Generalized conditional symmetry enhanced physics-informed neural network and application to the forward and inverse problems of nonlinear diffusion equations

Abstract

Physics-informed neural network (PINN) has been evolved into one of the most effective methods to solve the forward and inverse problems of partial differential equations (PDEs), but the limited accuracy of algorithm and the shortage of inserted inherent physical laws of PDEs are two main weaknesses of PINN. In this paper, we enforce the generalized conditional symmetry of PDEs into PINN, i.e. the generalized conditional symmetry enhanced PINN (gsPINN), to improve the accuracy of PINN. The gsPINN incorporates the inherent physical laws of PDEs and thus exerts high-efficiencies in solving the forward and inverse problems of PDEs, where the numerical experiments for the nonlinear diffusion equations with convection and source terms show that gsPINN performs better than PINN with fewer training points and simpler architecture. We discuss the computational overhead of gsPINN in terms of the relative computational cost to PINN and show that the training time of gsPINN has no obvious increases, even less than PINN for certain cases. Moreover, we perform three methods, PINN, gradient-enhanced PINN (gPINN) and the proposed gsPINN, for analyzing a non-integrable PDE and show that gsPINN has remarkable superiorities than the other two methods in terms of accuracy, robustness and training time.