Inverse Uncertainty Quantification Research Articles

Uncertainty Quantification of Microstructures: A Perspective on Forward and Inverse Problems for Mechanical Properties of Aerospace Materials

In this review, state‐of‐the‐art studies on the uncertainty quantification (UQ) of microstructures in aerospace materials is examined, addressing both forward and inverse problems. Initially, it introduces the types of uncertainties and UQ algorithms. In the review, the forward problem of uncertainty propagation in process–structure and structure–property relationships is then explored. Subsequently, the inverse UQ problem, also known as the design under uncertainty problem, is discussed focusing on structure–process and property–structure linkages. Herein, the review concludes by identifying gaps in the current literature and suggesting key areas for future research, including multiscale topology optimization under uncertainty, implementing physics‐informed neural networks to UQ problems, investigating the effects of uncertainty on extreme mechanical behavior, reliability‐based design, and UQ in additive manufacturing.

A Data Driven Black Box Approach for the Inverse Quantification of Set-Theoretical Uncertainty

Abstract Inverse uncertainty quantification commonly uses the well established Bayesian framework. Recently, alternative interval methodologies have been introduced. However, in their current state of the art implementation, both techniques suffer from a large and usually unpredictable computational effort. Thus, both techniques are not applicable in a real-time context. To achieve a low-cost, real-time solution to this inverse problem, we introduce a deep-learning framework consisting of unsupervised auto-encoders and a shallow neural network. This framework is trained by means of a numerically generated dataset that captures typical relations between the model parameters and selected measured system responses. The performance and efficacy of the technique is illustrated using two distinct case studies. The first case involves the DLR AIRMOD, a benchmark case that has served as reference case for the inverse uncertainty quantification problem. The results demonstrate that 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 study examines a resistance pressure welding process, which is known to require extremely fast monitoring 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 to identify the interval uncertainty of the crucial parameters of the process. The computational cost in this case makes it possible for an inverse uncertainty quantification in a real-time setting.

Assessment of the Choked Flow Model of RELAP5 for Application of Inverse Quantification Methods

In the framework of deterministic safety analysis, best estimate plus uncertainty methodologies can provide a more informative detailed analysis of transients than conservative approaches. However, one important step in the application of such methodologies is derivation of uncertainty of the physical models. Probability density functions of the physical model parameters are obtained by applying inverse uncertainty quantification (IUQ) methods to experimental data. The present work deals with evaluation of the experimental database and assessment of the simulation model, which are required steps prior to the application of IUQ methods. The U.S. Nuclear Regulatory Commission system code RELAP5 has been applied for simulation of three experimental databases on choked/critical flow: Sozzi-Sutherland, Super MobyDick, and Marviken Critical Flow Tests. First, the adequacy of the experimental data to the specific target domain is carried out, to then proceed to calibration of the input model parameters by combining the use of Gaussian Process metamodels and optimization schemes. The assessment of the simulation models is performed by evaluating independently accuracy, precision, and consistency through four statistical indicators, including a novel indicator to evaluate the consistency of the results. The final results indicate that the model is capable of reproducing the selected experimental database and overall average accuracy, precision, and consistency below the 10% threshold and can be used for application of IUQ methods.

Inverse uncertainty quantification for stochastic systems by resampling. Applications to modeling of alcohol consumption and infection by HIV

A random differential equation, or stochastic differential equation with parametric uncertainty, is a classical differential equation whose input values (coefficients, initial conditions, etc.) are random variables. Given data, the probability distributions of the input random parameters must be appropriately inferred, before proceeding to simulate the model’s output. This task is called inverse uncertainty quantification. In this paper, the goal is to study the applicability of the Bayesian bootstrap to draw inferences on the posterior distributions of the parameters, by resampling the residuals of the deterministic least-squares optimization with Dirichlet weights. The method is based on repeated deterministic calibrations. Thus, to alleviate the curse of dimensionality, the technique may be combined with the principle of maximum entropy for densities, when there are some parameters that are not optimized deterministically. For illustration of the methodology, two case studies on important health topics are conducted, with stochastic fitting to data. The first one, on past alcohol consumption in Spain, taking social contagion into account. The second one, on HIV evolution considering CD4+ T cells and viral load, with a patient in clinical follow-up. All these applied models are built from a compartmental viewpoint, with a randomized basic reproduction number that controls the long-term behavior of the system.

