The data-aware method of distributions (DA-MD) is a low-dimension data assimilation procedure to forecast the behavior of dynamical systems described by differential equations. It combines sequential Bayesian update with the MD, such that the former utilizes available observations while the latter propagates the (joint) probability distribution of the uncertain system state(s). The core of DA-MD is the minimization of a distance between an observation and a prediction in distributional terms, with prior and posterior distributions constrained on a statistical manifold defined by the MD. We leverage the information-geometric properties of the statistical manifold to reduce predictive uncertainty via data assimilation. Specifically, we exploit the information geometric structures induced by two discrepancy metrics, the Kullback-Leibler divergence and the Wasserstein distance, which explicitly yield natural gradient descent. To further accelerate optimization, we build a deep neural network as a surrogate model for the MD that enables automatic differentiation. The manifold's geometry is quantified without sampling, yielding an accurate approximation of the gradient descent direction. Our numerical experiments demonstrate that accounting for the information-geometry of the manifold significantly reduces the computational cost of data assimilation by facilitating the calculation of gradients and by reducing the number of required iterations. Both storage needs and computational cost depend on the dimensionality of a statistical manifold, which is typically small by MD construction. When convergence is achieved, the Kullback-Leibler and $L_2$ Wasserstein metrics have similar performances, with the former being more sensitive to poor choices of the prior.