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@@ -535,7 +535,7 @@ In contrast to standard MLP's, the loss term of a PINN comprises two
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components. The first term incorporates the aforementioned prior knowledge to pertinent the problem. As Raissi
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\etal~\cite{Raissi2017} propose, the residual of each differential equation in
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the system must be minimized in order for the model to optimize its output in accordance with the theory.
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-We obtain the residual $R_i$, with $i\in\{1, ...,N_d\}$, by rearranging the differential equation and
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+We obtain the residual $r_i$, with $i\in\{1, ...,N_d\}$, by rearranging the differential equation and
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calculating the difference between the left-hand side and the right-hand side
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of the equation. $N_d$ is the number of differential equations in a system. As
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Raissi \etal~\cite{Raissi2017} propose the \emph{physics
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@@ -616,10 +616,10 @@ Tenenbaum and Morris provide, there are three potential solutions to this
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issue. However only the \emph{underdamped case} results in an oscillating
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movement of the body, as illustrated in~\Cref{fig:spring}. In order to apply a
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PINN to this problem, we require a set of training data $x$. This consists of
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-pairs of timepoints and corresponding displacement measurements
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+pairs of time points and corresponding displacement measurements
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$(t^{(i)}, u^{(i)})$, where $i\in\{1, ..., N_t\}$. In this hypothetical case,
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we know the mass $m=1kg$, and the spring constant $k=200\frac{N}{m}$ and the
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-initial displacement $u^{(1)} = 1$ and $\frac{du(0)}{dt} = 0$, However, we do
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+initial displacement $u^{(1)} = 1$ and $\frac{du(0)}{dt} = 0$. However, we do
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not know the value of the friction $\mu$. In this case the loss function,
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\begin{equation}
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\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,
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@@ -631,5 +631,45 @@ parameter and the observation loss.
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\subsection{Disease Informed Neural Networks 2}
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\label{sec:pinn:dinn}
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-
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+In this section, we describe the capability of MLP's to solve systems of
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+differential equations. In~\Cref{sec:pandemicModel:sir}, we describe the SIR
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+model, which models the relations of susceptible, infectious and removed
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+individuals and simulates the progress of a disease in a population with a
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+constant size. A system of differential equations models these relations. Shaier
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+\etal~\cite{Shaier2021} propose a method to solve the equations of the SIR model
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+using a PINN, which they call a \emph{disease-informed neural network} (DINN).\\
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+
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+To solve~\Cref{eq:sir} we need to find the transmission rate $\beta$ and the
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+recovery rate $\alpha$. As Shaier \etal~\cite{Shaier2021} point out, there are
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+different approaches to solve this set of equations. For instance, building on
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+the assumption, that at the beginning one infected individual infects $-n$ other
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+people, concluding in $\frac{dS(0)}{dt} = -n$. Then,
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+\begin{equation}
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+ \beta=-\frac{\frac{dS}{dt}}{S_0I_0}
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+\end{equation}
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+would calculate the initial transmission rate using the initial size of the
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+susceptible group $S_0$ and the infectious group $I_0$. The recovery rate, then
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+could be defined using the amount of days a person between the point of
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+infection and the start of isolation $d$, $\alpha = \frac{1}{d}$. The analytical
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+solutions to the SIR models often use heuristic methods and require knowledge
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+like the sizes $S_0$ and $I_0$. A data-driven approach such as the one that
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+Shaier \etal~\cite{Shaier2021} propose does not have these problems. Since the
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+model learns the parameters $\beta$ and $\alpha$ while learning the training
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+data consisting of the time points $\boldsymbol{t}$, and the corresponding
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+measured sizes of the groups $\boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}$.
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+Let $\hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}}$ be the
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+model predictions of the groups and
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+$r_S=\frac{d\hat{\boldsymbol{S}}}{dt}+\beta \hat{\boldsymbol{S}}\hat{\boldsymbol{I}},
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+ r_I=\frac{d\hat{\boldsymbol{I}}}{dt}-\beta \hat{\boldsymbol{S}}\hat{\boldsymbol{I}}+\alpha \hat{\boldsymbol{I}}$
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+and $r_R=\frac{d \hat{\boldsymbol{R}}}{dt} - \alpha \hat{\boldsymbol{I}}$ the
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+residuals of each differential equation using the model predictions. Then,
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+\begin{equation}
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+ \begin{split}
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+ \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 &+\\
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+ ||\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}||^2 &+\\
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+ ||\hat{\boldsymbol{R}}^{(i)}-\boldsymbol{R}^{(i)}||^2 &,
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+ \end{split}
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+\end{equation}
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+is the loss function of a DINN, with $\alpha$ and $beta$ being learnable
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+parameters.
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% -------------------------------------------------------------------
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Phillip Rothenbeck