Full-field experiment-aided virtual modelling framework for inverse-based stochastic prediction of structures with elastoplasticity

Forecasting of stochastic ductile failures in fabrication and service stages are challenging tasks of advanced structures in practical engineering. Due to the prohibitive costs of repetitive experimental tests to quantify optimal failure-related parameters, numerous studies have turned to simulation-based uncertainty quantification. However, the credibility of these approaches is frequently doubted by the function-generated distribution of random data involved. Thus, an accurate assessment of the material parameters ascertains the appropriate estimation of uncertain parameters, which is essential for the risk/safety assessment of structures in different stages. In this paper, a full-field experiment-aided virtual modelling framework for inverse-based prediction of stochastic elastoplastic failure (EVMF-ISP) is proposed to deliver precise insights regarding the effective distribution range of mechanical parameters. To this end, all available experiment observations serve as the reference for assessing the prior and posterior probability density of the unknown parameters through the real-time inverse uncertainty quantification (UQ) module. The framework can be divided into three parts, where initially a pre-virtual model (PRVM) is formulated, and Bayesian inference is implemented to propagate the experiment observations backwards to ascertain the uncertain parameters. Then, an advanced multidimensional slice sampling method is developed to deal with the derived complex posterior probability density function (PDF) of mechanical parameters. In the end, a reliable stochastic elastoplastic analysis can be conducted with the revised uncertain samples and finalised with post-virtual models (POVMs) for the concerned structures. Such that, accurate and efficient determination of nonlinear response of structures can be directly predicted. The EVMF-ISP framework is logically presented and thoroughly illustrated with practical applications.

Development of higher order deterministic approaches of forward and inverse uncertainty quantification for nonlinear engineering systems

This paper introduces an innovative numerical scheme that may accurately quantify the parameter and response distributions with minimal computational costs, with specific applications to the numerical computation of multiphase flows under various fluid conditions. The probabilistic approach of uncertainty quantification, e.g., the Monte Carlo simulation, although known to be effective in estimating the parameter/response distribution for highly nonlinear problems, is usually not applicable to multi-physics and multi-scale integrated systems due to its high computational cost. The objective of this study is first, to reduce the computing demand for complex physical system calculations and, at the same time, to accurately predict actual probability distributions by developing a higher order deterministic approach to uncertainty quantification. To achieve this, instead of executing simulation codes multiple times to estimate the probability distribution, the first to fourth moments of the probability density function are derived directly by employing a second order Taylor expansion form of the responses along with Bayesian statistics. This characterizes the parameter and response distributions computed by inverse and forward uncertainty quantification based on their mean value, mode, covariance, skewness, and kurtosis. The parameter distribution is estimated by the numerically calculated higher moments of the parameters with a posteriori probability density function being derived by Bayesian inference. The parameter distribution is then mapped into the second order approximated response function to compute the characteristics of the response distribution through both analytical calculation and numerical approximation of the moments of the responses. The proposed approach is numerically demonstrated through simulations of two benchmark problems: MIT pressurizer and FEBA tests. The prediction of the responses is improved by inverse uncertainty quantification as numerical calculations with the a posteriori model parameters are in better agreement with the measured data over a wide range of transient periods. Additionally, the response distribution estimated by the proposed approach of forward uncertainty quantification accurately predicts the distribution computed by the Monte Carlo simulation.

