## 1 Introduction  
Over the past five decades, there have been substantial advances in developing computational fluid dynamics (CFD) techniques to numerically solve Navier-Stokes (N-S) equations for fluid mechanics in real-world applications [1, 2]. In conventional CFD methods, e.g., the finite-difference (FD) and finite-volume (FV) methods, sophisticated computational grids are required for domain discretization, especially when complex geometries are involved. The generation of high-quality grids makes CFD computations cumbersome and is always demanding for CFD engineers. On the other hand, due to the lack of high-performance generic algorithm libraries, developing flow solvers based on conventional CFD methods is always challenging and time-consuming for CFD developers. Machine learning, as a main branch of artificial intelligence (AI) and computer science, can be used to solve partial differential equations (PDEs) including N-S equations with avoidance of the above problems of CFD methods [3, 4], which creates new horizons and possibly even transformations for the current lines of fluid mechanics research. Depending on the manners by which the ML approaches are integrated with CFD solvers, ML approaches for fluid mechanics can be roughly categorized into three classes, i.e., data-fit models, projection-based models and physics-informed models. A data-fit model is usually based on the artificial neural network (ANN) [5, 6], radial basis function (RBF) [7] and support vector regression (SVR) [8, 9], and it treats a CFD solver as a black box and is essentially a fast, inexpensive but approximate model that extracts mechanismsunderlying the complex system from available data in a supervised manner. Data-fit models have been widely used as a substitute for expensive CFD simulations in design optimization or uncertainty quantification of fluid systems [10]. However, constructing such types of models requires additional overhead of data simulation, which could become expensive and even unaffordable for high-dimensional problems. As for projection-based models, an orthogonal linear transformation is first defined from physical coordinates into a modal basis in an unsupervised manner by means of principle orthogonal decomposition (POD) or other methods, and then the PDE operator is projected onto the subspace spanned by the reduced modal basis [11; 12]. Although the degree of freedom of solution can be significantly reduced, projected-based models need modifications of the original CFD codes, and the issues of stability and robustness are still not well addressed. Different from the above two types of models, physics-informed models embed a priori physics into the learning architecture, which not only remains code-nonintrusive but also reduces the dependency on data availability [13]. Owing to the rapid advances in deep learning, the current mainstream of physic-informed models is the physics-informed neural network (PINN) [14; 15], whereby the PDEs are directly incorporated in the loss function of the neural network by penalizing deviations from the target values. So far, PINN models have been successfully applied in solving incompressible and compressible flows [16], and there are basically two application scenarios, i.e., inverse and forward problems. In inverse problems, parameters of the equation (including definite solution conditions) and flow field are partly known while the remaining ones need to be solved, which is thus considered to be a semi-supervised problem. For example, [PERSON] et al. [17] proposed a \"hidden fluid mechanics\" framework based on PINN and successfully identified unknown equation parameters and inversed the velocity and pressure of the flow around a cylinder with the concentration field provided. [PERSON] et al. [17] solved the density, velocity and pressure of the Sod and Lax problems by PINN with the density gradient and pressure of some samples given in advance. As can be seen, the solution accuracy of the inverse problem is still dependent on prior information.