Physics-Informed Machine Learning for Universal Surrogate Modelling of Water Quality Parameters in Water Distribution Networks ^†

Abstract

Modelling and assessing water quality parameters in water distribution networks is essential for providing safe drinking water to end users. While simulation-based modeling approaches rely on costly differentiation for numerical solvers, surrogate models using Artificial Neural Networks (ANNs) can predict solutions with minimal computational effort. In this work, we formulate the idea of a universal surrogate model for predicting water quality dynamics that, once trained, will apply to all water distribution networks. To this end, we adapt the idea of meta-parameterized ANNs to account for variable boundary and initial conditions.

1. Introduction

Ensuring consistent and high-level water quality is paramount for water utilities to meet health requirements and attain customer satisfaction. To this end, water utilities must constantly surveil all relevant water quality parameters, such as free available chlorine and pH, and optimally control water age, chlorine dosage rates, and other control parameters in their water distribution networks (WDNs). While sampling experimental field data with high frequency and spatial density is fairly cost- and time-intensive, water utilities need fast and accurate prediction capabilities for these water quality parameters. Over the last decades, simulation models coupling WDN hydraulics and water quality concerning multi-species reactive-transport (MSRT) have been well-established and proven highly accurate [1]. However, their computational complexity hampers their direct applicability in simulation-based optimization studies, e.g., regarding system design or control parameters, which rely on several simulation runs.

Surrogate models employing machine learning techniques, e.g., Artificial Neural Networks (ANN), have proven effective and accurate in estimating water quality in WDNs [2,3]. However, their prediction capabilities are greatly constrained to the WDN that was utilized for their training and the boundaries of the corresponding data because an inherent limitation of machine learning methods, particularly ANNs, is their inability to generally extrapolate outside of the range of the training data. This results in the necessity to retrain the ANN whenever the input conditions transition outside of the predefined range, i.e., a change to the WDN is made regarding its extent, topology, operation, or water demands.

In this work, we formulate a machine learning-based surrogate modeling approach to predict the water quality dynamics in a WDN in a universal manner by reducing the problem to a single generic pipe and ultimately combining those parameter predictions by employing the general mixing model commonly used in simulation techniques. To account for the MSRT dynamics, we extend upon similar approaches in related domains, whose physics is likely founded in a system of partial differential equations. Such approaches have proven effective for fixed domain dimensions and stationary boundary conditions. In our case, the problem extends to a variant input boundary condition concerning the flow velocity and concentration parameters essential for the surrogate modeling of a single pipe as a constituting element of an entire WDN.

2. State-of-the-Art of Water Quality Modeling

Traditional water quality modeling in WDNs is composed of a two-step process involving (i) the calculation of its hydraulics, which is followed by (ii) the subsequent calculation of the processes controlling the dynamics of the water quality parameters, i.e., transport (advection and dispersion) and reactions [3]. In the first step, calculating the hydraulics involves solving a system of nonlinear equations to derive the distribution of nodal pressures and pipe flow velocities across the entire WDN. Then, the advective–dispersive–reactive (ADR) transport problem of water quality parameters within pipes in the second step requires the solution of a set of one-dimensional partial differential equations (PDEs), as well as the application of a mixing model at the WDN nodes, described in detail in [4].

In particular, solving the ADR transport problem within network pipes necessitates a numerical approach involving spatial and temporal discretization [5]. In the case of WDNs, desirable ranges for the latter span from approximately one minute to one hour, while the most critical cases regarding flow velocities may be encountered in dead ends, falling below 10⁻⁴ m/s. Under consideration of the Courant–Friedrichs–Lewy condition [6], these values entail very small discretization lengths in the order of 10⁻⁶ m. This is especially critical for very long pipes with a length of more than 100 m, resulting in an extremely high number of spatial partitions. Moreover, this bottleneck scales with the network dimension, i.e., the number of pipes in a WDN, leading to immense computational effort.

3. A Universal Surrogate Model for Water Quality Dynamics in WDNs

The primary target in constructing a surrogate model for modeling and solving the water quality in a WDN concerns the avoidance of any spatial discretization as the main bottleneck. In general, surrogate models for water quality transport may take the hydraulic conditions and dosage parameters as input, delivering very accurate prediction results for the distribution of water quality parameters at the network scale [3]. However, this type of surrogate models remains network-specific and requires retraining before being applied to other WDNs, or whenever WDN characteristics change.

However, the problem of network specificity can be tackled by deconstructing the transport problem, as displayed in Figure 1, into its base elements of (i) the water quality transport and reactions within individual pipes and (ii) the mixing of fluid portions from multiple pipes connecting at WDN nodes. As described in [3], the mixing process at the nodes is linear and subject to instantaneous and complete mixing with the further flow-weighted distribution of the fluid to connecting pipes; hence, it does not necessitate an iterative approach to its solution. Finally, to facilitate the success of a universal surrogate model for water quality transport in WDNs, a surrogate model that predicts water quality transport and reactions in individual pipes is needed.

Based on past work that elucidates the capability of Physics-Informed Neural Networks (PINNs) to accurately predict solutions of PDE-based transport problems—even in a parameterized fashion, as shown in [7]—we propose a similar approach to the prediction of water quality transport. Therein, the concentration $C_A^P$ of a compound $A$ in a pipe $P$ subject to ADR transport can be calculated at location $x$ along the pipe length using a multilayer perceptron (MLP). Furthermore, this MLP can be parameterized employing a further meta-MLP for the hydraulic boundary conditions, i.e., the flow velocity $u$, and the water quality boundary condition, i.e., $C_A^P\left(x=0\right)$, at a specified time step $t$, as well as the initial conditions $\left(t=0\right)$ at specified locations $\left\{x_1,\dots ,x_k\right\}$. This approach is illustrated in Figure 2.

4. Conclusions

This work presents an approach to develop a universal surrogate model for water quality transport and reactions in a WDN. Once trained, the surrogate model is universally applicable to every kind of WDN due to its compartmentalized formulation, separating the mixing procedure occurring in nodes and the transport and reaction processes occurring in pipes. Moreover, the processes within pipes are parameterized to account for variant time-step sizes, boundary conditions, and initial conditions. Further work will entail demonstrating and validating this idea on data generated from simulation models to provide a proof of concept in a controlled environment, as well as application to real-world case studies.