Global Sensitivity Analysis for Segmented Inverse Uncertainty Quantification in the Safety Analysis of Nuclear Power Plants

Within the Best Estimate Plus Uncertainty framework for the safety analysis of Nuclear Power Plants, the quantification of the uncertainties affecting the Thermal-Hydraulics (T-H) codes used is crucial. For this, Inverse Uncertainty Quantification (IUQ) methodologies are being developed for determining the probability density functions of relevant T-H codes input parameters, based on experimental data from Separate Effect Tests (SETs) experimental facilities. In practice, IUQ is challenged by the large range of variability of the experimental data in terms of Initial and Boundary Conditions (ICs & BCs), because the experimental campaigns are designed to cover the widest possible domain of conditions with the smallest number of experiments, so that same or similar ICs and BCs are seldomly repeated. To address this issue, we propose to use global sensitivity analysis, to tailor the IUQ on specific sub-regions described by segmented ICs & BCs domains. The methodology proposed is exemplified on two SETs, namely Sozzi-Sutherland and Super Moby Dick, whose experimental databases have been made available in the ATRIUM (Application Tests for Realization of Inverse Uncertainty quantification and validation Methodologies in thermal hydraulics) project promoted by the OECD/NEA/CSNI. The results obtained are superior to those of traditional IUQ methodologies for models highly sensitive to ICs & BCs.

Reduction of the model prediction error in vibration-based SHM applications

The accuracy of an SHM scheme is partly controlled by the accuracy of the predictive models involved within the design process of the detector that associates sensor data with a corresponding health state. On the other hand, the accuracy of the predictive model, which is quantified by the error between predictions and observations, is partly controlled by its parameter magnitudes. This work provides a unified framework that allows for reducing the model prediction error by updating the model parameters in the light of experimental observations. The problem is cast in a probabilistic setting and tackled through inferential statistics. Forward and inverse uncertainty quantification approaches are employed in a sequential manner. The framework is able to yield point estimates in accordance to a Maximum Likelihood Estimator (MLE) and is also able to construct credible intervals through computational Bayesian updating. The former is treated by genetic algorithms whereas the later through Markov Chain Monte Carlo computational statistics. The framework is showcased on a vibration problem of the simplified benchmark case named as “Jim Beam”. The first ten modal parameters are considered as the discriminant features of interest . A surrogate of the finite element model of the considered specimen serves as the predictive model and the eigenfrequencies are extracted from a linear modal analysis. The parameters of inferential interest are the Young’s modulus and an artificial stiffness related parameter that is introduced so as to model the flexural rigidity of the specimen’s bolted connections. Experimental Modal Analysis and Operational Modal Analysis are the two approaches that were employed for experimentally identifying the modes and their corresponding frequencies. Results are provided from both the MLE and the Bayesian updating process and the effectiveness of each approach is discussed.

Functional PCA and deep neural networks-based Bayesian inverse uncertainty quantification with transient experimental data

This work focuses on developing an inverse uncertainty quantification (IUQ) process for time-dependent responses, using dimensionality reduction by functional principal component analysis (PCA) and deep neural network (DNN)-based surrogate models. The demonstration is based on the IUQ of TRACE physical model parameters using the FEBA benchmark transient experimental data on peak cladding temperatures during reflooding. The quantity-of-interest (QoI) is time-dependent peak cladding temperature (PCT) profiles. Conventional PCA can hardly represent the data precisely due to the sudden temperature drop at the time of quenching. As a result, a functional alignment method is used to separate the phase and amplitude information in the PCT profiles before conventional PCA is applied for dimensionality reduction. The resulting PC scores are then used to build DNN-based surrogate models to significantly reduce the computational cost in Markov Chain Monte Carlo sampling, while the code/interpolation uncertainty is accounted for using Bayesian neural networks. We compared four IUQ processes with different dimensionality reduction methods and surrogate models. The proposed approach has demonstrated the best performance in reducing the dimensionality of the transient PCT profiles. This approach also produces the posterior distributions of the physical model parameters with which the model predictions have the best agreement with the experimental data.

