Uncertainty Quantification

Also known as UQ

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44 papers in the last four weeks, up 110% on the four weeks before. 0.4% of all new papers.

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Jul 22, 2026cs.LG

Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modeling

Constitutive modeling under uncertainty remains a central challenge for reliable mechanics simulations, particularly when the available stress-deformation data are sparse, noisy, or heterogeneous. We propose interval and fuzzy physics-augmented neural networks (iPANNs and fPANNs) for uncertainty-aware hyperelastic constitutive modeling. iPANNs learn sparse lower, mean, and upper free energy density branches whose stresses, obtained by automatic differentiation, ultimately enclose noisy stress observations. In contrast to this deterministic interval description, fPANNs embed the learned iPANN branches into a fuzzy-set representation through alpha-cut interpolation, yielding a nested family of admissible responses. iPANNs and fPANNs encode mechanistic constraints - preserving objectivity, consistency and promoting polyconvexity - and smoothed L0 regularization promotes interpretable energy representations. The bound models are trained through a two-stage transfer-learning procedure in which a sparse mean constitutive response is learned first and then fine-tuned into lower and upper energy branches. We evaluate the framework on synthetic isotropic hyperelastic data with heteroscedastic noise, varying random realizations, shifted noise means, and varying noise magnitudes. The results show that the learned bounds enclose noisy stress observations while generalizing to the test set. Further, we examine the propagation of uncertainty through the mean, upper and lower bound predictions of the learned iPANN models in a finite element setting. The proposed framework provides a compact, physics-consistent route for distribution-free aleatoric uncertainty quantification in hyperelastic constitutive modeling, and propagation in downstream finite element simulations.

Somesh Pratap Singh, Govinda Anantha Padmanabha, Jingye Tan +4

Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, 14853, NY, USA · Ecole Polytechnique Federale de Lausanne (EPFL), 1015, Lausanne, Switzerland · Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, 90007, CA, USA +2

Uncertainty QuantificationHyperelasticityPhysics-Informed MLUncertainty PropagationRegression Prediction Intervals
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Jul 22, 2026cs.CV

Factor-Informed Uncertainty Distillation for Gaze Estimation

Deep gaze estimation works well in controlled capture but degrades in unconstrained settings, where systems must reject unreliable predictions. Single-pass uncertainty (e.g., heteroscedastic regression) infers uncertainty from pixels without explicit input-validity cues, while sampling based methods are often too costly for real time use. We propose Factor-Informed Uncertainty Distillation (FIUD), a teacher-student framework that aligns uncertainty with interpretable image-quality failure modes. A gradient-boosting teacher predicts expected gaze error from factors such as illumination, sharpness, eye visibility and symmetry; a neural student distills these signals via curriculum learning and ranking supervision into a lightweight single-pass uncertainty head. Across ETH-XGaze, Gaze360, and MPIIFaceGaze (>300k samples), FIUD improves uncertainty, error rank correlation and selective prediction versus deterministic and sampling-based baselines, with the largest gains in unconstrained settings.

Mohammadreza Jamalifard, Yaxiong Lei, Javier Fumanal Idocin +3

University of Essex · Colchester, UK

Gaze EstimationUncertainty QuantificationSelective PredictionKnowledge Distillation
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Jul 22, 2026cs.LG

Good Practice Guide for quantifying uncertainties for machine learning models applied to photoplethysmography signals

This Good Practice Guide presents work done in the QUMPHY project (Uncertainty quantification for machine learning models applied to photoplethysmography signals) that considered both machine learning and uncertainty quantification for problems which used photoplethysmography (PPG) signals from wearable devices as input. It provides high-level guidance on what types of machine learning model might be used and how different models compare when applied to both regression and classification tasks. It provides guidance on the implementation of different methods for uncertainty quantification, covering both model-dependent and model-independent techniques, and on the validation of the results provided by those methods. It also describes six benchmark problems together with pointers to different benchmark datasets for each problem. Software is described that can assist practitioners in implementing the methods described herein and there is a brief consideration of ethical issues. It concludes with a summary and recommendations.

P. Harris, C. Bench, M. Rinkevičius +6

PTB Physikalisch-Technische Bundesanstalt · NPL National Physical Laboratory · CMI Cesky Metrologicky Institut +5

Uncertainty QuantificationPhotoplethysmography
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Jul 22, 2026cs.CV

SIINR: Structurally Informed Implicit Neural Representations for super-resolution with uncertainty quantification of clinical quality diffusion MRI datasets

Diffusion Magnetic Resonance Imaging (dMRI) is a powerful tool for probing brain microstructure, but clinical acquisitions are often limited by low out-of-plane resolution, resulting in degraded structural information and reduced utility for advanced analysis. We introduce SIINR (Structurally Informed Implicit Neural Representations), a general framework for super-resoltion of clinical dMRI datasets while quantifying uncertainty in the reconstructed outputs. SIINR utilizes a supervised 3D U-net as a prior and combines it with a self-supervised implicit neural representation (INR) that fuses the high-resolution prior and the original low-resolution data. The INR enables joint modeling across spatial and angular domains, enforces data consistency, and provides analytic approximate posterior distributions for downstream uncertainty quantification. We validate the framework on a diverse set of open-access dMRI datasets, demonstrating that SIINR outperforms standard interpolation methods in both quantitative error metrics and qualitative anatomical fidelity. Experiments on clinical cases, including subjects with multiple sclerosis and brain lesions, illustrate the framework its ability to propagate intensity changes and flag uncertain regions in challenging scenarios. SIINR is flexible, modular, and can be adapted to different upsampling ratios and downstream tasks, providing a principled approach for enhancing clinical dMRI and supporting robust interpretation of derived neuroimaging metrics.

