Neural network‐based approaches for building high dimensional and quantum dynamics‐friendly potential energy surfaces

Abstract

Development and applications of neural network (NN)-based approaches for representing potential energy surfaces (PES) of bound and reactive molecular systems are reviewed. Specifically, it is shown that when the density of ab initio points is low, NNs-based potentials with multibody or multimode structure are advantageous for representing high-dimensional PESs. Importantly, with an appropriate choice of the neuron activation function, PESs in the sum-of-products form are naturally obtained, thus addressing a bottleneck problem in quantum dynamics. The use of NN committees is also analyzed and it is shown that while they are able to reduce the fitting error, the reduction is limited by the nonrandom nature of the fitting error. The approaches described here are expected to be directly applicable in other areas of science and engineering where a functional form needs to be constructed in an unbiased way from sparse data.

Machine learning for the solution of the Schrödinger equation

“…The ML methods considered above are ignorant of symmetry. This can be addressed by explicit or implicit symmetrization [46,98,136,157]. ML typically requires relatively costly optimization of parameters (NN) or hyper-parameters (GPR, KRR) or extensive evolutionary searches (GA-inspired schemes).…”

“…We only consider the use of ML for the solution of the vibrational and electronic SE or the KS equation or the Hohenberg-Kohn (HK) equation and not methods that aim to avoid such solutions (such as those directly mapping the molecular structure to the spectrum or energy or properties without solving for the density or wavefunction [29][30][31][32][33][34][35][36]); we also do not consider uses of ML for other types of quantum mechanical modelling or other types of differential or integral equations [37][38][39][40][41][42].We do not aim to present a comprehensive review but rather a survey allowing similarities and differences of ML uses in all these applications, as well as promising directions for future research, to transpire. This is not a review of ML methods; the key machine learning techniques that found use in quantum chemistry are well reviewed elsewhere [3-5, 43, 44], their description will not be repeated there; the reader is advised to consult the literature for their introduction, such as [45,46] for neural networks, [47,48] for Gaussian process regression (GPR), [49] for kernel ridge regression (KRR; note the similarity in the form of the function representation between GPR and KRR), [50] for genetic algorithms (GA) and [51] for particle swarm optimization. These methods as applied to quantum mechanics have recently been reviewed [44].…”

“…We do not aim to present a comprehensive review but rather a survey allowing similarities and differences of ML uses in all these applications, as well as promising directions for future research, to transpire. This is not a review of ML methods; the key machine learning techniques that found use in quantum chemistry are well reviewed elsewhere [3-5, 43, 44], their description will not be repeated there; the reader is advised to consult the literature for their introduction, such as [45,46] for neural networks, [47,48] for Gaussian process regression (GPR), [49] for kernel ridge regression (KRR; note the similarity in the form of the function representation between GPR and KRR), [50] for genetic algorithms (GA) and [51] for particle swarm optimization. These methods as applied to quantum mechanics have recently been reviewed [44].…”

Fitting potential energy surfaces to sum-of-products form with neural networks using exponential neurons

“…As such, a variety of different approaches to generating analytical forms for PESs have been developed and applied to a diversity of molecules and molecular complexes. Hence, PES¯tting has been and remains a vigorously studied problem; see, for example, a number of recent (since 2010) reviews [1][2][3][4][5][6][7][8][9] and the references cited therein. In this paper, the focus is on tting PESs based on high-level ab initio energies in sum-of-products form, and in particular using neural networks (NN) with exponential neurons.…”

An adaptive interpolation scheme for molecular potential energy surfaces

“…The two main categories for the representation of PESs can be divided into fitting methods and interpolation methods. Fitting methods include simple polynomial representations, many-body polynomials 3 and a broad variety of neural networks 4,5 . Interpolation methods proa) Electronic mail: mkowalew@uci.edu vide the possibility to represent the PES without the necessity of prior knowledge of its shape, and reproduce the function exactly at the sample points.…”