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.

本文介绍了一种名为有限体积神经网络（FINN）的新方法，它采用了有限体积方法的数值结构来处理偏微分方程，同时容纳可学习参数。FINN 方法不仅能更好地处理控制体之间的通量，从而正确处理不同类型的数值边界条件，并且还能够明确提取和学习本构关系，同时在合成数据集和实际的稀疏实验数据上表现出优秀的泛化能力，因此作为一种数据驱动的建模工具具有相关性。