Perform Uncertainty Quantification Research Articles

Uncertainty quantification of the flexural strengthening of RC beams with UHPC using improved metaheuristic algorithm

This study presents an artificial intelligence-based model to predict the flexural strengthening of RC beam/slab strengthened with UHPC (RC-UHPC) and perform uncertainty quantification of such values. Improved eliminate particle swamp optimization hybridized artificial neural network (IEPANN) was a model developed on artificial neural network (ANN) optimized by improved-eliminate-particle-swamp optimization (IEPSO). Experimental datasets were used to establish the hybrid ML model. The efficiency of IEPANN was compared against support vector regression (SVR) and multiple linear regression (MLR). The IEPANN demonstrates outstanding performance over MLR and SVR. The flexural strengthening of RC-UHPC depends on materials and geometrical configurations, which have stochastic characteristics. Convergence studies, sensitivity, and probabilistic analyses were conducted on each random parameter. Simulation results indicated that strengthening configuration is the most sensitive to flexural strengthening. Systematic assessment of uncertainties in flexural strengthening may lead to a higher confidence level in model outcomes, and RC-UHPC is examined and designed. It enhances construction reliability.

Learning thermodynamically constrained equations of state with uncertainty

Numerical simulations of high energy-density experiments require equation of state (EOS) models that relate a material’s thermodynamic state variables—specifically pressure, volume/density, energy, and temperature. EOS models are typically constructed using a semi-empirical parametric methodology, which assumes a physics-informed functional form with many tunable parameters calibrated using experimental/simulation data. Since there are inherent uncertainties in the calibration data (parametric uncertainty) and the assumed functional EOS form (model uncertainty), it is essential to perform uncertainty quantification (UQ) to improve confidence in EOS predictions. Model uncertainty is challenging for UQ studies since it requires exploring the space of all possible physically consistent functional forms. Thus, it is often neglected in favor of parametric uncertainty, which is easier to quantify without violating thermodynamic laws. This work presents a data-driven machine learning approach to constructing EOS models that naturally captures model uncertainty while satisfying the necessary thermodynamic consistency and stability constraints. We propose a novel framework based on physics-informed Gaussian process regression (GPR) that automatically captures total uncertainty in the EOS and can be jointly trained on both simulation and experimental data sources. A GPR model for the shock Hugoniot is derived, and its uncertainties are quantified using the proposed framework. We apply the proposed model to learn the EOS for the diamond solid state of carbon using both density functional theory data and experimental shock Hugoniot data to train the model and show that the prediction uncertainty is reduced by considering thermodynamic constraints.

A novel uncertainty quantification method for determining deformations and reliabilities of stochastic laminated composite plates with geometric nonlinearity

Stochastic finite element method (SFEM) for uncertainty quantification is widely applied for analysis of structures with intrinsic randomness. For determining the geometrically nonlinear deformations of laminated composite plates with random fields, the existing intrusive SFEMs have the limitations of low applicability, insufficient accuracy, or low efficiency. To this end, this paper proposes a novel and efficient non-intrusive SFEM incorporating direct probability integral method to achieve probability density functions (PDFs) of stochastic responses and reliabilities of laminated composite plates with geometric nonlinearity. Firstly, the von Kármán strain-displacement relation based on the third-order shear deformation theory is employed to model the geometric nonlinearity of laminated plates. The random field is discretized via Karhunen–Loève expansion. Secondly, the probability density integral equation (PDIE) is derived from the new perspective of probability conservation. The proposed non-intrusive SFEM decouples the equilibrium equation and PDIE to compute the response PDFs and reliabilities of uncertain laminated composite plates in a unified way. Moreover, the criterion which can judge the applicability of geometrically nonlinear theory is suggested for performing uncertainty quantification. Finally, comparisons of the results in terms of Monte Carlo simulation and the literature demonstrate the high accuracy and efficiency of the proposed method. For stochastic laminated plates, the response statistical moments vary nonlinearly with linear increase of load amplitude due to geometric nonlinearity, the deflection variability increases and structural reliability decreases with the increase in variability and correlation length of random field, and the stacking angle significantly affects the stochastic nonlinear deflections and reliabilities.

