Data-driven modeling of spatiotemporal physical processes with general deep learning methods is a highly challenging task. It is further exacerbated by the limited availability of data, leading to poor generalizations in standard neural network models. To tackle this issue, we introduce a new approach called the Finite Volume Neural Network (FINN). The FINN method adopts the numerical structure of the well-known Finite Volume Method for handling partial differential equations, so that each quantity of interest follows its own adaptable conservation law, while it concurrently accommodates learnable parameters. As a result, FINN enables better handling of fluxes between control volumes and therefore proper treatment of different types of numerical boundary conditions. We demonstrate the effectiveness of our approach with a subsurface contaminant transport problem, which is governed by a non-linear diffusion-sorption process. FINN does not only generalize better to differing boundary conditions compared to other methods, it is also capable to explicitly extract and learn the constitutive relationships (expressed by the retardation factor). More importantly, FINN shows excellent generalization ability when applied to both synthetic datasets and real, sparse experimental data, thus underlining its relevance as a data-driven modeling tool.