Optimal control of PDEs using physics-informed neural networks (PINNs)

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 solution, PINNs can seamlessly blend measurement data with known physical constraints. Here, we extend this framework to optimal control problems, for which the governing PDEs are fully known except for a control variable that minimizes a desired cost objective. We show that by adding the cost objective to the training loss function of the PINN, we can find an optimal solution for the control variable. We validate the performance of our method by comparing it to adjoint-based nonlinear optimal control, which is purely based on the governing PDEs. We evaluate the pros and cons of the two approaches in light of several distributed control examples based on the Laplace, Burgers, Kuramoto-Sivashinski and Navier-Stokes equations.