## 1 Introduction
On the contrary, the forward problem could be an unsupervised one where only governing equations and definite solution conditions are known and all the flow quantities are to be solved. For example, by using the PINN model, [PERSON] et al. [18] accurately predicted the velocity field of Couette flow, stenosis flow and aneurysm flow. [PERSON] et al. [19] adopted a similar method to predict the turbulent channel flow without any physical information given in advance. Although solving the forward problem by PINN does not necessarily rely on any prior information, constraint enforcement should be paid particular attention to guarantee the model prediction accuracy at the specific definite condition. This is because PINN modeling is essentially an optimization problem, and the accuracy of the model is closely associated with the way the constraints are enforced. The constraint enforcement method of PINN can be generally divided into two categories, i.e., the \"soft\" constraint method and the \"hard\" constraint method. In the soft constraint method, the original optimization problem is transformed into an unconstrained one by adding penalty terms of initial conditions (ICs) and boundary conditions (BCs). [PERSON] et al. [20] constructed PINNs with soft constraints based on an adaptive activation function to solve PDEs including the nonlinear Klein-Gordon equation, the nonlinear Burgers equation and the Helmholtz equation. [PERSON] et al. [21] presented a learning rate annealing algorithm to solve the numerical stiffness problem in solving PDEs with PINN. [PERSON] and [PERSON] [22] integrated the space-time domain decomposition method into the soft-constraint-based PINN framework to parallelize the computations. Despite the inspiring progress in solving PDEs, the soft constraint method for PINN still has two major disadvantages, as [PERSON] et al. [19] pointed out. First, it can hardly guarantee that the network output fully conforms to ICs and BCs, thus giving rise to the loss of prediction accuracy of PINN; second, the involved penalty coefficients are often difficult to determine, either relying on experience or requiring trial and error, which is also detrimental to the modeling efficiency of PINN. The hard constraint method can circumvent the above problems by introducing a particular solution and a smooth function that enables the PINN prediction results to satisfy ICs and BCs in a mandatory manner. The particular solution can be solely determined according to ICs and BCs, while the smooth function connects the boundary to the internal domain, which is crucial to the PINN prediction accuracy. In the work by [PERSON] et al. [19], the PINN prediction results with hard constraint were in good agreement with FV-based CFD results, while those with soft constraint deviated significantly from FV-based CFD results. However, no principles were given in their work for the formulation of smooth functions in the hard constraints. To the best knowledge of the authors, no further study has been conducted on how to formulate hard constraints for PINN, which to some extent hinders the application of PINN in realistic engineering fields. The purpose of this study is twofold. The first is to investigate the effect of parameters in the smooth function on the PINN prediction accuracy, where several flows including Couette flow, plate shear flow and stenotic/aneurysmal flow at different Reynolds numbers are examined. The second is to provide guidelines for the formulation of hard constraints for PINN modeling. The rest of this paper is organized as follows. Section 2 describes the basis of the PINN model and constraint methods. In Section 3, some numerical tests are carried out, and formulation principles of hard constraints are summarized from the corresponding results. PINN solutions of Couette flow, plate shear flow and stenotic/aneurysmal flow under different values of degree and scaling factor are shown in Section 3. Conclusions are finally drawn in Section 4.