinsong Chen, Samson S. Yu, Kashem M. Muttaqi

School of Engineering, Deakin University, Melbourne, VIC 3216, Australia · ARC Training Centre in Energy Technologies for Future Grids, School of Engineering, University of Wollongong, Wollongong, NSW 2522, Australia

Uncertainty QuantificationTime Series ForecastingProbabilistic ForecastingAleatoric UncertaintyWind Power ForecastingUncertainty Decomposition+1
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May 20, 2026

Evaluating Neural Probabilistic Forecasts of Sparse Earthquake Counts: What Survives a Careful Evaluation

We revisit a neural negative binomial model for weekly earthquake counts on a 3 x 3 degree grid over the Tian Shan and Pamir, and ask which of its conclusions survive a careful evaluation. At the completeness magnitude of 4.5, the 144-week test period holds 2448 cell-weeks; six cell-weeks, in five distinct weeks, have three or more events. Two conclusions survive. First, Poisson forecasts, temporal ETAS included, understate the probability of such a busy week: ETAS expects 0.72 where six occurred, a shortfall by a factor of 8.3 (p = 0.00009, exact test). One dispersion parameter removes it (ETAS with dispersion expects 4.45, p = 0.58); simulating aftershock chains within the week also reproduces the tail but overshoots the total count by about a quarter. Second, the network's advantage over a linear model disappears once that model has an intercept per cell: the remaining gain is 3.4% of deviance, not significant (p = 0.57 over five test years), and it comes from one cell-week out of the 4420 scored; without it the network is 0.7% worse. A blind test on 2024 to September 2026, with every model fitted on data through 2023, confirms the tail result: Poisson forecasts are rejected (p from 3 x 10^-8 to 0.001), dispersion models are not (p between 0.25 and 0.84), except the NB GLM, whose level blew up out of sample and which is rejected at p = 0.003. Out of sample, model families differ in how they extrapolate beyond the training range, not in the per-cell term. We also measure the evaluation traps met on the way: a biased reference level, a single seed (five of six effects in v1's results table change sign), too few bootstrap resamples, and a threshold below completeness, which shrinks the ETAS shortfall from 8.3 to 2.8 times. This version replaces preprint v1; Section 7 lists the corrections.

Alim Igilik

[physics.geo-ph]

Probabilistic ForecastingSeismologySpatiotemporal Forecasting
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May 20, 2026

Deep Learning Surrogates for Emulating Stochastic Climate Tipping Dynamics

This work explores a dynamics-informed Temporal Fusion Transformer (TFT) as a data-driven surrogate for computationally intensive Earth system simulations. Focusing on multivariate time series describing global ocean transport, we demonstrate the surrogate's ability to forecast tip events across thousands of time steps. The data involve up to 21 non-stationary time series in addition to static covariates describing free parameters and initial conditions. Modifications to the architecture and objective function yield a surrogate that anticipates the timing of Atlantic and Pacific collapses to high fidelity and captures the stochastic uncertainty in transition timing across ensemble predictions. The learned surrogate achieves a 465x computational speedup over the numerical simulator while maintaining differentiability with respect to parameters and initial conditions.

Adeline Hillier, Jennifer Sleeman, Jay Brett +3

Johns Hopkins Applied Physics Laboratory Hopkins University · Johns

Time Series TransformersMultivariate Time Series ForecastingClimate ScienceNeural Surrogate ModelingProbabilistic ForecastingNon-Stationary Time Series Forecasting+2
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May 19, 2026

Improving Ensemble CAPE Forecasts with a Diffusion Model Incorporating Aerosol Information

