Uncertainty Quantification Algorithms Research Articles

A meta-weighted network equipped with uncertainty estimations for remaining useful life prediction of turbopump bearings

Uncertainty quantification and negative transfer characterization are significant challenges in the context of deep learning-based remaining useful life prediction with limited data. Conventional Bayesian methods lack scalability to deep learning algorithms for uncertainty quantification due to intricate network connections and extensive training parameters. Additionally, although transfer learning is a promising way to improve the generalization ability of prediction algorithms, its effectiveness is not always guaranteed when leveraging source data undesirably reduces the prediction performance, which is named negative transfer. This manuscript proposed a meta-weighted neural network equipped with uncertainty estimation to discern in-distribution from out-of-distribution data and an adaptive sample meta re-weighting strategy is designed by specifying the weighting function from prediction loss to sample weight according to the gradient direction. Performance evaluations on cryogenic bearings demonstrate that the proposed algorithm can quantitatively determine the weights of source data based on prediction error, ultimately leading to accurate interval remaining life prediction.

Uncertainty quantification of time-average quantities of chaotic systems using sensitivity-enhanced polynomial chaos expansion.

We consider the effect of multiple stochastic parameters on the time-average quantities of chaotic systems. We employ the recently proposed sensitivity-enhanced generalized polynomial chaos expansion, se-gPC, to quantify efficiently this effect. se-gPC is an extension of gPC expansion, enriched with the sensitivity of the time-averaged quantities with respect to the stochastic variables. To compute these sensitivities, the adjoint of the shadowing operator is derived in the frequency domain. Coupling the adjoint operator with gPC provides an efficient uncertainty quantification algorithm, which, in its simplest form, has computational cost that is independent of the number of random variables. The method is applied to the Kuramoto-Sivashinsky equationand is found to produce results that match very well with Monte Carlo simulations. The efficiency of the proposed method significantly outperforms sparse-grid approaches, such as Smolyak quadrature. These properties make the method suitable for application to other dynamical systems with many stochastic parameters.

Induced Seismicity Forecasting with Uncertainty Quantification: Application to the Groningen Gas Field

Abstract Reservoir operations for gas extraction, fluid disposal, carbon dioxide storage, or geothermal energy production are capable of inducing seismicity. Modeling tools exist for seismicity forecasting using operational data, but the computational costs and uncertainty quantification (UQ) pose challenges. We address this issue in the context of seismicity induced by gas production from the Groningen gas field using an integrated modeling framework, which combines reservoir modeling, geomechanical modeling, and stress-based earthquake forecasting. The framework is computationally efficient thanks to a 2D finite-element reservoir model, which assumes vertical flow equilibrium, and the use of semianalytical solutions to calculate poroelastic stress changes and predict seismicity rate. The earthquake nucleation model is based on rate-and-state friction and allows for an initial strength excess so that the faults are not assumed initially critically stressed. We estimate uncertainties in the predicted number of earthquakes and magnitudes. To reduce the computational costs, we assume that the stress model is true, but our UQ algorithm is general enough that the uncertainties in reservoir and stress models could be incorporated. We explore how the selection of either a Poisson or a Gaussian likelihood influences the forecast. We also use a synthetic catalog to estimate the improved forecasting performance that would have resulted from a better seismicity detection threshold. Finally, we use tapered and nontapered Gutenberg–Richter distributions to evaluate the most probable maximum magnitude over time and account for uncertainties in its estimation. Although we did not formally account for uncertainties in the stress model, we tested several alternative stress models, and found negligible impact on the predicted temporal evolution of seismicity and forecast uncertainties. Our study shows that the proposed approach yields realistic estimates of the uncertainties of temporal seismicity and is applicable for operational forecasting or induced seismicity monitoring. It can also be used in probabilistic traffic light systems.

