High dimensional model representation with machine-learned component functions: a powerful tool to learn multivariate functions from sparse data

Machine learning approaches including neural networks (NN) and Gaussian process regression (GPR) are finding widepread use to recover functional dependencies from multidimensional data. As powerful as these approaches are, they may fail when data density is low, which is always the case in highly-dimensional cases. Some methods like GPR also cannot easily work with large datasets. Using modified high dimensional model representation (HDMR) to represent a multivariate function with machine-learned lower-dimensional terms allows recovering functions from very sparse data, down to ~2 data per dimension. Sub-dimensional component functions are easier to fit and to use. Specifically here we present a HDMR-GPR combination where the use of GPR to represent component functions allows nonparametric (unbiased) representation and the possibility to work only with functions of desired dimensionality, obviating the need to build an expansion over orders of coupling. All component functions are determined from a single set of samples. We test the method by fitting potential energy surfaces of polyatomic molecules as well as by computing vibrational spectra.