Discovering causal structures from multivariate time series is a key problem because interactions span across multiple lags and possibly involve instantaneous dependencies. Additionally, the search space of the dynamic graphs is combinatorial in nature. In this study, we propose \textit{Stable Causal Dynamic Differentiable Discovery (SC3D)}, a two-stage differentiable framework that jointly learns lag-specific adjacency matrices and, if present, an instantaneous directed acyclic graph (DAG). In Stage 1, SC3D performs edge preselection through node-wise prediction to obtain masks for lagged and instantaneous edges, whereas Stage 2 refines these masks by optimizing a likelihood with sparsity along with enforcing acyclicity on the instantaneous block. Numerical results across synthetic and benchmark dynamical systems demonstrate that SC3D achieves improved stability and more accurate recovery of both lagged and instantaneous causal structures compared to existing temporal baselines.

翻译：从多元时间序列中发现因果结构是一个关键问题，因为交互作用可能跨越多个滞后阶并可能涉及瞬时依赖关系。此外，动态图的搜索空间本质上是组合性的。在本研究中，我们提出了\textit{稳定因果动态可微分发现（SC3D）}，这是一个两阶段的可微分框架，能够联合学习滞后特定邻接矩阵以及（若存在）瞬时有向无环图（DAG）。在第一阶段，SC3D通过节点级预测进行边预选，以获得滞后边与瞬时边的掩码；而在第二阶段，该框架通过优化带有稀疏性的似然函数，并对瞬时块施加无环性约束，来细化这些掩码。在合成与基准动力系统上的数值结果表明，与现有的时序基线方法相比，SC3D在滞后与瞬时因果结构的恢复上具有更高的稳定性和更准确的恢复能力。