As a typical application of deep learning, physics-informed neural network (PINN) has been successfully used to find numerical solutions of partial differential equations (PDEs) and has developed into one of the most effective methods to solve the forward and inverse problems of PDEs, but the limited accuracy of algorithm and the shortage of sufficient inherent physical laws of PDEs are two main weaknesses of PINN. Thus we first introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries or non-classical symmetries of PDEs are embedded into the loss function in PINN, to improve the accuracy of PINN for solving the forward and inverse problems of PDEs. Then motivated by the success of the above technique and the idea of the gradient-enhanced PINN (gPINN), we enforce the generalized conditional symmetry of PDEs to the loss function of PINN, i.e. the generalized conditional symmetry enhanced PINN (gsPINN), to improve the accuracy and reliability of solutions of PDEs. The SPINN and gsPINN methods incorporate the inherent physical laws of PDEs to PINN and exert high-efficiencies in solving the forward and inverse problems of PDEs. Then, we test the effectiveness of SPINN and gsPINN for the forward problem via two groups of ten independent numerical experiments using different numbers of collocation points and neurons per layer for the linear and nonlinear equations, and for the inverse problem by considering different layers and neurons as well as different numbers of training points with different levels of noise for the Burgers equation in potential form and a coupled system with two-component nonlinear diffusion equations. The numerical results show that SPINN and gsPINN perform better than PINN with fewer training points and simpler architecture of neural network, and the $L_2$ relative error of SPINN and gsPINN can reach $10^{-5}$ which is seldom in the literatures. Furthermore, we discuss the computational overhead of SPINN and gsPINN in terms of the relative computational cost to PINN and show that the training time of SPINN and gsPINN have no obvious increases, even less than PINN for certain cases. Moreover, by considering the Sawada-Kotera equation the SPINN method exhibits superiorities than the PINN method and the two-stage PINN method. The analysis of three methods, PINN, gPINN and gsPINN, on a non integrable PDE shows that gsPINN has significant superiorities in terms of accuracy, robustness and training time. The results further demonstrate that the inherent physical properties of PDEs can further improve the performances of PINN and thus is worthy of deep exploring.