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.

翻译：监督学习近年来在计算物理领域受到广泛关注，因其能够有效提取复杂模式，用于求解偏微分方程或预测材料属性等任务。传统上，此类数据集的输入为表示问题几何形状（视为图）且包含大量节点的网格，对应输出则由数值求解器获得。这意味着监督学习模型必须能够处理具有连续节点属性的大规模稀疏图。本文聚焦于高斯过程回归，并为此引入了切片Wasserstein Weisfeiler-Lehman（SWWL）图核。与现有图核不同，所提SWWL核具有正定性和显著降低的计算复杂度，从而能够处理以前无法处理的数据集。该新核首先在分子数据集上的图分类任务中验证，其中输入图仅包含几十个节点。随后，在计算流体动力学和固体力学中的图回归任务中展示了SWWL核的高效性，其中输入图由数万个节点组成。