cs.LGMar 26, 2026

Pure and physics-guided deep learning approaches for spatio-temporal groundwater level prediction

Authors: Matteo Salis, Gabriele Sartor, Rosa Meo, Stefano Ferraris, Abdourrahmane M. Atto

Organizations: Computer Science Department, University of Turin, Turin, Italy · LISTIC Laboratory, Université Savoie Mont Blanc, Annecy-le-vieux, France · Interuniversity Department of Regional and Urban Studies and Planning, Politecnico di Torino and University of Turin, Turin, Italy

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Abstract

Groundwater represents a key element of the water cycle, yet it exhibits complex and context-dependent relationships that make its modeling challenging. Theory-based models have been the cornerstone of scientific understanding. However, their computational cost, simplifying assumptions, and calibration requirements limit their use. In recent years, data-driven models have emerged as powerful alternatives. In particular, deep learning has proven to be a promising approach for its design flexibility and ability to learn complex relationships directly from the data without requiring extensive domain information. We proposed an attention-based pure deep learning model, named STAINet, to predict weekly groundwater levels in Piedmont (Italy), leveraging both irregular groundwater time series and weather image sequences. To enhance the model's trustworthiness and generalization ability, we merged the theory and data-driven approaches by considering physics-guided strategies to inject the groundwater flow equation into the model. Firstly, we restructured the tail of the architecture to predict the three terms of the governing equation, named the autoregressive, diffusion, and residual components - we thus obtained the PSTAINet-IB. Then, we further injected physics priors by adding loss terms related to the estimated equation components, obtaining the PSTAINet-ILB model. Lastly, we developed the PSTAINet-ILRB by imposing a loss term specific to the residual component, which forces the groundwater recharge to occur within the groundwater body recharge zone, which is identified by domain experts. The models were evaluated both by feeding true lagged values as input and by iterating their own predictions (rollouts) over the whole test set. The PSTAINet-ILB model performed the best, achieving remarkable test performance, and generating equation components in line with domain experts' expectations.

