Although Bayesian optimization (Bo) has been employed for accelerating materials design in computational materials engineering, existing works are restricted to problems with quantitative variables. However, real designs of materials systems involve both qualitative and quantitative design variables representing material compositions, microstructure morphology, and processing conditions. for mixed-variable problems, existing Bayesian optimization (Bo) approaches represent qualitative factors by dummy variables first and then fit a standard Gaussian process (GP) model with numerical variables as the surrogate model. this approach is restrictive theoretically and fails to capture complex correlations between qualitative levels. We present in this paper the integration of a novel latentvariable (LV) approach for mixed-variable GP modeling with the BO framework for materials design. LVGP is a fundamentally different approach that maps qualitative design variables to underlying numerical LV in GP, which has strong physical justification. It provides flexible parameterization and representation of qualitative factors and shows superior modeling accuracy compared to the existing methods. We demonstrate our approach through testing with numerical examples and materials design examples. The chosen materials design examples represent two different scenarios, one on concurrent materials selection and microstructure optimization for optimizing the light absorption of a quasirandom solar cell, and another on combinatorial search of material constitutes for optimal Hybrid organic-inorganic perovskite (Hoip) design. it is found that in all test examples the mapped LVs provide intuitive visualization and substantial insight into the nature and effects of the qualitative factors. though materials designs are used as examples, the method presented is generic and can be utilized for other mixed variable design optimization problems that involve expensive physics-based simulations. With advances in computational engineering, materials design and discovery have been increasingly viewed as optimization problems with the goal of achieving desired material properties or device performance 1-3 . One challenge of designing new materials systems is the co-existence of qualitative and quantitative design variables associated with material compositions, microstructure morphology, and processing conditions. While microstructure morphology can be described using quantitative, variables such as those associated with correlation function 4 , descriptors 3,5,6 , and spectral density functions 1,2 , many composition and processing conditions are discrete and qualitative by nature. For example, in polymer nanocomposite design, there are numerous choices of material constituents (e.g., the types of filler and matrix) and processing conditions (e.g., the type of surface treatment); each combination follows drastically different physical mechanisms with significant impact on the overall properties 3,7 . As illustrated in Fig. 1 , the existence of both quantitative and qualitative material design variables results in multiple disjointed regions in the property/performance space. The combinatorial nature poses additional challenges in materials modeling and the search for optimal solution. In contrast to the traditional trial-and-error based experiment approach to materials design, computational materials design methods have emerged as an efficient and effective alternative in the past decade, building upon the advancement of simulation techniques such as finite element analysis (FEA) and density functional theory (DFT) that can accurately model and predict material properties at different length scales 8, 9 . Recent years have
A recent work by Zhang et al. REF also studies Bayesian optimization in the context of qualitative and quantitative inputs.
Bayesian Optimization for Materials Design with Mixed Quantitative and Qualitative Variables
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