% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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:  20.08.2024
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Theoretical Background   12}
\label{chap:background}

This chapter introduces the theoretical foundations for the work presented in
this thesis. In~\Cref{sec:domain} and~\Cref{sec:differentialEq}, we describe
differential equations and the underlying theory. In these Sections both the
explanations and the approach are based on a book on analysis by
Rudin~\cite{Rudin2007} and a book about ordinary differential equations by
Tenenbaum and Pollard~\cite{Tenenbaum1985}. Subsequently, we employ this
knowledge to examine various pandemic models in~\Cref{sec:epidemModel}. Finally,
we address the topic of neural networks with a focus on the multilayer
perceptron in~\Cref{sec:mlp} and physics informed neural networks
in~\Cref{sec:pinn}.

% -------------------------------------------------------------------

\section{Mathematical Modelling using Functions   1}
\label{sec:domain}

To model a physical problem mathematically, it is necessary to define a set of
fundamental numbers\todo{meeting question 1} or quantities upon which the subsequent calculations will be
based. These sets may 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} is a changing entity living in a
certain domain. In this thesis, we will focus on domains of real numbers in
$\mathbb{R}$.\\

The mapping between variables enables the modeling of a physical process and may
depict semantics. We use functions in order to facilitate this mapping. Let
$A, B\subset\mathbb{R}$ be to subsets of the real numbers, then we define a
function as the mapping
\begin{equation}
  f: A\rightarrow B.
\end{equation}
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 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 step to the distance covered since a starting point.
In this case, time serves as the domain, while the distance is the codomain.

% -------------------------------------------------------------------

\section{Mathematical Modelling using Differential Equations   1}
\label{sec:differentialEq}

Often, the behavior of a variable or a quantity across a domain is more
interesting than its current state. Functions are able to give us the latter, \todo{meeting question 2}
but only passively give information about the change of a system. 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. The expression
\begin{equation}
  m = \frac{f(b) - f(a)}{a-b}
\end{equation}
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, \todo{look up in Rudin - cite (wordly)}
we need to narrow down, the interval to an infinitesimal. For each $x\in[a, b]$
we calculate
\begin{equation} \label{eqn:differential}
  \frac{df}{dx} = \lim_{t\to x} \frac{f(t) - f(x)}{t-x},
\end{equation}
if it exists. As the Tenenbaum and Pollard~\cite{Tenenbaum1985} define,
$\nicefrac{df}{dx}$ is the \emph{derivative}, which is ``the rate of change of a
variable with respect to another''. The relation between a variable and its
derivative is modeled in a \emph{differential equation}. The derivative of
$\nicefrac{df}{dx}$ yields $\nicefrac{d^2f}{dx^2}$, which is the function that
calculates the rate of change of the rate of change and is called the
\emph{second order derivative}. Iterating this $n$ times results in
$\nicefrac{d^nf}{dx^n}$, the derivative of the $n$'th order. A method for
obtaining a differential equation is to derive it from the semantics of a
problem. For example, in physics a differential equation can be derived from the
law of the conservation of energy~\cite{Demtroeder2021}. Differential equations \todo{is this good?}
find application in several areas such as engineering \eg, the Chua's
circuit~\cite{Matsumoto1984}, physics with, \eg, the Schrödinger
equation~\cite{Schroedinger1926}, economics, \eg, Black-Scholes
equation~\cite{Oksendal2000}, epidemiology, and beyond.\\

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, wherein each parameter represents one axis.
Another example would be a function, that maps its inputs of a location variable
and a time variable on a height. The term \emph{partial differential equations}
(\emph{PDE}'s) describes differential equations of such functions, which contain
partial derivatives with respect to each individual domain. In contrast, \emph{ordinary differential
  equations} (\emph{ODE}'s) are the single derivatives for a function having only
one domain~\cite{Tenenbaum1985}. In this thesis, we restrict ourselves to ODE's.\\

A \emph{system of differential equations} is the name for a set of differential
equations. The derivatives in a system of differential equations each have their
own codomain, which is part of the problem, while they all share the same
domain.\\

Tenenbaum and Pollard~\cite{Tenenbaum1985} provide many examples for ODE's,
including the \emph{Motion of a Particle Along a Straight Line}. Further,
Newton's second law states that ``the rate of change of 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 meters per second and $t$ the time in seconds.
Then, Newton's second law translates mathematically to
\begin{equation} \label{eq:newtonSecLaw}
  F = m\frac{dv}{dt}.
\end{equation}
It is evident that the acceleration, $a=\frac{dv}{dt}$, as the rate of change of
the velocity is part of the equation. Additionally, the velocity of a body is
the derivative of the distance traveled by that body. Based on these findings,
we can rewrite the~\Cref{eq:newtonSecLaw} to
\begin{equation}
  F=ma=m\frac{d^2s}{dt^2}.
\end{equation}\\

To conclude, note that this explanation of differential equations focuses on the
aspects deemed crucial for this thesis and is not intended to be a complete
explanation of the subject. To gain a better understanding of it, we recommend
the books mentioned above~\cite{Rudin2007,Tenenbaum1985}. In the following
section we describe the application of these principles in epidemiological
models.

