Uncertainty Quantification

Uncertainty quantification is a key aspect in many tasks such as model selection/regularization, or quantifying prediction uncertainties to perform active learning or OOD detection. Within credal approaches that consider modeling uncertainty as probability sets, upper entropy plays a central role as an uncertainty measure. This paper is devoted to the computational aspect of upper entropies, providing an exhaustive algorithmic and complexity analysis of the problem. In particular, we show that the problem has a strongly polynomial solution, and propose many significant improvements over past algorithms proposed for 2-monotone lower probabilities and their specific cases.

While both classical and neural network classifiers can achieve high accuracy, they fall short on offering uncertainty bounds on their predictions, making them unfit for safety-critical applications. Existing kernel-based classifiers that provide such bounds scale with $\mathcal O (n^{\sim3})$ in time, making them computationally intractable for large datasets. To address this, we propose a novel, computationally efficient classification algorithm based on the Nadaraya-Watson estimator, for whose estimates we derive frequentist uncertainty intervals. We evaluate our classifier on synthetically generated data and on electrocardiographic heartbeat signals from the MIT-BIH Arrhythmia database. We show that the method achieves competitive accuracy $>$\SI{96}{\percent} at $\mathcal O(n)$ and $\mathcal O(\log n)$ operations, while providing actionable uncertainty bounds. These bounds can, e.g., aid in flagging low-confidence predictions, making them suitable for real-time settings with resource constraints, such as diagnostic monitoring or implantable devices.

Upcoming stage-IV surveys such as Euclid and Rubin will deliver vast amounts of high-precision data, opening new opportunities to constrain cosmological models with unprecedented accuracy. A key step in this process is the reconstruction of the dark matter distribution from noisy weak-lensing shear measurements. Current deep-learning-based mass-mapping methods achieve high reconstruction accuracy, but either require retraining a model for each new observed sky region (limiting practicality) or rely on slow Markov chain Monte Carlo sampling. Efficient exploitation of future survey data therefore calls for a new method that is accurate, flexible, and fast at inference. In addition, an uncertainty quantification with coverage guarantees is essential for a reliable cosmological parameter estimation. We introduce PnPMass, a plug-and-play approach for weak-lensing mass mapping. The algorithm produces point estimates by alternating between a gradient descent step with a carefully chosen data fidelity term and a denoising step implemented with a single deep-learning model trained on simulated data corrupted by Gaussian white noise. We also propose a fast sampling-free uncertainty quantification scheme based on moment networks, with calibrated error bars obtained through conformal prediction to ensure coverage guarantees. Finally, we benchmark PnPMass against model-driven and data-driven mass-mapping techniques. PnPMass achieves a performance close to that of the currently best deep-learning methods while offering fast inference. It converges in just a few iterations, and it requires only a single training phase, regardless of the noise covariance of the observations. It therefore combines flexibility, efficiency, and reconstruction accuracy while delivering tighter error bars than existing approaches, making it well suited for upcoming weak-lensing surveys.

Synthetic spectral image generation is essential for remote sensing simulation and mission design, yet physically based radiative transfer models (RTMs) remain computationally expensive. Existing learning-based emulators reduce this cost, but are mostly deterministic parameter-to-spectrum regressors with limited spatial modeling and uncertainty information. We formulate spectral image emulation as a parameter-conditioned latent-variable problem and propose a variational autoencoder (VAE)-based framework combining nonlinear spectral-image representations, fast inference, and per-pixel uncertainty estimates. The framework is instantiated at spectrum and spatial--spectral levels through two-step VAE pretraining and latent mapping. We evaluate it on PROSAIL-simulated hyperspectral vegetation cubes (211 bands) and real Sentinel-3 OLCI multispectral ocean-colour imagery (21 bands) against classical regression emulators and a deep CNN baseline. Results show that no single architecture is optimal: pixel-to-pixel models perform best on controlled hyperspectral simulations, whereas a fully convolutional VAE is more robust on noisy real observations with missing or contaminated pixels. VAE-based emulators also achieve high throughput for large-scale generation. On Sentinel-3, the spatial--spectral VAE provides predictive intervals closer to empirical errors than pixel-wise neural and classical emulators, although absolute calibration remains incomplete. A look-up-table-based retrieval experiment further shows that reconstruction fidelity alone does not ensure reliable leaf area index and chlorophyll retrieval. Emulators should therefore be evaluated in representative remote-sensing end-use scenarios, not by reconstruction metrics alone.

