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.

翻译：在计算物理学中，机器学习现已发展成为一种强大的辅助工具，能够高效探索工程研究中的候选设计方案。此类监督学习问题的输出是定义在网格上的信号，一个自然的问题是如何将一般的标量输出回归模型推广至此类复杂输出。输入几何结构在尺寸与邻接关系两方面的变化，尤其使得这一转换过程变得非平凡。本研究提出了一种创新的高斯过程回归策略，其输入为具有连续节点属性的大型稀疏图，输出则为定义在对应输入图节点上的信号。该方法融合了正则化最优传输、降维技术以及基于图索引的高斯过程。除了实现信号预测外，本方案的核心价值在于能够提供节点值的置信区间，这对于不确定性量化与主动学习至关重要。数值实验验证了该方法在流体力学与固体力学实际问题上求解的有效性。