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

arXiv: https://arxiv.org/abs/2205.03377 · HTML: https://mlsanity.com/paper/2205.03377/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 44 references that are in the map.
- First posted on arXiv 6 May 2022.

## Builds on (most cited first, 22 of 22)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Physics-Informed Neural Nets for Control of Dynamical Systems](https://mlsanity.com/paper/2104.02556/) (2021)
- [Fast PDE-constrained optimization via self-supervised operator learning](https://mlsanity.com/paper/2110.13297/) (2021)
- [Physics Informed Neural Networks for Control Oriented Thermal Modeling of Buildings](https://mlsanity.com/paper/2111.12066/) (2021)
- [Physics-Informed Neural Networks for Minimising Worst-Case Violations in DC Optimal Power Flow](https://mlsanity.com/paper/2107.00465/) (2021)
- [Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators](https://mlsanity.com/paper/2109.10793/) (2021)
- [Importance of the Mathematical Foundations of Machine Learning Methods for Scientific and Engineering Applications](https://mlsanity.com/paper/1808.02213/) (2018)
- [Adjoint-Matching Neural Network Surrogates for Fast 4D-Var Data Assimilation](https://mlsanity.com/paper/2111.08626/) (2021)

## Built on by (most cited first, 19 of 19)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs](https://mlsanity.com/paper/2502.00963/) (2025)
- [Deep Operator Networks for Bayesian Parameter Estimation in PDEs](https://mlsanity.com/paper/2501.10684/) (2025)
- [Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems](https://mlsanity.com/paper/2307.00296/) (2023)
- [The Hard-Constraint PINNs for Interface Optimal Control Problems](https://mlsanity.com/paper/2308.06709/) (2023)
- [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (2024)
- [Physics-Informed Neural Networks with Trust-Region Sequential Quadratic Programming](https://mlsanity.com/paper/2409.10777/) (2024)
- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)
- [PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems](https://mlsanity.com/paper/2409.01899/) (2024)
- [KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems](https://mlsanity.com/paper/2409.06649/) (2024)
- [The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach](https://mlsanity.com/paper/2302.08309/) (2023)
- [Receding Hamiltonian-Informed Optimal Neural Control and State Estimation for Closed-Loop Dynamical Systems](https://mlsanity.com/paper/2411.01297/) (2024)
- [Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order models](https://mlsanity.com/paper/2412.09942/) (2024)
- [Employing Deep Neural Operators for PDE control by decoupling training and optimization](https://mlsanity.com/paper/2506.04742/) (2025)
- [Ensemble based Closed-Loop Optimal Control using Physics-Informed Neural Networks](https://mlsanity.com/paper/2510.18195/) (2025)
- [A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problems](https://mlsanity.com/paper/2601.04120/) (2026)
- [Dual-Network PINNs for Optimal Control: A Reproducible Benchmark on the Mass-Spring-Damper System](https://mlsanity.com/paper/2606.15271/) (2026)
- [Real-time optimal control with shallow recurrent decoder networks](https://mlsanity.com/paper/2607.19302/) (2026)

## 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.