Benchmarking Evaluation of Physics-Informed Neural Networks for Computer Agriculture Soil-Nitrogen Transport Under Sparse Lysimeter Calibration: An Artificial Intelligence Mass-Balance Violation Bound for Fertilizer Leaching Regulation

Physics-informed neural networks (PINNs) promise to embed mass-conservation laws into data-driven soil-nitrogen transport models, yet the reliability of that promise under sparse lysimeter calibration is unknown. The framework is evaluated through a replication-based design study of the 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 at cluster counts of 8, 16, 24 and 32 and effect sizes of 3, 5 and 8 percentage points.54 clusters are required for 80% power at a 5-percentage-point effect, and it formalises the PINN mass-balance violation as a bounded functional of the calibration residual. The derived bound converts a network's training residual into a worst-case leaching-load error, providing regulators with an auditable threshold rather than a point prediction. The contribution is a benchmark protocol and a violation bound that together make PINN-based nitrogen models accountable to the mass-balance requirement that fertilizer regulation presupposes.