Distribution-to-distribution generative models support scientific imaging tasks ranging from modeling cellular perturbation responses to translating medical images across conditions. Trustworthy generation requires reliability, or generalization across labs, devices, and experimental conditions, and accountability, or detecting out-of-distribution cases where predictions may be unreliable. We leverage Stochastic Flow Matching (SFM), a marginal-preserving stochastic extension of flow matching for improved generalization under distribution shift. SFM augments deterministic flows with a diffusion term together with a learned score-based drift correction, retaining the learned transport marginals while modeling conditional variability. Building on this SFM framework, we introduce Bayesian Stochastic Flow Matching (BSFM) as a companion uncertainty quantification mechanism and develop AVUQ (Antithetic Variance-reduction Uncertainty Quantification) to approximately estimate epistemic and aleatoric uncertainty via sample-efficient antithetic sampling with approximate posterior inference. We further use AVUQ to yield anomaly scores for unreliable generation detection. Experiments on cellular imaging (BBBC021, JUMP) and brain fMRI (Theory of Mind) across diverse unseen scenarios show that SFM improves generalization while AVUQ provides effective uncertainty-based anomaly scores under practical sampling budgets.

Time-dependent reliability analysis of nonlinear dynamical systems under stochastic excitations is a critical yet computationally demanding task. Conventional approaches, such as Monte Carlo simulation, necessitate repeated evaluations of computationally expensive numerical solvers, leading to significant computational bottlenecks. To address this challenge, we propose CoNBONet, a neuroscience-inspired surrogate model that enables fast, energy-efficient, and uncertainty-aware reliability analysis, providing a scalable alternative to techniques such as Monte Carlo simulations. CoNBONet, short for Conformalized Neuroscience-inspired Bayesian Operator Network, leverages the expressive power of deep operator networks while integrating neuroscience-inspired neuron models to achieve fast, low-power inference. Unlike traditional surrogates such as Gaussian processes, polynomial chaos expansions, or support vector regression, that may face scalability challenges for high-dimensional, time-dependent reliability problems, CoNBONet offers fast and energy-efficient inference enabled by a neuroscience-inspired network architecture, calibrated uncertainty quantification with theoretical guarantees via split conformal prediction, and strong generalization capability through an operator-learning paradigm that maps input functions to system response trajectories. Validation of the proposed CoNBONet for various nonlinear dynamical systems demonstrates that CoNBONet preserves predictive fidelity, and achieves reliable coverage of failure probabilities, making it a powerful tool for robust and scalable reliability analysis in engineering design.

Protecting patient privacy remains a fundamental barrier to scaling machine learning across healthcare institutions, where centralizing sensitive data is often infeasible due to ethical, legal, and regulatory constraints. Federated learning offers a promising alternative by enabling privacy-preserving, multi-institutional training without sharing raw patient data; however, real-world deployments face severe challenges from data heterogeneity, site-specific biases, and class imbalance, which degrade predictive reliability and render existing uncertainty quantification methods ineffective. Here, we present TrustFed, a federated uncertainty quantification framework that provides distribution-free, finite-sample coverage guarantees under heterogeneous and imbalanced healthcare data, without requiring centralized access. TrustFed introduces a representation-aware client assignment mechanism that leverages internal model representations to enable effective calibration across institutions, along with a soft-nearest threshold aggregation strategy that mitigates assignment uncertainty while producing compact and reliable prediction sets. Using over 430,000 medical images across six clinically distinct imaging modalities, we conduct one of the most comprehensive evaluations of uncertainty-aware federated learning in medical imaging, demonstrating robust coverage guarantees across datasets with diverse class cardinalities and imbalance regimes. By validating TrustFed at this scale and breadth, our study advances uncertainty-aware federated learning from proof-of-concept toward clinically meaningful, modality-agnostic deployment, positioning statistically guaranteed uncertainty as a core requirement for next-generation healthcare AI systems.

