Generative modeling typically seeks the path of least action via deterministic flows (ODE). While effective for in-distribution tasks, we argue that these deterministic paths become brittle under causal interventions, which often require transporting probability mass across low-density regions ("off-manifold") where the vector field is ill-defined. This leads to numerical instability and spurious correlations. In this work, we introduce the Causal Schrödinger Bridge (CSB), a framework that reformulates counterfactual inference as Entropic Optimal Transport. Unlike deterministic approaches that require strict invertibility, CSB leverages diffusion processes (SDEs) to robustly "tunnel" through support mismatches while strictly enforcing structural admissibility constraints. We prove the Structural Decomposition Theorem, showing that the global high-dimensional bridge factorizes into local, robust transitions. Empirical validation on high-dimensional interventions (Morpho-MNIST) demonstrates that CSB significantly outperforms deterministic baselines in structural consistency, particularly in regimes of strong, out-of-distribution treatments.

翻译：生成建模通常通过确定性流（常微分方程）寻求最小作用量路径。尽管对于分布内任务有效，我们认为这些确定性路径在因果干预下会变得脆弱，因为干预通常需要跨越低密度区域（“离流形”）传输概率质量，而该区域的向量场定义不良。这会导致数值不稳定性和伪相关性。本文提出因果薛定谔桥框架，将反事实推断重构为熵最优输运问题。与需要严格可逆性的确定性方法不同，CSB利用扩散过程（随机微分方程）鲁棒地“穿越”支撑集不匹配区域，同时严格强制执行结构可容性约束。我们证明了结构分解定理，表明全局高维桥可分解为局部鲁棒转移。在高维干预实验（Morpho-MNIST）中，CSB在结构一致性方面显著优于确定性基线方法，尤其在强分布外干预机制下表现突出。