Engineering design often involves qualitative and quantitative design variables, which requires systematic methods for the exploration of these mixed-variable design spaces. Expensive simulation techniques, such as those encountered in materials design, underline the need for efficient search strategies — Bayesian optimization being one of the most widely adopted. Although recent developments in mixed-variable Bayesian optimization have shown promise, the effects of dimensionality of qualitative variables have not been well studied. High dimensional qualitative variables, i.e., with many levels, impose a large design cost as they typically require a larger dataset to quantify the effect of each level on the optimization objective. We address this challenge by leveraging domain knowledge about underlying physical descriptors to infer the effect of unobserved levels that have not been sampled yet. We show that domain knowledge about physical descriptors can be intuitively embedded into the latent variable Gaussian process approach — a mixed-variable GP modeling technique — and used to selectively explore levels of qualitative variables in the Bayesian optimization framework. Our method is robust to certain types of incomplete domain knowledge and significantly reduces the design cost for problems with high-dimensional qualitative variables.

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