Deep Neural Networks (DNNs) have shown promising results in classifying known Low-probability-of-intercept (LPI) radar signals in noisy environments. However, regular DNNs produce low-quality confidence and uncertainty estimates, making them unreliable, which inhibit deployment in real-world settings. Hence, the need for robust uncertainty estimation methods has grown, and two categories emerged, Bayesian approximation and ensemble learning. As autonomous LPI radar classification is deployed in safety-critical environments, this study compares Bayesian Neural Networks (BNNs) and Deep Ensembles (DEs) as uncertainty estimation methods. We synthetically generate a training and test data set, as well as a shifted data set where subtle changes are made to the signal parameters. The methods are evaluated on predictive performance, relevant confidence and uncertainty estimation metrics, and method-related metrics such as model size, training, and inference time. Our results show that our DE achieves slightly higher predictive performance than the BNN on both in-distribution and shifted data with an accuracy of 74% and 32%, respectively. Further, we show that both methods exhibit more cautiousness in their predictions compared to a regular DNN for in-distribution data, while the confidence quality significantly degrades on shifted data. Uncertainty in predictions is evaluated as predictive entropy, and we show that both methods exhibit higher uncertainty on shifted data. We also show that the signal-to-noise ratio affects uncertainty compared to a regular DNN. However, none of the methods exhibit uncertainty when making predictions on unseen signal modulation patterns, which is not a desirable behavior. Further, we conclude that the amount of available resources could influence the choice of the method since DEs are resource-heavy, requiring more memory than a regular DNN or BNN. On the other hand, the BNN requires a far longer training time. / Tidigare studier har visat att djupa neurala nätverk (DNN) kan klassificera signalmönster för en speciell typ av radar (LPI) som är skapad för att vara svår att identifiera och avlyssna. Traditionella neurala nätverk saknar dock ett naturligt sätt att skatta osäkerhet, vilket skadar deras pålitlighet och förhindrar att de används i säkerhetskritiska miljöer. Osäkerhetsskattning för djupinlärning har därför vuxit och på senare tid blivit ett stort område med två tydliga kategorier, Bayesiansk approximering och ensemblemetoder. LPI radarklassificering är av stort intresse för försvarsindustrin, och tekniken kommer med största sannolikhet att appliceras i säkerhetskritiska miljöer. I denna studie jämför vi Bayesianska neurala nätverk och djupa ensembler för LPI radarklassificering. Resultaten från studien pekar på att en djup ensemble uppnår högre träffsäkerhet än ett Bayesianskt neuralt nätverk och att båda metoderna uppvisar återhållsamhet i sina förutsägelser jämfört med ett traditionellt djupt neuralt nätverk. Vi skattar osäkerhet som entropi och visar att osäkerheten i metodernas slutledningar ökar både på höga brusnivåer och på data som är något förskjuten från den kända datadistributionen. Resultaten visar dock att metodernas osäkerhet inte ökar jämfört med ett vanligt nätverk när de får se tidigare osedda signal mönster. Vi visar också att val av metod kan influeras av tillgängliga resurser, eftersom djupa ensembler kräver mycket minne jämfört med ett traditionellt eller Bayesianskt neuralt nätverk.

In response to the planned increase in operational speeds and axle loads of passengertrains that may lead to resonance-induced excessive vibrations in railway bridges,recent studies examined the reliability of bridges concerning train running safety andpassenger comfort limit states. In this respect, valuable information regarding theimportance of input variables can be obtained by conducting Sensitivity Analysis (SA).For instance, the determination of unimportant variables (where they can be treated asconstant) reduces the computational time, which is usually very high for probabilisticsimulations. In some of the previous studies, only deterministic SA has beenperformed. This thesis follows a stochastic approach using Global Sensitivity Analysis(GSA) methods. The considered performance functions are vertical acceleration anddeflection of single track ballasted simply supported reinforced concrete bridges.To reduce the computational time, available semi-analytical solution of a planarbeam under the passage of a series of moving loads is employed. To simulatethe bridge behaviour realistically, simplified methods to account for rail irregularityamplification, train-bridge interactions, and axle load redistribution were adopted.The considered random variables are train modal properties, number of train coaches,bogie spacing, axle spacing and loads, bridge mass, flexural stiffness and damping,and rail amplification factor. The analyses were carried out for a selected set of bridgelengths [10-30]m and a range of train speeds [100-400] km/hr. The study findingsshow that, in both acceleration and displacement, the dynamic response of the bridgeis sensitive to randomness in bridge mass, moment of inertia, coach length, and axleloads. Furthermore, the rail amplification factor and Young’s modulus are primarilyimportant for acceleration and displacement, respectively.

