E FFICIENT G ROUNDWATER F LOW M ODELING U SING D EEP
N EURAL O PERATORSMaria Luisa Taccari* 1 , He Wang†2 , Somdatta Goswami‡3 , Mario De Florio§4 , Jonathan Nuttall¶5 , XiaohuiChen||1 , and Peter K. Jimack** 6
1
School of Civil Engineering, University of Leeds, Leeds, UK
2
Department of Computer Science, University College London, London, UK
3
Department of Civil and Systems Engineering, Johns Hopkins University, Baltimore, U.S.A
4
Division of Applied Mathematics, Brown University, Providence, U.S.A
5
Deltares, Delft, The Netherlands
6
School of Computing, University of Leeds, Leeds, UK

January 8, 2024

A BSTRACTThis work introduces a cost-effective emulator for groundwater flow modeling using deep neuraloperators (DeepONet). We address the computational challenges in traditional groundwater modelingby proposing a novel approach that efficiently and accurately forecasts aquifer responses. The model
demonstrates excellent performance across various scenarios, including time-dependent and nonlinear
systems.
Keywords Deep neural operator · Groundwater flow · Surrogate modeling · Deep learning

1

Introduction

Traditional numerical models, such as MODFLOW [6], have been extensively used for simulating groundwater flows.
However, these models often face computational challenges, especially when dealing with large-scale or highly nonlinear
systems [2, 3, 9].
Recent progress in deep learning has introduced innovative approaches for functional approximation, making asignificant impact in various fields including fluid dynamics and subsurface flows [4, 11, 5]. Deep convolutional neuralnetworks (CNNs) have been successfully employed as surrogate models in dynamic multiphase flow problems [12, 10].
However, these methods have limitations, particularly in their dependency on lattice grid structures and the need forretraining for different mesh resolutions [14].
In response to these challenges, deep operator networks (DeepONets) have emerged as a promising solution. Introducedby Lu et al. [8], DeepONets leverage the universal approximation theorem for operators [1], offering a framework thatlearns mappings between infinite-dimensional Banach spaces. This approach enables the solution evaluation at any
arbitrary spatial and temporal location [7].
∗cnmlt@leeds.ac.uk
he_wang@ucl.ac.uk‡
sgoswam4@jhu.edu§
mario_de_florio@brown.edu¶
Jonathan.Nuttall@deltares.nl∥
X.Chen@leeds.ac.uk∗∗
P.K.Jimack@leeds.ac.uk†

International Conference on Scientific Computing and Machine Learning 2024 (SCML2024)

Efficient Groundwater Flow Modeling with DeepONet

This study introduces a deep neural operator-based approach as a surrogate model for groundwater flow, offering acomputationally efficient alternative to standard numerical methods. We aim to demonstrate its applicability throughvarious experimental setups, addressing both forward and inverse problems in subsurface flows.

2

Problem Statement and Methodology

The movement of groundwater is governed by a PDE that combines Darcy’s Law and the principle of mass conservation.
Traditional methods like the finite difference apporach based MODFLOW have been extensively used but face limitationsin handling large-scale, complex simulations efficiently [6, 3].
Deep neural operators learn complex, non-linear functions in spaces of unbounded dimensions, a concept discussed in[8] and [1], offering an alternative for groundwater level predictions. The traditioanl DeepONet framework comprises
two deep neural networks: the branch network and the trunk network. The branch network processes the input function,
denoted as u, at fixed sensor points {x1 , x2 , . . . , xm }. The trunk network encodes the spatiotemporal coordinatesζ = {xi , yi , ti }, at which the operator’s output is evaluated for loss calculation. This learning mechanism operates
under a general paradigm where the sensor points (xi mi=1 ) for u evaluations do not require uniform spacing, but theymust maintain consistency across evaluations. The branch network inputs [u(x1 ), u(x2 ), . . . , u(xm )]T and outputs[b1 , b2 , . . . , bq ]T ∈ Rq , whereas the Trunk Network inputs ζ and outputs [t1 , t2 , . . . , tq ]T ∈ Rq . The outputs of thesesubnetworks are integrated via a dot product, yielding the final output.P
An additional bias term b0 ∈ R is incorporatedq
to enhance expressiveness, resulting in the approximation G(u)(ζ) ≈ i=k bk tk + b0 . The trainable parameters θ are
optimized by minimizing a mean square error loss function.
The study comprises five computational experiments, each addressing a unique aspect of groundwater modeling:• E1: Learning the forward mapping from conductivity fields to hydraulic heads for a stationary well scenario.
• E2: Expanding to multiple input functions, including various well locations and varying conductivity fields.
• E3: Undertaking inverse modeling to estimate hydraulic conductivity from observed hydraulic heads.
• E4: Modeling time-dependent problems with changing pumping rates.
• E5: Solving nonlinear problems with head-dependent boundary conditions.
Each experiment demonstrates DeepONet’s adaptability and effectiveness in addressing various challenges in groundwater modeling. This work presents the results of the first three experiments. For details on the remaining experiments,
the readers are referred to our published work [13].

3Results

In this section, we demonstrate through experiments E1, E2 and E3 that DeepONet can be employed for approximatinga range of groundwater flow simulation problems, accurately and efficiently.
In Experiment E1, DeepONet is tasked to learn the mapping from hydraulic conductivity to the hydraulic head in a
scenario with a fixed well location. The model achieved a mean square error (MSE) of 1.8 × 10−5 during training
and 2.5 × 10−4 in testing over Ntrain = 1000 and Ntest = 200 samples respectively. Training took 283 seconds for24, 600 iterations. Figure 1 showcases the model’s capability to predict the hydraulic head around the well accurately
for a conductivity field unseen during training. DeepONet accurately predicts the pressure buildup and sharp increaseof the hydraulic head.

