## 2 Related Work
### Physics-Informed Neural Networks  
The physics-informed neural networks technique is introduced for solving problems related to partial differential equations. Through automatic differentiation, the PINNs embed PDEs into a neural network's loss function, enabling seamless integration of both the measurements and PDEs. As opposed to fitting a neural network to a set of state-value pairs, PINNs consider the underlying PDE or physics of the problem. [PERSON] revealed a promising way to leverage prior knowledge about a solution in earlier research about creating physics-informed machine learning [34]. For a variety of physical problems, [PERSON] et al. used Gaussian process regression to infer solutions and estimate uncertainty [35]. [PERSON] et al. conducted the first study that demonstrated the implementation of the PINNs approach for solving nonlinear PDEs such as Schrodinger, Burgers, and Allen-Cahn equations [7]. Their PINNs approach estimated the solutions to governing mathematical forward and inverse problems, where model parameters were fine-tuned using the data. In the following years, PINNs attracted considerable attention from scientific computing researchers because of their high flexibility and expressive ability. To solve spatiotemporal fractional equations, [PERSON] et al., 2019 extended PINNs to fractional PINNs (fPINNs) by using both automatic differentiation for integer-order operators as well as numerical discretization for fractional operators to construct residuals in the loss function. Since the standard chain rule in integer calculus is not valid in fractional calculus, their approach solved the difficulty of using automatic differentiation with fractional operators [36]. In 2020, [PERSON] et al. extended PINNs to solve time-dependent stochastic partial differential equations (SPDEs), such as the advection equation, stochastic Burger's equation, and nonlinear reaction-diffusion equation. To determine the loss function, they incorporated dynamically orthogonal (DO) constraints and borthogonal constraints (BO) [37]. [PERSON] et al., 2021 presented an hp-VPINN variant using least-squares residuals based on sub-domain Petrov-Galerkin methods, where trial space consists of neural network spaces and test space consists of localized non-overlapping high-order polynomials [38]. In order to solve the ill-posed inverse problems encountered in the evaluation of non-destructive testing, [PERSON] et al., 2022 employed PINNs to identify the material through its spatial variation in compliance coefficients. In this study, they analyzed the microstructure of polycrystalline nickel by using realistic ultrasonic surface acoustic wavefield data. They integrated in-plane and out-of-plane elastic wave equations and adaptive activation functions to physically inform the neural network and accelerate its convergence, respectively [39]. Leveraging the PINN's ability to integrate experimental data and the governing physical laws, [PERSON] et al., 2022 extended its application to strongly nonlinear and weakly dispersive surface water waves problems represented by asymptotic Serre-Green-Naghdi system. They also discussed how to determine the optimal gauge locations for the best predictive accuracy [40]. To accurately solve the problems where the gradient and solution changes swiftly, [PERSON] and [PERSON] presented a novel self-adaptive PINN (SA-PINNs) based on the conceptual soft self-attention mechanism used in Computer Vision where the model autonomously identifies the most important inputs during training [41]. [PERSON] et al., 2020 extended PINNs to one-dimensional and two-dimensional Euler equations using initial/boundary conditions to solve forward and inverse problems for high-speed aerodynamic flows [42]. They solved the situations with smooth solutions and discontinuous solutions and their results showed that the clustered training points, which are associated with more data surrounding discontinuities, led to more accurate results compared to random or uniform training points.