add finished DINN

chapters/chap02/chap02.tex

@@ -535,7 +535,7 @@ In contrast to standard MLP's, the loss term of a PINN comprises two
 components. The first term incorporates 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
+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
@@ -616,10 +616,10 @@ 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 timepoints and corresponding displacement measurements
+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
+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,
@@ -631,5 +631,45 @@ parameter and the observation loss.
 
 \subsection{Disease Informed Neural Networks   2}
 \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.
 % -------------------------------------------------------------------

thesis.bbl

@@ -88,6 +88,12 @@
 \newblock \emph{Analysis}.
 \newblock Oldenbourg Wissenschaftsverlag GmbH, 2007
 
+\bibitem[SRS21]{Shaier2021}
+\textsc{Shaier}, Sagi ; \textsc{Raissi}, Maziar  ; \textsc{Seshaiyer},
+  Padmanabhan:
+\newblock \emph{Data-driven approaches for predicting spread of infectious
+  diseases through DINNs: Disease Informed Neural Networks}
+
 \bibitem[TP85]{Tenenbaum1985}
 \textsc{Tenenbaum}, Morris ; \textsc{Pollard}, Harry:
 \newblock \emph{Ordinary Differential Equations}.