Tom Hendriks, William Consagra, Anna Vilanova +2

Department of Computer Science and Mathematics, Eindhoven University of Technology, Groene Loper 5, Eindhoven, 5612 AP, Noord-Brabant, The Netherlands. · Department of Statistics, University of South Carolina, Columbia, 29225, South Carolina, United States of America. · Psychiatry Neuroimaging Laboratory, Brigham and Women’s Hospital, Harvard Medical School, 399 Revolution Drive, Boston, 02215, MA, United States.

Implicit Neural RepresentationsUncertainty QuantificationMedical Image Super-ResolutionMagnetic Resonance ImagingDiffusion MRINeuroimaging
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Jul 21, 2026cs.LG

Probabilistic Physics-Aware Machine Learning Predictions of Electric Truck Energy Consumption with Field Data

In this work, we incorporate first principle physics into the construction of data-driven methods by considering a model that accounts for the different sources of energy losses during vehicle operations. Our results show that Bayesian linear regression based on this physics-aware model can improve the reliability of the expected energy consumption, as compared with standard linear regression. Further, it is shown that more complex machine learning models such as neural networks and gradient boosted regression trees, based on the same physical model, can further improve the accuracy in energy forecasting and significantly outperform standard versions of the same machine learning models. In addition to point predictions of the energy consumption, we develop a framework for estimating the corresponding uncertainty in the form of predicted standard deviation. Our results show that all of the models learn to estimate the uncertainty reasonably well.

Hannes Nilsson, Rafael Basso, Balázs Kulcsár +1

Volvo Technology AB Gothenburg, Sweden · Department of Electric Engineering Chalmers University of Technology Gothenburg, Sweden · Department of Computer Science and Engineering Chalmers University of Technology and University of Gothenburg Gothenburg, Sweden

Uncertainty QuantificationProbabilistic ForecastingPhysics-Informed ML
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Jul 21, 2026cs.LG

Variational meta-learning inference for low dimensional neural system identification

Deep learning has proven highly effective for nonlinear system identification, but heavily parameterized neural networks are prone to overfitting in low-data regimes and lack reliable uncertainty quantification. The recently developed manifold meta-learning framework addresses the data efficiency problem by restricting the model parameters to a meta-learned low-dimensional manifold. However, that method is purely deterministic. We propose a fully probabilistic extension of the manifold meta-learning framework, based on amortized Variational Inference, where a generative prior over the low-dimensional parameter manifold is learned. During task-specific adaptation, we combine Maximum A Posteriori estimation with the Laplace approximation to yield a mathematically grounded posterior approximation. Evaluated on a static regression task and the Bouc--Wen dynamical system benchmark, the proposed approach achieves predictive accuracy comparable to its deterministic counterpart while successfully providing calibrated uncertainty bounds in severely low-data regimes.

Matteo Rufolo, Dario Piga, Marco Forgione

SUPSI-DTI-IDSIA, Dalle Molle Institute for Artificial Intelligence, Lugano, Switzerland.

Uncertainty QuantificationMeta-LearningNonlinear System IdentificationSystem IdentificationVariational Inference
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Jul 20, 2026stat.ME

Calibrating Semantic Uncertainty from Observable Language-Model Probabilities

As generative artificial intelligence enters scientific and professional work, its uncertainty must be defined on the states that matter for inference and decision-making. Language models assign probabilities to words, whereas applications require uncertainty over meaningful states such as diagnoses, hypotheses or operational conditions. We introduce a \emph{semantic map}: a prespecified, testable bridge from probabilities over verbal responses to a posterior over declared finite states. The language distribution remains unrestricted; held-out calibration connects it to a reference posterior. We derive posterior-error bounds and conditions for existence, conditional uniqueness, presentation stability and stable inverse recovery. This distinction matters because language probabilities depend on prompt wording, while the target posterior should not change under information-equivalent rewording. Experiments use professional market text compiled from Federal Reserve economic and financial series, together with controlled simulations having exact posteriors. Across two fitted language models, language-derived probabilities outperform printed numerical confidence, recover held-out posteriors with valid uncertainty coverage, remain largely stable under paraphrase and respond appropriately to altered evidence. \textbf{Prompt engineering optimises a wording-dependent response; robust scientific use requires validated stability of application-relevant meaning.} The proposed map turns semantic uncertainty in generative systems into an identifiable and testable statistical measurement problem and, when its acceptance conditions hold, yields an auditable posterior estimate.

Matthew F. Dixon

Artificial Intelligence Finance Institute (AIFI), 69 Charlton St, New York, NY 10014, USA

Uncertainty QuantificationConfidence Estimation in Language ModelsLanguage Model CalibrationLLM Uncertainty EstimationUncertainty Calibration
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Jul 17, 2026cs.RO

Vision-Language-Motion Maps: An Open-Vocabulary, Uncertainty-Aware, Queryable Motion Attribute for 3D Scene Maps

Open-vocabulary 3D maps let robots answer language queries about what and where, but they assume a static world and cannot answer queries about how scene elements behave. We introduce Vision-Language-Motion Maps (VLMM), an open-vocabulary, language-queryable 3D map - queried through a rule-based intent router over open-vocabulary object nouns, not a general natural-language interface - in which each element carries a fused motion attribute: a VLM/LLM semantic movability prior combined with geometrically observed cross-frame motion, together with a per-element uncertainty. Queries reduce to attribute filters that distinguish what has been seen to move, what could move but has not, and what stays still. On a controlled simulator benchmark with exact ground truth (AI2-THOR, three scene types) we show through ablation that the schema fields are non-substitutable: a semantic-only baseline fails motion queries even with strong features, and neither motion field substitutes for the other (the prior cannot answer "what is moving," observed motion cannot answer "what could move"). On real dynamic RGB-D (TUM and Bonn, six sequences) we show the uncertainty channel - our key difference from prior fused-motion work - consistently improves moving-vs-static average precision and reduces false motion flags, and that it is robust to estimated (noisy) poses. The raw confidence is not calibrated, but post-hoc isotonic calibration reaches an expected calibration error of 0.10. VLMM is a representation contribution: the closest prior maps each lack at least one of the four properties - open-vocabulary, language-queryable, fused prior-and-observed motion, and per-element uncertainty - that our combination provides.

