Physics-informed neural network surrogate for reactive nitrate transport in groundwater

Predicting nitrate ( ) attenuation in groundwater requires solving the advection–dispersion–reaction dynamics governing donor limitation, nonlinear kinetics, and competition among electron acceptors. This study develops a physics-informed neural network (PINN) surrogate for reactive nitrate transport in saturated porous media, embedding the advection–dispersion–reaction equation, boundary conditions, non-negativity, and a redox-ordering constraint directly within the loss function. Four one-dimensional benchmarks of increasing geochemical complexity were constructed with PHREEQC , spanning linear denitrification, dual-linear redox competition, dual-substrate Monod kinetics, and fully coupled dual-Monod nitrate–Fe(III) reduction. The PINN was trained on full space–time fields using stochastic minibatches under a simulation-wise train–test split to avoid information leakage. Across all benchmarks, the surrogate reproduced the reference fields with high accuracy (test RMSE ∼ 1 ⋅ 1 0 − 5 mol kg w − 1 ; R 2 ≥ 0 . 99 in nonlinear cases) while maintaining ADR residuals, boundary condition violations, and non-negativity penalties several orders of magnitude below characteristic concentration scales. The dual-Monod regime, representing the strongest kinetic nonlinearity, was captured with robust fidelity, including DOC-limited tailing and nitrate–Fe(III) competition. Uncertainty quantification using a 50-member ensemble with split-conformal calibration produced sharp, statistically valid prediction intervals whose width increased only near reactive fronts, consistent with the underlying nonlinear sensitivity. These results demonstrate that PINNs can serve as accurate, physically consistent surrogates for geochemical reactive transport and offer a computationally efficient pathway for scenario analysis, uncertainty propagation, and incorporation into groundwater management workflows.