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

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.

💡 Research Summary

The paper tackles the challenge of predicting subsurface temperature fields generated by groundwater heat pumps (GWHP) in large‑scale, heterogeneous domains where training data are extremely scarce. Classical high‑fidelity simulations are computationally expensive, and data‑driven surrogate models typically require many labeled examples to capture both local diffusion and long‑range advection. The authors observe that the advection‑diffusion equation naturally separates into a locally dominated diffusion component and a globally dominated advection component. Leveraging this insight, they propose the Local‑Global Convolutional Neural Network (LGCNN), a modular architecture that explicitly couples a lightweight numerical solver for global transport with two convolutional neural networks (CNNs) that learn local processes.

LGCNN operates in three sequential steps: (1) a CNN takes as input the initial hydraulic pressure field p, the heterogeneous permeability field k, and a one‑hot map of heat‑pump locations i, and predicts a steady‑state velocity field (\hat{v}). This network captures the local relationship between pressure gradients, permeability, and source terms. (2) Using (\hat{v}) as a physical prior, a non‑learnable IVP (initial value problem) solver computes streamlines (\hat{s}) that represent the global advection pathways emanating from each pump. Because this step is purely physics‑based, it provides accurate long‑range coupling without requiring deep receptive fields or large datasets. (3) A second CNN receives k, i, the predicted velocity (\hat{v}), and the streamlines (\hat{s}) and outputs the temperature field T, effectively learning the local diffusion and plume widening around the advective paths.

The authors evaluate LGCNN on two datasets. The synthetic dataset uses Perlin‑noise generated permeability fields over a 12.8 km × 12.8 km domain discretized into 2560 × 2560 cells, with only three to five high‑fidelity PFLOTRAN simulations (3dp, 4dp) for training, validation, and testing. The real‑world dataset employs permeability maps derived from borehole measurements in the Munich region, again with only four training simulations and one larger test simulation. Each simulation runs for roughly 27–91 hours to reach a quasi‑steady state after ~27.5 years of simulated time.

Performance is measured using MAE, RMSE, Huber loss, SSIM, and a physics‑oriented Measure of Connectivity (MoC) that quantifies how many predicted plume cells are disconnected from any pump. Six baseline models are compared: UNet, DDUNet (domain‑decomposed UNet), dilated UNet, UNet++, Fourier Neural Operator (FNO), and Physics‑Informed Neural Network (PINN). In data‑scarce regimes (≤ 3 training simulations), all baselines struggle: UNet variants produce fragmented plumes with high MoC, FNO cannot fit the full domain due to memory constraints and periodic‑boundary assumptions, and PINNs collapse to predicting velocity‑like fields rather than temperature. When trained on a larger synthetic set (101 simulations), UNet and DDUNet improve but still require substantially more data than LGCNN.

LGCNN achieves MAE as low as 0.05 °C and MoC ≈ 0.09 when trained on only three simulations, outperforming the best baseline (UNet trained on 101 simulations) which attains MAE ≈ 0.05 °C but with higher MoC. Moreover, LGCNN generalizes to domains twice as large as those seen during training without retraining, demonstrating true scalability. Inference speed is accelerated by roughly 2,000× compared with full PFLOTRAN runs, making the approach suitable for real‑time decision support, uncertainty quantification, and optimization tasks.

The paper acknowledges limitations: the current implementation is restricted to two‑dimensional steady‑state problems and assumes that temperature‑induced flow perturbations are weak or spatially localized. Extending the framework to three‑dimensional, transient scenarios is conceptually straightforward but left for future work due to a lack of appropriate training data.

In summary, LGCNN showcases how embedding explicit physics modules within a neural architecture can dramatically reduce data requirements while preserving accuracy and scalability for complex, multiscale subsurface flow problems. This work represents a significant step toward data‑efficient, physics‑aware surrogate modeling in geosciences and related engineering fields.