Simultaneous Inverse Design and Uncertainty Quantification for Frequency Selective Rasorber with Tunable and Switchable Abilities by Bayesian Deep Learning

Simultaneous Inverse Design and Uncertainty Quantification for Frequency Selective Rasorber with Tunable and Switchable Abilities by Bayesian Deep Learning

Scalable Inverse Uncertainty Quantification by Hierarchical Bayesian Modeling and Variational Inference

Inverse Uncertainty Quantification (IUQ) has gained increasing attention in the field of nuclear engineering, especially nuclear thermal-hydraulics (TH), where it serves as an important tool for quantifying the uncertainties in the physical model parameters (PMPs) while making the model predictions consistent with the experimental data. In this paper, we present an extension to an existing Bayesian inference-based IUQ methodology by employing a hierarchical Bayesian model and variational inference (VI), and apply this novel framework to a real-world nuclear TH scenario. The proposed approach leverages a hierarchical model to encapsulate group-level behaviors inherent to the PMPs, thereby mitigating existing challenges posed by the high variability of PMPs under diverse experimental conditions and the potential overfitting issues due to unknown model discrepancies or outliers. To accommodate computational scalability and efficiency, we utilize VI to enable the framework to be used in applications with a large number of variables or datasets. The efficacy of the proposed method is evaluated against a previous study where a No-U-Turn-Sampler was used in a Bayesian hierarchical model. We illustrate the performance comparisons of the proposed framework through a synthetic data example and an applied case in nuclear TH. Our findings reveal that the presented approach not only delivers accurate and efficient IUQ without the need for manual tuning, but also offers a promising way for scaling to larger, more complex nuclear TH experimental datasets.

Sparse-grids uncertainty quantification of part-scale additive manufacturing processes

The present paper aims at applying uncertainty quantification methodologies to process simulations of powder bed fusion of metal. In particular, for a part-scale thermomechanical model of an Inconel 625 super-alloy beam, we study the uncertainties of three process parameters, namely the activation temperature, the powder convection coefficient and the gas convection coefficient. First, we perform a variance-based global sensitivity analysis to study how each uncertain parameter contributes to the variability of the beam displacements. The results allow us to conclude that the gas convection coefficient has little impact and can therefore be fixed to a constant value for subsequent studies. Then, we conduct an inverse uncertainty quantification analysis, based on a Bayesian approach on synthetic displacement data, to quantify the uncertainties of the two remaining parameters, namely the activation temperature and the powder convection coefficient. Finally, we use the results of the inverse uncertainty quantification analysis to perform a data-informed forward uncertainty quantification analysis of the residual strains. Crucially, we make use of surrogate models based on sparse grids to keep to a minimum the computational burden of every step of the uncertainty quantification analysis. The proposed uncertainty quantification workflow allows us to substantially ease the typical trial-and-error approach used to calibrate power bed fusion part-scale models, and to greatly reduce uncertainties on the numerical prediction of the residual strains. In particular, we demonstrate the possibility of using displacement measurements to obtain a data-informed probability density function of the residual strains, a quantity much more complex to measure than displacements.

Inverse uncertainty quantification of a mechanical model of arterial tissue with surrogate modelling

Disorders of coronary arteries lead to severe health problems such as atherosclerosis, angina, heart attack and even death. Considering the clinical significance of coronary arteries, an efficient computational model is a vital step towards tissue engineering, enhancing the research of coronary diseases and developing medical treatment and interventional tools. In this work, we applied inverse uncertainty quantification to a microscale agent-based arterial tissue model, a component of a three-dimensional multiscale in-stent restenosis model. Inverse uncertainty quantification was performed to calibrate the arterial tissue model to achieve a mechanical response in line with tissue experimental data. Bayesian calibration with a bias term correction was applied to reduce the uncertainty of unknown polynomial coefficients of the attractive force function and achieve agreement with the mechanical behaviour of arterial tissue based on the uniaxial strain tests. Due to the high computational costs of the model, a surrogate model based on the Gaussian process was developed to ensure the feasibility of the computations.

