Optimal control of PDEs using physics-informed neural networks

Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the residual of PDE into loss function network-based surrogate model unknown state, PINNs can seamlessly blend measurement data with physical constraints. Here, we extend this framework to PDE-constrained optimal control problems, which governing is fully known goal find variable that minimizes desired cost objective. We provide set guidelines obtaining good solution; first selecting an appropriate PINN architecture training parameters based on problem, second choosing best value critical scalar weight in using simple but effective two-step line search strategy. then validate performance comparing it adjoint-based nonlinear control, performs gradient descent discretized while satisfying PDE. This comparison carried out several distributed examples Laplace, Burgers, Kuramoto-Sivashinsky, Navier-Stokes equations. Finally, discuss advantages caveats approaches constrained PDEs.