Groundwater inverse modeling: Physics-informed neural network with disentangled constraints and errors

This study combines a physics-informed neural network (PINN) and Karhunen-Loeve expansion (KLE) (i.e., KLE-PINN) to solve the groundwater inverse problem. The hydraulic head (u) distribution is approximated by a deep neural network (DNN), while the hydraulic conductivity (K) field is constructed by KLE with given prior geostatistical information. KLE-PINN is applied to investigate the inversion using data from a single pumping test, natural gradient flow (NG), and hydraulic tomography (HT). The results from these cases demonstrate that our inverse method is robust. Our error analysis endeavors to quantify the sources of error by using two custom reference indicators, e_forward and e_coupling. Moreover, the study finds that the inversion using data from multiple pumping tests (HT) yields more accurate estimates, leads to faster training convergence, and maintains higher stability. In addition, by investigating cases with different outer boundary conditions (BCs), we find that KLE-PINN is more flexible. Precisely, in scenarios with missing BCs, our network still fits well with the observed data, and the estimates capture the approximate spatial pattern in the region where the observation wells are distributed. Even with incorrect BCs, our network still performs well because the observational data constraints are strongly enforced during training.