  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 the
manifold learning schemes Diffusion Maps and the associated Geometric
Harmonics. Diffusion Maps is used for discovering a reduced representation of
the available data, and Geometric Harmonics for extending functions defined on
the 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 DMAP"
approach. The presented methodology is easily transferable to application
domains beyond reactor engineering.