Dibyendu Ghosh, Ayushi Shakya

Rekise Marine, Bangalore, India

Uncertainty Quantification3D Scene RepresentationRobotic Perception
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Jul 17, 2026cs.CV

Disentangling Model and Human Data Uncertainty in Apparent Facial Age Estimation

Estimating the apparent age of individuals from facial images is challenging due to the subjective nature of perception and the inherent variability of the data. We investigate the role of uncertainty estimation, attributing uncertainty jointly to a lack of knowledge (epistemic) or inherent noise/chance (aleatoric). Leveraging the APPA-REAL dataset, we train Bayesian Neural Networks on datasets of varying sizes using three BNN approximations: MC-DropConnect, Flipout, and Deep Ensembles using supervision on human aleatoric uncertainty available in the APPA-REAL dataset. Each model outputs both the predicted apparent age and the amount of aleatoric and epistemic uncertainty. Our results confirm the hypothesis that the inherent aleatoric uncertainty remains stable across dataset sizes, while epistemic uncertainty increases as training data decreases. These findings demonstrate that different sources of uncertainty can be quantified in face age estimation.

Andrei Foitos, Ivo Pascal de Jong, Matias Valdenegro-Toro

Bernoulli Institute, University of Groningen, Groningen, NL

Bayesian Neural NetworksUncertainty QuantificationAleatoric UncertaintyEpistemic Uncertainty
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Jul 16, 2026stat.ML

Subjective Risk Decomposition: A New View for Uncertainty Quantification

We present a novel viewpoint for uncertainty quantification. Uncertainty measures are not primitives, in need of axioms and argumentation, but instead consequences, of higher-level modelling decisions. We show how epistemic and aleatoric uncertainty measures can be derived via decomposition of a subjective risk, based on a strictly proper loss. Reverse cross entropy provides a prominent example, where decomposition recovers the classic information-theoretic uncertainty terms. The same approach recovers numerous measures previously proposed across the UQ literature, providing them a common theoretical foundation. This suggests a new approach to UQ: given a modelling scenario and strictly proper loss, the corresponding epistemic and aleatoric terms are induced by the subjective-risk decomposition. We then extend our view to learning theory: we introduce and analyse subjective risk analogues of excess risk, approximation error and estimation error, and identify the connections to UQ. We consider this a first step towards a full learning-theoretic framework for uncertainty quantification.

Raghad Alamri, Michele Caprio, Gavin Brown

Department of Computer Science, The University of Manchester · Department of Computer Science, University of Warwick

Uncertainty QuantificationAleatoric UncertaintyUncertainty DecompositionEpistemic Uncertainty
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Jul 16, 2026physics.chem-ph

Full-data accuracy with fewer labels for training and fine-tuning machine-learning force fields

Machine-learning force fields (MLFFs) are reliable only near their training distribution, making efficient construction of diverse training sets a major bottleneck for both train-from-scratch and foundation fine-tuning workflows. Active learning can reduce this cost, but standard model-committee uncertainty is impractical for foundation MLFFs because each committee member requires a separate fine-tuning run. We present an active-learning workflow based on last-layer-projection regression (LLPR), a forward-pass-cheap per-configuration uncertainty estimator. Across molecular, condensed-phase, and electrolyte systems, LLPR identifies compact, high-value training sets that recover full-data accuracy using only a small fraction of electronic-structure labels. In foundation-model fine-tuning, LLPR-selected configurations reach the full-pool fine-tuning ceiling with substantially fewer labels than random selection. In iterative electrolyte fine-tuning, LLPR detects unphysical local coordination before DFT labelling, provides an absolute force-error threshold, and enables automatic termination of the learning loop. The resulting models reproduce reference density and ion-coordination structure, providing a scalable uncertainty-quantification strategy across MLFF training regimes.

Sheng Bi, Yi-Ze Wang, Jun Cheng

College of Materials, Xiamen University, Xiamen, China. · Laboratory of AI for Electrochemistry (AI4EC), IKKEM, Xiamen, China. · State Key Laboratory of Physical Chemistry of Solid Surfaces, iChEM, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, China.

Uncertainty QuantificationMachine Learning Interatomic PotentialsActive LearningData-Efficient Learning
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Jul 15, 2026cs.IT

Decision Making Needs Uncertainty Quantification [Lecture Notes]

Many signal processing systems ultimately exist to {act}. Whenever the state variable that determines the action to be taken by a decision maker, or agent, is uncertain, the way that uncertainty is represented decides how well the agent performs and how much its performance can be trusted. This lecture note develops, from first principles and within a single decision-theoretic setting, the link between the {objective} and the knowledge of an agent and the form of uncertainty representation that is sufficient to act optimally. To start, assuming a known environment distribution, we show that a risk-neutral agent needs the posterior distribution over the state, whereas a risk-averse agent can rely without loss of optimality on a {prediction set} and a worst-case decision rule. We then turn to the case in which the environment is unknown, and identify three complementary approaches to address the resulting epistemic uncertainty: calibration of a fixed predictor, credal (ambiguity) sets with distributionally robust optimization, and Bayesian inference over model parameters. The common thread is that reliable decisions require an uncertainty representation matched to the decision objective and to the knowledge profile of the agent, together with a guarantee that certifies the utility the agent will actually obtain.