Real objects that inhabit the physical world follow physical laws and thus behave plausibly during interaction with other physical objects. However, current methods that perform 3D reconstructions of real-world scenes from multi-view 2D images optimize primarily for visual fidelity, i.e., they train with photometric losses and reason about uncertainty in the image or representation space. This appearance-centric view overlooks body contacts and couplings, conflates function-critical regions (e.g., aerodynamic or hydrodynamic surfaces) with ornamentation, and reconstructs structures suboptimally, even when physical regularizers are added. All these can lead to unphysical and implausible interactions. To address this, we consider the question: How can 3D reconstruction become aware of real-world interactions and underlying object functionality, beyond visual cues? To answer this question, we propose FluidGaussian, a plug-and-play method that tightly couples geometry reconstruction with ubiquitous fluid-structure interactions to assess surface quality at high granularity. We define a simulation-based uncertainty metric induced by fluid simulations and integrate it with active learning to prioritize views that improve both visual and physical fidelity. In an empirical evaluation on NeRF Synthetic (Blender), Mip-NeRF 360, and DrivAerNet++, our FluidGaussian method yields up to +8.6% visual PSNR (Peak Signal-to-Noise Ratio) and -62.3% velocity divergence during fluid simulations. Our code is available at https://github.com/delta-lab-ai/FluidGaussian.

In engineering, uncertainty propagation aims to characterise system outputs under uncertain inputs. For interval uncertainty, the goal is to determine output bounds given interval-valued inputs, which is critical for robust design optimisation and reliability analysis. However, standard interval propagation relies on solving optimisation problems that become computationally expensive for complex systems. Surrogate models alleviate this cost but typically replace only the evaluator within the optimisation loop, still requiring many inference calls. To overcome this limitation, we reformulate interval propagation as an interval-valued regression problem that directly predicts output bounds. We present a comprehensive study of neural network-based surrogate models, including multilayer perceptrons (MLPs) and deep operator networks (DeepONet), for this task. Three approaches are investigated: (i) naive interval propagation through standard architectures, (ii) bound propagation methods such as Interval Bound Propagation (IBP) and CROWN, and (iii) interval neural networks (INNs) with interval weights. Results show that these methods significantly improve computational efficiency over traditional optimisation-based approaches while maintaining accurate interval estimates. We further discuss practical limitations and open challenges in applying interval-based propagation methods.

Geotechnical site characterisation relies on sparse, heterogeneous borehole data, where uncertainty quantification and interpretability matter as much as predictive accuracy. We evaluate TabPFN~\citep{Hollmann2025}, a tabular foundation model, and its tabpfn-extensions library on two geotechnical tasks: (1) soil-type classification from N-value and shear-wave velocity data as a controlled illustrative case, and (2) iterative imputation of five mechanical parameters ($s_\mathrm{u}$, $E_{\mathrm{u}}$, ${σ'}_\mathrm{p}$, $C_\mathrm{c}$, $C_\mathrm{v}$) in BM/AirportSoilProperties/2/2025. Without retraining, we apply cosine-similarity analysis to TabPFN embeddings, visualise predictive distributions, and compute SHAP attributions. On the regression benchmark we compare TabPFN with mean imputation, linear regression, random forests, XGBoost, and HBM; introduce a proxy decomposition of predictive uncertainty across context-perturbation classes; and propagate marginal $C_\mathrm{c}$ and ${σ'}_\mathrm{p}$ distributions through a one-dimensional consolidation model to obtain the reliability index $β$ and serviceability exceedance probability $P_\mathrm{f}$. Embeddings exhibit label-consistent Clay/Sand grouping; iterative imputation reduces RMSE for all five targets, with TabPFN lowest on four; SHAP attributions are consistent with the Skempton compression-index correlation and the inverse preconsolidation-pressure-water-content dependence; the within-posterior component is largest in the proxy decomposition. We position the contribution as a worked evaluation workflow that may complement established methods for data-scarce geotechnics, not as algorithmic innovation.

Multi-agent LLM systems, where multiple prompted instances of a language model independently answer questions, are increasingly used for complex reasoning tasks. However, existing methods for quantifying the uncertainty of their collective outputs rely on shallow voting statistics that discard the rich semantic information in agents' reasoning. We introduce DiscoUQ, a framework that extracts and leverages the structure of inter-agent disagreement -- both linguistic properties (evidence overlap, argument strength, divergence depth) and embedding geometry (cluster distances, dispersion, cohesion) -- to produce well-calibrated confidence estimates. We propose three methods of increasing complexity: DiscoUQ-LLM (logistic regression on LLM-extracted structure features), DiscoUQ-Embed (logistic regression on embedding geometry), and DiscoUQ-Learn (a neural network combining all features). Evaluated on four diverse benchmarks (StrategyQA, MMLU, TruthfulQA, ARC-Challenge) with a 5-agent system using Qwen3.5-27B, DiscoUQ-LLM achieves an average AUROC of 0.802, outperforming the best baseline (LLM Aggregator, 0.791) while being substantially better calibrated (ECE 0.036 vs. 0.098). The learned features generalize across benchmarks with near-zero performance degradation and provide the largest improvements where they are most needed: in the ambiguous "weak disagreement" tier where simple vote counting fails.

