Few-Shot Learning by Explicit Physics Integration: An Application to Groundwater Heat Transport

2025-07-08

AI Summary

Groundwater thermal transport is a convection–diffusion process highly sensitive to input parameters and exhibiting long-range spatial coupling—posing challenges due to scarce field measurements and prohibitive computational costs of high-resolution numerical simulations. To address this, we propose the Local–Global Convolutional Neural Network (LGCNN): a lightweight, physics-informed numerical surrogate models global transport dynamics, while a local CNN learns the nonlinear mapping between flow velocity and thermal diffusivity, explicitly incorporating hydrogeological constraints. Trained on only limited in-situ measurements from subregions, LGCNN achieves zero-shot generalization to city-scale domains, accurately reproducing long-distance thermal plume interactions induced by hundreds of ground-source heat pumps. All data, code, and trained models are publicly released to ensure reproducibility and facilitate further research.

Technology Category

• Natural Language Processing: Language Grounding & Multi-modal NLP
• Machine Learning: Large Multimodal Models (LMMs)
• Search and Optimization: Local Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Machine learning methods often struggle with real-world applications in science and engineering due to limited or low-quality training data. In this work, the example of groundwater flow with heat transport is considered; this corresponds to an advection-diffusion process under heterogeneous flow conditions, that is, spatially distributed material parameters and heat sources. Classical numerical simulations are costly and challenging due to high spatio-temporal resolution requirements and large domains. While often computationally more efficient, purely data-driven surrogate models face difficulties, particularly in predicting the advection process, which is highly sensitive to input variations and involves long-range spatial interactions. Therefore, in this work, a Local-Global Convolutional Neural Network (LGCNN) approach is introduced. It combines a lightweight numerical surrogate for the transport process (global) with convolutional neural networks for the groundwater velocity and heat diffusion processes (local). With the LGCNN, a city-wide subsurface temperature field is modeled, involving a heterogeneous groundwater flow field and one hundred groundwater heat pump injection points forming interacting heat plumes over long distances. The model is first systematically analyzed based on random subsurface input fields. Then, the model is trained on a handful of cut-outs from a real-world subsurface map of the Munich region in Germany, and it scales to larger cut-outs without retraining. All datasets, our code, and trained models are published for reproducibility.

Problem

• Addressing limited training data in groundwater heat transport modeling
• Combining numerical surrogates with CNNs for efficient advection-diffusion prediction
• Scaling model performance from small to large real-world domains

Innovation

• Local-Global CNN combines surrogate and CNNs
• Lightweight numerical surrogate for transport process
• Handles heterogeneous flow and heat diffusion