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<p>Terminal airspace around an airport is the biggest bottleneck for commercial operations in the National Airspace System (NAS). In order to prognosticate the safety status of the terminal airspace, effective prediction of the airspace evolution is necessary. While there are fixed procedural structures for managing operations at an airport, the confluence of a large number of aircraft and the complex interactions between the pilots and air traffic controllers make it challenging to predict its evolution. Modeling the high-dimensional spatio-temporal interactions in the airspace given different environmental and infrastructural constraints is necessary for effective predictions of future aircraft trajectories that characterize the airspace state at any given moment. A novel deep learning architecture using Graph Neural Networks is proposed to predict trajectories of aircraft 10 minutes into the future and estimate prog?nostic metrics for the airspace. The uncertainty in the future is quantified by predicting distributions of future trajectories instead of point estimates. The framework’s viability for trajectory prediction and prognosis is demonstrated with terminal airspace data from Dallas Fort Worth International Airport (DFW). </p>

<p dir="ltr">Uncertainty quantification is a science and engineering subject that aims to quantify and analyze the uncertainty arising from mathematical models, simulations, and measurement data. An uncertainty quantification analysis usually consists of conducting experiments to collect data, creating and calibrating mathematical models, predicting through numerical simulation, making decisions using predictive results, and comparing the model prediction with new experimental data.</p><p dir="ltr">The overarching goal of uncertainty quantification is to determine how likely some quantities in this analysis are if some other information is not exactly known and ultimately facilitate decision-making. This dissertation delivers a complete package, including theory, algorithms, and applications of information field theory, a Bayesian uncertainty quantification tool that leverages the state-of-the-art machine learning framework to accelerate solving the classical uncertainty quantification problems specified by differential equations.</p>

For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems and medical imaging problems such as greenhouse gas tracking and computational tomography reconstruction. This dissertation describes advancements in computational tools for solving large-scale inverse problems and for uncertainty quantification. Oftentimes, inverse problems are ill-posed and large-scale. Iterative projection methods have dramatically reduced the computational costs of solving large-scale inverse problems, and regularization methods have been critical in obtaining stable estimations by applying prior information of unknowns via Bayesian inference. However, by combining iterative projection methods and variational regularization methods, hybrid projection approaches, in particular generalized hybrid methods, create a powerful framework that can maximize the benefits of each method. In this dissertation, we describe various advancements and extensions of hybrid projection methods that we developed to address three recent open problems. First, we develop hybrid projection methods that incorporate mixed Gaussian priors, where we seek more sophisticated estimations where the unknowns can be treated as random variables from a mixture of distributions. Second, we describe hybrid projection methods for mean estimation in a hierarchical Bayesian approach. By including more than one prior covariance matrix (e.g., mixed Gaussian priors) or estimating unknowns and hyper-parameters simultaneously (e.g., hierarchical Gaussian priors), we show that better estimations can be obtained. Third, we develop computational tools for a respirometry system that incorporate various regularization methods for both linear and nonlinear respirometry inversions. For the nonlinear systems, blind deconvolution methods are developed and prior knowledge of nonlinear parameters are used to reduce the dimension of the nonlinear systems. Simulated and real-data experiments of the respirometry problems are provided. This dissertation provides advanced tools for computational inversion and uncertainty quantification. / Doctor of Philosophy / For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems such as greenhouse gas tracking where the problem of estimating the amount of added or removed greenhouse gas at the atmosphere gets more difficult. The number of observations has been increased with improvements in measurement technologies (e.g., satellite). Therefore, the inverse problems become large-scale and they are computationally hard to solve. Another example of an inverse problem arises in tomography, where the goal is to examine materials deep underground (e.g., to look for gas or oil) or reconstruct an image of the interior of the human body from exterior measurements (e.g., to look for tumors). For tomography applications, there are typically fewer measurements than unknowns, which results in non-unique solutions. In this dissertation, we treat unknowns as random variables with prior probability distributions in order to compensate for a deficiency in measurements. We consider various additional assumptions on the prior distribution and develop efficient and robust numerical methods for solving inverse problems and for performing uncertainty quantification. We apply our developed methods to many numerical applications such as greenhouse gas tracking, seismic tomography, spherical tomography problems, and the estimation of CO2 of living organisms.