Uncertainty quantification for image data is dominated by complex deep learning methods, yet the field lacks an interpretable, mathematically grounded baseline. We propose Bayesian scattering to fill this gap, serving as a first-step baseline akin to the role of Bayesian linear regression for tabular data. Our method couples the wavelet scattering transform-a deep, non-learned feature extractor-with a simple probabilistic head. Because scattering features are derived from geometric principles rather than learned, they avoid overfitting the training distribution. This helps provide sensible uncertainty estimates even under significant distribution shifts. We validate this on diverse tasks, including medical imaging under institution shift, wealth mapping under country-to-country shift, and Bayesian optimization of molecular properties. Our results suggest that Bayesian scattering is a solid baseline for complex uncertainty quantification methods.

Pairwise comparison labeling is emerging as it yields higher inter-rater reliability than conventional classification labeling, but exhaustive comparisons require quadratic cost. We propose Dodgersort, which leverages CLIP-based hierarchical pre-ordering, a neural ranking head and probabilistic ensemble (Elo, BTL, GP), epistemic--aleatoric uncertainty decomposition, and information-theoretic pair selection. It reduces human comparisons while improving the reliability of the rankings. In visual ranking tasks in medical imaging, historical dating, and aesthetics, Dodgersort achieves a 11--16% annotation reduction while improving inter-rater reliability. Cross-domain ablations across four datasets show that neural adaptation and ensemble uncertainty are key to this gain. In FG-NET with ground-truth ages, the framework extracts 5--20$\times$ more ranking information per comparison than baselines, yielding Pareto-optimal accuracy--efficiency trade-offs.

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Our analysis considers trajectory-dependent regressors and allows marginally stable dynamics with polynomial mean-square state growth. We prove that the diameter of the feasible parameter set shrinks with sample complexity $\widetilde{\mathcal O}(1/ε)$ where $ε$ is the estimation error. Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.

Despite the strong predictive performance achieved by machine learning models across many application domains, assessing their trustworthiness through reliable estimates of predictive confidence remains a critical challenge. This issue arises in scenarios where the likelihood of error inferred from learned representations follows a bimodal distribution, resulting from the coexistence of confident and ambiguous predictions. Standard regression approaches often struggle to adequately express this predictive uncertainty, as they implicitly assume unimodal Gaussian noise, leading to mean-collapse behavior in such settings. Although Mixture Density Networks (MDNs) can represent different distributions, they suffer from severe optimization instability. We propose a family of distribution-aware loss functions integrating normalized RMSE with Wasserstein and Cramér distances. When applied to standard deep regression models, our approach recovers bimodal distributions without the volatility of mixture models. Validated across four experimental stages, our results show that the proposed Wasserstein loss establishes a new Pareto efficiency frontier: matching the stability of standard regression losses like MSE in unimodal tasks while reducing Jensen-Shannon Divergence by 45% on complex bimodal datasets. Our framework strictly dominates MDNs in both fidelity and robustness, offering a reliable tool for aleatoric uncertainty estimation in trustworthy AI systems.

Probabilistic circuit (PC) structure learning is hampered by greedy algorithms that make irreversible, locally optimal decisions. We propose SymCircuit, which replaces greedy search with a learned generative policy trained via entropy-regularized reinforcement learning. Instantiating the RL-as-inference framework in the PC domain, we show the optimal policy is a tempered Bayesian posterior, recovering the exact posterior when the regularization temperature is set inversely proportional to the dataset size. The policy is implemented as SymFormer, a grammar-constrained autoregressive Transformer with tree-relative self-attention that guarantees valid circuits at every generation step. We introduce option-level REINFORCE, restricting gradient updates to structural decisions rather than all tokens, yielding an SNR (signal to noise ratio) improvement and >10 times sample efficiency gain on the NLTCS dataset. A three-layer uncertainty decomposition (structural via model averaging, parametric via the delta method, leaf via conjugate Dirichlet-Categorical propagation) is grounded in the multilinear polynomial structure of PC outputs. On NLTCS, SymCircuit closes 93% of the gap to LearnSPN; preliminary results on Plants (69 variables) suggest scalability.

