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% Part: theoretical background
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% Description:
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% summary of the content in this chapter
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-% Version: 26.07.2024
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+% Version: 30.07.2024
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\chapter{Theoretical Background}
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\label{chap:background}
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-This chapter is set to introduce the theoretical knowledge on which the work in this thesis is founded on. First we talk about domain mathematics and differential equations.
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-In these sections both the explanations and the approach are strongly based on the book on analysis by W. Rudin\cite{Rudin2007} and the book about ordinary differential equations
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-by M. Tenenbaum and H. Pollard. %\cite{Tenenbaum1985}. %TODO introduce other sections
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+This chapter introduces the theoretical knowledge that forms the foundation of the work presented in this thesis. In sections~\ref{sec:domain} and~\ref{sec:differentialEq}, we talk about
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+differential equations and the underlying theory. In these sections both the explanations and the approach are strongly based on the book on analysis by Rudin~\cite{Rudin2007}
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+and the book about ordinary differential equations by Tenenbaum and Pollard~\cite{Tenenbaum1985}. Subsequently, we employ this knowledge to examine various pandemic models in section~\ref{sec:pandemicModel}.
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+Finally, we address the topic of neural networks with a focus on the multilayer perceptron in section~\ref{sec:mlp} and physics informed neural networks in section~\ref{sec:pinn}.
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% -------------------------------------------------------------------
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-\section{Domain Mathematics and Functions}
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+\section{Mathematical Modelling using Functions}
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\label{sec:domain}
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In order to model a mathematical problem, it is necessary to define a set of fundamental numbers or quantities upon which the subsequent calculations will be based. These sets may
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-represent a specific time interval or a distance, for instance. The term "domain" is used to describe these fundamental sets of numbers or quantities. For the purpose this thesis
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-we will restrict ourselves to domains of real numbers in $\mathbb{R}$.\\
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+represent, for instance, a specific time interval or a distance. The term \emph{domain} describes these fundamental sets of numbers or quantities~\cite{Rudin2007}. A \emph{variable}
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+is a changing entity living in a certain domain. In this thesis,
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+we will focus on domains of real numbers in $\mathbb{R}$.\\
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-The mapping of a value from one domain to a value from another domain enables the modelling and depiction of semantics. In order to facilitate this mapping, the use of functions is
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-employed. The function $f$ is defined as
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+The mapping between variables enables the modeling of the process and depicts the semantics. We use functions in order to facilitate this mapping. Let $A, B\subset\mathbb{R}$ be to subsets
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+of the real numbers, then we define a function as the mapping
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\begin{equation}
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- f: A\rightarrow B
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+ f: A\rightarrow B.
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\end{equation}
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-In other words, $f$ assigns each value $x$ of an interval $A$ to a value $y$ of $B$. $A$ is referred to as the domain and $B$ as the codomain of $f$. In our case, both $A$ and $B$
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-are subsets of $\mathbb{R}$. Functions are capable of representing the state of a system as a value based on an input value from their domain. One illustrative example is a function
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-that maps a time point to the distance covered since the last time point. In this case, time serves as the domain, while the distance is the codomain.
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+In other words, the function $f$ maps elements $x\in A$ to values $f(x)\in B$. $A$ is the \emph{domain} of $f$, while $B$ is the \emph{codomain} of $f$. Functions are capable of representing
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+the state of a system as a value based on an input value from their domain. One illustrative example is a function that maps a time point to the distance covered since a starting point. In this
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+case, time serves as the domain, while the distance is the codomain.
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% -------------------------------------------------------------------
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@@ -38,28 +40,41 @@ that maps a time point to the distance covered since the last time point. In thi
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\label{sec:differentialEq}
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Often, the change of a system is more interesting than its current state. Functions are able to give us the latter, but only passively give information about the change of a system.
