  This paper presents a groundbreaking approach to causal inference by
integrating continuous normalizing flows (CNFs) with parametric submodels,
enhancing their geometric sensitivity and improving upon traditional Targeted
Maximum Likelihood Estimation (TMLE). Our method employs CNFs to refine TMLE,
optimizing the Cram\'er-Rao bound and transitioning from a predefined
distribution $p_0$ to a data-driven distribution $p_1$. We innovate further by
embedding Wasserstein gradient flows within Fokker-Planck equations, thus
imposing geometric structures that boost the robustness of CNFs, particularly
in optimal transport theory.
  Our approach addresses the disparity between sample and population
distributions, a critical factor in parameter estimation bias. We leverage
optimal transport and Wasserstein gradient flows to develop causal inference
methodologies with minimal variance in finite-sample settings, outperforming
traditional methods like TMLE and AIPW. This novel framework, centered on
Wasserstein gradient flows, minimizes variance in efficient influence functions
under distribution $p_t$. Preliminary experiments showcase our method's
superiority, yielding lower mean-squared errors compared to standard flows,
thereby demonstrating the potential of geometry-aware normalizing Wasserstein
flows in advancing statistical modeling and inference.