A Physics-Informed Spatial-Temporal Neural Network for Reservoir Simulation and Uncertainty Quantification

Summary Surrogate models play a vital role in reducing computational complexity and time burden for reservoir simulations. However, traditional surrogate models suffer from limitations in autonomous temporal information learning and restrictions in generalization potential, which is due to a lack of integration with physical knowledge. In response to these challenges, a physics-informed spatial-temporal neural network (PI-STNN) is proposed in this work, which incorporates flow theory into the loss function and uniquely integrates a deep convolutional encoder-decoder (DCED) with a convolutional long short-term memory (ConvLSTM) network. To demonstrate the robustness and generalization capabilities of the PI-STNN model, its performance was compared against both a purely data-driven model with the same neural network architecture and the renowned Fourier neural operator (FNO) in a comprehensive analysis. Besides, by adopting a transfer learning strategy, the trained PI-STNN model was adapted to the fractured flow fields to investigate the impact of natural fractures on its prediction accuracy. The results indicate that the PI-STNN not only excels in comparison with the purely data-driven model but also demonstrates a competitive edge over the FNO in reservoir simulation. Especially in strongly heterogeneous flow fields with fractures, the PI-STNN can still maintain high prediction accuracy. Building on this prediction accuracy, the PI-STNN model further offers a distinct advantage in efficiently performing uncertainty quantification, enabling rapid and comprehensive analysis of investment decisions in oil and gas development.

Uncertainty quantification in timber-like beams using sparse grids: Theory and examples with off-the-shelf software utilization

When dealing with timber structures, the characteristic strength and stiffness of the material are made highly variable and uncertain by the unavoidable, yet hardly predictable, presence of knots and other defects. In this work, we apply the sparse grids stochastic collocation method to perform uncertainty quantification for structural engineering in the scenario described above. Sparse grids have been developed by the mathematical community in the last decades, and their theoretical background has been rigorously and extensively studied. The document proposes a brief practice-oriented introduction with minimal theoretical background, provides detailed instructions for the use of the off-the-shelf Sparse Grid Matlab kit (freely available online and straightforward to use) and discusses two preliminary examples inspired from timber engineering problems that highlight how sparse grids exhibit superior performances compared to the plain Monte Carlo method.

A Functional Bayesian Model for Hydrogen-Deuterium Exchange Mass Spectrometry.

Proteins often undergo structural perturbations upon binding to other proteins or ligands or when they are subjected to environmental changes. Hydrogen-deuterium exchange mass spectrometry (HDX-MS) can be used to explore conformational changes in proteins by examining differences in the rate of deuterium incorporation in different contexts. To determine deuterium incorporation rates, HDX-MS measurements are typically made over a time course. Recently introduced methods show that incorporating the temporal dimension into the statistical analysis improves power and interpretation. However, these approaches have technical assumptions that hinder their flexibility. Here, we propose a more flexible methodology by reframing these methods in a Bayesian framework. Our proposed framework has improved algorithmic stability, allows us to perform uncertainty quantification, and can calculate statistical quantities that are inaccessible to other approaches. We demonstrate the general applicability of the method by showing it can perform rigorous model selection on a spike-in HDX-MS experiment, improved interpretation in an epitope mapping experiment, and increased sensitivity in a small molecule case-study. Bayesian analysis of an HDX experiment with an antibody dimer bound to an E3 ubiquitin ligase identifies at least two interaction interfaces where previous methods obtained confounding results due to the complexities of conformational changes on binding. Our findings are consistent with the cocrystal structure of these proteins, demonstrating a bayesian approach can identify important binding epitopes from HDX data. We also generate HDX-MS data of the bromodomain-containing protein BRD4 in complex with GSK1210151A to demonstrate the increased sensitivity of adopting a Bayesian approach.

A surrogate-assisted uncertainty quantification and sensitivity analysis on a coupled electrochemical–thermal battery aging model

High-fidelity physics-based models are required to comprehend battery behavior at various operating conditions. This paper proposes an uncertainty quantification analysis on a coupled electrochemical–thermal aging model to improve the reliability of a battery model, while also investigating the impact of parametric model uncertainties on battery voltage, temperature, and aging. The coupled model’s high computing cost, however, is a significant barrier to perform uncertainty quantification (UQ) and sensitivity analysis (SA). To address this problem, a surrogate model – i.e, by simulating the outcome of a quantity of interest that cannot be easily computed or measured – based on the Gaussian process regression (GPR) theory and principle component analysis (PCA) is built, using a small collection of finite element simulation results as synthetic training data. In total, 43 variable electrochemical–thermal parameters as well as 13 variable aging parameters are studied and estimated. Moreover, the trained surrogate model is also used in the parameterization of the electrochemical and thermal models. The results show that the uncertainties in the input parameters significantly affect the estimations of battery voltage, temperature, and aging. Based on this sensitivity analysis, the most influential parameters affecting the above mentioned battery outputs are reported. This approach is thereby helpful for developing robust and reliable high-fidelity battery aging models with potential applications in digital twins as well as for synthetic data generation.

Learning dynamics on invariant measures using PDE-constrained optimization.

