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.


  • Related News & Events

    •  NEWS    Saviz Mowlavi gave an invited talk at North Carolina State University
      Date: April 12, 2024
      MERL Contact: Saviz Mowlavi
      Research Areas: Control, Dynamical Systems, Machine Learning, Optimization
      • Saviz Mowlavi was invited to present remotely at the Computational and Applied Mathematics seminar series in the Department of Mathematics at North Carolina State University.

        The talk, entitled "Model-based and data-driven prediction and control of spatio-temporal systems", described the use of temporal smoothness to regularize the training of fast surrogate models for PDEs, user-friendly methods for PDE-constrained optimization, and efficient strategies for learning feedback controllers for PDEs.
  • Related Publications

  •  Mowlavi, S., Nabi, S., "Optimal Control of PDEs Using Physics-Informed Neural Networks", Advances in Neural Information Processing Systems (NeurIPS) workshop, December 2022.
    BibTeX TR2022-163 PDF
    • @inproceedings{Mowlavi2022dec,
    • author = {Mowlavi, Saviz and Nabi, Saleh},
    • title = {Optimal Control of PDEs Using Physics-Informed Neural Networks},
    • booktitle = {Advances in Neural Information Processing Systems (NeurIPS) workshop},
    • year = 2022,
    • month = dec,
    • url = {}
    • }
  •  Mowlavi, S., Nabi, S., "Optimal control of PDEs using physics-informed neural networks", arXiv, November 2021.
    BibTeX arXiv
    • @article{Mowlavi2021nov,
    • author = {Mowlavi, Saviz and Nabi, Saleh},
    • title = {Optimal control of PDEs using physics-informed neural networks},
    • journal = {arXiv},
    • year = 2021,
    • month = nov,
    • url = {}
    • }