Bayesian Optimization for Materials Design with Mixed Quantitative and Qualitative Variables (1910.01688v1)

Abstract: 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 latent-variable (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. 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.

• The paper introduces a latent-variable Gaussian Process model to seamlessly integrate mixed quantitative and qualitative variables.
• It demonstrates significantly faster convergence and enhanced prediction accuracy in optimizing materials for solar cells and HOIPs.
• The framework offers actionable insights into qualitative correlations, improving computational efficiency in complex design scenarios.

Bayesian Optimization for Materials Design with Mixed Quantitative and Qualitative Variables

The paper addresses a significant gap in Bayesian Optimization (BO) applications within computational materials engineering by proposing a method tailored to problems involving both quantitative and qualitative variables. Traditional approaches to BO have primarily focused on quantitative variables due to their natural ordering and distance metrics, leaving qualitative variables inadequately treated, typically through dummy variable representations. This conventional approach, while functional, has inherent limitations in capturing the nuanced correlations between different qualitative levels, particularly when the number of levels is large.

The authors introduce a novel Latent-Variable Gaussian Process (LVGP) modeling technique that seamlessly integrates within the BO framework. The LVGP model represents qualitative variables using underlying numerical latent variables (LVs), offering a methodologically sound alternative with robust physical justification. In this model, each qualitative factor is mapped into a latent space, providing not only an efficient parameterization but also facilitating a more accurate depiction of complex correlations between levels. This structural advancement enables the BO method to handle mixed variables with improved prediction accuracy and optimization efficiency.

The utility of the LVGP-BO framework is demonstrated through two distinct materials design scenarios. Firstly, it is applied to the concurrent optimization of material selection and microstructure for quasi-random solar cells, aiming to enhance light absorption. Secondly, the framework is employed in a combinatorial search for Hybrid Organic-Inorganic Perovskites (HOIPs) focusing on enhancing intermolecular binding energy, a problem featuring purely qualitative variables.

The results from the solar cell optimization substantiate the efficacy of LVGP-BO, achieving a significantly faster convergence than traditional methods. The design optimized using LVGP-BO outperformed the benchmark with a light absorption coefficient reaching 0.94. The latent variables assigned to different material constituents provided insights into the influence of qualitative factors on device performance, underlining the practical benefits of the LVGP method.

In the HOIP example, the LVGP-BO framework demonstrated superior robustness and accuracy in identifying the globally optimal solution compared to the dummy variable-based approach. The ability to efficiently navigate the qualitative design space and converge towards the best solution with less computational overhead highlights the practical implications of the proposed method in real-world applications.

In addition to these materials design examples, the paper presents mathematical examples involving the Branin and Gold-Stein Price functions. These examples further illustrate the robustness and efficiency of the LVGP-BO approach in handling mixed-variable optimization problems by consistently achieving faster convergence and reducing variance across multiple test replicates.

The implications of this research extend beyond materials design, offering a generic framework applicable to various engineering optimization problems characterized by the coexistence of qualitative and quantitative variables. Future work is anticipated to explore extensions of this framework for multi-objective optimization tasks, further enhancing its versatility in complex design scenarios. Through the introduction of LVGP, this research not only advances the theoretical understanding but also demonstrates practical improvements in modeling and optimization capabilities for mixed-variable design tasks.