Improved description of atomic environments using low-cost polynomial functions with compact support

Abstract

The prediction of chemical properties using machine learning techniques calls for a set of appropriate descriptors that accurately describe atomic and, on a larger scale, molecular environments. A mapping of conformational information on a space spanned by atom-centred symmetry functions (SF) has become a standard technique for energy and force predictions using high-dimensional neural network potentials (HDNNP). An appropriate choice of SFs is particularly crucial for accurate force predictions. Established atom-centred SFs, however, are limited in their flexibility, since their functional form restricts the angular domain that can be sampled without introducing problematic derivative discontinuities. Here, we introduce a class of atom-centred SFs based on polynomials with compact support called polynomial symmetry functions (PSF), which enable a free choice of both, the angular and the radial domain covered. We demonstrate that the accuracy of PSFs is either on par or considerably better than that of conventional, atom-centred SFs. In particular, a generic set of PSFs with an intuitive choice of the angular domain inspired by organic chemistry considerably improves prediction accuracy for organic molecules in the gaseous and liquid phase, with reductions in force prediction errors over a test set approaching 50% for certain systems. Contrary to established atom-centred SFs, computation of PSF does not involve any exponentials, and their intrinsic compact support supersedes use of separate cutoff functions, facilitating the choice of their free parameters. Most importantly, the number of floating point operations required to compute polynomial SFs introduced here is considerably lower than that of other state-of-the-art SFs, enabling their efficient implementation without the need of highly optimised code structures or caching, with speedups with respect to other state-of-the-art SFs reaching a factor of 4.5 to 5. This low-effort performance benefit substantially simplifies their use in new programs and emerging platforms such as graphical processing units. Overall, polynomial SFs with compact support improve accuracy of both, energy and force predictions with HDNNPs while enabling significant speedups compared to their well-established counterparts.

Systematic Identification of Atom-Centered Symmetry Functions for the Development of Neural Network Potentials

“…Using the shortlisted set of ACSFs and the training data, we develop BP-NNPs for water and Cu 2 S systems within the n2p2 package. ,,, The NN for both the systems had two hidden layers with 25 neurons each. A soft plus activation function was used in the hidden layers, and a linear activation was used for the output layer.…”

“…While the location of the maxima in the angular ACSFs can be moved by applying a phase shift within the cosine function, this introduces a nonzero force derivative. To overcome these issues, Bircher et al recently proposed polynomial-type SFs (PTSFs), whose functional forms are given in eqs –. f c poly = { true f poly 2 true( r r c true) , 0 < r ≤ r c 0 , r > r c where f poly 2 = x 3 ( x false( 15 − 6 x false) − 10 ) + 1 …”

New venues in electron density analysis

“…This is generally done in the form of atomic environment vectors (AEVs), which contain the desired information in a computer-understandable manner. Thus, a lot of effort has been made in recent years within the scientific community to develop suitable featurization approaches 100,101 such as the Bag of Bonds scheme, 36 Coulomb matrices, 46,102 Atom Centered Symmetry Functions (ACSF), 103,104 along with its many different flavors, [105][106][107] or the more recent Gaussian moments, 108 which would ultimately allow for the construction of reliable ML approaches. Despite being rigorous and useful, most strategies still recover little to no information about the actual chemistry, and are usually focused on encoding the radial and angular environments and the chemical composition of the system, just to mimic the external potential without further chemical insight.…”

Subsystem Density Functional Theory Augmented by a Delta Learning Approach to Achieve Kohn–Sham Accuracy

“…Weights and biases are determined by fitting the total energy and the individual atomic forces to a reference data set of configurations. The machine learning training has been carried out using the n2p2 code. − Further details about the specific implementation and the choice of the symmetry functions are described in the related literature. , …”