Convective available potential energy (CAPE) is an important variable for forecasting severe weather and understanding deep convection and precipitation. The latest versions of the Global Forecast System (GFS) and related Global Ensemble Forecast System (GEFS) have exhibited a bias towards underestimating CAPE values during the summertime. We train an artificial intelligence (AI) diffusion model to improve the skill and uncertainty quantification of afternoon 6-hour lead time ensemble forecasts over the United States. Our model takes a GFS CAPE forecast as input and outputs an ensemble that significantly outperforms both GFS and GEFS 6-hour forecasts on root mean square error, continuous ranked probability score, and Brier score. We propose a two-stage training pipeline to leverage both a larger historical GFS forecast dataset and a smaller historical GEFS dataset, despite the two using initialization and parameterization schemes that vary over time. We also show that classifier-free guidance can be used to control the skill and spread of the forecasts. We then demonstrate the versatility of our framework by adding aerosol optical depths (AODs) of black carbon, organic carbon, dust, sea salt, and sulfates as additional input features. Aerosols can invigorate or suppress convection depending on atmospheric conditions. Our AI models effectively incorporate aerosols to produce improved CAPE forecasts. We interpret the model components by using permutation feature importance to rank the influence of the different AODs and find that black carbon, organic carbon, and sulfate aerosols have a greater impact on the model's CAPE predictions than sea salt and dust aerosols.

Zachary James, Joseph Guinness, Arthur DeGaetano

Cornell University, Department of Statistics and Data Science · Washington University in St. Louis, Department of Statistics and Data Science · Cornell University, Department of Earth and Atmospheric Sciences

Probabilistic ForecastingDiffusion ModelsWeather Forecasting
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May 19, 2026

Probabilistic Multivariate Time Series Forecasting with Diffusion Copulas

Accurately assessing financial risk requires capturing both individual asset volatility and the complex, asymmetric dependence structures that emerge during extreme market events. While modern diffusion-based models have advanced multivariate forecasting, they often suffer from a "normality bias" when trained end-to-end, sacrificing marginal calibration for joint coherence and consistently underestimating tail risk. To address this, we propose a Diffusion-Copula framework that explicitly decouples the learning of marginal distributions from their dependence structure. We employ deep Mixture Density Networks to capture heavy-tailed asset dynamics, followed by a Classification-Diffusion Copula to model the joint dependence. Applied to cryptocurrency markets, our approach demonstrates superior performance over state-of-the-art baselines in forecasting systemic extremes of both marginal and joint events. Crucially, we demonstrate that while baseline models classify simultaneous market crashes as statistically impossible "Black Swans" (high surprise), our framework identifies them as "Expected Crashes" (low surprise), successfully preserving the correlation structure necessary for robust risk management during contagion events.

David Huk, Dongshan Wang, Miha Bresar

Department of Statistics The University of Warwick · School of Data Science The Chinese University of Hong Kong, Shenzhen

Multivariate Time Series ForecastingCopula ModelsQuantitative FinanceProbabilistic ForecastingProbabilistic Time Series ForecastingDiffusion Models
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May 19, 2026

Accurate, Efficient, and Explainable Deep Learning Approaches for Environmental Science Problems

Environmental science plays a pivotal role in safeguarding ecosystems, a domain driven by large-scale, heterogeneous data. In the big data era, artificial intelligence (AI) has emerged as a transformative tool for learning patterns and supporting decision-making. This dissertation develops AI-based approaches tailored to complex environmental science problems to achieve Environmental Intelligence, studying three specific challenges. First, we focus on flood prediction and management in coastal river systems. Conventional physics-based models are computationally intensive, limiting real-time application. To overcome this, we propose a deep learning (DL)-based model, WaLeF, for water level forecasting, and a forecast-informed DL model, FIDLAr, to manage water levels. Evaluated in a flood-prone coastal system in South Florida characterized by extreme rainfall and sea level fluctuations, FIDLAr outperforms baselines in accuracy and efficiency while providing interpretable outputs. Second, we target global weather prediction, which is challenged by massive data scale. Traditional physics methods are deterministic and computationally heavy. We propose CoDiCast, a conditional diffusion model tailored for probabilistic weather forecasting. Adapted from generative AI for predictive tasks, experiments show CoDiCast achieves accurate, efficient forecasts with explicit uncertainty quantification. Lastly, we address scientific question-answering in environmental science. When answering in-domain questions, large language models (LLMs) often suffer from hallucinations due to out-of-date or limited knowledge. While retrieval-augmented generation (RAG) retrieves domain-specific knowledge, existing methods trade off accuracy, efficiency, or explainability. We propose Hypercube-RAG, built on a structured text cube framework, which successfully exhibits all three properties simultaneously.

