Uncertainty Quantification

Non-intrusive reduced-order models (NIROMs) have become a standard tool for approximating parametric partial differential equations from computer design of experiments while significantly reducing computational costs. However, assessing the reliability of their predictions remains a major challenge, particularly in extrapolation regimes or under limited training data. In this work, we introduce a framework for quantifying model-form uncertainty in NIROMs by combining a perturbative stochastic representation of reduced bases with distribution-free conformal-type methods. Starting from a deterministic reduced basis constructed from snapshot matrices, we model uncertainty through random perturbations defined on the Stiefel manifold, directed along the discarded modes, yielding stochastic reduced-order approximations whose induced variance reflects the basis-truncation error. A transport approximation gives a closed-form posterior variance that separates basis-induced from regression-induced uncertainty, without re-training the underlying Gaussian processes. We include this posterior variance within a conformal risk control calibration framework, that provides prediction sets with coordinate miscoverage guarantees. The calibration factor produced by this framework is itself an interpretable, scalar diagnostic of the quality of the uncertainty estimate. The methodology is evaluated on parametric PDE benchmarks and an industrial tire-manufacturing calendering process. Numerical experiments demonstrate reliable, locally informative uncertainty quantification that goes beyond the Gaussian predictive variance.

Asked to animate the same modern artwork under different random seeds, a text to video model returns visibly different films, one reading per seed. Because modern art is ambiguous by intent, this disagreement is signal, not noise. Yet prevailing uncertainty quantification (UQ) collapses a set of generations to a dispersion scalar that says how much the seeds differ but not how: it cannot tell a compact interpretation from a dominant reading plus an outlier, two competing modes, or diffuse instability, nor whether the set still contains a rendering faithful to the original. We present the first study of the structure of generative uncertainty for modern art animation, and a reusable protocol for identifying source blind multiseed uncertainty: a suite of seven source blind and six reference aware estimators; a distributional profile (robust spread, outlier influence, explicit topology, multimodality, anisotropy, leave one seed influence, reference coverage); a distribution model ablation (vMF, Kent, ACG, Student t, kernel, mixture); eight identification questions; and an artwork level statistical protocol. We build the first corpus: 250 modern artwork captions rendered by Wan2.1 14B under four seeds (1000 videos) across 4 encoders, artworks withheld from generation. As a diagnostic the protocol succeeds: it classifies seed set topology at balanced accuracy 0.98 (chance 0.25), isolates the outlier configuration at AUROC 1.00 where a scalar reaches only 0.35, and splits high uncertainty artworks into reference covering (n=97) and reference missing (n=56) diversity, reliably from three seeds and across encoders.

With AI systems gaining more access to individuals' information, it is important to protect privacy when reporting statistical answers. Equally important is to privatize the reporting of uncertainty in such answers. To this end, we adopt a Bayesian likelihood-free framework and make simulation from the posterior private. In particular, we propose a new private instantiation of the Bayesian bootstrap using a blocking strategy. Rather than assigning idiosyncratic random weights to each individual, we randomly group individuals and assign a single weight to each group. By concealing individuals' contributions within a group, we fortify differential privacy gates. We harness amortized inference that decouples private learning from posterior sampling. A push-forward map from observation weights to posterior samples is learned privately by adding calibrated noise during training. Subsequent posterior draws require no additional privacy and computation budget. We call the resulting method the Private Generative Bayesian Bootstrap (PGBB). We establish a differential privacy guarantee, analyze convergence to the non-private blocked-bootstrap target, and quantify the discrepancy between the ordinary and blocked Bayesian-bootstrap posteriors. In addition, we derive data-free tuning of the block Dirichlet concentration parameter that restores posterior dispersion asymptotically. We also show a single fit of PGBB can support a family of loss-based decision rules simultaneously without additional privacy cost. In simulations and in applications to U.S. Census returns to schooling and U.S. natality birthweight quantiles, PGBB gives competitive private uncertainty quantification and improves over private Bayesian alternatives that require a specified data-generating model in common settings.

Sim-to-real policies are designed under nominal dynamics, but target-system trials may yield only a few isolated one-step transitions. We study pre-execution certification of a fixed control sequence, such as an action chunk produced by a learned policy. If the sequence reaches an unobserved state-input region, the observations remain consistent with target systems whose trajectories separate along it by an arbitrarily large amount. Any deterministic certifier sound for all of them must then decline to certify or return a reachable tube with arbitrarily large projected width. For bounded smooth classes of the target-nominal model error, we derive a finite plan-dependent projected-width lower bound. These results expose a trilemma among uniform trajectory containment, finite projected width, and unrestricted model-error behavior beyond the observations. ForeReach requires a supplied componentwise Lipschitz bound on the model error. Observed transition pairs can refute this declaration but cannot establish it outside the observed locations. Conditional on a valid declaration, our method constructs a set-membership envelope for the model error, propagates a zonotopic reachable tube, and certifies only when propagation remains within the certification domain and every projected tube slice avoids the unsafe set. In two benchmark systems, calibration baselines may remain narrow after losing trajectory containment outside data support, whereas our method declines to certify unsupported sequences and recovers certification when relevant target data and sufficient obstacle clearance are available.

