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@@ -141,6 +141,57 @@ employed by the PINN models, which we describe in the subsequent section.
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\section{PINN for the SIR Model 3}
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\label{sec:pinn:sir}
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+In the last section we present the methods, we use to transform the RKI data
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+(see~\Cref{sec:preprocessing}) into the format that is used by the PINNs to seek
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+a solution for the SIR models. In this section we lay out the methodology we
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+employ for this thesis concerning PINNs for SIR models.\\
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+
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+The data, which is yielded by the preprocessing, is in the structure of pairs of
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+$(\boldsymbol{t^{(i)}}, (\boldsymbol{S^{(i)}},\boldsymbol{I^{(i)}},\boldsymbol{R^{(i)}}))$,
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+which contain the sizes of the susceptible, infectious, and removed compartments
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+together with their respective time point with the index $i$. This means that
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+this training data contains the measured solutions of the functions $S(t)$,
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+$I(t),$ and $R(t)$, which a neural network may use to approximate these
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+functions. Furthermore, a PINN can carry out this task with a higher precision
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+for more complex problems were the unknown function is more complex and just a
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+system of differential equations is given.\\
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+
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+In this thesis we want to find the solutions of the SIR models belonging to the
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+cases of the datasets. The SIR model is given through the system of differential
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+equations (see~\Cref{eq:sir}), which describes the relations and fluctuations of
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+the three compartments through transition rates $\beta$ and $\alpha$. As we
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+explain in~\Cref{sec:pandemicModel:sir}, these parameters influence course of
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+the pandemic, which is described by their respective model. Mathematically, when
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+we find a pair of parameters for a dataset, these parameters describe a
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+function, that solves the system of differential equations for our data set. A
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+PINN finds parameters for a given set of differential equations by solving the
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+inverse problem. As Shaier \etal~\cite{Shaier2021} propose, a DINN solves inverse
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+problems by setting the parameters $\beta$ and $\alpha$ to trainable parameters
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+$\widehat{\beta}$ and $\widehat{\alpha}$. As described in~\Cref{sec:pinn}, the
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+DINN learns the parameters to optimize its model predictions $\hat{\boldsymbol{S}}$,
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+$\hat{\boldsymbol{I}}$, and $\hat{\boldsymbol{R}}$, to fit the differential
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+equations through the usage of their residuals and the given data.\\
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+
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+The PINN uses the loss function to determine how far it is away from the true
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+solution. For the DINN~\cite{Shaier2021} this loss function includes the mean
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+squared error of each residual in addition to the mean squared error of the
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+model predictions concerning their respective true solutions. On the contrary to
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+Shaier \etal, who use the set of differential equations of~\Cref{eq:sir} for
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+their loss function, we use~\Cref{eq:modSIR}. The reason for this choice is that
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+we encountered a better practical performance during our work than when using
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+the equation, used by Shaier \etal. Let $N$ be the size of the population and
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+$N_t$ the number of training point of the used dataset then,
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+
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+\begin{equation}
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+ \begin{split}
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+ \mathcal{L}_{\text{SIR}}(\boldsymbol{t}, \boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}, \hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}}) = &\bigg\|\frac{d\hat{\boldsymbol{S}}}{d\boldsymbol{t}}+ \widehat{\beta}\frac{\hat{\boldsymbol{S}}\hat{\boldsymbol{I}}}{N}\bigg\|^2\\ + &\bigg\|\frac{d\hat{\boldsymbol{I}}}{d\boldsymbol{t}} - \widehat{\beta}\frac{\hat{\boldsymbol{S}}\hat{\boldsymbol{I}}}{N} + \widehat{\alpha}\hat{\boldsymbol{I}}\bigg\|^2\\ + &\bigg\|\frac{d\hat{\boldsymbol{R}}}{d\boldsymbol{t}} + \widehat{\alpha}\hat{\boldsymbol{I}}\bigg\|^2\\
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+ + &\frac{1}{N_t}\sum_{i=1}^{N_t} \Big\|\hat{\boldsymbol{S}}^{(i)}-\boldsymbol{S}^{(i)}\Big\|^2 + \Big\|\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}\Big\|^2 + \Big\|\hat{\boldsymbol{R}}^{(i)}-\boldsymbol{R}^{(i)}\Big\|^2,
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+ \end{split}
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+\end{equation}
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+
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+is the loss function, that employ to find the transition parameters $\beta$ and
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+$alpha$ for the given dataset.
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+
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% -------------------------------------------------------------------
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\section{PINN for the reduced SIR Model 2}
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Phillip Rothenbeck