Jimeng Shi

FLORIDA INTERNATIONAL UNIVERSITY Miami, Florida

Retrieval-Augmented GenerationExplainable Artificial IntelligenceProbabilistic ForecastingQuestion AnsweringSpatiotemporal ForecastingFlood Forecasting
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May 17, 2026

Bayesian-Monte Carlo Schedule Updating for Construction Digital Twins: A Probabilistic Framework for Dynamic Project Forecasting

Construction projects frequently experience schedule delays and forecasting uncertainty due to variability in labor productivity, material availability, weather conditions, and project coordination. Conventional deterministic scheduling methods such as the Critical Path Method (CPM) assume fixed activity durations and therefore cannot adequately represent dynamic project uncertainty. This study presents a Bayesian-Monte Carlo probabilistic schedule updating framework for construction digital twin environments. The proposed methodology integrates stochastic activity-duration modeling, Bayesian recursive updating, Monte Carlo simulation, and uncertainty propagation within a unified computational framework for adaptive schedule forecasting. Activity durations are modeled using lognormal probability distributions and continuously updated through Bayesian inference as new project observations become available. Monte Carlo simulation is then used to propagate updated uncertainty throughout project networks and generate probabilistic completion-time forecasts, delay-risk estimates, and activity criticality measures. Simulation experiments using PSPLIB benchmark project networks demonstrate that the proposed framework improves forecasting accuracy and uncertainty representation compared with deterministic CPM and static probabilistic scheduling approaches. The framework further supports adaptive project forecasting through integration of BIM reports, drone observations, IoT telemetry, productivity logs, and site monitoring data.

Atena Khoshkonesh, Mohsen Mohammadagha, Vinayak Kaushal +1

1* M.S. Graduate, Department of Civil Engineering, The University of Texas at Arlington, Arlington, TX 76019, USA. · 2Ph.D. Researcher, Department of Civil Engineering, The University of Texas at Arlington, Arlington, TX 76019, USA. · 3Associate Professor of Instruction, Department of Civil Engineering, The University of Texas at Arlington, Arlington, TX 76019, USA.

Probabilistic ForecastingDigital TwinsDecision-Making under Uncertainty
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May 15, 2026

SwAIther-Precip: Lead-Time-Aware Bias Correction Enables Kilometer-Scale Downscaling of Global AI Precipitation Forecasts over Switzerland

Skillful medium-range precipitation forecasting at kilometer scale remains challenging over complex terrain because precipitation arises from multiscale nonlinear processes that global models cannot explicitly resolve at affordable cost. Global AI weather models can produce skillful medium-range forecasts, but their native 0.25 degrees resolution limits direct use for local hazard applications. Statistical downscaling can help bridge this gap, yet existing approaches often struggle with state-dependent, and especially lead-time-dependent, biases in global forecasts. We introduce SwAIther-Precip, a lead-time-aware downscaling framework that converts coarse-resolution AIFS forecasts into probabilistic km-scale precipitation fields over Switzerland. First, a U-Net conditioned on lead time via feature-wise linear modulation deterministically corrects systematic biases at coarse resolution. This targeted correction enables a cheaper super-resolution stage conditioned only on corrected precipitation, allowing direct training on observations rather than on the full atmospheric state. A diffusion-based model then generates fine-scale spatial variability independently of lead time. Using AIFS forecasts and CombiPrecip radar-gauge observations, SwAIther-Precip reduces CRPS by 48% relative to raw AIFS. The generated fields reproduce observed spatial variability with spectral fidelity above 0.85 at large scales and 0.88 at small scales, corresponding to an effective resolution of approximately 4 km on a 1 km grid for lead times up to 5 days. Training across lead times further improves long-range performance, yielding a 13% CRPS reduction at 6 days relative to lead-time-specific models. These results show that explicitly correcting lead-time-dependent biases before generative super-resolution is key to efficient km-scale probabilistic downscaling of global AI precipitation forecasts.