Comparing stochastically trained models requires estimating both a performance difference and its uncertainty from repeated runs. We study whether training logs from those same runs can make such comparisons more precise. Because training-log covariates are produced during training rather than measured before it, we use arm-specific covariate adjustment: each model is adjusted only with statistics from its own runs, and the raw mean difference remains the reported effect. In a vision study spanning three architectures and three datasets, simple adjustments based on early training logs often reduce uncertainty in model comparisons. The main limitation is covariate selection. Broadly searching the log pool for the most correlated statistic often adds more noise than it removes, even when useful statistics exist in hindsight. Training logs therefore appear useful for more precise model comparisons, but only when the adjustment avoids large selection noise.

Classical neural networks frequently produce overconfident predictions on ambiguous or out-of-distribution (OOD) data, a liability that grows with each AI system deployed in safety-critical real-world scenarios. Bayesian neural networks (BNNs) provide a principled framework for uncertainty-aware prediction by replacing deterministic parameters with probability distributions, but repeated sampling increases latency, memory traffic, and energy consumption. Photonic probabilistic computing offers a promising alternative by exploiting intrinsic optical stochasticity for fast and parallel sampling. However, photonic BNNs are not ideal samplers: analog constraints on quantization, programming error, dynamic range, and representable mean and variance restrict the variational families that can be implemented in hardware. In this work, we study which hardware-imposed constraints limit scalable photonic BNN inference, how these constraints can be represented, and which ranges can be tolerated by photonic BNNs beyond small proof-of-concept networks. We formulate photonic BNN inference as constrained stochastic variational inference and perform a systematic ablation study over stochasticity location, stochasticity modality, quantization, programming error, and mean/variance bounds. From these results, we derive concrete co-design guidelines that distinguish hardware constraints that can be compensated by training from those requiring hardware or architecture intervention. We validate these guidelines under coupled, hardware-realistic constraints on Dirty-MNIST, CIFAR-10, and CINIC-10, using Fashion-MNIST and SVHN as OOD benchmarks, showing that hardware-aware training recovers predictive performance and uncertainty quality whenever the required variational family remains representable, whereas violations of representational limits require targeted hardware modifications.

Accurate estimation of forest height from satellite imagery is essential for applications such as carbon accounting, biodiversity monitoring, and ecosystem management. While recent deep learning approaches provide accurate predictions, they typically do not quantify predictive uncertainty. This limitation is particularly relevant in geospatial settings characterized by sparse supervision and geographic distribution shift. In this work, we investigate Deep Evidential Regression (DER) for forest height estimation on the TreeUQ benchmark, a large-scale dataset designed for the joint estimation of tree count and average tree height at 10 m resolution, based on Sentinel-1/-2 data as well as tree inventory data over the federal state of Bavaria. To account for the extreme label sparsity of the tree inventory data, we introduce a masked evidential loss for dense geospatial prediction. Using a U-Net architecture with multimodal Sentinel-1 and Sentinel-2 inputs, the proposed approach jointly predicts tree height and associated uncertainty estimates in a single forward pass. Experimental results show that DER achieves predictive performance comparable to a deterministic U-Net while additionally providing well-calibrated uncertainty estimates. These findings demonstrate the potential of evidential learning as an efficient framework for uncertainty-aware forest structure estimation from Earth observation data.

Drifting objectives compare a target and model distribution through a vector field observed noisily at finitely many locations. We ask what distributional conclusion such a frozen measurement system warrants. For integrable antisymmetric interactions and absolutely continuous laws in a declared finite density basis, the unnormalized sampled numerator satisfies $\operatorname{vec}(V_X)=Mc$, where $c$ is an antisymmetric mismatch and $M$ is probe-dependent. This identity yields an a posteriori total-variation (TV) upper confidence bound accounting for held-out field noise, estimated-operator error, and externally validated $L^1$ residual radii around normalized density approximants in the span; a nonpositive observability margin returns the trivial TV bound and abstains. The audit recomputes this numerator from held-out samples; a normalized drift statistic requires a separate joint numerator--denominator analysis. For Gaussian-RBF interactions, a global envelope supports distribution-free and empirical-Bernstein radii without truncation, with companion bounds for the Laplace similarity in the original drifting objective. We characterize random-probe observability by a population Gram matrix, identify rank and symmetry degeneracies, and prove large-bandwidth collapse toward mean matching. Synthetic studies exercise Gaussian and Laplace numerators, separately prespecified bounded-vector and variance-adaptive radii, Monte Carlo-calibrated operators, nonzero residual radii around normalized finite-basis approximants, outward-rounded observability bounds, and designed abstention. A joint basis-size/dimension stress path extends evaluation through $m=8$. The result is a conditional diagnostic for a finite density class, or for normalized finite-basis density approximants with external residual radii, not a universal guarantee from small training drift.

