# Physics-Informed Neural Networks for Minimising Worst-Case Violations in DC Optimal Power Flow

arXiv: https://arxiv.org/abs/2107.00465 · HTML: https://mlsanity.com/paper/2107.00465/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 28 Jun 2021.

## Builds on (most cited first, 15 of 15)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [Learning Optimal Solutions for Extremely Fast AC Optimal Power Flow](https://mlsanity.com/paper/1910.01213/) (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)
- [Learning to Solve the AC-OPF using Sensitivity-Informed Deep Neural Networks](https://mlsanity.com/paper/2103.14779/) (2021)
- [A Convex Neural Network Solver for DCOPF with Generalization Guarantees](https://mlsanity.com/paper/2009.09109/) (2020)
- [Efficient Database Generation for Data-driven Security Assessment of Power Systems](https://mlsanity.com/paper/1806.01074/) (2018)
- [Lecture Notes on Optimal Power Flow (OPF)](https://mlsanity.com/paper/1811.00943/) (2018)

## Built on by (most cited first, 23 of 23)

- [Physics-Informed Neural Networks for AC Optimal Power Flow](https://mlsanity.com/paper/2110.02672/) (2021)
- [End-to-End Feasible Optimization Proxies for Large-Scale Economic Dispatch](https://mlsanity.com/paper/2304.11726/) (2023)
- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [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)
- [Deep Operator Networks for Bayesian Parameter Estimation in PDEs](https://mlsanity.com/paper/2501.10684/) (2025)
- [A Physics-Informed Machine Learning for Electricity Markets: A NYISO Case Study](https://mlsanity.com/paper/2304.00062/) (2023)
- [Minimizing Worst-Case Violations of Neural Networks](https://mlsanity.com/paper/2212.10930/) (2022)
- [Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN)](https://mlsanity.com/paper/2401.02300/) (2024)
- [Self-supervised Equality Embedded Deep Lagrange Dual for Approximate Constrained Optimization](https://mlsanity.com/paper/2306.06674/) (2023)
- [Scalable Exact Verification of Optimization Proxies for Large-Scale Optimal Power Flow](https://mlsanity.com/paper/2405.06109/) (2024)
- [Compact Optimality Verification for Optimization Proxies](https://mlsanity.com/paper/2405.21023/) (2024)
- [Physics Informed Piecewise Linear Neural Networks for Process Optimization](https://mlsanity.com/paper/2302.00990/) (2023)
- [End-to-End Reinforcement Learning of Curative Curtailment with Partial Measurement Availability](https://mlsanity.com/paper/2405.03262/) (2024)
- [Adaptive Informed Deep Neural Networks for Power Flow Analysis](https://mlsanity.com/paper/2412.02659/) (2024)
- [Scaling Laws of Machine Learning for Optimal Power Flow](https://mlsanity.com/paper/2601.02706/) (2026)
- [Enriching Neural Network Training Dataset to Improve Worst-Case Performance Guarantees](https://mlsanity.com/paper/2303.13228/) (2023)
- [AI Driven Near Real-time Locational Marginal Pricing Method: A Feasibility and Robustness Study](https://mlsanity.com/paper/2306.10080/) (2023)
- [A Hybrid Approach of Transfer Learning and Physics-Informed Modeling: Improving Dissolved Oxygen Concentration Prediction in an Industrial Wastewater Treatment Plant](https://mlsanity.com/paper/2401.11217/) (2024)
- [Differentiable Projection-based Learn to Optimize in Wireless Network-Part I: Convex Constrained (Non-)Convex Programming](https://mlsanity.com/paper/2502.00053/) (2025)
- [Limitations of Physics-Informed Neural Networks: a Study on Smart Grid Surrogation](https://mlsanity.com/paper/2508.21559/) (2025)
- [Self-Certifying Primal-Dual Optimization Proxies for Large-Scale Batch Economic Dispatch](https://mlsanity.com/paper/2510.15850/) (2025)

## Abstract (arXiv)

Physics-informed neural networks exploit the existing models of the underlying physical systems to generate higher accuracy results with fewer data. Such approaches can help drastically reduce the computation time and generate a good estimate of computationally intensive processes in power systems, such as dynamic security assessment or optimal power flow. Combined with the extraction of worst-case guarantees for the neural network performance, such neural networks can be applied in safety-critical applications in power systems and build a high level of trust among power system operators. This paper takes the first step and applies, for the first time to our knowledge, Physics-Informed Neural Networks with Worst-Case Guarantees for the DC Optimal Power Flow problem. We look for guarantees related to (i) maximum constraint violations, (ii) maximum distance between predicted and optimal decision variables, and (iii) maximum sub-optimality in the entire input domain. In a range of PGLib-OPF networks, we demonstrate how physics-informed neural networks can be supplied with worst-case guarantees and how they can lead to reduced worst-case violations compared with conventional neural networks.