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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. • Machine learning motivates revisiting post-processing Galerkin ROMs. • Diffusion Maps & autoencoders discover relations between sets of latent variables. • Data-driven workflows produce ROM closures to enhance accuracy. • "Gray box" models improve first principles ROMs when they are inaccurate. [ABSTRACT FROM AUTHOR]

A data-driven framework is presented, that enables the prediction of quantities, either observations or parameters, given sufficient partial data. The framework is illustrated via a computational model of the deposition of Cu in a Chemical Vapor Deposition (CVD) reactor, where the reactor pressure, the deposition temperature and feed mass flow rate are important process parameters that determine the outcome of the process. The sampled observations are high-dimensional vectors containing the outputs of a detailed CFD steady-state model of the process, i.e. the values of velocity, pressure, temperature, and species mass fractions at each point in the discretization. A machine learning workflow is presented, able to predict out-of-sample (a) observations (e.g. mass fraction in the reactor), given process parameters (e.g. inlet temperature); (b) process parameters, given observation data; and (c) partial observations (e.g. temperature in the reactor), given other partial observations (e.g. mass fraction in the reactor). The proposed workflow relies on two manifold learning schemes: Diffusion Maps and the associated Geometric Harmonics. Diffusion Maps are used for discovering a reduced representation of the available data, and Geometric Harmonics for extending functions defined on the discovered manifold. In our work a special use case of Geometric Harmonics is formulated and implemented, which we call Double Diffusion Maps, to map from the reduced representation back to (partial) observations and process parameters. A comparison of our manifold learning scheme to the traditional Gappy-POD approach is provided: ours can be thought of as a "Gappy DMAPs" approach. The presented methodology is easily transferable to application domains beyond reactor engineering. • Nonlinear manifold learning is implemented for out-of-sample predictions. • Chemical Vapor Deposition reactor state variables predicted for new inputs. • Double DMAPs enables predictions of full observables even from partial data. • Efficient and accurate prediction of inputs from scarce observables. • Gappy DMAPS outperforms linear Gappy POD. [ABSTRACT FROM AUTHOR]