Discrete physics-informed neural networks for multi-species pesticide reaction–diffusion and uncertainty quantification in soil

High-fidelity simulations of pesticide transport and degradation in soil-column systems are computationally demanding, and continuous-residual physics-informed neural networks (PINNs) often struggle with stiff reaction–diffusion systems. Building on existing discrete-residual PINN ideas, this study develops and evaluates a dPINN workflow with a scheme-aligned discrete transport–reaction residual constructed on the same grid and time step as the FiPy finite-volume reference model for multi-species pesticide reaction–diffusion in a two-dimensional soil column. Backward-Euler time stepping and a five-point cell-centered Laplacian are embedded in the loss so that the surrogate is aligned with the grid, time stepping and reaction structure of the finite-volume benchmark and can be used for forward prediction, parameter inversion, and uncertainty propagation. Sensitivity analysis identifies the effective diffusion coefficient D e and the first two reaction rates K 1 and K 2 as dominant; when they are treated as trainable parameters, relative errors of order 1% are obtained from sparse synthetic data. Monte Carlo dPINN inversions with 5% noisy observations yield narrow, unimodal parameter distributions, a strong positive correlation between D e and K 1 , and narrow ± 2 σ prediction bands that cover most reference values at environmentally relevant locations. A four-way benchmark (MLP/KAN and continuous/discrete residuals) further confirms that residual discretization is the dominant factor governing accuracy for this stiff multi-species system. These results indicate that the discrete-residual PINN formulation approaches the finite-volume reference accuracy while providing a differentiable and identifiable surrogate for risk-oriented modelling and uncertainty quantification of pesticide pollution in soil.