Enhance clarity in manuscript by refining surrogate model evaluation and validation results

Manuscript/RESILINK_surrogate_optimization_manuscript.tex

@@ -324,63 +324,63 @@ The supervised-learning comparison shows a clear hierarchy among the candidate s
   \label{fig:surrogate_selection_summary_barplot}
 \end{figure}
 
-These results indicate that kernel-based models are particularly well suited to the present surrogate task. SVR provides the best compromise between accuracy and computational cost, while GPR is the second most competitive supervised strategy, especially in some higher-dimensional cases. Tree-based models, although robust, are less frequently selected, and MLP models are not competitive in terms of computational efficiency for the dataset sizes considered here. This behaviour motivated the additional evaluation of RBF interpolation as a simpler surrogate alternative. In contrast to the supervised models, which require Bayesian hyperparameter optimization and cross-validation, RBF models were trained in less than one second per output, making them especially attractive for repeated surrogate updates within the adaptive optimization loop.
+These results indicate that kernel-based models are particularly well suited to the present surrogate task. SVR provides the best compromise between accuracy and computational cost, while GPR is the second most competitive supervised strategy, especially in some higher-dimensional cases. Tree-based models, although robust, are less frequently selected, and MLP models are not competitive in terms of computational efficiency for the dataset sizes considered here. As mentioned in previous sections, this behaviour motivated the additional evaluation of RBF interpolation as a simpler surrogate alternative. In contrast to the supervised models, which require Bayesian hyperparameter optimization and cross-validation, RBF models were trained in less than one second per output, making them especially attractive for repeated surrogate updates within the adaptive optimization loop.
 
-The FEM validation of the optimized geometries is summarized in Table~\ref{tab:final_surrogate_comparison}. Only the final adaptive iteration of each geometry family is reported. The maximum variable error, $e_{\max}$, is computed as the largest relative error among all predicted quantities entering the objective function, namely ${\Exy}_i$, $\TFD_i$ and $\TFD_f$. The objective-function error, $|e_J|$, is reported in absolute value. Both supervised ML and RBF surrogates satisfy the adopted validation criteria in the final iteration for all geometry families, with $e_{\max}<5\%$ and $|e_J|<10$.
+The FEM validation of the optimized geometries is summarized in Table~\ref{tab:final_surrogate_comparison}. For each geometry family and surrogate strategy, the table reports the final accepted adaptive iteration, the optimized window thicknesses, the surrogate-predicted objective value, the corresponding FEM-recomputed objective value, and the associated validation errors. The optimization process required between two and three adaptive iterations depending on the geometry family and surrogate type, with most cases converging after three iterations. No systematic difference in the number of iterations was observed between RBF and supervised ML surrogates. The maximum variable error, $e_{\max}$, is defined as the largest relative error among all quantities entering the objective function, namely ${\Exy}_i$, $\TFD_i$ and $\TFD_f$, whereas the objective-function error, $|e_J|$, is reported in absolute value.
 
 \begin{table*}[ht!]
 \centering
-\caption{Final FEM validation of the optimized geometries obtained with supervised ML and RBF surrogates. The maximum variable error $e_{\max}$ is computed over the quantities entering the objective function, namely ${\Exy}_i$, $\TFD_i$ and $\TFD_f$. The objective error $|e_J|$ is reported in absolute value.}
+\caption{Final FEM validation of the optimized geometries obtained with supervised ML and RBF surrogates.}
 \label{tab:final_surrogate_comparison}
 \scriptsize
-\begin{tabular}{llccccc}
+\begin{tabular}{llcccccc}
 \toprule
-Family & Surrogate & $\mathbf{t}_w^{\star}$ [mm] & $J_{\mathrm{surr}}$ & $J_{\mathrm{FEM}}$ & $|e_J|$ & $e_{\max}$ [\%] \\
+Family & Surrogate & Iterations & $\mathbf{t}_w^{\star}$ [mm] & $J_{\mathrm{surr}}$ & $J_{\mathrm{FEM}}$ & $|e_J|$ & $e_{\max}$ [\%] \\
 \midrule
-H30\_B29 & RBF
+H30\_B29 & RBF & 3
 & $[12.56,\;14.79]$
 & 0.00 & 0.41 & 0.41 & 0.29 \\
 
