  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.
