Gaussian Processes on Directed Metric Graphs

2026-09-01

AI Summary

本文提出了一种基于随机微分方程的统计框架，用于解决在有向度量图上任意边位置处的高斯场问题，并展示了该方法在温度建模和交通速度预测中的应用。

Abstract

We introduce a statistical framework for Gaussian fields indexed at arbitrary edge locations on general compact directed metric graphs. The construction is based on a stochastic differential equation with a first-order operator and conditions at the vertices. We characterise well-posedness and identify the covariance reproducing kernel Hilbert space. We also connect the proposed framework to earlier stream-network models, showing that these arise from the same system under particular boundary conditions, and introduce new boundary conditions that yield more physically realistic processes. The differential-equation representation enables computationally efficient inference and prediction. This makes the method applicable to large data sets without approximation. Applications to temperature modelling on river networks and traffic speeds on road networks illustrate the framework, including the computational efficiency and improved performance under physically informed vertex conditions.

Problem

• Gaussian Processes
• Directed Metric Graphs
• Stochastic Differential Equation
• Covariance Kernel

Innovation

• Gaussian Processes
• Directed Metric Graphs
• Stochastic Differential Equation
• Covariance Reproducing Kernel Hilbert Space
• Physically Informed Boundary Conditions