Publications
Much of my current research involves the design and analysis of efficient machine learning algorithms that are tailor-made for scientific and other types of continuum data. I study ways to achieve better accuracy with fewer training data and develop principled uncertainty quantification techniques for operator learning. My work is motivated by scientific computing tasks that involve complex physical systems or inverse problems, where the data is often heterogeneous, noisy, incomplete, and limited in number. I deploy the methodologies arising from my research in several application areas, including medical imaging, climate modeling, and materials science. Please refer to my curriculum vitae to learn more about my background and research experience.
In addition to my published work below, my Google Scholar profile may be found here and my ORCID iD here.
Journal Articles
6. Hyperparameter optimization for randomized algorithms: a case study on random features
Oliver R. A. Dunbar, Nicholas H. Nelsen, and Maya Mutic
Statistics and Computing, Vol. 35, No. 56, 2025.
[Download .pdf] [Official Version] [arXiv:2407.00584 cs.LG] [Code 1] [Code 2] [Code 3]
5. An operator learning perspective on parameter-to-observable maps
Daniel Zhengyu Huang, Nicholas H. Nelsen, and Margaret Trautner
Foundations of Data Science, Vol. 7, No. 1, pp. 163–225, 2025.
[Download .pdf] [Official Version] [arXiv:2402.06031 cs.LG] [Code] [Data]
4. Operator learning using random features: a tool for scientific computing
Nicholas H. Nelsen and Andrew M. Stuart
SIAM Review (SIGEST award section), Vol. 66, No. 3, pp. 535–571, 2024.
[Download .pdf] [Official Version] [arXiv:2408.06526 cs.LG] [Code] [Data] [Short Video] [Long Video] [Poster]
3. Convergence rates for learning linear operators from noisy data
Maarten V. de Hoop, Nikola B. Kovachki, Nicholas H. Nelsen, and Andrew M. Stuart
SIAM/ASA Journal on Uncertainty Quantification, Vol. 11, No. 2, pp. 480–513, 2023.
[Download .pdf] [Official Version] [arXiv:2108.12515 math.ST] [Video]
2. The random feature model for input-output maps between Banach spaces
Nicholas H. Nelsen and Andrew M. Stuart
SIAM Journal on Scientific Computing, Vol. 43, No. 5, pp. A3212–A3243, 2021.
[Download .pdf] [Official Version] [arXiv:2005.10224 math.NA] [Code] [Data] [Short Video] [Long Video] [Poster]
1. Diastolic vortex alterations with reducing left ventricular volume: an in vitro study
Milad Samaee, Nicholas H. Nelsen, Manikantam G. Gaddam, and Arvind Santhanakrishnan
Journal of Biomechanical Engineering, Vol. 142, No. 12, 2020.
Peer Reviewed Conference Papers
1. Error bounds for learning with vector-valued random features
Samuel Lanthaler and Nicholas H. Nelsen
Advances in Neural Information Processing Systems, Vol. 36, pp. 71834–71861, 2023 (NeurIPS 2023 spotlight paper).
[Download .pdf] [Official Version] [arXiv:2305.17170 stat.ML] [Code] [Video] [Poster]
Lecture Notes
1. Operator-valued kernels
Nicholas H. Nelsen
Chap. III.3, pp. 286–297, in ACM 204: Matrix Analysis by Joel A. Tropp, Caltech CMS Lectures Notes Winter 2022.
Theses
3. Statistical foundations of operator learning
Nicholas H. Nelsen
Ph.D. Thesis, California Institute of Technology, 2024.
W.P. Carey and Co. Prize for Best Thesis in Applied & Computational Mathematics and Centennial Prize for Best Thesis in Mechanical & Civil Engineering
[Download .pdf] [Official Version]
2. On partial differential equations modified with fractional operators and integral transformations
Nicholas H. Nelsen
Bachelor's Honors Thesis, Oklahoma State University, Department of Mathematics, 2018.
[Download .pdf] [Official Version]
1. A reduced order framework for optimal control of nonlinear partial differential equations
Nicholas H. Nelsen
Bachelor's Honors Thesis, Oklahoma State University, School of Mechanical and Aerospace Engineering, 2018.
Miscellaneous
2. Lagrangian particle methods for the shallow water equations in varied geometries
Nicholas H. Nelsen and Peter A. Bosler
Sandia National Laboratories Center for Computing Research Summer Proceedings 2018, pp. 163–182, SAND2019-5093R, 2019.
[Download .pdf] [Official Version]
1. Advanced and exploratory shock sensing mechanisms
Nicholas H. Nelsen et al.
Sandia National Laboratories Technical Report SAND2017-10221, 2017.