Application Of Uncertainty Quantification Research Articles

Democratizing uncertainty quantification

Uncertainty Quantification (UQ) is vital to safety-critical model-based analyses, but the widespread adoption of sophisticated UQ methods is limited by technical complexity. In this paper, we introduce UM-Bridge (the UQ and Modeling Bridge), a high-level abstraction and software protocol that facilitates universal interoperability of UQ software with simulation codes. It breaks down the technical complexity of advanced UQ applications and enables separation of concerns between experts. UM-Bridge democratizes UQ by allowing effective interdisciplinary collaboration, accelerating the development of advanced UQ methods, and making it easy to perform UQ analyses from prototype to High Performance Computing (HPC) scale.In addition, we present a library of ready-to-run UQ benchmark problems, all easily accessible through UM-Bridge. These benchmarks support UQ methodology research, enabling reproducible performance comparisons. We demonstrate UM-Bridge with several scientific applications, harnessing HPC resources even using UQ codes not designed with HPC support.

Multi-fidelity wavelet neural operator surrogate model for time-independent and time-dependent reliability analysis

Operator learning frameworks have recently emerged as an effective scientific machine learning tool for learning complex nonlinear operators of differential equations. Since neural operators learn an infinite-dimensional functional mapping, it is useful in applications requiring rapid prediction of solutions for a wide range of input functions. A task of a similar nature arises in many applications of uncertainty quantification, including reliability estimation and design under uncertainty, each of which demands thousands of samples subjected to a wide range of possible input conditions, an aspect to which neural operators are specialized. Although the neural operators are capable of learning complex nonlinear solution operators, they require an extensive amount of data for successful training. Unlike the applications in computer vision, the computational complexity of the numerical simulations and the cost of physical experiments contributing to the synthetic and real training data compromise the performance of the trained neural operator model, thereby directly impacting the accuracy of uncertainty quantification results. We aim to alleviate the data bottleneck by using multi-fidelity learning in neural operators, where a neural operator is trained by using a large amount of inexpensive low-fidelity data along with a small amount of expensive high-fidelity data. We propose the multi-fidelity wavelet neural operator, capable of learning solution operators from a multi-fidelity dataset, for efficient and effective data-driven reliability analysis of dynamical systems. We illustrate the performance of the proposed framework on bi-fidelity data simulated on coarse and refined grids for spatial and spatiotemporal systems.

Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

Stochastic Spin‐Orbit‐Torque Synapse and its Application in Uncertainty Quantification

AbstractStochasticity plays a significant role in the low‐power operation of a biological neural network. In an artificial neural network, stochasticity also contributes to critical functions such as the uncertainty quantification (UQ) for estimating the probability for the correctness of prediction. This UQ is vital for cutting‐edge applications, including medical diagnostics, autopilots, and large language models. Thanks to nonlinear variation of analogous Hall resistance with high computing velocity and low dissipation, a stochastic spin‐orbit‐torque (SOT) device exhibits significant potential for implementing the UQ. However, up until now, the application of UQ for stochastic SOT devices remains unexplored. In this study, based on SOT‐induced stochastic magnetic domain wall (DW) motion with varying velocity, a SOT synapse is fabricated that can emulate stochastic weight update following the Spike‐Timing‐Dependent‐Plasticity (STDP) rule. Furthermore, a stochastic SNN is set up, which, when compared to its deterministic counterpart, demonstrates a clear advantage in quantifying uncertainty for diagnosing the type of breast tumor (benign or malignant).

Quality measures for the evaluation of machine learning architectures on the quantification of epistemic and aleatoric uncertainties in complex dynamical systems

Machine learning methods for the construction of data-driven reduced order model models are used in an increasing variety of engineering domains, especially as a supplement to expensive computational fluid dynamics for design problems. An important check on the reliability of surrogate models is Uncertainty Quantification (UQ), a self assessed estimate of the model error. Accurate UQ allows for cost savings by reducing both the required size of training data sets and the required safety factors, while poor UQ prevents users from confidently relying on model predictions. We introduce two quality measures for the quantification of uncertainty that are suitable for high dimensional problems: the distribution of normalized residuals on validation data and the distribution of estimated uncertainties. While the distribution of estimated uncertainties evaluates how confident the model is in making predictions, the distribution of normalized residuals evaluates how accurate these uncertainty estimates are We examine several machine learning techniques, including both Gaussian processes and a family UQ-augmented neural networks: Ensemble neural networks (ENN), Bayesian neural networks (BNN), Dropout neural networks (D-NN), and Gaussian neural networks (G-NN). We evaluate UQ accuracy (distinct from model accuracy) We apply these metrics to two model data sets representative of complex dynamical systems: an ocean engineering problem in which a ship traverses irregular wave episodes, and a dispersive wave turbulence system with extreme events, the Majda–McLaughlin–Tabak model. We present conclusions concerning model architecture and hyperparameter tuning for use in applications of UQ to dynamical systems, such as active learning schemes.

