Physics Informed Neural Networks are a relatively new subject of study in the area of numerical mathematics. In this thesis, we take a look at part of the work that has been done in this area up until now, with the ultimate goal to develop a new type of PINN that improves upon the old concept. We introduce the concept of parameterized PINNs, which allow a single trained network to solve multiple partial differential equations for multiple boundary conditions and geometries by parametrizing these variables as an input for the network. Two methods are tested: one using global basis functions, and one using B-splines. The proposed methods are tested for Laplace’s equation and Poisson’s equation in multiple dimensions, most of which show that these methods are viable alternatives for the current style of collocation-based PINNs.

In this thesis, we take a look Tarski's circle squaring problem: Are an open disk and an open square equidecomposable by using finitely many Borel pieces? The proof is a special case of the one given by Marks and Unger and as such will be based on their proof. We have distilled the original proof to make it understandable for a bachelor student without requiring extra knowledge. We give an example of another case that can be solved without using the method used in the proof. We also take a look at what Borel complexity is, calculate the complexity of the sets used in the example and go over the process of calculating the complexity of the pieces used by Marks and Unger in broad strokes.
Since the proofs of the used lemmas are very long and/or technical, they have not been included in this report, but can be found in their respective source material.