A PINN Approach for the Online Identification and Control of Unknown PDEs

Physics-Informed Neural Networks (PINNs) have revolutionized solving differential equations by integrating physical laws into neural networks training. This paper explores PINNs for open-loop optimal control problems with incomplete information, such as sparse initial and boundary data and partially unknown system parameters. We derive optimality conditions from the Lagrangian multipliers and use PINNs to predict the state, adjoint, and control variables. In contrast with previous methods, our approach integrates these elements into a single neural network and addresses scenarios with consistently limited data. In addition, we address the study of partially unknown equations identifying underlying parameters online by searching for the optimal solution recurring to a 2-in-series architecture of PINNs, in which scattered data of the uncontrolled solution is used. Numerical examples show the effectiveness of the proposed method even in scenarios characterized by a considerable lack of information.