# Refining Graphical Neural Network Predictions Using Flow Matching for Optimal Power Flow with Constraint-Satisfaction Guarantee

arXiv: https://arxiv.org/abs/2512.11127 · HTML: https://mlsanity.com/paper/2512.11127/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 20 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 11 of 11)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Learning Optimal Solutions for Extremely Fast AC Optimal Power Flow](https://mlsanity.com/paper/1910.01213/) (2019)
- [Physics-Informed Neural Networks for AC Optimal Power Flow](https://mlsanity.com/paper/2110.02672/) (2021)
- [DeepOPF: A Deep Neural Network Approach for Security-Constrained DC Optimal Power Flow](https://mlsanity.com/paper/1910.14448/) (2019)
- [Leveraging power grid topology in machine learning assisted optimal power flow](https://mlsanity.com/paper/2110.00306/) (2021)
- [QCQP-Net: Reliably Learning Feasible Alternating Current Optimal Power Flow Solutions Under Constraints](https://mlsanity.com/paper/2401.06820/) (2024)
- [Physics-Informed Gradient Estimation for Accelerating Deep Learning based AC-OPF](https://mlsanity.com/paper/2502.16373/) (2025)
- [Data-driven AC Optimal Power Flow with Physics-informed Learning and Calibrations](https://mlsanity.com/paper/2404.09128/) (2024)
- [PowerGNN: A Topology-Aware Graph Neural Network for Electricity Grids](https://mlsanity.com/paper/2503.22721/) (2025)
- [Optimal Power Flow in Highly Renewable Power System Based on Attention Neural Networks](https://mlsanity.com/paper/2311.13949/) (2023)

## Abstract (arXiv)

The DC Optimal Power Flow (DC-OPF) problem is fundamental to power system operations, requiring rapid solutions for real-time grid management. While traditional optimization solvers provide optimal solutions, their computational cost becomes prohibitive for large-scale systems requiring frequent recalculations. Machine learning approaches offer promise for acceleration but often struggle with constraint satisfaction and cost optimality. We present a novel two-stage learning framework that combines physics-informed Graph Neural Networks (GNNs) with Continuous Flow Matching (CFM) for solving DC-OPF problems. Our approach embeds fundamental physical principles--including economic dispatch optimality conditions, Kirchhoff's laws, and Karush-Kuhn-Tucker (KKT) complementarity conditions--directly into the training objectives. The first stage trains a GNN to produce feasible initial solutions by learning from physics-informed losses that encode power system constraints. The second stage employs CFM, a simulation-free continuous normalizing flow technique, to refine these solutions toward optimality through learned vector field regression. Evaluated on the IEEE 30-bus system across five load scenarios ranging from 70% to 130% nominal load, our method achieves near-optimal solutions with cost gaps below 0.1% for nominal loads and below 3% for extreme conditions, while maintaining 100% feasibility. Our framework bridges the gap between fast but approximate neural network predictions and optimal but slow numerical solvers, offering a practical solution for modern power systems with high renewable penetration requiring frequent dispatch updates.