extrapolation assessment

This work addresses the challenge of obtaining well-calibrated uncertainty estimates from neural PDE surrogates under limited or partially observed data, where existing methods often fail to adaptively quantify uncertainty. The authors propose a cross-regularized uncertainty learning framework that jointly optimizes the predictor and low-dimensional uncertainty control parameters during training, enabling calibration-aware uncertainty modeling without post-processing or fixed noise assumptions. The approach supports learning continuous noise levels at multiple model components—including output layers, hidden features, or operator-specific elements such as spectral modes—by integrating a Fourier Neural Operator (FNO) with a gradient routing mechanism. Prediction accuracy and uncertainty calibration are optimized separately on the training and regularization sets. Evaluated on the APEBench benchmark, the method consistently achieves superior calibration across varying observation ratios and training scales, with uncertainty fields effectively highlighting high-error regions.

Neural networks suffer not only accuracy degradation but also severe deterioration in uncertainty calibration—measured by metrics such as the Brier score and Expected Calibration Error (ECE)—under adversarial perturbations, undermining trustworthy decision-making in safety-critical applications. This work formally introduces the problem of *certified calibration*, deriving worst-case theoretical robustness bounds for both the Brier score and ECE: an analytical upper bound for the Brier score and a mixed-integer programming (MIP)-based approximate upper bound for ECE. We further design a novel calibration-directed adversarial attack and propose an adversarial calibration training paradigm. Experiments demonstrate that calibration degradation is quantitatively characterizable, and the derived certified bounds tightly reflect empirical deterioration. The proposed training method substantially improves calibration robustness across diverse adversarial attacks, with consistent efficacy and generalizability validated on multiple benchmark datasets.

This work addresses the prohibitive computational cost of classical probabilistic Richardson extrapolation in high-dimensional settings, where the curse of dimensionality leads to a super-exponential growth in the number of required simulations as tolerance parameters increase. To overcome this limitation, the authors propose a sparse probabilistic Richardson extrapolation framework that reformulates numerical computation as an extrapolation problem with respect to tolerance parameters. By incorporating a sparsity assumption, the method effectively reduces the effective dimensionality of the extrapolation task and integrates multi-fidelity simulations with optimal experimental design. The resulting approach substantially decreases the number of simulations needed in high-dimensional scenarios while preserving accuracy, offering a computationally efficient solution that is both theoretically well-founded and empirically effective, without sacrificing simplicity.

Existing calibration error estimation lacks differentiable, optimizable estimators, hindering end-to-end calibration optimization. Method: We formulate the squared calibration error estimation as a regression task over i.i.d. sample pairs, adopting mean-squared error (MSE) as the risk criterion. Leveraging the bilinear structure of the squared calibration error, we employ kernel ridge regression with joint hyperparameter optimization within a novel train-validation-test estimation pipeline. Contribution/Results: This work establishes the first unified risk-based framework for calibration error estimation; reformulates canonical calibration error estimation as a learnable, differentiable regression problem; and introduces a principled three-stage estimation protocol. Evaluated on standard image classification benchmarks, our estimator achieves significantly higher accuracy than state-of-the-art methods. It is the first practical, end-to-end optimizable estimator for canonical calibration error, enabling gradient-based calibration refinement.

This study addresses the robustness of global post-hoc interpretability methods—such as Partial Dependence Plots (PDP) and Accumulated Local Effects (ALE)—for black-box predictive models on tabular data under data and model perturbations. Motivated by reliability deficits arising from their fragility in model debugging and scientific discovery, we propose the first theoretical bounds quantifying explanation deviation under worst-case and best-case perturbations. Our approach integrates rigorous theoretical analysis with extensive empirical evaluation across multiple real-world and synthetic datasets, systematically assessing explanation sensitivity to controlled perturbations. Results demonstrate that even minor perturbations can substantially distort global feature effect estimates—up to complete reversal of effect direction in adversarial cases. This work establishes the first formal robustness framework for global interpretability, filling a critical gap in the theoretical foundations of trustworthy XAI. It provides the first quantifiable diagnostic benchmark for evaluating the stability and fidelity of global explanations.

This work addresses the challenge that existing machine learning methods struggle to distinguish between epistemic and aleatoric uncertainty under extrapolation scenarios and often lack rigorous coverage guarantees. The authors propose a model-agnostic conformal fuzzy inference framework that, for the first time, integrates conformal prediction with fuzzy reasoning. By incorporating a distance-aware mechanism, the method generates fuzzy predictions—formally represented as probability boxes—with theoretical validity guarantees. This approach enables adaptive uncertainty quantification under distributional shift and demonstrates superior robustness and reliable coverage compared to conventional probabilistic methods on both synthetic and benchmark datasets, with particularly pronounced advantages in data-scarce regimes.

This work addresses the challenge of selecting the regularization parameter (nugget) in ill-posed linear systems arising in machine learning, where existing adaptive methods lack compatibility with automatic differentiation and suffer from computational inefficiency. To overcome these limitations, we introduce autonugget, a lightweight Python package fully compatible with JAX’s automatic differentiation framework. Our approach uniquely integrates Richardson extrapolation with Tikhonov regularized solutions computed across multiple nugget values, thereby preserving end-to-end differentiability while avoiding the information loss inherent in single-solution strategies. Experimental results demonstrate that autonugget significantly enhances solution accuracy and training stability without compromising rapid prototyping capabilities.

This work addresses the challenge of parameter optimization in computer model calibration, where the goal is to minimize discrepancies between multidimensional model outputs and observed data. The authors propose a novel root-finding paradigm that reformulates calibration as a root-search problem by constructing signed-residual-based Kriging or stochastic Kriging surrogates. Their approach integrates a sequential search space reduction strategy with a new acquisition function compatible with first-order optimizers. Notably, it guarantees algorithmic robustness even when roots may not exist—a scenario unaddressed by prior methods. Empirical evaluations demonstrate that the proposed method significantly outperforms conventional calibration techniques across both data-driven and physics-based modeling tasks, achieving higher computational efficiency while enhancing solution robustness.

This study addresses the unreliable uncertainty estimation of cuffless photoplethysmography (PPG)-based blood pressure prediction under out-of-distribution (OOD) scenarios. The authors systematically evaluate various deep learning architectures and uncertainty quantification strategies using an XResNet1D-50 backbone, combined with deep ensembles (DE), Monte Carlo dropout (MCD), Gaussian negative log-likelihood (GNLL), and mean squared error (MSE) losses. Post-hoc calibration techniques—including conformal prediction (CP), temperature scaling (TS), and isotonic regression (IR)—are further integrated to refine predictive uncertainty. The work reveals, for the first time, that DE exhibits superior robustness to domain shift compared to MCD, GNLL inherently yields higher-quality uncertainty estimates, and post-calibration is particularly crucial for MSE-trained models. Experimental results demonstrate that GNLL combined with DE, followed by CP or TS calibration, achieves the best-calibrated uncertainty, with CP and TS consistently delivering the most substantial performance improvements.

This study addresses the stability of the solution operator with respect to perturbations in the input parameter distribution within the framework of nonparametric Bayesian computer model calibration. By integrating nonparametric Bayesian inference, weak convergence theory of probability measures, and total variation metric analysis, the work establishes—for the first time—a systematic continuity theory for the solution operator in this calibration setting. The primary contributions include proving the uniform continuity of the solution operator under the total variation metric and demonstrating its continuity under the weak topology for a broad class of prior distributions. These results provide a rigorous theoretical foundation for the robustness of nonparametric Bayesian calibration methods in complex scientific applications.