Date of Award

8-2020

Degree Type

Creative Project

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

Jia Zhao

Second Advisor

Joseph Koebbe

Third Advisor

Heng Da Cheng

Abstract

Designing numerical algorithms for solving partial differential equations (PDEs) is one of the major research branches in applied and computational mathematics. Recently there has been some seminal work on solving PDEs using the deep neural networks. In particular, the Physics Informed Neural Network (PINN) has been shown to be effective in solving some classical partial differential equations. However, we find that this method is not sufficient in solving all types of equations and falls short in solving phase-field equations. In this thesis, we propose various techniques that add to the power of these networks. Mainly, we propose to embrace the adaptive idea in both space and time and introduce various sampling strategies to improve the efficiency and accuracy of the PINN. The improved PINN can solve a broader set of PDEs, and in particular, the phase-field equations. The improved PINN sheds light on numerical approximations of other PDEs in general.

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