E-PINN: extended physics informed neural network for the forward and inverse problems of high-order nonlinear integro-differential equations

Abstract

Physics informed neural network (PINN) is a new deep learning paradigm, which embeds the physical information delineated by PDEs in the loss function and optimizes the weights in the neural network. Based on PINN, an extended PINN(E-PINN) is proposed, which is a mixture of the polynomial function approximation method and PINN's learning framework. A preprocess layer is added before the classical PINN, using Legendre polynomials as the polynomial basis function. Therefore, E-PINN not only has the excellent approximation ability of the polynomial basis function, but also inherits the learning framework of the neural network method. In numerical experiments, the proposed E-PINN algorithms have high accuracy in solving 1D, 2D high-order nonlinear Fredholm equations and equations system, including the forward and inverse problems.