Vibration-based structural health monitoring (SHM) of (large) infrastructure through operational modal analysis (OMA) is a commonly adopted strategy. This is typically a four-step process, comprising estimation, tracking, data normalization, and decision-making. These steps are essential to ensure structural modes are correctly identified, and results are normalized for environmental and operational variability (EOV). Other challenges, such as nonstructural modes in the OMA, for example, rotor harmonics in (offshore) wind turbines (OWTs), further complicate the process. Typically, these four steps are considered independently, making the method simple and robust, but rather limited in challenging applications, such as OWTs. Therefore, this study aims to combine tracking, data normalization, and decision-making through a single machine learning (ML) model. The presented SHM framework starts by identifying a “healthy” training dataset, representative of all relevant EOV, for all structural modes. Subsequently, operational and weather data are used for feature selection and a comparative analysis of ML models, leading to the selection of tree-based learners for natural frequency prediction. Uncertainty quantification (UQ) is introduced to identify out-of-distribution instances, crucial to guarantee low modeling error and ensure only high-fidelity structural modes are tracked. This study uses virtual ensembles for UQ through the variance between multiple truncated submodel predictions. Practical application to monopile-supported OWT data demonstrates the tracking abilities, separating structural modes from rotor dynamics. Control charts show improved decision-making compared to traditional reference-based methods. A synthetic dataset further confirms the approach’s robustness in identifying relevant natural frequency shifts. This study presents a comprehensive data-driven approach for vibration-based SHM.

We introduce a comprehensive data-driven framework aimed at enhancing the modeling of physical systems, employing inference techniques and machine-learning enhancements. As a demonstrative application, we pursue the modeling of cathodic electrophoretic deposition, commonly known as e-coating. Our approach illustrates a systematic procedure for enhancing physical models by identifying their limitations through inference on experimental data and introducing adaptable model enhancements to address these shortcomings. We begin by tackling the issue of model parameter identifiability, which reveals aspects of the model that require improvement. To address generalizability, we introduce modifications, which also enhance identifiability. However, these modifications do not fully capture essential experimental behaviors. To overcome this limitation, we incorporate interpretable yet flexible augmentations into the baseline model. These augmentations are parameterized by simple fully-connected neural networks, and we leverage machine-learning tools, particularly neural ordinary differential equations, to learn these augmentations. Our simulations demonstrate that the machine-learning-augmented model more accurately captures observed behaviors and improves predictive accuracy. Nevertheless, we contend that while the model updates offer superior performance and capture the relevant physics, we can reduce off-line computational costs by eliminating certain dynamics without compromising accuracy or interpretability in downstream predictions of quantities of interest, particularly film thickness predictions. The entire process outlined here provides a structured approach to leverage data-driven methods by helping us comprehend the root causes of model inaccuracies and by offering a principled method for enhancing model performance.

High-resolution simulations such as the ICOsahedral Non-hydrostatic Large-Eddy Model (ICON-LEM) can be used to understand the interactions among aerosols, clouds, and precipitation processes that currently represent the largest source of uncertainty involved in determining the radiative forcing of climate change. Nevertheless, due to the exceptionally high computing cost required, this simulation-based approach can only be employed for a short period within a limited area. Despite the potential of machine learning to alleviate this issue, the associated model and data uncertainties may impact its reliability. To address this, we developed a neural network (NN) model powered by evidential learning, which is easy to implement, to assess both data (aleatoric) and model (epistemic) uncertainties applied to satellite observation data. By differentiating whether uncertainties stem from data or the model, we can adapt our strategies accordingly. Our study focuses on estimating the autoconversion rates, a process in which small droplets (cloud droplets) collide and coalesce to become larger droplets (raindrops). This process is one of the key contributors to the precipitation formation of liquid clouds, crucial for a better understanding of cloud responses to anthropogenic aerosols and, subsequently, climate change. We demonstrate that incorporating evidential regression enhances the model’s credibility by accounting for uncertainties without compromising performance or requiring additional training or inference. Additionally, the uncertainty estimation shows good calibration and provides valuable insights for future enhancements, potentially encouraging more open discussions and exploration, especially in the field of atmospheric science.