Dan Assouline, Erwan Koch, Federico Amato +5

Swiss Data Science Center, ETH, Zürich, Switzerland · Expertise Center for Climate Extremes, University of Lausanne, Lausanne, Switzerland · Faculty of Geosciences and Environment, University of Lausanne, Lausanne, Switzerland +1

Precipitation ForecastingTime Series ForecastingProbabilistic ForecastingPrecipitation DownscalingWeather Forecasting
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May 14, 2026

Njord: A Probabilistic Graph Neural Network for Ensemble Ocean Forecasting

Ocean dynamics are inherently chaotic, yet existing machine learning ocean models produce only deterministic forecasts. We introduce Njord, a probabilistic data-driven model for ocean forecasting, applicable to both global and regional domains. Njord combines a deep latent variable framework with a graph neural network architecture, enabling sampling each forecast step in a single forward pass. We apply Njord globally at 0.25° resolution and regionally to the Baltic Sea at 2 km resolution. To scale to these large ocean grids we introduce K-means cluster meshes that adapt to irregular sea surface geometry. Experiments demonstrate strong performance on both domains compared to deterministic machine learning baselines, while also providing uncertainty estimates from the sampled ensemble forecasts. On the global OceanBench benchmark, Njord achieves the lowest errors on average across upper-ocean variables when evaluated against real-world observations, with the largest improvements in surface temperature prediction.

Daniel Holmberg, Joel Oskarsson, Erik Wikingsson +2

University of Helsinki · ETH AI Center · Linköping University

Graph Neural NetworksUncertainty QuantificationProbabilistic ForecastingSpatiotemporal Forecasting
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May 14, 2026

VMU-Diff: A Coarse-to-fine Multi-source Data Fusion Framework for Precipitation Nowcasting

Precipitation nowcasting is a vital spatio-temporal prediction task for meteorological applications but faces challenges due to the chaotic property of precipitation systems. Existing methods predominantly rely on single-source radar data to build either deterministic or probabilistic models for extrapolation. However, the single deterministic model suffers from blurring due to MSE convergence. The single probabilistic model, typically represented by diffusion models, can generate fine details but suffers from spurious artifacts that compromise accuracy and computational inefficiency. To address these challenges, this paper proposes a novel coarse-to-fine Vision Mamba Unet and residual Diffusion (VMU-Diff) based precipitation nowcasting framework. It realizes precipitation nowcasting through a two-stage process, i.e., a deterministic model-based coarse stage to predict global motion trends and a probabilistic model-based fine stage to generate fine prediction details. In the coarse prediction stage, rather than single-source radar data, both radar and multi-band satellite data are taken as input. A spatial-temporal attention block and several Vision mamba state-space blocks realize multi-source data fusion, and predict the future echo global dynamics. The fine-grained stage is realized by a spatio-temporal refine generator based on residual conditional diffusion models. It first obtains spatio-temporal residual features based on coarse prediction and ground truth, and further reconstructs the residual via conditional Mamba state-space module. Experiments on Jiangsu SWAN datasets demonstrate the improvements of our method over state-of-the-art methods, particularly in short-term forecasts.

Chunlei Shi, Hao Li, Yufeng Zhu +6

Department of Automation, Southeast University, Nanjing 210096, China

Precipitation ForecastingProbabilistic ForecastingPrecipitation NowcastingDiffusion Models
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May 14, 2026

When Individually Calibrated Models Become Collectively Miscalibrated

Probabilistic prediction systems often aggregate probability estimates from multiple models into a single decision. A common assumption is that if each model is individually calibrated, the aggregate prediction will also be well calibrated. We show that this assumption fails in multi-agent settings: individually calibrated predictors can become collectively miscalibrated when their predictions interact strategically, in the game-theoretic sense of Brier-optimal local response, even without deliberate coordination. This phenomenon arises naturally when agents are independently trained on overlapping data. We prove that under Brier-score-based aggregation with positively correlated beliefs, each agent's individually optimal report systematically underestimates the positive-class probability, yielding a Price of Anarchy greater than one whenever Cov(b_i, b_j) > 0. In a canonical setting (n = 5 agents, pairwise correlation = 0.5, base rate = 0.3), the empirically measured PoA in false-negative rate reaches 7.25x. In contrast, VCG-based aggregation aligns incentives by rewarding marginal contribution, achieving dominant-strategy incentive compatibility and near-optimal performance. Experiments on three real-world datasets (NSL-KDD, UNSW-NB15, Credit Card Fraud) show that VCG provides strong robustness while maintaining comparable accuracy. It performs particularly well in data-sparse and adversarial settings, and adaptive weighting further improves performance under distribution shift.

