Nicholas H. Nelsen
NSF Graduate Research Fellow and Ph.D. Candidate
California Institute of Technology
About Me
Welcome! I am a fifth year graduate student in the Division of Engineering and Applied Science at Caltech, where I work with my advisor Prof. Andrew M. Stuart. My research interests are in theory and algorithms for high-dimensional scientific and data-driven computation. Within applied and computational mathematics, some particular areas that I work in include scientific machine learning, inverse problems, uncertainty quantification, and statistical inference.
My current work is centered on operator learning—regressing, from (noisy) data, operators that map between infinite-dimensional (function) spaces—with application to forward and inverse problems, especially those arising from parametric partial differential equations (PDEs) that model complex physical systems. To this end, I develop and utilize tools from machine learning, model reduction, numerical analysis, and statistics. Please refer to my curriculum vitae and my publications page to learn more about my background and research experience.
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.
nnelsen [at] caltech [dot] edu
Recent News
2022/11 (New): Our paper on the theory of linear operator learning was accepted for publication in the SIAM/ASA Journal on Uncertainty Quantification.
2022/09 (New): Professor Joel Tropp's course lecture notes on "Matrix Analysis" are now publicly available and include chapter III.8 that I wrote on the topic of "Operator-Valued Kernels."
2022/09: I am giving an invited talk about "Scalable Uncertainty Quantification with Random Features" in MS85: Recent Advances in Kernel Methods for Computing and Learning, part of SIAM MDS22 in San Diego, CA. There, I am also co-organizing MS81: Provable Guarantees for Learning Dynamical Systems.
2022/06: An improved version of my work on linear operator learning is now available on arXiv. In it, three fundamental principles reveal the types of linear operators, types of training data, and types of distribution shift that lead to reduced sample size requirements for supervised learning in infinite dimensions.
2022/01: This year I am co-organizing the Caltech Department of Computing and Mathematical Sciences CMX Student/Postdoc Seminar.