Publications

Preprints

4. Extension and neural operator approximation of the electrical impedance tomography inverse map

Maarten V. de Hoop, Nikola B. Kovachki, Matti Lassas, and Nicholas H. Nelsen

Submitted November 2025.

[arXiv:2511.20361 math.NA] [Code] [Data]

3. Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes

Nicholas H. Nelsen, Houman Owhadi, Andrew M. Stuart, Xianjin Yang, and Zongren Zou

Submitted October 2025.

[arXiv:2510.05568 stat.ML] [Code]

2. Operator learning meets inverse problems: A probabilistic perspective

Nicholas H. Nelsen and Yunan Yang

Submitted August 2025, revised December 2025.

Accepted in Handbook of Numerical Analysis, Vol. 27: Machine Learning Solutions for Inverse Problems (Part B), Elsevier, 2026.

[arXiv:2508.20207 math.NA]

1. Learning where to learn: Training data distribution optimization for scientific machine learning

Nicolas Guerra, Nicholas H. Nelsen, and Yunan Yang

Submitted May 2025, revised December 2025.

[arXiv:2505.21626 cs.LG] [Code]

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. 

[Download .pdf] [Official Version]

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]

Book Chapters

 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.

[Download .pdf] [Official Version]

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 the Best Thesis in MCE

[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 MAE, 2018. 

[Download .pdf] [Official Version] 

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. 

[Download .pdf] [Official Version]