## 2 Methodology
### Artificial Neural Network (ANN)
#### 2.2.2 Error Back Propagation Training (BP)-ANN  
Neural network algorithms can be classified into the Error Back Propagation Training (BP)-ANN model, the perceptron neural network model, the Radial Basis Function (RBF) neural network model, the Hopfield feedback neural network model, the self-organizing neural network model, and so on. Since it successfully solved the weight adjustment problem of the multilayer feedforward neural network for non-linear continuous functions, the BP-ANN is widely used. Figure 2 shows a mathematical model of the BP-ANN, in which \(j\) is the index of the nodes. BP-ANN is based on gradient search technology with two processes: forward propagation process of information and back propagation process of errors. In forward propagation, the input signal passes through the hidden layers to the output layer. If desired output appears, the learning algorithm ends; otherwise, it turns to back propagation. In back propagation, the error transits through the output layer, and the weights of each layer of neurons are adjusted according to the gradient descent method to reduce the error. Such loop computation continues to make the output of the network approach the expected values as close as possible. Due to its strong nonlinear mapping ability, BP-ANN is used in this study to establish the relationships between the reservoir flushed sediment and its influence factors, the calculation steps of which are illustrated below. Assuming the input vector is u, the number of neurons in input layer is n, the output vector is y, the number of neurons in the output layer is m, the length of the input/output sample pair is \(\overline{L}\), and the steps of the BP-ANN algorithm include:  
1. Setting the initial weight \(\omega\left(0\right)\), which is relatively small random nonzero value;
2. Giving the input/output sample pair, and calculating the output of the neural network:  
Assuming the input of the \(p\)-th sample is \(u_{p}=\left(u_{1p},u_{2p},\ldots,u_{np}\right)\), the output of the \(p\)-th sample is \(d_{p}=\left(d_{1p},d_{2p},\ldots,d_{mp}\right)\), \(p=1,2,\)..., \(\overline{L}\), the output of node i with p-th sample is \(y_{ip}\):  
\[y_{ip}\left(t\right)=f\left[x_{ip}\left(t\right)\right]=f\left[\sum_{j}\omega_{ ij}\left(t\right)I_{jp}\right], \tag{11}\]  
Figure 2: Calculation principle diagram of Back Propagation Training Artificial Neural Network (BP-ANN). where \(I_{jp}\) is the \(j\)-th input of node \(i\) when inputting the \(p\)-th sample, and \(f(\bullet)\) is the activation transfer function as shown in Equation (10);
3. Calculating the objective function \(J\) of the network: Assuming \(E_{p}\) is the objective function of the network when inputting the \(p\)-th sample, then: \[E_{p}=\frac{1}{2}\sum_{k}\left[d_{kp}-y_{kp}\left(t\right)\right]^{2}=\frac{1} {2}\sum_{k}\epsilon_{kp}^{2}\left(t\right),\] (12) where \(y_{kp}\) is the network output after \(t\) times of weight adjustment when inputting the \(p\)-th sample and \(k\) is the index of the node in output layer; The overall objective function of the network is used to estimate the training level of the network, as shown in Equation (13): \[J\left(t\right)=\sum_{p}E_{p}\left(t\right).\] (13)
4. Discriminating whether the algorithm should stop: if \(J\left(t\right)<\varepsilon\), it stops; otherwise, it turns to step (5), where \(\varepsilon>\ 0\) is preset;
5. Back propagation calculating: Starting from the output layer, referring to \(J\), do the calculation according to the gradient descent algorithm to adjust the value of weight.