## 2 Methodology
### Extreme Learning Machine  
The extreme learning machine (ELM) [20] is adopted to implement the reconstructing of the interlayer pressure of MLIs in this study. ELM is a type of feedforward neural network that is characterized by its simplicity and efficiency in training. As is shown in Figure 2, it consists of a single hidden layer with a large number of hidden units, and the weights of the hidden units are randomly initialized and fixed during training. The output weights of the ELM are then learned using a simple linear regression algorithm, making the training process very fast. ELM is particularly useful for tasks that require fast training and prediction times, such as online learning and real-time applications. It has been applied to a wide range of tasks in machine learning, including classification, regression, clustering, and feature learning. One of the key advantages of ELM is that it can learn complex nonlinear relationships in the data without the need for complex optimization algorithms, such as back-propagation. This makes it particularly well-suited for tasks where the data are noisy or have a high dimensionality. For a given set of training samples \(S=\left\{\left(T_{\text{dis}}{}^{(i)},p_{\text{dis}}{}^{(i)}\right)\mid i=1,.., N\right\}\subset\mathbb{R}^{d}\times\mathbb{R}\), where \(T_{\text{dis}}\) and \(p_{\text{dis}}\) have the same form:  
\[T_{\text{dis}}= \begin{bmatrix}T_{1,1}&\cdots&T_{1,M}\\ \vdots&\ddots&\vdots\\ T_{N,1}&\cdots&T_{N,M}\end{bmatrix}  \tag{10}\]ELM is a unified three-layer frontward network whose output has \(L\) hidden nodes. It has an excellent ability to approximate any complex function. In the sample set, a group of interlayer pressure distributions \(p\) could be represented as,  
\[p(T)=\sum_{i=1}^{L}h_{i}(T)\beta_{i}=h(T)\beta,T\in\mathbb{R}^{d} \tag{11}\]  
where the \(h(T)=\left[h_{1}(T),h_{2}(T),\ldots,h_{L}(T)\right]\) and \(\beta=\left[\beta_{1},\beta_{2},\ldots,\beta_{L}\right]^{\top}h_{i}(T)\) are the hidden layer function \(g(\mathrm{a}_{i},b_{i},T)\) between the input layer and the \(i\)th hidden node. \(A_{i}\) and \(b_{i}\) are randomly generated independent of the training data. \(\beta_{i}\) is the output weight between the \(i\)th hidden node to the output node. The computational hidden nodes can be obtained from the sigmoid function, additive and radial basis function, hinging functions, wavelets, and so on, e.g., \(g(\mathrm{a}_{i},b_{i},T)=\sigma(\mathrm{a}_{i}^{\top}T+b_{i})\) where \(\sigma\) is the sigmoid function. To minimize the least square error [21], the objective loss function is  
\[\min_{\beta}\ :\ \parallel\mathrm{H}\beta-p\parallel^{2} \tag{12}\]  
where,  
\[H=\left[h\Big{(}T^{(1)}\Big{)}^{\top},h\Big{(}T^{(2)}\Big{)}^{\top},\ldots,h \Big{(}T^{(N)}\Big{)}^{\top}\right]^{\top},p=\left[p^{(1)},p^{(2)},\ldots,p^{( N)}\right]^{\top} \tag{13}\]  
where \(N\) represents the number of samples. The solution for Equation (13) is  
\[\beta= \begin{cases}\left(H^{\top}H\right)^{-1}H^{\top}p,&N\geq L\\ H^{\top}\big{(}HH^{\top}\big{)}^{-1}p,&N<L\end{cases}  \tag{14}\]  
When the measurement temperature \(T_{\mathrm{obj}}\) is given, the corresponding interlayer pressure \(p_{\mathrm{obj}}\) can be predicted by ELM as shown in Figure 3.  
Random connection weights between the input layer and the hidden layer, \(a_{ij}\), and the threshold of the hidden layer neurons, \(b_{ij}\), are generated by the ELM algorithms. The reconstructed interlayer pressure distribution \(p_{\mathrm{obj}}^{\cdot}\) is diverse for each training case. Therefore, we adopt the method of multiple trainings. Firstly, give a number of neurons (e.g., 20) and sigmoid function type (e.g., radial basis function), then repeat the training 100 times by inputting the temperature distribution \(T_{\mathrm{obj}}\) determined by the abovementioned LBL model. A total of 100 pressure distributions  
Figure 3: The neural network structure of ELM. are reconstructed by the ELM model.