Solving Differential Equations With Deep Neural Networks (DNNs)
Open Access
Author:
Grafton, Jaysa
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
Leonid V Berlyand, Thesis Supervisor Victoria V Sadovskaya, Thesis Honors Advisor
Keywords:
Differential Equations Deep Neural Networks DNNs Machine Learning ODEs
Abstract:
Overall, the goal of this project is to make use of the machine learning algorithm of deep neural networks (DNNs) to solve differential equations. Specifically, this project aims to solve two different second-order differential equations: Poisson and Ginzburg-Landau equations. Results for the Poisson equation show an accurate solution can be acquired using a single layer network with no activation function due to the linearity of the equation. These results demonstrate that finding solutions to differential equations is possible through the use of deep neural networks. For the Ginzburg-Landau equation, two different loss functions are utilized with adjustments being made to account for boundary conditions and derivatives. Results indicate an accurate approximation for various mesh sizes (i.e., coarse versus fine mesh) and allow for the comparison of network architectures for each mesh size in order to determine the parameters necessary for an accurate solution.
