Easy representation of multivariate functions with low-dimensional terms via Gaussian process regression kernel design: applications to machine learning of potential energy surfaces and kinetic energy densities from sparse data

Abstract

We show that Gaussian process regression (GPR) allows representing multivariate functions with low-dimensional terms via kernel design. When using a kernel built with HDMR (High-dimensional model representation), one obtains a similar type of representation as the previously proposed HDMR-GPR scheme while being faster and simpler to use. We tested the approach on cases where highly accurate machine learning is required from sparse data by fitting potential energy surfaces and kinetic energy densities.

Computational vibrational spectroscopy of molecule–surface interactions: what is still difficult and what can be done about it

“…Replacing the N-mode representation of the potential, which is a particular case of high-dimensional model representation (HDMR), 129,130 the so-called cut-HDMR, with random-sampling (RS-) HDMR, 131,132 in which all mode terms are computable from one and the same set of ab initio data, could simplify the applicability of VSCF and VCI methods to molecule-surface systems. In particular, combinations of HDMR with machine learning algorithms [133][134][135][136][137] simplify constructing of the PES in the RS-HDMR form. The RS-HDMR-ML approach has been demonstrated on free molecules 135 and is yet to be explored for the spectroscopy of molecules on surfaces.…”

Advanced Machine Learning Methods for Learning from Sparse Data in High-Dimensional Spaces: A Perspective on Uses in The Upstream of Development of Novel Energy Technologies

“…and are equal to multidimensional integrals (specifically, (D -d) dimensional integrals need to be computed for d-dimensional component functions) which are quite difficult to compute [113,114]; further, any lower-dimensional component functions need to be constructed first before constructing d-dimensional functions. We proposed an extension of HMDR whereby we do not build the entire expansion but directly represent the function f with d-dimensional functions [105,[116][117][118][119].…”

“…HDMR component functions can be built with machine learning methods like neural networks [116,118,120] or Gaussian process regressions [105,106,119]. Practically this can be done by fitting component functions one at a time to the difference of the value of the function f at the training points and the sum of all other component functions:…”

“…(3.3.1.1) is that existing NN engines with simple architecture can be used and work well. When using GPR, one can achieve an HDMR form of also by using an HDMR-type kernel [119,121]: �. It can be used, in particular, to prune unnecessary component functions and thereby alleviate the issue of combinatorial scaling of the number of HDMR terms with D and d [106,119].…”

“…When using GPR, one can achieve an HDMR form of also by using an HDMR-type kernel [119,121]: �. It can be used, in particular, to prune unnecessary component functions and thereby alleviate the issue of combinatorial scaling of the number of HDMR terms with D and d [106,119]. This scaling is a major bottleneck in highdimensional problems.…”