Deep ensembles (DE) have emerged as a powerful approach for quantifying predictive uncertainty and distinguishing its aleatoric and epistemic components, thereby enhancing model robustness and reliability. However, their high computational and memory costs during inference pose significant challenges for wide practical deployment. To overcome this issue, we propose credal ensemble distillation (CED), a novel framework that compresses a DE into a single model, CREDIT, for classification tasks. Instead of a single softmax probability distribution, CREDIT predicts class-wise probability intervals that define a credal set, a convex set of probability distributions, for uncertainty quantification. Empirical results on out-of-distribution detection benchmarks demonstrate that CED achieves superior or comparable uncertainty estimation compared to several existing baselines, while substantially reducing inference overhead compared to DE.

Epistemic uncertainty in neural networks is commonly modeled using two second-order paradigms: distribution-based representations, which rely on posterior parameter distributions, and set-based representations based on credal sets (convex sets of probability distributions). These frameworks are often regarded as fundamentally non-comparable due to differing semantics, assumptions, and evaluation practices, leaving their relative merits unclear. Empirical comparisons are further confounded by variations in the underlying predictive models. To clarify this issue, we present a controlled comparative study enabling principled, like-for-like evaluation of the two paradigms. Both representations are constructed from the same finite collection of predictive distributions generated by a shared neural network, isolating representational effects from predictive accuracy. Our study evaluates each representation through the lens of 3 uncertainty measures across 8 benchmarks, including selective prediction and out-of-distribution detection, spanning 6 underlying predictive models and 10 independent runs per configuration. Our results show that meaningful comparison between these seemingly non-comparable frameworks is both feasible and informative, providing insights into how second-order representation choices impact practical uncertainty-aware performance.

Solving PDE-governed physical problems is computationally expensive, limiting the availability of high-fidelity (HF) data for training neural operators, which typically require large datasets. Multi-fidelity learning addresses this by combining abundant low-fidelity (LF) data with limited HF samples. To evaluate several multi-fidelity neural operator strategies, including two-step, residual, intermediate, and transfer learning, we conduct a systematic assessment across four PDE test cases: two standard grid-based problems, a modified Darcy flow with LF-HF governing equation discrepancies, and an unsteady smoke inflow with temporal dynamics. Most prior studies focus on grid-based LF-HF differences, but real-world discrepancies, such as those between RANS and LES models in CFD, can be more complex. To capture these more challenging scenarios, we introduce two new test cases specifically designed to emulate nontrivial LF-HF mismatch beyond simple grid-resolution differences, enabling a more realistic assessment of multi-fidelity strategies under substantial model discrepancies. We find that methods directly feeding LF predictions into HF models are prone to error amplification under large LF-HF discrepancies, particularly in autoregressive settings. In contrast, transfer learning, which uses LF-trained weights to initialise the HF model, consistently improves accuracy by providing a robust prior refined with HF data. These results highlight the importance of establishing transfer learning as a reliable approach for efficient multi-fidelity PDE surrogate modelling.

Effectively estimating the uncertainty attached to neural network predictions thus becomes essential to improve robustness, reliability, and trustworthiness. This paper provides an overview of various methodologies for representing, quantifying, and distinguishing two major types of uncertainties (namely, ‘aleatoric’ and ‘epistemic’ uncertainty) in neural networks. The review covers classical probabilistic techniques such as Bayesian neural networks and deep ensembles, methods from generalized probability that leverage uncertainty representations such as Dirichlet distributions, belief functions, random sets, probability intervals, and credal sets, among others. Additionally, interval-based approaches employing interval models are also examined. We discuss the strengths and limitations of various methodologies and identify promising research directions for potential future exploration.

Credal predictors are epistemic-uncertainty-aware models that produce a convex set of probabilistic predictions. They provide a principled framework for quantifying predictive epistemic uncertainty (EU) and have been shown to improve model robustness across a range of settings. However, most state-of-the-art (SOTA) methods primarily define EU as disagreement induced by random training initializations, which mainly reflects sensitivity to optimization randomness rather than uncertainty from more substantive sources. In response, we formulate EU as disagreement between models trained under different degrees of relaxation of the i.i.d. assumption between the training and test distributions. Building on this idea, we propose , which learns an ensemble of plausible models via distributionally robust optimization. As a result, CreDRO captures EU arising not only from training randomness but also from informative disagreement due to potential train–test distribution shifts. Empirically, CreDRO consistently outperforms SOTA credal approaches on downstream tasks, including out-of-distribution detection on extensive benchmarks and selective classification in medical settings.

