Identifying a compact manifold on which the local solution of a nonlinear, possibly chaotic, dynamical system lies can solve many problems including inference, uncertainty quantification and model reduction. The manifold serves as a reduced model, it can be used in Bayesian inference, and enable effective re-sampling for propagating uncertainty. In this talk, we used randomized algorithms to identify the manifold produced from snapshots or initial perturbations, and show that a diffeomorphic realignment of this manifold in the presence of observations is effective in a Bayesian inference procedure and offers improvements over mixture and kernel-based methods for non-Gaussian inference. We then discuss a targeted re-sampling on the manifold to further propagate uncertainty. Examples from chaotic and nonlinear dynamical systems suggests that this approach is promising for solving the inference problems entailed by the feedback loops in DDDAS.