Modeling Transient Flow in Heterogeneous Aquifers With the Mixed Pressure‐Velocity Formulation of Physics Informed Neural Networks

Physics‐Informed Neural Networks (PINNs) have emerged as a powerful framework for modeling groundwater flow using deep learning neural networks, particularly in scenarios where traditional data‐driven approaches are limited by the scarcity of data. Despite their promise, PINNs often encounter significant challenges when applied to systems characterized by strong heterogeneity or discontinuous material properties. Furthermore, PINNs require a large training time when used for simulating time‐dependent processes. The primary objective of this study is to develop a robust implementation of PINNs for modeling transient groundwater flow in heterogeneous unconfined aquifers. The proposed implementation addresses key limitations of conventional PINNs by adopting the mixed pressure‐velocity formulation of the governing equations and coupling this formulation with a discrete‐time approach based on a higher‐order Runge‐Kutta method. The newly developed PINN implementation is used to simulate different scenarios of groundwater flow in unconfined aquifers based on a small‐scale hypothetical test case and a field inspired case in Eastern France. Its accuracy is evaluated by comparing its results to standard PINNs with the finite element solutions serving as a reference and its efficiency is assessed by comparing its training time and memory requirements to standard PINNs. The new PINN implementation demonstrates a strong capability for simulating transient flow in aquifers with diverse and complex heterogeneities, including sharp discontinuities and abrupt transitions in hydraulic conductivity. Its performance becomes increasingly advantageous as the problem complexity increases. This new PINN approach provides a more reliable, efficient, and physically consistent means of applying deep learning to subsurface flow modeling.