Data-informed uncertainty quantification for wave scattering by heterogeneous media

We present an efficient data-driven offline/online Bayesian algorithm for uncertainty quantification (UQ) in the induced scattered field when a time-harmonic incident wave interacts with an uncertain heterogeneous medium. The incident wave of interest need not be known in advance, and the uncertainty is informed by noisy scattering data obtained from other incident waves impinging on the medium. Our UQ algorithm is accelerated by a novel stochastic reduced order model (ROM) based on the T-matrix, and the ROM is independent of both the incident wave, and other incident waves used to generate the data. This important property allows the model to be set up offline.

PyPESTO: a modular and scalable tool for parameter estimation for dynamic models.

Mechanistic models are important tools to describe and understand biological processes. However, they typically rely on unknown parameters, the estimation of which can be challenging for large and complex systems. pyPESTO is a modular framework for systematic parameter estimation, with scalable algorithms for optimization and uncertainty quantification. While tailored to ordinary differential equation problems, pyPESTO is broadly applicable to black-box parameter estimation problems. Besides own implementations, it provides a unified interface to various popular simulation and inference methods. pyPESTO is implemented in Python, open-source under a 3-Clause BSD license. Code and documentation are available on GitHub (https://github.com/icb-dcm/pypesto).

Implementation of Asynchronous Distributed Gauss-Newton Optimization Algorithms for Uncertainty Quantification by Conditioning to Production Data

Summary Previous implementation of the distributed Gauss-Newton (DGN) optimization algorithm ran multiple optimization threads in parallel, employing a synchronous running mode (S-DGN). As a result, it waits for all simulations submitted in each iteration to complete, which may significantly degrade performance because a few simulations may run much longer than others, especially for time-consuming real-field cases. To overcome this limitation and thus improve the DGN optimizer’s execution, we propose two asynchronous DGN (A-DGN) optimization algorithms in this paper. The two A-DGN optimization algorithms are (1) the local-search algorithm (A-DGN-LS) to locate multiple maximum a-posteriori (MAP) estimates and (2) the integrated global-search algorithm with the randomized maximum likelihood (RML) method (A-DGN + RML) to generate hundreds of RML samples in parallel for uncertainty quantification. We propose using batch together with a checking time interval to control the optimization process. The A-DGN optimizers check the status of all running simulations after every checking time frame. The iteration index of each optimization thread is updated dynamically according to its simulation status. Thus, different optimization threads may have different iteration indices in the same batch. A new simulation case is proposed immediately once the simulation of an optimization thread is completed, without waiting for the completion of other simulations. We modified the training data set updating algorithm using each thread’s dynamically updated iteration index to implement the asynchronous running mode. We apply the modified QR decomposition method to estimate the sensitivity matrix at the best solution of each optimization thread by linear interpolation of all or a subset of the training data to avoid the issue of solving a linear system with a singular matrix because of insufficient training data points in early batches. A new simulation case (or search point) is generated by solving the Gauss-Newton (GN) trust-region subproblem (GNTRS) using the estimated sensitivity matrix. We developed a more efficient and robust GNTRS solver using eigenvalue decomposition (EVD). The proposed A-DGN optimization methods are tested and validated on a 2D analytical toy problem and a synthetic history-matching problem and then applied to a real-field deepwater reservoir model. Numerical tests confirm that the proposed A-DGN optimization methods can converge to solutions with matching quality comparable to those obtained by the S-DGN optimizers, saving on the time required for the optimizer to converge by a factor ranging from 1.3 to 2 when compared to the S-DGN optimizer depending on the problem. The new A-DGN optimization algorithms improve efficiency and robustness in solving history-matching or inversion problems, especially for uncertainty quantification of subsurface model parameters and production forecasts of real-field reservoirs by conditioning production data.