Figure 1: Hydraulic head predictions for two test cases with unseen conductivity fields in Experiment E1.
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Efficient Groundwater Flow Modeling with DeepONet

In Experiment E2, we addressed the limitations of the vanilla DeepONet in scenarios with varying well locations.
The vanilla architecture struggled to accurately capture sharp gradients near the pumping wells when the position of
the well was changed for each sample across training and testing dataset. This was evident as the model produced
either inaccurate predictions of the location of the pressure front or overly smoothed gradients near the source terms.
Multiple approaches to encode the well’s location in the branch network were explored, including Cartesian coordinatesand Gaussian functions centered at the well, but these methods did not yield satisfactory results. It became clear that
informing the trunk network about the well location was crucial for accurate learning. The incorporation of wellcoordinates into the trunk network input improved the model’s ability to localize the basis functions effectively aroundthe wells.
However, in more complex scenarios involving multiple wells or non-point features, this method proves impractical.
Consequently, we adopted a novel approach, illustrated in Figure 2, that interlinks the hidden layers of the branch andthe trunk networks. In this design, we encode the well location as a binary map, which is then concatenated to thehydraulic conductivity field in the branch network, and outputs from the branch network’s pooling layers are integrated
with those from each layer of the trunk network. The output from the branch network undergoes flattening and isprocessed through a dense neural network with a Sigmoid activation function. The resultant vector, interpretable as aset of coefficients, matches the dimensionality of the trunk network’s corresponding hidden layer, allowing for their
combination via an inner product. This combined output then advances through subsequent layers of both the trunk andbranch networks, following reshaping and concatenation with the pooling layer’s output. Our findings show that thisarchitecture provides precise predictions of the hydraulic head’s steep gradients in Experiment E2, an improvement
over the more generalized predictions of the vanilla DeepONet.

Figure 2: Schematic of the novel DeepONet architecture proposed for experiment E2.

Figure 3 demonstrates the results obtained using the proposed DeepONet architecture, where the model accurately
captures sharp gradients the well locations, that are typically situated at distinct locations for each training and testing
sample. The training process for this experiment took approximately 324 seconds, and the error metrics on the training
and testing datasets were 6.6 × 10−5 and 2.6 × 10−4 , respectively. Extended analyses included scenarios with two
randomly located pumping wells, as described in [13], where the proposed architecture continued to demonstratesuperior performance.
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Efficient Groundwater Flow Modeling with DeepONet

Figure 3: Hydraulic head predictions with varying well locations using vanilla and modified DeepONet architectures inExperiment E2.

Finally, Experiment E3 delves into solving inverse problems, where DeepONet estimates the conductivity field basedon known hydraulic head information. The operator network demonstrates trustworthy predictions, with a MSE test
error 1.46 × 10−2 and a standard deviation 5.3 × 10−3 , obtained by conducting five experiments. Incorporating 20% of
the hydraulic conductivity data as supplementary input led to a 29.5% reduction in the MSE error for test samples. This
improvement was even more pronounced, reaching 37.6%, with 40% of hydraulic conductivity data as additional input.
We also observed that providing the network with observations of hydraulic head at different time intervals further
refined the predictions.
Figure 4 displays a test case that highlights the model’s proficiency in predicting the conductivity field from hydraulichead data. Specifically, the first column shows the unknown hydraulic conductivity field and the location of sparse
observations used as additional input. The second column presents the input hydraulic head. The third column depicts
the outcomes of the simulated inverse procedure, while the final column illustrates the hydraulic head corresponding to
the predicted conductivity, as calculated using a conventional solver.
It is important to consider the ill-posed nature of this problem, particularly when based on sparse hydraulic headobservations. Multiple conductivity maps could lead to an identical set of sparse Hydraulic head observations, alikelihood that increases with greater sparsity. Therefore, the results should be interpreted within this context.

Figure 4: Inverse analysis in Experiment E3: Estimating conductivity fields from hydraulic head observations.

4

Summary and Discussion

The DeepONet framework has shown remarkable capabilities in simulating groundwater flow. Our experiments covered
scenarios with fixed and variable pumping well locations. The integration of the branch network with the trunk network
was crucial in predicting solutions for variable well locations. This novel approach allowed for greater flexibility andaccuracy, overcoming limitations in the model’s original design.
While the current study primarily focused on single and two-well configurations, future work aims to expand this tomultiple wells and to scenarios with varying abstraction rates, aquifer geometries, and interactions with surface water
systems. The integration of multi-fidelity and multi-modality data sources also presents an exciting frontier, considering
the varied and sparse nature of real-world data.
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Efficient Groundwater Flow Modeling with DeepONet

Acknowledgments
This work was supported by the Leeds-York-Hull Natural Environment Research Council (NERC) Doctoral Training
Partnership (DTP) Panorama under grant NE/S007458/1. We would like to acknowledge the support provided by
Deltares and we express our sincere gratitude to Bennie Minnema for his invaluable contribution to designing thenon-linear test case. Furthermore, we express our gratitude to Professor George Karniadakis for generously hosting thefirst author at Brown University within the CRUNCH group. His mentorship and insightful inputs have significantly
contributed to the research development and understanding in this study.

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