% -------------------------------------------------------------------

\section{Epidemiological Models   4}
\label{sec:epidemModel}

Pandemics, like \emph{COVID-19}, which have resulted in a significant
number of fatalities. Hence, the question arises: How should we analyze a \todo{Better?}
pandemic effectively? It is essential to study whether the employed
countermeasures are efficacious in combating the pandemic. Given the unfavorable
public response to measures such as lockdowns, it is imperative to investigate
that their efficacy remains commensurate with the costs incurred to those
affected. In the event that alternative and novel technologies were in use, such
as the mRNA vaccines in the context of COVID-19, it is needful to test the
effect and find the optimal variant. In order to shed light on the
aforementioned events, we need a method to quantify the pandemic along with its
course of progression.\\

The real world is a highly complex system, which presents a significant
challenge attempting to describe it fully in a mathematical model. Therefore,
the model must reduce the complexity while retaining the essential information.
Furthermore, it must address the issue of limited data availability. For
instance, during COVID-19 institutions such as the Robert Koch Institute
(RKI)\footnote[1]{\url{https://www.rki.de/EN/Home/homepage_node.html}} were only
able to collect data on infections and mortality cases. Consequently, we require
a model that employs an abstraction of the real world to illustrate the events
and relations that are pivotal to understanding the problem.

% -------------------------------------------------------------------

\subsection{SIR Model   3}
\label{sec:pandemicModel:sir}

In 1927, Kermack and McKendrick~\cite{1927} introduced the \emph{SIR Model},
which subsequently became one of the most influential epidemiological models.
This model enables the modeling of infections during epidemiological events such as pandemics.
The book \emph{Mathematical Models in Biology}~\cite{EdelsteinKeshet2005}
reiterates the model and serves as the foundation for the following explanation
of SIR models.\\

The SIR model is capable of illustrating diseases, which are transferred through
contact or proximity of an individual carrying the illness and a healthy
individual. This is possible due to the distinction between infected individuals
who are carriers of the disease and the part of the population, which is
susceptible to infection. In the model, the mentioned groups are capable to
change, \eg,  healthy individuals becoming infected.  The model assumes the
size $N$ of the population remains constant throughout the duration of the
pandemic. The population $N$ comprises three distinct compartments: the
\emph{susceptible} group $S$, the \emph{infectious} group $I$ and the
\emph{removed} group $R$ (hence SIR model). Let $\mathcal{T} = [t_0, t_f]\subseteq
  \mathbb{R}_{\geq0}$ be the time span of the pandemic, then,
\begin{equation}
  S: \mathcal{T}\rightarrow\mathbb{N}, \quad I: \mathcal{T}\rightarrow\mathbb{N}, \quad R: \mathcal{T}\rightarrow\mathbb{N},
\end{equation}
give the values of $S$, $I$ and $R$ at a certain point of time
$t\in\mathcal{T}$. For $S$, $I$, $R$ and $N$ applies:
\begin{equation} \label{eq:N_char}
  N = S + I + R.
\end{equation}
The model makes another assumption by stating that recovered people are immune
to the illness and infectious individuals can not infect them. The individuals
in the $R$ group are either recovered or deceased, and thus unable to transmit
or carry the disease.
\begin{figure}[h]
  \centering
  \includegraphics[width=\textwidth]{sir_graph.pdf}
  \caption{A visualization of the SIR model, illustrating $N$ being split in the
    three groups $S$, $I$ and $R$.}
  \label{fig:sir_model}
\end{figure}
As visualized in the~\Cref{fig:sir_model} the
individuals may transition between groups based on transition rates. The
transmission rate $\beta$ is responsible for individuals becoming infected,
while the rate of removal or recovery rate $\alpha$ (also referred to as
$\delta$ or $\nu$, \eg,~\cite{EdelsteinKeshet2005,Millevoi2023}) moves
individuals from $I$ to $R$.\\

