mlsanityData to 7 Oct 2026

Physics-informed neural networks for PDE-constrained optimization and control

arXiv 2205.03377 ↗First posted 6 May 2022cs.LG math.OCcited by 19

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Abstract (arXiv)

A fundamental problem in science and engineering is designing optimal control policies that steer a given system towards a desired outcome. This work proposes Control Physics-Informed Neural Networks (Control PINNs) that simultaneously solve for a given system state, and for the optimal control signal, in a one-stage framework that conforms to the underlying physical laws. Prior approaches use a two-stage framework that first models and then controls a system in sequential order. In contrast, a Control PINN incorporates the required optimality conditions in its architecture and in its loss function. The success of Control PINNs is demonstrated by solving the following open-loop optimal control problems: (i) an analytical problem, (ii) a one-dimensional heat equation, and (iii) a two-dimensional predator-prey problem.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). Builds on 22 of its 44 references that are in the map. First posted on arXiv 6 May 2022.

Builds on

22 of 44 references.

References · 22

  1. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  2. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  3. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  4. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  5. 2020Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systemscited by 160
  6. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  7. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  8. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 134
  9. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  10. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  11. 2018Optimal Control Via Neural Networks: A Convex Approachcited by 82
  12. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
10 more
  1. 2019Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transportcited by 60
  2. 2021Optimal control of PDEs using physics-informed neural networkscited by 51
  3. 2021Solving PDE-constrained Control Problems Using Operator Learningcited by 41
  4. 2021Physics-Informed Neural Nets for Control of Dynamical Systemscited by 31
  5. 2021Fast PDE-constrained optimization via self-supervised operator learningcited by 24
  6. 2021Physics Informed Neural Networks for Control Oriented Thermal Modeling of Buildingscited by 24
  7. 2021Physics-Informed Neural Networks for Minimising Worst-Case Violations in DC Optimal Power Flowcited by 23
  8. 2021Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulatorscited by 21
  9. 2018Importance of the Mathematical Foundations of Machine Learning Methods for Scientific and Engineering Applicationscited by 4
  10. 2021Adjoint-Matching Neural Network Surrogates for Fast 4D-Var Data Assimilationcited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 4; 2024: 7; 2025: 4; 2026: 3.

19 papers.

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  1. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  2. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  3. 2025PDE-Controller: LLMs for Autoformalization and Reasoning of PDEscited by 23
  4. 2025Deep Operator Networks for Bayesian Parameter Estimation in PDEscited by 6
  5. 2023Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problemscited by 4
  6. 2023The Hard-Constraint PINNs for Interface Optimal Control Problemscited by 4
  7. 2024Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order modelscited by 4
  8. 2024Physics-Informed Neural Networks with Trust-Region Sequential Quadratic Programmingcited by 4
  9. 2023Solving Elliptic Optimal Control Problems via Neural Networks and Optimality Systemcited by 3
  10. 2024PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problemscited by 3
  11. 2024KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problemscited by 3
  12. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
7 more
  1. 2024Receding Hamiltonian-Informed Optimal Neural Control and State Estimation for Closed-Loop Dynamical Systemscited by 0
  2. 2024Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order modelscited by 0
  3. 2025Employing Deep Neural Operators for PDE control by decoupling training and optimizationcited by 0
  4. 2025Ensemble based Closed-Loop Optimal Control using Physics-Informed Neural Networkscited by 0
  5. 2026A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problemscited by 0
  6. 2026Dual-Network PINNs for Optimal Control: A Reproducible Benchmark on the Mass-Spring-Damper Systemcited by 0
  7. 2026Real-time optimal control with shallow recurrent decoder networkscited by 0