# Physics-Informed Neural Networks for AC Optimal Power Flow

arXiv: https://arxiv.org/abs/2110.02672 · HTML: https://mlsanity.com/paper/2110.02672/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 20 references that are in the map.
- First posted on arXiv 6 Oct 2021.

## Builds on (most cited first, 12 of 12)

- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Predicting AC Optimal Power Flows: Combining Deep Learning and Lagrangian Dual Methods](https://mlsanity.com/paper/1909.10461/) (2019)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms](https://mlsanity.com/paper/1908.02788/) (2019)
- [DeepOPF: Deep Neural Network for DC Optimal Power Flow](https://mlsanity.com/paper/1905.04479/) (2019)
- [Learning for DC-OPF: Classifying active sets using neural nets](https://mlsanity.com/paper/1902.05607/) (2019)
- [Learning Optimal Power Flow: Worst-Case Guarantees for Neural Networks](https://mlsanity.com/paper/2006.11029/) (2020)
- [Verification of Neural Network Behaviour: Formal Guarantees for Power System Applications](https://mlsanity.com/paper/1910.01624/) (2019)
- [Physics-Informed Neural Networks for Minimising Worst-Case Violations in DC Optimal Power Flow](https://mlsanity.com/paper/2107.00465/) (2021)
- [Efficient Database Generation for Data-driven Security Assessment of Power Systems](https://mlsanity.com/paper/1806.01074/) (2018)
- [Efficient Creation of Datasets for Data-Driven Power System Applications](https://mlsanity.com/paper/1910.01794/) (2019)

## Built on by (most cited first, 30 of 54)

- [Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2407.17611/) (2024)
- [End-to-End Feasible Optimization Proxies for Large-Scale Economic Dispatch](https://mlsanity.com/paper/2304.11726/) (2023)
- [Topology-aware Graph Neural Networks for Learning Feasible and Adaptive ac-OPF Solutions](https://mlsanity.com/paper/2205.10129/) (2022)
- [Foundation Models for the Electric Power Grid](https://mlsanity.com/paper/2407.09434/) (2024)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [Compact Optimization Learning for AC Optimal Power Flow](https://mlsanity.com/paper/2301.08840/) (2023)
- [Dual Conic Proxies for AC Optimal Power Flow](https://mlsanity.com/paper/2310.02969/) (2023)
- [Closing the Loop: A Framework for Trustworthy Machine Learning in Power Systems](https://mlsanity.com/paper/2203.07505/) (2022)
- [Ensuring DNN Solution Feasibility for Optimization Problems with Convex Constraints and Its Application to DC Optimal Power Flow Problems](https://mlsanity.com/paper/2112.08091/) (2021)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [QCQP-Net: Reliably Learning Feasible Alternating Current Optimal Power Flow Solutions Under Constraints](https://mlsanity.com/paper/2401.06820/) (2024)
- [Global Performance Guarantees for Neural Network Models of AC Power Flow](https://mlsanity.com/paper/2211.07125/) (2022)
- [A Physics-Informed Machine Learning for Electricity Markets: A NYISO Case Study](https://mlsanity.com/paper/2304.00062/) (2023)
- [GPU-Accelerated Verification of Machine Learning Models for Power Systems](https://mlsanity.com/paper/2306.10617/) (2023)
- [Towards graph neural networks for provably solving convex optimization problems](https://mlsanity.com/paper/2502.02446/) (2025)
- [Minimizing Worst-Case Violations of Neural Networks](https://mlsanity.com/paper/2212.10930/) (2022)
- [Optimal Power Flow Based on Physical-Model-Integrated Neural Network with Worth-Learning Data Generation](https://mlsanity.com/paper/2301.03766/) (2023)
- [Power Flow Analysis Using Deep Neural Networks in Three-Phase Unbalanced Smart Distribution Grids](https://mlsanity.com/paper/2401.07465/) (2024)
- [Learning to Solve the Constrained Most Probable Explanation Task in Probabilistic Graphical Models](https://mlsanity.com/paper/2404.11606/) (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)
- [Aligning Perception, Reasoning, Modeling and Interaction: A Survey on Physical AI](https://mlsanity.com/paper/2510.04978/) (2025)
- [Model-Informed Generative Adversarial Network (MI-GAN) for Learning Optimal Power Flow](https://mlsanity.com/paper/2206.01864/) (2022)
- [Physics-Guided Graph Neural Networks for Real-time AC/DC Power Flow Analysis](https://mlsanity.com/paper/2305.00216/) (2023)
- [Self-supervised Equality Embedded Deep Lagrange Dual for Approximate Constrained Optimization](https://mlsanity.com/paper/2306.06674/) (2023)
- [A Survey of Physics-Informed AI for Complex Urban Systems](https://mlsanity.com/paper/2506.13777/) (2025)
- [LUMINA: A Grid Foundation Model for Benchmarking AC Optimal Power Flow Surrogate Learning](https://mlsanity.com/paper/2605.02133/) (2026)
- [Learning-based AC-OPF Solvers on Realistic Network and Realistic Loads](https://mlsanity.com/paper/2205.09452/) (2022)
- [Grid-SiPhyR: An end-to-end learning to optimize framework for combinatorial problems in power systems](https://mlsanity.com/paper/2206.06789/) (2022)
- [Physics Informed Piecewise Linear Neural Networks for Process Optimization](https://mlsanity.com/paper/2302.00990/) (2023)
- [Correctness Verification of Neural Networks Approximating Differential Equations](https://mlsanity.com/paper/2402.07621/) (2024)

## Abstract (arXiv)

This paper introduces, for the first time to our knowledge, physics-informed neural networks to accurately estimate the AC-OPF result and delivers rigorous guarantees about their performance. Power system operators, along with several other actors, are increasingly using Optimal Power Flow (OPF) algorithms for a wide number of applications, including planning and real-time operations. However, in its original form, the AC Optimal Power Flow problem is often challenging to solve as it is non-linear and non-convex. Besides the large number of approximations and relaxations, recent efforts have also been focusing on Machine Learning approaches, especially neural networks. So far, however, these approaches have only partially considered the wide number of physical models available during training. And, more importantly, they have offered no guarantees about potential constraint violations of their output. Our approach (i) introduces the AC power flow equations inside neural network training and (ii) integrates methods that rigorously determine and reduce the worst-case constraint violations across the entire input domain, while maintaining the optimality of the prediction. We demonstrate how physics-informed neural networks achieve higher accuracy and lower constraint violations than standard neural networks, and show how we can further reduce the worst-case violations for all neural networks.