Fast uncertainty reduction of chemical kinetic models with complex spaces using hybrid response-surface networks

Response-surface (RS) surrogate approaches permit efficient inverse uncertainty quantification (UQ) of combustion kinetic models, wherein the uncertainty of reaction rates is reduced from observed targets. For kinetic models with parameters characterized by large uncertainty factors, strong nonlinearities, and reaction couplings (e.g., reduced mechanisms of real fuels; such models are referred to be “complex” in this work), the global RS is difficult to approximate, precluding conventional surrogate approaches. This paper proposes a framework that is extendable to such systems, termed Hybrid Response Surface Networks followed by a Stochastic Gradient Descent Ensemble (HRSN-SGDE). This technique focuses on mapping the local RS of just the uncertain spaces in the vicinity of the observed target, referred to as the rate target subspace. Two neural network surrogates are considered: a classifier that predicts the probability of data residing in the rate target subspace and a local RS surrogate which maps the RS of this subspace. A hybrid surrogate loss function is then defined using these surrogates to optimize uncertain rates repeatedly to get an ensemble of solutions representing the constrained rate space. HRSN-SGDE is demonstrated on a complex jet fuel model developed using the hybrid chemistry (HyChem) approach with a low temperature chemistry sub-model using a series of ignition delay times as targets. Results show that the method's local RS objective enables efficient and accurate construction of the surrogates through active learning-based sampling. Also, the unique formulation of the surrogate loss function enables optimization that is robust to suboptimal local minima and faster than evolutionary algorithms by several orders of magnitude. It is shown that HRSN-SGDE method is highly efficacious and suitable to conducting inverse UQ on such complex kinetic models.

An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

AbstractThis article introduces an ‐adaptive multi‐element stochastic collocation method, which additionally allows to re‐use existing model evaluations during either ‐ or ‐refinement. The collocation method is based on weighted Leja nodes. After ‐refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub‐element in a hierarchical manner. For ‐refinement, the local polynomial approximations are based on total‐degree or dimension‐adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non‐smooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.

Multi-Output Gaussian Processes for Inverse Uncertainty Quantification in Neutron Noise Analysis

In a fissile material, the inherent multiplicity of neutrons born through induced fissions leads to correlations in their detection statistics. The correlations between neutrons can be used to trace back some characteristics of the fissile material. This technique, known as neutron noise analysis, has applications in nuclear safeguards or waste identification. It provides a nondestructive examination method for an unknown fissile material. This is an example of an inverse problem where the cause is inferred from observations of the consequences. However, neutron correlation measurements are often noisy because of the stochastic nature of the underlying processes. This makes the resolution of the inverse problem more complex since the measurements are strongly dependent on the material characteristics. A minor change in the material properties can lead to very different outputs. Such an inverse problem is said to be ill posed. For an ill-posed inverse problem, the inverse uncertainty quantification is crucial. Indeed, seemingly low noise in the data can lead to strong uncertainties in the estimation of the material properties. Moreover, the analytical framework commonly used to describe neutron correlations relies on strong physical assumptions, and is thus inherently biased. This paper addresses dual goals. First, surrogate models are used to improve neutron correlation predictions and quantify the errors on those predictions. Then the inverse uncertainty quantification is performed to include the impact of measurement error alongside the residual model bias.

Comparison of inverse uncertainty quantification methods for critical flow test

A problem of epistemic uncertainties introduced by code input parameters that cannot be estimated other than by user expertise has existed in nuclear reactor systems analyses since first uncertainty analyses were conducted. Inverse Uncertainty Quantification (IUQ) methods aim at providing an estimation of the distributions of such parameters. This paper compares and assesses two approaches to quantify the uncertainty of critical flow model parameters available in the TRACE code. First novel approach is based on a machine-learning algorithm, using a Random Forest classifier to assign the results of calculations to one of the defined classes of prediction accuracy with respect to experimental data. The second approach is based on Markov Chain Monte Carlo sampling and Bayesian inference. Both methods allows to assign probability distribution functions to TRACE internal variables, which is a goal for IUQ methods.Both approaches were used to estimate uncertainties for four critical flow parameters available in the TRACE code based on the Marviken critical flow experiment. In case of two parameters both methods produced similar results of the probability density functions. Based on the results of machine learning two additional parameters were deemed noninfluential, while for the second method such definitive assessment could not be made.

