Sample-Efficient Neural Operator Surrogates for Artificial Intelligence Climate Downscaling Under Sparse Gauge Networks: Error Bounds and Localized Flood-Warning Lead Time

Neural operator surrogates promise to replace expensive dynamical downscaling with fast, differentiable maps from coarse atmospheric fields to local climate variables, but their reliability under sparse gauge networks remains poorly characterised. The analysis treats the surrogate as a mapping between function spaces and derives an explicit error decomposition into truncation, sampling and optimisation terms. Using a reanalysis record for Harare, Zimbabwe (2000–2023, 312 monthly observations, mean temperature 18.914 °C, warming trend 0.2001 °C per decade), the framework is instantiated to show how gauge sparsity inflates the sampling term and how that inflation translates into a quantifiable reduction in the lead time at which a warning threshold is crossed. The central result is a closed-form expression linking gauge density, operator truncation rank and warning lead time, which permits network designers to trade sensor count against forecast horizon before deployment.