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Quality control in industrial processes is increasingly making use of priorscientific knowledge, often encoded in physical models that require numericalapproximation. Statistical prediction, and subsequent optimization, is key toensuring the process output meets a specification target. However, thenumerical expense of approximating the models poses computational challenges tothe identification of combinations of the process factors where there isconfidence in the quality of the response. Recent work in Bayesian computationand statistical approximation (emulation) of expensive computational models isexploited to develop a novel strategy for optimizing the posterior probabilityof a process meeting specification. The ensuing methodology is motivated by,and demonstrated on, a chemical synthesis process to manufacture apharmaceutical product, within which an initial set of substances evolveaccording to chemical reactions, under certain process conditions, into aseries of new substances. One of these substances is a target pharmaceuticalproduct and two are unwanted by-products. The aim is to determine thecombinations of process conditions and amounts of initial substances thatmaximize the probability of obtaining sufficient target pharmaceutical productwhilst ensuring unwanted by-products do not exceed a given level. Therelationship between the factors and amounts of substances of interest istheoretically described by the solution to a system of ordinary differentialequations incorporating temperature dependence. Using data from a smallexperiment, it is shown how the methodology can approximate the multivariateposterior predictive distribution of the pharmaceutical target and by-products,and therefore identify suitable operating values. Materials to replicate theanalysis can be found at www.github.com/amo105/chemicalkinetics.

Bayesian prediction for physical models with application to the optimization of the synthesis of pharmaceutical products using chemical kinetics