We present DynActiveGS, a dynamic-aware active reconstruction framework based on 3D Gaussian Splatting (3DGS) for autonomous exploration in dynamic environments. The framework incrementally reconstructs a 3D Gaussian scene representation while suppressing motion-corrupted observations through online uncertainty prediction and uncertainty-weighted Gaussian optimization. A key component of DynActiveGS is the explicit decomposition of uncertainty into structural uncertainty and motion-induced uncertainty, which enables the system to distinguish under-reconstructed static regions from dynamically unreliable areas. Based on these uncertainty fields, DynActiveGS performs dynamic-aware viewpoint selection and dynamic-constrained path planning to favor informative yet stable observations during exploration. The resulting system forms a unified closed-loop pipeline for robust active reconstruction in dynamic scenes. Extensive experiments on challenging dynamic benchmarks demonstrate consistent improvements over existing active reconstruction baselines in reconstruction accuracy, completeness, rendering quality, and exploration efficiency.

Low-Rank Adaptation (LoRA) enables parameter-efficient fine-tuning, but standard LoRA produces a single deterministic model and does not directly support predictive uncertainty estimation. We introduce EulerLoRA, a stochastic extension of LoRA that generates multiple predictive trajectories by sampling structured variations along the rank-one components of shared low-rank adapters, while preserving the deterministic LoRA transformation in expectation. We evaluate EulerLoRA with vision transformers on CIFAR-10, CIFAR-100, and HAM10000, together with out-of-distribution detection on SVHN. Across these benchmarks, EulerLoRA achieves comparable or improved performance relative to strong LoRA-Ensemble baselines. Using two rank-20 adapters, EulerLoRA requires approximately 3 million trainable adapter parameters, compared with about 10 million for a rank-8, 16-adapter LoRA-Ensemble, corresponding to roughly 69% fewer trainable parameters. These results show that useful predictive diversity can be obtained from a small number of shared adapters.

Autoregressive models accumulate error over long rollouts, yet at deployment there is no ground truth to measure it against. We train a single conditional latent diffusion model that steps a dynamical system forward or backward in time via a direction flag, and show that this bidirectionality supplies a measurement-free test-time error signal: rolling forward $i$ steps and then backward $i$ steps must return the model to its start, so the round-trip discrepancy $\mathcal{C}_i$ is a self-supervised proxy for the unobservable rollout error: no ensembles, no held-out data, no governing equations, for one extra rollout. We validate on compressible magnetohydrodynamics (MHD), an astrophysical turbulent radiative mixing layer, and natural face videos (CelebV-HQ). On held-out MHD trajectories, $\mathcal{C}_i$ ranks rollout error (Spearman $0.91$-$0.98$ at fixed depth; $0.69 \pm 0.16$ within trajectories), and a simple calibrator fit on training rollouts predicts its magnitude to within $1.14\times$ ($68\%$) and $1.29\times$ ($95\%$) with near-nominal coverage - one nat beyond a depth-only predictor, transferring to all six decoded physical fields. The same signal flags the out-of-distribution Orszag-Tang vortex (AUROC $0.98$; $1.0$ by depth $10$) exactly where sampling-dispersion baselines invert, and it cuts incurred error by $15\%$ at $80\%$ coverage - three times the depth-only baseline. Bidirectional training comes at negative cost, beating direction specialists in both directions, and the backward direction doubles as a fast inverse solver. On LE-PDE-UQ's turbulent Navier-Stokes benchmark, a single bidirectional model reaches accuracy within $1.3\times$ of their ten-model ensemble at a tenth of the training cost, with the best training-free pixel-level calibration. Round-trip consistency turns reversibility into a practical trust signal for generative models.

