# AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems

arXiv: https://arxiv.org/abs/2302.02076 · HTML: https://mlsanity.com/paper/2302.02076/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2023.

## Built on by (most cited first, 13 of 13)

- [Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEs](https://mlsanity.com/paper/2305.18702/) (2023)
- [Neural Network Approaches for Parameterized Optimal Control](https://mlsanity.com/paper/2402.10033/) (2024)
- [Deep adaptive sampling for surrogate modeling without labeled data](https://mlsanity.com/paper/2402.11283/) (2024)
- [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (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)
- [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)
- [Systematic construction of continuous-time neural networks for linear dynamical systems](https://mlsanity.com/paper/2403.16215/) (2024)
- [Solving Functional Optimization with Deep Networks and Variational Principles](https://mlsanity.com/paper/2410.06277/) (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)
- [Real-time optimal control with shallow recurrent decoder networks](https://mlsanity.com/paper/2607.19302/) (2026)

## Abstract (arXiv)

Parametric optimal control problems governed by partial differential equations (PDEs) are widely found in scientific and engineering applications. Traditional grid-based numerical methods for such problems generally require repeated solutions of PDEs with different parameter settings, which is computationally prohibitive especially for problems with high-dimensional parameter spaces. Although recently proposed neural network methods make it possible to obtain the optimal solutions simultaneously for different parameters, challenges still remain when dealing with problems with complex constraints. In this paper, we propose AONN, an adjoint-oriented neural network method, to overcome the limitations of existing approaches in solving parametric optimal control problems. In AONN, the neural networks are served as parametric surrogate models for the control, adjoint and state functions to get the optimal solutions all at once. In order to reduce the training difficulty and handle complex constraints, we introduce an iterative training framework inspired by the classical direct-adjoint looping (DAL) method so that penalty terms arising from the Karush-Kuhn-Tucker (KKT) system can be avoided. Once the training is done, parameter-specific optimal solutions can be quickly computed through the forward propagation of the neural networks, which may be further used for analyzing the parametric properties of the optimal solutions. The validity and efficiency of AONN is demonstrated through a series of numerical experiments with problems involving various types of parameters.