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This research advances groundwater modeling by integrating deep learning architectures with physics-informed constraints to improve prediction reliability and generalization. By embedding the governing groundwater flow equation directly into the neural network's loss function, the study bridges the gap between data-driven flexibility and scientific rigor. The findings demonstrate that hybrid models can outperform pure data-driven approaches while maintaining alignment with domain expert expectations.
Pure and physics-guided deep learning approaches for spatio-temporal groundwater level prediction
arXiv:2603.25779v2 Announce Type: replace 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.
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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2603.25779 [cs.LG]
(or arXiv:2603.25779v2 [cs.LG] for this version)
Salis, M., Sartor, G., Meo, R., Ferraris, S., and Atto, A. M. Pure and physics-guided deep learning approaches for spatio-temporal groundwater level prediction. Machine Learning: Science and Technology, 7(5):055016, 2026
https://doi.org/10.1088/2632-2153/ae9af0 DOI(s) linking to related resources
Full text under the publisher's open licence · 316 words
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