A Conservative FV-Residual PINN Framework for Solute Transport through Subsurface Media with Dispersion Uncertainty for Data-Scarce Environments

Water quality monitoring in subsurface environments is often limited by sparse, irregular, and uncertain measurements, complicating the calibration process and reliability of transport models. In this study, we propose a Finite Volume (FV) residual Physics Informed Neural Network (PINN) framework for contaminant transport through subsurface media governed by the advection-dispersion equation (ADE), with a focus on generating predictions considering parameteric uncertainty for data-scarce environments. The core idea is to replace the strong-form PDE residual typically used in PINNs with a control-volume conservation imbalance derived from a discrete FV balance. Neural network predictions are used to evaluate advective and dispersive numerical fluxes at cell faces, and training minimizes the resulting cell-wise flux imbalance while enforcing initial and boundary conditions. This conservative formulation enables transport-specific numerical flux treatments (e.g., upwind/TVD advection and consistent boundary fluxes), and we assess performance for advection-dominated systems with sharp concentration fronts. 

To represent heterogeneity and uncertainty in dispersion, we parameterize the dispersion coefficient as a strictly positive random field using a low-dimensional basis. Uncertainty is propagated through the learned surrogate using Monte Carlo sampling to obtain prediction intervals and monitoring-relevant risk metrics such as threshold exceedance probabilities at selected locations. We outline two uncertainty workflows: (i) an ensemble strategy that trains FV-PINN models across sampled dispersion realizations, and (ii) a prospective conditional FV-PINN that takes random-field coefficients as additional inputs, enabling efficient Monte Carlo evaluation after a single training stage. The application of the methodology is demonstrated on simple benchmark examples designed to represent sparse monitoring data, showing how conservative learning and random-field uncertainty propagation can support reliable transport predictions when observations are limited.