Type Ia supernovae (SNe Ia) enrich galaxies with iron group and some intermediatemass elements while also serving as standardizable candles for testing cosmological models. Despite their importance in understanding the evolution of the universe, the progenitors of SNe Ia remain elusive. Cosmic distance estimation and the chemical evolution of the universe depend on the exact progenitor mechanism, therefore there is a need to identify their origins. A myriad of models have been developed over the past several decades to explain their unique observational features, all involving the thermonuclear disruption of a carbon- oxygen (C/O) white dwarf (WD) in a binary system in one of three major regimes. 1) A WD accretes mass from a companion, approaching the Chandrasekhar-mass, and initiates a thermonuclear runaway 2) A helium layer formed through accretion ignites and generates a thermonuclear burning front on the surface that drives a converging shock into the core 3) A merger between a pair of WDs initiates a carbon detonation as accreted material produces a hot spot on the surface. The nature of the binary interaction has a large effect on which channel may lead to the explosion including the composition of the accreted material, the accretion rate, and the nature of the binary evolution of the system. Each progenitor channel assumes a specific flame propagation mechanism that imprints itself on the stratification of abundances and densities within the ejecta. Inferring the stratification of these elements and their density distribution allows for making testable predictions regarding their origins. The elemental composition within the ejecta can be determined by modeling spectral observations with radiative transfer simulations. Supernova radiative transfer is very costly taking at a minimum tens of minutes to evaluate a simple spectrum. Thus exploring the large parameter space with tens of dimensions is out of the realm of current and future computational facilities. Thus, traditionally, such methods have relied on qualitative metrics of model fits and manual adjustments of elemental compositions. The results have then lacked information on the uncertainties and parameter degeneracies not unambiguously identifying progenitors. This thesis presents a novel methodology for rapid probabilistic reconstructions of SNe Ia through the application of deep-learning accelerated radiative transfer simulations under parametric ejecta models. This methodology is applied to explore the progenitors of SNe Ia in three different projects. First, analysis of the elemental composition of the outer ejecta of the archetypal SN Ia SN 2002bo shows that the parameter space is complex with multiple parameter degeneracies and multi-modalities but is overall inconsistent with traditional pure-deflagration models. Second, modeling the outer ejecta of a population of the super-luminous silicon-deficient 1991T-like thermonuclear supernovae finds that they appear as an extension or extreme case of the normal SN Ia population with their unique observational signatures primarily dictated by small deviations in production of intermediate-mass elements with higher ionization rates. Finally, progenitor channel probabilities are prescribed to the well observed SN Ia SN 2011fe by sampling a space of high-dimensional hydrodynamical models corresponding to a variety of SN Ia progenitor channels showing that it is best described by a core-detonation model of a sub-Chandrasekhar mass WD. These results both elucidate the progenitors of SNe Ia as well as provide insight regarding the limitations of current models to solve more detailed questions about their origins.
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