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

arXiv: https://arxiv.org/abs/2410.15721 · HTML: https://mlsanity.com/paper/2410.15721/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 74 references that are in the map.
- First posted on arXiv 21 Oct 2024.

## Builds on (most cited first, 30 of 30)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [A Survey on Graph Kernels](https://mlsanity.com/paper/1903.11835/) (2019)
- [Wasserstein Weisfeiler-Lehman Graph Kernels](https://mlsanity.com/paper/1906.01277/) (2019)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Graph signal processing for machine learning: A review and new perspectives](https://mlsanity.com/paper/2007.16061/) (2020)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [Graph Kernels: A Survey](https://mlsanity.com/paper/1904.12218/) (2019)
- [Accurate Point Cloud Registration with Robust Optimal Transport](https://mlsanity.com/paper/2111.00648/) (2021)
- [Gaussian Processes on Graphs via Spectral Kernel Learning](https://mlsanity.com/paper/2006.07361/) (2020)
- [A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems](https://mlsanity.com/paper/2202.04648/) (2022)
- [Template based Graph Neural Network with Optimal Transport Distances](https://mlsanity.com/paper/2205.15733/) (2022)
- [MMGP: a Mesh Morphing Gaussian Process-based machine learning method for regression of physical problems under non-parameterized geometrical variability](https://mlsanity.com/paper/2305.12871/) (2023)
- [Gaussian Processes Over Graphs](https://mlsanity.com/paper/1803.05776/) (2018)
- [Metrics for Benchmarking and Uncertainty Quantification: Quality, Applicability, and a Path to Best Practices for Machine Learning in Chemistry](https://mlsanity.com/paper/2010.00110/) (2020)
- [Conformal Approach To Gaussian Process Surrogate Evaluation With Coverage Guarantees](https://mlsanity.com/paper/2401.07733/) (2024)
- [Gaussian process regression with Sliced Wasserstein Weisfeiler-Lehman graph kernels](https://mlsanity.com/paper/2402.03838/) (2024)
- [Gaussian Processes on Distributions based on Regularized Optimal Transport](https://mlsanity.com/paper/2210.06574/) (2022)
- [Bayesian neural networks for predicting uncertainty in full-field material response](https://mlsanity.com/paper/2406.14838/) (2024)
- [Functional principal component analysis for global sensitivity analysis of model with spatial output](https://mlsanity.com/paper/2005.10285/) (2020)

## Built on by (most cited first, 1 of 1)

- [PLAID: A Unified Data Model for Machine Learning on Heterogeneous Physics Simulations](https://mlsanity.com/paper/2505.02974/) (2025)

## Abstract (arXiv)

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.