Design-space dimensionality reduction in global optimization of functional surfaces: recent developments and way forward

ABSTRACT In shape optimization of complex industrial products (such as, but not limited to, hull forms, rudder and appendages, propellers), there exists an inherent similarity between global optimization (GO) and uncertainty quantification (UQ): they rely on an extensive exploration of the design and operational spaces, respectively; often, they need local refinements to ensure accurate identification of optimal solutions or probability density regions (such as distribution tails), respectively; they both are dramatically affected by the curse of dimensionality as GO and UQ algorithms' complexity and especially computational cost rapidly increase with the problem dimension. Therefore, there exists a natural ground for transferring dimensionality reduction methods from UQ to GO. These enable the efficient exploration of large design spaces in shape optimization, which, in turn, enable global optimization (possibly in a multidisciplinary and stochastic setting). The paper reviews and discusses recent techniques for design-space dimensionality reduction in shape optimization, based on the Karhunen-Loève expansion (equivalent to proper orthogonal decomposition and, at the discrete level, principal component analysis). An example is shown and discussed for the hydrodynamic optimization of a ship hull.

Estimation of eruption source parameters for the 2021 La Soufrière eruption (St Vincent): implications for quantification of eruption magnitude on volcanic islands

Abstract Eruption source parameters (ESPs) used to characterize explosive eruptions are estimated from tephra deposit data using different models (statistical or numerical) and inversion approaches. The ESPs thus derived are subject to substantial uncertainties when the bulk of the tephra deposit, including information about its full spatial extent and spatial variation in grain-size distribution is missing due to geographical and environmental conditions. We use an advection–diffusion model coupled with a Bayesian inversion and uncertainty quantification algorithm to investigate how ESPs can be robustly estimated given such conditions. The 2021 eruption of La Soufrière volcano (St Vincent and the Grenadines) is our case study. An inversion is conducted for the first two explosive phases of this eruption (U1 and U2). We estimate: an erupted mass of 3.3 × 10 10 ± 1 × 10 10 kg for U1 and 3.1 × 10 10 ± 1.9 × 10 9 kg for U2, with an average particles release height of c. 13.5 km a.s.l. ±0.5 km for both phases. Given the efficiency of the proposed approach and the plausibility of the stochastic inversion results, we recommend this procedure for estimating ESPs for explosive eruptions for which the bulk of the deposit is missing or is inaccessible.

Towards an automatic uncertainty compiler

An uncertainty compiler is a tool that automatically translates original computer source code lacking explicit uncertainty quantification into code containing appropriate uncertainty representations and uncertainty propagation algorithms. It handles the specifications of input uncertainties, and inserts calls to intrusive uncertainty quantification algorithms. In theory, one could create an uncertainty compiler for any scientific programming language. The uncertainty compiler can apply intrusive uncertainty propagation methods to codes or parts of codes and, therefore, more comprehensively and flexibly address epistemic and aleatory uncertainties. This paper explores the concept and the practicalities of creating such a compiler.

A new algorithm for uncertainty quantification for thermal conductivity measurement on polymers with the Haakvoort method using differential scanning calorimetry considering specimen height and real contact area

A new algorithm for the quantification of uncertainty in thermal conductivity measurements on polymers according to the Haakvort method is presented. This fast and convenient method using differential scanning calorimetry has been established as DIN EN ISO Standard 11357–8 with an error margin of 5–10%, which is a rather large value when considering that this is an important material parameter for many applications and is often used in combined quantities, such as thermal diffusivity or thermal effusivity. Unfortunately, the DIN EN ISO standard does not provide useful information on the dependence of the error range on the number of specimens or important parameters, such as the height of the specimens or their real contact area. Applying a rigorous statistical approach, based on the law of large numbers (LLN) and different techniques which are also used in well-known methods, such as Monte-Carlo- or Markov chain Monte Carlo (MCMC) algorithms, we establish and investigate a method to optimize the experimental effort to a specific target, especially the number of specimens, the aspect ratio and the real contact surface of the specimen.

