Probabilistic inference, especially in the Bayesian framework, is a foundation for quantifying uncertainties in geophysical inversion applications. However, due to the presence of high-dimensional datasets and the large-scale nature of geophysical inverse problems, the applicability and scalability of probabilistic inference face significant challenges for such applications. This thesis is dedicated to improving the probabilistic inference algorithms' scalability and demonstrating their applicability for large-scale geophysical inversion applications. In this thesis, I delve into three leading applied approaches in computing the Bayesian posterior distribution in geophysical inversion applications: Laplace's approximation, Markov chain Monte Carlo (MCMC), and variational Bayesian inference. The first approach, Laplace's approximation, is the simplest form of approximation for intractable Bayesian posteriors. However, its accuracy relies on the estimation of the posterior covariance matrix. I study the visualization of the misfit landscape in low-dimensional subspace and the low-rank approximations of the covariance for full waveform inversion (FWI). I demonstrate that a non-optimal Hessian's eigenvalues truncation for the low-rank approximation will affect the approximation accuracy of the standard deviation, leading to a biased statistical conclusion. Furthermore, I also demonstrate the propagation of uncertainties within the Bayesian physics-informed neural networks for hypocenter localization applications through this approach. For the MCMC approach, I develop approximate Langevin MCMC algorithms that provide fast sampling at efficient computational costs for large-scale Bayesian FWI; however, this inflates the variance due to asymptotic bias. To account for this asymptotic bias and assess their sample quality, I introduce the kernelized Stein discrepancy (KSD) as a diagnostic tool. When larger computational resources are available, exact MCMC algorithms (i.e., with a Metropolis-Hastings criterion) should be favored for an accurate posterior distribution statistical analysis. For the variational Bayesian inference, I propose a regularized variational inference framework that performs posterior inference by implicitly regularizing the Kullback-Leibler divergence loss with a deep denoiser through a Plug-and-Play method. I also developed Plug-and-Play Stein Variational Gradient Descent (PnP-SVGD), a novel algorithm to sample the regularized posterior distribution. The PnP-SVGD demonstrates its ability to produce high-resolution, trustworthy samples representative of the subsurface structures for a post-stack seismic inversion application.

<p>Extrusion Deposition Additive Manufacturing is a process by which short fiber-reinforced polymers are extruded in a screw and deposited onto a build platform using a set of instructions specified in the form of a machine code. The highly non-isothermal process can lead to undesired effects in the form of residual deformation and part delamination. Process simulations that can predict residual deformation and part delamination have been a thrust area of research to prevent the repeated trial and error process before a useful part has been produced. However, populating the material properties required for the process simulations require extensive characterization efforts. Tackling this experimental bottleneck is the focus of the first half of this research.</p><p>The first contribution is a method to infer the fiber orientation state from only tensile tests. While measuring fiber orientation state using computed tomography and optical microscopy is possible, they are often time-consuming, and limited to measuring fibers with circular cross-sections. The knowledge of the fiber orientation is extremely useful in populating material properties using micromechanics models. To that end, two methods to infer the fiber orientation state are proposed. The first is Bayesian methodology which accounts for aleatoric and epistemic uncertainty. The second method is a deterministic method that returns an average value of the fiber orientation state and polymer properties. The inferred orientation state is validated by performing process simulations using material properties populated using the inferred orientation state. A different challenge arises when dealing with multiple extrusion systems. Considering even the same material printed on different extrusion systems requires an engineer to redo the material characterization efforts (due to changes in microstructure). This, in turn, makes characterization efforts expensive and time-consuming. Therefore, the objective of the second contribution is to address this experimental bottleneck and use prior information about the material manufactured in one extrusion system to predict its properties when manufactured in another system. A framework that can transfer thermal conductivity data while accounting for uncertainties arising from different sources is presented. The predicted properties are compared to experimental measurements and are found to be in good agreement.</p><p>While the process simulations using finite element methods provide a reliable framework for the prediction of residual deformation and part delamination, they are often computationally expensive. Tackling the fundamental challenges regarding this computational bottleneck is the focus of the second half of this dissertation. To that end, as the third contribution, a neural network based solver is developed that can solve parametric partial differential equations. This is attained by deriving the weak form of the governing partial differential equation. Using this variational form, a novel loss function is proposed that does not require the evaluation of the integrals arising out of the weak form using Gauss quadrature methods. Rather, the integrals are identified to be expectation values for which an unbiased estimator is developed. The method is tested for parabolic and elliptical partial differential equations and the results compare well with conventional solvers. Finally, the fourth contribution of this dissertation involves using the new solver to solve heat transfer problems in additive manufacturing, without the need for discretizing the time domain. A neural network is used to solve the governing equations in the evolving geometry. The weak form based loss is altered to account for the evolving geometry by using a novel sequential collocation sampling method. This work forms the foundational work to solve parametric problems in additive manufacturing.</p>

<p>Nuclear engineering hosts diverse domains including, but not limited to, power plant automation, human-machine interfacing, detection and identification of special nuclear materials, modeling of reactor kinetics and dynamics that most frequently are described by systems of differential equations (DEs), either ordinary (ODEs) or partial ones (PDEs). In this work we study multiple problems related to safety and Special Nuclear Material detection, and numerical solutions for partial differential equations using neural networks. More specifically, this work is divided in six chapters. Chapter 1 is the introduction, in Chapter</p> <p>2 we discuss the development of a gamma-ray radionuclide library for the characterization</p> <p>of gamma-spectra. In Chapter 3, we present a new approach, the ”Variance Counterbalancing”, for stochastic</p> <p>large-scale learning. In Chapter 4, we introduce a systematic approach for constructing proper trial solutions to partial differential equations (PDEs) of up to second order, using neural forms that satisfy prescribed initial, boundary and interface conditions. Chapter 5 is about an alternative, less imposing development of neural-form trial solutions for PDEs, inside rectangular and non-rectangular convex boundaries. Chapter 6 presents an ensemble method that avoids the multicollinearity issue and provides</p> <p>enhanced generalization performance that could be suitable for handling ”few-shots”- problems frequently appearing in nuclear engineering.</p>