Atmospheric models used for weather and climate prediction are traditionally formulated in a deterministic manner. In other words, given a particular state of the resolved scale variables, the most likely forcing from the subgrid scale processes is estimated and used to predict the evolution of the large-scale flow. However, the lack of scale separation in the atmosphere means that this approach is a large source of error in forecasts. Over recent years, an alternative paradigm has developed: the use of stochastic techniques to characterize uncertainty in small-scale processes. These techniques are now widely used across weather, subseasonal, seasonal, and climate timescales. In parallel, recent years have also seen significant progress in replacing parametrization schemes using machine learning (ML). This has the potential to both speed up and improve our numerical models. However, the focus to date has largely been on deterministic approaches. In this position paper, we bring together these two key developments and discuss the potential for data-driven approaches for stochastic parametrization. We highlight early studies in this area and draw attention to the novel challenges that remain.

In this rejoinder we discuss the following aspects of our approach to model discrepancy: the interpretations of the two populations and adventitious error, the choice of inverse Wishart distribution, the perceived danger of justifying a model with bad fit, the relationship among our new approach, Chen’s (J R Stat Soc Ser B, 41:235–248, 1979) approach and the existing RMSEA-based approach, and the Pitman drift assumption.

Wu and Browne (Psychometrika, 79, 2015) have proposed an innovative approach to modeling discrepancy between a covariance structure model and the population that the model is intended to represent. Their contribution is related to ongoing developments in the field of Uncertainty Quantification (UQ) on modeling and quantifying effects of model discrepancy. We provide an overview of basic principles of UQ and some relevant developments and we examine the Wu–Browne work in that context. We view the Wu–Browne contribution as a seminal development providing a foundation for further work on the critical problem of model discrepancy in statistical modeling in psychological research.

Carefully designing blade geometric parameters is necessary as they determine the aerodynamic performance of a rotor. However, manufacturing inaccuracies cause the blade geometric parameters to deviate randomly from the ideal design. Therefore, it is essential to quantify uncertainty and analyse the sensitivity of the blade geometric deviations on the compressor performance. This work considers a subsonic compressor rotor stage and examines samples with different geometry features using three-dimensional Reynolds-averaged Navier-Stokes simulations. A method to combine Halton sequence and non-intrusive polynomial chaos is adopted to perform the uncertainty quantitative (UQ) analysis. The Sobol’ index and Spearman correlation coefficient help analyse the sensitivity and correlation between the compressor performance and blade geometric deviations, respectively. The results show that the fluctuation amplitude of the compressor performance decreases for lower mass flow rates, and the sensitivity of the compressor performance to the blade geometrical parameters varies with the working conditions. The effects of various blade geometric deviations on the compressor performance are independent and linearly superimposed, and the combined effects of different geometric deviations on the compressor performance are small.

Space-borne passive microwave (PMW) data provide rich information on atmospheric state, including cloud structure and underlying surface properties. However, PMW data are sparse and limited due to low Earth orbit collection, resulting in coarse Earth system sampling. This study demonstrates that Bayesian deep learning (BDL) is a promising technique for predicting synthetic microwave (MW) data and its uncertainties from more ubiquitously available geostationary infrared observations. Our BDL models decompose predicted uncertainty into aleatoric (irreducible) and epistemic (reducible) components, providing insights into uncertainty origin and guiding model improvement. Low and high aleatoric uncertainty values are characteristic of clear sky and cloudy regions, respectively, suggesting that expanding the input feature vector to allow richer information content could improve model performance. The initially high average epistemic uncertainty metrics quantified by most models indicate that the training process would benefit from a greater data volume, leading to improved performance at most studied MW frequencies. Using quantified epistemic uncertainty to select the most useful additional training data (a training dataset size increase of 3.6%), the study reduced the mean absolute error and root mean squared error by 1.74% and 1.38%, respectively. The broader impact of this study is the demonstration of how predicted epistemic uncertainty can be used to select targeted training data. This allows for the curation of smaller, more optimized training datasets and also allows for future active learning studies.

