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。

使用插值物理信息图神经网络 (InterPIGNN) 的美国陆地热地球模型

Thermal Earth Model for the Conterminous United States Using an  Interpolative Physics-Informed Graph Neural Network (InterPIGNN)

Physics-informed Graph Neural Networks have achieved remarkable performance
in learning through graph-structured data by mitigating common GNN challenges
such as over-smoothing, over-squashing, and heterophily adaption. Despite these
advancements, the development of a simple yet effective paradigm that
appropriately integrates previous methods for handling all these challenges is
still underway. In this paper, we draw an analogy between the propagation of
GNNs and particle systems in physics, proposing a model-agnostic enhancement
framework. This framework enriches the graph structure by introducing
additional nodes and rewiring connections with both positive and negative
weights, guided by node labeling information. We theoretically verify that GNNs
enhanced through our approach can effectively circumvent the over-smoothing
issue and exhibit robustness against over-squashing. Moreover, we conduct a
spectral analysis on the rewired graph to demonstrate that the corresponding
GNNs can fit both homophilic and heterophilic graphs. Empirical validations on
benchmarks for homophilic, heterophilic graphs, and long-term graph datasets
show that GNNs enhanced by our method significantly outperform their original
counterparts.

通过在图结构数据中学习，物理启发图神经网络在解决常见图神经网络挑战（如平滑过度、压缩过度和异质适应）方面取得了显著的性能。本文通过在传播过程中引入额外的节点和重新连接具有正负权重的连接来丰富图结构，在理论上验证了通过我们的方法增强的图神经网络可以有效地解决平滑过度问题，并对压缩过度具有鲁棒性。此外，我们对重连后的图进行谱分析，证明相应的图神经网络可以适应同质化和异质化的图。在同质化、异质化图以及长期图数据集的基准验证中，我们通过我们的方法增强的图神经网络显著优于原始版本。