A generalized Bayes framework for probabilistic clustering

Summary Loss-based clustering methods, such as k-means clustering and its variants, are standard tools for finding groups in data. However, the lack of quantification of uncertainty in the estimated clusters is a disadvantage. Model-based clustering based on mixture models provides an alternative approach, but such methods face computational problems and are highly sensitive to the choice of kernel. In this article we propose a generalized Bayes framework that bridges between these paradigms through the use of Gibbs posteriors. In conducting Bayesian updating, the loglikelihood is replaced by a loss function for clustering, leading to a rich family of clustering methods. The Gibbs posterior represents a coherent updating of Bayesian beliefs without needing to specify a likelihood for the data, and can be used for characterizing uncertainty in clustering. We consider losses based on Bregman divergence and pairwise similarities, and develop efficient deterministic algorithms for point estimation along with sampling algorithms for uncertainty quantification. Several existing clustering algorithms, including k-means, can be interpreted as generalized Bayes estimators in our framework, and thus we provide a method of uncertainty quantification for these approaches, allowing, for example, calculation of the probability that a data point is well clustered.

Data coverage assessment on neural network based digital twins for autonomous control system

In a recently developed Nearly Autonomous Management and Control (NAMAC) system, neural networks (NNs) are used to develop digital twins for diagnosis (DT-Ds). However, NNs are not usually considered extrapolation models and may result in large errors if they are applied to unseen data outside the training data (uncovered). In this study, we propose a data coverage assessment (DCA) to determine if the NN-based DT-Ds are extrapolated based on their epistemic uncertainty. The uncertainty quantification algorithms and uncertainty thresholds are selected based on the confusion matrix of classifying evaluation data into covered or uncovered data. To demonstrate the adaptability of the proposed framework, we applied it to a basic feedforward neural network and a more advanced recurrent neural network based on a more nonlinear database. Case studies show that the proposed framework can distinguish unseen data for both basic and advanced applications with proper uncertainty quantification algorithms and thresholds.

Uncertainty quantification of metallic microstructures using principal image moments

The present work uses Markov Random Field (MRF) algorithm to construct large-scale and statistically-equivalent samples from small-scale experimental data of metallic microstructures. While the MRF method can build such digital material representations in large computational domains, its algorithmic stochasticity (epistemic uncertainty) causes variations in the resulting microstructural features, such as the texture and grain topology. This work addresses the effects of the epistemic uncertainty on homogenized mechanical properties by characterizing the variations in the microstructural features using a shape descriptor based on the concept of moment invariants. In particular, 2D and 3D synthetic microstructure data for Titanium-7wt%Aluminum (Ti-7Al) alloy is generated with the MRF method using smaller-scale 2D experimental data. To quantify the uncertainty of the reconstructed synthetic samples, a graphical method and five different metrics of statistical variability are developed. Next, the propagation of the microstructural uncertainty on homogenized properties is studied using an analytical uncertainty quantification (UQ) algorithm and Gaussian Process Regression (GPR).

Analytical methods for superresolution dislocation identification in dark-field X-ray microscopy

In this work, we develop several inference methods to estimate the position of dislocations from images generated using dark-field X-ray microscopy (DFXM) -- achieving superresolution accuracy and principled uncertainty quantification. Using the framework of Bayesian inference, we incorporate models of the DFXM contrast mechanism and detector measurement noise, along with initial position estimates, into a statistical model coupling DFXM images with the dislocation position of interest. We motivate several position estimation and uncertainty quantification algorithms based on this model. We then demonstrate the accuracy of our primary estimation algorithm on synthetic realistic DFXM images of edge dislocations in single crystal aluminum. We conclude with a discussion of our methods' impact on future dislocation studies and possible future research avenues.