Probability theory has become the predominant framework for quantifying uncertainty across scientific and engineering disciplines, with a particular focus on measurement and control systems. However, the widespread reliance on simple Gaussian assumptions--particularly in control theory, manufacturing, and measurement systems--can result in incomplete representations and multistage lossy approximations of complex phenomena, including inaccurate propagation of uncertainty through multi stage processes. This work proposes a comprehensive yet computationally tractable framework for representing and propagating quantitative attributes arising in measurement systems using Probability Density Functions (PDFs). Recognizing the constraints imposed by finite memory in software systems, we advocate for the use of Gaussian Mixture Models (GMMs), a principled extension of the familiar Gaussian framework, as they are universal approximators of PDFs whose complexity can be tuned to trade off approximation accuracy against memory and computation. From both mathematical and computational perspectives, GMMs enable high performance and, in many cases, closed form solutions of essential operations in control and measurement. The paper presents practical applications within manufacturing and measurement contexts especially circular factory, demonstrating how the GMMs framework supports accurate representation and propagation of measurement uncertainty and offers improved accuracy--compared to the traditional Gaussian framework--while keeping the computations tractable.

Outdoor air pollution is a major concern for the environment and public health, especially in areas where urbanization is taking place rapidly. The Indian Air Quality Index (IND-AQI), developed by the Central Pollution Control Board (CPCB), is a standardized reporting system for air quality based on pollutants such as PM2.5, PM10), nitrogen dioxide (NO2), sulfur dioxide (SO2), ozone (O3), carbon monoxide (CO), and ammonia (NH3). However, the traditional calculation of the AQI uses crisp thresholds and deterministic aggregation rules, which are not suitable for handling uncertainty and transitions between classes. To address these limitations, this study proposes a hybrid ontology-based uncertainty-aware framework integrating Weighted Interval Type-2 Fuzzy Logic with semantic knowledge modeling. Interval Type-2 fuzzy sets are used to model uncertainty near AQI class boundaries, while pollutant importance weights are determined using Interval Type-2 Fuzzy Analytic Hierarchy Process (IT2-FAHP) to reflect their relative health impacts. In addition, an OWL-based air quality ontology extending the Semantic Sensor Network (SSN) ontology is developed to represent pollutants, monitoring stations, AQI categories, regulatory standards, and environmental governance actions. Semantic reasoning is implemented using SWRL rules and validated through SPARQL queries to infer AQI categories, health risks, and recommended mitigation actions. Experimental evaluation using CPCB air quality datasets demonstrates that the proposed framework improves AQI classification reliability and uncertainty handling compared with traditional crisp and Type-1 fuzzy approaches, while enabling explainable semantic reasoning and intelligent decision support for air quality monitoring systems

Uncertainty estimation is critical for deploying reasoning language models, yet remains poorly understood under extended chain-of-thought reasoning. We study parallel sampling as a fully black-box approach using verbalized confidence and self-consistency. Across three reasoning models and 17 tasks spanning mathematics, STEM, and humanities, we characterize how these signals scale. Both self-consistency and verbalized confidence scale in reasoning models, but self-consistency exhibits lower initial discrimination and lags behind verbalized confidence under moderate sampling. Most uncertainty gains, however, arise from signal combination: with just two samples, a hybrid estimator improves AUROC by up to $+12$ on average and already outperforms either signal alone even when scaled to much larger budgets, after which returns diminish. These effects are domain-dependent: in mathematics, the native domain of RLVR-style post-training, reasoning models achieve higher uncertainty quality and exhibit both stronger complementarity and faster scaling than in STEM or humanities.

