Physics-Informed Neural Network Surrogates for Agriculture Drainage Design Under Uncertain Soil Permeability: A Computer Calibration-Reliability Evaluation

Physics-informed neural networks (PINNs) promise fast surrogates for the Richards equation in agricultural drainage design, but their reliability under uncertain soil permeability is rarely quantified. The framework separates epistemic calibration error from aleatoric permeability uncertainty and derives a reliability bound on the surrogate's predicted water-table drawdown. A formal analysis establishes conditions under which the PINN residual loss bounds the expected design error, and a variance decomposition identifies the intraclass correlation of the permeability field as the dominant driver of surrogate reliability. The analysis is theoretical and methodological: no observational data are used or claimed. The framework is operationalised through a design-based evaluation that characterises the operating characteristics of a before-after-control-impact estimator under a declared variance structure (between-cluster SD = 6.0, residual SD = 4.5, intraclass correlation = 0.64), with power estimated across 2000 design cells. The results indicate that 54 clusters are required for 80% power at a 5 percentage-point effect, and that the intraclass correlation of 0.64 inflates the minimum detectable effect substantially at small cluster counts. The contribution is a reliability-aware calibration protocol that lets drainage engineers decide when a PINN surrogate is trustworthy enough to replace a numerical solver in design loops.