We can describe this problem mathematically using a system of differential
equations (see ~\Cref{sec:differentialEq}). Thus, Kermack and
McKendrick~\cite{1927} propose the following set of differential equations:
\begin{equation}\label{eq:sir}
  \begin{split}
    \frac{dS}{dt} &= -\beta S I,\\
    \frac{dI}{dt} &= \beta S I - \alpha I,\\
    \frac{dR}{dt} &= \alpha I.
  \end{split}
\end{equation}
This set of differential equations, is based on the following assumption:
``The rate of transmission of a microparasitic disease is proportional to the
rate of encounter of susceptible and infective individuals modelled by the
product ($\beta S I$)'', according to Edelstein-Keshet~\cite{EdelsteinKeshet2005}.
The system shows the change in size of the groups per time unit due to
infections, recoveries, and deaths.\\

The term $\beta SI$ describes the rate of encounters of susceptible and infected
individuals. This term is dependent on the size of $S$ and $I$, thus Anderson
and May~\cite{Anderson1991} propose a modified model:
\begin{equation}\label{eq:modSIR}
  \begin{split}
    \frac{dS}{dt} &= -\beta \frac{SI}{N},\\
    \frac{dI}{dt} &= \beta \frac{SI}{N} - \alpha I,\\
    \frac{dR}{dt} &= \alpha I.
  \end{split}
\end{equation}
In~\Cref{eq:modSIR} $\beta SI$ gets normalized by $N$, which is more correct in
a real world aspect~\cite{Anderson1991}.\\

The initial phase of a pandemic is characterized by the infection of a small
number of individuals, while the majority of the population remains susceptible.
The infectious group has not yet infected any individuals thus
neither recovery nor mortality is possible. Let $I_0\in\mathbb{N}$ be
the number of infected individuals at the beginning of the disease. Then,
\begin{equation}\label{eq:startCond}
  \begin{split}
    S(0) &= N - I_{0},\\
    I(0) &= I_{0},\\
    R(0) &= 0,
  \end{split}
\end{equation}
describes the initial configuration of a system in which a disease has just
emerged.\\

\begin{figure}[h]
  %\centering
  \setlength{\unitlength}{1cm} % Set the unit length for coordinates
  \begin{picture}(12, 9.5) % Specify the size of the picture environment (width, height)
    % reference
    \put(0, 1.75){
      \begin{subfigure}{0.4\textwidth}
        \centering
        \includegraphics[width=\textwidth]{reference_params_synth.pdf}
        \caption{$\alpha=0.35$, $\beta=0.5$}
        \label{fig:synth_norm}
      \end{subfigure}
    }
    % 1. row, 1.image (low beta)
    \put(5.5, 5){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{low_beta_synth.pdf}
        \caption{$\alpha=0.25$, $\beta=0.5$}
        \label{fig:synth_low_beta}
      \end{subfigure}
    }
    % 1. row, 2.image (high beta)
    \put(9.5, 5){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{high_beta_synth.pdf}
        \caption{$\alpha=0.45$, $\beta=0.5$}
        \label{fig:synth_high_beta}
      \end{subfigure}
    }
    % 2. row, 1.image (low alpha)
    \put(5.5, 0){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{low_alpha_synth.pdf}
        \caption{$\alpha=0.35$, $\beta=0.4$}
        \label{fig:synth_low_alpha}
      \end{subfigure}
    }
    % 2. row, 2.image (high alpha)
    \put(9.5, 0){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{high_alpha_synth.pdf}
        \caption{$\alpha=0.35$, $\beta=0.6$}
        \label{fig:synth_high_alpha}
      \end{subfigure}
    }
  \end{picture}
  \caption{Synthetic data, using~\Cref{eq:modSIR} and $N=7.9\cdot 10^6$, $I_0=10$ with different sets of parameters.
    We visualize the case with the reference parameters in (a). In (b) and (c) we keep $\alpha$ constant, while varying
    the value of $\beta$. In contrast, (d) and (e) have varying values of $\alpha$.}
  \label{fig:synth_sir}
\end{figure}

In the SIR model the temporal occurrence and the height of the peak (or peaks)
of  the infectious group are of paramount importance for understanding the
dynamics of a pandemic. A low peak occurring at a late point in time indicates
that the disease is unable to keep pace with the rate of recovery, resulting
in its demise before it can exert a significant influence on the population. In
contrast, an early and high peak means that the disease is rapidly transmitted
through the population, with a significant proportion of individuals having been
infected.~\Cref{fig:sir_model} illustrates this effect by varying the values of
$\beta$ or $\alpha$ while simulating  a pandemic using a model such
as~\Cref{eq:modSIR}. It is evident that both the transmission rate $\beta$
and the recovery rate $\alpha$ influence the height and time of the peak of $I$.
When the number of infections exceeds the number of recoveries, the peak of $I$
will occur early and will be high. On the other hand, if recoveries occur at a
faster rate than new infections the peak will occur later and will be low. Thus,
it is crucial to know both $\beta$ and $\alpha$, as these parameters
characterize how the pandemic evolves.\\

