Mathematically Rigorous Deep Learning Paradigms for Data-Driven Scientific Modeling

Author

Owen Nicholas DavisFollow

Publication Date

Spring 4-12-2024

Abstract

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.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Mohammad Motamed

Second Committee Member

Gianluca Geraci

Third Committee Member

Jacob Schroder

Fourth Committee Member

Raul Tempone

Language

English

Keywords

multifidelity modeling, Fourier features, neural networks, approximation theory, deep learning

Document Type

Dissertation

Recommended Citation

Davis, Owen Nicholas. "Mathematically Rigorous Deep Learning Paradigms for Data-Driven Scientific Modeling." (2024). https://digitalrepository.unm.edu/math_etds/204