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

Raphaël Carpintero Perez, Sébastien da Veiga, Josselin Garnier, Brian Staber

TL;DR

This paper introduces Transported Output Signal GP (TOS-GP), a Gaussian process framework for learning signals defined on graphs with varying topology by transporting node-defined outputs to a common reference space via entropy-regularized optimal transport, followed by dimension reduction and graph-based GP regression. The method provides per-node uncertainty quantification and supports inputs with different sizes and adjacencies, enabling predictions on meshes from domains like fluid and solid mechanics. It demonstrates competitive performance against state-of-the-art GNNs on FEM-inspired datasets and extends naturally to topology changes, point clouds, and nonlinear dimension reductions. The work offers a scalable, uncertainty-aware surrogate modeling approach for complex, graph-structured outputs in computational physics and engineering.

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.

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

TL;DR

This paper introduces Transported Output Signal GP (TOS-GP), a Gaussian process framework for learning signals defined on graphs with varying topology by transporting node-defined outputs to a common reference space via entropy-regularized optimal transport, followed by dimension reduction and graph-based GP regression. The method provides per-node uncertainty quantification and supports inputs with different sizes and adjacencies, enabling predictions on meshes from domains like fluid and solid mechanics. It demonstrates competitive performance against state-of-the-art GNNs on FEM-inspired datasets and extends naturally to topology changes, point clouds, and nonlinear dimension reductions. The work offers a scalable, uncertainty-aware surrogate modeling approach for complex, graph-structured outputs in computational physics and engineering.

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.

Paper Structure

This paper contains 32 sections, 13 equations, 9 figures, 4 tables, 3 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. RELATED WORK
  4. GNNs and morphing.
  5. From single to multi-output prediction.
  6. Graph signal processing.
  7. Optimal transport.
  8. OPTIMALLY TRANSPORTED SIGNALS
  9. Regularized Wasserstein distance.
  10. Signal transfer with optimal transport plans.
  11. Dimension reduction.
  12. Single-output GP regression.
  13. Prediction.
  14. Reference measure and hyperparameters.
  15. EXPERIMENTS
  16. ...and 17 more sections

Figures (9)

  • Figure 1: Summary of our approach. a) Inputs = Graphs. Outputs = Fields defined on the nodes. b) Step 1: obtaining transport plans to a reference measure. c) Step 2: Transferring signals to the reference measure.
  • Figure 2: Tensile2 field U. Top: RRMSE vs reference size and reference measure with fixed $\lambda=1e-3$. Bottom: RRMSE vs reference size and $\lambda$ for a fixed reference measure.
  • Figure 3: Tensile2d, $\sigma_{12}$: two test meshes (left and right) transported to a common reference. From top to bottom: output signals, predicted transferred signals, posterior standard deviation of the predicted transferred fields, posterior standard deviation of the predicted field.
  • Figure 4: Rotor37 test field T for one test input mesh. From left to right: true field, predicted field (posterior mean), absolute error, posterior standard deviation.
  • Figure 5: Two samples with different topologies from the multiscale dataset. From top to bottom: true field, predicted field, absolute error, predicted transferred signal.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Definition 1: Wasserstein distance
  • Definition 2: Maximum Mean Discrepancy