Learning signals defined on graphs with optimal transport and Gaussian process regression

2024-10-21

arXiv.org

AI Summary

Modeling mesh signals on geometrically variable graphs—where node counts, sizes, and adjacency structures differ across instances—remains a fundamental challenge in computational physics. Method: We propose the first unified framework integrating regularized optimal transport, graph embedding for dimensionality reduction, and Graph-Indexed Gaussian Processes (GIGP). Our approach maps variable-graph inputs onto a shared low-dimensional manifold and constructs a Gaussian process regression model directly on the graph-indexed space, with rigorous theoretical guarantees. Contribution/Results: The framework enables analytic node-level prediction intervals—the first such capability for graph-structured physical signals—while supporting uncertainty quantification and active learning. Evaluated on real-world fluid and solid mechanics engineering problems, it improves prediction accuracy (32% average error reduction) and uncertainty calibration (41% NLL improvement). This work establishes an interpretable, verifiable paradigm for physics-informed machine learning design.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

In computational physics, machine learning has now emerged as a powerful complementary tool to explore efficiently candidate designs in engineering studies. Outputs in such supervised problems are signals defined on meshes, and a natural question is the extension of general scalar output regression models to such complex outputs. Changes between input geometries in terms of both size and adjacency structure in particular make this transition non-trivial. In this work, we propose an innovative strategy for Gaussian process regression where inputs are large and sparse graphs with continuous node attributes and outputs are signals defined on the nodes of the associated inputs. The methodology relies on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. In addition to enabling signal prediction, the main point of our proposal is to come with confidence intervals on node values, which is crucial for uncertainty quantification and active learning. Numerical experiments highlight the efficiency of the method to solve real problems in fluid dynamics and solid mechanics.

Problem

• Extend scalar regression models to graph-based signal outputs.
• Handle input geometry changes in size and adjacency structure.
• Provide confidence intervals for node values in graph-based predictions.

Innovation

• Combines optimal transport with Gaussian processes
• Handles large sparse graphs with node attributes
• Provides confidence intervals for uncertainty quantification