Date of Award
Master of Science (MS)
Mathematics and Statistics
Heng Da Cheng
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.
Wight, Colby, "Numerical Approximations of Phase Field Equations with Physics Informed Neural Networks" (2020). All Graduate Plan B and other Reports. 1461.