CAREER: Gaussian Processes for Scientific Machine Learning: Theoretical Analysis and Computational Algorithms

Machine learning and artificial intelligence are increasingly changing our lives at every scale, from our personal day-to-day activities to large scale shifts in our society, economy, and geopolitics. These technologies have also profoundly transformed sciences with new ideas and algorithms being developed at an immense speed. However, our mathematical understanding of these algorithms is still far beyond their practical development and widespread adoption. Put simply, in many cases we do not know how or why machine learning algorithms work so well, which in turn limits our ability to safely deploy them in safety critical engineering and scientific scenarios. The goal of this project is to develop mathematical formalism and understanding of problems at the intersection of machine learning and science (i.e., scientific machine learning) using rigorous mathematics, leading to novel algorithms and computational methodologies that are interpretable, supported by rigorous theory, and aware of uncertainties. The project is focused on the development of novel Gaussian Process (GP) based computational frameworks for scientific machine learning that are provably well-posed, robust, and stable, thereby meeting the high standards of scientific computing. The developed methodologies will be capable of rigorous uncertainty quantification and inherit the desirable properties of machine learning algorithms such as flexibility, generalizability, and applicability in high-dimensions. The efforts of the project are directed in three primary directions: (1) GPs for solving nonlinear, high-dimensional and parametric PDEs; (2) GPs for operator learning, emulation, and physics discovery; and (3) GPs for high-dimensional sampling, inference, and generative modeling. Each research direction focuses on the development of algorithms, foundational theory, and concrete applications in engineering and science. The project also contains an extensive education plan focused on machine learning and data science education from high-school through graduate levels with extensive opportunities for training of graduate and undergraduate students as well as local and international outreach. 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.