Uncertainty quantification of phase transitions in magnetic materials lattices

This Perspective article aims to emphasize the crucial role of uncertainty quantification (UQ) in understanding magnetic phase transitions, which are pivotal in various applications, especially in the transportation and energy sectors [D. C. Jiles, Acta Mater. 51, 5907–5939 (2003) and Gutfleisch et al., Adv. Mater. 23, 821–842 (2011)]. Magnetic materials undergoing phase transitions, particularly due to high temperatures, pose challenges related to the loss of their inherent properties. However, pinpointing a definitive phase transition temperature proves challenging due to the diverse and uncertain nanostructure of materials. Deterministic approaches are limited when seeking a precise threshold. As a result, there is a need to develop probabilistic methods to improve the understanding of this physical problem. In this study, UQ is explored within the context of magnetic phase transitions. In addition, the broader applications of UQ in relation to microstructures and Hamiltonian systems are discussed to highlight its significance in materials science. Furthermore, this study discusses the potential future work on the integration of quantum computing to achieve more efficient UQ solutions of magnetic phase transitions using Ising models.

Uncertainty Quantification of the Effect of Variable Conductivity in Ventricular Fibrotic Regions on Ventricular Tachycardia.

Ventricular tachycardia (VT) is a life-threatening cardiac arrhythmia for which a common treatment pathway is electroanatomical mapping and ablation. Recent advances in both noninvasive ablation techniques and computational modeling have motivated the development of patient-specific computational models of VT. Such models are parameterized by a wide range of inputs, each of which is associated with an often unknown amount of error and uncertainty. Uncertainty quantification (UQ) is a technique to assess how variability in the inputs to a model affects its outputs. UQ has seen increased attention in computational cardiology as an avenue to further improve, understand, and develop patient-specific models. In this study we applied polynomial chaos-based UQ to explore the effect of varying the tissue conductivity of fibrotic border zones in a patient-specific model on the resulting VT simulation. We found that over a range of inputs, the model was most sensitive to fibrotic sheet direction, and uncertainty in fibrotic conductivity resulted in substantial variability in the VT reentry duration and cycle length. Overall, this study paves the way for future UQ applications to improve and understand VT models.

Main outcomes of the Phebus FPT1 uncertainty and sensitivity analysis in the EU-MUSA project

The Management and Uncertainties of Severe Accidents (MUSA) project was funded in HORIZON 2020 and is coordinated by CIEMAT (Spain). The project aims at consolidating a harmonized approach for the analysis of uncertainties and sensitivities associated with Severe Accidents (SAs) analysis, focusing on source term figures of merit. The Application of Uncertainty Quantification (UQ) Methods against Integral Experiments (AUQMIE – Work Package 4 (WP4)), led by ENEA (Italy), was devoted to apply and test UQ methodologies adopting the internationally recognized PHEBUS FPT1 test. FPT1 was chosen to test UQ methodologies because, even though it is a simplified SA scenario, it was representative of the in-vessel phase of a severe accident initiated by a break in the cold leg of a PWR primary circuit.WP4 served as a platform to identify and discuss the issues encountered in the application of UQ methodologies to SA analyses (e.g. discuss the UQ methodology, perform the coupling between the SA codes and the UQ tools, define the results post-processing methods, etc.). The purpose of this paper is to describe the MUSA PHEBUS FPT1 uncertainty application exercise with the related specifications and the methodologies used by the partners to perform the UQ exercise. The main outcomes and lessons learned of the analysis are: scripting was in general needed for the SA code and uncertainty tool coupling and to have more flexibility; particular attention should be devoted to the proper choice of the input uncertain parameters; outlier values of figures of merit should be carefully analyzed; the computational time is a key element to perform UQ in SA; the large number of uncertain input parameters may complicate the interpretation of correlation or sensitivity analysis; there is the need for a statistically solid handling of failed calculations.

