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