## 3 Results and Discussion
Similarly, in scenarios with partial observability--where up to 20% of the input power injections are masked--the model still yields over 93% improvement in current RMSE for large-scale feeders such as IEEE123. In some cases, the PINN exhibits slightly higher voltage errors compared to the baseline, resulting in negative improvement percentages. This is expected, as the addition of the physics-based loss prioritizes current \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Noise Level (\%)**} & \multirow{2}{*}{**Network**} & \multicolumn{2}{c}{**Baseline**} & \multicolumn{2}{c}{**PINN**} & \multicolumn{2}{c}{**Improvement (\%)**} \\ \cline{3-8}  & & \multicolumn{2}{c}{**MAE**} & \multicolumn{2}{c}{**MAE**} & \multicolumn{2}{c}{**MAE**} \\ \cline{3-8}  & & **V** & **I** & **V** & **I** & **V** & **I** \\ \hline \multirow{8}{*}{**0**} & IEEE13 & 0.26 & 23.44 & 0.17 & 0.60 & 35.13 & 97.43 \\  & IEEE34 & 1.30 & 16.78 & 1.18 & 2.93 & 9.83 & 82.56 \\  & IEEE37 & 0.78 & 7.27 & 0.22 & 0.42 & 72.27 & 94.22 \\  & IEEE123 & 0.14 & 5.32 & 0.13 & 0.44 & 7.05 & 91.82 \\ \hline \multirow{8}{*}{**1**} & IEEE13 & 1.06 & 38.27 & 1.23 & 2.03 & \(-\)16.04 & 94.70 \\  & IEEE34 & 9.44 & 20.36 & 9.95 & 3.84 & \(-\)5.40 & 81.14 \\  & IEEE37 & 1.32 & 7.96 & 1.22 & 0.70 & 7.58 & 91.21 \\  & IEEE123 & 0.74 & 5.45 & 0.79 & 0.48 & \(-\)6.76 & 91.19 \\ \hline \multirow{8}{*}{**5**} & IEEE13 & 4.93 & 144.92 & 5.88 & 9.43 & \(-\)19.27 & 93.49 \\  & IEEE34 & 43.24 & 58.34 & 46.07 & 12.48 & \(-\)6.54 & 78.61 \\  & IEEE37 & 5.10 & 17.14 & 5.80 & 2.72 & \(-\)13.73 & 84.13 \\  & IEEE123 & 3.49 & 7.72 & 3.74 & 0.98 & \(-\)7.16 & 87.31 \\ \hline \multirow{8}{*}{**1**} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**RMSE**} \\ \cline{3-8}  & & **V** & **I** & **V** & **I** & **V** & **I** \\ \hline \multirow{8}{*}{**0**} & IEEE13 & 0.39 & 50.94 & 0.25 & 1.08 & 35.63 & 97.89 \\  & IEEE34 & 1.90 & 50.94 & 1.70 & 7.32 & 10.14 & 85.64 \\  & IEEE37 & 0.97 & 26.26 & 0.28 & 0.54 & 70.87 & 97.95 \\  & IEEE123 & 0.20 & 21.09 & 0.17 & 0.62 & 14.52 & 97.08 \\ \hline \multirow{8}{*}{**1**} & IEEE13 & 1.47 & 80.58 & 1.73 & 3.32 & \(-\)17.69 & 95.88 \\  & IEEE34 & 12.44 & 60.80 & 13.19 & 8.86 & \(-\)6.03 & 85.43 \\  & IEEE37 & 1.58 & 28.36 & 1.44 & 0.86 & 8.86 & 96.97 \\  & IEEE123 & 0.92 & 21.57 & 0.98 & 0.69 & \(-\)6.52 & 96.80 \\ \hline \multirow{8}{*}{**5**} & IEEE13 & 6.87 & 306.76 & 8.20 & 15.48 & \(-\)19.36 & 94.95 \\  & IEEE34 & 57.09 & 173.26 & 61.25 & 26.98 & \(-\)7.29 & 84.43 \\ \cline{1-1}  & IEEE37 & 6.08 & 59.65 & 6.87 & 3.35 & \(-\)12.99 & 94.38 \\ \cline{1-1}  & IEEE123 & 4.33 & 29.98 & 4.67 & 1.60 & \(-\)7.85 & 94.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance metrics under noisy data input. consistency, which may occasionally compromise voltage accuracy. Nevertheless, the observed trade-off remains small and is outweighed by the substantial improvements in current estimation, which is a critical aspect of physically meaningful state estimation. In addition to its predictive accuracy, the PINN shows faster convergence compared to the baseline model, rapidly minimizing both voltage and current loss within the first few training epochs. This accelerated learning suggests that the inclusion of physics-based constraints not only improves generalization but also guides the model more efficiently through the solution space. Moreover, the qualitative assessment on the IEEE13 system confirms the model's ability to replicate the complex voltage and current profiles with high precision, even at buses with near-zero injections, where angle and relative errors are prone to instability. These results collectively highlight the practical utility of the PINN framework for real-world distribution system state estimation.