In stream finishing, the wear experienced by a workpiece depends strongly on its orientation within the rotating abrasive media. Determining suitable orientations to achieve uniform wear requires evaluating the wear-rate field over all feasible orientations. Although the discrete element method (DEM) accurately resolves particle interactions, simulating hundreds of feasible orientations for a new geometry is computationally expensive. We present an uncertainty-guided surrogate framework that predicts, directly from geometry, the three fields governing erosion: per-triangle normal impact velocity, tangential impact velocity, and particle impact flux. These fields are combined through the Finnie wear model to reconstruct the wear-rate distribution. The surrogate employs a deep ensemble whose disagreement estimates epistemic uncertainty, enabling an active-learning strategy that selectively performs DEM simulations for the most uncertain orientations. Trained using only $13\%$ of the $696$ feasible orientations, the surrogate achieves Spearman rank correlations of $0.93$, $0.89$, and $0.93$ for the normal impact velocity, tangential impact velocity, and particle impact flux, respectively. Moreover, the predicted uncertainty is well calibrated, reliably anticipating prediction error and the fidelity of the reconstructed wear field, which matches DEM with a Spearman rank correlation of up to $0.97$ for low-uncertainty orientations and degrades in a controlled manner as uncertainty increases.

A calibrated stochastic world model can reveal how uncertain a future is without revealing why it branches. The same conditional future law can arise because an observation aliases physical states or because dynamics remain random after the declared full state is fixed. We prove that ordinary transitions cannot identify these two sources, even for a perfect probabilistic predictor. ClosurePairs makes them identifiable by crossing compatible microstates with repeated exogenous disturbances and estimating state, noise, and state-noise interaction variance. The central consequence is operational: under finite hierarchical sampling, forecast difficulty governs the useful compute scale, while the alias/process composition provides complementary information about its direction-resolving the current state or sampling future randomness. ClosurePairs recovers source attribution at unchanged likelihood, reduces equal-budget decomposition error in a nonlinear interaction benchmark, and supports observation-only routing. On exact-marginal MetaWorld twins, an output-only allocator is at chance while a Closure-supervised probe on frozen JEPA-WM features routes 89.8-100%. In an independent ManiSkill PushCube confirmation, a stochastic RSSM's outputs and latents remain at chance, whereas an RGB-only Closure probe routes 100% under both ID and geometry/camera OOD over five seeds, matching direct allocation rather than exceeding it. Across five unseen allocation menus, the same Closure probe routes 92.5%/90.4% ID/OOD with no new oracle labels, versus 37.9%/32.9% for a frozen direct allocator. ClosurePairs is therefore an identifiable, reusable mechanism target that cannot be recovered from forecast quality alone.

Although large language models (LLMs) can produce fluent spatial reasoning traces, their intermediate relations may fail to support the final conclusion, making token-level confidence insufficient for final-answer reliability estimation. Existing formal verifiers provide stronger semantic evidence, but their applicability is partial: a parsed claim need not yield a definite semantic verdict. To address this issue, we introduce SymboUQ, a symbolic uncertainty quantification framework that estimates final-answer reliability from reasoning traces by distinguishing symbolizability, whether a claim can be represented in the verifier's formal language, from semantic determinacy, whether its execution yields an entailed or contradicted verdict rather than an unknown or not-evaluable outcome. SymboUQ comprises (i) a Layout Auditor that executes ordered spatial claims and extracts feasibility, conflict, and repair evidence; (ii) a label-free Determinacy Profile that characterizes effective executable coverage; and (iii) a Determinacy-Aware Reliability Composer that integrates constraint-based, representation-based, and decoding-based scores according to verifier applicability. Extensive experiments on five spatial reasoning benchmarks with four frozen LLM backbones show that SymboUQ achieves approximately an 8% relative improvement in AUROC and a 7% relative reduction in class-balanced Brier loss over the strongest baseline.

Double Machine Learning (DML) is a popular approach for treatment effect estimation in various settings, which allows a wide range of flexible machine learning methods to be used for nuisance parameter estimation while preserving valid inference. In practice, however, applied researchers must choose among many machine learning algorithms for nuisance models, and the impact of this choice on the variance estimation of DML is not well characterized. We conduct a comprehensive simulation study to compare the coverage probability of DML confidence intervals across different machine learning algorithms. In this study, we compare (1) analytical confidence intervals derived by DML theory versus (2) bootstrap confidence interval. We use a set of learners including ordinary least squares, LASSO, Random Forest, LightGBM, and Neural Networks under different data generation settings. We evaluate the performance across difference settings by bias, confidence interval width, and most importantly, coverage probability. Our results show substantial variability in coverage performance across analytical and bootstrap confidence intervals, highlighting that learner choice plays a critical role in reliable DML inference. Surprisingly, we find that in many settings, when sample size increases, the coverage probability of both DML analytical and bootstrap confidence interval decreases. We further investigate coverage probabilities using a real dataset on rural urban differences among U.S. counties. The real data analysis discovers that (1) the model performance still varies by the learner choices and (2) greater rurality has a statistically significant increasing effect on county level obesity prevalence.

