## 3 Results and Discussion
Even as the proportion of missing inputs increases, the PINN maintains relatively low MAE and RMSE values, suggesting that its ability to incorporate physical constraints enables more reliable extrapolation under sparse data conditions. For instance, in the IEEE123 network with 20% input dropout, the PINN achieves a 94.05% reduction in current RMSE compared to the baseline. In contrast, \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Bus**} & \multicolumn{2}{c}{**Voltage Magnitude**} & \multicolumn{2}{c}{**Voltage Angle (\({}^{\circ}\))**} & \multicolumn{2}{c}{**Current Magnitude**} & \multicolumn{2}{c}{**Current Angle (\({}^{\circ}\))**} \\ \cline{2-9}  & **Actual** & **Pred.** & **Actual** & **Pred.** & **Actual** & **Pred.** & **Actual** & **Pred.** \\ \hline
**611.3** & 2364 & 2365 & 115.90 & 115.90 & 62.92 & 62.19 & \(-\)85.62 & \(-\)85.59 \\
**632.1** & 2472 & 2472 & \(-\)2.31 & \(-\)2.30 & 0.00 & 1.22 & 90.00 & 20.20 \\
**632.2** & 2506 & 2506 & \(-\)121.53 & \(-\)121.53 & 0.00 & 2.08 & \(-\)153.43 & \(-\)98.98 \\
**632.3** & 2455 & 2455 & 117.82 & 117.82 & 0.00 & 1.13 & \(-\)101.31 & 82.33 \\
**670.1** & 2453 & 2453 & \(-\)3.16 & \(-\)3.16 & 8.91 & 8.93 & 144.83 & 148.14 \\
**670.2** & 2513 & 2514 & \(-\)121.69 & -121.69 & 36.60 & 37.72 & 24.81 & 26.42 \\
**670.3** & 2424 & 2424 & 117.19 & 117.19 & 52.56 & 52.94 & \(-\)93.79 & \(-\)93.99 \\
**671.1** & 2414 & 2414 & \(-\)4.95 & \(-\)4.94 & 150.08 & 160.71 & 151.18 & 143.44 \\
**671.2** & 2537 & 2537 & \(-\)121.99 & \(-\)121.98 & 150.08 & 126.94 & 31.21 & 39.13 \\
**671.3** & 2370 & 2371 & 116.13 & 116.13 & 150.14 & 203.73 & \(-\)88.81 & \(-\)87.49 \\
**680.1** & 2414 & 2415 & \(-\)4.95 & \(-\)4.94 & 0.00 & 0.50 & 104.04 & \(-\)80.56 \\
**680.2** & 2537 & 2537 & \(-\)121.99 & \(-\)121.98 & 0.00 & 0.08 & \(-\)36.87 & \(-\)67.06 \\
**680.3** & 2370 & 2371 & 116.13 & 116.14 & 0.00 & 0.67 & 0.00 & 89.30 \\
**645.3** & 2450 & 2450 & 117.86 & 117.86 & 0.00 & 0.91 & 0.00 & \(-\)130.72 \\
**645.2** & 2485 & 2485 & \(-\)121.70 & \(-\)121.70 & 75.67 & 76.26 & 22.66 & 23.06 \\
**646.3** & 2445 & 2445 & 117.90 & 117.90 & 64.04 & 63.29 & \(-\)122.26 & \(-\)122.20 \\
**646.2** & 2481 & 2481 & \(-\)121.78 & \(-\)121.77 & 64.04 & 63.31 & 57.74 & 57.81 \\
**692.1** & 2413 & 2414 & \(-\)4.95 & \(-\)4.95 & 46.16 & 59.28 & 105.75 & 160.54 \\
**692.2** & 2537 & 2537 & \(-\)121.99 & \(-\)121.99 & 0.00 & 28.87 & 90.00 & \(-\)3.53 \\
**692.3** & 2370 & 2370 & 116.13 & 116.14 & 46.16 & 11.77 & \(-\)74.25 & 44.47 \\
**684.1** & 2408 & 2409 & \(-\)4.97 & \(-\)4.96 & 0.00 & 0.37 & \(-\)135.00 & \(-\)16.60 \\
**684.3** & 2367 & 2368 & 116.04 & 116.04 & 0.00 & 0.70 & 180.00 & 149.88 \\
**652.1** & 2394 & 2395 & \(-\)4.91 & \(-\)4.90 & 66.88 & 67.02 & 143.55 & 143.79 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._the baseline model exhibits sharp increases in both voltage and current errors, especially in larger networks. Although some performance deterioration is observed in certain cases, the PINN consistently yields better overall state estimates than the baseline. These results demonstrate the superior robustness and generalization capability of the PINN under reduced observability--a critical feature for real-world deployment where sensor coverage is often limited. The presented results confirm the effectiveness of the PINN approach across a range of distribution network configurations and operational scenarios. Compared to the baseline model, the PINN achieves consistently significantly lower voltage and current estimation errors across all test feeders. In noise-free conditions, the proposed model reduces current MAE by up to 97.4% and voltage RMSE by up to 63.7%, demonstrating strong fidelity to the underlying power system behavior. Under noisy conditions (e.g., 5% perturbation), the PINN maintains robust performance, achieving over 95% improvement in current RMSE for the IEEE 37-bus system.