## 2 Method  
### Physics-Informed Neural Networks  
PINNs or Physics-Informed Neural Networks are a specific kind of neural network trained to approximate the solution to any given law of physics defined by a partial differential equation (PDE) or a system of PDEs [3]. The most significant benefit of a PINN over other methods is that it is a mesh-free method. The classical PINN follows the collocation-based approach, implying that the neural network aims to approximate the strong form of the governing equation at a set of collocation points. As the collocation points can be distributed randomly within the domain, and no mesh is required, this approach belongs to the category of mesh-free methods [41]. The core of PINN implementation is to calculate partial derivatives, and this task can be completed through automatic differentiation algorithms in mainstream deep learning frameworks like Pytorch or Tensorflow [42]. Several open-source libraries such as DeepXDE [43], SimNet [44], and SciANN [45] have been developed, making PINNs easier to apply in practice. The architecture of a PINN can vary depending on the specific problem. While many PINNs still use the feedforward fully connected neural network (FCN) as part of their architecture, the FCN is the basic architecture used in deep learning algorithms [46]. A fully connected neural network with \(L\) layers is a function \(f_{\theta}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}\) described by Equation (1):  
\[f_{\theta}(x)=W^{[L-1]}\sigma\circ(\ldots\sigma\circ(W^{[0]}x+b^{[0]})+\ldots) +b^{[L-1]}, \tag{1}\]  
where \(\sigma\) is an entry-wise activation function, \(W^{[l]}\) and \(b^{[l]}\) are, respectively, the weight matrices and the bias corresponding to each layer \(l\), and \(\theta\) is the set of weights and biases (Equation (2)):  
\[\theta=(W^{[0]},\ldots,W^{[L-1]},b^{[0]},\ldots,b^{[L-1]}). \tag{2}\]  
The activation function is a crucial component of a neural network, and several favoured choices are available, including the sigmoid function, hyperbolic tangent function (tanh), and rectified linear unit (ReLU). It is worth mentioning that we have implemented the hyperbolic tangent function as part of the neural network for PINN. The hyperbolic tangent activation function is defined as Equation (3):  
\[\tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}. \tag{3}\]  
The smoothness and overall S-shape of this function are similar to that of the sigmoid function. However, unlike the sigmoid function, the range of the outputs is centered at 0 and falls between (\(-1\), \(1\)). This makes the tanh activation function more appropriate for deep neural networks as it avoids creating a bias towards positive outputs [41]. ReLUis more commonly used as an activation function in neural networks. However, it is unsuitable for PINNs due to its second derivative being zero. PINNs harness the PDEs to guide and constrain the training process of neural networks. PINNs incorporate the PDEs' residuals, initial conditions, and boundary conditions into their loss function. The neural networks are tasked with fitting the observed data while simultaneously minimizing the PDE residuals. The neural network is thus trained to approximate the solution of the PDE by minimizing this loss function. This approach reduces the need for additional observational data, making it a powerful and efficient technique to solve PDEs. However, the loss function is highly dimensional and non-convex with competing loss terms. It is essential to weigh these loss terms; otherwise, the optimizer might train only one term and create a bias. Later in the 1D wave section, we discuss the temporal loss weighting technique that we used to assign the highest weight to temporal loss terms in the beginning. We consider a scalar function \(u(x,t)\) on the domain \(\Lambda\times[0,\infty)\); with the boundary \(\partial\Lambda\), where \(\Lambda\subset\mathbb{R}^{d}\). \(u(x,t)\) satisfies the following PDE (Equation (4)):  
\[