Nicholas H. Nelsen
NSF Graduate Research Fellow and Ph.D. Candidate
California Institute of Technology
About Me
Welcome! I am a fourth year graduate student in the Division of Engineering and Applied Science at Caltech, where I work with my advisor Prof. Andrew M. Stuart, and my research interests are in theory and algorithms for high-dimensional scientific and data-driven computation.
My current work is centered on operator regression—learning, from (noisy) data, operators that map between infinite-dimensional (function) spaces—with application to forward and inverse problems arising from parametric partial differential equations (PDEs) that model physical systems. To this end, I develop and utilize tools from machine learning, model reduction, and numerical/statistical analysis. To learn more about my background and research experience, please refer to my curriculum vitae and my publications page.
I am fortunate to be supported by a NSF Graduate Research Fellowship. In 2020, I obtained my M.Sc. from Caltech, and before starting doctoral study in the fall of 2018, I worked on Lagrangian particle methods for PDEs as a summer research intern in the Center for Computing Research at Sandia National Laboratories. I obtained my B.Sc. (Mathematics), B.S.M.E., and B.S.A.E. degrees from Oklahoma State University in 2018.
Recent News
2021/10: I am virtually attending the Statistical Aspects of Non-Linear Inverse Problems workshop hosted by the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) as an invited participant from October 31st to November 5th.
2021/10: I am giving an invited talk at the Caltech CMX Student Seminar on some of my Ph.D. work on operator regression.
2021/09: My paper on Banach space random feature methods, joint work with A.M. Stuart, was published in the SIAM Journal on Scientific Computing.
2021/09: I am virtually attending the Deep Learning and Inverse Problems workshop as a part of the Mathematics of Deep Learning programme at the Isaac Newton Institute for Mathematical Sciences, Cambridge UK.
2021/08: My new preprint on "Convergence Rates for Learning Linear Operators from Noisy Data," joint with M.V. de Hoop, N.B. Kovachki, and A.M. Stuart, is now available. In it, we prove that a class of compact, bounded, and even unbounded operators can be stably estimated from noisy input-output pairs.