## Appendix A Surrogate Models
### PINN
\begin{table} \
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**SIMULATION 12**} \\ \hline  & & \multicolumn{2}{c}{**MLP**} & \multicolumn{2}{c}{**PINN**} & \multicolumn{2}{c}{**SVR**} & \multicolumn{2}{c}{**GBDTR**} & \multicolumn{2}{c}{**KNNR**} & \multicolumn{2}{c}{**GPR**} \\  & **mean** & **std** & **mean** & **std** & **mean** & **std** & **std** & & \\ \hline \multirow{5}{*}{R2} & \(c_{xx}^{2}\) & 0.7511 & \(2.314\times 10^{-1}\) & \(-4.3817\) & 2.685 & \(-0.2933\) & 0.5204 & \(9.591\times 10^{-4}\) & 0.7164 & 0.0857 \\  & \(c_{xy}^{2}\) & 0.5801 & \(3.853\times 10^{-1}\) & \(-0.9114\) & \(1.065\times 10^{-1}\) & \(-0.1797\) & 0.2623 & \(2.829\times 10^{-3}\) & 0.6314 & 0.0485 \\  & \(c_{xy}^{2}\) & 0.7517 & \(2.319\times 10^{-1}\) & 0.1125 & \(9.052\times 10^{-2}\) & \(-0.2646\) & 0.5280 & \(1.380\times 10^{-3}\) & 0.7186 & 0.0873 \\  & \(c_{xx}^{2}\) & 0.7547 & \(2.273\times 10^{-1}\) & \(-2.1130\) & \(9.956\times 10^{-1}\) & \(-0.2740\) & 0.5357 & \(4.099\times 10^{-4}\) & 0.7131 & 0.0826 \\  & \(c_{xy}^{2}\) & 0.5616 & \(4.021\times 10^{-1}\) & 0.5356 & \(2.972\times 10^{-2}\) & \(-0.1116\) & 0.2250 & \(7.643\times 10^{-3}\) & 0.6277 & 0.0480 \\  & \(c_{xy}^{2}\) & 0.7556 & \(2.269\times 10^{-1}\) & \(-0.4174\) & \(3.678\times 10^{-1}\) & \(-0.2741\) & 0.5217 & \(8.630\times 10^{-4}\) & 0.7142 & 0.0830 \\  & \(c_{xz}^{2}\) & 0.6959 & \(2.525\times 10^{-1}\) & 0.4548 & \(2.042\times 10^{-1}\) & \(-0.4013\) & 0.4990 & \(7.038\times 10^{-5}\) & 0.7267 & 0.0912 \\  & \(c_{xy}\) & 0.4251 & \(5.887\times 10^{-1}\) & 0.9077 & \(3.046\times 10^{-3}\) & \(-0.4988\) & 0.7847 & \(3.553\times 10^{-4}\) & 0.8620 & 0.2942 \\  & \(c_{yy}\) & \(-0.0088\) & \(8.334\times 10^{-1}\) & 0.8314 & \(3.051\times 10^{-3}\) & 0.1802 & 0.4255 & \(6.447\times 10^{-4}\) & 0.6805 & 0.1513 \\  & \(c_{yy}\) & \(6.0402\) & \(4.473\times 10^{-1}\) & 0.6678 & \(3.865\times 10^{-4}\) & \(-0.0250\) & 0.7608 & \(5.706\times 10^{-6}\) & 0.8418 & 0.1470 \\  & \(c_{zz}\) & 0.6356 & \(3.848\times 10^{-1}\) & 0.9351 & \(1.247\times 10^{-3}\) & 0.0788 & 0.6999 & \(1.686\times 10^{-4}\) & 0.8282 & 0.1321 \\  & \(u\) & 0.9627 & \(2.725\times 10^{-2}\) & 0.9392 & \(4.615\times 10^{-3}\) & 0.3676 & 0.9467 & \(1.345\times 10^{-4}\) & 0.9625 & 0.4940 \\  & \(v\) & 0.9478 & \(5.094\times 10^{-2}\) & 0.9805 & \(3.346\times 10^{-3}\) & 0.7270 & 0.9521 & \(1.989\times 10^{-5}\) & 0.9697 & 0.5572 \\ \hline mean & 0.6475 & \(3.326\times 10^{-1}\) & \(-0.1122\) & \(3.265\times 10^{-1}\) & \(-0.0745\) & 0.8984 & \(1.579\times 10^{-4}\) & 0.7687 & 0.1771 \\ \hline \multirow{5}{*}{MSE} & \(c_{xx}^{2}\) & \(1.4412\times 10^{-3}\) & \(1.313\times 10^{-3}\) & \(3.054\times 10^{-2}\) & \(1.523\times 10^{-2}\) & \(7.339\times 10^{-3}\) & \(2.721\times 10^{-3}\) & \(5.442\times 10^{-6}\) & \(1.609\times 10^{-3}\) & \(5.188\times 10^{-3}\) \\  & \(c_{xy}^{2}\) & \(2.253\times 10^{-3}\) & \(2.067\times 10^{-3}\) & \(1.025\times 10^{-2}\) & \(5.715\times 10^{-4}\) & \(6.329\times 10^{-3}\) & \(3.958\times 10^{-3}\) & \(3.518\times 10^{-5}\) & \(1.519\times 10^{-3}\) & \(5.1977\times 10^{-3}\) & \(5.105\times 10^{-3}\) \\  & \(c_{xy}^{2}\) & \(1.422\times 10^{-3}\) & \(1.328\times 10^{-3}\) & \(5.080\times 10^{-3}\) & \(5.812\times 10^{-4}\) & \(7.229\times 10^{-3}\) & \(2.7\end{tabular}\end{table} 
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