# Modeling Advection on Directed Graphs using Mat'ern Gaussian Processes for Traffic Flow

arXiv: https://arxiv.org/abs/2201.00001 · HTML: https://mlsanity.com/paper/2201.00001/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 20 references that are in the map.
- First posted on arXiv 14 Dec 2021.

## Builds on (most cited first, 6 of 6)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Machine Learning of Space-Fractional Differential Equations](https://mlsanity.com/paper/1808.00931/) (2018)
- [PDE Approaches to Graph Analysis](https://mlsanity.com/paper/1505.00185/) (2015)
- [Numerical Gaussian process Kalman filtering](https://mlsanity.com/paper/1912.01234/) (2019)

## Built on by (most cited first, 2 of 2)

- [Mat'ern Noise for Triangulation-Agnostic Flow Matching on Meshes](https://mlsanity.com/paper/2605.19305/) (2026)
- [INDEQS: Informed Neural controlled Differential EQuationS](https://mlsanity.com/paper/2606.19138/) (2026)

## Abstract (arXiv)

The transport of traffic flow can be modeled by the advection equation. Finite difference and finite volumes methods have been used to numerically solve this hyperbolic equation on a mesh. Advection has also been modeled discretely on directed graphs using the graph advection operator [4, 18]. In this paper, we first show that we can reformulate this graph advection operator as a finite difference scheme. We then propose the Directed Graph Advection Mat'ern Gaussian Process (DGAMGP) model that incorporates the dynamics of this graph advection operator into the kernel of a trainable Mat'ern Gaussian Process to effectively model traffic flow and its uncertainty as an advective process on a directed graph.