This thesis establishes methods to quantify and explain uncertainty through high-order moments in time series data, along with first principal-based improvements on the standard autoencoder and variational autoencoder. While the first-principal improvements on the standard variational autoencoder provide additional means of explainability, we ultimately look to non-variational methods for quantifying uncertainty under the autoencoder framework. We utilize Shannon's differential entropy to accomplish the task of uncertainty quantification in a general nonlinear and non-Gaussian setting. Together with previously established connections between autoencoders and principal component analysis, we motivate the focus on differential entropy as a proper abstraction of principal component analysis to this more general framework, where nonlinear and non-Gaussian characteristics in the data are permitted. Furthermore, we are able to establish explicit connections between high-order moments in the data to those in the latent space, which induce a natural latent space decomposition, and by extension, an explanation of the estimated uncertainty. The proposed methods are intended to be utilized in economic and financial factor models in state space form, building on recent developments in the application of neural networks to factor models with applications to financial and economic time series analysis. Finally, we demonstrate the efficacy of the proposed methods on high frequency hourly foreign exchange rates, macroeconomic signals, and synthetically generated autoregressive data sets. / Master of Science / This thesis establishes methods to quantify and explain uncertainty in time series data, along with improvements on some latent variable neural networks called autoencoders and variational autoencoders. Autoencoders and varitational autoencodes are called latent variable neural networks since they can estimate a representation of the data that has less dimension than the original data. These neural network architectures have a fundamental connection to a classical latent variable method called principal component analysis, which performs a similar task of dimension reduction but under more restrictive assumptions than autoencoders and variational autoencoders. In contrast to principal component analysis, a common ailment of neural networks is the lack of explainability, which accounts for the colloquial term black-box models. While the improvements on the standard autoencoders and variational autoencoders help with the problem of explainability, we ultimately look to alternative probabilistic methods for quantifying uncertainty. To accomplish this task, we focus on Shannon's differential entropy, which is entropy applied to continuous domains such as time series data. Entropy is intricately connected to the notion of uncertainty, since it depends on the amount of randomness in the data. Together with previously established connections between autoencoders and principal component analysis, we motivate the focus on differential entropy as a proper abstraction of principal component analysis to a general framework that does not require the restrictive assumptions of principal component analysis. Furthermore, we are able to establish explicit connections between high-order moments in the data to the estimated latent variables (i.e., the reduced dimension representation of the data). Estimating high-order moments allows for a more accurate estimation of the true distribution of the data. By connecting the estimated high-order moments in the data to the latent variables, we obtain a natural decomposition of the uncertainty surrounding the latent variables, which allows for increased explainability of the proposed autoencoder. The methods introduced in this thesis are intended to be utilized in a class of economic and financial models called factor models, which are frequently used in policy and investment analysis. A factor model is another type of latent variable model, which in addition to estimating a reduced dimension representation of the data, provides a means to forecast future observations. Finally, we demonstrate the efficacy of the proposed methods on high frequency hourly foreign exchange rates, macroeconomic signals, and synthetically generated autoregressive data sets. The results support the superiority of the entropy-based autoencoder to the standard variational autoencoder both in capability and computational expense.

<p dir="ltr">This research aims to address these critical challenges in uncertainty quantification. The objective is to employ data-driven approaches for UQ with physics models.</p>

Simulations and modeling of large-scale systems are vital to understanding real world phenomena. However, even advanced numerical models can only approximate the true physics. The discrepancy between model results and nature can be attributed to different sources of uncertainty including the parameters of the model, input data, or some missing physics that is not included in the model due to a lack of knowledge or high computational costs. Uncertainty reduction approaches seek to improve the model accuracy by decreasing the overall uncertainties in models. Aiming to contribute to this area, this study explores uncertainty quantification and reduction approaches for complex physical problems. This study proposes several novel probabilistic and statistical approaches for identifying the sources of uncertainty, modeling the errors, and reducing uncertainty to improve the model predictions for large-scale simulations. We explore different computational models. The first class of models studied herein are inherently stochastic, and numerical approximations suffer from stability and accuracy issues. The second class of models are partial differential equations, which capture the laws of mathematical physics; however, they only approximate a more complex reality, and have uncertainties due to missing dynamics which is not captured by the models. The third class are low-fidelity models, which are fast approximations of very expensive high-fidelity models. The reduced-order models have uncertainty due to loss of information in the dimension reduction process. We also consider uncertainty analysis in the data assimilation framework, specifically for ensemble based methods where the effect of sampling errors is alleviated by localization. Finally, we study the uncertainty in numerical weather prediction models coming from approximate descriptions of physical processes. / Ph. D.