In engineering applications, uncertainty propagation refers to the systematic characterisation of a system output under uncertain inputs. Specifically for interval uncertainty, the objective is to determine the lower and upper bounds of the output, given interval-valued inputs. Such uncertainty propagation plays a crucial role in engineering tasks, including robust design optimisation and reliability analysis, where accurate characterisation of uncertainty is essential for safe and reliable decision-making. However, standard interval propagation requires solving optimisation problems that can be computationally expensive, particularly when dealing with complex physical systems. To address this challenge, surrogate models have been developed to enable efficient interval propagation. Although surrogate models are computationally more efficient, standard surrogate-based approaches typically only replace the evaluator function within the optimisation loop, which still requires a large number of inference calls. Therefore, we propose to directly estimate the output interval by reframing the problem as an interval-valued regression task. In this work, we present a comprehensive study of strategies for direct interval propagation using NN-based surrogate models, including standard multilayer perceptrons (MLPs) and deep operator networks (DeepONet). We investigate and compare three distinct approaches: (i) naive interval propagation through standard architectures, (ii) bound propagation techniques such as Interval Bound Propagation (IBP) and CROWN, and (iii) interval neural networks (INNs) with interval weights. Our results demonstrate that these methods are significantly more efficient compared to traditional optimisation-based interval propagation and are able to provide accurate interval estimates. We also discuss the limitations and open challenges associated with implementing interval-based propagation in practice.

In Hydrogen Fuel Cell Electric Vehicles, hydrogen is stored as compressed gas in tanks made from carbon fiber composite. Tanks are designed to withstand 2.25 times the nominal working pressure. If low variability in tank strength can be demonstrated, this margin of safety could be reduced. This requires accurate prediction of the strength of the vessel and quantification of its uncertainty. In this work, several composite pressure vessels are manufactured using filament winding, and the parameters that control the winding and curing recorded as time signals. Process parameters include fiber tension, winding speed, liner pressure, fiber volume fraction, winding time and curing pressure. The vessels are tested for burst pressure. Several configurations are defined, where each manufacturing parameter is changed. The recorded manufacturing data are fed to machine learning(ML) algorithms and trained to predict the vessel’s burst pressure. These ML algorithms include neural networks, Gaussian process regression GPR, Kernel Principal Component Analysis-Lasso (kPCA-Lasso), and an Ensemble regressor. The KPCA-Lasso and GPR models show good correlation. The Ensemble regressor yields better results by combining the KPCA-Lasso and GPR models. The study demonstrates the potential of ML algorithms in predicting the burst pressure of carbon fiber vessels from the monitored manufacturing data.

Uncertainty is an intrinsic aspect of many scientific experiments and stochastic simulations. In these settings, observation noise can vary across the input space, leading to heteroscedasticity. Heteroscedastic Gaussian process (HGP) regression has been widely used as a surrogate model in various applications due to its capability to handle noise-varying problems. However, the computational cost of HGP models remains high. In this paper, we propose a novel approach to reduce this computational burden. Our method follows a post-modelling learning strategy akin to the most likely heteroscedastic Gaussian process (MLHGP) algorithm. Unlike the MLHGP and its variations, which require multiple GP models, our proposed methodology only requires one GP model to fit the main function and uses the trained kernel parameters to estimate the noise level via kernel smoothing regression. Our proposed method is able to reduce the computational complexity from O(2.N3) to O(.N3 + .N2) and cut the requirements of training two models to only one. This approach translates to roughly 2x speed-up during training in our test cases. We also found that the proposed method is able to achieve better and more stable performance metrics. Additionally, in our Bayesian optimisation test case, the result shows that the proposed method outperforms MLHGP in case of a low number of initial observations and remains competitive in the medium and high initial observation settings, all while being faster in every case.

Aerospace design relies on computational models to manage the scarcity of test data and to quantify various input parameters. Still, the reliability of these estimates with respect to their robustness against unseen data remains uncertain. To address this, the present study introduces certified interval model updating using scenario optimization. This approach determines a lower bound for this reliability against missing data without having to characterize the underlying distribution of the data. Additionally, it allows calculating the required amount of data to achieve a predefined lower bound reliability before data gathering. Furthermore, scenario optimization allows for an efficient forward propagation of the interval uncertainty. The method's efficacy is illustrated using a simple four-degree-of-freedom mass-spring system, and its performance is benchmarked against other interval model-updating techniques using the DLR AIRMOD case.

