# Reconstructing nodal pressures in water distribution systems with graph neural networks

arXiv: https://arxiv.org/abs/2104.13619 · HTML: https://mlsanity.com/paper/2104.13619/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 57 references that are in the map.
- First posted on arXiv 28 Apr 2021.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Kernel-based Reconstruction of Graph Signals](https://mlsanity.com/paper/1605.07174/) (2016)
- [Deep Reinforcement Learning for Real-Time Optimization of Pumps in Water Distribution Systems](https://mlsanity.com/paper/2010.06460/) (2020)
- [Learning Optimal Control of Water Distribution Networks through Sequential Model-based Optimization](https://mlsanity.com/paper/2003.04268/) (2020)
- [Sensor selection on graphs via data-driven node sub-sampling in network time series](https://mlsanity.com/paper/2004.11815/) (2020)

## Built on by (most cited first, 6 of 6)

- [Physics-Informed Graph Neural Networks for Water Distribution Systems](https://mlsanity.com/paper/2403.18570/) (2024)
- [Graph Neural Networks for Pressure Estimation in Water Distribution Systems](https://mlsanity.com/paper/2311.10579/) (2023)
- [Spatial Graph Convolution Neural Networks for Water Distribution Systems](https://mlsanity.com/paper/2211.09587/) (2022)
- [Graph neural network-based surrogate modelling for real-time hydraulic prediction of urban drainage networks](https://mlsanity.com/paper/2404.10324/) (2024)
- [Algorithm-Informed Graph Neural Networks for Leakage Detection and Localization in Water Distribution Networks](https://mlsanity.com/paper/2408.02797/) (2024)
- [Scalable and Robust Physics-Informed Graph Neural Networks for Water Distribution Systems](https://mlsanity.com/paper/2502.12164/) (2025)

## Abstract (arXiv)

Knowing the pressure at all times in each node of a water distribution system (WDS) facilitates safe and efficient operation. Yet, complete measurement data cannot be collected due to the limited number of instruments in a real-life WDS. The data-driven methodology of reconstructing all the nodal pressures by observing only a limited number of nodes is presented in the paper. The reconstruction method is based on K-localized spectral graph filters, wherewith graph convolution on water networks is possible. The effect of the number of layers, layer depth and the degree of the Chebyshev-polynomial applied in the kernel is discussed taking into account the peculiarities of the application. In addition, a weighting method is shown, wherewith information on friction loss can be embed into the spectral graph filters through the adjacency matrix. The performance of the proposed model is presented on 3 WDSs at different number of nodes observed compared to the total number of nodes. The weighted connections prove no benefit over the binary connections, but the proposed model reconstructs the nodal pressure with at most 5% relative error on average at an observation ratio of 5% at least. The results are achieved with shallow graph neural networks by following the considerations discussed in the paper.