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Mathematically Rigorous Deep Learning Paradigms for Data-Driven Scientific Modeling /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Mathematically Rigorous Deep Learning Paradigms for Data-Driven Scientific Modeling /Owen N Davis.
Author:	Davis, Owen N.,
Description:	1 electronic resource (207 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 86-02, Section: B.
Notes:	Advisors: Motamed, Mohammad Committee members: Schroder, Jacob; Tempone, Raul; Geraci, Gianluca.
Contained By:	Dissertations Abstracts International86-02B.
Subject:	Applied mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31148643
ISBN:	9798383596715

Mathematically Rigorous Deep Learning Paradigms for Data-Driven Scientific Modeling /
Davis, Owen N.,

Mathematically Rigorous Deep Learning Paradigms for Data-Driven Scientific Modeling /Owen N Davis. - 1 electronic resource (207 pages)

Source: Dissertations Abstracts International, Volume: 86-02, Section: B.

This dissertation explores the crucial role of data-driven modeling in science and engineering, with a focus on developing surrogate models to accelerate large-scale computational tasks, aiding in both outer-loop functions like uncertainty quantification and expensive inner-loop tasks within broader computational frameworks. Challenges arise with increased problem dimension and sparse, noisy training data, particularly significant when constructing surrogates for very expensive computational models where acquiring sufficient high-fidelity training data is unfeasible. In such scenarios, training surrogates from an ensemble of multifidelity information sources of varying accuracy and cost becomes essential. We emphasize neural network-based modeling paradigms, which are flexible in integrating diverse information sources and have demonstrated advantage in scaling to high-dimensional problems, yet generally lack comprehensive mathematical theory elucidating their optimal performance. This thesis aims to advance neural network-based modeling paradigms for both single-fidelity and multifidelity tasks while enhancing the supporting mathematical theory. We present results for both ReLU neural networks and random Fourier feature residual networks. For ReLU networks, we derive approximation error estimates for a broad array of bounded target functions with minimal regularity assumptions and further exploit these estimates to craft a rigorously supported multifidelity computational framework. Regarding random Fourier feature residual networks, we devise a global optimization-free training algorithm to circumvent prevalent neural network training issues and additionally illustrate their application in a bi-fidelity context.

English

ISBN: 9798383596715Subjects--Topical Terms:

377601
Applied mathematics.
Subjects--Index Terms:

Approximation theory

Mathematically Rigorous Deep Learning Paradigms for Data-Driven Scientific Modeling /
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