## 2 Materials and Methods
### Deep Learning Neural Network  
Recent developments in the Deep Learning Neural Network (DNN) are increasingly opening up great opportunities in flood mapping research. Deep Learning as one of the Machine Learning models has shown promising results in image processing and pattern recognition. Therefore, this research will propose mapping the potential flood areas using the DNN algorithm. DNN is based on Artificial Neural Network and generally consists of an input layer, with more than one hidden layer and one output layer [52]. Figure 6 shows the conceptual structure of the DNN model used for flood vulnerability mapping. The input layer is the factor that affects flood (\(F1\)-\(Fn\)). The information is processed and analyzed in the hidden layer to determine the weight and classification of each pixel. The final result of the classification is an indication that there is a flood in the output layer with two possible labels: Flood (positive class) and Others (negative class).  
DNN is a feed-forward network and is trained using the back-propagation method. However, more hidden layers will make the network challenging to train because of the different adjustment speeds in the hidden layer. DNN was implemented successfully in various applications, especially in automatic image recognition, speech recognition, language processing, and some applications in remote sensing. There is no rule of thumb about the number of hidden layers and neurons in each layer since it depends on the complexity of the problem and the conditions of the dataset.  
The number of hidden layers in DNN has the advantage of representing a very complex relationship between factors. The hidden layer on DNN has neurons that are activated with the Rectified Linear Unit (ReLU) function as an alternative whose computation is more straightforward when compared to the sigmoid. Because DNN is trained on the principle of back-propagation, ReLU can minimize the decrease in learning gradient, hindering the learning process. Mathematically, the ReLU activation function can be expressed as \(h^{\prime}(x)\) in Equation (1).  
\[h^{\prime}(x)=\left\{\begin{array}{c}1 if_{x}>0\\ 0 if_{x}\leq 0\end{array}\right. \tag{1}\]  
Hidden layers in DNN perform increasingly complex feature transformations to produce a more discriminatory feature abstraction. The classification results displayed in the output layer are based on the most abstract features obtained in the last hidden layer. During the DNN learning phase, the connection weights between layers are adjusted to reduce the difference between observed and predicted results. The back-propagation process trains DNN by providing feedback on the error results to the hidden layer. The deviation between the observed and predicted results is expressed in the loss function between entropies, as expressed in Equation (2).  
\[L=-\frac{1}{N_{D}}\underset{n=1}{\overset{N_{D}}{\sum}}Tln(Y)+(1-T)\ln(1-Y) \tag{2}\]  
where \(N_{D}\) is the number of training data points, \(T\) represents the observed output, and \(Y\) represents the predicted output. The back-propagation learning gradient used for the training sample of \(m\) is formulated in Equation (3):  
\[g=\frac{1}{m}\underset{i=1}{\overset{C}{\sum}}\frac{\partial L}{\partial w} \tag{3}\]  
where \(L\) is the loss function, \(w\) represents the network weight, and \(C=2\) represents the number of output classes used (flood and others).  
Figure 6: DNN structure concept for mapping flood potential.