# Resolving Extreme Data Scarcity by Explicit Physics Integration: An Application to Groundwater Heat Transport

arXiv: https://arxiv.org/abs/2507.06062 · HTML: https://mlsanity.com/paper/2507.06062/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 8 Jul 2025.

## Builds on (most cited first, 17 of 17)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [UNet++: A Nested U-Net Architecture for Medical Image Segmentation](https://mlsanity.com/paper/1807.10165/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [How Much Position Information Do Convolutional Neural Networks Encode?](https://mlsanity.com/paper/2001.08248/) (2020)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [Physics-guided Convolutional Neural Network (PhyCNN) for Data-driven Seismic Response Modeling](https://mlsanity.com/paper/1909.08118/) (2019)
- [Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems](https://mlsanity.com/paper/2308.07051/) (2023)
- [Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networks](https://mlsanity.com/paper/2308.02137/) (2023)
- [DDU-Net: A Domain Decomposition-Based CNN for High-Resolution Image Segmentation on Multiple GPUs](https://mlsanity.com/paper/2407.21266/) (2024)
- [A Staged Deep Learning Approach to Spatial Refinement in 3D Temporal Atmospheric Transport](https://mlsanity.com/paper/2412.10945/) (2024)

## Built on by (most cited first, 1 of 1)

- [Domain-Decomposed Graph Neural Network Surrogate Modeling for Ice Sheets](https://mlsanity.com/paper/2512.01888/) (2025)

## Abstract (arXiv)

Real-world flow applications in complex scientific and engineering domains, such as geosciences, challenge classical simulation methods due to large spatial domains, high spatio-temporal resolution requirements, and potentially strong material heterogeneities that lead to ill-conditioning and long runtimes. While machine learning-based surrogate models can reduce computational cost, they typically rely on large training datasets that are often unavailable in practice. To address data-scarce settings, we revisit the structure of advection-diffusion problems and decompose them into multiscale processes of locally and globally dominated components, separating spatially localized interactions and long-range effects. We propose a Local-Global Convolutional Neural Network (LGCNN) that combines a lightweight numerical model for global transport with two convolutional neural networks addressing processes of a more local nature. We demonstrate the performance of our method on city-scale geothermal heat pump interaction modeling and show that, even when trained on fewer than five simulations, LGCNN generalizes to arbitrarily larger domains, and can be successfully transferred to real subsurface parameter maps from the Munich region, Germany.