Causal graph discovery for space-time systems is challenging in high-dimensional gridded data, which often has many more grid cells than temporal observations per cell. The Causal Space-Time Stencil Learning (CaStLe) meta-algorithm was developed to address that niche under space-time locality and stationarity assumptions, but it is currently limited to univariate analyses. In this work, we present M-CaStLe. M-CaStLe generalizes the local embedding and parent-identification phases of CaStLe to jointly model local within-variable and cross-variable space-time causal structures in gridded data. Like CaStLe, by constraining candidate parents to a constant-size space-time neighborhood and pooling spatial replicates, M-CaStLe increases effective sample size to make discovery tractable in high-dimensional settings. We further decompose the resulting multivariate stencil graph into reaction and spatial graphs to aid interpretation in complex settings. We study M-CaStLe in four settings: a multivariate space-time vector autoregression benchmark with known ground truth, an advective-diffusive-reaction partial differential equation verification problem with derived physical reference structure, an atmospheric chemistry case study in a low-temporal-sample regime, and an El Niño Southern Oscillation study on reanalysis data, identifying phase-dependent ocean--atmosphere coupling. Across these settings, M-CaStLe more accurately recovers multivariate causal structure in controlled settings and identifies important physical dynamics in real-world case studies. Overall, M-CaStLe advances causal discovery for multivariate space-time systems while retaining interpretability at the grid level.

翻译：时空系统的因果图发现面临高维网格数据的挑战，这类数据通常存在网格单元数量远大于每个单元时间观测值的问题。尽管基于时空局部性与平稳性假设的CaStLe（因果时空模板学习）元算法已针对该领域问题被提出，但其目前局限于单变量分析。本文提出M-CaStLe方法，将CaStLe的局部嵌入与父节点识别阶段推广至网格数据中变量内与跨变量的联合建模。与CaStLe类似，通过将候选父节点限制在恒定大小的时空邻域内并汇集空间重复样本，M-CaStLe可增加有效样本量，使得高维场景下的发现过程具有可行性。我们进一步将所得多变量模板图分解为反应图与空间图，以辅助复杂场景下的解释。我们在四个场景中验证M-CaStLe：具有已知真值的多变量时空向量自回归基准试验、基于物理参考结构的平流-扩散-反应偏微分方程验证问题、低时间采样率条件下的大气化学案例研究，以及对再分析数据开展的厄尔尼诺-南方涛动研究（识别出阶段依赖性海洋-大气耦合效应）。实验表明，M-CaStLe能在受控场景中更准确地恢复多变量因果结构，并在真实世界案例中识别关键物理动力学过程。总体而言，M-CaStLe在保持网格级可解释性的同时，推进了多变量时空系统的因果发现研究。