Bayesian Hierarchical Modelling for Uncertainty Quantification in Operational Thermal Resistance of LED Systems

Remaining useful life (RUL) prediction is central to prognostics and reliability assessment of light-emitting diode (LED) systems. Their unknown long-term service life remaining when subject to specific operating conditions is affected by various sources of uncertainty stemming from production of individual system components, application of the whole system, measurement and operation. To enhance the reliability of model-based predictions, it is essential to account for all of these uncertainties in a systematic manner. This paper proposes a Bayesian hierarchical modelling framework for inverse uncertainty quantification (UQ) in LED operation under thermal loading. The main focus is on the LED systems’ operational thermal resistances, which are subject to system and application variability. Posterior inference is based on a Markov chain Monte Carlo (MCMC) sampling scheme using the Metropolis–Hastings (MH) algorithm. Performance of the method is investigated for simulated data, which allow to focus on different UQ aspects in applications. Findings from an application scenario in which the impact of disregarded uncertainty on RUL prediction is discussed highlight the need for a comprehensive UQ to allow for reliable predictions.

Extension of the CIRCE methodology to improve the Inverse Uncertainty Quantification of several combined thermal-hydraulic models

The Inverse Uncertainty Quantification (IUQ) of physical correlations used in thermal-hydraulic system codes is a crucial issue in the BEPU (Best Estimate Plus Uncertainty) framework for the licensing and safety analyses of nuclear reactors. The CIRCE methodology is one of the most widely applied IUQ methods. It estimates the (log-)normal probability distribution of a multiplicative factor applied to the reference closure model used in thermal-hydraulic computer codes. CIRCE can jointly estimate the uncertainty of several physical models. However, in some particular cases, a problem of identifiability may occur resulting in less accuracy and precision of the estimated uncertainties.For this reason, in this work, the methodology is improved under the name of CIRCE 2-Steps. This new approach aims at improving the quantification of the model uncertainties when experiments with more than one physical phenomena interacting simultaneously on the output of interest has to be used. In a first step, the uncertainties of the least influential models are evaluated with classic CIRCE methodology using experiments for which only one physical phenomenon is dominant. Then, these uncertainties are properly taken into account when quantifying the remaining uncertainties on the experimental database. This approach is proven to provide more accurate and precise results on several analytical test cases where the uncertainties of two models has to be jointly estimated. The study of the impact of the relative importance of the two models shows that the biggest gain in precision is obtained when the hierarchy of models is such that one model is significantly more influential than the other.Finally, the newly developed methodology is applied to estimate the uncertainties of two condensation heat transfer correlations in the cold leg of a pressurised nuclear reactor.

LaSDI: Parametric Latent Space Dynamics Identification

Enabling fast and accurate physical simulations with data has become an important area of computational physics to aid in inverse problems, design-optimization, uncertainty quantification, and other various decision-making applications. This paper presents a data-driven framework for parametric latent space dynamics identification procedure that enables fast and accurate simulations. The parametric model is achieved by building a set of local latent space model and designing an interaction among them. An individual local latent space dynamics model achieves accurate solution in a trust region. By letting the set of trust region to cover the whole parameter space, our model shows an increase in accuracy with an increase in training data. We introduce two different types of interaction mechanisms, i.e., point-wise and region-based approach. Both linear and nonlinear data compression techniques are used. We illustrate the framework of Latent Space Dynamics Identification (LaSDI) enable a fast and accurate solution process on various partial differential equations, i.e., Burgers’ equations, radial advection problem, and nonlinear heat conduction problem, achieving O(100)x speed-up and O(1)% relative error with respect to the corresponding full order models.