## 5 The PINN-Based Surrogate Model  
To achieve precise predictions of the beam's displacement across an extensive parameter plane, a PINN-based surrogate model is constructed, as depicted in Figure 8. The outputs from the 15 PINN models presented in the previous section serve as the comprehensive training dataset for this surrogate model. The methodology and essential steps are outlined as follows:  
The input data comprise 50 evenly spaced coordinate points selected within the range \(y\in[-5\;\mathrm{m},5\;\mathrm{m}]\), along with the corresponding beam displacement predictions generated by the 15 pre-established target-domain PINN models. These inputs are denoted as \(y_{j}\) (coordinate values) and \(w_{j}^{i}(y_{j})\) (displacement predictions), where \(i=1,\ldots,15\) and \(j=1,\ldots,50\). The output layer of the surrogate model represents the beam displacement predictions, matching the dimensionality of the input displacement values and denoted as \(w_{p}^{i}(y_{j})\), with \(i=1,\ldots,15\) and \(j=1,\ldots,50\). To construct the feedforward neural network architecture, a structured weight constraint approach is employed to define the  
Figure 8: Schematic diagram of the PINN-based surrogate model. \begin{table}
\begin{tabular}{c c c c} \hline
**Source Domain** & **Target Domain** & **Parameters (\(c,v\))** & \(\boldsymbol{R^{2}}\) \\ \hline \multirow{6}{*}{\(1\)} & \(1\) & \((2.00\times 10^{6},\;\;15)\) & 0.9992 \\  & \(2\) & \((2.00\times 10^{6},\;\;20)\) & 0.9995 \\  & \(3\) & \((2.00\times 10^{6},\;\;25)\) & 0.9995 \\  & \(4\) & \((2.00\times 10^{6},\;\;30)\) & 0.9997 \\  & \(5\) & \((2.00\times 10^{6},\;\;35)\) & 0.9997 \\ \hline \multirow{6}{*}{\(2\)} & \(1\) & \((2.25\times 10^{6},\;\;15)\) & 0.9995 \\  & \(2\) & \((2.25\times 10^{6},\;\;20)\) & 0.9996 \\  & \(3\) & \((2.25\times 10^{6},\;\;25)\) & 0.9997 \\  & \(4\) & \((2.25\times 10^{6},\;\;30)\) & 0.9997 \\  & \(5\) & \((2.25\times 10^{6},\;\;35)\) & 0.9996 \\ \hline \multirow{6}{*}{\(3\)} & \(1\) & \((2.50\times 10^{6},\;\;15)\) & 0.9996 \\  & \(2\) & \((2.50\times 10^{6},\;\;20)\) & 0.9995 \\  & \(3\) & \((2.50\times 10^{6},\;\;25)\) & 0.9997 \\  & \(4\) & \((2.50\times 10^{6},\;\;30)\) & 0.9996 \\  & \(5\) & \((2.50\times 10^{6},\;\;35)\) & 0.9994 \\ \hline \end{tabular}
\end{table} 
Table 3: Accuracy values of the PINN models in the target domains. mapping relationship between the output layer \(w_{p}^{i}(y_{j})\) and the input layers \(y_{j}\) and \(w_{t}^{i}(y_{j})\). Sparsity constraints are imposed on the network's weight matrix \(M\) and bias vector \(b\) to enforce selective connectivity during training. This ensures that each neuron's output is influenced exclusively by its corresponding input channel, while non-essential connections are pruned by constraining their weights to zero. This optimization strategy can be formally represented as  
\[w_{p}^{i}(y_{j})=\tanh\cdot(M_{ii}\cdot\left[y_{j},w_{t}^{i}(y_{j})\right]+b_{ i}),\quad i=1,\cdots,15;\,j=1,\cdots,50, \tag{32}\]  
where \(\tanh\) is the activation function, and \(M_{ii}\) denotes the diagonal elements of the weight matrix. The loss function for the PINN surrogate model is formulated as an ensemble of loss functions derived from multiple local PINN models. This formulation is expressed as  
\[ \begin{split} L_{data}&=\frac{1}{50}\cdot\sum_{i=1 }^{15}\sum_{j=1}^{50}\left|w_{p}^{i}(y_{j})-w_{t}^{i}(y_{j})\right|^{2}\\ L_{\triangledown}&=\frac{1}{50}\cdot\sum_{i=1}^{15 }\sum_{j=1}^{50}\left|f^{i}(y_{j})\right|^{2}\end{split}  \tag{33}\]  
The PINN surrogate model differs from local PINN models in that it must handle a significantly larger dataset, leading to considerably higher computational costs during training. To address this, the surrogate model constructed in this study adopts a three-layer fully connected architecture (20 neurons per layer) with the \"tanh\" activation function. For data partitioning, a 4:1 ratio is implemented to divide the training set and validation set.