-H30\_B29 & Supervised ML
+H30\_B29 & Supervised ML & 2
 & $[12.53,\;14.75]$
 & 0.00 & 0.08 & 0.08 & 3.14 \\
 \midrule
-H30\_B34 & RBF
+H30\_B34 & RBF & 3
 & $[14.77,\;18.95]$
 & 688.53 & 691.06 & 2.53 & 0.10 \\
 
-H30\_B34 & Supervised ML
+H30\_B34 & Supervised ML & 3
 & $[15.20,\;20.00]$
 & 796.29 & 802.64 & 6.35 & 0.64 \\
 \midrule
-H45\_B29 & RBF
+H45\_B29 & RBF & 3
 & $[5.81,\;7.88,\;8.98]$
 & 0.00 & 0.93 & 0.93 & 1.70 \\
 
-H45\_B29 & Supervised ML
+H45\_B29 & Supervised ML & 3
 & $[5.72,\;7.87,\;8.96]$
 & 0.00 & 7.47 & 7.47 & 2.85 \\
 \midrule
-H45\_B34 & RBF
+H45\_B34 & RBF & 2
 & $[6.81,\;9.02,\;9.65]$
 & 0.00 & 1.39 & 1.39 & 4.21 \\
 
-H45\_B34 & Supervised ML
+H45\_B34 & Supervised ML & 3
 & $[6.84,\;9.05,\;9.73]$
 & 0.00 & 1.35 & 1.35 & 2.91 \\
 \midrule
-H60\_B34 & RBF
+H60\_B34 & RBF & 3
 & $[5.77,\;7.38,\;8.37,\;6.18,\;5.00]$
 & 17.74 & 19.26 & 1.51 & 2.66 \\
 
-H60\_B34 & Supervised ML
+H60\_B34 & Supervised ML & 3
 & $[5.74,\;7.46,\;8.50,\;6.37,\;5.00]$
 & 16.90 & 26.87 & 9.97 & 3.81 \\
 \bottomrule
 \end{tabular}
 \end{table*}
 
-The final validation results show that both surrogate strategies provide FEM-consistent optimized geometries. RBF surrogates generally lead to lower objective-function discrepancies, with an average $|e_J|$ of 1.36, compared with 5.04 for the supervised ML surrogates. The average maximum variable error is also lower for RBF, with 1.79\% compared with 2.67\% for supervised ML. The RBF surrogate provides particularly accurate predictions for the two-window devices, with $e_{\max}$ below 0.3\%, while remaining below 4.3\% for the three- and five-window families. The supervised ML surrogates also satisfy all validation criteria, although they show larger discrepancies in some cases, especially for the H60\_B34 family, where the objective-function error reaches 9.97.
+The final validation results show that both surrogate strategies provide FEM-consistent optimized geometries after few adaptive iterations. RBF surrogates generally lead to lower objective-function discrepancies, with an average $|e_J|$ of 1.36, compared with 5.04 for the supervised ML surrogates. The average maximum variable error is also lower for RBF, with 1.79\% compared with 2.67\% for supervised ML. The RBF surrogate provides particularly accurate predictions for the two-window devices, with $e_{\max}$ below 0.3\%, while remaining below 4.3\% for the three- and five-window families. The supervised ML surrogates also satisfy all validation criteria, although they show larger discrepancies in some cases, especially for the H60\_B34 family, where the objective-function error reaches 9.97.
 
 The adaptive validation loop is essential to reach these levels of agreement. In several cases, the first surrogate-optimized candidate did not satisfy the error tolerances, particularly for the three- and five-window devices. After incorporating the additional FEM results and retraining the surrogates, the prediction errors decreased significantly. For instance, the maximum variable error of the RBF surrogate decreased from 13.70\% to 1.70\% in the H45\_B29 family, and from 11.44\% to 2.66\% in the H60\_B34 family. A similar behaviour was observed for the supervised ML surrogates, whose final candidates also satisfied all acceptance criteria. This confirms that the adaptive loop reduces the risk of accepting geometries that appear optimal only because of surrogate extrapolation errors.