# Employing Deep Neural Operators for PDE control by decoupling training and optimization

arXiv: https://arxiv.org/abs/2506.04742 · HTML: https://mlsanity.com/paper/2506.04742/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 44 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 18 of 18)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [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 networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [An extended physics informed neural network for preliminary analysis of parametric optimal control problems](https://mlsanity.com/paper/2110.13530/) (2021)
- [AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems](https://mlsanity.com/paper/2302.02076/) (2023)
- [Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay](https://mlsanity.com/paper/2307.11436/) (2023)
- [Neural Network Approaches for Parameterized Optimal Control](https://mlsanity.com/paper/2402.10033/) (2024)
- [A PINN approach for the online identification and control of unknown PDEs](https://mlsanity.com/paper/2408.03456/) (2024)
- [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (2024)
- [Optimal control of district heating networks using a reduced order model](https://mlsanity.com/paper/1907.05255/) (2019)
- [Nonlinear Optimization of District Heating Networks](https://mlsanity.com/paper/1910.06453/) (2019)
- [Optimal Control of a Reaction-Diffusion Epidemic Model with Noncompliance](https://mlsanity.com/paper/2407.17298/) (2024)

## Abstract (arXiv)

Neural networks have been applied to control problems, typically by combining data, differential equation residuals, and objective costs in the training loss or by incorporating auxiliary architectural components. Instead, we propose a streamlined approach that decouples the control problem from the training process, rendering these additional layers of complexity unnecessary. In particular, our analysis and computational experiments demonstrate that a simple neural operator architecture, such as DeepONet, coupled with an unconstrained optimization routine, can solve tracking-type partial differential equation (PDE) constrained control problems with a single physics-informed training phase and a subsequent optimization phase. We achieve this by adding a penalty term to the cost function based on the differential equation residual to penalize deviations from the PDE constraint. This allows gradient computations with respect to the control using automatic differentiation through the trained neural operator within an iterative optimization routine, while satisfying the PDE constraints. Once trained, the same neural operator can be reused across different tracking targets without retraining. We benchmark our method on scalar elliptic (Poisson's equation), nonlinear transport (viscous Burgers' equation), and flow (Stokes equation) control problems. For the Poisson and Burgers problems, we compare against adjoint-based solvers: for the time-dependent Burgers problem, the approach achieves competitive accuracy with iteration times up to four times faster, while for the linear Poisson problem, the adjoint method retains superior accuracy, suggesting the approach is best suited to nonlinear and time-dependent settings. For the flow control problem, we verify the feasibility of the optimized control through a reference forward solver.