The SIR model makes a number of assumptions that are intended to reduce the
model's overall complexity while simultaneously increasing its divergence from
actual reality. One such assumption is that the size of the population, $N$,
remains constant, as the daily change is negligible to the total population.
This depiction is not an accurate representation of the actual relations \todo{other assumptions in a bad light?}
observed in the real world, as the size of a population is subject to a number
of factors that can contribute to change. The population is increased by the
occurrence of births and decreased by the occurrence of deaths. Other examples
are the impossibility for individuals to be susceptible again, after having
recovered, or the possibility for the transition rates to change due to new
variants or the implementation of new countermeasures. We address this latter
option in the next~\Cref{sec:pandemicModel:rsir}.

% -------------------------------------------------------------------

\subsection{Reduced SIR Model and the Reproduction Number   1}
\label{sec:pandemicModel:rsir}
The~\Cref{sec:pandemicModel:sir} presents the classical SIR model. This model
contains two scalar parameters $\beta$ and $\alpha$, which describe the course
of a pandemic over its duration. This is beneficial when examining the overall
pandemic; however, in the real world, disease behavior is dynamic, and the
values of the parameters $\beta$ and $\alpha$ change throughout the course of
the disease. The reason for this is due to events such as the implementation of
countermeasures that reduce the contact between the infectious and susceptible
individuals, the emergence of a new variant of the disease that increases its
infectivity or deadliness, or the administration of a vaccination that provides
previously susceptible individuals with immunity without ever being infected. \todo{sai correction -> is this point not already included?}
To address this Millevoi \etal~\cite{Millevoi2023} introduce a model that \todo{are there older sources}
simultaneously reduces the size of the system of differential equations and
solves the problem of time scaling at hand.\\

First, they alter the definition of $\beta$ and $\alpha$ to be dependent on the time interval
$\mathcal{T} = [t_0, t_f]\subseteq \mathbb{R}_{\geq0}$,
\begin{equation}
  \beta: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}, \quad\alpha: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}.
\end{equation}
Another crucial element is $D(t) = \frac{1}{\alpha(t)}$, which represents the initial time
span an infected individual requires to recuperate. Subsequently, at the initial time point
$t_0$, the \emph{reproduction number},
\begin{equation}
  \RO = \beta(t_0)D(t_0) = \frac{\beta(t_0)}{\alpha(t_0)},
\end{equation}
represents the number of susceptible individuals, that one infectious individual
infects at the onset of the pandemic. In light of the effects of $\beta$ and
$\alpha$ (see~\Cref{sec:pandemicModel:sir}), $\RO < 1$ indicates that the
pandemic is emerging. In this scenario $\alpha$ is relatively low due to the
limited number of infections resulting from $I(t_0) << S(t_0)$.\\ Further,
$\RO > 1$ leads to the disease spreading rapidly across the population, with an
increase in $I$ occurring at a high rate. Nevertheless, $\RO$ does not cover
the entire time span. For this reason, Millevoi \etal~\cite{Millevoi2023}
introduce $\Rt$ which has the same interpretation as $\RO$, with the exception
that $\Rt$ is dependent on time. The time-dependent reproduction number is
defined as,
\begin{equation}\label{eq:repr_num}
  \Rt=\frac{\beta(t)}{\alpha(t)}\cdot\frac{S(t)}{N},
\end{equation}
on the time interval $\mathcal{T}$. This definition includes the transition
rates for information about the spread of the disease and information of the
decrease of the ratio of susceptible individuals in the population. In contrast
to $\beta$ and $\alpha$, $\Rt$ is not a parameter but \todo{Sai comment - earlier?}
a state variable in the model and enabling the following reduction of the SIR
model.\\