Uncertainty quantification of the effect of cardiac position variability in the inverse problem of electrocardiographic imaging

Objective. Electrocardiographic imaging (ECGI) is a functional imaging modality that consists of two related problems, the forward problem of reconstructing body surface electrical signals given cardiac bioelectric activity, and the inverse problem of reconstructing cardiac bioelectric activity given measured body surface signals. ECGI relies on a model for how the heart generates bioelectric signals which is subject to variability in inputs. The study of how uncertainty in model inputs affects the model output is known as uncertainty quantification (UQ). This study establishes develops, and characterizes the application of UQ to ECGI. Approach. We establish two formulations for applying UQ to ECGI: a polynomial chaos expansion (PCE) based parametric UQ formulation (PCE-UQ formulation), and a novel UQ-aware inverse formulation which leverages our previously established ‘joint-inverse’ formulation (UQ joint-inverse formulation). We apply these to evaluate the effect of uncertainty in the heart position on the ECGI solutions across a range of ECGI datasets. Main results. We demonstrated the ability of our UQ-ECGI formulations to characterize the effect of parameter uncertainty on the ECGI inverse problem. We found that while the PCE-UQ inverse solution provided more complex outputs such as sensitivities and standard deviation, the UQ joint-inverse solution provided a more interpretable output in the form of a single ECGI solution. We find that between these two methods we are able to assess a wide range of effects that heart position variability has on the ECGI solution. Significance. This study, for the first time, characterizes in detail the application of UQ to the ECGI inverse problem. We demonstrated how UQ can provide insight into the behavior of ECGI using variability in cardiac position as a test case. This study lays the groundwork for future development of UQ-ECGI studies, as well as future development of ECGI formulations which are robust to input parameter variability.

Accelerated multiscale mechanics modeling in a deep learning framework

Microstructural heterogeneity affects the macro-scale behavior of materials. Conversely, load distribution at the macro-scale changes the microstructural response. These up-scaling and down-scaling relations are often modeled using multiscale finite element (FE) approaches such as FE-squared (FE2). However, FE2 requires numerous calculations at the micro-scale, which often renders this approach intractable. This paper reports an enormously faster machine learning (ML) based approach for multiscale mechanics modeling. The proposed ML-driven multiscale analysis approach uses an ML-model that predicts the local stress tensor fields in a linear elastic fiber-reinforced composite microstructure. This ML-model, specifically a U-Net deep convolutional neural network (CNN), is trained separately to perform the mapping between the spatial arrangement of fibers and the corresponding 2D stress tensor fields. This ML-model provides effective elastic material properties for up-scaling and local stress tensor fields for subsequent down-scaling in a multiscale analysis framework. Several numerical examples demonstrate a substantial reduction in computational cost using the proposed ML-driven approach when compared with the traditional multiscale modeling approaches such as full-scale FE analysis, and homogenization based FE2 analysis. This approach has tremendous potential in efficient multiscale analysis of complex heterogeneous materials, with applications in uncertainty quantification, design, and optimization.

Uncertainty Quantification of Laminar–Turbulent Transition on Airfoil and Fuselage

The effect of uncertainty in Reynolds-averaged Navier–Stokes (RANS) simulations is determined through the application of uncertainty quantification (UQ). In this study, the sensitivity of aerodynamic coefficients to uncertainty in the freestream turbulence intensity (FSTI) and surface roughness is computed for an airfoil (SC1095) and a rotorcraft fuselage model (Rotor-Body-Interaction-Mod7). A laminar–turbulent transition model has been extended to consider roughness-induced transition by incorporating an existing roughness-induced transition model. The current UQ analysis is based on the Monte Carlo method with the Gaussian distributions of uncertain input parameters. A surrogate model allows for incorporating the results of intensive RANS simulations into the Monte Carlo method. The surrogate model is generated using a cubic interpolation for a single-uncertain parameter or a radial basis function for multiple-uncertain parameters. The stochastic standard deviation is measured using combined FSTI and roughness uncertainties. The standard deviation of each aerodynamic coefficient depends on flow conditions (angle of attack, mean FSTI, and mean roughness) and the transition type (natural or separation-induced). The main source of output uncertainty is identified between the unknown FSTI and roughness under various flow conditions.

