Theoretical Guarantees of Machine Learning Methods for High Dimensional Partial Differential Equations: Numerical Analysis and Uncertainty Quantification

Machine learning algorithms have achieved tremendous empirical successes in providing practical answers to various applications in our everyday life, such as face recognition and autonomous driving. This project will develop theoretical foundations of machine learning methods for applied problems in science and engineering. The research will play a principal role in determining predictive power and quantifying the robustness and stability of the machine learning methodology in applications. The investigator will mentor graduate and undergraduate students to work on both theoretical and applied aspects of the project. The investigator will provide outreach to high school students with an introductory course on Data Science and develop new mathematical machine learning courses at both graduate and advanced undergraduate levels. The project will develop a systematic mathematical framework for analyzing neural network-based methods for solving partial differential equations (PDEs), emphasizing their high-dimensional performance and uncertainty quantification. The investigator will work on two projects. The first is to derive new dimension-explicit convergence estimates on the generalization error and training dynamics of neural network solutions. This relies on establishing a new regularity theory for PDEs in new complexity-based function spaces tied to neural networks. The second objective is to quantify the uncertainty in the neural network prediction in a Bayesian framework. The research will focus on studying the frequentist performance and the scalable posterior computation of the Bayesian neural networks for solving high dimensional PDEs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.