Variance-based global sensitivity analysis (GSA) plays a key role in uncertainty quantification by identifying the contributions of uncertain inputs to the variability of the model response. The repeated model evaluations required for these tasks are often prohibitively expensive; surrogate models provide an efficient alternative by constructing inexpensive approximations of the underlying system response. Constructing surrogate models that combine scalability and interpretability for systems with high-dimensional stochastic inputs and functional responses remains challenging, particularly when sensitivity estimates are required across spatial or temporal domains. Polynomial chaos expansion (PCE) provides an effective framework for uncertainty propagation and sensitivity analysis due to its orthogonal structure and direct relationship with variance-based sensitivity measures. However, PCE suffers from the curse of dimensionality, whose computational burden is amplified for problems with functional responses. In this work, we introduce the Structured Neural Chaos (sNC) expansion as a surrogate modeling framework for variance-based GSA, inspired by the interpretability and orthogonal structure of PCE. The proposed framework retains the interpretability of structured decompositions while leveraging the expressive power of neural networks. The sNC expansion mirrors a truncated functional ANOVA decomposition, where each interaction component admits a separable low-rank approximation whose basis functions and coefficients are parameterized by neural networks. The expansion is constructed sequentially, adaptively identifying the dominant modes within each ANOVA subspace and determining the effective complexity of the representation. The resulting structure enables the extraction of statistical and sensitivity quantities directly from the coefficients of the sNC expansion at negligible cost.

The deployment of deep neural networks in safety-critical domains demands reliable estimates of predictive confidence, yet conventional architectures lack principled uncertainty quantification. This survey provides a structured, critical review of methods for Uncertainty Quantification (UQ) in deep learning, scoped to ensemble-based and approximate Bayesian approaches and the measures used to summarize their outputs. Relative to existing UQ surveys, our contribution is depth on efficient ensemble approximations and single-pass methods, and a unified treatment that separates the method producing a predictive distribution from the measure that summarizes its uncertainty. We organize methods into five families: Bayesian neural networks, Monte Carlo Dropout, deep ensembles, efficient ensemble approximations, and last-layer or single-pass approaches. We situate adjacent work on evidential and prior networks, conformal prediction, and post-hoc calibration, together with the decision-time tasks of out-of-distribution detection and selective prediction. For each, we examine theoretical motivation, implementation, empirical performance, and limitations. We then review ensemble diversity theory and uncertainty measures and their decompositions, contrasting the entropy decomposition with pairwise divergence measures, and consolidate evaluation methodology so that our qualitative comparisons share a common basis. We close with a brief treatment of uncertainty in large language models and open research directions, including efficient epistemic measures for classification, last-layer diversity, diversity and calibration under shift, and hybrid architectures.

YOLO-Pose models provide efficient keypoint localization, but do not quantify the associated spatial uncertainty. We introduce a lightweight post-hoc probabilistic extension that augments a trained YOLO-Pose model with calibrated bivariate predictive distributions over keypoint locations, centered at the model's original predictions. Concretely, we train additional probabilistic heads with an importance-weighted negative log-likelihood to predict an input-dependent $2\times2$ dispersion matrix for each keypoint, followed by Gaussian calibration for broad downstream compatibility or Student-$t$ calibration for distributional fidelity. Complementing this, we propose an evaluation protocol that combines a suite of distributional calibration diagnostics with average keypoint precision (AKP), a keypoint-level extension of the COCO AP protocol for assessing reliability rankings. Experiments on COCO show that the learned uncertainty estimates enable effective keypoint-level reliability ranking, Student-$t$ calibration best captures the empirical residual distribution, and uncertainty-based pruning removes unreliable keypoints. A central application-level demonstration is vision-based aircraft landing, where calibrated covariances for runway keypoints support uncertainty-aware aircraft position estimation and downstream sensor fusion.

