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.