Gradient auxiliary physics-informed neural network for nonlinear biharmonic equation

Abstract

Physics-informed neural network (PINN) has gained wide attention for solving forward and inverse problems of partial differential equation (PDE) from both data-driven and model-driven perspectives. In a short time, various machine learning methods based on PINN have been developed for solving a broad range of PDE. To improve the training accuracy of PINN, the popular approach is to introduce penalty parameters to correct the imbalance among different parts of loss function during model training. However, this approach generally fails to eliminate the model error generated by the boundary condition. To eliminate this issue, we propose a gradient auxiliary physics-informed neural network (GA-PINN) for nonlinear biharmonic equation. The key idea of GA-PINN is to split original biharmonic equation subjected to clamped or simply supported boundary conditions into third order or second order differential equations by introducing gradient auxiliary functions. As a consequence, we can rewrite clamped or simply supported boundary conditions as Dirichlet boundary conditions, then conveniently construct the neural network composite functions to satisfy those Dirichlet boundary conditions. We introduce the dynamic weight method to automatically balance the contributions of different loss terms during training. The capabilities of the proposed GA-PINN by solving several nonlinear biharmonic problems are demonstrated.