continue mlp

chapters/chap02/chap02.tex

@@ -7,7 +7,7 @@
 %         summary of the content in this chapter
 % Version:  05.08.2024
 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\chapter{Theoretical Background}
+\chapter{Theoretical Background   12}
 \label{chap:background}
 
 This chapter introduces the theoretical knowledge that forms the foundation of
@@ -24,7 +24,7 @@ in~\Cref{sec:pinn}.
 
 % -------------------------------------------------------------------
 
-\section{Mathematical Modelling using Functions}
+\section{Mathematical Modelling using Functions   1}
 \label{sec:domain}
 
 To model a physical problem using mathematical tools, it is necessary to define
@@ -51,7 +51,7 @@ In this case, time serves as the domain, while the distance is the codomain.
 
 % -------------------------------------------------------------------
 
-\section{Basics of Differential Equations}
+\section{Basics of Differential Equations   1}
 \label{sec:differentialEq}
 
 Often, the change of a system is more interesting than its current state.
@@ -123,7 +123,7 @@ describe the application of these principles in epidemiological models.
 
 % -------------------------------------------------------------------
 
-\section{Epidemiological Models}
+\section{Epidemiological Models   3}
 \label{sec:epidemModel}
 
 Pandemics, like \emph{COVID-19}, which has resulted in a significant
@@ -150,7 +150,7 @@ and relations that are pivotal to understanding the problem.
 
 % -------------------------------------------------------------------
 
-\subsection{SIR Model}
+\subsection{SIR Model   2}
 \label{sec:pandemicModel:sir}
 
 In 1927, Kermack and McKendrick~\cite{1927} introduced the \emph{SIR Model},
@@ -328,7 +328,7 @@ next~\Cref{sec:pandemicModel:rsir}.
 
 % -------------------------------------------------------------------
 
-\subsection{Reduced SIR Model and the Reproduction Number}
+\subsection{Reduced SIR Model and the Reproduction Number   1}
 \label{sec:pandemicModel:rsir}
 The~\Cref{sec:pandemicModel:sir} presents the classical SIR model. The model
 comprises two parameters $\beta$ and $\alpha$, which describe the course of a
@@ -406,7 +406,7 @@ systems, as we describe in~\Cref{sec:mlp}.
 
 % -------------------------------------------------------------------
 
-\section{Multilayer Perceptron}
+\section{Multilayer Perceptron   2}
 \label{sec:mlp}
 In~\Cref{sec:differentialEq} we show the importance of differential equations to
 systems, being able to show the change of it dependent on a certain parameter of
@@ -460,19 +460,38 @@ Hornik \etal~\cite{Hornik1989} shows, MLP's are universal approximators.\\
   \label{fig:mlp_example}
 \end{figure}
 
-The process of optimizing the parameters $\theta$ is called \emph{learning}.
-Here, we define a metric for the quality of the results, of our neural network.
-This metric is called a loss function
+The process of optimizing the parameters $\theta$ is called \emph{training}.
+For trainning we will have 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 the
+\emph{loss function} $\Loss{ }$, using the model prediction
+$\hat{\boldsymbol{y}}$ and the true value $\boldsymbol{y}$, which will act as a
+metric of how far the model is away from the correct answer. One of the most
+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}
+  \Loss{MSE} = \frac{1}{N}\sum_{n=1}^{N} ||\hat{\boldsymbol{y}}^{(i)}-\boldsymbol{y}^{(i)}||^2,
+\end{equation}
+calculates the squared differnce between each model prediction and true value
+of a training and takes the mean across the whole training data. Now, we only
+need a way to change the parameters using our loss. For this gradient descent
+is used in gradient-based learning.
+- gradient descent/gradient-based learning
+- backpropagation
+- optimizers
+- learning rate
+- scheduler
 
 
 % -------------------------------------------------------------------
 
-\section{Physics Informed Neural Networks}
+\section{Physics Informed Neural Networks   5}
 \label{sec:pinn}
 
 % -------------------------------------------------------------------
 
-\subsection{Disease Informed Neural Networks}
+\subsection{Disease Informed Neural Networks   2}
 \label{sec:pinn:dinn}
 
 % -------------------------------------------------------------------

macros.tex

@@ -3,12 +3,12 @@
 %\newcommand\todo[1]{\textcolor{red}{TODO: #1}}
 %\newcommand\fixme[1]{\textcolor{green}{FIXME: #1}}
 
-\newcommand\ie{\textit{i.e.}\xspace}
-\newcommand\eg{\textit{e.g.}\xspace}
-\newcommand\etal{\textit{et al.}\xspace}
+\newcommand{\ie}{\textit{i.e.}\xspace}
+\newcommand{\eg}{\textit{e.g.}\xspace}
+\newcommand{\etal}{\textit{et al.}\xspace}
 
 
 \newcommand{\RO}{\ensuremath{\mathcal{R}_0}}
 \newcommand{\Rt}{\ensuremath{\mathcal{R}_t}}
 
-% do loss command\newcommand{}
+\newcommand{\Loss}[1]{\ensuremath{\mathcal{L}_{#1}(\hat{\boldsymbol{y}}, \boldsymbol{y})}}