# Finite Volume Neural Network: Modeling Subsurface Contaminant Transport

arXiv: https://arxiv.org/abs/2104.06010 · HTML: https://mlsanity.com/paper/2104.06010/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 14 references that are in the map.
- First posted on arXiv 13 Apr 2021.

## Builds on (most cited first, 8 of 8)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Neural Machine Translation in Linear Time](https://mlsanity.com/paper/1610.10099/) (2016)
- [Deep Learning of Subsurface Flow via Theory-guided Neural Network](https://mlsanity.com/paper/1911.00103/) (2019)
- [A Distributed Neural Network Architecture for Robust Non-Linear Spatio-Temporal Prediction](https://mlsanity.com/paper/1912.11141/) (2019)

## Built on by (most cited first, 15 of 15)

- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Composing Partial Differential Equations with Physics-Aware Neural Networks](https://mlsanity.com/paper/2111.11798/) (2021)
- [Finite volume method network for acceleration of unsteady computational fluid dynamics: non-reacting and reacting flows](https://mlsanity.com/paper/2105.03332/) (2021)
- [NeuralPDE: Modelling Dynamical Systems from Data](https://mlsanity.com/paper/2111.07671/) (2021)
- [Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2405.14099/) (2024)
- [DynaBench: A benchmark dataset for learning dynamical systems from low-resolution data](https://mlsanity.com/paper/2306.05805/) (2023)
- [Efficient hybrid modeling and sorption model discovery for non-linear advection-diffusion-sorption systems: A systematic scientific machine learning approach](https://mlsanity.com/paper/2303.13555/) (2023)
- [On the Relationships between Graph Neural Networks for the Simulation of Physical Systems and Classical Numerical Methods](https://mlsanity.com/paper/2304.00146/) (2023)
- [FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates](https://mlsanity.com/paper/2602.01941/) (2026)
- [TaylorPDENet: Learning PDEs from non-grid Data](https://mlsanity.com/paper/2306.14511/) (2023)
- [GrINd: Grid Interpolation Network for Scattered Observations](https://mlsanity.com/paper/2403.19570/) (2024)
- [Inferring Underwater Topography with FINN](https://mlsanity.com/paper/2408.10649/) (2024)
- [A nudge to the truth: atom conservation as a hard constraint in models of atmospheric composition using an uncertainty-weighted correction](https://mlsanity.com/paper/2408.16109/) (2024)
- [Label-Free Finite-Volume-Residual Training of Attention Graph Neural Networks for Coupled Thermo-Fluid Fields](https://mlsanity.com/paper/2607.20321/) (2026)

## Abstract (arXiv)

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.