  Smoothing is a specialized form of Bayesian inference for state-space models
that characterizes the posterior distribution of a collection of states given
an associated sequence of observations. Ramgraber et al. (2023) proposes a
general framework for transport-based ensemble smoothing, which includes linear
Kalman-type smoothers as special cases. Here, we build on this foundation to
realize and demonstrate nonlinear backward ensemble transport smoothers. We
discuss parameterization and regularization of the associated transport maps,
and then examine the performance of these smoothers for nonlinear and chaotic
dynamical systems that exhibit non-Gaussian behavior. In these settings, our
nonlinear transport smoothers yield lower estimation error than conventional
linear smoothers and state-of-the-art iterative ensemble Kalman smoothers, for
comparable numbers of model evaluations.