Research on the Application of Uncertainty Quantification (UQ) Method in High-Voltage (HV) Cable Fault Location

In HV cable fault location technology, line parameter uncertainty has an impact on the location criterion and affects the fault location result. Therefore, it is of great significance to study the uncertainty quantification of line parameters. In this paper, an impedance-based fault location criterion was used for an uncertainty study. Three kinds of uncertainty factors, namely the sheath resistivity per unit length, the equivalent grounding resistance on both sides, and the length of the cable section, were taken as random input variables without interaction. They were subject to random uniform distribution within a 50% amplitude variation. The relevant statistical information, such as the mean value, standard deviation and probability distribution, of the normal operation and fault state were calculated using the Monte Carlo simulation (MCS) method, the polynomial chaos expansion (PCE) method, and the univariate dimension reduction method (UDRM), respectively. Thus, the influence of uncertain factors on fault location was analyzed, and the calculation results of the three uncertainty quantification methods compared. The results indicate that: (1) UQ methods are effective for simulation analysis of fault locations, and UDRM has certain application prospects for HV fault location in practice; (2) the quantification results of the MCS, PCE, and UDRM were very close, while the mean convergence rate was significantly higher for the UDRM; (3) compared with the MCS, PCE, and UDRM, the PCE and UDRM had higher accuracy, and MCS and UDRM required less running time.

Stabilized Scalar Auxiliary Variable Ensemble Algorithms for Parameterized Flow Problems

Computing a flow system a number of times with different samples of flow parameters is a common practice in many uncertainty quantification applications, which can be prohibitively expensive for complex nonlinear flow problems. This report presents two second order, stabilized, scalar auxiliary variable (SAV) ensemble algorithms for fast computation of the Navier--Stokes flow ensembles: Stab-SAV-CN and Stab-SAV-BDF2. The proposed ensemble algorithms are based on the ensemble timestepping idea which makes use of a quantity called the ensemble mean to construct a common coefficient matrix for all realizations at the same time step after spatial discretization, in which case efficient block solvers, e.g., block GMRES, can be used to significantly reduce both storage and computational time. The adoption of a recently developed SAV approach that treats the nonlinear term explicitly results in a constant shared coefficient matrix among all realizations at different time steps, which further cuts down the computational cost, yielding an extremely efficient ensemble algorithm for simulating nonlinear flow ensembles with provable long time stability without any time step conditions. The SAV approach for the Navier--Stokes equations for a single realization was proved to be unconditionally stable in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1--22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465--2491]. However we found the SAV approach has very low accuracy that compromises its stability in our initial numerical investigations for several commonly tested benchmark flow problems. In this report, we propose to use the stabilization $-\alpha h \Delta (u^{n+1}-u^n)$ in Stab-SAV-CN and $-\alpha h \Delta (3u^{n+1}-4 u^{n}+u^{n-1})$ in Stab-SAV-BDF2 to address this issue. We prove that both of our ensemble algorithms are long time stable under one parameter fluctuation condition, without any time step constraints. For a single realization, both algorithms are unconditionally stable and have better accuracy than the SAV methods studied in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1--22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465--2491] for our test problems. Extensive numerical experiments are performed to show the efficiency of the proposed ensemble algorithms and the effectiveness of the stabilization for increasing accuracy and stability.

Sparse Harmonic Transforms: A New Class of Sublinear-Time Algorithms for Learning Functions of Many Variables

