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

arXiv: https://arxiv.org/abs/2403.09961 · HTML: https://mlsanity.com/paper/2403.09961/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 130 references that are in the map.
- First posted on arXiv 15 Mar 2024.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Relation-Shape Convolutional Neural Network for Point Cloud Analysis](https://mlsanity.com/paper/1904.07601/) (2019)
- [Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs](https://mlsanity.com/paper/1704.02901/) (2017)
- [PointNetLK: Robust & Efficient Point Cloud Registration using PointNet](https://mlsanity.com/paper/1903.05711/) (2019)
- [PPFNet: Global Context Aware Local Features for Robust 3D Point Matching](https://mlsanity.com/paper/1802.02669/) (2018)
- [The Perfect Match: 3D Point Cloud Matching with Smoothed Densities](https://mlsanity.com/paper/1811.06879/) (2018)
- [PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors](https://mlsanity.com/paper/1808.10322/) (2018)
- [Pointwise Convolutional Neural Networks](https://mlsanity.com/paper/1712.05245/) (2017)
- [DropGNN: Random Dropouts Increase the Expressiveness of Graph Neural Networks](https://mlsanity.com/paper/2111.06283/) (2021)
- [DensePoint: Learning Densely Contextual Representation for Efficient Point Cloud Processing](https://mlsanity.com/paper/1909.03669/) (2019)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Notes on the Behavior of MC Dropout](https://mlsanity.com/paper/2008.02627/) (2020)

## Built on by (most cited first, 2 of 2)

- [Tackling extreme urban heat: a machine learning approach to assess the impacts of climate change and the efficacy of climate adaptation strategies in urban microclimates](https://mlsanity.com/paper/2411.05952/) (2024)
- [In-context learning enables continental-scale subsurface temperature prediction from sparse local observations](https://mlsanity.com/paper/2605.16665/) (2026)

## Abstract (arXiv)

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 km2 per grid cell. Our model showed superior temperature, surface heat flow and thermal conductivity mean absolute errors of 4.8deg C, 5.817 mW/m2 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.