Positional Encoder Graph Neural Networks (PE-GNNs) are a leading approach for modeling continuous spatial data. However, they often fail to produce calibrated predictive distributions, limiting their effectiveness for uncertainty quantification. We introduce the Positional Encoder Graph Quantile Neural Network (PE-GQNN), a novel method that integrates PE-GNNs, Quantile Neural Networks, and recalibration techniques in a fully nonparametric framework, requiring minimal assumptions about the predictive distributions. We propose a new network architecture that, when combined with a quantile-based loss function, yields accurate and reliable probabilistic models without increasing computational complexity. Our approach provides a flexible, robust framework for conditional density estimation, applicable beyond spatial data contexts. We further introduce a structured method for incorporating a KNN predictor into the model while avoiding data leakage through the GNN layer operation. Experiments on benchmark datasets demonstrate that PE-GQNN significantly outperforms existing state-of-the-art methods in both predictive accuracy and uncertainty quantification.

翻译：位置编码器图神经网络（PE-GNNs）是建模连续空间数据的主流方法。然而，它们往往无法生成校准良好的预测分布，这限制了其在不确定性量化中的有效性。本文提出位置编码器图分位数神经网络（PE-GQNN），这是一种新颖的方法，将PE-GNNs、分位数神经网络和重校准技术整合在一个完全非参数化的框架中，仅需对预测分布做最小化假设。我们设计了一种新的网络架构，当与基于分位数的损失函数结合时，可在不增加计算复杂度的前提下产生准确可靠的概率模型。该方法为条件密度估计提供了一个灵活、鲁棒的框架，其应用范围可扩展至空间数据以外的场景。我们还提出了一种结构化方法，将KNN预测器集成到模型中，同时通过GNN层操作避免数据泄露。在基准数据集上的实验表明，PE-GQNN在预测准确性和不确定性量化方面均显著优于现有的最先进方法。