Nonlinear dimensionality reduction then and now: AIMs for dissipative
  PDEs in the ML era
Eleni D. Koronaki Nikolaos Evangelou Cristina P. Martin-Linares Edriss S. Titi Ioannis G. Kevrekidis
This study presents a collection of purely data-driven workflows for
constructing reduced-order models (ROMs) for distributed dynamical systems. The
ROMs we focus on, are data-assisted models inspired by, and templated upon, the
theory of Approximate Inertial Manifolds (AIMs); the particular motivation is
the so-called post-processing Galerkin method of Garcia-Archilla, Novo and
Titi. Its applicability can be extended: the need for accurate truncated
Galerkin projections and for deriving closed-formed corrections can be
circumvented using machine learning tools. When the right latent variables are
not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a
manifold learning scheme) can be used to discover good sets of latent variables
and test their explainability. The proposed methodology can express the ROMs in
terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD
modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both
Black-Box and (theoretically-informed and data-corrected) Gray-Box models are
described; the necessity for the latter arises when truncated Galerkin
projections are so inaccurate as to not be amenable to post-processing. We use
the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative
partial differential equations to illustrate and successfully test the overall
framework.