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

ICLR 2026 Conference Submission20731 Authors

Keywords: UNet, CNN, physics-aware, scalability, real-world data, few-shot learning, domain transfer, surrogate modeling, heat transport, groundwater flow, heat pumps

TL;DR: We propose modeling complex scenarios like groundwater flow simulations around dozens of heat pumps with a hybrid, physics-inspired, staged CNN-based approach "Local Global CNN", that is trained on just a few data points.

Abstract: Machine learning methods often struggle with real-world applications in science and engineering due to an insufficient amount or quality of training data. In this work, the example of subsurface porous media flow is considered; this corresponds to advection-diffusion processes under heterogeneous flow conditions, i.e., for spatially varying material parameters, and a large number of spatially distributed source terms. This challenge comes at high computing costs for classical simulation methods due to the required high spatio-temporal resolution and large domains. Machine learning-based surrogate models seem to offer a computationally efficient alternative. However, faced with real-world data-limitations, purely data-driven approaches face difficulties in predicting the advection process, which is highly sensitive to input variations and involves long-range interactions. Therefore, in this work, a Local-Global Convolutional Neural Network (LGCNN) approach is introduced, that combines a lightweight numerical surrogate for the global transport process with convolutional neural networks (CNNs) for the local processes. With the LGCNN, we model a city-wide subsurface temperature field, involving a heterogeneous groundwater flow field and one hundred groundwater heat pump injection points forming interacting heat plumes over long distances. In order to first systematically analyze the method, random subsurface input fields are employed. Then, the model is trained on a few cut-outs from a real-world subsurface map of the Munich region in Germany. Our model scales to larger cut-outs without retraining by accounting for the global effects with numerical physics models. All datasets, our code, and trained models are published for reproducibility.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 20731