In this paper we develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification (UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases (Chkifa et al. in Polynomial approximation via compressed sensing of high-dimensional functions on lower sets. arXiv:1602.05823 , 2016; Rauhut and Schwab in Math Comput 86(304):661–700, 2017). We expect that our results provide a starting point for a new line of research on sublinear-time solution techniques for UQ applications of the type above which will eventually be able to scale to significantly higher-dimensional problems than what are currently computationally feasible. More concretely, let $${\mathcal {B}}$$ be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality $$|{\mathcal {B}}| = N$$ . Herein we will develop methods that rapidly approximate any function f that is sparse in the BOPB, that is, $$f: {\mathcal {D}} \subset {\mathbb {R}}^D \rightarrow {\mathbb {C}}$$ of the form $$\begin{aligned} f(\varvec{x}) = \sum _{b \in {\mathcal {S}}} c_b \cdot b(\varvec{x}) \end{aligned}$$ with $${\mathcal {S}} \subset {\mathcal {B}}$$ of cardinality $$|{\mathcal {S}}| = s \ll N$$ . Our method adapts the CoSaMP algorithm (Needell and Tropp in Appl Comput Harmon Anal 26(3):301–321, 2009) to use additional function samples from f along a randomly constructed grid $${\mathcal {G}} \subset {\mathbb {R}}^D$$ with universal approximation properties in order to rapidly identify the multi-indices of the most dominant basis functions in $${\mathcal {S}}$$ component by component during each CoSaMP iteration. It has a runtime of just $$(s \log N)^{{\mathcal {O}}(1)}$$ , uses only $$(s \log N)^{{\mathcal {O}}(1)}$$ function evaluations on the fixed and nonadaptive grid $${\mathcal {G}}$$ , and requires not more than $$(s \log N)^{{\mathcal {O}}(1)}$$ bits of memory. We emphasize that nothing about $${\mathcal {S}}$$ or any of the coefficients $$c_b \in {\mathbb {C}}$$ is assumed in advance other than that $${\mathcal {S}} \subset {\mathcal {B}}$$ has $$|{\mathcal {S}}| \le s$$ . Both $${\mathcal {S}}$$ and its related coefficients $$c_b$$ will be learned from the given function evaluations by the developed method. For $$s\ll N$$ , the runtime $$(s \log N)^{{\mathcal {O}}(1)}$$ will be less than what is required to simply enumerate the elements of the basis $${\mathcal {B}}$$ ; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as sublinear-time. This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size $$\varOmega (N)$$ during the solution process.

ON TRANSFER LEARNING OF NEURAL NETWORKS USING BI-FIDELITY DATA FOR UNCERTAINTY PROPAGATION

Due to their high degree of expressiveness, neural networks have recently been used as surrogate models for mapping inputs of an engineering system to outputs of interest. Once trained, neural networks are computationally inexpensive to evaluate and remove the need for repeated evaluations of computationally expensive models in uncertainty quantification applications. However, given the highly parameterized construction of neural networks, especially deep neural networks, accurate training often requires large amounts of simulation data that may not be available in the case of computationally expensive systems. In this paper, to alleviate this issue for uncertainty propagation, we explore the application of transfer learning techniques using training data generated from both high- and low-fidelity models. We explore two strategies for coupling these two datasets during the training procedure, namely, the standard transfer learning and the bi-fidelity weighted learning. In the former approach, a neural network model mapping the inputs to the outputs of interest is trained based on the low-fidelity data. The high-fidelity data is then used to adapt the parameters of the upper layer(s) of the low-fidelity network, or train a simpler neural network to map the output of the low-fidelity network to that of the high-fidelity model. In the latter approach, the entire low-fidelity network parameters are updated using data generated via a Gaussian process model trained with a small high-fidelity dataset. The parameter updates are performed via a variant of stochastic gradient descent with learning rates given by the Gaussian process model. Using three numerical examples, we illustrate the utility of these bi-fidelity transfer learning methods where we focus on accuracy improvement achieved by transfer learning over standard training approaches.

Importance of uncertainty quantification in nuclear fuel behaviour modelling and simulation

Fuel performance codes describe the physical phenomena occurring in fuel during its irradiation in steady-state and transient conditions. In this paper, the principles of the modelling of nuclear fuel behaviour are presented. Many reasons show that it is also important to consider uncertainties in the fuel modelling process. It can be experimental reasons (validation database and material properties determination), uncertainties in the modelling process itself (lack of knowledge for example). This paper also presents several applications of uncertainty quantification and sensitivity analysis applied to fuel behaviour modelling. The first example is about the validation process of the fuel performance code, which results in a quantification of the uncertainty of the calculated quantities. The second application is about the use of the code in safety analysis to determine probabilities of failure considering epistemic and random uncertainties. In the last application, we show how important it is to quantify modelling uncertainties in the calibration process of complex physical models.