Inverse uncertainty quantification commonly uses the well established Bayesian framework.Recently, alternative interval methodologies have been introduced. However, in theircurrent state of the art implementation, both techniques suffer from a large and usuallyunpredictable computational effort. Thus, both techniques are not applicable in a real-timecontext. To achieve a low-cost, real-time solution to this inverse problem, we introduce adeep-learning framework consisting of unsupervised auto-encoders and a shallow neuralnetwork. This framework is trained by means of a numerically generated dataset thatcaptures typical relations between the model parameters and selected measured systemresponses. The performance and efficacy of the technique is illustrated using two distinctcase studies. The first case involves the DLR AIRMOD, a benchmark case that has served asreference case for the inverse uncertainty quantification problem. The results demonstratethat the achieved accuracy is on par with the existing interval method found in literature,while requiring only a fraction of its computational resources. The second case studyexamines a resistance pressure welding process, which is known to require extremely fastmonitoring and control due to the high process throughput. Based on the proposed method,and with only a limited selection of simulated responses of the process, it is possible toidentify the interval uncertainty of the crucial parameters of the process. The computationalcost in this case makes it possible for an inverse uncertainty quantification in a real-timesetting.[DOI: 10.1115/1.4066619]

Effective uncertainty estimation is becoming increasingly attractive for enhancing the reliability of neural networks. This work presents a novel approach, termed Credal-Set Interval Neural Networks (CreINNs), for classification. CreINNs retain the fundamental structure of traditional Interval Neural Networks, capturing weight uncertainty through deterministic intervals. CreINNs are designed to predict an upper and a lower probability bound for each class, rather than a single probability value. The probability intervals can define a credal set, facilitating estimating different types of uncertainties associated with predictions. Experiments on standard multiclass and binary classification tasks demonstrate that the proposed CreINNs can achieve superior or comparable quality of uncertainty estimation compared to variational Bayesian Neural Networks (BNNs) and Deep Ensembles. Furthermore, CreINNs significantly reduce the computational complexity of variational BNNs during inference. Moreover, the effective uncertainty quantification of CreINNs is also verified when the input data are intervals.

This paper presents an innovative approach, called credal wrapper, to formulating a credal set representation of model averaging for Bayesian neural networks (BNNs) and deep ensembles (DEs), capable of improving uncertainty estimation in classification tasks. Given a finite collection of single predictive distributions derived from BNNs or DEs, the proposed credal wrapper approach extracts an upper and a lower probability bound per class, acknowledging the epistemic uncertainty due to the availability of a limited amount of distributions. Such probability intervals over classes can be mapped on a convex set of probabilities (a credal set) from which, in turn, a unique prediction can be obtained using a transformation called intersection probability transformation. In this article, we conduct extensive experiments on several out-of-distribution (OOD) detection benchmarks, encompassing various dataset pairs (CIFAR10/100 vs SVHN/Tiny-ImageNet, CIFAR10 vs CIFAR10-C, CIFAR100 vs CIFAR100-C and ImageNet vs ImageNet-O) and using different network architectures (such as VGG16, ResNet-18/50, EfficientNet B2, and ViT Base). Compared to the BNN and DE baselines, the proposed credal wrapper method exhibits superior performance in uncertainty estimation and achieves a lower expected calibration error on corrupted data.

Decision Focused Learning has emerged as a critical paradigm for integrating machine learning with downstream optimisation. Despite its promise, existing methodologies predominantly rely on probabilistic models and focus narrowly on task objectives, overlooking the nuanced challenges posed by epistemic uncertainty, non-probabilistic modelling approaches, and the integration of uncertainty into optimisation constraints. This paper bridges these gaps by introducing innovative frameworks: (i) a non-probabilistic lens for epistemic uncertainty representation, leveraging intervals (the least informative uncertainty model), Contamination (hybrid model), and probability boxes (the most informative uncertainty model); (ii) methodologies to incorporate uncertainty into constraints, expanding Decision-Focused Learning's utility in constrained environments; (iii) the adoption of Imprecise Decision Theory for ambiguity-rich decision-making contexts; and (iv) strategies for addressing sparse data challenges. Empirical evaluations on benchmark optimisation problems demonstrate the efficacy of these approaches in improving decision quality and robustness and dealing with said gaps.

