Physics-informed conditional generative adversarial network for hydraulic conductivity inversion study

Physics-informed neural network (PINN) models have been widely applied to forward groundwater modeling problems, such as groundwater head and flow simulations. However, for hydrogeological parameter inversion, the performance of standalone PINN model is often limited by sparse observations and the well-known issue of equifinality, whereby different parameter combinations can produce similar hydraulic responses. These limitations introduce substantial uncertainty into inversion results. To address these challenges and improve the interpretability of hydrogeological parameter inversion, this study integrated PINN with conditional generative adversarial network (CGAN) to develop a physics-informed conditional generative adversarial network (PICGAN) framework. A two-dimensional heterogeneous transient arithmetic model was designed to evaluate the applicability and performance of the proposed model. Under the joint constraints of physical conditions and observed groundwater level, the discriminator of the PICGAN model continuously required the generator to update the global hydrogeological parameter field more in line with the reality until the convergence criteria of the generator and the discriminator were satisfied. At this point, it could be considered that the PICGAN model has completed the inverse solution of the hydraulic conductivity field of the heterogeneous transient confined aquifer, and could also simultaneously simulate and predict the water level. The results show that in the case simulation with 5% sampling rate, the root-mean-square error of water head simulated by the PICGAN model could be stabilized at about 0.95 m, with an accuracy of 89%. The root-mean-square error of hydraulic conductivity field could be stabilized at about 0.69 m/d, with an accuracy of up to 95%, and the distribution form was highly consistent with the reference field. As the sampling rate increased, the global error of the model would be further reduced. After the sampling rate reached 10% or more, the inversion error of the global hydraulic conductivity was reduced to 0.35 m/d, with an accuracy of 97%. In conclusion, the PICGAN model proposed in this study can provide a novel and effective method for bidirectional solution of groundwater problems under small sample conditions, especially for the inversion of heterogeneous hydrogeological parameter fields.