Neural networks vs Gaussian process regression for representing potential energy surfaces: A comparative study of fit quality and vibrational spectrum accuracy

Abstract

For molecules with more than three atoms, it is difficult to fit or interpolate a potential energy surface (PES) from a small number of (usually ab initio) energies at points. Many methods have been proposed in recent decades, each claiming a set of advantages. Unfortunately, there are few comparative studies. In this paper, we compare neural networks (NNs) with Gaussian process (GP) regression. We re-fit an accurate PES of formaldehyde and compare PES errors on the entire point set used to solve the vibrational Schrödinger equation, i.e., the only error that matters in quantum dynamics calculations. We also compare the vibrational spectra computed on the underlying reference PES and the NN and GP potential surfaces. The NN and GP surfaces are constructed with exactly the same points, and the corresponding spectra are computed with the same points and the same basis. The GP fitting error is lower, and the GP spectrum is more accurate. The best NN fits to 625/1250/2500 symmetry unique potential energy points have global PES root mean square errors (RMSEs) of 6.53/2.54/0.86 cm, whereas the best GP surfaces have RMSE values of 3.87/1.13/0.62 cm, respectively. When fitting 625 symmetry unique points, the error in the first 100 vibrational levels is only 0.06 cm with the best GP fit, whereas the spectrum on the best NN PES has an error of 0.22 cm, with respect to the spectrum computed on the reference PES. This error is reduced to about 0.01 cm when fitting 2500 points with either the NN or GP. We also find that the GP surface produces a relatively accurate spectrum when obtained based on as few as 313 points.

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 for the solution of the Schrödinger equation

“…Only 5000 collocation points and on the order of 50 neurons were used. This distribution was found to be more efficient than a uniform grid, similar to the experience when fitting potential energy surfaces or solving the electronic Schrödinger equation [19,28,68,69]. This approach was also used by the same authors to represent vibrational wavefunctions of a real molecule, H 2 O [70].…”

“…This has the advantage that the wavefunction expansion need not be integrable or differentiable everywhere, allowing for use of discontinuous or non-integrable functions (the common sigmoidal functions e.g. can have infinite integrals over their infinite support) which only need to be smooth at the collocation points [23,28,60]. Lagaris et al also solved the variational problem with a NN wavefunction.…”

“…We have shown, however, that this issue is easily circumvented by the application of the exact KEO numerically [73]. Sub-cm −1 accuracy for a four-atomic molecule (d = 6) can be obtained with equation (4) when using about 100 000 collocation points and about 30 000 basis functions without ML optimization of the basis or of the points [28,73,74]. It is expected that significant gains can be achieved for computational vibrational spectroscopy by basis set and point optimization with ML.…”

“…This followed a series of developments since the mid-2000s from the first spectroscopically accurate molecular potential energy surfaces by Manzhos and Carrington [19] to accurate potentials suitable for large-scale solid state and nanoparticles simulations by Behler et al [20][21][22]. These developments were achieved by combining general properties of ML methods, mostly NN (neural networks) and GPR (Gaussian process regression), with modifications specific to the application, such as sum-of-product representations [23] or expansions of energy over orders of coupling [24,25] or its representation as a sum of atomic energies dependent on the chemical environment [26,27] as well as data selection schemes specific to the application [19,28]. The field of machine learning for interatomic potentials has been active and diverse and will not be reviewed here.…”