Osvaldo Simeone

Institute for Intelligent Networked Systems, Northeastern University London, One Portsoken Street, London, E1 8PH, UK

Uncertainty QuantificationDecision-Making under UncertaintyEpistemic Uncertainty
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Jul 15, 2026cs.CV

PlumeQuant: Uncertainty-aware consistency assessment of methane plume masks and emission-rate estimates

Imaging spectrometers increasingly distribute source-resolved methane plume products in which the plume mask, integrated mass enhancement (IME), plume length, emission rate, and uncertainty are physically and algorithmically linked. Using 63 EMIT-derived Carbon Mapper plume records from 27 scenes, we show that these published scalar quantities do not uniquely constrain the plume boundary: substantially different yet plausible masks reproduce the same IME, plume length, and emission rate. Genetic-algorithm (GA) ensembles conditioned on the published IME and plume length make this equifinality explicit: the high-confidence core selected by nearly all target-consistent masks covers a median of 13% of the plausible footprint envelope, and ambiguity is largest for weak, low-overlap plumes. The diagnostics come from PlumeQuant, which recomputes IME, plume length, emission rate, and five-term uncertainty from distributed product components under stated conventions and evaluates four mask representations: the distributed reference mask, a transparent Carbon Mapper-informed analogue (CM-like), the GA ensemble, and optional expert edits. The CM-like mask is generated per plume without access to the reference mask or published quantities, with settings fixed once on a scene-disjoint 44-plume development split. It reproduced published IME with +0.72% median difference and emission rate with +0.16% (6.98% mean absolute), reached 0.843 median intersection-over-union against the reference masks, and matched the published uncertainty scale (median ratio 1.01). Holdout mean absolute errors were 7.6% (IME), 9.5% (length), and 6.1% (rate). These are product-level consistency diagnostics, not independent validation. They flag weak, offset, or ambiguous plumes for expert review.

Parisa Masnadi Khiabani, Wolfgang Jentner, Alireza Rangrazjeddi +4

Data Institute for Societal Challenges, University of Oklahoma, Norman, OK 73019, USA · Data Science & Analytics Institute, University of Oklahoma, Norman, OK 73019, USA · School of Industrial & Systems Engineering, University of Oklahoma, Norman, OK 73019, USA +4

Uncertainty QuantificationRemote SensingSegmentation Uncertainty Estimation
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Jul 15, 2026cs.CV

Posterior Variance Is a Constraint Map, Not an Error Map: Closed-Form Uncertainty for Radiative Gaussian Splatting in Sparse-View CT

Radiative Gaussian splatting reconstructs sparse-view CT fast and accurately, and recent work attaches per-Gaussian posteriors to yield per-voxel uncertainty maps. We ask what such a map actually measures: posterior variance is a data-constraint map, not an error map -- its alarms are trustworthy, its all-clears are not. Exploiting the strict linearity of X-ray rendering in the per-Gaussian densities, we derive a clamp-aware closed form that the unchanged rasterizer evaluates exactly in one forward pass, in volume and projection space: the infinite-sample limit of the sampling estimator of concurrent work, at ~8x lower cost. On the official 15-scene benchmark this uncertainty ranks true error on 14 of 15 scenes. Restricted to the object interior -- the tissue a clinician reads -- the ranking collapses (median Spearman 0.11, 0/15 pass), identically for a deep ensemble and for a strictly positive log-normal posterior: three constructions, two estimator families, no survivors. The mechanism is structural: about 90% of in-object error is bias that reproduces across retrainings, invisible to model disagreement; 73-81% of the full-volume correlation is carried by object/surround contrast; and an exactly solvable control puts the observed in-object ranking 4-5x below what a perfectly calibrated posterior with the same sigma-spread would score. The error scale, by contrast, is an engineering problem, and we solve it: reparameterizing the posterior contracts the cross-scene temperature spread from 19.3x to 2.6x, one scene-agnostic temperature transfers to unseen scenes (10/15 leave-one-scene-out), and the repaired scale tracks photon count at the Poisson-predicted -1/2 power. We distill evaluation practice that would have caught the illusion -- masked calibration, seed-wise bias decomposition, an exact-posterior reference -- and release all protocols, seeds and per-run evidence.

Chulin Zhao, Yiran Xu, Shu Liu

1Dundee International Institute, Central South University · 2Central South University

3D Gaussian SplattingUncertainty QuantificationSparse-View CT ReconstructionUncertainty Calibration
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Jul 14, 2026cs.SE

Code-MUE: Measuring Code LLMs' Uncertainty through Execution-based Semantic Interaction Graphs

As Code Large Language Models (LLMs) become central to modern software engineering, their inherent stochasticity poses significant real-world risks, where even minor errors can lead to severe functional, security, or safety consequences. Reliable automation, therefore, demands the ability to distinguish between confident, well-supported predictions and stochastic guessing. However, existing uncertainty estimation methods face a critical gap: white and grey-box techniques are often inapplicable to closed-source models, while standard "black-box" text metrics fail to capture the unique fragility of code, where syntactic variation does not always imply semantic divergence. To bridge this syntax-semantics gap, we introduce Code-MUE, a purely black-box framework that measures uncertainty through execution-based Semantic Interaction Graphs. Unlike prior approaches that rely on superficial textual similarity, Code-MUE grounds uncertainty in observable runtime behavior, calculating the Von Neumann entropy of the solution space to quantify global semantic diversity. A large-scale empirical study across eight state-of-the-art LLMs demonstrates that Code-MUE achieves a strong negative correlation with functional correctness (Spearman's correlation up to -0.98), significantly outperforming lexical and embedding-based baselines while enabling robust risk detection and selective prediction in practical workflows.