Modeling and forecasting nonlinear dynamics under distribution shifts is essential for robust decision-making in real-world systems. In this work, we propose MetaKoopman, a Bayesian meta-learning framework for modeling nonlinear dynamics through linear latent representations. MetaKoopman learns a Matrix Normal-Inverse Wishart (MNIW) prior over the Koopman operator, enabling closed-form Bayesian updates conditioned on recent trajectory segments. Moreover, it provides a closed-form posterior predictive distribution over future state trajectories, capturing both epistemic and aleatoric uncertainty in the learned dynamics. We evaluate MetaKoopman on a full-scale autonomous truck and trailer system across a wide range of adverse winter scenarios, including snow, ice, and mixed-friction conditions, as well as in simulated control tasks with diverse distribution shifts. MetaKoopman consistently outperforms prior approaches in multi-step prediction accuracy, uncertainty calibration, and robustness to distributional shifts. Field experiments further demonstrate its effectiveness in dynamically feasible motion planning, particularly during evasive maneuvers and operation at the limits of traction. Project website: https://mahmoud-selim.github.io/MetaKoopman/

This paper investigates how multi-agent systems (MAS)-based on large language models (LLMs) can support actuarial risk modelling, with a particular focus on uncertainty quantification. Actuarial workflows represent a high-stakes decision-support setting where unreliable outputs may lead to incorrect risk assessment, unfair pricing, and regulatory non-compliance. To address uncertainty introduced by the probabilistic nature of LLMs and dependencies between agents, a multi-agent framework is proposed in which specialised agents perform data preparation, modelling, review, and explanation tasks under a central hub. The main contribution is a novel approach to uncertainty propagation using token-level log-probabilities and a Bayesian Network. Importantly, log probabilities are not treated as direct probabilities of correctness or task success. Instead, length-normalised log-probability summaries are transformed into calibrated task-level confidence estimates before incorporation into the Bayesian Network. Results show that the framework reproduces baseline actuarial performance while providing additional insight into workflow stability and runtime uncertainty propagation.

Sampling-based methods offer a principled approach to uncertainty quantification in Bayesian neural networks. Their practical use, however, is often challenged by the computational cost of exploring high-dimensional and multimodal posterior distributions. To overcome these difficulties, Bayesian Deep Ensembles, i.e., warmstarting the sampling from several optimized solutions, have proven to be an effective strategy. In this paper, we demonstrate that curvature estimates computed during the warmstart as a byproduct in adaptive optimizers such as AdamW can inform the sampling phase at negligible additional cost. Specifically, our proposed preconditioned sampling strategy based on optimizer-derived geometries can substantially reduce or even eliminate the need for a lengthy sampling burn-in phase and leads to greater numerical stability. This approach consistently maintains or improves predictive performance and uncertainty quantification without any additional computational costs. We confirm the consistency of our findings across various datasets and network architectures.

A Last-Layer Ensemble (LLE), $K$ linear units on one shared frozen feature map, is an efficient single-pass approach to the disagreement-based epistemic uncertainty for out-of-distribution (OOD) detection. Its weakness is that members share the backbone gradient and can converge toward the same function, collapsing the inter-member diversity the signal depends on. Whether last-layer diversity can be restored, and what mitigates the collapse, is an open question. The weight-orthonormality defining Orthonormal Certificates (OC), the weight-orthonormal special case of the LLE, is only an indirect correction; it decorrelates the weights of the members, not their predictions. Here, we instead target the collapse directly in function space, with a Covariance Last-Layer Ensemble (cov-LLE) that places a direct covariance penalty on member activations. Cov-LLE restores the function-space diversity that weight-orthonormality cannot, and at matched $K$ recovers much of the diversity and calibration of a deep ensemble at $1\times$ backbone cost (in-distribution prediction variance $0.05\!\to\!9.3$ vs. $22.1$ ($\times10^{-3}$), and ECE $0.135\!\to\!0.090$ vs. $0.035$, for a $K\times$-cost deep ensemble), at no cost to accuracy. Viewing OC as a last-layer ensemble also organizes detectors into a two-axis taxonomy (by how their units are trained and how their outputs are scored) and exposes the OC score as a magnitude, motivating a scale-invariant, label-free direction score that repairs its near-OOD failure, adding $+0.16$ to $+0.18$ ROC AUC on every backbone.

The use of variable well perforation and injection strategies can improve the efficiency of geological carbon storage operations. We develop a new multimodal auto-regressive transformer surrogate to model these operations under geological uncertainty. A modified SEAM CO2 geomodel, which involves a faulted system with three stacked aquifers, is considered. The two injection wells are perforated in stages, from bottom to top, with the stage durations and individual well injection rates treated as control variables. The surrogate model processes three input modalities - the 3D geomodel, scalar parameters characterizing relative permeability functions, and control variables - through separate encoders. These are fused via self-attention in a transformer encoder, and a temporal decoder generates predictions auto-regressively through encoder-decoder cross-attention. The surrogate is trained, using 4000 GEOS flow simulations, to predict saturation and pressure at monitoring locations, total injected and mobile CO2 mass, and saturation footprints. For a new test set, involving randomly sampled geomodels and control variables, the surrogate achieves a median saturation MAE of 0.028 and median relative errors of 0.2-5% for the other quantities of interest. Importantly, it captures the switch from rate to bottom-hole-pressure control. The surrogate model is used within a hierarchical Markov chain Monte Carlo data assimilation procedure for a synthetic true model under three operational strategies. Substantial uncertainty reduction is achieved for key metaparameters, particularly the fault permeabilities. Posterior predictions for saturation footprints and total injected and mobile CO2 mass are also shown to be generally consistent with true model results.

