Development of backward compatible physics-informed neural networks to reduce error accumulation based on a nested framework

Physical information neural network (PINN) provides an effective method for solving partial differential equations, and many variants have been derived, the most representative of which is backward compatible physical information neural network (BC-PINN). The core of BC-PINN is to use the prediction of the previous time period as the label data of the current time period, which leads to error accumulation in the process of backward compatibility. To solve this problem, a nested backward compatible physical information neural network (NBC-PINN) is proposed in this paper. NBC-PINN has an overlap region between the computation domain of the previous time period and the computation domain of the current time period, which is trained twice in total. Numerical experiments on four representative time-varying partial differential equations show that NBC-PINN can effectively reduce error accumulation, improve computational efficiency and accuracy, and improve the L2 relative error of the numerical solution with fewer residual allocation points. The development of NBC-PINN provides a theoretical basis for the scientific calculation of partial differential equations, and promotes the progress of PINN to a certain extent.