The finite element method (FEM) is widely used to simulate a variety of physics phenomena. Approaches that integrate FEM with neural networks (NNs) are typically leveraged as an alternative to conducting expensive FEM simulations in order to reduce the computational cost without significantly sacrificing accuracy. However, these methods can produce biased predictions that deviate from those obtained with FEM, since these hybrid FEM-NN approaches rely on approximations trained using physically relevant quantities. In this work, an uncertainty estimation framework is introduced that leverages ensembles of Bayesian neural networks to produce diverse sets of predictions using a hybrid FEM-NN approach that approximates internal forces on a deforming solid body. The uncertainty estimator developed herein reliably infers upper bounds of bias/variance in the predictions for a wide range of interpolation and extrapolation cases using a three-element FEM-NN model of a bar undergoing plastic deformation. This proposed framework offers a powerful tool for assessing the reliability of physics-based surrogate models by establishing uncertainty estimates for predictions spanning a wide range of possible load cases.

Data-informed predictive maintenance planning largely relies on stochastic deterioration models. Monitoring information can be utilized to update sequentially the knowledge on model parameters. In this context, on-line (recursive) Bayesian filtering algorithms typically fail to properly quantify the full posterior uncertainty of time-invariant model parameters. Off-line (batch) algorithms are—in principle—better suited for the uncertainty quantification task, yet they are computationally prohibitive in sequential settings. In this work, we adapt and investigate selected Bayesian filters for parameter estimation: an on-line particle filter, an on-line iterated batch importance sampling filter, which performs Markov Chain Monte Carlo (MCMC) move steps, and an off-line MCMC-based sequential Monte Carlo filter. A Gaussian mixture model approximates the posterior distribution within the resampling process in all three filters. Two numerical examples provide the basis for a comparative assessment. The first example considers a low-dimensional, nonlinear, non-Gaussian probabilistic fatigue crack growth model that is updated with sequential monitoring measurements. The second high-dimensional, linear, Gaussian example employs a random field to model corrosion deterioration across a beam, which is updated with sequential sensor measurements. The numerical investigations provide insights into the performance of off-line and on-line filters in terms of the accuracy of posterior estimates and the computational cost, when applied to problems of different nature, increasing dimensionality and varying sensor information amount. Importantly, they show that a tailored implementation of the on-line particle filter proves competitive with the computationally demanding MCMC-based filters. Suggestions on the choice of the appropriate method in function of problem characteristics are provided.

Uncertainty quantification (UQ) plays a crucial role in data assimilation (DA) since it impacts both the quality of the reconstruction and near-future forecast. However, traditional UQ approaches are often limited in their ability to handle complex datasets and may have a large computational cost. In this paper, we present a new ensemble-based approach to extend the 4DVarNet framework, an end-to-end deep learning scheme backboned on variational DA used to estimate the mean of the state along a given DA window. We use conditional 4DVarNet simulations compliant with the available observations to estimate the 4DVarNet probability density function. Our approach enables to combine both the efficiency of 4DVarNet in terms of computational cost and validation performance with a fast and memory-saving Monte-Carlo based post-processing of the reconstruction, leading to the so-called En4DVarNet estimation of the state pdf. We demonstrate our approach in a case study involving the sea surface height: 4DVarNet is pretrained on an idealized Observation System Simulation Experiment (OSSE), then used on real-world dataset (OSE). The sampling of independent realizations of the state is made among the catalogue of model-based data used during training. To illustrate our approach, we use a nadir altimeter constellation in January 2017 and show how the uncertainties retrieved by combining 4DVarNet with the statistical properties of the training dataset lead to a relevant information providing in most cases a confidence interval compliant with the Cryosat-2 nadir alongtrack dataset kept for validation.

While finite element (FE) modeling is widely used for ultimate strength assessments of structural systems, incorporating complex distortions and imperfections into FE models remains a challenge. Conventional methods typically rely on assumptions about the periodicity of distortions through spectral or modal methods. However, these approaches are not viable under the many realistic scenarios where these assumptions are invalid. Research efforts have consistently demonstrated the ability of point cloud data, generated through laser scanning or photogrammetry-based methods, to accurately capture structural deformations at the millimeter scale. This enables the updating of numerical models to capture the exact structural configuration and initial imperfections without the need for unrealistic assumptions. This research article investigates the use of point cloud data for updating the initial distortions in a FE model of a stiffened ship deck panel, for the purposes of ultimate strength estimation. The presented approach has the additional benefit of being able to explicitly account for measurement uncertainty in the analysis. Calculations using the updated FE models are compared against ground truth test data as well as FE models updated using standard spectral methods. The results demonstrate strength estimation that is comparable to existing approaches, with the additional advantages of uncertainty quantification and applicability to a wider range of application scenarios.

