用最优传输与高斯过程，实现复杂图上信号的精准预测与不确定性评估。
Learning signals defined on graphs with optimal transport and Gaussian process regression
- 结合最优传输与降维，处理带节点属性的大规模稀疏图输入
- 在流体力学与固体力学问题中实现高精度信号预测并给出置信区间
- 适合需要不确定性量化和主动学习的工程设计场景
在计算物理中，机器学习已成为高效探索工程设计候选方案的强大辅助工具。此类监督问题的输出是定义在网格上的信号，关键挑战在于将通用标量输出回归模型推广至这类复杂输出。由于输入几何在尺寸和邻接结构上的变化，这一迁移极具挑战性。本文提出一种创新的高斯过程回归方法，适用于具有连续节点属性的大规模稀疏图输入，输出为关联图节点上的信号。该方法融合正则化最优传输、降维技术及基于图的高斯过程。除实现信号预测外，核心优势在于提供节点值的置信区间，对不确定性量化和主动学习至关重要。数值实验验证了该方法在流体力学与固体力学真实问题中的高效性。
原文摘要 · Abstract (English)
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.
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