Xiaoning Ren, Yinxing Xue, Lei Ma +1

Xi’an Jiaotong University, China · Institute of AI for Industries, Chinese Academy of Sciences, China · The University of Tokyo, Japan +1

Uncertainty QuantificationLLM Uncertainty EstimationCode Language Models
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Jul 13, 2026cs.CV

Uncertainty Quantification for EO Regression Tasks: Building Height, Tree Canopy Height and Above-ground Biomass Estimation

Earth Observation regression tasks such as building height, canopy height, and above-ground biomass estimation underpin critical applications in urban planning, forest monitoring, and climate policy, where both accuracy and reliability are critical. Yet most deep learning models yield only deterministic predictions, providing no indication of per-pixel reliability. These regression tasks are inherently challenging due to heterogeneous land surfaces, skewed target distributions, sensor noise, and signal saturation at high target values, making uncertainty (UC) estimation essential for reliable inference. We address this gap by modeling aleatoric uncertainty using year-long Sentinel-1 SAR and Sentinel-2 MSI time series, proposing two complementary approaches: (i) Gaussian UC, which jointly predicts mean and standard deviation under a Gaussian assumption, and (ii) Quantile UC, which estimates the 10th, 50th, and 90th quantiles to capture asymmetric and heteroscedastic error distributions. Both models are evaluated on three representative EO regression tasks at 10 m spatial resolution. Results show that both approaches match or surpass deterministic benchmarks and existing global products, while delivering well-calibrated, interpretable, and operationally useful confidence estimates. Notably, both models outperform the current 10 m state-of-the-art uncertainty-aware model for canopy height estimation. Our implementation will be available at: https://github.com/RituYadav92/EO-Regression-Uncertainty-Estimation

Ritu Yadav, Andrea Nascetti, Yifang Ban

Ritu Yadav, Andrea Nascetti and Yifang Ban is with Division of Geoinformatics, KTH Royal Institute of Technology, Sweden.

Uncertainty QuantificationRemote SensingRegression Prediction Intervals
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Jul 12, 2026math.NA

When is the combined load identifiable from a stress-intensity profile? A coupled forward-inverse study on SIFBench finite-element data

This work studies the inverse problem of recovering the relative magnitudes of the tension, bending, and bearing loads acting on a crack from its stress-intensity-factor profile along the crack front, using the public SIFBench finite-element data. The central claim is not forensic load recovery on field cases, but a rigorous characterization of when the combined load is identifiable at all, together with an estimator that returns calibrated uncertainty precisely in the regimes where it is not. For a known geometry the forward map from loads to profile is exactly linear, and identifiability reduces to a single geometric question: whether the three elementary load profiles are linearly independent as functions along the front. When they are nearly dependent, many different load combinations produce almost the same profile and the inverse problem is illposed; the analysis shows that the degree of ill-posedness is controlled by an intrinsic stability margin, not by the conditioning number alone. A single crack-front operator serves both as a structured forward surrogate and as the differentiable map required by a simplex-constrained, set-valued inverse estimator. On the SIFBench corner-crack scenario the empirical behaviour matches the theory: the typical geometry is well posed while a sizable minority is genuinely ill-posed, so a point estimate is reliable on the majority and provably uninformative on the rest. Validation is on controlled synthetic noise; no real fracture cases are used or claimed.

Giansalvo Cirrincione, Filippo Grassia

Laboratory of Innovative Technologies (LTI, EA 3899), Université de Picardie Jules Verne, Amiens, 80025, France

Parameter IdentifiabilityUncertainty QuantificationStructural MechanicsInverse Problems
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Jul 12, 2026cs.LG

Hierarchical Bayesian Quadrature

Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning. Bayesian Quadrature uses Gaussian process surrogates that explicitly encode structural assumptions about the integrand to obtain integral estimates with quantified uncertainty. These surrogates are predominantly based on stationary covariance functions, which results in model misspecification for integrands exhibiting nonstationary behavior. We tackle this issue through an adaptively growing, tree-based partition of the integration domain into local stationary models. Our method recombines the local integral estimates through a hierarchy of GP conditioning that reintroduces cross-subdomain correlations, while model selection criteria control the tree growth to avoid unnecessary partitioning. The resulting algorithm is simple, requires no MCMC, and adapts its evaluation budget to local integrand complexity. On benchmark integration problems and a model evidence computation for an epidemiological model, Hierarchical Bayesian Quadrature achieves substantial gains over standard Bayesian Quadrature on nonstationary integrands while matching its performance on stationary ones.

Tim Weiland, Toni Karvonen, Philipp Hennig

Tübingen AI Center, University of Tübingen, Tübingen, Germany · School of Engineering Sciences, Lappeenranta–Lahti University of Technology LUT, Lappeenranta, Finland

Uncertainty QuantificationBayesian InferenceGaussian Process Regression
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Jul 11, 2026cs.LG

BattVAE-GP: Generative Modeling of Long-Horizon Battery Degradation with Uncertainty Quantification

Long-horizon physics-based simulations of battery degradation provide mechanistic insight but remain computationally expensive, limiting their use for dense exploration of operating conditions over extended cycle life. Here, we propose a hybrid physics-probabilistic learning framework for surrogate modeling of lithium-ion battery degradation trajectories at unseen charging rates. Cycle-resolved degradation data generated with a DFN/P2D electrochemical model in PyBaMM are first transformed into capacity-aligned voltage and derivative features and encoded using a Variational Autoencoder (VAE). The resulting two-dimensional latent space organizes degradation trajectories according to both cycle progression and charging protocol. A sparse multitask Gaussian process (GP) is then trained in this latent space using cycle number and C-rate as input variables, providing continuous interpolation of latent degradation dynamics together with posterior uncertainty estimates. Under protocol-level holdout evaluation, the latent-space GP accurately recovers unseen C-rate trajectories and exhibits uncertainty behavior consistent with the support of the training data. When queried at unseen interior C-rates, the model generates latent trajectories that remain coherently positioned between neighboring simulated protocols. Decoding the GP-predicted latent states through the frozen VAE decoder yields smooth voltage-capacity evolution, while Monte Carlo propagation of the GP latent posterior through an auxiliary latent to State of Health (SOH) predictor provides uncertainty-aware SOH estimates. The proposed BattVAE-GP framework therefore offers a computationally efficient and uncertainty-aware surrogate for long-horizon degradation modeling, providing a structured basis for extending battery health prediction toward richer operating conditions and future simulation-experiment fusion.

