In this presentation, eigenvalue problems are investigates. With particular focus on the eigenvalues of Laplace operators, we explore the analytical and numerical approaches to finding eigenfunction-eigenvalue pairs. While considerable detail is devoted to the analytical methods in their relation to how the solution is derived we use this as a means of demonstrating how numerical methods deviate from this practice. Lastly, the effectiveness of Machine Learning techniques, such as PINNS, is discussed and a few test problems are presented.
