Stable Partial Order Constraints for Temporal Causal Structure Learning

Temporal causal structure learning aims to recover lag-aware causal relations from multivariate time series. While prior knowledge has been widely used to improve static structure learning, its integration into lag-aware structure learning remains limited, because available priors are often lag-agnostic. Recent studies have explored lag-agnostic edge presence priors, but partial order priors, which can prune the ordering space and improve structure learning, remain underexplored. A key challenge is that variables may influence each other at different lags, leading to cyclic variable-level relations where ordering is no longer naturally defined. Moreover, directly applying static differentiable formulations of partial order constraints can be unstable due to repeated-walk accumulation on cycles. To address this gap, we represent partial order priors on the summary graph and follow the natural idea of the order relation that only allows a single direction of possible causality between two variables, which is equivalent to forbidding all reversed causality, direct or indirect. This converts partial order constraints into path prohibition constraints that can be extended to cyclic summary structures. For stable differentiable characterization, we propose a decoupled partial order constraint method. The method preserves the original lag-aware structure for data fitting, imposes partial order constraints through a normalized summary representation, and captures higher-order reachability with a closed-form connectivity characterization. Experiments further demonstrate its effectiveness in nonstationary settings.