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

Mohammad J. Aljubran, Roland N. Horne

TL;DR

The paper tackles the lack of nationwide, high-resolution temperature-at-depth maps by introducing InterPIGNN, a physics-informed graph neural network that interpolates subsurface temperature, surface heat flow, and rock thermal conductivity across the conterminous US. By combining a novel interpolative graph module with three PDE-consistent GNNs and a suite of physics-informed losses, the approach achieves accurate predictions from 0 to 7 km depth on an 18 km^2 grid and provides uncertainty estimates via Monte Carlo Dropout and feature-attribution via Integrated Gradients. Compared to prior regional models and baselines, InterPIGNN delivers lower mean absolute errors (temperature: 4.8°C; heat flow: 5.817 mW/m^2; conductivity: 0.022 W/(C·m)) and produces physically plausible depth profiles while highlighting the most influential input features. The work advances geothermal resource assessment and subsurface engineering by delivering a scalable, data-integrated, and interpretable thermal Earth model for the United States.

Abstract

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° 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.

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

TL;DR

The paper tackles the lack of nationwide, high-resolution temperature-at-depth maps by introducing InterPIGNN, a physics-informed graph neural network that interpolates subsurface temperature, surface heat flow, and rock thermal conductivity across the conterminous US. By combining a novel interpolative graph module with three PDE-consistent GNNs and a suite of physics-informed losses, the approach achieves accurate predictions from 0 to 7 km depth on an 18 km^2 grid and provides uncertainty estimates via Monte Carlo Dropout and feature-attribution via Integrated Gradients. Compared to prior regional models and baselines, InterPIGNN delivers lower mean absolute errors (temperature: 4.8°C; heat flow: 5.817 mW/m^2; conductivity: 0.022 W/(C·m)) and produces physically plausible depth profiles while highlighting the most influential input features. The work advances geothermal resource assessment and subsurface engineering by delivering a scalable, data-integrated, and interpretable thermal Earth model for the United States.

Abstract

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 per grid cell. Our model showed superior temperature, surface heat flow and thermal conductivity mean absolute errors of 4.8° C, 5.817 mW/m 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.
Paper Structure (15 sections, 11 equations, 33 figures)

This paper contains 15 sections, 11 equations, 33 figures.

Table of Contents

  1. Introduction
  2. Data Collection
  3. Thermal Quantities
  4. Elevation and Sediment Thickness
  5. Magnetic and Gravity Anomalies
  6. Radioactive Elements
  7. Seismicity and Electric Conductivity
  8. Methods
  9. Interpolative Convolutional Module
  10. Three-Dimensional Heat Conduction
  11. Physics-Informed Graph Neural Networks
  12. Uncertainty Quantification
  13. Prediction Explainability
  14. Results
  15. Conclusion

Figures (33)

  • Figure 1: Log-scale plot of BHT record count per data source.
  • Figure 2: BHT records projected on the conterminous US, colored based on data source.
  • Figure 3: BHT records projected on the conterminous US, colored based on magnitude. Note that we set 250$^{\circ} \ C$ as an upper threshold for visual convenience.
  • Figure 4: Heat flow records for the conterminous US, colored based on magnitude. Note that we set $300 \ mW/m^2$ as an upper threshold for visual convenience.
  • Figure 5: Thermal conductivity records for the conterminous US, colored based on magnitude.
  • ...and 28 more figures