Uncertainty estimation for ensemble particle image velocimetry

We present a novel approach to estimate the uncertainty in ensemble particle image velocimetry (PIV) measurements. The ensemble PIV technique is widely used when the cross-correlation signal-to-noise ratio is insufficient to perform a reliable instantaneous velocity measurement. Despite the utility of ensemble PIV, uncertainty quantification for this type of measurement has not been studied. Here, we propose a method for estimating the uncertainty directly from the probability density function of displacements found by deconvolving the ensemble cross-correlation from the ensemble autocorrelation. We then find the second moment of the probability density function and apply a scaling factor to report the uncertainty in the velocity measurement. We call this method the moment of probability of displacement (MPD). We assess MPD’s performance with synthetic and experimental images. We show that predicted uncertainties agree well with the expected root mean square (RMS) of the error in the velocity measurements over a wide range of image and flow conditions. MPD shows good sensitivity to various PIV error sources with around 86% accuracy in matching the RMS of the error in the baseline data sets. So, MPD establishes itself as a reliable uncertainty quantification algorithm for ensemble PIV. We compared the results of MPD against one of the existing instantaneous PIV uncertainty approaches, moment of correlation (MC). We adapted the MC approach for ensemble PIV, however, its primary limitations remain the assumption of the Gaussian probability density function of displacements and the Gaussian particles’ intensity profile. In addition, our analysis shows that ensemble MC consistently underestimates the uncertainty, while MPD outperforms that and removes the limiting Gaussian assumption for the particle and probability density function, thus overcoming the limitations of MC.

The design of an open-source carbonate reservoir model

This work presents a new open-source carbonate reservoir case study, the COSTA model, that uniquely considers significant uncertainties inherent to carbonate reservoirs, providing a far more challenging and realistic benchmarking test for a range of geo-energy applications. The COSTA field is large, with many wells and large associated volumes. The dataset embeds many interacting geological and petrophysical uncertainties in an ensemble of model concepts with realistic geological and model complexity levels and varying production profiles. The resulting number of different models and long run times creates a more demanding computational challenge than current benchmarking models. The COSTA model takes inspiration from the shelf-to-basin geological setting of the Upper Kharaib Member (Early Cretaceous), one of the most prolific aggradational parasequence carbonate formations sets in the world. The dataset is based on 43 wells and the corresponding fully anonymised data from the northeastern part of the Rub Al Khali basin, a sub-basin of the wider Arabian Basin. Our model encapsulates the large-scale geological setting and reservoir heterogeneities found across the shelf-to-basin profile, into one single model, for geological modelling and reservoir simulation studies. The result of this research is a semi-synthetic but geologically realistic suite of carbonate reservoir models that capture a wide range of geological, petrophysical, and geomodelling uncertainties and that can be history-matched against an undisclosed, synthetic ‘truth case’. The models and dataset are made available as open-source to analyse several issues related to testing new numerical algorithms for geological modelling, uncertainty quantification, reservoir simulation, history matching, optimization and machine learning. Supplementary material: supplementary data associated with this article can be found in the data repository for the COSTA model available at https://doi.org/10.17861/6e36e28d-50d9-4e31-9790-18db4bce6e5d and supplementary material is available at https://doi.org/10.6084/m9.figshare.c.5823571

Model Regeneration Scheme Using a Deep Learning Algorithm for Reliable Uncertainty Quantification of Channel Reservoirs

Abstract Reservoir characterization is one of the essential procedures for decision makings. However, conventional inversion methods of history matching have several inevitable issues of losing geological information and poor performances, when it is applied to channel reservoirs. Therefore, we propose a model regeneration scheme for reliable uncertainty quantification of channel reservoirs without conventional model inversion methods. The proposed method consists of three parts: feature extraction, model selection, and model generation. In the feature extraction part, drainage area localization and discrete cosine transform are adopted for channel feature extraction in near-wellbore area. In the model selection part, K-means clustering and an ensemble ranking method are utilized to select models that have similar characteristics to a true reservoir. In the last part, deep convolutional generative adversarial networks (DCGAN) and transfer learning are applied to generate new models similar to the selected models. After the generation, we repeat the model selection process to select final models from the selected and the generated models. We utilize these final models to quantify uncertainty of a channel reservoir by predicting their future productions. After applying the proposed scheme to three different channel fields, it provides reliable models for production forecasts with reduced uncertainty. The analyses show that the scheme can effectively characterize channel features and increase a probability of existence of models similar to a true model.

