analytic variance propagation

This work addresses the limitations of existing uncertainty estimation methods, which often incur high computational costs in real-time settings and struggle to model input-dependent, asymmetric, and heavy-tailed error distributions. The authors extend the ACCRUE framework by introducing input-dependent non-Gaussian distributions—such as two-piece Gaussian and asymmetric Laplace—into the calibration of deterministic predictions for the first time. A neural network is employed to learn these complex uncertainty structures, and an end-to-end training procedure is developed using a novel loss function that jointly optimizes predictive accuracy and calibration reliability. Experimental results on both synthetic and real-world datasets demonstrate that the proposed approach effectively captures the skewness and heavy-tailed nature of prediction errors, yielding significantly improved probabilistic forecasting performance.

This paper addresses uncertainty propagation in feedforward neural networks under stochastic input perturbations, with a focus on architectures employing the Leaky ReLU activation function. We propose a high-accuracy analytical method: first, piecewise linearization of Leaky ReLU preserves its essential nonlinearity; second, a Gaussian Copula-based joint distribution surrogate model is constructed to yield closed-form expressions for the output probability density function and arbitrary-order statistical moments. The approach circumvents the prohibitive computational cost of Monte Carlo simulation while achieving excellent agreement between theoretical predictions and numerical experiments. Its robustness to large input perturbations is further validated in modeling nonlinear integral-differential operators within polynomial function spaces. The core contribution is the first rigorous, analytically tractable, high-fidelity, and robust theoretical framework for uncertainty propagation in Leaky ReLU networks.

In Bayesian inverse problems, surrogate models—constrained by limited simulation budgets and approximation errors—often induce biased parameter estimates and overconfident posterior distributions. To address this, we propose the first scalable Bayesian surrogate modeling framework that rigorously quantifies and propagates uncertainty across the entire pipeline: surrogate construction, posterior inference, and model validation. Methodologically, we integrate Gaussian process surrogate modeling, probabilistic programming, and posterior calibration techniques to design three novel Bayesian inversion algorithms, overcoming classical analytical assumptions and computational bottlenecks. We validate the framework on three real-world linear and nonlinear inverse problems. Results demonstrate substantial improvements in posterior calibration and parameter estimation reliability, leading to reduced decision risk. The framework establishes a new paradigm for robust uncertainty quantification under resource constraints.

This work addresses the lack of reliable uncertainty quantification in non-intrusive reduced-order models (NIROMs) under data-scarce or extrapolative conditions. The authors propose a novel approach that introduces stochastic perturbations to the reduced basis on the Stiefel manifold to model structural uncertainty, combined with a distribution-free conformal risk control framework to construct prediction sets with coordinate-wise miscoverage guarantees. By innovatively integrating manifold-based perturbations with conformal calibration, the method achieves, for the first time, a separation of uncertainties arising from basis truncation and regression errors. A scalar, interpretable calibration factor is introduced to assess uncertainty quality. Evaluated on parametric PDE benchmarks and a tire calendering process, the approach significantly outperforms conventional Gaussian process variance estimates, delivering more reliable and spatially refined uncertainty quantification.

Machine learning (ML) surrogate models in engineering simulation suffer from unreliable predictions due to the coupling of model uncertainty and input variability. To address this, we propose a unified uncertainty quantification framework based on polynomial chaos expansion (PCE). This method jointly maps the predictive distribution of Gaussian process regression and the input random variables onto a standard orthogonal polynomial basis, enabling synergistic modeling of both sources of uncertainty. The framework enables efficient computation of output statistical moments (e.g., mean and variance) and global sensitivity analysis, allowing quantitative decomposition of the individual contributions of each input variable and model uncertainty to the total output variability. Compared with conventional Monte Carlo methods, our approach achieves comparable accuracy while drastically reducing computational cost. As a result, it enhances the reliability and interpretability of ML surrogates in engineering design applications.

This study addresses the quantification of sources of predictive uncertainty and their contributions to prediction interval width. Building upon the law of total variance, the work proposes several conservative decompositions of posterior predictive variance, systematically characterizing the components of uncertainty and their interdependencies through conditional expectation and conditional variance terms. Experimental evaluations across multiple canonical models demonstrate that the proposed approach effectively identifies the dominant sources of uncertainty and reveals coherent patterns of co-variation among decomposition terms. These insights offer a novel perspective for model assessment and refinement, enhancing interpretability and guiding targeted improvements in predictive reliability.

This work addresses the limitations of traditional Gaussian assumptions in accurately representing complex uncertainties, which often lead to information loss and reduced accuracy in multi-stage measurement and control processes. To overcome these challenges, the paper proposes a scalable precision framework based on Gaussian Mixture Models (GMMs), leveraging GMMs as universal approximators of probability density functions. The approach integrates closed-form uncertainty propagation algorithms with memory-efficient computational strategies, thereby transcending the representational constraints of Gaussian methods while maintaining computational tractability. Experimental evaluations in manufacturing and metrology scenarios—such as circular factories—demonstrate that the proposed method significantly enhances the fidelity of uncertainty characterization and propagation, outperforming conventional Gaussian-based techniques.

This work addresses the challenge of parameter estimation in physical system simulation models arising from complex residual distributions. The authors propose an end-to-end estimation framework that employs embedded normalizing flows to map intricate residuals onto a simple base distribution. Indirect constraints are imposed on this base distribution through empirical likelihood under moment conditions. Model and flow parameters are jointly optimized via implicit differentiation combined with first-order gradient methods. By innovatively integrating normalizing flows with constrained empirical likelihood, the approach establishes an information-theoretically interpretable and computationally tractable framework. The resulting inverse transformation serves as an invertible surrogate model, enhancing both the accuracy and efficiency of parameter estimation while enabling quantification of model bias and sensitivity analysis.

This work addresses the unreliability of deep learning models in high-stakes scenarios due to overconfidence, a limitation inadequately mitigated by existing Bayesian approaches that require multiple forward passes at test time and incur substantial computational overhead. To overcome this, the authors propose Calibrated Variance Propagation (CVP), an efficient method that estimates predictive uncertainty in a single forward pass and is readily applicable to modern architectures such as Transformers and CNNs. CVP introduces a novel variance propagation mechanism through normalization layers, incorporates approximations for activation functions, and applies a lightweight posterior calibration step to absorb residual errors. Experiments demonstrate that CVP substantially improves prediction coverage: on NLVR2 and VQAv2, BEiT-3 achieves coverage of 14.6% at a 0.5% risk level, up from 8.2%, while ViLT improves from 2.6% to 10.8%.