Mapping spatiotemporal soil temperature using physics-informed neural networks with the heat transfer equation

Soil temperature (ST) plays a critical role in regulating biogeochemical processes in soils, yet its spatiotemporal mapping remains a challenge. Existing ST maps are largely derived from land-surface models and are typically available only at coarse spatial resolutions, limiting their usefulness for agricultural applications. This study aims to predict daily ST at 10-cm depth intervals from the surface to 120 cm and to generate maps at 80 m spatial resolution. We developed a differentiable model (DM) that integrates neural networks (NN) estimated surface soil temperature (SST) with a physics-based heat-transfer model. The model was trained and validated using three years of spatiotemporal ST observations from 33 monitoring sites across Tasmania, Australia (25,191 records). Our results show that the DM improved prediction accuracy relative to the purely physics-based model (PBM) and enhanced the stability and generalisability of the fully data-driven NN. Compared with fully data-driven NN models, the DM reduced prediction error by up to 25%, exhibited low variability, and performed well in soils with high soil organic carbon. By explicitly estimating SST, the DM provided a mathematically optimised upper boundary condition, resulting in smoother, more physically consistent ST profiles across depth and more reliable seasonal spatial patterns than the NN models. The SST estimates also showed coherent temporal behaviour, with lower day-to-day variability and warmer summer conditions than air temperature. Furthermore, our findings demonstrate that coupling machine learning with process-based heat transfer can improve the accuracy, physical realism, and scalability of soil temperature prediction.