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Physics-informed neural networks (PINN) have recently become attractive tosolve partial differential equations (PDEs) that describe physics laws. Byincluding PDE-based loss functions, physics laws such as mass balance areenforced softly. In this paper, we examine how well mass balance constraintsare satisfied when PINNs are used to solve the resulting PDEs. We investigatePINN's ability to solve the 1D saturated groundwater flow equations forhomogeneous and heterogeneous media and evaluate the local and global massbalance errors. We compare the obtained PINN's numerical solution andassociated mass balance errors against a two-point finite volume numericalmethod and the corresponding analytical solution. We also evaluate the accuracyof PINN to solve the 1D saturated groundwater flow equation with and withoutincorporating hydraulic head as training data. Our results showed that oneneeds to provide hydraulic head data to obtain an accurate solution for thehead for the heterogeneous case. That is, only after adding data at somecollocation points the PINN methodology was able to give a solution close tothe finite volume and analytical solutions. In addition, we demonstrate thateven after adding hydraulic head data at some collocation points, the resultinglocal mass balance errors are still significant at the other collocation pointscompared to the finite volume approach. Tuning the PINN's hyperparameters suchas the number of collocation points, epochs, and learning rate did not improvethe solution accuracy or the mass balance errors compared to the finite volumesolution. Mass balance errors may pose a significant challenge to PINN'sutility for applications where satisfying physical and mathematical propertiesis crucial.

Do Physics-informed Neural Networks Satisfy Local and Global Mass Balance?