## 1 Introduction  
Machine learning, especially deep learning, has received increasing attention due to its breakthrough performance in various cognitive applications. In recent years, the modeling of partial differential equations has been subjected to neural networks (NNs), which can be regarded as universal approximators of nonlinear functions. On this basis, [PERSON] et al. [1] proposed physics-informed neural networks (PINNs), which changed the conventional method of modeling physical systems. When dealing with physical systems, PINNs utilize available measurement data to obtain a parameterized partial differential Equation (PDE), which provides additional information for NN training [1, 2, 3, 4, 5]. PINNs can solve difficult problems in which local boundary conditions are missing, such as the thermal boundary conditions in heat transfer problems [6], or that rely on small displacements [7] to detect holes and defects in materials when performing inference tasks. Recently, many researchers have used PINNs in the field of fluid dynamics. In applied research, [PERSON] et al. [8] developed hidden fluid dynamics using PINNs to directly extract the velocity and pressure field of a fluid flow from images. [PERSON] et al. [9] used PINNs to precisely code the Navier-Stokes governing equations into deep NNs, overcoming the challenges faced by traditional numerical discrete solutions with incompressible laminar and turbulent flows, such as strong grid dependence, absence of boundary conditions,and the high computational cost of inverse problems. Moreover, using velocity-pressure and vorticity-velocity forms, Navier-Stokes flow networks were developed. [PERSON] et al. [10] applied PINNs in the modeling of three-dimensional metal processing and required only a small amount of labeled data to learn the dynamic changes in temperature and melt distribution. [PERSON] et al. [11] presented a Physics-Informed Neural Networks (PINNs) framework for identifying constitutive parameters in soft materials, achieving under 5% error in complex geometries and noisy conditions. Their approach leverages multi-modal synthetic datasets to ensure robust performance across various testing scenarios. For more research findings, please refer to reference [12]. [PERSON] et al. [13] used adaptive activation functions to approximate the solutions of linear and nonlinear partial differential equations. This method has high solution accuracy and fast convergence speed, especially in the early training stage. [PERSON] et al. [14] observed that the loss function composed of a weighted combination of multiple loss sub-functions plays an important role in training PINNs. Therefore, they proposed an adaptive loss function method based on maximum likelihood estimation, which automatically assigns the weight of the loss by updating the noise parameters at each iteration step. [PERSON] et al. [15] proposed the use of the minimax algorithm to adaptively adjust the weights of the loss function based on this foundation. [PERSON] et al. [16] proposed a gradient-enhanced PINN that embeds the gradient information of PDE residuals into the loss function when training PINN models and proved the effectiveness of this approach in forward and reverse PDE problems. To improve the modeling efficiency, [PERSON] et al. [17] proposed a conservative PINN over discrete domains based on the nonlinear conservation law. This enables the solution process of PINNs to be parallelized, thus enhancing the solution efficiency. [PERSON] et al. [18] then generalized this method to adapt to problems with various complex computational domains. Although PINNs have achieved remarkable effectiveness, they rely on specific boundary conditions, initial conditions, and special source terms during the training stage. This approach requires extensive expertise and incurs significant costs in the inference process.