# Optimal control of PDEs using physics-informed neural networks

arXiv: https://arxiv.org/abs/2111.09880 · HTML: https://mlsanity.com/paper/2111.09880/

## Key facts

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

## Built on by (most cited first, 30 of 52)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks](https://mlsanity.com/paper/2205.02902/) (2022)
- [PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs](https://mlsanity.com/paper/2502.00963/) (2025)
- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradients](https://mlsanity.com/paper/2209.07075/) (2022)
- [Higher-order-ReLU-KANs (HRKANs) for solving physics-informed neural networks (PINNs) more accurately, robustly and faster](https://mlsanity.com/paper/2409.14248/) (2024)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial Differential Equations](https://mlsanity.com/paper/2405.11401/) (2024)
- [Gradient-free training of neural ODEs for system identification and control using ensemble Kalman inversion](https://mlsanity.com/paper/2307.07882/) (2023)
- [Neural Network Approaches for Parameterized Optimal Control](https://mlsanity.com/paper/2402.10033/) (2024)
- [CL-DiffPhyCon: Closed-loop Diffusion Control of Complex Physical Systems](https://mlsanity.com/paper/2408.03124/) (2024)
- [Control of Dual-Sourcing Inventory Systems using Recurrent Neural Networks](https://mlsanity.com/paper/2201.06126/) (2022)
- [A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints](https://mlsanity.com/paper/2310.02286/) (2023)
- [Deep Learning for Continuous-Time Stochastic Control with Jumps](https://mlsanity.com/paper/2505.15602/) (2025)
- [A Framework for Strategic Discovery of Credible Neural Network Surrogate Models under Uncertainty](https://mlsanity.com/paper/2403.08901/) (2024)
- [DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling](https://mlsanity.com/paper/2502.06250/) (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)
- [TSONN: Time-stepping-oriented neural network for solving partial differential equations](https://mlsanity.com/paper/2310.16491/) (2023)
- [Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularization](https://mlsanity.com/paper/2402.15636/) (2024)
- [Control of Medical Digital Twins with Artificial Neural Networks](https://mlsanity.com/paper/2403.13851/) (2024)
- [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)
- [From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Control](https://mlsanity.com/paper/2502.02205/) (2025)
- [PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methods](https://mlsanity.com/paper/2604.04920/) (2026)

## 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.