  We introduce a data-driven approach to building reduced dynamical models
through manifold learning; the reduced latent space is discovered using
Diffusion Maps (a manifold learning technique) on time series data. A second
round of Diffusion Maps on those latent coordinates allows the approximation of
the reduced dynamical models. This second round enables mapping the latent
space coordinates back to the full ambient space (what is called lifting); it
also enables the approximation of full state functions of interest in terms of
the reduced coordinates. In our work, we develop and test three different
reduced numerical simulation methodologies, either through pre-tabulation in
the latent space and integration on the fly or by going back and forth between
the ambient space and the latent space. The data-driven latent space simulation
results, based on the three different approaches, are validated through (a) the
latent space observation of the full simulation through the Nystr\"om Extension
formula, or through (b) lifting the reduced trajectory back to the full ambient
space, via Latent Harmonics. Latent space modeling often involves additional
regularization to favor certain properties of the space over others, and the
mapping back to the ambient space is then constructed mostly independently from
these properties; here, we use the same data-driven approach to construct the
latent space and then map back to the ambient space.
