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This paper presents a new Gaussian process (GP) surrogate modeling forpredicting the outcome of a physical experiment where some experimental inputsare controlled by other manipulating factors. Particularly, we are interestedin the case where the control precision is not very high, so the input factorvalues vary significantly even under the same setting of the correspondingmanipulating factors. The case is observed in our main application to carbonnanotube growth experiments, where one experimental input among many ismanipulated by another manipulating factors, and the relation between the inputand the manipulating factors significantly varies in the dates and times ofoperations. Due to this variation, the standard GP surrogate that directlyrelates the manipulating factors to the experimental outcome does not provide agreat predictive power on the outcome. At the same time, the GP model relatingthe main factors to the outcome directly is not appropriate for the predictionpurpose because the main factors cannot be accurately set as planned for afuture experiment. Motivated by the carbon nanotube example, we propose atwo-tiered GP model, where the bottom tier relates the manipulating factors tothe corresponding main factors with potential biases and variation independentof the manipulating factors, and the top tier relates the main factors to theexperimental outcome. Our two-tier model explicitly models the propagation ofthe control uncertainty to the experimental outcome through the two GP modelingtiers. We present the inference and hyper-parameter estimation of the proposedmodel. The proposed approach is illustrated with the motivating example of aclosed-loop autonomous research system for carbon nanotube growth experiments,and the test results are reported with the comparison to a benchmark method,i.e. a standard GP model.

Gaussian process surrogate modeling with manipulating factors for carbon nanotube growth experiments