Deep learning has achieved remarkable success across a wide range of tasks, but its models often suffer from instability and vulnerability: small changes to the input may drastically affect predictions, while optimization can be hindered by sharp loss landscapes. This thesis addresses these issues through the unifying perspective of sensitivity analysis, which examines how neural networks respond to perturbations at both the input and parameter levels. We study Lipschitz networks as a principled way to constrain sensitivity to input perturbations, thereby improving generalization, adversarial robustness, and training stability. To complement this architectural approach, we introduce regularization techniques based on the curvature of the loss function, promoting smoother optimization landscapes and reducing sensitivity to parameter variations. Randomized smoothing is also explored as a probabilistic method for enhancing robustness at decision boundaries. By combining these perspectives, we develop a unified framework where Lipschitz continuity, randomized smoothing, and curvature regularization interact to address fundamental challenges in stability. The thesis contributes both theoretical analysis and practical methodologies, including efficient spectral norm computation, novel Lipschitz-constrained layers, and improved certification procedures.

Stochastic gradient descent (SGD) provides a simple and yet stunningly efficient way to solve a broad range of machine learning problems.

We live in a time when it's never been easier or less expensive to sequence a plant's complete genome. But knowing all a plant's genes is not the same thing as knowing what all those genes do. Michigan State experts in plant biology and computer science plan to close that gap with the help of artificial intelligence and a new $1.4 million grant from the National Science Foundation. Ultimately, the goal is to help farmers grow crops with genes that give their plants the best chance to withstand threats such as drought and disease. To get to that point, though, researchers still need to reveal the fundamental role of many of the genes found in plants.

East Lansing, MI (July 13, 2021) - We live in a time when it's never been easier or less expensive to sequence a plant's complete genome. But knowing all of a plant's genes is not the same thing as knowing what all those genes do. Michigan State experts in plant biology and computer science plan to close that gap with the help of artificial intelligence and a new $1.4 million grant from the National Science Foundation. Ultimately, the goal is to help farmers grow crops with genes that give their plants the best chance to withstand threats such as drought and disease. To get to that point, though, researchers still need to reveal the fundamental role of many of the genes found in plants.

Shiu is a professor in the College of Natural Science's Department of Plant Biology and in Computational Mathematics, Science and Engineering, a department jointly administered by the College of Natural Science and the College of Engineering. The researchers believe that AI can provide the assistance researchers need to crack those tough cases, which represent a sizable fraction of plant genes. To "teach" the AI, the team will program in available data, describing what scientists do know about plant genes and their functions. That's why the team also recruited machine learning ace Yuying Xie, assistant professor in the Department of Computational Mathematics, Science and Engineering, and expert experimentalist Melissa Lehti-Shiu, a research assistant professor in the Department of Plant Biology. In reaching its scientific goals -- using machine learning to predict gene functions -- the team is also aiming to demonstrate the power of AI as a research tool to the plant science community.

We live in a time when it's never been easier or less expensive to sequence a plant's complete genome. But knowing all a plant's genes is not the same thing as knowing what all those genes do. Michigan State experts in plant biology and computer science plan to close that gap with the help of artificial intelligence and a new $1.4 million grant from the National Science Foundation. Ultimately, the goal is to help farmers grow crops with genes that give their plants the best chance to withstand threats such as drought and disease. To get to that point, though, researchers still need to reveal the fundamental role of many of the genes found in plants.

IMAGE: Be "decoding " plant genetics, the MSU researchers hope to help farmers grow crops with genes that give their plants the best chance to withstand threats like drought and disease. We live in a time when it's never been easier or less expensive to sequence a plant's complete genome. But knowing all a plant's genes is not the same thing as knowing what all those genes do. Michigan State experts in plant biology and computer science plan to close that gap with the help of artificial intelligence and a new $1.4 million grant from the National Science Foundation. Ultimately, the goal is to help farmers grow crops with genes that give their plants the best chance to withstand threats such as drought and disease.

Neural network-based machine learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the same time, machine learning has also acquired the reputation of being a set of "black box" type of tricks, without fundamental principles. This has been a real obstacle for making further progress in machine learning. In this article, we try to address the following two very important questions: (1) How machine learning has already impacted and will further impact computational mathematics, scientific computing and computational science? (2) How computational mathematics, particularly numerical analysis, {can} impact machine learning? We describe some of the most important progress that has been made on these issues. Our hope is to put things into a perspective that will help to integrate machine learning with computational mathematics.