Promising directions of machine learning for partial differential equations
SL Brunton, JN Kutz - Nature Computational Science, 2024 - nature.com
Partial differential equations (PDEs) are among the most universal and parsimonious
descriptions of natural physical laws, capturing a rich variety of phenomenology and …
descriptions of natural physical laws, capturing a rich variety of phenomenology and …
Operator learning: Algorithms and analysis
Operator learning refers to the application of ideas from machine learning to approximate
(typically nonlinear) operators mapping between Banach spaces of functions. Such …
(typically nonlinear) operators mapping between Banach spaces of functions. Such …
Operator learning using random features: A tool for scientific computing
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 powerful …
pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful …
Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning
One of the main challenges in molecular dynamics is overcoming the 'timescale barrier': in
many realistic molecular systems, biologically important rare transitions occur on timescales …
many realistic molecular systems, biologically important rare transitions occur on timescales …
Machine learning for partial differential equations
SL Brunton, JN Kutz - arXiv preprint arXiv:2303.17078, 2023 - arxiv.org
Partial differential equations (PDEs) are among the most universal and parsimonious
descriptions of natural physical laws, capturing a rich variety of phenomenology and multi …
descriptions of natural physical laws, capturing a rich variety of phenomenology and multi …
Error bounds for learning with vector-valued random features
S Lanthaler, NH Nelsen - Advances in Neural Information …, 2024 - proceedings.neurips.cc
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 …
features (RF). The theory is developed for RF ridge regression in a fully general infinite …
An operator learning perspective on parameter-to-observable maps
Computationally efficient surrogates for parametrized physical models play a crucial role in
science and engineering. Operator learning provides data-driven surrogates that map …
science and engineering. Operator learning provides data-driven surrogates that map …
Towards optimal Sobolev norm rates for the vector-valued regularized least-squares algorithm
We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a
continuous scale of norms that interpolate between $ L_2 $ and the hypothesis space, which …
continuous scale of norms that interpolate between $ L_2 $ and the hypothesis space, which …
Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms
We study theoretical properties of a broad class of regularized algorithms with vector-valued
output. These spectral algorithms include kernel ridge regression, kernel principal …
output. These spectral algorithms include kernel ridge regression, kernel principal …
Nonparametric Control-Koopman Operator Learning: Flexible and Scalable Models for Prediction and Control
Linearity of Koopman operators and simplicity of their estimators coupled with model-
reduction capabilities has lead to their great popularity in applications for learning dynamical …
reduction capabilities has lead to their great popularity in applications for learning dynamical …