Using symbolic machine learning to assess and model substance transport and decay in water distribution networks

Sep 17, 2026

Drinking water infrastructures are treated as networked pipe systems that transport clean water, where water quality analysis evaluates advective diffusion of substances that may cause contamination or support disinfection. The study models substance transport and decay by integrating pipe-network differential equations using kinetics within a Lagrangian scheme driven by hydraulic velocities under steady-state assumptions. Symbolic machine learning via Evolutionary Polynomial Regression generates synthetic formulas to predict node concentrations from travel time, and identifies kinetic reaction-model structures that accurately approximate differential-equation integration across real and test networks.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

AccessFree to download

Overview

Drinking water infrastructures and water quality

Modeling problem in distribution networks

Differential-equation integration and Lagrangian scheme

Hydraulic analysis assumptions

Symbolic machine learning approach

[www. nature.com/scientificreports](www. nature.com/scientificreports) OPEN Using symbolic machine learning to assess and model substance transport and decay in water distribution networks Daniele Biagio Laucelli1, Laura Enríquez1, Juan Saldarriaga2 & Orazio Giustolisi3* Drinking water infrastructures are systems of pipes which are generally networked. They play a crucial role in transporting and delivering clean water to people. The water quality analysis refers to the evaluation ofthe advective diffusion of any substance in drinking water infrastructures from source nodes. Such substances could be a contamination for the system or planned for the disinfection, e.g., chlorine. The water quality analysis is performed by integrating the differential equation in the pipes network domain using the kinetics ofthe substance decay and the Lagrangian scheme. The kinetics can be formulated using a specific reaction order depending on the substance characteristics. The basis for the integration is the pipes velocity field calculated by means of hydraulic analysis. The aim of the present work is to discover the intrinsic mechanism of the substance transport in drinking water infrastructures, i.e., their pipes network domain, using the symbolic machine learning, named Evolutionary Polynomial Regression, which provides “synthetic” models (symbolic formulas) from data. We demonstrated, using one real network and two test networks, that the concentration at each node of the network can be predicted using the travel time along the shortest path(s) between the source and each node. Additionally, the formula models provided by symbolic machine learning allowed discovering that a unique formula based on kinetic reaction model structure allows predicting the residual substance concentration at each node, given the source node concentration, surrogating with a good accuracy the integration of the differential equations. Drinking water infrastructures (DWIs) are assets playing a crucial role in transporting and delivering clean water to people, thus providing an essential service to billions of people worldwide. They are required to efficiently supply water with adequate quality for people health, thereby ensuring equitable access to water for all citizens. DWIs encompass both Water Transmission (WTS) and Water Distribution (WDS) Systems. The former, generally characterized by long pipelines with large diameters, have few or no connections to the consumers property, while the latter are characterized by shorter pipelines with smaller diameters and several connections to properties to feed consumers. WTSs facilitate the transfer of water volumes to consumption centres (towns or cities) . WDSs, also called Water Distribution Networks (WDNs), because of their networked structure, transfer water to end consumers. Therefore, the issue of water quality with respect to the contamination and disinfection for people health is mainly related to WDNs. Water quality modelling in WDNs deals with the evaluation of water age, water trace and transport of the reactant substances considering the decay due to chemical reactions. A substance concentration over time is assumed entering from a source node and the calculation aims at determining the concentration of such substance in each node of the network allowing the assessment of the concentration to consumers. To the purpose, assuming the advective diffusion, the differential equations based on a specific kinetic model need to be integrated in the pipes network domain to calculate the concentration of the substance over time to any node. Therefore, the calculation is based on the pipes velocity field deriving from the hydraulic analysis of the system. The hydraulic analysis is based on a steady-state assumption, i.e. neglecting the unsteady flow. Therefore, assuming a time interval mainly for consumers’ demand stationarity, the basis of the hydraulic analysis is the daily scale (i.e., one or more

FAQs

What is the core goal of the work on water distribution networks?

To discover intrinsic mechanisms of substance transport and decay in drinking water networks by building symbolic (formula-based) models from data.

How are substance concentrations modeled in the pipe network?

By integrating differential equations in the network domain using substance decay kinetics within a Lagrangian scheme driven by pipe velocity fields from hydraulic analysis.

What does Evolutionary Polynomial Regression contribute?

It produces synthetic symbolic formulas that predict node concentration and reveals which kinetic reaction-model structure best supports accurate residual concentration predictions.

Sep 17, 2026

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