Gaussian process model of 51-dimensional potential energy surface for protonated imidazole dimer

Abstract

The goal of the present work is to obtain accurate potential energy surfaces (PESs) for high-dimensional molecular systems with a small number of ab initio calculations in a system-agnostic way. We use probabilistic modeling based on Gaussian processes (GPs). We illustrate that it is possible to build an accurate GP model of a 51-dimensional PES based on 5000 randomly distributed ab initio calculations with a global accuracy of <0.2 kcal/mol. Our approach uses GP models with composite kernels designed to enhance the Bayesian information content and represents the global PES as a sum of a full-dimensional GP and several GP models for molecular fragments of lower dimensionality. We demonstrate the potency of these algorithms by constructing the global PES for the protonated imidazole dimer, a molecular system with 19 atoms. We illustrate that GP models thus constructed can extrapolate the PES from low energies (<10 000 cm−1), yielding a PES at high energies (>20 000 cm−1). This opens the prospect for new applications of GPs, such as mapping out phase transitions by extrapolation or accelerating Bayesian optimization, for high-dimensional physics and chemistry problems with a restricted number of inputs, i.e., for high-dimensional problems where obtaining training data is very difficult.

Gaussian Processes with Spectral Delta kernel for higher accurate Potential Energy surfaces for large molecules

“…These kernel structure discovery methods have been demonstrated to extrapolate physical observables accurately enough to detect phase transitions 34,35 . Additionally, GPs with complex kernels have shown the possibility to predict accurate energies for PES trained only with low energy points 17,18 . The computational complexity of Gaussian processes compounded with the fact that kernel search requires training many such models has resulted in an inability to use full available datasets [17][18][19] .…”

“…Interpolation of potential energy surfaces (PESs) is one of the most common applications of supervised machine learning (ML) methods. Commonly, this is done with deep neural networks (DNNs) 1-13 , parametric models, or kernel models, such as Gaussian Processes (GPs) or kernel ridge regression (KRR) 1, [14][15][16][17][18][19][20][21][22][23][24][25] . While both methodologies have proven to be flexible enough, GPs require less "tunning" compared to NNs, where the search for an optimal architecture could be computationally demanding 26,27 .…”

“…Although kernel structure search has been shown to accurately interpolate of PESs, they are not often applied to study high-dimensional physical systems due to a lack of accuracy and significant computational cost of O(N 3 ), where N is the number of training points. [16][17][18][19] . Typically, approximations must be used to achieve the required scalability, which can degrade the predictive performance of GP models even further.…”

“…In the following sections, we briefly introduce GPs and review how complex kernels can be constructed by using Bochners' theorem, a strategy known as spectral kernel learning. We interpolate the PES of four different chemical systems, e.g., protonated Imidazole dimer 18 , Benzene , Malonaldehyde, and Ethanol 41,42 . We also illustrate that by using GPytorch 46 and KEOPS 47 , modern deeplearning libraries, combined with GPUs we can efficiently train full GPs with 20 000 training points without relying on the Nyström approximation.…”

Gaussian process models of potential energy surfaces with boundary optimization

“…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).…”