# Interpolation and extrapolation of global potential energy surfaces for polyatomic systems by Gaussian processes with composite kernels

## TL;DR

This paper introduces an improved Gaussian process approach with composite kernels for accurately modeling global potential energy surfaces of polyatomic molecules, enabling effective interpolation and extrapolation with fewer data points.

## Contribution

The study presents an automated kernel selection method, iterative kernel complexity enhancement, and demonstrates physical extrapolation of PES using Gaussian processes with composite kernels.

## Key findings

- Achieved 65.8 cm$^{-1}$ RMS error with 500 points for H$_3$O$^+$ PES.
- Extended PES accuracy to 21,000 cm$^{-1}$ using only low-energy points.
- Validated PES extrapolation at high energies based on low-energy data.

## 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 {\it 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$^{-1}$.