Analyzing nuclear reactor simulation data and uncertainty with the group method of data handling

Abstract Group method of data handling (GMDH) is considered one of the earliest deep learning methods. Deep learning gained additional interest in today's applications due to its capability to handle complex and high dimensional problems. In this study, multi-layer GMDH networks are used to perform uncertainty quantification (UQ) and sensitivity analysis (SA) of nuclear reactor simulations. GMDH is utilized as a surrogate/metamodel to replace high fidelity computer models with cheap-to-evaluate surrogate models, which facilitate UQ and SA tasks (e.g. variance decomposition, uncertainty propagation, etc.). GMDH performance is validated through two UQ applications in reactor simulations: (1) low dimensional input space (two-phase flow in a reactor channel), and (2) high dimensional space (8-group homogenized cross-sections). In both applications, GMDH networks show very good performance with small mean absolute and squared errors as well as high accuracy in capturing the target variance. GMDH is utilized afterward to perform UQ tasks such as variance decomposition through Sobol indices, and GMDH-based uncertainty propagation with large number of samples. GMDH performance is also compared to other surrogates including Gaussian processes and polynomial chaos expansions. The comparison shows that GMDH has competitive performance with the other methods for the low dimensional problem, and reliable performance for the high dimensional problem.

Multi-fidelity uncertainty quantification method with application to nonlinear structural response analysis

• A multi-fidelity method is proposed for statistical moment calculations of nonlinear structural responses. • The proposed formulation requires only small numbers of high fidelity solution and simple statistical techniques. • High-cost material damage analysis can be limited in local areas without biasing results of uncertainty quantification . • A computational cost reduction of 52.7% has been achieved in the numerical example. The application of uncertainty quantification (UQ) in complex structural response analysis is limited by solution efficiency. A multi-fidelity (MF) method for UQ is proposed in this paper, in which statistical moments are first evaluated using low cost low-fidelity (LF) model first, and then calibrated with a small number of high-fidelity (HF) samples. Only the error distribution of LF solutions and the covariance between the errors and the LF solutions are employed to derive a simple and straight forward MF formulation. The proposed method is demonstrated in the UQ of damage analysis of a C/SiC plate with a hole, where the HF model is a nonlinear global model considering C/SiC material damage, and the LF model is a linear global model driven nonlinear sub model. Uncertainty propagation is carried out using a sparse polynomial chaos expansion method. Evaluations of the MF method based on four factors: correctness, efficiency, precision and reliability, are carried out. The results show that the MF method can estimate the statistical moments of nonlinear strain responses unbiasedly. Computational cost is reduced by 52.7% compared to that utilizing HF model alone. MF methods can reduce computational cost significantly while maintaining accuracy and can be used for wide range of applications.

The need for uncertainty quantification in machine-assisted medical decision making

Medicine, even from the earliest days of artificial intelligence (AI) research, has been one of the most inspiring and promising domains for the application of AI-based approaches. Equally, it has been one of the more challenging areas to see an effective adoption. There are many reasons for this, primarily the reluctance to delegate decision making to machine intelligence in cases where patient safety is at stake. To address some of these challenges, medical AI, especially in its modern data-rich deep learning guise, needs to develop a principled and formal uncertainty quantification (UQ) discipline, just as we have seen in fields such as nuclear stockpile stewardship and risk management. The data-rich world of AI-based learning and the frequent absence of a well-understood underlying theory poses its own unique challenges to straightforward adoption of UQ. These challenges, while not trivial, also present significant new research opportunities for the development of new theoretical approaches, and for the practical applications of UQ in the area of machine-assisted medical decision making. Understanding prediction system structure and defensibly quantifying uncertainty is possible, and, if done, can significantly benefit both research and practical applications of AI in this critical domain. Arguably one of the most promising as well as critical applications of deep learning is in supporting medical sciences and decision making. It is time to develop methods for systematically quantifying uncertainty underlying deep learning processes, which would lead to increased confidence in practical applicability of these approaches.

Model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification

In this work, we present a model order reduction (MOR) technique for hyperbolic conservation laws with applications in uncertainty quantification (UQ). The problem consists of a parametrized time dependent hyperbolic system of equations, where the parameters affect the initial conditions and the fluxes in a non- linear way. The procedure utilized to reduce the order is a combination of a Greedy algorithm in the parameter space, a proper orthogonal decomposition (POD) in time and empirical interpolation method (EIM) to deal with non-linearities (Drohmann, 2012). We provide under some hypothesis an error bound for the reduced solution with respect to the high order one. The algorithm shows small errors and savings of the computational time up to 90% in the UQ simulations, which are performed to validate the algorithm.