Zhaohui Wang

USC Viterbi School of Engineering, University of Southern California

Probability CalibrationGame TheoryProbabilistic ForecastingMulti-Agent SystemsPrice of Anarchy
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May 9, 2026

LEAF: A Living Benchmark for Event-Augmented Forecasting

Large Language Models (LLMs) are increasingly applied to forecasting. To evaluate this capability while mitigating pre-training data contamination, several living benchmarks have been proposed. However, existing benchmarks either lack the multidimensional events essential for accurate forecasting due to data scarcity, or focus on relatively closed environments. To assess the predictive capabilities of LLMs in complex, real-world scenarios, we propose LEAF, the first living benchmark for event-augmented forecasting tasks, including future event probabilities, trend and time series forecasting. LEAF utilizes a recursive retrieval agent system paired with dual-agent cross-validation to provide comprehensive and relevant auxiliary text for forecasting. Evaluating state-of-the-art proprietary and open-weight LLMs, we find that these models can leverage signals extracted from complex events to enhance predictive performance. In the stock domain, we find that LLMs achieve better performance on equities they confidently identify as more predictable. Furthermore, the events demonstrate a strong correlation with the target equities. To this end, LEAF provides a necessary, dynamically updating testbed to continuously track and drive progress in event-driven forecasting tasks.

Mingtian Tan, Mihir Parmar, Palash Goyal +5

Google · University of Virginia

Financial ForecastingTime Series ForecastingForecasting BenchmarksProbabilistic ForecastingLLM-Based Time Series Forecasting
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May 7, 2026

Tyche: One Step Flow for Efficient Probabilistic Weather Forecasting

Probabilistic weather forecasting requires not only accurate trajectories, but calibrated distributions over plausible atmospheric futures. Recent data-driven systems have achieved remarkable deterministic skill, and diffusion-based ensemble forecasters have substantially improved sample realism and uncertainty quantification. However, their inference cost scales with forecast horizon, ensemble size, and the number of denoising steps required for each transition, making large operational ensembles expensive. To address this, we present Tyche, a one-step conditional flow model for efficient probabilistic weather forecasting. Tyche models the conditional forecast distribution with a destination-aware average-velocity flow that maps Gaussian noise directly to future weather states in a single function evaluation (1-NFE). To make this one-step transport learnable in high-dimensional geophysical fields, we derive a JVP-regularized rectification objective that enforces temporal self-consistency across source and destination flow timesteps without explicitly forming Jacobians. The transport field is parameterized by an isotropic Swin-style transformer that preserves fine-scale spatial structure while remaining scalable on global grids. To improve ensemble reliability under autoregressive forecasting, we further introduce a rollout-based finetuning stage with curriculum CRPS calibration supervision. Experiments on ERA5 at 1.5∘^\circ and 6-hour resolution show that our Tyche, using merely a single NFE, matches or exceeds the forecast skill and calibration of state-of-the-art multi-step generative baselines and the operational ECMWF IFS ensemble.

Fan Xu, Yuan Gao, Kun Wang +4

1USTC · 2SLAI · 3THU +3

Flow MatchingProbabilistic ForecastingOne-Step Generative ModelingConditional Generative Modeling
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May 6, 2026

Two Hebrew folk meteorological proverbs tested: rainfall on Rosh Chodesh and Shabbat Mevarechim as predictors of monthly precipitation (Israel, 1950-2024)

Folk meteorological proverbs encode centuries of empirical observation by agricultural communities. Two Hebrew proverbs link lunar calendar anchor days to monthly winter rainfall: (i) "If Rosh Chodesh is rainy, the whole month is rainy" and (ii) "If it rains on Shabbat Mevarechim, the whole month is rainy." Shabbat Mevarechim is the last Saturday before each new Hebrew month, preceding Rosh Chodesh by one to seven days. The first proverb is widely known; the second circulates in Hasidic oral tradition with no identified written source. Both have never been formally tested. We analyse 75 years (1950-2024) of daily precipitation data from seven Israeli cities across three climatic regions, comprising 191,758 station-days and 2,422 Hebrew-month observations during the winter rainy season (Marcheshvan-Adar). A rainy Rosh Chodesh increases the probability of a rainy month from 22.2% to 38.6% (lift +16.4 percentage points; chi-square = 57.8, p = 2.9e-14; Bayes factor 1.81). A rainy Shabbat Mevarechim produces a similar effect (lift +16.5 percentage points, p = 8.0e-13), despite preceding Rosh Chodesh by up to seven days. The effect decays with lag and mirrors daily rainfall autocorrelation (r = 0.35-0.44 at lag 1; ~0 at lag 7), consistent with Mediterranean cyclone persistence. A bootstrap permutation test (p < 1e-4) and a 15-year rolling analysis show declining predictive power (-0.20 percentage points per year, p < 0.001), consistent with shortening precipitation events under warming climate conditions. Both proverbs encode real but probabilistic meteorological signals whose reliability is decreasing over time.

