Physics-Informed Neural Networks for Groundwater Flow Modeling with Uncertainty Quantification

Abstract

Physics-Informed Neural Networks (PINNs) offer a novel paradigm for solving partial differential equations governing groundwater flow by embedding physical constraints directly into neural network training. This work presents a comprehensive framework for applying PINNs to the groundwater equation, demonstrating both forward simulation and uncertainty quantification capabilities. We compare three uncertainty quantification methods: ensemble learning, Monte Carlo Dropout, and residualbased approaches. Results show that PINNs can effectively solve transient groundwater problems while providing reliable uncertainty estimates, with ensemble methods achieving high predictive accuracy (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R^{2}=0.9933$</tex>) and mean uncertainty of 0.1478 m. The framework demonstrates computational efficiency and flexibility for hydrogeological applications.