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better nn explaination

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Lukas Winkler 2019-08-01 15:40:59 +02:00
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2
20_simulations.tex
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@ -7,7 +7,7 @@ For a realistic model of two gravitationally colliding bodies the SPH (\textit{s
In the simulation two celestial bodies are placed far enough apart so that tidal forces can affect the collision. Both objects consist of a core with the physical properties of basalt rocks and an outer mantle made of water ice.
To keep the simulation time short and make it possible to do many simulations with varying parameters 20k SPH particles are used and each simulation is ran for 300 timesteps of each \SI{144}{\second} so that a whole day of collision is simulated.
To keep the simulation time short and make it possible to do many simulations with varying parameters 20k SPH particles are used and each simulation is ran for 300 timesteps of each \SI{144}{\second} so that 12 hours of collision is simulated.
\section{Parameters}

3
30_results.tex
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@ -3,12 +3,11 @@
For the large set of simulations we can now extract the needed value. The output of the relaxation program (\texttt{spheres\_ini\_log}) gives us the precise values for impact angle and velocity and the exact masses of all bodies. As theses values slightly differ from the parameters explained in Section \ref{sec:parameters} due to the setup of the simulation, in the following steps only the precise values are considered. From the \texttt{aggregates.txt} explained in Section \ref{sec:postprocessing} the final masses and water fractions of the two largest fragments are extracted. From these the main output considered in this analysis, the water retention of the two fragments can be calculated.
\section{Correlations}
\label{sec:cov}
One very easy, but sometimes flawed\footnote{\todo[inline]{explain issues with pearson}} way to look at the whole dataset at once is calculating the \textit{Pearson correlation coefficient} between the input parameters and the output water fraction (Figure \ref{fig:cov}). This shows the expected result that a higher collision angle (so a more hit-and-run like collision) has a higher water retention and a higher collision speed results in far less water left on the two largest remaining fragments. In addition higher masses seem to result in less water retention. The initial water fractions of the two bodies does seem to have very little influence on the result of the simulations.
\begin{figure}[h] % TODO: h is temporary
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{images/cov.pdf}
\caption{TODO}

64
43_nn.tex
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@ -1,9 +1,21 @@
% !TeX spellcheck = en_US
\begin{figure}[h] % TODO: h is temporary
\begin{figure}[h] % also temporary
\centering
\includegraphics[width=0.4\linewidth]{images/graphviz/general.pdf}
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{images/graphviz/general.pdf}
\caption{a simple neural network}
\label{fig:neuralnetwork-general}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{images/graphviz/graph.pdf}
\caption{the network used for interpolation}
\label{fig:neuralnetwork-graph}
\end{subfigure}
\caption{}
\label{fig:neuralnetwork-general}
\end{figure}
\section{Neural Networks}
@ -32,25 +44,51 @@ After this first step (the \textit{feedforward}) is done, the weights can be mod
To update the weights the derivative of the Loss function with respect to the weights is calculated and added to the existing weights.\todo{more details?}\footcite{NN-python}
\subsection{Implementation}
As building a neural network from scratch gets complex very quickly, it is easier to use \texttt{Keras}\footnote{\url{https://keras.io}} which provides easy to use high-level functions over the calculations provided by \texttt{TensorFlow}\footnote{\url{https://www.tensorflow.org/}}. To build our network, we only need to specify the structure of the layers, take our input and let the network train for 200 epochs (iterations of feedforward and backpropagation).\todo{explain more once it is finished}
As building a neural network from scratch gets complex very quickly, it is easier to use \texttt{Keras}\footnote{\url{https://keras.io}} which provides easy to use high-level functions over the calculations provided by \texttt{TensorFlow}\footnote{\url{https://www.tensorflow.org/}}. To build our network, we only need to specify the structure of the layers, take our input and let the network train for 200 epochs (iterations of feedforward and backpropagation).
The network needs six nodes in the input layer for the input parameters and one node in the output layer for the prediction. In between are two layers with decreasing numbers of nodes as this seems to give the best results. (Figure \ref{fig:neuralnetwork-graph})
\begin{lstlisting}[language=Python,caption=the used model as Keras code,label=lst:model]
from keras import Sequential
from keras.layers import Dense
\begin{lstlisting}[language=Python]
model = Sequential()
model.add(Dense(6, input_dim=6, kernel_initializer='normal', activation='relu'))
model.add(Dense(5, kernel_initializer='normal', activation='relu'))
model.add(Dense(6, input_dim=6, activation='relu'))
model.add(Dense(4, kernel_initializer='normal', activation='relu'))
model.add(Dense(1, kernel_initializer='normal'))
model.add(Dense(3, kernel_initializer='normal', activation='relu'))
model.add(Dense(1, kernel_initializer='normal', activation="sigmoid"))
model.compile(loss='mean_squared_error', optimizer='adam')
model.fit(X, Y, epochs=200)
\end{lstlisting}\todo{just a placeholder}
model.fit(x, Y, epochs=200, validation_data=(x_test, Y_test))
\end{lstlisting}
\subsection{Training}
To find the ideal parameters to use the simulation data (excluding the data from Section \ref{sec:tests}) is split into two groups: The original set of simulations and \SI{80}{\percent} of the new simulation set is used to train the neural network while the remaining \SI{20}{\percent} are used for validation. This means that after every epoch not only the loss function for the training data is calculated, but also for the separate validation data (Figure \ref{fig:loss_val}). Finally the model with the lowest loss on the validation data set was chosen (Listing \ref{lst:model}).
\todo{more text on how exactly the data is split up once it is decided}
\begin{figure}[h] % also temporary
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{images/loss.pdf}
\caption{loss function on the training data}
\label{fig:loss}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{images/val_loss.pdf}
\caption{loss function on the validation data}
\label{fig:val_loss}
\end{subfigure}
\caption{During training the loss function (mean squared error) decreases with every epoch until it converges to a final value}
\label{fig:loss_val}
\end{figure}
After the training the resulting model can be saved in a small \texttt{HDF5} file which can be used to evaluate the model very quickly (about \SI{100}{\milli\second} for \num{10000} interpolations).

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images/graphviz/general.dot
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@ -44,6 +44,11 @@ To understand how the water transport works exactly one has to find an estimatio
\appendix
\chapter{Mass retention}
\todo[inline]{Just mention quickly that everything can also be applied to mass retention and show a few plots}
\chapter{Placeholder}