Operator-Splitting Error Bounds for Physics-Informed Neural Emulation of Coupled Land–Atmosphere Fluxes Under Diurnal Forcing: Convergence Guarantees and Drought-Onset Advisory Reliability

Physics-informed neural networks (PINNs) that emulate coupled land–atmosphere fluxes must reconcile two disparate time scales: the stiff, sub-hourly relaxation of the surface energy balance and the slow, multi-day accumulation of moisture deficits that underpins drought-onset advisories. Operator splitting offers a principled route to this reconciliation, but its interaction with neural-network approximation error has not been bounded in a form usable by advisory systems. the present analysis derives a posteriori error bounds for a Lie–Trotter splitting of the coupled surface energy and moisture budget equations when each sub-step is emulated by a physics-informed network. The analysis separates splitting error, network approximation error and accumulated forcing error, and shows that the drought-onset advisory reliability is governed by the accumulated moisture residual rather than by instantaneous flux accuracy. The bounds are stated as explicit inequalities involving the Lipschitz constants of the flux parameterisations and the network training residual, and they yield a certified advisory horizon beyond which a drought-onset warning cannot be trusted. The framework is illustrated with a reanalysis-derived temperature record for Lagos, Nigeria (NASA POWER MERRA-2, 2000–2023, 312 monthly observations; mean 26.414 °C, standard deviation 0.943 °C, warming trend 0.1553 °C per decade), which supplies the diurnal-forcing envelope within which the bounds are evaluated. The contribution is a certified reliability criterion for drought-onset advisories built on split PINN emulators, together with the conditions under which that criterion can be tightened by refining the moisture sub-step alone.