A data-free partial differential equation (PDE) solver in the framework of physics-informed neural networks (PINN)

Physics-informed neural networks (PINN) have been proposed to solve partial differential equation (PDE) given laws of physics and sparse training data. With PINN, the training data, laws of physics, boundary condition (BC) and initial condition (IC) are treated as parts of the total loss function, which is optimized. The derivative terms in PDEs are modeled by the auto-differential (AD) technique for easy implementation, and the BCs and ICs are imposed in a soft manner by optimization. Provided sparse solutions, the neural network is trained. However, the derivative term modeled by AD technique has lower accuracy than modeled by finite difference (FD) scheme, which will be demonstrated in this talk. The BCs and ICs could not be satisfied exactly due to the nature of the optimization of the total loss function, which affects the accuracy of the solution. In many cases, the training data are hard to obtain. In this talk, combining the advantages of finite difference method (FDM) and PINN, a new data-free PDE solver called PINN-FDM is introduced. In FDM-PINN, the derivative terms are modeled by FD instead of AD, and BCs and ICs are imposed exactly. FDM-PINN could solve the complicated PDE without training data, and the accuracy of the solution could be improved significantly.