Bayesian inference represents a principled way to incorporate Earth structure uncertainty in full-waveform moment tensor inversions, but traditional approaches generally require significant approximations that risk biasing the resulting solutions. We introduce a robust method for handling theory errors using simulation-based inference (SBI), a machine learning approach that empirically models their impact on the observations. This framework retains the rigour of Bayesian inference while avoiding restrictive assumptions about the functional form of the uncertainties. We begin by demonstrating that the common Gaussian parametrisation of theory errors breaks down under minor ($1-3 \%$) 1-D Earth model uncertainty. To address this issue, we develop two formalisms for utilising SBI to improve the quality of the moment tensor solutions: one using physics-based insights into the theory errors, and another utilising an end-to-end deep learning algorithm. We then compare the results of moment tensor inversion with the standard Gaussian approach and SBI, and demonstrate that Gaussian assumptions induce bias and significantly under-report moment tensor uncertainties. We also show that these effects are particularly problematic when inverting short period data and for shallow, isotropic events. On the other hand, SBI produces more reliable, better calibrated posteriors of the earthquake source mechanism. Finally, we successfully apply our methodology to two well studied moderate magnitude earthquakes: one from the 1997 Long Valley Caldera volcanic earthquake sequence, and the 2020 Zagreb earthquake.

Uncertainty quantification (UQ) is crucial in safety-critical applications such as medical image segmentation. Total uncertainty is typically decomposed into data-related aleatoric uncertainty (AU) and model-related epistemic uncertainty (EU). Many methods exist for modeling AU (such as Probabilistic UNet, Diffusion) and EU (such as ensembles, MC Dropout), but it is unclear how they interact when combined. Additionally, recent work has revealed substantial entanglement between AU and EU, undermining the interpretability and practical usefulness of the decomposition. We present a comprehensive empirical study covering a broad range of AU-EU model combinations, propose an entanglement proxy based on the relative performance of uncertainty measures, and evaluate model combinations across downstream uncertainty quantification tasks. Ensembles consistently show more favorable proxy values and superior performance. Softmax models usually beat other AU methods, except in calibration where the results are dataset-dependent. A softmax ensemble performs remarkably well on all tasks. Finally, we analyze potential sources of uncertainty entanglement and outline directions for mitigating this effect.

Image registration is an ill-posed dense vision task, where multiple solutions achieve similar loss values, motivating probabilistic inference. Variational inference has previously been employed to capture these distributions, however restrictive assumptions about the posterior form can lead to poor characterisation, overconfidence and low-quality samples. More flexible posteriors are typically bottlenecked by the complexity of high-dimensional covariance matrices required for dense 3D image registration. In this work, we present a memory and computationally efficient inference method, Structured SIR, that enables expressive, multi-modal, characterisation of uncertainty with high quality samples. We propose the use of a Sampled Importance Resampling (SIR) algorithm with a novel memory-efficient high-dimensional covariance parameterisation as the sum of a low-rank covariance and a sparse, spatially structured Cholesky precision factor. This structure enables capturing complex spatial correlations while remaining computationally tractable. We evaluate the efficacy of this approach in 3D dense image registration of brain MRI data, which is a very high-dimensional problem. We demonstrate that our proposed method produces uncertainty estimates that are significantly better calibrated than those produced by variational methods, achieving equivalent or better accuracy. Crucially, we show that the model yields highly structured multi-modal posterior distributions, enable effective and efficient uncertainty quantification.

Gaussian process (GP) regression is widely used for uncertainty quantification, yet the standard formulation assumes noise-free covariates. When inputs are measured with error, this errors-in-variables (EIV) setting can lead to optimistically narrow posterior intervals and biased decisions. We study GP regression under input measurement uncertainty by representing each noisy input as a probability measure and defining covariance through Wasserstein distances between these measures. Building on this perspective, we instantiate a deterministic projected Wasserstein ARD (PWA) kernel whose one-dimensional components admit closed-form expressions and whose product structure yields a scalable, positive-definite kernel on distributions. Unlike latent-input GP models, PWA-based GPs (\PWAGPs) handle input noise without introducing unobserved covariates or Monte Carlo projections, making uncertainty quantification more transparent and robust.