\Cref{eq:N_char} allows for the calculation of the value of the group $R$ using
$S$ and $I$, with the term $R(t)=N-S(t)-I(t)$. Thus,
\begin{equation}
  \begin{split}
    \frac{dS}{dt} &= \alpha(\Rt-1)I(t),\\
    \frac{dI}{dt} &= -\alpha\Rt I(t),
  \end{split}
\end{equation}
is the reduction of~\Cref{eq:sir} on the time interval $\mathcal{T}$ using this
characteristic and the reproduction number $\Rt$ (see ~\Cref{eq:repr_num}).
Another issue that Millevoi \etal~\cite{Millevoi2023} seek to address is the
extensive range of values that the SIR groups can assume. Accordingly, they
initially scale the time interval $\mathcal{T}$ using its borders to calculate
the scaled time $t_s = \frac{t - t_0}{t_f - t_0}\in[0, 1]$. Subsequently, they
calculate the scaled groups,
\begin{equation}
  S_s(t_s) = \frac{S(t)}{C},\quad I_s(t_s) = \frac{I(t)}{C},\quad R_s(t_s) = \frac{R(t)}{C},
\end{equation}
using a large constant scaling factor $C\in\mathbb{N}$. Applying this to the
variable $I$, results in,
\begin{equation}\label{eq:reduced_sir_ODE}
  \frac{dI_s}{dt_s} = \alpha(t_f - t_0)(\Rt - 1)I_s(t_s),
\end{equation}
which is a further reduced version of~\Cref{eq:sir}. This less complex
differential equation results in a less complex solution, as it entails the
elimination of a parameter ($\beta$) and the two state variables ($S$ and $R$).
The reduced SIR model, is more precise in applications with a worse data
situation, due to its fewer input variables.

% -------------------------------------------------------------------

\section{Multilayer Perceptron   2}
\label{sec:mlp}
In~\Cref{sec:differentialEq}, we demonstrate the significance of differential
equations in systems, illustrating how they can be utilized to elucidate the
impact of a specific parameter on the system's behavior.
In~\Cref{sec:epidemModel}, we show specific applications of differential
equations in an epidemiological context. The final objective is to solve these
equations by finding a function that fits. Fitting measured data points to
approximate such a function, is one of the multiple methods to achieve this
goal. The \emph{Multilayer Perceptron} (MLP)~\cite{Rumelhart1986} is a
data-driven function approximator. In the following section, we provide a brief
overview of the structure and training of these \emph{neural networks}. For
reference, we use the book \emph{Deep Learning} by Goodfellow
\etal~\cite{Goodfellow-et-al-2016} as a foundation for our explanations.\\

The objective is to develop an approximation method for any function $f^{*}$,
which could be a mathematical function or a mapping of an input vector to the
desired output. Let $\boldsymbol{x}$ be the input vector and $\boldsymbol{y}$
the label, class, or result. Then, $\boldsymbol{y} = f^{*}(\boldsymbol{x})$,
is the function to approximate. In the year 1958,
Rosenblatt~\cite{Rosenblatt1958} proposed the perceptron modeling the concept of
a neuron in a neuroscientific sense. The perceptron takes in the input vector
$\boldsymbol{x}$ performs an operation and produces a scalar result. This model
optimizes its parameters $\theta$ to be able to calculate $\boldsymbol{y} =
  f(\boldsymbol{x}; \theta)$ as accurately as possible. As Minsky and
Papert~\cite{Minsky1972} demonstrate, the perceptron is only capable of
approximating a specific class of functions. Consequently, there is a necessity
for an expansion of the perceptron.\\

As Goodfellow \etal~\cite{Goodfellow-et-al-2016} proceed, the solution to this issue is to decompose $f$ into
a chain structure of the form,
\begin{equation} \label{eq:mlp_char}
  f(\boldsymbol{x}) = f^{(3)}(f^{(2)}(f^{(1)}(\boldsymbol{x}))).
\end{equation}
This nested version of a perceptron is a multilayer perceptron. Each
sub-function, designated as $f^{(i)}$, is represented in the structure of an
MLP as a \emph{layer}, which contains a linear mapping and a nonlinear mapping
in form of an \emph{activation function}. A multitude of
\emph{Units} (also \emph{neurons}) compose each layer. A neuron performs the
same vector-to-scalar calculation as the perceptron does. Subsequently, a
nonlinear activation function transforms the scalar output into the activation
of the unit. The layers are staggered in the neural network, with each layer
being connected to its neighbors, as illustrated in~\Cref{fig:mlp_example}. The
input vector $\boldsymbol{x}$ is provided to each unit of the first layer
$f^{(1)}$, which then gives the results to the units of the second layer
$f^{(2)}$, and so forth. The final layer is the \emph{output layer}. The
intervening layers, situated between the first and the output layers are the
\emph{hidden layers}. The term \emph{forward propagation} describes the
process of information flowing through the network from the input layer to the
output layer, resulting in a scalar loss. The alternating structure of linear
and nonlinear calculation enables MLP's to approximate any function. As Hornik
\etal~\cite{Hornik1989} proves, MLP's are universal approximators.\\

\begin{figure}[h]
  \centering
  \includegraphics[width=\textwidth]{MLP.pdf}
  \caption{A illustration of an MLP with two hidden layers. Each neuron of a layer
    is connected to every neuron of the neighboring layers. The arrow indicates
    the direction of the forward propagation.}
  \label{fig:mlp_example}
\end{figure}

