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

TL;DR

This paper introduces a novel Gaussian process regression method for signals on graphs, combining optimal transport and dimension reduction, enabling accurate predictions and confidence intervals for complex engineering data.

Contribution

The work presents an innovative approach integrating optimal transport and Gaussian processes for graph-based signals, with uncertainty quantification capabilities.

Findings

Effective in fluid dynamics applications

Provides confidence intervals for node predictions

Outperforms existing methods in complex graph scenarios

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.