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A data-driven framework is presented, that enables the prediction ofquantities, either observations or parameters, given sufficient partial data.The framework is illustrated via a computational model of the deposition of Cuin a Chemical Vapor Deposition (CVD) reactor, where the reactor pressure, thedeposition temperature and feed mass flow rate are important process parametersthat determine the outcome of the process. The sampled observations arehigh-dimensional vectors containing the outputs of a detailed CFD steady-statemodel of the process, i.e. the values of velocity, pressure, temperature, andspecies mass fractions at each point in the discretization. A machine learningworkflow is presented, able to predict out-of-sample (a) observations (e.g.mass fraction in the reactor) given process parameters (e.g. inlettemperature); (b) process parameters given observation data; and (c) partialobservations (e.g. temperature in the reactor) given other partial observations(e.g. mass fraction in the reactor). The proposed workflow relies on themanifold learning schemes Diffusion Maps and the associated GeometricHarmonics. Diffusion Maps is used for discovering a reduced representation ofthe available data, and Geometric Harmonics for extending functions defined onthe manifold. In our work a special use case of Geometric Harmonics isformulated and implemented, which we call Double Diffusion Maps, to map fromthe reduced representation back to (partial) observations and processparameters. A comparison of our manifold learning scheme to the traditionalGappy-POD approach is provided: ours can be thought of as a "Gappy DMAP"approach. The presented methodology is easily transferable to applicationdomains beyond reactor engineering.

From partial data to out-of-sample parameter and observation estimation with Diffusion Maps and Geometric Harmonics