The term \emph{training} describes the process of optimizing the parameters
$\theta$. In order to undertake training, it is necessary to have a set of
\emph{training data}, which is a set of pairs (also called training points) of
the input data $\boldsymbol{x}$ and its corresponding true solution
$\boldsymbol{y}$ of the function $f^{*}$. For the training process we must
define a \emph{loss function} $\Loss{ }$, using the model prediction
$\hat{\boldsymbol{y}}$ and the true value $\boldsymbol{y}$, which will act as a
metric for evaluating the extent to which the model deviates from the correct
answer. One common loss function is the \emph{mean square error}
(MSE) loss function. Let $N$ be the number of points in the set of training
data. Then,
\begin{equation} \label{eq:mse}
  \Loss{MSE} = \frac{1}{N}\sum_{i=1}^{N} ||\hat{\boldsymbol{y}}^{(i)}-\boldsymbol{y}^{(i)}||^2,
\end{equation}
calculates the squared difference between each model prediction and true value
of a training and takes the mean across the whole training data. \\

Ultimately, the objective is to utilize this information to optimize the parameters, in order to minimize the
loss. One of the most fundamental optimization strategy is \emph{gradient
  descent}. In this process, the derivatives are employed to identify the location
of local or global minima within a function, which lie where the gradient is
zero. Given that a positive gradient
signifies ascent and a negative gradient indicates descent, we must move the
variable by a \emph{learning rate} (step size) in the opposite
direction to that of the gradient. The calculation of the derivatives in respect
to the parameters is a complex task, since our functions is a composition of
many functions (one for each layer). We can address this issue taking advantage
of~\Cref{eq:mlp_char} and employing the chain rule of calculus. Let
$\hat{\boldsymbol{y}} = f(\boldsymbol{x}; \theta)$ be the model prediction with the
decomposed version $f(\boldsymbol{x}; \theta) = f^{(3)}(w; \theta_3)$ with
$w = f^{(2)}(z; \theta_2)$ and $z = f^{(1)}(\boldsymbol{x}; \theta_1)$.
$\boldsymbol{x}$ is the input vector and $\theta_3, \theta_2, \theta_1\subset\theta$.
Then,
\begin{equation}\label{eq:backprop}
  \nabla_{\theta_3} \Loss{ } = \frac{d\mathcal{L}}{d\hat{\boldsymbol{y}}}\frac{d\hat{\boldsymbol{y}}}{df^{(3)}}\nabla_{\theta_3}f^{(3)},
\end{equation}
is the gradient of $\Loss{ }$ in respect of the parameters $\theta_3$. To obtain
$\nabla_{\theta_2} \Loss{ }$, we have to derive $\nabla_{\theta_3} \Loss{ }$ in
respect to $\theta_2$. The name of this method in the context of neural
networks is \emph{back propagation}~\cite{Rumelhart1986}, as it propagates the
error backwards through the neural network.\\

In practical applications, an optimizer often accomplishes the optimization task
by executing back propagation in the background. Furthermore, modifying the
learning rate during training can be advantageous. For instance, making larger \todo{leave whole paragraph out? - Niklas}
steps at the beginning and minor adjustments at the end. Therefore, schedulers
are implementations algorithms that employ diverse learning rate alteration
strategies.\\

For a more in-depth discussion of practical considerations and additional
details like regularization, we direct the reader to the book
\emph{Deep Learning} by Goodfellow \etal~\cite{Goodfellow-et-al-2016}. The next
section will demonstrate the application of neural networks in approximating
solutions to differential systems.

% -------------------------------------------------------------------

\section{Physics Informed Neural Networks   4}
\label{sec:pinn}

In~\Cref{sec:mlp}, we describe the structure and training of MLP's, which are
wildely recognized tools for approximating any kind of function. In this
section, we apply this capability to create a solver for ODE's and PDE's
as Legaris \etal~\cite{Lagaris1997} describe in their paper. In this approach,
the model learns to approximate a function using provided data points while
leveraging the available knowledge about the problem in the form of a system of
differential equations. The \emph{physics-informed neural network} (PINN)
learns the system of differential equations during training, as it optimizes
its output to align with the equations.\\

