Functional Output Regression for Machine Learning in Materials Science

In recent years, there has been a rapid growth in the use of machine learning in material science. Conventionally, a trained predictive model describes a scalar output variable, such as thermodynamic, electronic, or mechanical properties, as a function of input descriptors that vectorize the compositional or structural features of any given material, such as molecules, chemical compositions, or crystalline systems. In machine learning of material data, on the other hand, the output variable is often given as a function. For example, when predicting the optical absorption spectrum of a molecule, the output variable is a spectral function defined in the wavelength domain. Alternatively, in predicting the microstructure of a polymer nanocomposite, the output variable is given as an image from an electron microscope, which can be represented as a two- or three-dimensional function in the image coordinate system. In this study, we consider two unified frameworks to handle such multidimensional or functional output regressions, which are applicable to a wide range of predictive analyses in material science. The first approach employs generative adversarial networks, which are known to exhibit outstanding performance in various computer vision tasks such as image generation, style transfer, and video generation. We also present another type of statistical modeling inspired by a statistical methodology referred to as functional data analysis. This is an extension of kernel regression to deal with functional outputs, and its simple mathematical structure makes it effective in modeling even with small amounts of data. We demonstrate the proposed methods through several case studies in materials science.