Assessing Gaussian Process Regression and Permutationally Invariant Polynomial Approaches To Represent High-Dimensional Potential Energy Surfaces

Abstract

The mathematical representation of large data sets of electronic energies has seen substantial progress in the past 10 years. The so-called Permutationally Invariant Polynomial (PIP) representation is one established approach. This approach dates from 2003, when a global potential energy surface (PES) for CH was reported using a basis of polynomials that are invariant with respect to the 120 permutations of the five equivalent H atoms. More recently, several approaches from "machine learning" have been applied to fit these large data sets. Gaussian Process (GP) regression is such an approach. Here, we consider the implementation of the (full) GP due to Krems and co-workers, with a modification that renders it permutationally invariant, which we denote by PIP-GP. This modification uses the approach of Guo and co-workers and later extended by Zhang and co-workers, to achieve permutational invariance for neural-network fits. The PIP, GP, and PIP-GP approaches are applied to four case studies for fitting data sets of electronic energies: HO, OCHCO, and HCO/ cis-HCOH/ trans-HCOH with the goal of assessing precision, accuracy in normal-mode analysis and barrier heights, and timings. We also report an application to (HCOOH), where the full PIP approach is possible but where the PIP-GP one is not feasible. However, by replicating data, which is feasible in this case, the GP approach is able to represent the data with precision comparable to that of the PIP approach. We examine these assessments for varying sizes of data sets in each case to determine the dependence of properties of the fits on the training data size. We conclude with some comments on the different aspects of computational effort of the PIP, GP, and PIP-GP approaches and also challenges these methods face for more "rugged" PESs, exemplified here by HCO/ cis-HCOH/ trans-HCOH.

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

“…In all previous studies [21][22][23][24][25][26][27][28][29][30], the GP models of PES were constructed with a fixed kernel function, such as one of the kernel functions (3) - (5). In the present work, we follow…”

“…In previous work [18][19][20][21][22][23][24][25][26][27][28][29][30], GP models of PES for polyatomic systems were constructed with one of the simple kernels (2) - (5). The focus has been on improving the accuracy of GP models by increasing the number of potential energy points in the training set y and selecting an optimal distribution of these points in the configuration space.…”

“…The present work reports several significant results. We first consider the interpolation problem, as in previous work [18][19][20][21][22][23][24][25][26][27][28][29][30], and show that the accuracy of the resulting GP models of the PES can be significantly enhanced by increasing the complexity of GP model kernels.…”

Bayesian optimization for the inverse scattering problem in quantum reaction dynamics

“…where E i are the ab initio energy points and the weights (w i ) are determined by the two-tiered GP model to yield the best outcome of the quantum scattering calculation. GPs have been previously used for interpolating PES for molecular dynamics applications [12][13][14][15][16], spectroscopic line calculations [17,18] and molecular scattering calculations [19,20]. We emphasize that equation (3) is not a fit of the PES but a non-parametric regression.…”

Machine Learning Configuration Interaction for ab Initio Potential Energy Curves

“…This includes recent work using Gaussian process regression [1][2][3] or artificial neural networks (ANNs). 4 Alternatively one may employ machine learning, often trained on density-functional theory (DFT) results, to predict properties from electronic structure data.…”