Accurate uncertainty quantification (UQ) in digital image correlation (DIC) deformations is essential for quantitative DIC-based finite element (FE) model validation. DIC UQ is well-studied in the current literature, both from a theoretical as well as experimental point-of-view, but rarely from the model validation perspective. Moreover, the DIC uncertainties are usually considered as spatial averages over the whole field of view while local contrast variations generally lead to spatially-varying noise floors. This paper investigates how DIC UQ should be performed when validating FE models. UQ was performed using experimental stationary images of a test sample. Spatial maps of point-wise temporal standard deviation (noise) and mean (bias) were constructed to be used in the model validation process. The effectiveness of reference image averaging at reducing bias and noise was also studied. Specular reflection ('hotspots') was given special attention, an important additional source of uncertainty not simulated by the Digital Twin (DT) used to perform the validation. As expected, image noise was found to be the most dominant source of DIC uncertainty. The spatially-random noise on the reference stationary image was found to be responsible for the temporal bias of the displacement distribution, as the copy of noise from that initial image affects all displacement maps since this image is used for all displacement maps. Spatially-random noise on the deformed stationary images was found to be responsible for the temporal standard deviation (noise). Both temporal noise and bias were found to be comparable in magnitude, highlighting the necessity for a spatially heterogeneous model validation criterion that accounts for both. The impact of specular reflection was difficult to quantify and exhibits potential for significantly increasing DIC uncertainties. The use of polarized lights and polarizing filters can mitigate this issue but more work is needed to allow for a realistic error budget to be established for this. Heat haze (refraction from warm air flow between camera and object) and camera heating are additional effects that are difficult to error-budget for. Finally, the effect of stereo-DIC calibration errors needs to be studied further.

This paper introduces a formulation of the robust topology optimization problem that is tailored for designing fiber-reinforced composite structures with spatially varying principal mechanical properties. Specifically, a methodology is developed that incorporates the spatial variability in the engineering constants of the composite lamina into the concurrent topology (i.e., material distribution) and morphology (i.e., fiber orientation distribution) optimization problem for the minimization of the robust compliance function. The spatial variability in the mechanical properties of the lamina is modeled as a homogeneous random field within the design domain by means of the Karhunen-Loe´ve series expansion, and is thereafter intrusively propagated into the stochastic finite element analysis of the composite structure. To carry out the stochastic finite element analysis per iteration of the optimization cycle, the first-order perturbation method is utilized for approximating the current state variables of the physical system. The resulting robust topology and fiber orientation optimization problem is formulated step-by-step for the minimization of the robust compliance function. With the view of solving the optimization problem at hand by means of gradient-based solution algorithms, the first-order derivatives of the involved design functions w.r.t. the associated design variables are analytically derived. The present work concludes with a series of numerical examples, focusing on the benchmark academic case studies of the 2D cantilever and the half part of the Messerschmitt-Bölkow-Blohm beam, aiming to demonstrate the developed methodology as well as to explore the effect that different parameterization instances of the random field bear on the predicted topology and morphology of the beams.

Full-field optical experimental techniques such as Digital Image Correlation (DIC) enable measurement of information-rich heterogeneous strain states. The aim of Material Testing 2.0 is to capitalise on this and design experimental setups for efficient material characterisation using inverse identification techniques such as the Virtual Fields Method (VFM). In that case, however, a priori knowledge of the constitutive model is required, which sometimes is not a trivial task. This limitation can be overcome by identifying multiple constitutive models using the same DIC strain fields and comparing their performance. In this work, we present a methodology to evaluate model performance and compare different constitutive models based on quantitative metrics. DIC experiments were performed on an S-shaped high density polyethylene test sample under uniaxial load. The heterogeneous DIC strain fields were used in combination with the VFM to identify parameters of hyperelastic constitutive models with increasing complexity. Two deterministic metrics, the equilibrium gap indicator (EGI) and the reconstructed axial force ratio (RAFR), were defined and used to compare the performance of these constitutive models. The impact of spatial smoothing on DIC strain fields was studied by calculating the EGI and RAFR using a digital twin of the DIC experiments. EGI and RAFR metrics provide complementary information regarding the material behaviour, and both are necessary to make an informed choice. In this case study, hyperelastic constitutive models were found not to capture the material stiffness well in the small strain regime. A linear-elastic contribution exponentially decaying with the equivalent shear stress was added to the hyperelastic models and the corresponding model parameters identified using the VFM. The elastic-hyperelastic constitutive models were found to perform better than their purely hyperelastic counterparts.