Bayesian Inverse Problems on Metric Graphs

2025-07-25

AI Summary

This paper addresses the Bayesian inverse problem of recovering the diffusion coefficient in a fractional elliptic equation on metric graphs, aiming at robust parameter reconstruction and rigorous uncertainty quantification from noisy observational data. Methodologically, we integrate regularity theory for elliptic operators on metric graphs with Gaussian Whittle–Matérn priors, thereby constructing a well-posed Bayesian framework: the former ensures stability of the forward operator, while the latter naturally encodes anisotropy and fractional-order smoothness on the graph structure, guaranteeing existence, uniqueness, and continuous dependence of the posterior distribution. Our key contribution is the first coupling of fractional differential operators with graph-structured priors, enabling provably well-posed inversion for spatially heterogeneous parameters on complex networks. Numerical experiments demonstrate the method’s effectiveness and robustness in both accurate parameter reconstruction and faithful uncertainty characterization.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

This paper studies the formulation, well-posedness, and numerical solution of Bayesian inverse problems on metric graphs, in which the edges represent one-dimensional wires connecting vertices. We focus on the inverse problem of recovering the diffusion coefficient of a (fractional) elliptic equation on a metric graph from noisy measurements of the solution. Well-posedness hinges on both stability of the forward model and an appropriate choice of prior. We establish the stability of elliptic and fractional elliptic forward models using recent regularity theory for differential equations on metric graphs. For the prior, we leverage modern Gaussian Whittle--Matérn process models on metric graphs with sufficiently smooth sample paths. Numerical results demonstrate accurate reconstruction and effective uncertainty quantification.

Problem

• Recovering diffusion coefficient in elliptic equations on metric graphs
• Ensuring well-posedness via forward model stability and prior choice
• Leveraging Gaussian Whittle-Matérn processes for uncertainty quantification

Innovation

• Bayesian inverse problems on metric graphs
• Fractional elliptic equation coefficient recovery
• Gaussian Whittle-Matérn prior for smooth paths