Emma Hart

Ph.D. Student
Emory University - Department of Mathematics

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Current and ongoing work with Julianne and Matthias Chung. In this work, we describe a new data-driven approach for inverse problems that exploits technologies from machine learning (e.g., neural networks and autoencoder networks) and dimensionality reduction (e.g., low-rank and latent representations). We consider a paired autoencoder framework, where two autoencoders are used to efficiently represent the input and target spaces separately, and optimal mappings are learned between latent spaces. Similar to end-to-end approaches, the paired approach creates a surrogate model for forward propagation and for regularized inversion, but our approach can outperform existing approaches in scenarios where training data for unsupervised learning are readily available but labeled training pairs are scarce. We focus on interpretations using Bayes risk and empirical Bayes risk minimization, and we provide various theoretical results and connections to existing works on low-rank matrix approximations. Moreover, we show that cheaply computable evaluation metrics are available through this framework and can be used to predict whether the solution for a new sample should be predicted well. Work with Yijin Wang, Annan Yu, Soon Hoe Lim, Michael Mahoney, Xiaoye Sherry Li, and Ben Erichson. Flow matching has recently emerged as a powerful paradigm for generative modeling and has been extended to probabilistic time series forecasting in latent spaces. However, the impact of the specific choice of probability path model on forecasting performance remains under-explored. In this work, we demonstrate that forecasting spatio-temporal data with flow matching is highly sensitive to the selection of the probability path model, and we propose a novel probability path model designed to improve forecasting performance. Current and ongoing work with Julianne Chung, Matthias Chung, Bas Peters, and Eldad Haber). We consider the solution of nonlinear inverse problems where the forward problem is a discretization of a partial differential equation. Such problems are notoriously difficult to solve in practice and require minimizing a combination of a data-fit term and a regularization term. The main computational bottleneck of typical algorithms is the direct estimation of the data misfit. Therefore, likelihood-free approaches have become appealing alternatives. Nonetheless, difficulties in generalization and limitations in accuracy have hindered their broader utility and applicability. In this work, we use a paired autoencoder framework as a likelihood-free estimator (LFE) for inverse problems. We show that the use of such an architecture allows us to construct a solution efficiently and to overcome some known open problems when using LFEs. In particular, our framework can assess the quality of the solution and improve on it if needed. We demonstrate the viability of our approach using examples from full waveform inversion and inverse electromagnetic imaging. Work completed at NSF REU with Lars Ruthotto, Elle Buser, and Ben Huenemann. We compared two segmentation methods, one atlas-based and one neural-network-based, to see how well they can each automatically segment the brain stem and cerebellum in Displacement Encoding with Stimulated Echoes Magnetic Resonance Imaging (DENSE-MRI) data. The segmentation is a pre-requisite for estimating the average displacements in these regions, which have recently been proposed as biomarkers in the diagnosis of Chiari Malformation type I (CMI). In numerical experiments, the segmentations of both methods were similar to manual segmentations provided by trained experts. Overall, the neural-network-based method alone produced more accurate segmentations than the atlas-based method did alone, but that a combination of the two methods, in which the atlas-based method is used for the segmentation of the brain stem and the neural-network is used for the segmentation of the cerebellum, may be the most successful. A semester long exploration completed as an undergraduate thesis with Dan Schult. Smoldering combustion is characterized by the slow, low temperature, flameless burning of solid fuel and is the most persistent type of combustion. Flaming combustion, in contrast, involves a higher temperature burning of gaseous fuel and is rather limited in how long it can be sustained. Smoldering and flaming combustion are very interrelated, often occurring simultaneously in nature and seeming to feed into each other. Despite this inter-relatedness, the literatures of the two are somewhat sparsely connected. Better understanding the mechanics of transition between smoldering and flaming combustion, and how the two work to sustain each other, is an especially relevant topic of study in fire safety, engineering, ecology, and earth science contexts alike, among others. This project expands on previously developed models, complicating the combustion reaction scheme in order to be able to support both smoldering and flaming solutions in a single model.