Abraham Itzhak Weinberg

AI-WEINBERG, AI Experts [physics.ao-ph]

Precipitation ForecastingProbabilistic ForecastingWeather Forecasting
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May 5, 2026

Towards accurate extreme event likelihoods from diffusion model climate emulators

ML climate model emulators are useful for scenario planning and adaptation, allowing for cost-efficient experimentation. Recently, the diffusion model Climate in a Bottle (cBottle) has been proposed for generation of atmospheric states compatible with boundary conditions of solar position and sea surface temperatures. Crucially, cBottle can be guided to generate extreme events such as Tropical Cyclones (TCs) over locations of interest. Diffusion models such as cBottle work by approximating the probability density of the training data. Here, we show use cases of the probability density estimates of atmospheric states obtained from this climate emulator. Most importantly, these estimates allow us to calculate likelihoods of extreme events under guidance. When guiding the model towards states including TCs, comparing the probability density under the guided and unguided model enables us to quantify how much more likely the guidance has made the TC. We show how these odds ratios allow us to importance-sample from the TC distribution, reducing the standard error of the probability estimate compared to simple Monte Carlo sampling. Furthermore, we discuss results and limitations of the application of model probability densities to extreme event attribution-like experiments. We present these early but encouraging results hoping they will spur more research into probabilistic information that can be gained from diffusion models of the atmosphere.

Peter Manshausen, Noah Brenowitz, Julius Berner +2

NVIDIA, Santa Clara, CA, USA

Diffusion Model GuidanceClimate ScienceClimate EmulationProbabilistic ForecastingClimate ModelingRare-Event Estimation
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May 1, 2026

Foresight Arena: An On-Chain Benchmark for Evaluating AI Forecasting Agents

Evaluating the true forecasting ability of AI agents requires environments that are resistant to environments resistant to overfitting, free from centralized trust, and grounded in incentive-compatible scoring. Existing benchmarks either rely on static datasets vulnerable to training-data contamination, or measure trading PnL -- a metric conflating predictive accuracy with timing, sizing, and risk appetite. We introduce Foresight Arena, the first permissionless, on-chain benchmark for evaluating AI forecasting agents on real-world prediction markets. Agents submit probabilistic forecasts on binary Polymarket markets via a commit-reveal protocol enforced by Solidity smart contracts on Polygon PoS; outcomes are resolved trustlessly through the Gnosis Conditional Token Framework. Performance is measured by the Brier Score and a novel Alpha Score -- proper scoring rules that incentivize honest probability reporting and isolate predictive edge over market consensus. We provide a formal analysis: closed-form variance for per-market Alpha, the connection to Murphy's classical Brier decomposition, and a power analysis characterizing the number of rounds required to reliably distinguish agents of different skill levels. We show that detecting a true edge of α∗=0.02α^* = 0.02 at 80% power requires approximately 350 resolved binary predictions (50 rounds of 7 markets), while α∗=0.01α^* = 0.01 requires four times more. We complement these analytical results with a deterministic, seed-controlled simulation study calibrated to literature-reported Brier-score ranges, illustrating how Murphy decomposition distinguishes well-calibrated agents from market-tracking agents that fail through reduced resolution. Live results from the deployed benchmark will be reported in a future revision. All smart contracts and evaluation infrastructure are open-source.

Maksym Nechepurenko, Pavel Shuvalov

Director of Research, Devnull, Dubai, UAE · Chief Technology Officer, Devnull, Dubai, UAE

AI Agent EvaluationForecasting BenchmarksProbabilistic ForecastingProper Scoring RulesPrediction Markets
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