# Residual Power Flow for Neural Solvers

arXiv: https://arxiv.org/abs/2601.09533 · HTML: https://mlsanity.com/paper/2601.09533/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 14 Jan 2026.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Physics-Informed Neural Networks for AC Optimal Power Flow](https://mlsanity.com/paper/2110.02672/) (2021)
- [Learning Optimal Power Flow: Worst-Case Guarantees for Neural Networks](https://mlsanity.com/paper/2006.11029/) (2020)
- [Foundation Models for the Electric Power Grid](https://mlsanity.com/paper/2407.09434/) (2024)
- [PowerFlowNet: Power Flow Approximation Using Message Passing Graph Neural Networks](https://mlsanity.com/paper/2311.03415/) (2023)
- [Power Flow Balancing with Decentralized Graph Neural Networks](https://mlsanity.com/paper/2111.02169/) (2021)
- [PowerGraph: A power grid benchmark dataset for graph neural networks](https://mlsanity.com/paper/2402.02827/) (2024)
- [Physics-Informed Neural Networks for Time-Domain Simulations: Accuracy, Computational Cost, and Flexibility](https://mlsanity.com/paper/2303.08994/) (2023)
- [KCLNet: Physics-Informed Power Flow Prediction via Constraints Projections](https://mlsanity.com/paper/2506.12902/) (2025)
- [Optimal Power Flow Pursuit](https://mlsanity.com/paper/1601.07263/) (2016)
- [Large-scale Grid Optimization: The Workhorse of Future Grid Computations](https://mlsanity.com/paper/2501.04623/) (2025)
- [Sample-Based Piecewise Linear Power Flow Approximations Using Second-Order Sensitivities](https://mlsanity.com/paper/2501.13825/) (2025)
- [Geometric Contact Flows: Contactomorphisms for Dynamics and Control](https://mlsanity.com/paper/2506.17868/) (2025)
- [Parameterized Linear Power Flow for High Fidelity Voltage Solutions in Distribution Systems](https://mlsanity.com/paper/2209.04508/) (2022)
- [AC Power Flow Feasibility Restoration via a State Estimation-Based Post-Processing Algorithm](https://mlsanity.com/paper/2304.11418/) (2023)
- [Dynamic Slack Bus](https://mlsanity.com/paper/2405.20100/) (2024)
- [PRIME: Fast Primal-Dual Feedback Optimization for Markets with Application to Optimal Power Flow](https://mlsanity.com/paper/2504.16048/) (2025)
- [Machine Learning for Physical Simulation Challenge Results and Retrospective Analysis: Power Grid Use Case](https://mlsanity.com/paper/2505.01156/) (2025)

## Built on by (most cited first, 1 of 1)

- [LUMINA: Foundation Models for Topology Transferable ACOPF](https://mlsanity.com/paper/2603.04300/) (2026)

## Abstract (arXiv)

The energy transition challenges operational tasks based on simulations and optimisation. These computations need to be fast and flexible as the grid is ever-expanding, and renewables' uncertainty requires a flexible operational environment. Learned approximations, proxies or surrogates -- we refer to them as Neural Solvers -- excel in terms of evaluation speed, but are inflexible with respect to adjusting to changing tasks. Hence, neural solvers are usually applicable to highly specific tasks, which limits their usefulness in practice; a widely reusable, foundational neural solver is required. Therefore, this work proposes the Residual Power Flow (RPF) formulation. RPF formulates residual functions based on Kirchhoff's laws to quantify the infeasibility of an operating condition. The minimisation of the residuals determines the voltage solution; an additional slack variable is needed to achieve AC-feasibility. RPF forms a natural, foundational subtask of tasks subject to power flow constraints. We propose to learn RPF with neural solvers to exploit their speed. Furthermore, RPF improves learning performance compared to common power flow formulations. To solve operational tasks, we integrate the neural solver in a Predict-then-Optimise (PO) approach to combine speed and flexibility. The case study investigates the IEEE 9-bus system and three tasks (AC Optimal Power Flow (OPF), power-flow and quasi-steady state power flow) solved by PO. The results demonstrate the accuracy and flexibility of learning with RPF.