Deep Learning and Scientific Machine Learning For Medical Imaging Applications

Medical imaging technologies, such as magnetic resonance imaging (MRI) and computed tomography (CT), are invaluable tools in modern medicine - allowing for safe, non-invasive diagnosis and treatment planning. However, leveraging these imaging techniques for scientific modeling often involves a complex process that includes image acquisition, labeling (i.e., segmenting) regions of interest, generating a computational mesh, and numerical simulation. The contributions of this thesis focus on two areas within this pipeline: deep learning-based methods for efficient, scalable, and accurate 3D medical image segmentation and scientific machine learning (SciML) for solving partial differential equations (PDEs). In the realm of medical imaging segmentation, this thesis presents two innovative architectures that are based on geometric multigrid methods: the FMG and W-Nets. These novel architectures improve the segmentation accuracy compared to current state-of-the-art architectures with fewer parameters and converge to lower loss function values in fewer iterations. This thesis goes on to present the Generalized Surface Loss function, a novel, boundary-aware loss function that improves the accuracy of deep learning-based medical imaging segmentation. These innovations are incorporated into a scalable, open-source framework called the Medical Imaging Segmentation Toolkit, created to empower researchers to develop and test new concepts, such as novel architectures and loss functions, in a simple and reproducible pipeline. The presentation of work related to SciML begins by introducing a novel adaptive collocation point selection strategy for Physics-Informed Neural Networks (PINNs) using the QR Discrete Empirical Interpolation Method (QR-DEIM). This method improves both the training efficiency (i.e., convergence to lower loss values in fewer iterations) and predictive accuracy of PINNs by enabling more informed and targeted sampling of collocation points, particularly in regions of high solution complexity. Additionally, this work demonstrates how convolutional neural networks (CNNs) can approximate finite difference and discontinuous Galerkin (DG) solutions to elliptic and parabolic PDEs, both in supervised and unsupervised settings. These results highlight how SciML techniques can enhance traditional numerical solvers. The thesis culminates in the development of graph neural networks (GNNs) for approximating DG solutions over general domains. By representing computational meshes as graphs, these GNNs can generalize across varying resolutions and topologies, offering a flexible tool for scientific computing in irregular domains. Together, the contributions in this dissertation bridge the gap between deep learning, scientific machine learning, and classical numerical methods, offering a cohesive framework for deploying deep learning and SciML in medical imaging and broader computational physics applications.