  Scientific and engineering problems often require the use of artificial
intelligence to aid understanding and the search for promising designs. While
Gaussian processes (GP) stand out as easy-to-use and interpretable learners,
they have difficulties in accommodating big datasets, categorical inputs, and
multiple responses, which has become a common challenge for a growing number of
data-driven design applications. In this paper, we propose a GP model that
utilizes latent variables and functions obtained through variational inference
to address the aforementioned challenges simultaneously. The method is built
upon the latent variable Gaussian process (LVGP) model where categorical
factors are mapped into a continuous latent space to enable GP modeling of
mixed-variable datasets. By extending variational inference to LVGP models, the
large training dataset is replaced by a small set of inducing points to address
the scalability issue. Output response vectors are represented by a linear
combination of independent latent functions, forming a flexible kernel
structure to handle multiple responses that might have distinct behaviors.
Comparative studies demonstrate that the proposed method scales well for large
datasets with over 10^4 data points, while outperforming state-of-the-art
machine learning methods without requiring much hyperparameter tuning. In
addition, an interpretable latent space is obtained to draw insights into the
effect of categorical factors, such as those associated with building blocks of
architectures and element choices in metamaterial and materials design. Our
approach is demonstrated for machine learning of ternary oxide materials and
topology optimization of a multiscale compliant mechanism with aperiodic
microstructures and multiple materials.
