Document Type

Thesis - Open Access

Award Date

2023

Degree Name

Master of Science (MS)

Department / School

Mathematics and Statistics

First Advisor

Jung-Han Kimn

Abstract

Solving higher order partial differential equations (PDEs) can often prove to be a challenging task due to the involvement of higher-order derivatives of the unknown function, particularly for complex problems. The higher the order of the PDE, the more challenging it becomes to obtain an analytical solution. In such cases, alternative numerical methods are often used, such as finite element method or finite difference method. However, these methods can be computationally expensive and require a significant amount of mathematical expertise to implement. In recent times, there has been significant progress in applying neural networks to various fields, including the solution of physical problems that are governed by PDEs and their corresponding constraints. To achieve this, various techniques have been developed for solving PDEs using neural networks which include the use of deep neural networks, residual neural networks, and autoencoder-based neural networks like Physics-Informed Neural Networks (PINNs) and Physics-Constrained Generative Adversarial Networks (PC-GANs). For my thesis, I will be solving a 2D(Space)+1(Time) 4th order PDE, known as the Cahn Hilliard (CH) Equation, through the utilization of a local deep Galerkin method (LDGM). This method involves introducing intermediate variables and reformulating the fourth order PDE into a system of first-order equations. Therefore, this approach avoids the need for calculating high-order derivatives, making it a more efficient method. Then, a multi-output deep neural network is used to approximate the intermediate variables as well as the solutions to the PDE.

Library of Congress Subject Headings

Galerkin methods.
Differential equations, Partial.
Neural networks (Computer science)

Publisher

South Dakota State University

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Rights Statement

In Copyright