We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269-310 (2023)] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.

Modelling the Impact of Human Population and Its Associated Pressure on Forest Biomass and Forest-Dependent Wildlife Population

Mathematical models have been widely used to explain the system originating from human-nature interaction, investigate the impacts of various components, and forecast system behaviour. This paper provides a profound reference to the current state of the art regarding the application of mathematical models to study the impact of human population and population pressure on forest biomass and forest-dependent wildlife. The review focused on two aspects, namely, model formulation and model analysis. In model formulation, the review revealed that socioeconomic status influences forest resource consumption patterns, thus, stratification of the human population based on economic status is a critical phenomenon in modelling human-nature interactions; however, this component has not been featured in the reviewed models. Regarding model analysis, in most of the reviewed work, single parameter approach was utilized to perform uncertainty quantification of the model parameter; this approach has been proven to be inadequate in measuring the uncertainty and sensitivity of the parameter. Thus, the use of correlation or variance based methods, which are multidimensional parameter space methods are of significant importance. Generally, despite the limitations of many assumptions in mathematical modelling, it is revealed that mathematical models demonstrate the ability to handle complex systems originating from interactions between humans and nature.

Evaluation of global sensitivity analysis methods for computational structural mechanics problems

Abstract The curse of dimensionality confounds the comprehensive evaluation of computational structural mechanics problems. Adequately capturing complex material behavior and interacting physics phenomenon in models can lead to long run times and memory requirements resulting in the need for substantial computational resources to analyze one scenario for a single set of input parameters. The computational requirements are then compounded when considering the number and range of input parameters spanning material properties, loading, boundary conditions, and model geometry that must be evaluated to characterize behavior, identify dominant parameters, perform uncertainty quantification, and optimize performance. To reduce model dimensionality, global sensitivity analysis (GSA) enables the identification of dominant input parameters for a specific structural performance output. However, many distinct types of GSA methods are available, presenting a challenge when selecting the optimal approach for a specific problem. While substantial documentation is available in the literature providing details on the methodology and derivation of GSA methods, application-based case studies focus on fields such as finance, chemistry, and environmental science. To inform the selection and implementation of a GSA method for structural mechanics problems for a nonexpert user, this article investigates five of the most widespread GSA methods with commonly used structural mechanics methods and models of varying dimensionality and complexity. It is concluded that all methods can identify the most dominant parameters, although with significantly different computational costs and quantitative capabilities. Therefore, method selection is dependent on computational resources, information required from the GSA, and available data.

Bayesian Variable Selection in Double Generalized Linear Tweedie Spatial Process Models

Double generalized linear models provide a flexible framework for modeling data by allowing the mean and the dispersion to vary across observations. Common members of the exponential dispersion family including the Gaussian, Poisson, compound Poisson-gamma (CP-g), Gamma and inverse-Gaussian are known to admit such models. The lack of their use can be attributed to ambiguities that exist in model specification under a large number of covariates and complications that arise when data display complex spatial dependence. In this work we consider a hierarchical specification for the CP-g model with a spatial random effect. The spatial effect is targeted at performing uncertainty quantification by modeling dependence within the data arising from location based indexing of the response. We focus on a Gaussian process specification for the spatial effect. Simultaneously, we tackle the problem of model specification for such models using Bayesian variable selection. It is effected through a continuous spike and slab prior on the model parameters, specifically the fixed effects. The novelty of our contribution lies in the Bayesian frameworks developed for such models. We perform various synthetic experiments to showcase the accuracy of our frameworks. They are then applied to analyze automobile insurance premiums in Connecticut, for the year of 2008.

Data‐Driven Stochastic Lie Transport Modeling of the 2D Euler Equations

AbstractIn this paper, we propose and assess several stochastic parametrizations for data‐driven modeling of the two‐dimensional Euler equations using coarse‐grid SPDEs. The framework of Stochastic Advection by Lie Transport (SALT) (Cotter et al., 2019, https://doi.org/10.1137/18m1167929) is employed to define a stochastic forcing that is decomposed in terms of a deterministic basis (empirical orthogonal functions, EOFs) multiplied by temporal traces, here regarded as stochastic processes. The EOFs are obtained from a fine‐grid data set and are defined in conjunction with corresponding deterministic time series. We construct stochastic processes that mimic properties of the measured time series. In particular, the processes are defined such that the underlying probability density functions (pdfs) or the estimated correlation time of the time series are retained. These stochastic models are compared to stochastic forcing based on Gaussian noise, which does not use any information of the time series. We perform uncertainty quantification tests and compare stochastic ensembles in terms of mean and spread. Reduced uncertainty is observed for the developed models. On short timescales, such as those used for data assimilation (Cotter et al., 2020a, https://doi.org/10.1007/s10955-020-02524-0), the stochastic models show a reduced ensemble mean error and a reduced spread. Particularly, using estimated pdfs yields stochastic ensembles which rarely fail to capture the reference solution on small time scales, whereas introducing correlation into the stochastic models improves the quality of the coarse‐grid predictions with respect to Gaussian noise.