In contrast to standard MLP's, PINNs are not only data-driven. The loss term of a PINN comprises two
components. The first term incorporates the equations of the aforementioned prior knowledge to pertinent the problem. As Raissi
\etal~\cite{Raissi2017} propose, the residual of each differential equation in
the system must be minimized in order for the model to optimize its output in accordance with the theory.
We obtain the residual $r_i$, with $i\in\{1, ...,N_d\}$, by rearranging the differential equation and
calculating the difference between the left-hand side and the right-hand side
of the equation. $N_d$ is the number of differential equations in a system. As
Raissi \etal~\cite{Raissi2017} propose the \emph{physics
  loss} of a PINN,
\begin{equation}
  \mathcal{L}_{physics}(\boldsymbol{x},\hat{\boldsymbol{y}}) = \frac{1}{N_d}\sum_{i=1}^{N_d} ||r_i(\boldsymbol{x},\hat{\boldsymbol{y}})||^2,
\end{equation}
takes the input data and the model prediction to calculate the mean square
error of the residuals. The second term, the \emph{observation loss}
$\Loss{obs}$, employs the mean square error of the distances between the
predicted and the true values for each training point. Additionally, the
observation loss may incorporate extra terms of inital and boundary conditions. Let $N_t$
denote the number of training points. Then,
\begin{equation}
  \mathcal{L}_{PINN}(\boldsymbol{x}, \boldsymbol{y},\hat{\boldsymbol{y}}) = \frac{1}{N_d}\sum_{i=1}^{N_d} ||r_i(\boldsymbol{x},\hat{\boldsymbol{y}})||^2 + \frac{1}{N_t}\sum_{i=1}^{N_t} ||\hat{\boldsymbol{y}}^{(i)}-\boldsymbol{y}^{(i)}||^2,
\end{equation}\\
represents the comprehensive loss function of a physics-informed neural network. \\

Given the nature of residuals, calculating the loss term of
$\mathcal{L}_{physics}(\boldsymbol{x},\hat{\boldsymbol{y}})$ requires the
calculation of the derivative of the output with respect to the input of
the neural network. As we outline in~\Cref{sec:mlp}, during the process of
back-propagation we calculate the gradients of the loss term in respect to a
layer-specific set of parameters denoted by $\theta_l$, where $l$ represents
the index of the respective layer. By employing
the chain rule of calculus, the algorithm progresses from the output layer
through each hidden layer, ultimately reaching the first layer in order to
compute the respective gradients. The term,
\begin{equation}
  \nabla_{\boldsymbol{x}} \hat{\boldsymbol{y}} = \frac{d\hat{\boldsymbol{y}}}{df^{(2)}}\frac{df^{(2)}}{df^{(1)}}\nabla_{\boldsymbol{x}}f^{(1)},
\end{equation}
illustrates that, in contrast to the procedure described in~\cref{eq:backprop},
this procedure the \emph{automatic differenciation} goes one step further and
calculates the gradient of the output with respect to the input
$\boldsymbol{x}$. In order to calculate the second derivative
$\frac{d\hat{\boldsymbol{y}}}{d\boldsymbol{x}}=\nabla_{\boldsymbol{x}} (\nabla_{\boldsymbol{x}} \hat{\boldsymbol{y}} ),$
this procedure must be repeated.\\

Above we present a method for approximating functions through the use of
systems of differential equations. As previously stated, we want to find a
solver for systems of differential equations. In problems, where we must solve
an ODE or PDE, we have to find a set of parameters, that satisfies the system
for any input $\boldsymbol{x}$. In terms of the context of PINN's this is the
inverse problem, where we have a set of training data from measurements, for
example, is available along with the respective differential equations but
information about the parameters of the equations is lacking. To address this
challenge, we set these parameters as distinct learnable parameters within the
neural network. This enables the network to utilize a specific value, that
actively influences the physics loss $\mathcal{L}_{physics}(\boldsymbol{x},\hat{\boldsymbol{y}})$.
During the training phase the optimizer aims to minimize the physics loss,
which should ultimately yield an approximation of the true value.\\

