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 (