Adversarial physics-informed neural networks with hard constraints for optimal control of PDEs

Physics-informed neural networks (PINNs) have emerged as a promising deep learning approach for solving optimal control problems of partial differential equations (PDEs). Balancing the trade-offs between competing loss terms remains a significant challenge when using PINNs to solve optimal control problems of PDEs. The balance between different competing loss terms is crucial for control performance. Generative adversarial networks (GANs) have been proven to significantly improve the accuracy of PINNs in solving PDEs by introducing adversarial training methods to handle the weight relationships of different loss terms. In order to resolve the challenge, we propose the hard-constrained PINN adversarial method (hPINN-Adver), an innovative approach that integrates the PINN framework with GANs. The method aims to “learn the loss function” and dynamically adjust the weight relationships of different loss terms through adversarial training, thereby optimizing the balance between competing loss terms. We conduct detailed and comprehensive experiments to compare hPINN-Adver with soft-constrained PINN line search method (sPINN-Line), hard-constrained PINN line search method (hPINN-Line), hard-constrained PINN penalty method (hPINN-Penalty) and hard-constrained PINN augmented Lagrangian method (hPINN-Augmented) in solving five typical and representative optimal control problems of PDEs. Extensive numerical experiments demonstrate the great potential of hPINN-Adver method in solving optimal control problems of PDEs.