This paper presents a physics-inspired graph-structured kernel designed for power flow learning using Gaussian Process (GP). The kernel, named the vertex-degree kernel (VDK), relies on latent decomposition of voltage-injection relationship based on the network graph or topology. Notably, VDK design avoids the need to solve optimization problems for kernel search. To enhance efficiency, we also explore a graph-reduction approach to obtain a VDK representation with lesser terms. Additionally, we propose a novel network-swipe active learning scheme, which intelligently selects sequential training inputs to accelerate the learning of VDK. Leveraging the additive structure of VDK, the active learning algorithm performs a block-descent type procedure on GP's predictive variance, serving as a proxy for information gain. Simulations demonstrate that the proposed VDK-GP achieves more than two fold sample complexity reduction, compared to full GP on medium scale 500-Bus and large scale 1354-Bus power systems. The network-swipe algorithm outperforms mean performance of 500 random trials on test predictions by two fold for medium-sized 500-Bus systems and best performance of 25 random trials for large-scale 1354-Bus systems by 10%. Moreover, we demonstrate that the proposed method's performance for uncertainty quantification applications with distributionally shifted testing data sets.

翻译：本文提出了一种受物理启发的图结构核函数，用于基于高斯过程的潮流学习。该核函数被称为顶点度核（VDK），其核心思想是基于网络图或拓扑结构对电压-注入关系进行隐式分解。值得注意的是，VDK的设计避免了求解优化问题来进行核函数搜索。为提升效率，我们还探索了一种图约简方法，以获得项数更少的VDK表示。此外，我们提出了一种新颖的网络扫描主动学习方案，该方案智能地选择序贯训练输入以加速VDK的学习。利用VDK的加性结构，该主动学习算法对高斯过程的预测方差执行类块下降过程，作为信息增益的代理指标。仿真结果表明，在中等规模500节点和大规模1354节点电力系统中，与完整高斯过程相比，所提出的VDK-GP实现了超过两倍的样本复杂度降低。网络扫描算法在中等规模500节点系统上的测试预测表现优于500次随机试验的平均性能两倍，并在大规模1354节点系统上超越25次随机试验的最佳性能10%。此外，我们还证明了所提方法在分布偏移测试数据集上的不确定性量化应用性能。