Learning with constraints: A physics-guided Bayesian framework for reliable geothermal temperature mapping in northeastern Italy

Reliable deep-temperature estimation remains a central challenge in geothermal exploration, particularly in sedimentary–carbonate provinces where borehole data are sparse and vertically imbalanced. These conditions lead to unreliable extrapolation and poor uncertainty characterization in classical interpolation and purely data-driven models. This study develops a physics-guided Bayesian neural network (BNN) framework that integrates heteroscedastic probabilistic learning with conductive heat-transfer constraints to reconstruct the 3D geothermal structure of the Lower Friulian Plain (NE Italy) using 849 irregularly distributed temperature measurements. To explicitly address depth imbalance and evaluate extrapolation performance, a depth-aware training and validation strategy was implemented, including controlled withholding of deep observations (1000–1200 m). The architecture combines Monte Carlo-dropout Bayesian inference with two physics-informed formulations, a monotonicity constraint enforcing non-negative vertical gradients and a gradient-band constraint bounding geothermal slopes within regionally plausible conductive ranges. The resulting models achieve high fidelity in the data-supported domain (<1000 m), with coefficient of determination (R 2 ) ≈ 0.95–0.96 and root mean square error ≈ 2.6–2.8 °C, while classical and data-driven models show reduced reliability under extrapolation conditions. The monotonic BNN attains R 2 = 0.877 and RMSE = 1.33 °C, and the gradient-band BNN produces physically consistent geothermal gradients (≈0.014–0.018 °C/m), aligning with regional heat-flow constraints. Both physics-guided models deliver near-ideal uncertainty calibration (≈97–99% prediction-interval coverage) and eliminate nonphysical deep-zone oscillations. Independent validation against the Grado-1 well confirms agreement within ±1 °C across the full 100–1200 m profile. The proposed framework enables extrapolation-robust and uncertainty-aware 3D geothermal mapping, providing a transferable approach for geothermal resource assessment in data-limited subsurface systems.