Figures & tables

Figure 1 : On the left, the Piedmont region with the ROI box and the selected piezometers with a colormap representing the elevation; bold names represent Piedmont’s Provinces. On the right, a zoom over the ROI with an additional colormap representing the mean groundwater levels; bold names represent cities.
Figure 2 : Time series of the sensors in a) Cuneo, b) Racconigi, c) Scalenghe, and d) Vottignasco.
Figure 3 : The adopted modeling pipeline. Each of our models takes as input the weather and groundwater data from a window of length T+1\mathcal{T}+1 and T\mathcal{T} , respectively, from the target time t∗t^{*} . The predictions are compared with true measurement data, and through backpropagation, the weights of the model are updated to lower such differences. What differs between models is whether or not the physics knowledge is leveraged (dashed lines). This could be done by applying an inductive bias or a learning bias that acts on the model architecture and on the learning algorithm (backpropagation), respectively.
Figure 4 : a) Multi-Head Attention block adopted in the models. b) STSA Module c) STAI Module that takes as values the hidden representation of the input points’ value embeddings (ST-VEmb), as keys the input points’ spatio-temporal coordinates embeddings (ST-CEmb), and as queries the prediction points’ spatio-temporal coordinates embeddings (ST-CEmb).
Figure 5 : STAINet architecture.
Figure 6 : a) D\mathcal{D} Module in which Act & Norm stand for activation and normalization respectively; b) ΔGW\Delta_{GW} Module.
Figure 7 : PSTAINet-IB architecture.
Modelαortho\alpha_{\text{ortho}}αcoh\alpha_{\text{coh}}αdiff\alpha_{\text{diff}}α∣∣R∣∣1\alpha_{||\mathcal{R}||_{1}}α∣∣R∣∣2\mathbf{\alpha_{||\mathcal{R}||_{2}}}αRCH\alpha_{\text{RCH}}
STAINet10−310^{-3}-----
PSTAINet-IB10−310^{-3}-----
PSTAINet-ILB10−310^{-3}112.5⋅10−22.5\cdot 10^{-2}5⋅10−45\cdot 10^{-4}10−410^{-4}-
PSTAINet-ILRB10−310^{-3}112.5⋅10−22.5\cdot 10^{-2}5⋅10−45\cdot 10^{-4}10−410^{-4}5⋅10−45\cdot 10^{-4}
Table 1 : Loss terms weights.
 STAINetPSTAINet-IBPSTAINet-ILBPSTAINet-ILRB
 True DataRolloutTrue DataRolloutTrue DataRolloutTrue DataRollout
NBIAS0.10420.17130.05940.0862-0.00930.01240.01770.0685
(0.2297)(0.2297)(0.1741)(0.2028)(0.1137)(0.1351)(0.2212)(0.1636)
RMSE [m]0.83511.02920.62960.73360.50580.58510.65210.6783
(0.8635)(1.2064)(0.8175)(0.9495)(0.3031)(0.4485)(0.5347)(0.8296)
MAPE [%]0.28340.28680.23190.26060.14790.16040.19000.1953
Table 2 : Median performance metrics on the test set feeding true data (True Data) and in the rollout setting (Rollout). Numbers in parentheses represent the Interquartile Range (IQR).
Figure 8 : Violin plots representing MAPE values for the proposed models on the test set. Q1 , Q2 , Q3 represent the first, second (median), and third quartiles, respectively. The shapes around the box plots are the KDE density estimates of the MAPE values over the sensors.
Figure 11
Figure 9 : STAINet, PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB predictions on the test set feeding true data in a) , c) and in the rollout setting b) , d) in Racconigi and Scalenghe sensors.
Figure 10 : STAINet, PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB groundwater level prediction maps at 1.51.5 km/pixel for 2022-07-17.
Figure 11 : Predicted equation components by PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB for the weeks starting on 2022-07-17 ( a ), and 2022-11-13 ( b ). The first column reports Δ^GWt∗\hat{\Delta}_{GW_{t^{*}}} , the second R^t∗\hat{\mathcal{R}}_{t^{*}} , and the last D^\hat{\mathcal{D}} . Green lines identify the recharge zones. For Δ^GWt∗\hat{\Delta}_{GW_{t^{*}}} , a blue pixel means inflow while a red one means outflow; for R^t∗\hat{\mathcal{R}}_{t^{*}} , a blue pixel means recharge while red represents a loss.
 NBIASRMSE [m]MAPE [%]NSEKGE
PSTAINet-ILB-0.02780.44900.14040.63140.7547
Table 3 : PSTAINet-ILB time series reconstruction median Metrics, in the rollout setting with a 26 time step forecast horizon.
Figure 12 : PSTAINet-ILB time series reconstruction in the rollout setting with a 26 time step forecast horizon for a) Racconigi and b) Scalenghe piezometers.
Figure 13 : Loss plots a) training L\mathcal{L} , b) test Ldata\mathcal{L}_{\text{{data}}} , c) training Lcoh\mathcal{L}_{\text{{coh}}} , d) training Ldiff\mathcal{L}_{\text{{diff}}} , e) training LRCH\mathcal{L}_{RCH} .
Figure 14 : Correlation plots between MAPE and missing values percentage for STAINet, PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB; in all plots, the red line is the bisector, r is Pearson’s correlation coefficient, and pvalue is the corresponding p-value.
Figure 15 : Monthly average MAPE over all sensors in the rollout setting on the test set. Vertical lines represent the corresponding monthly standard deviations.
Figure 16 : Correlation plots between MAPE and Trend Strength ST\mathcal{S}_{T} for STAINet, PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB; in all plots, the red line is the bisector, r is Pearson’s correlation coefficient, and pvalue is the corresponding p-value.
Figure 17 : Correlation plots between KGE and Trend Strength ST\mathcal{S}_{T} for STAINet, PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB; in all plots, the red line is the bisector, r is Pearson’s correlation coefficient, and pvalue is the corresponding p-value.
Figure 18 : Correlation plots between MAPE and Seasonal Strength SS\mathcal{S}_{S} for STAINet, PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB; in all plots, the red line is the bisector, r is Pearson’s correlation coefficient, and pvalue is the corresponding p-value.
Figure 19 : Correlation plots between KGE and Seasonal Strength SS\mathcal{S}_{S} for STAINet, PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB; in all plots, the red line is the bisector, r is Pearson’s correlation coefficient, and pvalue is the corresponding p-value.
Figure 20 : NBIAS for all sensors in the rollout setting on the test set.
Figure 21 : MAPE for all sensors in the rollout setting on the test set.
Figure 22 : Box plots representing predictions for Δ^GWt∗\hat{\Delta}_{GW_{t^{*}}} ( a) ) and R^t∗\hat{\mathcal{R}}_{t^{*}} ( b ) generated by PSTAINet-IB, PSTAINet-ILB, and PSTAINet-ILRB. The star in each box plot represents the mean, while the horizontal line inside the box is the median.
Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1 : Bar plot representing the percentage of missing data for each sensor considered in the study. In the x-axis, sensors are labeled using both the municipality where they are placed and the identification code (sensor ID).
Figure S2 : Bar plot representing the number of observations for each sensor in the training, validation, and test sets (also reported by the vertical numbers for each bar, respectively). In the x-axis, sensors are labeled using both the municipality where they are placed and the identification code (sensor ID).
Figure 29
Figure 30
Figure S3 : Groundwater level time series.
Figure 32
Figure 33
Figure S4 : Test set predictions in the rollout setting.
Figure 35
Figure 36
Figure 37
Figure S5 : Test set predictions in the rollout setting.

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