## 4 Results
### Discussion
#### 4.3.2 Ablation Test for Weight-Updating Algorithm  
While training the PI_DeepONet for the beam with two ends clamped, the multi-span beam with the restrictions of clamped-simply supported-clamped, and the plate with two sides clamped, as described above, Algorithm 1 was used to perform gradient descent iterations. To demonstrate the superiority of Algorithm 1, ablation tests are performed in this section. The histograms of the back-propagated gradients of each physics-driven loss with respect to the parameters of the PI_DeepONet at the first layer of the trunk net are shown in Figures 24-26. They were monitored after 40,000 iterations. The results indicate that the gradients of \(L_{B_{2}}(\theta)\) have significantly higher overall values than the gradients of other losses when the updated weights are not applied. This makes it challenging for the PI_DeepONet to evenly fit each loss. Using the weights calculated in Algorithm 1 as the multipliers of each loss, the imbalance in the gradients of each loss is alleviated. Figure 24: Histograms of back-propagated gradients of each physics-driven loss for the beam with restrictions of clamped–clamped at the first layer of the trunk net after 40,000 training iterations of a PI_DeepONet: (**a**) histograms of back-propagated gradients with the updated weights applied and (**b**) histograms of back-propagated gradients with all weights set as 1. Figure 25: Histograms of back-propagated gradients of each physics-driven loss for the multi-span beam with the restrictions of clamped–simply supported–clamped at the first layer of trunk net after 40,000 iterations of training for PI_DeepONet: (**a**) histograms of back-propagated gradients with the updated weights applied and (**b**) histograms of back-propagated gradients with all weights set as 1. For comparison purposes, a PL_DeepONet trained using conventional gradient descent iterations was constructed for each structure with more than one Neumann boundary condition. Additionally, a PI_DeepONet trained using the adaptive weighting method based on Gaussian probabilistic models [38] was also constructed, with the loss function expressed as follows:  
\[L(\theta)=\frac{1}{2\beta_{p}^{2}}L_{p}(\theta)+\sum_{i=1}^{H+1}\frac{1}{2\beta_ {B_{i}}^{2}}L_{B_{i}}(\theta)+\log\left(\beta_{p}\prod_{i=1}^{H+1}\beta_{B_{i}}\right) \tag{42}\]  
where the \(\beta_{p}\) and \(\left\{\beta_{B_{i}}\right\}_{i=1}^{H+1}\) describe the adaptive weights of the loss terms and are tuned via the Adam optimizer before updating the parameters of the network in each gradient descent iteration. The relative errors in the maximum response calculated by the PL_DeepONet trained using the three methods are summarized in Tables 11-13. The results show that the algorithm based on Gaussian probabilistic models can improve the reconstruction accuracy of single-span and multi-span beams, but leads to the incorrect convergence of the plate. In contrast, Algorithm 1 has broader applicability and can significantly reduce the relative errors by an order of magnitude. The variables used to update the parameters of the network via gradient descent are gradients of loss terms rather than magnitudes. The gradients and magnitudes of loss terms tend to exhibit completely different distributions. Therefore, it is more logical to balance the interplay among loss terms from the perspective of gradients. This is why our proposed algorithm demonstrates superiority. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Relative Errors**} \\ \cline{2-4}  & \(T_{P}\) & \(T_{U}\) & \(T_{S}\) \\ \hline Algorithm 1 & 1.6\% & 0.64\% & 3.7\% \\ \hline Gaussian probabilistic models & 1.0\% & 3.4\% & 33\% \\ Normal gradient descent iterations & 31\% & 40\% & 55\% \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Relative errors at the maximum response calculated by the PL_DeepONet for the beam with the restrictions of damped–clamped.