Gaussian process regression with Sliced Wasserstein Weisfeiler-Lehman graph kernels

TL;DR

This paper introduces the Sliced Wasserstein Weisfeiler-Lehman (SWWL) graph kernel for Gaussian process regression, enabling efficient processing of large, sparse graphs in computational physics applications like fluid dynamics and solid mechanics.

Contribution

The paper presents a novel positive definite graph kernel with reduced complexity, allowing Gaussian process regression on large-scale graph datasets previously infeasible to analyze.

Findings

SWWL kernel achieves positive definiteness and lower computational complexity.

Validated on molecular graph classification with small graphs.

Demonstrated on large graphs in fluid dynamics and solid mechanics.

Abstract

Supervised learning has recently garnered significant attention in the field of computational physics due to its ability to effectively extract complex patterns for tasks like solving partial differential equations, or predicting material properties. Traditionally, such datasets consist of inputs given as meshes with a large number of nodes representing the problem geometry (seen as graphs), and corresponding outputs obtained with a numerical solver. This means the supervised learning model must be able to handle large and sparse graphs with continuous node attributes. In this work, we focus on Gaussian process regression, for which we introduce the Sliced Wasserstein Weisfeiler-Lehman (SWWL) graph kernel. In contrast to existing graph kernels, the proposed SWWL kernel enjoys positive definiteness and a drastic complexity reduction, which makes it possible to process datasets that were previously impossible to handle. The new kernel is first validated on graph classification for molecular datasets, where the input graphs have a few tens of nodes. The efficiency of the SWWL kernel is then illustrated on graph regression in computational fluid dynamics and solid mechanics, where the input graphs are made up of tens of thousands of nodes.