An Efficient Surrogate-based Multi-objective Optimisation Framework with Novel Sampling Strategy for Sustainable Island Groundwater Management

In groundwater pumping optimization (GPO), offline-trained data-driven surrogates can be used to replace numerical-intensive simulators in order to save computing time. The traditional offline training approach involves building surrogates prior to optimization, fitting training datasets that cover the input space uniformly or randomly, which can prove inefficient due to the potential oversampling of low-gradient areas and under-sampling of high-gradient areas. This study proposes an offline machine-learning (ML) algorithm that ranks candidate training points by scoring them based on their distance to the closest training point and on the local gradient of the surrogate estimate and then choosing the highest-rank point. This method is applied to develop surrogates for solving a two-objective GPO problem formulated on a three-dimensional (3D) island aquifer, using hydrogeological conditions representative of San Salvador Island, Bahamas. The objectives are to minimise the supply cost (f_OC) resulting from groundwater pumping and desalination and maximise fresh groundwater supply (Q_p), subject to constraints on seawater intrusion (SWI) control expressed in terms of aquifer drawdown Δs at pumping locations and aquifer salt mass increase ΔSM. Gaussian Process (GP) is the technique applied to construct surrogates of objectives and constraints, alongside the estimation of uncertainties. Using GP models, it is possible to estimate the probability of “Pareto optimality” for each pumping scheme by Monte Carlo simulation. Pareto optimal pumping schemes (POPS) are then characterized by a probability of occurrence, which can be verified by numerical simulation. The GP training strategy's effectiveness in generating POPS is compared to traditional training approaches, showing that such a strategy can efficiently identify reliable POPS.

1 Introduction

Limited by recharge rates and land area, freshwater in island aquifers is typically in the shape of a lens, with a thickness between a few meters to a few tens of meters, which makes it particularly vulnerable to SWI (Kourakos and Mantoglou, 2015; Gulley et al., 2016; Coulon et al., 2022). Island aquifers are often the main freshwater source for local communities, making groundwater pumping inevitable. Yu et al. (2023) have shown that while pumping from the island center at shallow depths is cost-effective for meeting demand, it increases the risk of seawater intrusion (SWI), highlighting a conflict between supply costs and SWI control.

GPO is usually adopted to investigate the trade off between sustainability and/or SWI control against cost. GPO is formulated by optimizing management objectives subject to constraints, both of which are typically functions of decision variables (DVs) and state variables (SVs). Applying the simulation-optimization (SO) method to derive POPS may incur huge computational costs due to repeated calls to numerically-intensive simulators. To reduce this cost, fast data-driven surrogates can be employed instead, forming a surrogate-based SO framework. Traditionally, these surrogates are built by “offline” training, that is, prior to the optimization, by fitting training datasets that cover the DV space either uniformly or randomly. Traditional training methods may result in inefficiency due to oversampling low-gradient and under-sampling high-gradient areas of the DV space. We introduce an offline ML algorithm for selecting training points, applied in an island aquifer multi-objective GPO. This efficient sampling strategy helps develop surrogate models to quickly identify reliable POPS and promote a sustainable coastal hydrogeological environment.

Figure 1 (a) The conceptualization of the island aquifer domain. (b) Offline MOOA framework based on the proposed strategy.

2 Methods

2.1 Study area

An illustrative rectangular aquifer, loosely representing the lens-shaped freshwater system of San Salvador Island (Bahamas) (Gulley et al., 2016), is chosen as a test site. This aquifer, depicted in Fig. 1a, is discretized into a 3D finite-difference grid of 60 000 cells. The four lateral sides are at a constant salt concentration of 35 g L⁻¹ and a constant head of 0 m, representing seaward boundary conditions. Hydrogeological details specific to the study area can be found in Yu et al. (2023). It is important to note that this model, though computationally substantial, does not fully account for real-world site conditions, such as the presence of surface water bodies. This simplification approach aims to provide qualitative insights applicable to similar island aquifer settings.

2.2 SWI modelling and surrogate-based GPO

Pumping is operated by five wells, one at the island centre, and the other four positioned symmetrically relative to the “central” well, located at (0,0). The central well is defined by two DVs, the pumping depth D₁ and the pumping rate Q₁. The pumping depth D₂, the pumping rate Q₂, and the coordinates (±X₂,±Y₂), are the DVs for the other four wells (Fig. 1a).

