## 2 Materials and Methods
### Optimization Strategies  
In order to obtain more accurate results for the HWPINN proposed in this paper, we preprocessed the data and adopted multiple optimization strategies when training the network.  
In the application of PINN to solve PDE problems, utilizing min-max normalization holds significant importance [35]. This normalization method ensures that input and output data fall within similar numerical ranges, enhancing the numerical stability of the neural network. When the numerical disparity between input and output data is significant, unnormalized data might lead to numerical instability during the training process, affecting the model's convergence and accuracy. Min-max normalization helps to speed up the training process. After normalization, the optimization algorithms for neural networks converge more easily. Methods such as gradient descent can find appropriate weights and biases faster, thus improving training efficiency. Using the normalization scheme not only improves the numerical stability of the model but also speeds up the training process. This approach provides an effective numerical treatment for solving complex PDE problems in scientific research.  
The equation for min-max normalization is as follows:  
\[Y_{normalized}=\frac{Y-min(Y)}{max(Y)-min(Y)} \tag{10}\]  
where \(Y\) represents the original data; \(Y_{normalized}\) is the normalized data; \(max(Y)\) and \(min(Y)\) are the minimum and maximum values of the original data, respectively. Subtracting the minimum value from the original data and then dividing by the difference between the maximum and minimum values yields the normalized value. This process scales the data range to [0, 1].  
For initializing the neural network weights and biases, we used Xavier initialization. The hyperbolic tangent function (Tanh) was chosen for the activation function. Xavier initialization, also known as Glorot initialization [36], is a method specifically designed for initializing neural network weights, especially when using S-shaped activation functions like Tanh. The advantage of Xavier initialization lies in its ability to mitigate the issues of vanishing and exploding gradients associated with these activation functions. When employing Tanh as the activation function, its output range is limited to between \(-1\) and \(1\). Initializing weights too large results in Tanh saturating to \(1\) or \(-1\), causing gradients to approach zero and leading to the vanishing gradient problem. Initializing weights too small makes the activations close to zero, again resulting in very small gradients and the same vanishing gradient problem.  
Xavier initialization intelligently adjusts the initial weight range based on the number of input and output units in the preceding layer. By initializing weights from a uniform or Gaussian distribution with a standard deviation related to the input and output unit numbers, Xavier initialization ensures that activations are in a suitable range, preventing vanishing and exploding gradients. Therefore, Xavier initialization, when applied with Tanh or similar activation functions, accelerates network convergence, enhances training efficiency, and effectively addresses the challenges associated with gradient-related problems, making neural networks more trainable.  
In our cases, training is achieved by minimizing the error using a variant of the stochastic gradient descent (SGD) method of adaptive moment estimation weight-decay (AdamW) [37] and Broyden-Fletcher-Goldfarb-Shanno (BFGS) [38]. Training begins by applying the AdamW optimizer to the constructed model. This training phase is stopped according to a predetermined number of training calendar elements. Subsequently, another training phase is performed using the L-BFGS-B optimization method. This optimization sequence set up compensates for the limited amount of training data by reducing computational loss and is expected to achieve faster convergence [20].