## 2 Materials and Methods
### Training  
Training the PINN for distribution system state estimation involves a structured pipeline consisting of data preprocessing, optimization, and evaluation. The full dataset is first randomly partitioned into training and test subsets, typically using an 80/20 split. Preprocessing steps, including normalization, are applied only to the training set to avoid data leakage and ensure fair evaluation of generalization.  
All complex-valued inputs and targets are normalized to stabilize training and prevent numerical instabilities. This is especially important to mitigate exploding values, which can occur during the initial epochs due to inaccurate current estimates derived from poorly initialized voltage states. A robust normalization technique is applied separately to the real and imaginary parts of the data. For a given complex tensor \(x=x_{re}\,+\,jx_{im}\), the normalized components are computed as:  
\[x_{re,norm}=\frac{x_{re}-\mu_{re}}{\sigma_{re}},\,\,\,x_{in,norm}=\frac{x_{im}- \mu_{im}}{\sigma_{im}} \tag{7}\]  
where \(\mu_{re},\,\,\mu_{im}\) are the means, and \(\sigma_{re},\,\,\sigma_{im}\) are the standard deviations of the real and imaginary parts, respectively, computed across the batch.  
Although this normalization is based on the standard z-score approach, it incorporates several important modifications to enhance robustness for complex-valued power system data:  
* The standard deviations are clamped from below using a small multiple of the corresponding mean absolute value to avoid division by near-zero values that could lead to numerical instability.
* The resulting normalized values are clipped to a predefined range (e.g., \([-10,10]\)) to limit the influence of outliers and prevent gradient explosions.
* The process is applied independently to each component of the complex values, which is not typical in standard real-valued z-score normalization.  
These enhancements make the method more reliable for use in physics-informed training where both scale sensitivity and complex-valued operations are involved.  
The network parameters are optimized using the Adam algorithm [26], a first-order gradient-based optimizer. The learning rate is adjusted dynamically using a \"reduce on plateau\" scheduler, which reduces the learning rate when the validation loss ceases to improve. Specifically, if the loss does not decrease for a defined number of epochs (patience), the learning rate \(\eta\) is updated according to:  
\[\eta_{new}\;=\;\eta_{old}\;\times\;factor \tag{8}\]  
where \(factor\) is a multiplicative constant (e.g., 0.5). This adaptive strategy allows the model to make large updates when learning is active and finer updates as it approaches convergence. To further enhance stability, gradient norms are clipped to a maximum value of 1.0 during backpropagation.  
Key hyperparameters--including the learning rate, batch size, number of training epochs, and the regularization factor \(\lambda\) between data and physics losses--are manually tuned to achieve a good balance between data fitting and physical consistency. During training, the model state with the lowest total validation loss is checkpointed and used for final evaluation.