SEAWAT (Langevin et al., 2008) is used to model aquifer response under different pump schemes and construct training data. GP with a square exponential covariance (kernel) function is used to build surrogates (Rasmussen and Williams, 2006) of objectives and constraints in the 6D DV space. GP_OC, GP${}_{{\mathrm{Q}}_{\mathrm{p}}}$, GP${}_{\mathrm{\Delta }{s}_{\mathrm{1}}}$, GP${}_{\mathrm{\Delta }{s}_{\mathrm{2}}}$ and GP_ΔSM denote the GPs for estimating f_OC, Q_p, Δs₁, Δs₂, and ΔSM, respectively. The GP training relies on an offline ML training strategy, in which an initial set of training points is progressively reinforced by adding new points, selected by identifying the DV set with a maximum value of a score function $F=\mathrm{0.5}\cdot R\left(\mathrm{dc}\right)+\mathrm{0.5}\cdot R\left(\mathrm{\nabla }G\right)$. R is a “normalization” operator ($R\left(\blacksquare \right)\in$[0,1]), “dc” is the distance of a candidate point from the closest training point, and ∇G is the local gradient based on GP model estimates. Figure 1b shows a flow chart of the offline multi-objective optimisation (MOOA) framework based on the proposed strategy.

For any DV set, GP models enable the estimation of f_OC, Q_p, Δs₁, Δs₂ and ΔSM and quantify the uncertainty on it. The problem solution is represented by Pareto optimal (PO) sets of “non-dominated” solutions, each of which is characterized by a probability of Pareto optimality PPO_i, estimated by Monte Carlo simulation with a sample size of 100. The reliability of the derived POPS is assessed by means of the average probability of Pareto optimality of a trade-off set ($\stackrel{\mathrm{‾}}{\mathrm{PPO}}$), and the normalized root mean square errors (NRMSE), εOC, εQ_P, εΔs₁, εΔs₂ and εΔSM, which quantify the difference between GP model predictions and their “true” values calculated by SEAWAT simulations over the 30 PO solutions with higher PPO.

Figure 2 PO sets of rate of freshwater produced Q_p vs. supply cost f_OC along with values of the probability of Pareto-optimality and estimation uncertainty under training scenarios (a) T1, (b) T2, (c) T3 and (d) T4.

3 Results and Discussion

Preliminary tests (not presented here) indicate that the performance of the offline training approach depends on the number of allowed training points N_M. In these tests, $\stackrel{\mathrm{‾}}{\mathrm{PPO}}$ is seen to increase with N_M, as larger training datasets enhance surrogate estimates, but gains become marginal for N_M approaching 400. We thus investigate the case, denoted as T1, in which N_M=400, against the case T2 in which 700 training points are uniformly distributed within the DV space, and two other cases T3 and T4, which consist of two different realizations of 1000 training points randomly distributed across the DV space.

Figure 2 presents the PO sets obtained in (a) T1, (b) T2, (c) T3 and (c) T4. Each point is plotted with a colour depending on PPO_i (see scale), and horizontal and vertical error bars indicative of the surrogate uncertainty. POPS in T1 are characterized by a remarkably smaller uncertainty than those in T2, T3 and T4, indicating that the former are more robust and more reliable than the latter. $\stackrel{\mathrm{‾}}{\mathrm{PPO}}$ in T1 is significantly higher than in T2, T3 and T4, suggesting that the training approach is more effective, other than computationally less intensive, than traditional methods. NMRSE values reported in Table 1 confirm the better reliability of results obtained in T1.

Table 1 Normalized root mean square errors of POPS in the training scenarios T1–T4.

4 Conclusions

The performance of a novel surrogate sampling strategy, using a novel ML algorithm for training point selection, was assessed for an island multi-objective GPO problem. The findings revealed that, compared to traditional training approaches, the proposed strategy can generate more reliable POPS and consume a lower computational cost. For the formulated GPO problem, the results in Fig. 2a indicate that supply cost f_OC increases gradually with a fresh groundwater supply Q_p up to 950 m³ d⁻¹. Beyond this point, f_OC increases sharply, suggesting that 950 m³ d⁻¹ is a sustainability threshold for balancing economic viability and SWI control.