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Optimal control of PDEs using physics-informed neural networks

arXiv 2111.09880 ↗First posted 18 Nov 2021math.OC physics.comp-phcited by 52

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Abstract (arXiv)

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.

Key facts. Cited by 52 papers in the map (data through 14 Oct 2026). First posted on arXiv 18 Nov 2021.

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Citing papers per year: 2022: 5; 2023: 10; 2024: 18; 2025: 11; 2026: 8.

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  1. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  2. 2023AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identificationcited by 29
  3. 2024Wavelet Diffusion Neural Operatorcited by 29
  4. 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
  5. 2025PDE-Controller: LLMs for Autoformalization and Reasoning of PDEscited by 23
  6. 2022Physics-informed neural networks for PDE-constrained optimization and controlcited by 19
  7. 2022Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradientscited by 17
  8. 2024Higher-order-ReLU-KANs (HRKANs) for solving physics-informed neural networks (PINNs) more accurately, robustly and fastercited by 13
  9. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 11
  10. 2024PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial Differential Equationscited by 10
  11. 2023Gradient-free training of neural ODEs for system identification and control using ensemble Kalman inversioncited by 9
  12. 2024Neural Network Approaches for Parameterized Optimal Controlcited by 8
38 more
  1. 2024CL-DiffPhyCon: Closed-loop Diffusion Control of Complex Physical Systemscited by 8
  2. 2022Control of Dual-Sourcing Inventory Systems using Recurrent Neural Networkscited by 6
  3. 2023A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraintscited by 6
  4. 2025Deep Learning for Continuous-Time Stochastic Control with Jumpscited by 6
  5. 2024A Framework for Strategic Discovery of Credible Neural Network Surrogate Models under Uncertaintycited by 5
  6. 2025DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modelingcited by 5
  7. 2023Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problemscited by 4
  8. 2023The Hard-Constraint PINNs for Interface Optimal Control Problemscited by 4
  9. 2024Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order modelscited by 4
  10. 2024Physics-Informed Neural Networks with Trust-Region Sequential Quadratic Programmingcited by 4
  11. 2023Solving Elliptic Optimal Control Problems via Neural Networks and Optimality Systemcited by 3
  12. 2023TSONN: Time-stepping-oriented neural network for solving partial differential equationscited by 3
  13. 2024Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularizationcited by 3
  14. 2024Control of Medical Digital Twins with Artificial Neural Networkscited by 3
  15. 2024PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problemscited by 3
  16. 2024KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problemscited by 3
  17. 2025From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Controlcited by 3
  18. 2026PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methodscited by 3
  19. 2023Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networkscited by 2
  20. 2026Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Controlcited by 2
  21. 2024Physics informed cell representations for variational formulation of multiscale problemscited by 1
  22. 2024Multi-modal Policies with Physics-informed Representations in Complex Fluid Environmentscited by 1
  23. 2024Safe PDE Boundary Control with Neural Operatorscited by 1
  24. 2025Learning Discontinuous Galerkin Solutions to Elliptic Problems via Small Linear Convolutional Neural Networkscited by 1
  25. 2025Physics-Informed Neural Networks for Optimal Vaccination Plan in SIR Epidemic Modelscited by 1
  26. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
  27. 2023Detecting hidden structures from a static loading experiment: topology optimization meets physics-informed neural networkscited by 0
  28. 2024Statistical Mechanics and Artificial Neural Networks: Principles, Models, and Applicationscited by 0
  29. 2024Solving Functional Optimization with Deep Networks and Variational Principlescited by 0
  30. 2024Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order modelscited by 0
  31. 2025Learning task-specific predictive models for scientific computingcited by 0
  32. 2025Employing Deep Neural Operators for PDE control by decoupling training and optimizationcited by 0
  33. 2025Physics-Informed Neural Networks for Control of Single-Phase Flow Systems Governed by Partial Differential Equationscited by 0
  34. 2025Control of dynamical systems with neural networkscited by 0
  35. 2026A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problemscited by 0
  36. 2026INEUS: Iterative Neural Solver for High-Dimensional PIDEscited by 0
  37. 2026Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problemscited by 0
  38. 2026Mechanical Analysis of Parachute Suspension Line Deployment with Binding Tapes Using PINNcited by 0

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