Gray-Box Machine Learning Framework for Extracting Groundwater–Irrigation Response Functions and Inverting Hydrogeological Parameters

Groundwater-fed irrigation sustains global food production but drives chronic aquifer depletion, creating an urgent need for quantitative tools that link irrigation intensity to groundwater response. This study proposes a gray-box machine learning (ML) framework that learns the parametric coefficients of polynomial irrigation–groundwater response functions—rather than predicting state variables directly—thereby embedding physical interpretability into the ML output. Using a well-validated SWAT-GW model of a representative over-exploited piedmont plain in the North China Plain as the training data generator, gradient irrigation scenarios were constructed for 70 hydrological response units over 20 years, producing 21,000 paired records of winter-wheat irrigation intensity versus three groundwater response variables: vertical recharge, aquifer storage change, and water table depth change. Quadratic polynomials were identified as the optimal functional form through joint evaluation of fitting accuracy (R2 > 0.994) and ML learnability. Ensemble boosting algorithms predicted the three quadratic coefficients, with R2 ranging from 0.74 to 0.97, and retained acceptable accuracy even when input features were restricted to readily available meteorological and soil data. Four management-critical hydrogeological parameters—the precipitation infiltration coefficient (α), irrigation infiltration coefficient (β), natural recharge (R_nat), and recharge–abstraction equilibrium point (IRR_eq)—were successfully inverted from the predicted coefficients and validated against independent regional groundwater resource assessments. The SHapley Additive exPlanations and Causal Forest analyses confirmed that the learned relationships are governed by physically interpretable drivers. The framework advances groundwater machine learning from state-variable prediction toward functional-structure extraction, offering a transferable approach for deriving irrigation–groundwater response curves and sustainability thresholds in over-exploited aquifer systems.