System Stabilization of PDEs using Physics-Informed Neural Networks (PINNs)

As a popular neural network model for solving forward and inverse problems in partial differential equation (PDE) control, Physics-Informed Neural Networks (PINNs) have received extensive attention in recent years and have made break-throughs in various fields. With the application of PINNs being extended to optimal control problems constrained by PDEs, where the control PDE is fully known, the problem objective is to find a control variable to minimize the desired cost objective. In this paper, with the idea of using PINNs to solve optimal control problems, we investigated effective methods to find boundary control and distributed control which can drive the PDE state towards unstable zero-point solutions. We also demonstrated the effectiveness of boundary control and distributed control through numerous numerical experiments on Reaction-Diffusion equations and Burgers' equations.