Uncertainty quantification calculations of nuclide number densities of Gd-bearing fuel rods in light water reactors

The Post-Irradiation Examination (PIE) data are quite useful to validate the evaluated nuclear data. Recently, a new experimental program, REGAL (the Rod-Extremity and Gadolinia AnaLysis), has been proposed and launched. During this program, the PIE data are acquired for the Gd-bearing fuel rods during (or after) the initial Gd burnout. These PIE data would be different from those for the normal UOX or MOX fuel rods from a viewpoint of nuclide transmutation process. In the present work, we attempt to quantify this difference through calculating sensitivity coefficients of nuclide number densities of the Gd-bearing rods using the depletion perturbation theory and performing uncertainty quantification calculations with the sensitivity coefficients and the nuclear data covariance data. Through the numerical analyses, we found that the impact of the uncertainties of the nuclear data of Gd isotopes on the number densities of actinoids is negligible. The nuclear data-induced uncertainties of Gd isotopes number densities are relatively large during the Gd depletion, and those become small after the Gd burnout. After the Gd burnout, the relative standard deviations of Gd-155 and -157 number densities are approximately 6% and 13%, respectively, and the nuclear data of the parent nuclides of these Gd isotopes, Gd-154 and -156, are also important as well as those of Gd-155 and -157.

A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors

Hierarchical models with gamma hyperpriors provide a flexible, sparsity-promoting framework to bridge and regularizations in Bayesian formulations to inverse problems. Despite the Bayesian motivation for these models, existing methodologies are limited to maximum a posteriori estimation. The potential to perform uncertainty quantification has not yet been realized. This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement. In addition, it lends itself naturally to conduct model selection for the choice of hyperparameters. We illustrate the performance of our methodology in several computed examples, including a deconvolution problem and sparse identification of dynamical systems from time series data.

Exascale integrated modeling of low-order wavefront sensing and control for the Roman Coronagraph instrument.

Astronomical instruments to detect exoplanets require extreme wavefront stability. For these missions to succeed, comprehensive and precise modeling is required to design and analyze suitable coronagraphs and their wavefront control systems. In this paper, we describe techniques for integrated modeling at scale that is, to the best of our knowledge, 1000 times faster than previously published works. We show how this capability has been used to validate performance and perform uncertainty quantification for the Roman Coronagraph instrument. Finally, we show how this modeling capacity may be necessary to design and build the next generation of space-based coronagraph instruments.

Non-Probabilistic Uncertainty Quantification of Fiber-Reinforced Composite Laminate Based on Micro- and Macro-Mechanical Analysis

In this paper, the main aim is to study and predict macro elastic mechanical parameters of fiber-reinforced composite laminates by combining micro-mechanical analysis models and the non-probabilistic set theory. It deals with uncertain input parameters existing in quantification models as interval variables. Here, several kinds of micro-mechanical mathematical models are introduced, and the parameter vertex solution theorem and the Monte Carlo simulation method can be used to perform uncertainty quantification of macro elastic properties for composites. In order to take the correlations between macro elastic properties into consideration, the obtained limited sample points or experimental data are utilized on the basis of the grey mathematical modeling theory, where correlated uncertain macro parameters can be treated as a closed and bounded convex polyhedral model. It can give out a clear analytical description for feasible domains of correlated macro elastic properties in the process of uncertainty quantification. Finally, two numerical examples are carried out to account for the validity and feasibility of the proposed quantification method. The results show that the proposed method can become a powerful and meaningful supplement for uncertainty quantification of composite laminates and provide data support for further uncertainty propagation analysis.

