Nicholas H. Nelsen
Ph.D. Candidate
Amazon AI4Science Fellow and NSF Graduate Research Fellow
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 centers 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 the Amazon/Caltech AI4Science Fellows Program and 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
2023/06 (new): I am participating in the INdAM Learning for Inverse Problems workshop in Rome, Italy and the BIRS workshop on Scientific Machine Learning at the Banff Centre in Alberta, Canada.
2023/05 (new): Our new preprint establishes state-of-the-art Error Bounds for Learning with Vector-Valued Random Features (joint work with Samuel Lanthaler). The theory holds in a general infinite-dimensional setting (applying to operator learning in particular) and is developed with a matrix-free analysis. This leads to the sharpest known rates (free of log factors) for random feature ridge regression to date.
2023/05 (new): My paper on linear operator learning was published in the SIAM/ASA Journal on Uncertainty Quantification.
2023/05: I am giving an invited talk in the Level Set Seminar at the UCLA Department of Mathematics.
2023/04: I am speaking at the Oden Institute's inaugural Workshop on Scientific Machine Learning at UT Austin, the Workshop on Establishing Benchmarks for Data-Driven Modeling of Physical Systems at USC, and the Southern California Applied Mathematics Symposium (SoCAMS) at UC Irvine.
2023/03: I have been selected as a 2022-2023 Amazon/Caltech AI4Science Fellow! The program recognizes researchers that have had a remarkable impact in artificial intelligence and machine learning, and in their application to fields beyond computer science.