Thesis/chapters/chap02/chap02.tex

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% Author:   Phillip Rothenbeck
% Title:    Investigating the Evolution of the COVID-19 Pandemic in Germany Using Physics-Informed Neural Networks
% File:     chap02/chap02.tex
% Part:     theoretical background
% Description:
%         summary of the content in this chapter
% Version:  25.07.2024
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\chapter{Theoretical background}
\label{chap:background}

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\section{Domain Mathematics}
\label{sec:domain}

The mathematics of domains work with spaces of numbers on which several mathematical operations are performed to retrieve a number in a different space.
For the definition of a function is essential. Which is described by Rudin\cite{Rudin2007}. A function $f$ assigns each element $x$ of the set $A$ to an
element of the set $B$.
\begin{equation}
  f: A \rightarrow B
\end{equation}
In this case $A$ is called the domain, while $B$ is called the codomain. For further explanations we refer to $f$ as:
\begin{equation}
  f: \mathbb{R} \rightarrow \mathbb{R}
\end{equation}
Functions are able to describe the condition of a system given certain parameters.

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\section{Basics of Differential Equations}
\label{sec:differentialEq}

- different information about the domain (domain -> codomain) -> (domain -> change of the funcion)
- example

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\section{Pandemic Models}
\label{sec:pandemicModel}
- modelation in a mathematical sense
- what do pandemic models have to do?

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\subsection{SIR Model}
\label{sec:pandemicModel:sir}

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\subsection{reduced SIR Model}
\label{sec:pandemicModel:rsir}

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\section{Multilayer Perceptron}
\label{sec:mlp}

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\section{Physics Informed Neural Networks}
\label{sec:pinn}

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\subsection{Disease Informed Neural Networks}
\label{sec:pinn:dinn}

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