Raghvender Raghvender, Mahdi Abid, Ferran Brosa Planella +2

Laboratoire de Réactivité et de Chimie des Solides (LRCS), UMR CNRS 7314-Université de Picardie Jules Verne, 15 Rue Baudelocque, 80000, Amiens, France · Réseau sur le Stockage Electrochimique de l’Energie (RS2E), CNRS FR 3459, Hub de l’Energie, 15 Rue Baudelocque, 80000, Amiens, France · Predeeption (INRIA Startup Studio / CNRS Innovation), Hub de l’energie, 80000, Amiens, France +2

Surrogate ModelingUncertainty QuantificationLatent Dynamics ModelingRUL EstimationLithium-Ion Batteries
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Jul 11, 2026cs.LG

Error Aware Distribution Prediction for Lightweight Implicit Neural Representations

Implicit neural representations (INRs) offer compact encoding of volumes, but as lossy approximators, inevitably have prediction errors. We consider INRs that can simultaneously encode relative error scales by predicting distributions using tools from uncertainty estimation. Typically, uncertainty estimation relies on computationally expensive approaches or on predefined parametric assumptions about the predictive distribution (e.g., Gaussian). In this study, we propose a lightweight method that reformulates regression-based INR training as a classification task by discretizing continuous targets into bins, enabling flexible distribution modeling to capture complex multimodal behaviors. We analyze the trade-off between regression and classification for INR training and demonstrate that the classification setting tends to achieve high reconstruction quality and competitive error awareness through uncertainty estimation, compared to regression-based approaches.

Zhimin Li, Jake D. Balla, Joshua A. Levine

Vanderbilt University · University of Arizona

Implicit Neural RepresentationsUncertainty Quantification
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Jul 9, 2026eess.IV

Calibrated Hybrid CNN-Transformer for Retinal OCT Classification

Deep models for retinal optical coherence tomography (OCT) classification report high accuracy but rarely report whether their confidence can be trusted -- a gap that matters when a wrong-but-confident reading delays sight-saving treatment. We pair a hybrid convolutional-Transformer encoder with a gradient-boosting (XGBoost) classification head and a three-part clinical safety layer: confidence calibration, out-of-distribution (OOD) rejection, and per-prediction uncertainty flagging. On four-class OCT (84,495 scans) the model reaches 95.4% accuracy while cutting calibration error twelve-fold (expected calibration error, ECE = 0.0024), so the confidence it reports tracks its true accuracy. To our knowledge this is the first OCT classifier to validate all three safety mechanisms jointly, with public weights and reproducible multi-seed evaluation.

Animesh Kumar

School of Computing, Newcastle University, UK

Uncertainty QuantificationOptical Coherence TomographyUncertainty Calibration
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Jul 8, 2026cs.LG

Hypergraph Neural Stochastic Diffusion: An SDE Framework for Uncertainty Estimation

Hypergraph neural networks have shown powerful capability in modeling higher-order relations, yet their predictive uncertainty remains underexplored. Unlike pairwise graphs, uncertainty in hypergraphs arises not only from noisy attributes and ambiguous labels, but also from variations in node-hyperedge incidence structures and complex higher-order dependencies. Existing approaches mainly estimate uncertainty from final predictions or rely on computationally expensive ensembles and Bayesian inference, limiting their ability to capture uncertainty evolution during representation learning. In this paper, we propose Hypergraph Neural Stochastic Diffusion(HyperNSD), a stochastic differential equation framework for uncertainty estimation on hypergraphs. HyperNSD models hypergraph representations as stochastic processes evolving over node-hyperedge incidence structures. A learnable drift function captures deterministic higher-order diffusion dynamics, while a learnable stochastic forcing function characterizes structural ambiguity and representation noise. Predictive uncertainty is directly quantified through the variability of stochastic representation trajectories, providing an intrinsic uncertainty measure beyond post-hoc confidence scores. We formulate HyperNSD with neural drift and diffusion networks, enabling joint learning of prediction and uncertainty propagation. Theoretical analyses establish well posedness, perturbation stability,permutation equivariance, and numerical convergence of the proposed stochastic dynamics. Experiments on multiple hypergraph benchmarks demonstrate that HyperNSD achieves reliable uncertainty estimation for out-of-distribution and misclassification detection while preserving competitive prediction accuracy. These results provide a principled stochastic-dynamical framework for trustworthy higher-order representation learning.

Zhiheng Zhou, Mengyao Zhou, Dengyi Zhao +2

Stochastic Differential EquationsUncertainty QuantificationHypergraph Neural Networks
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Jul 7, 2026cs.LG

Efficient Bayesian Deep Ensembles via Analytic Predictive Inference

We introduce an efficient Bayesian deep ensemble method for predictive regression designed to enhance interpretability while maintaining competitive predictive performance and computational efficiency. Our method combines the statistical rigor of Bayesian inference with the scalability of deep ensembles, providing calibrated uncertainty estimates that enable its use not only for standalone prediction but also as a component within broader learning systems. To achieve these goals, our work relies on three key design components: (i) low-dimensional ensemble representation: predictions are expressed as a combination of a small number of trained neural predictors, enabling scalable inference whose cost depends on ensemble size rather than dataset size; (ii) closed-form Bayesian aggregation: ensemble predictions are combined using Bayesian linear regression, yielding interpretable posterior weights and calibrated uncertainty without approximate inference; and (iii) Independent ensemble training: multiple neural networks are trained separately, producing diverse predictive representations that improve robustness and uncertainty calibration. Empirical results on standard regression benchmarks demonstrate that the proposed approach achieves competitive predictive performance while maintaining reliable uncertainty estimates across settings.

