A Data-Free Partial Differential Equation Solver Based on Physics-Informed Neural Networks (PINN): FDM-PINN

Solving partial differential equations (PDEs) based on machine learning methods has attracted significant attention recently. One of the representatives is Physics-informed Neural Network (PINN), which is based on deep learning and incorporates physical laws into its loss function. However, PINN has limited performance when the training data vanishes, leading to a challenge in developing a data-free PDE solver in the framework of PINN. In this talk, we propose a novel data-free PDE solver (FDM-PINN), which combines the superiorities of both FDM and PINN. The total loss function in FDM-PINN only contains one term related to a set of mesh-based difference equations, which allows that the boundary condition (BC) and initial condition (IC) are imposed exactly instead of treating them as parts of the loss function in the original PINN. Furthermore, auto-differential (AD) technique in original PINN, which has lower accuracy verified by other researchers, has been replaced by finite difference (FD) technique when modeling the derivative terms. We demonstrate the advantages of FDM-PINN over other PINN-based methods quantitatively through examples of various PDE types.