Fuzzy Integral (FI) based aggregation provides a powerful mechanism for nuanced aggregation, for example, in ensemble approaches or decision-level fusion more generally. The main challenge of this approach is the appropriate parametrization of the Fuzzy Measure (FM), which captures the worths of the individual components--and their combinations--which are being fused. Here, widely used approaches including the Sugeno-$λ$ and Decomposable FMs, parametrize the FM by extrapolating from the densities, i.e. the weights associated with individual sources, while respecting the FM's monotonicity constraint. This paper articulates that this information is, in general, insufficient to uniquely identify a discrete FM; but shows how an interval-valued FM can indeed be determined uniquely. We proceed to show how the incorporation of additional information beyond the above, such as the choice of a specific FI and a dataset, then allows for obtaining even more specific interval-valued FMs. In practice, establishing the quality of an empirically determined FM is not trivial. To help address this, we show how the likelihood with which a resulting interval FM encompasses the ideal', i.e. the commonly intangible, best, or ground-truth numeric FM, can be determined, producing a confidence interval at a given confidence level. Finally, based on a series of experiments, we demonstrate empirically that the Choquet FI output based on this FM can also be regarded as the confidence interval for the ideal' information fusion result, providing a novel means to characterize FI fusion outcomes a priori and charting a pathway for future research.

Nonstationary Gaussian process (GP) models are powerful tools for capturing input-dependent variability by adapting to observed data. However, with limited sampling and highly parameterized covariance structure, they are often prone to overfitting and overconfident uncertainty estimates, potentially leading to misleading predictions in safety-critical applications. Motivated by ionospheric modeling for satellite-based augmentation systems (SBAS), this paper proposes a Covariance-Boosted Gaussian Process (CBGP) framework centered upon boosting covariance priors to discover nonstationary latent functions for signal and observation variation that capture irregularities in the input domain. An additional layer of GP modeling of "partially-whitened" observations guides latent function relative error estimation that is used to iteratively update weak priors in a gradient descent-like procedure. Following boosting, restrictions are imposed upon prior covariances to prevent overfitting while posterior uncertainties are inflated to prevent model overconfidence. CBGP model efficacy and robustness are demonstrated through out-of-sample testing of both simulated and real-world applications that meet a three-nines integrity standard. The modeling of an extensive ionospheric storm dataset over South America suggests accurate and reliable means to compute SBAS ionospheric corrections in the most challenging space weather environment using regional models that are more informed and responsive than local fitting performed by currently-operating SBAS.

AC optimal power flow determines the minimum-cost generation dispatch under nonlinear power balance constraints and is solved thousands of times daily in electricity market operations. Learning a direct mapping from load conditions to OPF solutions can accelerate this computation, yet with deepening renewable penetration, a single optimal dispatch is no longer sufficient. Operators require a characterization of the distribution of feasible near-optimal solutions for risk quantification, sensitivity analysis, and multi-objective trade-off assessment. Supervised neural networks provide fast point predictions but cannot capture this conditional distribution. Diffusion-based generative models can sample diverse solutions in principle, yet existing methods operating in the raw state space exhibit degraded solution quality and fail to scale beyond medium-sized systems. We identify the root cause as the conflation of two distinct tasks within a single model. Compressing the high-dimensional OPF solution manifold is one task, and learning the conditional mapping from loads to that manifold is another. This paper presents FMOPF, a framework that resolves this conflation by decoupling compression from generation through latent flow matching and by explicitly modeling load-state coupling through a Constraint-Aware Interaction Prior Network. Experiments on four IEEE test systems demonstrate that FMOPF provides the most effective Newton-Raphson warm starts, achieves the lowest tail risk among generative methods, and is the first such method to scale to systems with several hundred buses while preserving full feasibility. Ablation studies confirm that the latent generation pipeline is a necessary condition for physical feasibility and that the interaction prior functions as a late-stage tail-risk controller.

