## 3 Physics-Informed Neural Networks (PINNs)
### Foundations of PINNs  
Physics-informed neural networks (PINNs) significantly advance computational physics and machine learning, combining deep learning's powerful approximation capabilities with the rigorous constraints of physical laws [11]. Their essential feature lies in embedding governing physical equations, often in the form of differential equations, directly into the training process via the loss function. Physics-informed neural networks (PINN) are models designed to observe physical laws specified by nonlinear partial dif  
Figure 2: SAG’s features involved in an overload. * features identified but are unavailable for analysis in this study. ferential equations (PDEs). In this context, the function \(u(z)\) is computationally estimated by a neural net (NN), parameterized by a set of parameters \(\mathbf{\theta}\in\Theta\), where \(\Theta\subseteq\mathbb{R}^{n}\). The variable \(z\in\mathcal{Z}\) represents the inputs to the model, which could include spatial coordinates, temporal variables, or other relevant physical quantities, with \(\mathcal{Z}\subseteq\mathbb{R}^{d}\). \[\hat{u}_{\mathbf{\theta}}(z)\approx u(z), \tag{1}\]  
where \(\hat{u}_{\mathbf{\theta}}\) refers to the NN approximation with parameters \(\mathbf{\theta}\in\Theta\)[18]. In this scenario, the neural network must learn to approximate the differential equations by identifying the parameters \(\mathbf{\theta}\) that define the neural network. This is done by minimizing a loss function that includes the differential equation \(\mathcal{L}_{F}\), the boundary conditions \(\mathcal{L}_{B}\), and possibly some known data \(\mathcal{L}_{\text{data}}\):  
\[\mathbf{\theta}^{*}=\operatorname*{arg\,min}_{\mathbf{\theta}\in\Theta}(\lambda_{d} \mathcal{L}_{\text{data}}(\mathbf{\theta})+\lambda_{F}\mathcal{L}_{F}(\mathbf{\theta} )+\lambda_{B}\mathcal{L}_{B}(\mathbf{\theta})), \tag{2}\]  
where \(\lambda_{d}\), \(\lambda_{F}\), and \(\lambda_{B}\) are the respective weights of each loss function, constrained such that \(\lambda_{d},\lambda_{F},\lambda_{B}\geq 0\) and \(\lambda_{d}+\lambda_{F}+\lambda_{B}=1\). These constraints ensure that the weights form a convex combination. PINNs can be considered unsupervised learning approaches when trained exclusively with physical equations and boundary conditions for forward problems. However, for inverse problems or when inferring some physical properties from potentially noisy data, PINNs can be considered supervised learning methods. In the context of a regression problem, we define the target values as \(\mathbf{y}\) and the input features as \(\mathbf{X}\). Here, \(\mathbf{X}=\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N}\}\) represents the independent variables or predictors, where each \(\mathbf{x}_{1}\in\mathcal{X}\subseteq\mathbb{R}^{d}\) is a \(d\)-dimensional vector of features relevant to the problem at hand. The target \(\mathbf{y}=\{y_{1},y_{2},\ldots,y_{N}\}\) represents the dependent variable or response, with each \(y_{i}\in\mathcal{Y}\subseteq\mathbb{R}\), which we aim to predict based on the inputs. In a physics-informed neural network (PINN), \(\mathbf{X}\) and \(\mathbf{y}\) are used to train the network while incorporating physical laws and constraints through additional loss terms, ensuring that the model's predictions are not only data-driven but also consistent with underlying physical principles. Likewise, the mean squared error (MSE) is a commonly used data loss function in regression problems.