133 citations · 198 across the 12 of their papers we have counts for
14 papers
Stability of Measure-to-Measure Transformers on Sub-Gaussian Data
Frank Cole, Nicholas H. Nelsen, Takashi Furuya
Transformers have exhibited impressive empirical success across various domains, but their theoretical foundations remain less developed. This work constitutes a mathematical study…
Universal Approximation of Measure-to-Measure Operators by Pushforwards
Takashi Furuya, Nicholas H. Nelsen, Frank Cole
Many learning tasks map an input distribution to an output distribution. A natural way to model such an operator is to transform each input sample using a continuous function that…
One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators
Panos Tsimpos, Edoardo Calvello, Ayoub Belhadji +1
Probabilistic conditioning is concerned with the identification of a distribution of a random variable given a random variable . It is a cornerstone of scientific and engine…
A short tour of operator learning theory: Convergence rates, statistical limits, and open questions
Simone Brugiapaglia, Nicola Rares Franco, Nicholas H. Nelsen
This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical…
Extension and neural operator approximation of the electrical impedance tomography inverse map
Maarten V. de Hoop, Nikola B. Kovachki, Matti Lassas +1
This paper considers the problem of noise-robust neural operator approximation for the solution map of Calderón's inverse conductivity problem. In this continuum model of electrica…
Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes
Nicholas H. Nelsen, Houman Owhadi, Andrew M. Stuart +2
Methods for solving scientific computing and inference problems, such as kernel- and neural network-based approaches for partial differential equations (PDEs), inverse problems, an…
Operator learning meets inverse problems: A probabilistic perspective
Nicholas H. Nelsen, Yunan Yang
Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems…
Learning where to learn: Training data distribution optimization for scientific machine learning
Nicolas Guerra, Nicholas H. Nelsen, Yunan Yang
In scientific machine learning, models are routinely deployed with parameter values or boundary conditions far from those used in training. This paper studies the learning-where-to…
Operator Learning Using Random Features: A Tool for Scientific Computing
Nicholas H. Nelsen, Andrew M. Stuart
Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a power…
Hyperparameter Optimization for Randomized Algorithms: A Case Study on Random Features
Oliver R. A. Dunbar, Nicholas H. Nelsen, Maya Mutic
Randomized algorithms exploit stochasticity to reduce computational complexity. One important example is random feature regression (RFR) that accelerates Gaussian process regressio…
An operator learning perspective on parameter-to-observable maps
Daniel Zhengyu Huang, Nicholas H. Nelsen, Margaret Trautner
Computationally efficient surrogates for parametrized physical models play a crucial role in science and engineering. Operator learning provides data-driven surrogates that map bet…
Error Bounds for Learning with Vector-Valued Random Features
Samuel Lanthaler, Nicholas H. Nelsen
This paper provides a comprehensive error analysis of learning with vector-valued random features (RF). The theory is developed for RF ridge regression in a fully general infinite-…
Convergence Rates for Learning Linear Operators from Noisy Data
Maarten V. de Hoop, Nikola B. Kovachki, Nicholas H. Nelsen +1
This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces. The training data comprises pairs of random input vectors in a Hilbert space and th…
The Random Feature Model for Input-Output Maps between Banach Spaces
Nicholas H. Nelsen, Andrew M. Stuart
Well known to the machine learning community, the random feature model is a parametric approximation to kernel interpolation or regression methods. It is typically used to approxim…