Integration of Deep Learning and Information Theory for Designing Monitoring Networks in Heterogeneous Aquifer Systems

AbstractGroundwater monitoring networks are direct sources of information for revealing subsurface system dynamic processes. However, designing such networks is difficult due to uncertainties in the spatial heterogeneity of aquifer parameters such as permeability (k). This study combines deep learning and information theory with an optimization framework to address network design problems in heterogeneous aquifer systems. The framework first employs a generative adversarial network to parameterize heterogeneous k distribution using a low‐dimensional latent representation. Then, surrogate models are developed based on the deep neural networks to perform uncertainty quantification of pressure heads and solute concentrations at locations of pre‐designed candidate monitoring stations. The monitoring stations are then ranked using the greedy search algorithm based on the maximum information minimum redundancy (MIMR) criterion. In order to depict the importance of each candidate monitoring location, the hotspot maps of the selection probability (Ps) are derived from MIMR repetition results. Comprehensive monitoring networks derived from the hotspot maps are then conducted as the final monitoring stations to improve monitoring information compared to MIMR results. Additionally, nine entropy quantization strategies are compared to evaluate their effects on monitoring network optimization results. Results indicate that caution should be taken when selecting entropy quantization strategies to achieve the accuracy required for model calibration and to improve the efficiency of monitoring optimization. Considering high‐dimensional uncertainties associated with aquifer parameters, the developed framework can provide important insights for monitoring network designs in various earth observational projects.

Acceleration of a Physics-Based Machine Learning Approach for Modeling and Quantifying Model-Form Uncertainties and Performing Model Updating

Abstract The nonparametric probabilistic method (NPM) for modeling and quantifying model-form uncertainties is a physics-based, computationally tractable, machine learning method for performing uncertainty quantification and model updating. It extracts from data information not captured by a deterministic, high-dimensional model (HDM) of dimension N and infuses it into a counterpart stochastic, hyperreduced, projection-based reduced-order model (SHPROM) of dimension n ≪ N. Here, the robustness and performance of NPM are improved using a two-pronged approach. First, the sensitivities of its stochastic loss function with respect to the hyperparameters are computed analytically, by tracking the complex web of operations underlying the construction of that function. Next, the theoretical number of hyperparameters is reduced from O(n2) to O(n), by developing a network of autoencoders that provides a nonlinear approximation of the dependence of the SHPROM on the hyperparameters. The robustness and performance of the enhanced NPM are demonstrated using two nonlinear, realistic, structural dynamics applications.

Assessments of epistemic uncertainty using Gaussian stochastic weight averaging for fluid-flow regression

We use Gaussian stochastic weight averaging (SWAG) to assess the epistemic uncertainty associated with neural-network-based function approximation relevant to fluid flows. SWAG approximates a posterior Gaussian distribution of each weight, given training data, and a constant learning rate. Having access to this distribution, it is able to create multiple models with various combinations of sampled weights, which can be used to obtain ensemble predictions. The average of such an ensemble can be regarded as the ‘mean estimation’, whereas its standard deviation can be used to construct ‘confidence intervals’, which enable us to perform uncertainty quantification (UQ) with regard to the training process of neural networks. We utilize representative neural-network-based function approximation tasks for the following cases: (i) a two-dimensional circular-cylinder wake; (ii) the DayMET dataset (maximum daily temperature in North America); (iii) a three-dimensional square-cylinder wake; and (iv) urban flow, to assess the generalizability of the present idea for a wide range of complex datasets. SWAG-based UQ can be applied regardless of the network architecture, and therefore, we demonstrate the applicability of the method for two types of neural networks: (i) global field reconstruction from sparse sensors by combining convolutional neural network (CNN) and multi-layer perceptron (MLP); and (ii) far-field state estimation from sectional data with two-dimensional CNN. We find that SWAG can obtain physically-interpretable confidence-interval estimates from the perspective of epistemic uncertainty. This capability supports its use for a wide range of problems in science and engineering.

Rank-based Bayesian variable selection for genome-wide transcriptomic analyses.

Variable selection is crucial in high-dimensional omics-based analyses, since it is biologically reasonable to assume only a subset of non-noisy features contributes to the data structures. However, the task is particularly hard in an unsupervised setting, and a priori ad hoc variable selection is still a very frequent approach, despite the evident drawbacks and lack of reproducibility. We propose a Bayesian variable selection approach for rank-based unsupervised transcriptomic analysis. Making use of data rankings instead of the actual continuous measurements increases the robustness of conclusions when compared to classical statistical methods, and embedding variable selection into the inferential tasks allows complete reproducibility. Specifically, we develop a novel extension of the Bayesian Mallows model for variable selection that allows for a full probabilistic analysis, leading to coherent quantification of uncertainties. Simulation studies demonstrate the versatility and robustness of the proposed method in a variety of scenarios, as well as its superiority with respect to several competitors when varying the data dimension or data generating process. We use the novel approach to analyze genome-wide RNAseq gene expression data from ovarian cancer patients: several genes that affect cancer development are correctly detected in a completely unsupervised fashion, showing the usefulness of the method in the context of signature discovery for cancer genomics. Moreover, the possibility to also perform uncertainty quantification plays a key role in the subsequent biological investigation.