Date of Award

Spring 2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics & Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

Yuesheng Xu

Committee Member

Gordon Melrose

Committee Member

Ruhai Zhou

Committee Member

Guohui Song

Committee Member

Yuan Zhang

Abstract

To address computational challenges in learning deep neural networks, properties of deep RELU networks were studied to develop a multi-scale learning model. The multi-scale model was compared to the multi-grade learning models. Unlike the deep neural network learned from the standard single-scale, single-grade model, the multi-scale neural networks use low scale information from all hidden layers, and thusly provide a robust approximation method that requires fewer parameters, lower computational time, and is resistant to noise. It is shown that the multiscale method is not subject to issues arising from the vanishing gradient problem. This allows very deep multi-scale networks to be effectively trained. It is proven that the collection of multi-scale neural networks are universal approximators in the space of continuous functions. The neural network learned from a multi-grade model is the superposition of the neural networks, in a stair-shape, each of which is learned from one grade of the learning. Three proof-of-concept numerical examples presented in the paper demonstrate that the multi-scale and multi-grade methods are superior to the single-scale, single-grade networks. The extended analysis on reconstructing kinematic observables in deep inelastic scattering kinematics with multi-scale neural networks shows that not only are those models effective for real world problems, but the power of the approximation is sufficiently great, that it can outperform reconstruction methods based on physical laws.

Rights

In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).

DOI

10.25777/nvkw-dk64

ISBN

9798384444121

Download

Share

COinS