## 2 Proposed Feed-Forward NN Structure-Based Estimation Methods
### Proposed NN-Based Estimation Method 1: FFNN Structure  
#### 2.2.1 Response Function Estimation Using FFNN  
According to [PERSON]'s philosophy, there are two output responses of interest, namely, the process mean and standard deviation. These output responses can be estimated simultaneously from the output layer in a single FFNN. Besides, as demonstrated by [PERSON] [32], any continuous function of n input variables can be approximated by an FFNN with only one hidden layer. A similar notion is also discussed by [PERSON] [31], [PERSON] et al. [33], and [PERSON] et al. [53]. The proposed FFNN-based RD modeling method with one hidden layer is illustrated in Figure 2.  
The input layer has \(k\) control factors, \(x_{1}\),..., \(x_{i}\),..., \(x_{k}\). The input for each hidden neuron is the weighted sum of these \(k\) factors plus their associated bias, expressed as \(\sum\limits_{i=1}^{k}w_{ji}x_{i}+a_{j}^{hid}\). This calculated value will be transformed by a transformation function (\(f_{1}\)), also called an activation function (expressed by Equation (5)). The transformed value \(y_{j}^{hid}\) is both the output of the hidden neuron and the input for the next layer (in our case, the output layer). Subsequently, the ultimate transformed outcome \(\hat{y}_{FFNN}\) is the final result. The outputs of the hidden neuron and FFNN can then be expressed as  
\[y_{j}^{hid}=f_{1}\left(\sum\limits_{i=1}^{k}w_{ji}x_{i}+a_{j}^{hid}\right) \tag{5}\]  
where \(w_{ji}\) is the weight connecting input factor \(x_{i}\) to hidden node \(j\); \(a_{j}^{hid}\) is the associated bias of the hidden layer node \(j\), and \(f_{1}\) is the transfer function of the hidden layer.  
\[\hat{y}_{FFNN}=g_{1}\left\{\sum\limits_{j=1}^{h}w_{j}y_{j}^{hid}+a^{out} \right\}=g_{1}\left\{\sum\limits_{j=1}^{h}t_{j}\left[f_{1}\left(\sum\limits_{ i=1}^{k}w_{ji}x_{i}+a_{j}^{hid}\right)\right]+b^{out}\right\} \tag{6}\]  
where \(g_{1}\) is the activation function of the output layer, \(h\) is the number of hidden nodes, \(t_{j}\) is the weight connecting each hidden node \(y_{j}^{hid}\) to the output layer, and \(b^{out}\) is the output bias.  
Figure 2: Proposed FFNN-based RD modeling method.  
The typical transfer functions for the hidden and output layers are the hyperbolic tangent sigmoid and linear functions, respectively. The estimated mean and standard deviation functions in this case are  
\[\hat{\eta}_{FFNN}(\mathbf{x})=\sum_{j=1}^{h_{\_}mean}\quad\left\{t_{j}^{mean} \left[\frac{2}{1+exp\left(-2\left(\sum_{i=1}^{k}w_{ij}^{mean}x_{i}+a_{\_}{j}^{ mean}\right)\right)}-1\right]\right\} \tag{7}\]  
\[\hat{\sigma}_{FFNN}(\mathbf{x})=\sum_{j=1}^{h_{\_}std}\left\{t_{j}^{std}\left[ \frac{2}{1+exp\left(-2\left(\sum_{i=1}^{k}w_{ij}^{std}x_{i}+a_{\_}{j}^{std} \right)\right)}-1\right]\right\}+b_{std}^{FFNN} \tag{8}\]  
where \(h_{\_}mean\) and \(h_{\_}std\) represent the number of hidden neurons in the mean and standard deviation estimation model, respectively. Moreover, \(a_{\_}{j}^{mean}\) and \(a_{\_}{j}^{std}\) represent the bias for hidden node \(j\) in both models, while \(b_{\_}{mean}^{FFNN}\) and \(b_{\_}{std}^{FFNN}\) denote the bias of the output neuron. Similarly, \(t_{\_}{j}^{mean}\) and \(t_{\_}{j}^{std}\) signify the weight connecting a hidden node \(j\) to the output for both models, while \(w_{\_}{ij}^{mean}\) and \(w_{\_}{ij}^{std}\) refer to the weight connecting the input factors to a hidden node \(j\) of the FFNN estimation model for mean and standard deviation, respectively.