## 1 Introduction  
Numerical modeling methods based on the finite element method (FEM) [1], finite difference method (FDM) [2], and finite volume method (FVM) [3] have made remarkable progress in solving physical problems. These methods are commonly used to describe partial differential equations (PDEs) for natural phenomena, relying on polynomials, segmented polynomials, and other elementary functions for modeling. Although traditional methods are considered to provide efficient and reliable solutions, their applicability is limited as the dimensionality of the problem increases, a phenomenon known as the \"curse of dimensionality\" (CoD) [4]. The CoD means that computational operations and resource requirements grow exponentially as the problem dimension increases, limiting the scope of application of these methods. In addition, grid-based methods suffer from discretization errors that fail to capture the required resolution of the modeling system when the grid size is not small enough, leading to inaccurate results. These challenges make it more difficult to solve PDEs in high-dimensional problems. When some physical problems with special properties are involved, such as chemical phase transition coupled porous media flow [5], lattice Boltzmann method coupled temperature field prediction [6], and microscopic fluid flow [7]. Complex chemical phase transitions and physical variability between the macroscopic and microscopic levels can lead to a dramatic increase in the computational cost of traditional numerical modeling methods. Faced with these physical problems with phase transitions and the obstacles in solving PDEs, researchers are actively looking for any potential solutions. The rapid development of machine learning in recent decades has introduced new opportunities in various fields, having a significant impact on science and engineering and providing an opportunity for a shift in the solution paradigm. Machine learning offers brand new possibilities for numerical modeling, prediction, and forward problem solving for the study of physics problems. Deep learning is a new and highly emphasized technique in the field of machine learning, one of the most important branches, and deep learning algorithms have been extremely effective in the fields of engineering and science, such as computer vision, speech recognition, natural language processing, assisted driving [8; 9; 10; 11], etc. They can process large-scale data based on the special structure of their networks, extract high-level features, and achieve excellent performance and accuracy in various tasks such as image classification, target detection, semantic segmentation, machine translation [12; 13; 14; 15], etc. Typical DNN models include multilayer perceptron (MLP) [16], convolutional neural network [17], and recurrent neural network (RNN) [18]. Physics-informed neural networks (PINNs), as new deep learning algorithms, can integrate physics law constraints with neural network algorithms for PDEs. They provide new methods for solving PDEs in addition to the traditional numerical modeling methods. PINNs predict the results of PDEs without constructing a detailed grid, thus overcoming the CoD problem experienced by traditional numerical modeling methods. Instead of using a grid for spatiotemporal stepping, PINNs collect points irregularly from the defined domain through different collection distributions [19]. The PDEs are embedded in PINNs in the form of a loss function, rather than an algebraic matrix as in traditional numerical modeling methods. The gradient optimizer is a residual minimizer in PINNs [20], which is quite different from the linear solver in traditional numerical modeling methods. The paradigm shift of PINNs provides a new approach to solving PDEs for practical physical applications where they are needed. [PERSON] et al. [21; 22] proposed and used PINNs to solve several cases of PDEs. They applied PINNs to solve NS equations to simulate the flow around the circular cylinder case. The SRT-LBM-PINN model was proposed by [PERSON] et al.