Radar cross-section (RCS) modeling is foundational to advancing the utility and sensitivity of spaceborne radar systems. This study introduces a deep sigma-point process (DSPP) model for predicting RCS in synthetic aperture radar (SAR) imagery using a RADARSAT-2 dataset containing 208,191 verified ships. The DSPP model not only strives for predictive accuracy but also characterizes the uncertainty inherent in the intricate relationships among radar signals, ship parameters, and environmental conditions. Unlike traditional approaches that rely on deterministic equations with static parameters, the DSPP uses a hierarchical Gaussian process framework with Bayesian inference to capture variability and uncertainty in RCS predictions. By generating predictive distributions rather than single estimates, the model accounts for the complex dynamics governing radar returns. Using a Matern kernel with automatic relevance determination, the DSPP identifies and ranks critical features across radar, operational, and environmental domains, thereby supporting transparency and interpretability. Performance evaluations demonstrate the model's superiority over linear regression baselines, with a 20.83 percent reduction in root mean squared error, a 25.89 percent increase in R-squared, and a 44.4 percent reduction in both the residual interquartile range and median absolute deviation on the test data. By providing calibrated uncertainty bounds, the DSPP enhances prediction reliability and supports robust decision-making. This work represents a shift toward probabilistic models that incorporate the inherent uncertainty of complex phenomena. By transitioning from fixed equations to distributions over outcomes, the DSPP fosters a deeper understanding of RCS behavior and enables systems to operate effectively in dynamic environments.

Where truths are scarce (e.g., seismic and medical imaging), learned priors in ill-posed inverse problems are trained on archives of legacy reconstructions---i.e., an older method's outputs---and their reported uncertainty is taken as data-driven. We show that this prior is, in the population limit, exactly the regularizer that produced its archive of posterior samples, advanced one expectation--maximization step toward the truth. While the step improves the regularizer on the directions the measurements resolve, it leaves the regularizer's assumption on the operator's blind subspace unchanged. An archive of single-best reconstructions collapses the blind interval to zero width. Neither error is detectable in practice, as truths differing only on the blind subspace share the data law, and simulation-based calibration is neutral by construction. We identify from the operator alone which directions the measurements do not inform, and, given a handful of ground-truth models, build intervals there that contain the truth as often as they claim to. We validate these findings on a two-dimensional example with closed-form predictions and in controlled experiments on seismic-imaging and groundwater-flow operators, against priors trained on the truth.

Neural network compression and interpretability remain open challenges in modern deep learn- ing, where billion-parameter architectures deliver impressive accuracy at the cost of trans- parency, computational efficiency, and reliable uncertainty quantification. This paper introduces Neural Atom Prevalence (NAP), a principled Bayesian framework for structured node-level model selection in feedforward neural networks. NAP introduces the neural atom (activation unit) and functions as a hybrid method operating through a four-phase pipeline: Bayesian Lottery Ticket (BLT) identification via Iterative Magnitude Pruning (IMP), soft variational training of the Spike and Slab Independent Gaussian (SS-IG) model, Poisson-Binomial (PB) optimal layer-size selection, and Bayesian fine-tuning to produce a sparse, stable, interpretable, and accurate model. Extensive empirical validation across simulated nonlinear regression, two UCI benchmark datasets (Concrete, YearPredictionMSD), and the MNIST image classification task demonstrates that NAP achieves state-of-the-art structural sparsity, reducing active nodes to as few as 8% of the original dense architecture on MNIST, while well-calibrated probabilisti- cally: the aleatoric-epistemic uncertainty decomposition reveals that model ignorance accounts for only 3 to 4% of total predictive variance across all experiments, and regression reliability diagrams confirm a near-nominal predictive interval coverage (93.4% observed against a 95% target). These results establish NAP as a reliable, theoretically grounded, and computation- ally tractable solution to the simultaneous pursuit of sparsity, accuracy, interpretability, and uncertainty quantification in Bayesian neural networks.

Multimodal learning is a robust approach to improve predictive performance in applications such as medical prognosis. However, the clinical applicability of models that use multimodal learning is hampered by their poor performance under noisy or uninformative data streams. Present fusion approaches often lack robust mechanisms for the dynamic assessment of data quality and for the provision of a trustable confidence score on the final prediction. This dissuades their deployment in safety-critical settings. To address these limitations, we introduce Adaptive Confidence-weighted Expansion (ACE), a novel framework to enhance the trustworthiness of multimodal fusion models. ACE first enhances the multimodal space by generating new, complementary modalities from intra-modality correlations. It then employs a dual-level confidence mechanism that (1) adaptively reweighs all modalities by their reliability before fusion and (2) estimates a global trust score over the fused, final decision. To evaluate ACE, we used four challenging multi-omics datasets (BRCA, KIPAN, LGG, and ROSMAP). ACE significantly outperforms existing state-of-the-art algorithms in both classification performance and confidence calibration. Our framework provides a more stable and robust data fusion method that facilitates the use of multimodal learning in addressing high-stakes problems.