## 3 Proposed Approach  
Given the criticality of the detection task in the context of tax under-reporting risk. This research aims to introduce an under-reporting detection approach based on deep learning techniques. Specifically, it formulates the prediction of Zakat under-reporting cases as a classification task. Moreover, it estimates the revenue at risk as the outcome through a regression task, which yields an objective prioritization of the auditing operations to be conducted by tax administrations. In other words, the proposed model classifies Zakat declarations into the \"under-reporting\" and \"actual declaration\" categories, which represent the positive class and the negative class, respectively. Furthermore, it determines which cases among those assigned to the positive class should be given higher priority for auditing by the tax administration. In fact, the priority sorting is conducted based on the confidence degree generated by the proposed model. Specifically, the proposed model assigns a high confidence value to under-reporting which corresponds to an expected high revenue at risk. In the following, we depict the design details of the proposed system. Figure 1 overviews the proposed network designed to extract relevant low-level features and learn their mapping into the predefined classes. As such, a collection of Zakat declarations including applicable attributes is fed into the system for the training phase. Note that these training instances are labelled, which makes them suitable for the training of the proposed deep neural networks. Note that the supervision information (labels) represents previous auditing results. Precisely, each training instance is associated with a ground truth class label as well as a Zakat revenue calculated as the difference between the pre-audit and post-audit Zakat amounts. The considered deep neural network consists of an input layer, a set of hidden layers, and two output layers. The latter are designed to perform simultaneously: (i) a regression task to predict the expected revenue at risk, and (ii) a classification task to assign each declaration to \"under-reporting\" or \"actual declaration\" category using a sigmoid layer fed with the same input. The proposed dual classification and regression task yields an objective prioritization of the auditing effort based on the confidence value generated by the designed model. The training of the proposed model relies on the optimization of the following loss function:  
\[L=\beta(MSE)+(1-\beta)(Binary\ cross\ entropy) \tag{1}\]  
where \(\beta\) represents the coefficient of the loss function that controls the tradeoff between linear regression layer and the sigmoid layer loss functions. Particularly, the loss function corresponding to the regression layer is formulated as:  
\[MSE=\frac{1}{N}{\sum_{i=1}^{N}}{(h(x_{i})-y_{i})}^{2} \tag{2}\]  
where \(N\) is the number of instances, \(h(x_{i})\) expresses the predicted value for input \(x_{i}\), and \(y_{i}\) represents the actual value corresponding to the input \(x_{i}\). Further, the binary cross entropy is exploited for the optimization of the classification model:  
\[Binary\ cross\ entropy=\frac{1}{N}{\sum_{i=1}^{N}}-(y_{i}*\log{(p_{i})}+(1-y_{i})* \log(1-p_{i})) \tag{3}\]  
where \(p_{i}\) represents the probability of the first category, and \((1-p_{i})\) represents the probability of the second category. The backpropagation algorithm considered for optimizing the loss function while training the proposed model and updating the network weights is detailed Algorithm 1.