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.

We review progress in neural network (NN) based methods for the construction of interatomic potentials from discrete samples (such as ab initio energies) for applications in classical and quantum dynamics including reaction dynamics and computational spectroscopy. The main focus is on methods for building molecular potential energy surfaces (PES) in internal coordinates that explicitly include all many-body contributions, even though some of the methods we review limit the degree of coupling, due either to a desire to limit computational cost or to limited data. Explicit and direct treatment of all many-body contributions is only practical for sufficiently small molecules, which are therefore our primary focus. This includes small molecules on surfaces. We consider direct, single NN PES fitting as well as more complex methods which impose structure (such as a multi-body representation) on the PES function, either through the architecture of one NN or by using multiple NNs. We show how NNs are effective in building representations with low-dimensional functions, including dimensionality reduction. We consider NN based approaches to build PESs in the sums-of-product form important for quantum dynamics, ways to treat symmetry, and issues related to sampling data distributions and the relation between PES errors and errors in observables. We highlight combinations of NNs with other ideas such as permutationally invariant polynomials or sums of environment-dependent atomic contributions which have recently emerged as powerful tools for building highly accurate PESs for relatively large molecular and reactive systems.

We combine the high dimensional model representation (HDMR) idea of Rabitz and co-workers [J. Phys. Chem. 110, 2474 (2006)] with neural network (NN) fits to obtain an effective means of building multidimensional potentials. We verify that it is possible to determine an accurate many-dimensional potential by doing low dimensional fits. The final potential is a sum of terms each of which depends on a subset of the coordinates. This form facilitates quantum dynamics calculations. We use NNs to represent HDMR component functions that minimize error mode term by mode term. This NN procedure makes it possible to construct high-order component functions which in turn enable us to determine a good potential. It is shown that the number of available potential points determines the order of the HDMR which should be used.

It is shown that neural networks (NNs) are efficient and effective tools for fitting potential energy surfaces. For H2O, a simple NN approach works very well. To fit surfaces for HOOH and H2CO, we develop a nested neural network technique in which we first fit an approximate NN potential and then use another NN to fit the difference of the true potential and the approximate potential. The root-mean-square error (RMSE) of the H2O surface is 1 cm(-1). For the 6-D HOOH and H2CO surfaces, the nested approach does almost as well attaining a RMSE of 2 cm(-1). The quality of the NN surfaces is verified by calculating vibrational spectra. For all three molecules, most of the low-lying levels are within 1 cm(-1) of the exact results. On the basis of these results, we propose that the nested NN approach be considered a method of choice for both simple potentials, for which it is relatively easy to guess a good fitting function, and complicated (e.g., double well) potentials for which it is much harder to deduce an appropriate fitting function. The number of fitting parameters is only moderately larger for the 6-D than for the 3-D potentials, and for all three molecules, decreasing the desired RMSE increases only slightly the number of required fitting parameters (nodes). NN methods, and in particular the nested approach we propose, should be good universal potential fitting tools.

We summarize the development and utilization of organic interfacial materials in solar cells, photodetectors and light-emitting diodes based on organic–inorganic halide perovskites.