Solving 2nd Order Volterra Integro-Differential Equations by Using IDRLnet Library and Physics-informed neural networks

Abstract

Physics-informed neural networks (PINNs) are a novel deep learning model that excels at solving both inverse and forward non-linear partial differential equation (PDE) problems. This paper presents a new algorithm implemented using Python and the IDRLnet library. The new algorithm of the open-source deep learning package is designed to solve PDEs using PINNs and is mainly used for physics-informed deep learning. This paper uses PINN to solve the 2nd-order Volterra integro-differential equations (VIDE) for the first time, in which a series of iterative solutions are established based on the Gauss-Legendre quadratic method after converting the 2nd-order VIDE into a PDE. In addition, three different examples are presented and solved using the new algorithm to highlight the accuracy and efficiency of the proposed method. It is found that the L2 relative error is between 0.1e-3 and 0.1e-4, but the absolute error is between 0.1e-3 and 0.1e-5, which is a good result., and the “training” time for all cases is 414.08, 724.2, 1339.31, and 2260.8 seconds for all the cases taken.