VT-PINN:Variable transformation improves physics-informed neural networks for approximating partial differential equations

A novel approach was proposed, that is, physics-informed neural networks combined with variable transformations(VT-PINN). This work demonstrates a technique that can greatly improve the approximation of physics- informed neural networks(PINN). We construct alternative equations by introducing variable transformations into partial differential equations(PDEs), and the solutions of alternative equations have smoother geometric properties. The alternative equations and the original equations are interrelated by variable transformations. The PINN loss function of the associated alternative equations is rederived. Several numerical examples have been used to verify the effectiveness of our proposed method, including the Poisson equation (RMSE: 0.17%), the wave equation (RMSE: 0.034%, example 3), and the advection-diffusion equation(0.25%, kappa: L-2 relative error), etc. Compared with the standard PINN(2.8%, 3.6%, 1.9%), the efficiency and accuracy of VT-PINN are demonstrated by numerical examples. The approximation accuracy of VT-PINN has generally increased by an order of magnitude. The numerical results reveal that the proposed method greatly improves the approximation of PINN. At the same time, VT-PINN does not add additional time cost. In addition, the relevant numerical experiments show that this method greatly improves the effect of PINN in estimating the unknown parameters of partial differential equations, that is, it has good performance for solving inverse problems.