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-Now the problem at hand is a way of calculating the change of a function across its domain. Given a function $f$ and an interval $[a, b]\subset \mathbb{R}$,
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+The objective is to determine an effective method for calculating the change of a function across its domain. Let $f$ be a function and $[a, b]\subset \mathbb{R}$ an interval of real numbers,
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+the expression
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\begin{equation}
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m = \frac{f(b) - f(a)}{a-b}
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\end{equation}
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-is the average rate of change across $[a, b]$. While the average change of rate is helpful in many cases, the momentary rate of change is more accurate. To find this the interval is
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-narrowed down, to be infinitesimal small. For each $x\in[a, b]$ we calculate
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+gives the average rate of change. While the average rate of change is useful in many cases, the momentary rate of change is more accurate. To calculate this, we need to
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+narrow down, the interval to an infinitesimal. For each $x\in[a, b]$ we calculate
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\begin{equation} \label{eqn:differential}
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- \frac{df}{dx} = \lim_{t\to x} \frac{f(t) - f(x)}{t-x}
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+ \frac{df}{dx} = \lim_{t\to x} \frac{f(t) - f(x)}{t-x},
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+\end{equation}
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+if it exists. $\frac{df}{dx}$ is the \emph{derivative}, or \emph{differential equation}, it returns the momentary rate of change of $f$ for each value $x$ of $f$'s
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+domain. Repeating this process on $\frac{df}{dx}$ yields $\frac{d^2f}{dx^2}$, which is the function that calculates the rate of change of the rate of change and is called the second order
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+derivative. Iterating this $n$ times results in $\frac{d^nf}{dx^n}$, the derivative of the $n$'th order. Another method for obtaining a differential equation is to create it from the semantics
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+of a problem. This method is useful if no basic function exists for a system.\\
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+
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+In the context of functions, it is possible to have multiple domains, meaning that function has more than one parameter. To illustrate, consider a function operating in two-dimensional space,
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+wherein each parameter represents one axis or one that, employs with time and locations as inputs. The term \emph{partial differential equations} (\emph{PDE}'s) describes differential equations
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+of such functions, which require a derivative for each of their domains. In contrast, \emph{ordinary differential equations} (\emph{ODE}'s) are the single derivatives for a function having only
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+one domain. In this thesis, we only need ODE's.\\
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+
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+A \emph{differential system} is the name for a collective of differential equations. The derivatives in a differential system each have their own codomain, which is part of the problem, while they
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+all share the same domain.\\
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+
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+Tenenbaum and Pollard~\cite{Tenenbaum1985} provides many examples for ODE's, including the \emph{Motion of a Particle Along a Straight Line}. Newton's second law states that ``the rate of change of
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+the momentum of a body ($momentum = mass \cdot velocity$) is proportional to the resultant external force $F$ acted upon it''~\cite{Tenenbaum1985}. Let $m$ be the mass of the body in kilograms, $v$ its velocity in seconds
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+per meter and $t$ the time in seconds. Then, Newton's second law translates mathematically to
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+\begin{equation} \label{eq:newtonSecLaw}
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+ F = m\frac{dv}{dt}.
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+\end{equation}
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+It is evident that the acceleration, $a=\frac{dv}{dt}$, as the rate of change of the velocity is part of the equation. Additionally, is the velocity of a body the derivative of the distance traveled
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+by that body. Based on these findings, we can rewrite the equation~\ref{eq:newtonSecLaw} to
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+\begin{equation}
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+ F=ma=m\frac{d^2s}{dt^2}.
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\end{equation}
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-if it exists. $\frac{df}{dx}$ is called the derivative or differential equation, it returns the momentary rate of change of $f$ for each value $x$ of $f$'s
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-domain. When repeating this process on $\frac{df}{dx}$ we get $\frac{d^2f}{dx^2}$ which is the function that calculates the rate of change of the rate of change and is called the second
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-derivative. Iterating this $n$ times results in $\frac{d^nf}{dx^n}$, the derivative of the $n$'th order. Another way to get a differential equation is to create it from the semantics of
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-a problem. This way is often chosen if no basic function exists for a system.\\
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-
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-For functions, it is possible to have multiple domains, meaning that function has more than one parameter. For example a function working with 2D parameters or a function that, works with
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-time and locations as its inputs. These functions need differential equations for each of their domains, which are called "partial differential equations" (PDE's). The ones for functions with one
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-single domain are called "ordinary differential equations" (ODE's). For this thesis only ODE's will be of interest.\\
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-
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-If a system has several codomains, it will have a differential equation for each codomain all sharing the same domain. The collective of these equations is called a differential system.\\
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-
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-For illustrating the functionality of a derivative we will look upon the specific problem. For this we
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% -------------------------------------------------------------------
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Phillip Rothenbeck