Physics-informed neural networks combined with polynomial interpolation to solve nonlinear partial differential equations

In this paper, we utilise the physics-informed neural networks (PINN) combined with interpolation polynomials to solve nonlinear partial differential equations and for simplicity, the resulted neural network is termed as polynomial interpolation physics-informed neural networks (PI-PINN). Classically, the neural network is expressed as a power series by optimization of the coefficients to get an approximate solution of the partial differential equations (PDEs). Due to well-defined approximate properties of orthogonal polynomials, orthogonal polynomials are used to construct the neural network. Compared with PINN, PI-PINN clearly has a simple structure and is easy to be understood. We carry out some numerical experiments, including parabolic partial differential equations, hyperbolic partial differential equations and an application in fluid mechanics. By these investigations, it is further demonstrated that the PI-PINN structure is effective in solving nonlinear partial differential equations. Further, investigations on reverse problems are taken and accurate results are obtained.