Interpolation and Extrapolation of Global Potential Energy Surfaces for Polyatomic Systems by Gaussian Processes with Composite Kernels

Abstract

Gaussian process regression has recently emerged as a powerful, system-agnostic tool for building global potential energy surfaces (PES) of polyatomic molecules. While the accuracy of GP models of PES increases with the number of potential energy points, so does the numerical difficulty of training and evaluating GP models. Here, we demonstrate an approach to improve the accuracy of global PES without increasing the number of energy points. The present work reports four important results. First, we show that the selection of the best kernel function for GP models of PES can be automated using the Bayesian information criterion as a model selection metric.Second, we demonstrate that GP models of PES trained by a small number of energy points can be significantly improved by iteratively increasing the complexity of GP kernels. The composite kernels thus obtained maximize the accuracy of GP models for a given distribution of potential energy points. Third, we show that the accuracy of the GP models of PES with composite kernels can be further improved by varying the training point distributions. Fourth, we show that GP models with composite kernels can be used for physical extrapolation of PES. We illustrate the approach by constructing the six-dimensional PES for H 3 O + . For the interpolation problem, we show that this algorithm produces a global six-dimensional PES for H 3 O + in the energy range between zero and 21, 000 cm −1 with the root mean square error 65.8 cm −1 using only 500 randomly selected ab initio points as input. To illustrate extrapolation, we produce the PES at high energies using the energy points at low energies as input. We show that one can obtain an accurate global fit of the PES extending to 21, 000 cm −1 based on 1500 potential energy points at energies below 10, 000 cm −1 . arXiv:1907.08717v1 [physics.chem-ph]

Full-dimensional potential energy surface for acetylacetone and tunneling splittings

“…A strength of the GPR approach is that it does produce a quality PES from a relatively small amount of data. 3,15 On the other hand this approach becomes computationally very demanding for datasets with more than 10 4 energies. By contrast, PIP/FI-NN PESs with 4-7 atoms typically use more than 10 4 energies.…”

Gaussian process models of potential energy surfaces with boundary optimization

“…For any intermolecular interaction the PES can be thought of as a multivariate function f (x), x ∈ R Z , where x is a vector of inputs and Z is the number of elements in x. Here the inputs are inverse interatomic separations, though promising results have also been obtained with Morse variables 24,55 . Only pair intermolecular interactions are considered, although GP models have been applied successfully to non-additive interactions 17,19,40 and the methodology outlined here can be extended to such cases straightforwardly.…”

“…For two training strategies that achieve the same error, that which does so with fewer training points is more computationally efficient. Attempts to develop more computationally efficient training strategies have involved active learning or sequential design methods 19,29,59 , composite kernels 24 and new sampling methods 27,60 .…”

“…This approach proceeds by training a statistical technique on a relatively small set of data from ab initio calculations on the PES of interest, known as the training set. Many such techniques have been employed to predict the energy in these algorithms, including neural networks [9][10][11][12][13] , moment tensors [14][15][16] and Gaussian processes [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] (GPs).…”

Exchange Spin Coupling from Gaussian Process Regression

“…Experience so far shows that it is suitable for rather small training sets, which is a favorable property given that reference data for J are relatively costly to generate. GPR has been used in chemistry for, e.g., fitting repulsive potentials in tight-binding DFT 77 , for correcting empirical dispersion models 78 , for evaluating work functions 79 , for calculating vibrational Raman spectra 80 , for transition-state 81 and molecular-structure optimization [82][83][84][85] , for fitting potential-energy surfaces 86,87 , and for the error-controlled exploration of reaction networks 88,89 . We will employ GPR here for asking to what extent it is possible to machine-learn J, as compared with other molecular properties.…”