Solving groundwater flow equation using physics-informed neural networks

In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained wide popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning frameworks for solving forward and inverse problems with non-linear PDEs. Recently, PINNs have shown promising results in different application domains. In this paper, we approach the groundwater flow equations numerically by searching for the unknown hydraulic head. Since singular terms in differential equations are very challenging from a numerical point of view, we approximate the Dirac distribution by different regularization terms. Furthermore, from a computational point of view, this study investigate how a PINN can solve higher-dimensional flow equations. In particular, we analyze the approximation error for one and two-dimensional cases in a statistical learning framework. The numerical experiments discussed include one and two-dimensional cases of a single or multiple pumping well in an infinite aquifer, demonstrating the effectiveness of this approach in the hydrology application domain. Lastly, we compare our results with the Finite Difference Method (FDM) to emphasize the several advantages of PINNs in solving PDEs without the need for discretization.