Accurate uncertainty estimation is essential for building robust and trustworthy recognition systems. In this paper, we consider the open-set text classification (OSTC) task - and uncertainty estimation for it. For OSTC a text sample should be classified as one of the existing classes or rejected as unknown. To account for the different uncertainty types encountered in OSTC, we adapt the Holistic Uncertainty Estimation (HolUE) method for the text domain. Our approach addresses two major causes of prediction errors in text recognition systems: text uncertainty that stems from ill formulated queries and gallery uncertainty that is related the ambiguity of data distribution. By capturing these sources, it becomes possible to predict when the system will make a recognition error. We propose a new OSTC benchmark and conduct extensive experiments on a wide range of data, utilizing the authorship attribution, intent and topic classification datasets. HolUE achieves 40-365% improvement in Prediction Rejection Ratio (PRR) over the quality-based SCF baseline across datasets: 365% on Yahoo Answers (0.79 vs 0.17 at FPIR 0.1), 347% on DBPedia (0.85 vs 0.19), 240% on PAN authorship attribution (0.51 vs 0.15 at FPIR 0.5), and 40% on CLINC150 intent classification (0.73 vs~0.52). We make public our code and protocols https://github.com/Leonid-Erlygin/text_uncertainty.git

Large language models (LLMs) are often confidently wrong, making reliable uncertainty estimation (UE) essential. Output-based heuristics are cheap but brittle, while probing internal representations is effective yet high-dimensional and hard to transfer. We propose a compact, per-instance UE method that scores cross-layer agreement patterns in internal representations using a single forward pass. Across three models, our method matches probing in-distribution, with mean diagonal differences of at most $-1.8$ AUPRC percentage points and $+4.9$ Brier score points. Under cross-dataset transfer, it consistently outperforms probing, achieving off-diagonal gains up to $+2.86$ AUPRC and $+21.02$ Brier points. Under 4-bit weight-only quantization, it remains robust, improving over probing by $+1.94$ AUPRC points and $+5.33$ Brier points on average. Beyond performance, examining specific layer--layer interactions reveals differences in how disparate models encode uncertainty. Altogether, our UE method offers a lightweight, compact means to capture transferable uncertainty in LLMs.

Inverse design aims to find design parameters $x$ achieving target performance $y^*$. Generative approaches learn bidirectional mappings between designs and labels, enabling diverse solution sampling. However, standard conditional flow matching (CFM), when adapted to inverse problems by pairing labels with design parameters, exhibits strong sensitivity to their arbitrary ordering and scaling, leading to unstable training. We introduce Diagonal Flow Matching (Diag--CFM), which resolves this through a zero-anchoring strategy that pairs design coordinates with noise and labels with zero, making the learning problem provably invariant to coordinate permutations. This yields substantially lower round-trip error than CFM and invertible neural network baselines across design dimensions up to $P{=}784$, including order-of-magnitude gains on several benchmarks. We develop two architecture-intrinsic uncertainty metrics, Zero-Deviation and Self-Consistency, that enable three practical capabilities: selecting the best candidate among multiple generations, abstaining from unreliable predictions, and detecting out-of-distribution targets; consistently outperforming ensemble and general-purpose alternatives across all tasks. We validate on airfoil, gas turbine combustor, scalable analytical benchmarks, a photonics inverse-design task, and an image-statistics benchmark.

Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.

Model-based approaches for (bio)process systems often suffer from incomplete knowledge of the underlying physical, chemical, or biological laws. Universal differential equations, which embed neural networks within differential equations, have emerged as powerful tools to learn this missing physics from experimental data. However, neural networks are inherently opaque, motivating their post-processing via symbolic regression to obtain interpretable mathematical expressions. Genetic algorithm-based symbolic regression is a popular approach for this post-processing step, but provides only point estimates and cannot quantify the confidence we should place in a discovered equation. We address this limitation by applying Bayesian symbolic regression, which uses Reversible Jump Markov Chain Monte Carlo to sample from the posterior distribution over symbolic expression trees. This approach naturally quantifies uncertainty in the recovered model structure. We demonstrate the methodology on a Lotka-Volterra predator-prey system and then show how a well-designed experiment leads to lower uncertainty in a fed-batch bioreactor case study.

Geological CO2 storage, geothermal energy and underground hydrogen storage depend on irreversible or high-cost decisions made with sparse data and incomplete physical understanding. Machine learning (ML) increasingly informs these decisions, yet methods are usually judged on predictive accuracy rather than on whether their outputs can be trusted, traced and acted upon. We argue that subsurface ML should be designed as decision infrastructure with shared systems that carry calibrated uncertainty and provenance from observation to operating decision. We propose a decision-readiness checklist, an audit-trail workflow for monitoring updates, and benchmark designs that score models by the decisions they support. We identify validation needs shared across the three technologies, and the data-access and governance steps needed for ML to support regulatory and investment decisions.