Soft causal constraints in groundwater machine learning: a new way to balance accuracy and physical consistency

Physics-informed machine learning (PIML) seeks to integrate scientific knowledge into conventional machine learning models to mitigate the black-box nature of the latter and prevent them from producing physically inconsistent results. Recently, Adombi et al. (2024) [a causal physics-informed deep learning formulation for groundwater flow modeling and climate change effect analysis] have shown that incorporating scientific knowledge into machine learning models is not enough to make them obey certain fundamental principles of physics, such as causality. They then derived certain constraints, called causal relationship constraints (CRC), to force PIML to obey the principle of causality. However, in some situations, CRC constraints in PIML prioritize the satisfaction of the principle of causality to the detriment of performance. In this study, we propose new CRC conditions and a new architecture for PIML, with the aim of testing the hypothesis that these conditions improve the performance of PIML models without transgressing the principle of causality. The models were tasked with simulating groundwater levels in six piezometers located in Quebec, Canada. A conventional machine learning model (convolutional neural network, 1D-CNN), a PIML model based on Adombi et al. (2024) (H-Lin) and a PIML model based on the architecture proposed in this work (H-LinC) were trained and subsequently compared. The results show that 1D-CNN outperforms H-LinC, which in turn outperforms H-Lin in terms of accuracy, with median NSE and KGE of 0.76 and 0.87 for 1D-CNN, 0.68 and 0.76 fir H-LinC, and 0.53 and 0.59 fir H-Lin. However, only H-LinC and H-Lin satisfy the principle of causality.