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.

In this paper we propose a new quadrature scheme for computing vibrational spectra and apply it, using a Lanczos algorithm, to CH(3)CN. All 12 coordinates are treated explicitly. We need only 157'419'523 quadrature points. It would not be possible to use a product Gauss grid because 33 853 318 889 472 product Gauss points would be required. The nonproduct quadrature we use is based on ideas of Smolyak, but they are extended so that they can be applied when one retains basis functions θ(n(1))(r(1))···θ(n(D))(r(D)) that satisfy the condition α(1)n(1) + ··· + α(D)n(D) ≤ b, where the α(k) are integers. We demonstrate that it is possible to exploit the structure of the grid to efficiently evaluate the matrix-vector products required to use the Lanczos algorithm.

By using exponential activation functions with a neural network (NN) method we show that it is possible to fit potentials to a sum-of-products form. The sum-of-products form is desirable because it reduces the cost of doing the quadratures required for quantum dynamics calculations. It also greatly facilitates the use of the multiconfiguration time dependent Hartree method. Unlike potfit product representation algorithm, the new NN approach does not require using a grid of points. It also produces sum-of-products potentials with fewer terms. As the number of dimensions is increased, we expect the advantages of the exponential NN idea to become more significant.

We present a parallelized contracted basis-iterative method for calculating numerically exact vibrational energy levels of CH(5)(+) (a 12-dimensional calculation). We use Radau polyspherical coordinates and basis functions that are products of eigenfunctions of bend and stretch Hamiltonians. The bend eigenfunctions are computed in a nondirect product basis with more than 200x10(6) functions and the stretch functions are computed in a product potential optimized discrete variable basis. The basis functions have amplitude in all of the 120 equivalent minima. Many low-lying levels are well converged. We find that the energy level pattern is determined in part by the curvature and width of the valley connecting the minima and in part by the slope of the walls of this valley but does not depend on the height or shape of the barriers separating the minima.

The size of the quadrature grid required to compute potential matrix elements impedes solution of the vibrational Schrodinger equation if the potential does not have a simple form. This quadrature grid-size problem can make computing (ro)vibrational spectra impossible even if the size of the basis used to construct the Hamiltonian matrix is itself manageable. Potential matrix elements are typically computed with a direct product Gauss quadrature whose grid size scales as N(D), where N is the number of points per coordinate and D is the number of dimensions. In this article we demonstrate that this problem can be mitigated by using a pruned basis set and a nonproduct Smolyak grid. The constituent 1D quadratures are designed for the weight functions important for vibrational calculations. For the SF(6) stretch problem (D=6) we obtain accurate results with a grid that is more than two orders of magnitude smaller than the direct product Gauss grid. If D>6 we expect an even bigger reduction.

We propose a method for fitting potential energy surfaces with a sum of component functions of lower dimensionality. This form facilitates quantum dynamics calculations. We show that it is possible to reduce the dimensionality of the component functions by introducing new and redundant coordinates obtained with linear transformations. The transformations are obtained from a neural network. Different coordinates are used for different component functions and the new coordinates are determined as the potential is fitted. The quality of the fits and the generality of the method are illustrated by fitting reference potential surfaces of hydrogen peroxide and of the reaction OH+H(2)-->H(2)O+H.

In this paper we report transition frequencies and line strengths computed for H(2)O-H(2) and compare with the experimental observations of [M. J. Weida and D. J. Nesbitt, J. Chem. Phys. 110, 156 (1999)]. To compute the spectra we use a symmetry adapted Lanczos algorithm and an uncoupled product basis set. Our results corroborate the assignments of Weida and Nesbitt and there is good agreement between calculated and observed transitions. Possible candidates for lines that Weida and Nesbitt were not able to assign are presented. Several other bands that may be observable are also discovered. Although all the observed bands are associated with states localized near the global potential minimum, at which H(2)O acts as proton acceptor, a state with significant amplitude near the T-shape secondary potential minimum at which H(2)O acts as proton donor is identified by examining many different probability density plots.

In recent years, we have been witnessing a paradigm shift in computational materials science. In fact, traditional methods, mostly developed in the second half of the XXth century, are being complemented, extended, and sometimes even completely replaced by faster, simpler, and often more accurate approaches. The new approaches, that we collectively label by machine learning, have their origins in the fields of informatics and artificial intelligence, but are making rapid inroads in all other branches of science. With this in mind, this Roadmap article, consisting of multiple contributions from experts across the field, discusses the use of machine learning in materials science, and share perspectives on current and future challenges in problems as diverse as the prediction of materials properties, the construction of force-fields, the development of exchange correlation functionals for density-functional theory, the solution of the many-body problem, and more. In spite of the already numerous and exciting success stories, we are just at the beginning of a long path that will reshape materials science for the many challenges of the XXIth century.