  Anomalous behavior is ubiquitous in subsurface solute transport due to the
presence of high degrees of heterogeneity at different scales in the media.
Although fractional models have been extensively used to describe the anomalous
transport in various subsurface applications, their application is hindered by
computational challenges. Simpler nonlocal models characterized by integrable
kernels and finite interaction length represent a computationally feasible
alternative to fractional models; yet, the informed choice of their kernel
functions still remains an open problem. We propose a general data-driven
framework for the discovery of optimal kernels on the basis of very small and
sparse data sets in the context of anomalous subsurface transport. Using
spatially sparse breakthrough curves recovered from fine-scale particle-density
simulations, we learn the best coarse-scale nonlocal model using a nonlocal
operator regression technique. Predictions of the breakthrough curves obtained
using the optimal nonlocal model show good agreement with fine-scale simulation
results even at locations and time intervals different from the ones used to
train the kernel, confirming the excellent generalization properties of the
proposed algorithm. A comparison with trained classical models and with
black-box deep neural networks confirms the superiority of the predictive
capability of the proposed model.
