Double Diffusion Maps and their Latent Harmonics for scientific computations in latent space

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 Nyström 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.