## 3 Results and Discussion  
To evaluate the performance of the proposed PINN-based state estimation algorithm, two scenarios are considered:  
* Baseline--a model trained without incorporating physical constraints (no current loss);
* PINN--a model trained with the current loss term to enforce physical consistency. To ensure consistency across experiments, the same set of hyperparameters was used for training all models. Table 1 summarizes the key training configurations applied in this work. The models were trained using the Adam optimizer, which offers adaptive learning rate adjustment and typically ensures faster convergence for deep networks. A modest architecture was employed, consisting of two hidden layers with 256 units each and tanh activation functions, balancing model expressiveness and overfitting risk. The total dataset comprised 10,000 time-indexed samples, split into training and testing subsets using an 80-20 ratio. The learning rate was initially set to 0.01 and adjusted dynamically during training through a scheduler with a patience of 50 epochs and a reduction factor of 0.5 when no improvement was observed. Training lasted up to 2000 epochs, although early stopping was effectively managed through the scheduler. Finally, the weights of the composite loss terms--associated with voltage and current equations--were set empirically, with \(\lambda\) values on the order of 0.001 to balance physical consistency and prediction accuracy. \begin{table}
\begin{tabular}{c c} \hline \hline
**Hyperparameter** & **Value** \\ \hline Optimizer & Adam \\ Activation Function & Tanh \\ Hidden Layers & 2 \\ Hidden Units Per Layer & 256 \\ Timestamps & 10,000 \\  & Learning Rate (Initial) & 0.01 \\ Scheduler & Patience & 50 epochs \\  & Factor & 0.5 \\  & Epochs & 2000 \\  & Train–Test Split & 0.8–0.2 \\  & \(\lambda\) Loss Weights & \(\sim\)0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of training hyperparameters used for all models. Initially, both models are evaluated under noise-free conditions. They are trained using identical hyperparameters, and the resulting performance on the test set is presented in Table 2. As shown, the PINN significantly outperforms the baseline by leveraging the physics-based constraint, leading to substantially lower current estimation errors while maintaining or improving voltage accuracy. This demonstrates that, under the same training conditions, the PINN is capable of learning the underlying physical behavior of the network and generating voltage estimates that yield highly accurate current predictions. In addition to the test set performance, the training behavior of the models provides further insights. Figure 2 illustrates the training metrics over the first 100 epochs for each network. It is evident that the PINN exhibits significantly faster convergence compared to the baseline model. In all cases, the PINN rapidly reduces both voltage and current errors within the first few epochs, indicating more efficient learning of the system's underlying physics. In contrast, the baseline model demonstrates a slower, more gradual reduction in error, requiring many more epochs to approach comparable levels of accuracy. This accelerated convergence of the PINN underlines the benefit of embedding domain knowledge directly into the learning process. By enforcing physical constraints during training, the model not only learns faster but also generalizes more robustly. Moreover, this advantage becomes increasingly pronounced as the network size scales up. Observing the progression from IEEE13 to IEEE123, the gap between the baseline and PINN performance--particularly in terms of current loss (red curves)--widens substantially. The expanding area between the solid and dashed red lines signifies the growing benefit of physics integration in larger, more complex systems, where conventional data-driven approaches struggle to efficiently capture the nonlinear interactions inherent in distribution networks.