Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, current neural network-based PDE solvers often rely on extensive training data or labeled input-output pairs, making them prone to challenges in generalizing to out-of-distribution examples. To mitigate the generalization gap encountered by conventional neural network-based methods in estimating PDE solutions, we formulate a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small convolutional neural networks. Our proposed algorithms demonstrate a comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.

FMG-Net and W-Net: Multigrid Inspired Deep Learning Architectures For Medical Imaging Segmentation

Accurate medical imaging segmentation is critical for precise and effective medical interventions. However, despite the success of convolutional neural networks (CNNs) in medical image segmentation, they still face challenges in handling fine-scale features and variations in image scales. These challenges are particularly evident in complex and challenging segmentation tasks, such as the BraTS multi-label brain tumor segmentation challenge. In this task, accurately segmenting the various tumor sub-components, which vary significantly in size and shape, remains a significant challenge, with even state-of-the-art methods producing substantial errors. Therefore, we propose two architectures, FMG-Net and W-Net, that incorporate the principles of geometric multigrid methods for solving linear systems of equations into CNNs to address these challenges. Our experiments on the BraTS 2020 dataset demonstrate that both FMG-Net and W-Net outperform the widely used U-Net architecture regarding tumor subcomponent segmentation accuracy and training efficiency. These findings highlight the potential of incorporating the principles of multigrid methods into CNNs to improve the accuracy and efficiency of medical imaging segmentation.

A Weighted Normalized Boundary Loss for Reducing the Hausdorff Distance in Medical Imaging Segmentation

Within medical imaging segmentation, the Dice coefficient and Hausdorff-based metrics are standard measures of success for deep learning models. However, modern loss functions for medical image segmentation often only consider the Dice coefficient or similar region-based metrics during training. As a result, segmentation architectures trained over such loss functions run the risk of achieving high accuracy for the Dice coefficient but low accuracy for Hausdorff-based metrics. Low accuracy on Hausdorff-based metrics can be problematic for applications such as tumor segmentation, where such benchmarks are crucial. For example, high Dice scores accompanied by significant Hausdorff errors could indicate that the predictions fail to detect small tumors. We propose the Weighted Normalized Boundary Loss, a novel loss function to minimize Hausdorff-based metrics with more desirable numerical properties than current methods and with weighting terms for class imbalance. Our loss function outperforms other losses when tested on the BraTS dataset using a standard 3D U-Net and the state-of-the-art nnUNet architectures. These results suggest we can improve segmentation accuracy with our novel loss function.

PocketNet: A Smaller Neural Network for 3D Medical Image Segmentation

Overparameterized deep learning networks have shown impressive performance in the area of automatic medical image segmentation. However, they achieve this performance at an enormous cost in memory, runtime, and energy. A large source of overparameterization in modern neural networks results from doubling the number of feature maps with each downsampling layer. This rapid growth in the number of parameters results in network architectures that require a significant amount of computing resources, making them less accessible and difficult to use. By keeping the number of feature maps constant throughout the network, we derive a new CNN architecture called PocketNet that achieves comparable segmentation results to conventional CNNs while using less than 3% of the number of parameters.