A simple method for adding uncertainty to neural network regression tasks in earth science via estimation of a general probability distribution is described. Specifically, we highlight the sinh-arcsinh-normal distributions as particularly well suited for neural network uncertainty estimation. The methodology supports estimation of heteroscedastic, asymmetric uncertainties by a simple modification of the network output and loss function. Method performance is demonstrated by predicting tropical cyclone intensity forecast uncertainty and by comparing two other common methods for neural network uncertainty quantification (i.e., Bayesian neural networks and Monte Carlo dropout). The simple approach described here is intuitive and applicable when no prior exists and one just wishes to parameterize the output and its uncertainty according to some previously defined family of distributions. The authors believe it will become a powerful, go-to method moving forward.

Understanding the subsurface is crucial in building a sustainable future, particularly for urban centers. Importantly, the thermal effects that anthropogenic infrastructure, such as buildings, tunnels, and ground heat exchangers, can have on this shared resource need to be well understood to avoid issues, such as overheating the ground, and to identify opportunities, such as extracting and utilizing excess heat. However, obtaining data for the subsurface can be costly, typically requiring the drilling of boreholes. Bayesian statistical methodologies can be used towards overcoming this, by inferring information about the ground by combining field data and numerical modeling, while quantifying associated uncertainties. This work utilizes data obtained in the city of Cardiff, UK, to evaluate the applicability of a Bayesian calibration (using GP surrogates) approach to measured data and associated challenges (previously not tested) and to obtain insights on the subsurface of the area. The importance of the data set size is analyzed, showing that more data are required in realistic (field data), compared to controlled conditions (numerically-generated data), highlighting the importance of identifying data points that contain the most information. Heterogeneity of the ground (i.e., input parameters), which can be particularly prominent in large-scale subsurface domains, is also investigated, showing that the calibration methodology can still yield reasonably accurate results under heterogeneous conditions. Finally, the impact of considering uncertainty in subsurface properties is demonstrated in an existing shallow geothermal system in the area, showing a higher than utilized ground capacity, and the potential for a larger scale system given sufficient demand.

As a Bayesian approach to fitting motorway traffic flow models remains rare in the literature, we empirically explore the sampling challenges this approach offers which have to do with the strong correlations and multimodality of the posterior distribution. In particular, we provide a unified statistical model to estimate using motorway data both boundary conditions and fundamental diagram parameters in a motorway traffic flow model due to Lighthill, Whitham, and Richards known as LWR. This allows us to provide a traffic flow density estimation method that is shown to be superior to two methods found in the traffic flow literature. To sample from this challenging posterior distribution, we use a state-of-the-art gradient-free function space sampler augmented with parallel tempering.

This paper develops theoretical foundations for extending Gauss–Hermite quadrature to robust design with computer experiments. When the proposed method is applied with m noise variables, the method requires 4m + 1 function evaluations. For situations in which the polynomial response is separable, this paper proves that the method gives exact transmitted variance if the response is a fourth-order separable polynomial response. It is also proven that the relative error mean and variance of the method decrease with the dimensionality m if the response is separable. To further assess the proposed method, a probability model based on the effect hierarchy principle is used to generate sets of polynomial response functions. For typical populations of problems, it is shown that the proposed method has less than 5% error in 90% of cases. Simulations of five engineering systems were developed and, given parametric alternatives within each case study, a total of 12 case studies were conducted. A comparison is made between the cumulative density function for the hierarchical probability models and a corresponding distribution function for case studies. The data from the case-based evaluations are generally consistent with the results from the model-based evaluation.

The Lee–Carter model has become a benchmark in stochastic mortality modeling. However, its forecasting performance can be significantly improved upon by modern machine learning techniques. We propose a convolutional neural network (NN) architecture for mortality rate forecasting, empirically compare this model as well as other NN models to the Lee–Carter model and find that lower forecast errors are achievable for many countries in the Human Mortality Database. We provide details on the errors and forecasts of our model to make it more understandable and, thus, more trustworthy. As NN by default only yield point estimates, previous works applying them to mortality modeling have not investigated prediction uncertainty. We address this gap in the literature by implementing a bootstrapping-based technique and demonstrate that it yields highly reliable prediction intervals for our NN model.