This study presents a data-driven spatial interpolation algorithm based on
physics-informed graph neural networks used to develop national
temperature-at-depth maps for the conterminous United States. The model was
trained to approximately satisfy the three-dimensional heat conduction law by
simultaneously predicting subsurface temperature, surface heat flow, and rock
thermal conductivity. In addition to bottomhole temperature measurements, we
incorporated other physical quantities as model inputs, such as depth,
geographic coordinates, elevation, sediment thickness, magnetic anomaly,
gravity anomaly, gamma-ray flux of radioactive elements, seismicity, and
electric conductivity. We constructed surface heat flow, and temperature and
thermal conductivity predictions for depths of 0-7 km at an interval of 1 km
with spatial resolution of 18 km$^2$ per grid cell. Our model showed superior
temperature, surface heat flow and thermal conductivity mean absolute errors of
4.8{\deg} C, 5.817 mW/m$^2$ and 0.022 W/(C-m)$, respectively. The predictions
were visualized in two-dimensional spatial maps across the modeled depths. This
thorough modeling of the Earth's thermal processes is crucial to understanding
subsurface phenomena and exploiting natural underground resources.

该研究基于数据驱动的空间插值算法，利用物理信息图神经网络开发了美国本土深度温度图。该模型通过同时预测地下温度、地表热流和岩石导热率，近似满足三维热传导定律。在模型输入中，除了底孔温度测量值外，还包括其他物理量，如深度、地理坐标、海拔、沉积厚度、磁异常、重力异常、放射性元素的伽马射线通量、地震活动和电导率等。我们构建了 0-7 km 深度范围内以 1 km 间隔的表面热流和温度、导热率预测值，每个网格单元的空间分辨率为 18 km²。我们的模型显示了温度、地表热流和导热率的平均绝对误差分别为 4.8°C、5.817 mW/m² 和 0.022 W/(°C-m)。预测结果在模拟深度上的二维空间地图中可视化。这种对地球热过程的全面建模对于理解地下现象和开发地下自然资源至关重要。