\begin{figure}[h]
  \centering
  \includegraphics[width=\textwidth]{oscilator.pdf}
  \caption{Illustration of of the movement of an oscillating body in the
    underdamped case. With $m=1kg$, $\mu=4\frac{Ns}{m}$ and $k=200\frac{N}{m}$.}
  \label{fig:spring}
\end{figure}
One illustrative example of a potential application for PINN's is the
\emph{damped harmonic oscillator}~\cite{Demtroeder2021}. In this problem, we
displace a body, which is attached to a spring, from its resting position. The
body is subject to three forces: firstly, the inertia exerted by the
displacement $u$, which points in the direction the displacement $u$; secondly
the restoring force of the spring, which attempts to return the body to its
original position and thirdly, the friction force, which points in the opposite
direction of the movement. In accordance with Newton's second law and the
combined influence of these forces, the body exhibits oscillatory motion around
its position of rest. The system is influenced by $m$ the mass of the body,
$\mu$ the coefficient of friction and $k$ the spring constant, indicating the
stiffness of the spring. The residual of the differential equation,
\begin{equation}
  m\frac{d^2u}{dx^2}+\mu\frac{du}{dx}+ku=0,
\end{equation}
shows relation of these parameters in reference to the problem at hand. As
Tenenbaum and Morris provide, there are three potential solutions to this
issue. However only the \emph{underdamped case} results in an oscillating
movement of the body, as illustrated in~\Cref{fig:spring}. In order to apply a
PINN to this problem, we require a set of training data $x$. This consists of
pairs of time points and corresponding displacement measurements
$(t^{(i)}, u^{(i)})$, where $i\in\{1, ..., N_t\}$. In this hypothetical case,
we know the mass $m=1kg$, and the spring constant $k=200\frac{N}{m}$ and the
initial displacement $u^{(1)} = 1$ and $\frac{du(0)}{dt} = 0$. However, we do
not know the value of the friction $\mu$. In this case the loss function,
\begin{equation}
  \mathcal{L}_{osc}(\boldsymbol{x}, \boldsymbol{u}, \hat{\boldsymbol{u}}) = (u^{(1)}-1)+\frac{du(0)}{dt}+||m\frac{d^2u}{dx^2}+\mu\frac{du}{dx}+ku||^2 + \frac{1}{N_t}\sum_{i=1}^{N_t} ||\hat{\boldsymbol{u}}^{(i)}-\boldsymbol{u}^{(i)}||^2,
\end{equation}
includes the border conditions, the residual, in which $\mu$ is a learnable
parameter and the observation loss.

% -------------------------------------------------------------------

\subsection{Disease Informed Neural Networks   1}
\label{sec:pinn:dinn}
In this section, we describe the capability of MLP's to solve systems of
differential equations. In~\Cref{sec:pandemicModel:sir}, we describe the SIR
model, which models the relations of susceptible, infectious and removed
individuals and simulates the progress of a disease in a population with a
constant size. A system of differential equations models these relations. Shaier
\etal~\cite{Shaier2021} propose a method to solve the equations of the SIR model
using a PINN, which they call a \emph{disease-informed neural network} (DINN).\\

To solve~\Cref{eq:sir} we need to find the transmission rate $\beta$ and the
recovery rate $\alpha$. As Shaier \etal~\cite{Shaier2021} point out, there are
different approaches to solve this set of equations. For instance, building on
the assumption, that at the beginning one infected individual infects $-n$ other
people, concluding in $\frac{dS(0)}{dt} = -n$. Then,
\begin{equation}
  \beta=-\frac{\frac{dS}{dt}}{S_0I_0}
\end{equation}
would calculate the initial transmission rate using the initial size of the
susceptible group $S_0$ and the infectious group $I_0$. The recovery rate, then
could be defined using the amount of days a person between the point of
infection and the start of isolation $d$, $\alpha = \frac{1}{d}$. The analytical
solutions to the SIR models often use heuristic methods and require knowledge
like the sizes $S_0$ and $I_0$. A data-driven approach such as the one that
Shaier \etal~\cite{Shaier2021} propose does not have these problems. Since the
model learns the parameters $\beta$ and $\alpha$ while learning the training
data consisting of the time points $\boldsymbol{t}$,  and the corresponding
measured sizes of the groups $\boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}$.
Let $\hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}}$ be the
model predictions of the groups and
$r_S=\frac{d\hat{\boldsymbol{S}}}{dt}+\beta \hat{\boldsymbol{S}}\hat{\boldsymbol{I}},
  r_I=\frac{d\hat{\boldsymbol{I}}}{dt}-\beta \hat{\boldsymbol{S}}\hat{\boldsymbol{I}}+\alpha \hat{\boldsymbol{I}}$
and $r_R=\frac{d \hat{\boldsymbol{R}}}{dt} - \alpha \hat{\boldsymbol{I}}$ the
residuals of each differential equation using the model predictions. Then,
\begin{equation}
  \begin{split}
    \mathcal{L}_{SIR}() = ||r_S||^2 + ||r_I||^2 + ||r_R||^2 + \frac{1}{N_t}\sum_{i=1}^{N_t} ||\hat{\boldsymbol{S}}^{(i)}-\boldsymbol{S}^{(i)}||^2 &+\\
    ||\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}||^2 &+\\
    ||\hat{\boldsymbol{R}}^{(i)}-\boldsymbol{R}^{(i)}||^2 &,
  \end{split}
\end{equation}
is the loss function of a DINN, with $\alpha$ and $beta$ being learnable
parameters.
% -------------------------------------------------------------------