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.

翻译：现实世界中复杂科学与工程领域（如地球科学）的流动应用，由于空间域广阔、时空分辨率要求高以及可能导致病态条件和长计算时间的强材料异质性，对传统模拟方法提出了挑战。虽然基于机器学习的代理模型能够降低计算成本，但它们通常依赖于大型训练数据集，而这些数据在实践中往往难以获取。针对数据稀缺场景，我们重新审视了平流-扩散问题的结构，将其分解为局部主导和全局主导的多尺度过程，从而分离空间局部相互作用与长程效应。我们提出了一种局部-全局卷积神经网络（LGCNN），该方法将用于全局传输的轻量数值模型与两个处理局部性质过程的卷积神经网络相结合。我们在城市尺度地源热泵相互作用建模中验证了该方法的性能，结果表明：即使在少于五次模拟的训练数据下，LGCNN仍能泛化至任意更大规模的计算域，并成功迁移应用于德国慕尼黑地区的真实地下参数场。