Sina Aghaee Dabaghan Fard, Marie Maros, Jaesung Lee

Wm Michael Barnes ’64 Department of Industrial and Systems Engineering Texas A&M University College Station, TX, USA

Uncertainty QuantificationInterpretable MLDeep Ensembles
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Jul 7, 2026stat.ME

tsbootstrap: Distribution-Free Uncertainty Quantification and Conformal Prediction for Time Series

Finance, sensing, and demand streams violate the exchangeability that IID conformal prediction and the IID bootstrap assume, and existing libraries implement either a general resampling engine or conformal calibration without the other. tsbootstrap provides block, residual, sieve, and wild resampling, classical bootstrap confidence intervals, and adaptive conformal calibrators (EnbPI, ACI, NexCP, AgACI) through a single typed API in which a specification object selects each method. In a controlled coverage study the IID bootstrap undercovers sharply under dependence; dependence-aware methods reduce the coverage deficit, the sieve nearest to nominal under short-memory linear dependence. On the shared fixed-statistic path a compiled backend runs several times faster than arch, and a streaming reduce avoids materializing the O(Bn)O(Bn) replicate tensor, limiting peak extra memory to O(B)O(B) for the statistic array. The software is MIT licensed (v0.6.1).

Sankalp Gilda

DeepThought Solutions

Uncertainty QuantificationConformal PredictionAdaptive Conformal Inference
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Jul 6, 2026cs.CV

Rendering-Aware Bayesian 3D Gaussian Splatting with Native Uncertainty and Adaptive Complexity Control

3D Gaussian splatting (3DGS) is a strong representation for real-time novel-view synthesis, but its standard training pipeline relies on point estimates and hand-tuned heuristics, providing no native uncertainty or principled complexity control. This is most limiting under sparse views or fixed acquisition budgets, where a model must identify weakly supported geometry and select informative views. We introduce a rendering-aware Bayesian 3DGS framework that tracks Gaussian geometry with a Normal-Inverse-Wishart posterior over means and covariances using renderer-derived surrogate summaries. An optional Dirichlet-process extension adds a probabilistic component-usage signal, and the training schedule makes the closed-form versus approximate inference boundary explicit. Re-rendering posterior geometry samples yields native predictive uncertainty for interval calibration and active view selection. In a fixed-budget 16-to-32 active-view task, native NIW acquisition improves PSNR by +0.453 dB and LPIPS by -0.0146 over a scoring-only 3-member standard-ensemble baseline, winning 29/39 scene-seed pairs and 10/13 scene means; it also improves over PPU-style (+0.355 dB) and NIW-proxy (+0.401 dB) acquisition. NIW native intervals reduce 95% coverage error by about 17x relative to a shared proxy (0.046 vs. 0.796) and are about 10x closer to nominal coverage than a 3-member deep ensemble (0.047 vs. 0.454) at roughly one-third the training cost. As a reconstruction compatibility check, paired NIW-vs-standard analysis over 39 scene-seed runs yields +0.030 dB PSNR with 1.6% additional training time. These results position Bayesian 3DGS as a practical probabilistic scene representation for decision-facing tasks such as active view selection.

Gaoxiang Jia, Vikram Appia, Junzhou Huang +1

Advanced Micro Devices, Inc. · Dept. of Computer Science and Engineering University of Texas at Arlington · Dept. of Mathematics & Division of Data Science University of Texas at Arlington

3D Gaussian SplattingActive PerceptionUncertainty QuantificationSparse-View 3D ReconstructionActive View Selection
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Jul 6, 2026cs.LG

Uncertainty-aware damage identification in short-span bridges via physics-informed variational autoencoder

Vibration-based damage identification in civil infrastructure is a challenging, ill-posed inverse problem due to measurement noise, sparse sensor arrays, and environmental variability. While deep learning is powerful for system identification, deterministic approaches lack reliable uncertainty quantification and can yield physically inconsistent results. This work proposes a robust probabilistic Scientific Machine Learning (SciML) framework: a physics-informed Gaussian copula variational autoencoder (PI-GCVAE) for structural health monitoring (SHM). First, we eliminate the need for data-driven surrogates by embedding a differentiable numerical eigenvalue solver directly into the VAE architecture. This ensures that latent space samples satisfy the governing equations of structural dynamics, reducing the trainable parameter space and improving generalization. Second, we replace the conventional independence assumption of latent variables with a Gaussian copula. This model captures complex, physics-dependent spatial cross-correlations between adjacent structural elements, defining feasible solutions while accounting for inherent system variability and measurement errors. Third, compared with alternatives such as Gaussian mixtures, our copula-based VAE provides an efficient distributional model for high-dimensional, strongly correlated latent spaces. We validate the approach using a synthetic dataset of a simply supported bridge subjected to various damage scenarios and corrupted with stochastic Gaussian noise. Synthetic data enables exhaustive validation against ground-truth stiffness values unavailable in practice. Results demonstrate that the PI-GCVAE accurately recovers the true posterior distribution, achieving 77.2% coverage. The proposed framework provides a reliable, scalable tool for early-stage damage diagnosis in operating bridges.

Ana Fernandez-Navamuel, A. Javier Omella, Diego Zamora-Sanchez +1

Basque Center for Applied Mathematics (BCAM), Bilbao, Spain. · Departamento de Matematica Aplicada, Universidad de Zaragoza/IUMA, Zaragoza, Spain. · TECNALIA, Basque Research and Technology Alliance (BRTA), Parque Cientifico y Tecnologico de Bizkaia, Astondo bidea, Edificio 700, Derio, E-48160, Spain. +1

Variational AutoencodersUncertainty QuantificationBayesian Inverse ProblemsPhysics-Informed MLStructural Health Monitoring
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Jul 6, 2026cs.AI

STAPO: Selective Trajectory-Aware Policy Optimization for LLM Agent Training

Reinforcement Learning (RL) is the dominant paradigm for training Large Language Model (LLM) agents on long-horizon tasks. However, sparse and delayed rewards often lead to trajectory neglect, in which agents lose focus on the task goal and interaction history at intermediate steps. Prior work has explored step-level supervision using Shannon-entropy-based uncertainty signals, which conflate inherent state complexity with agent confidence and therefore provide unreliable estimates of decision reliability. To address this issue, we propose normalized entropy, which measures confidence deviations relative to an agent's average behavior under a given state, thereby strengthening the association between low-quality actions and trajectory neglect. Building on this insight, we introduce Selective Trajectory-Aware Policy Optimization (STAPO), a hierarchical group-based RL framework. STAPO leverages normalized entropy to locate outlier steps associated with trajectory neglect and optimizes them via a joint mechanism of trajectory-aware reward and trajectory-independent penalty, enhancing trajectory awareness while preserving training stability. Extensive experiments on ALFWorld, WebShop, and Search-Augmented QA demonstrate that STAPO achieves state-of-the-art performance while substantially alleviating trajectory neglect, validating its effectiveness and robustness for agentic tasks.