A long short-term memory based deep learning algorithm for seismic response uncertainty quantification

The application of metamodeling technique to overcome computational challenge of Monte Carlo simulation (MCS) technique for response uncertainty quantification under stochastic earthquake load is a difficult task due to the high-dimensional nature of stochastic load. Recent developments in the sequential models for forecasting and prediction have opened a new avenue in this regard. Various deep learning algorithms, particularly the convolutional neural network and recurrent neural network are quite suitable for response uncertainty quantification of nonlinear stochastic dynamic system. However, most of the existing studies consider stochastic load as the only source of uncertainty assuming the parameters characterizing a structure as deterministic. The present study proposes a long short-term memory (LSTM) based deep learning algorithm for seismic response uncertainty quantification by duly addressing both the stochastic nature of dynamic load and structural system parameter uncertainty. The functional application program interface feature of Keras that allows layers sharing to form more complex model is explored to form a response approximation model. It incorporates more than one input source i.e., stochastic dynamic excitation sequence as well as structural system parameter uncertainty. The proposed algorithm is elucidated through two numerical examples i.e., a proof-of-concept example and one realistic structural engineering problem by considering the direct MCS based results as the benchmark. The results of accuracy matrices, regression analysis results, comparison of seismic response statistics and reliability results with the results of direct MCS technique clearly revealed the enhanced prediction capability of the proposed LSTM model.

Uncertainty quantification of input matrices and transfer function in input/output subspace system identification

The transfer function of a linear mechanical system can be defined in terms of the quadruplet of state-space matrices (A,B,C,D) that can be identified from input and output measurements with subspace-based system identification methods. The estimation of the quadruplet has been well studied in the literature from both theoretical and practical viewpoints. Nonetheless, a practical algorithm for uncertainty quantification of its estimation errors and the uncertainty of the resultant parametric transfer function is missing in the context of subspace identification. For several output-only and input/output subspace methods, the covariance related to the matrices (A,C) and to the resulting modal parameters can be effectively obtained with recently developed first-order perturbation-based schemes, while the corresponding uncertainty quantification for the input-related matrices (B,D) is missing. In this paper, explicit expressions for the covariance related to matrices (B,D) are developed, and applied to the covariance estimation of the resulting transfer function. The proposed schemes are validated on a simulated data of a mechanical system and are applied to laboratory measurements of a plate.

Modified Dimension Reduction-Based Polynomial Chaos Expansion for Nonstandard Uncertainty Propagation and Its Application in Reliability Analysis

This paper presents an algorithm for efficient uncertainty quantification (UQ) in the presence of many uncertainties that follow a nonstandard distribution (e.g., lognormal). Using the polynomial chaos expansion (PCE), the algorithm builds surrogate models of uncertainty as functions of a standard distribution (e.g., Gaussian variables). The key to build these surrogate models is to calculate PCE coefficients of model outputs, which is computationally challenging, especially when dealing with models defined by complex functions (e.g., nonpolynomial terms) under many uncertainties. To address this issue, an algorithm that integrates the PCE with the generalized dimension reduction method (gDRM) is utilized to convert the high-dimensional integrals, required to calculate the PCE coefficients of model predictions, into several lower-dimensional ones that can be rapidly solved with quadrature rules. The accuracy of the algorithm is validated with four examples in structural reliability analysis and compared to other existing techniques, such as Monte Carlo simulations and the least angle regression-based PCE. Our results show our algorithm provides accurate UQ results and is computationally efficient when dealing with many uncertainties, thus laying the foundation to address UQ in complex control systems.