Qiuyi Qi, Tian Liang, Mutian Bao +8

Zhejiang University · Ant Group · City University of Hong Kong

Uncertainty QuantificationSparse-Reward RLRL for LLM Agents
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Jul 6, 2026cs.SD

Towards Robust Uncertainty-Aware Speaker Modeling

Speaker embeddings aggregate frame-level acoustic features into compact representations for speaker recognition. Recent uncertainty-aware speaker modeling approaches further characterize the reliability of speaker embeddings by estimating their associated uncertainty. However, existing methods often suffer from inaccurate uncertainty estimation and uncertainty miscalibration under domain shifts. To address these challenges, we propose a robust uncertainty modeling framework from both estimation and adaptation perspectives. Specifically, we introduce an Inter- and Intra-Speaker-Aware Uncertainty Softmax that incorporates both inter-speaker separability and intra-speaker variability into uncertainty learning, enabling uncertainty estimates to better capture the reliability of speaker embeddings. Furthermore, we propose an Uncertainty-Calibrated Domain Adaptation (UCDA) framework to mitigate uncertainty miscalibration caused by domain mismatch. Extensive experiments on both in-domain and cross-domain benchmarks demonstrate that the proposed approach consistently improves uncertainty reliability and speaker recognition robustness.

Junjie Li, Yang Xiao, Kong Aik Lee

Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University, Hong Kong SAR · The University of Melbourne, Australia

Speaker VerificationUncertainty QuantificationDomain AdaptationUncertainty Calibration
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Jul 5, 2026cs.LG

Physics-Informed Graph Learning with Uncertainty Awareness for Open-Set Domain Generalization in Fault Diagnosis

Intelligent industrial maintenance critically relies on reliable fault diagnosis of rotating machinery. However, it faces formidable challenges from unknown fault types and domain shifts induced by varying operating conditions, which is formally formulated as the open-set domain generalization (OSDG) problem. Existing methods are mainly data-driven, thereby overlooking the cascaded propagation of uncertainty across feature extraction, topological learning, and decision-making stages.To tackle this challenge, we propose PGU-OD, a novel Physics-Informed Graph Learning framework with Uncertainty Awareness for Open-set Domain generalization. First, it designs a physics-informed spectral attention module to extract condition-robust fault features, thereby suppressing perceptual uncertainty caused by frequency shifts. Further, it constructs an uncertainty aware adaptive graph learning mechanism to dynamically adjust the edge weights of the sample graph guided by class-scale Gaussian distribution parameters, which mitigates the structural propagation of uncertainty. Finally, a Gaussian-distribution-based adaptive boundary loss function and a dual-criteria open-set inference strategy are developed to optimize decision boundaries and reliably reject unknown faults. Extensive experimental evaluations on two public and widely used rotating machinery fault datasets demonstrate that the proposed PGU-OD outperforms state-of-the-art baselines in both known fault classification and unknown fault rejection under domain shifts.

Jinfeng Zhu, Shiyu Long, Ye Yuan

College of Computer and Information Science, Southwest University, Tiansheng Road, Chongqing, 400715, China

Uncertainty QuantificationDomain GeneralizationDynamic Graph LearningPhysics-Informed MLFault Diagnosis
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Jul 4, 2026stat.ME

Significance-First Splitting: Aligning Treatment Heterogeneity Detection with Honest Estimation

Estimating heterogeneous treatment effects (CATE) requires simultaneously detecting effect modification and quantifying estimation uncertainty. Existing tree-based methods make an uneasy trade-off: significance-based approaches (Radcliffe and Surry 2011) identify subgroup interactions directly but lack valid inference; honest causal trees (Athey and Imbens 2016) deliver nominal confidence interval coverage but use outcome-agnostic splitting criteria that sacrifice interaction sensitivity. We introduce a hybrid algorithm that fuses significance-based splitting with honest sample-splitting and cross-validation. Our splitting criterion uses the squared tt-statistic for the treatment ×\times side interaction (t2t^2), which is shown to be directly aligned with the honest EMSEτ\text{EMSE}_τ criterion when the interaction is strong. Post-hoc honest cross-validation selects the cost-complexity penalty, giving a single principled estimator with nominal CI coverage at the leaf level. For forests, we retain bootstrap count vectors to enable an infinitesimal jackknife (IJ) variance estimate of Monte-Carlo convergence rather than formal pointwise inference. On the three synthetic designs from (Athey and Imbens 2016) the single tree achieves approximately 90% leaf-average CI coverage at the 90% nominal level across all three designs (200 replications each); on the Criteo, Hillstrom and Starbucks uplift datasets we match Qini coefficient performance of S-, T-learner and GRF baselines. An open-source Python package with reproducible seeds, sklearn-compatible API, and full test coverage accompanies this work (https://codeberg.org/hadjipantelis/rattus).

Pantelis Z. Hadjipantelis, Weng Man Chiang, Karthik Nagesh

TripAdvisor, UK

Causal Effect EstimationConfidence Region EstimationUncertainty QuantificationConditional Average Treatment Effect EstimationCausal InferenceIndividualized Treatment Effect Estimation
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