# WACSF - Weighted Atom-Centered Symmetry Functions as Descriptors in Machine Learning Potentials

arXiv: https://arxiv.org/abs/1712.05861 · HTML: https://mlsanity.com/paper/1712.05861/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Dec 2017.

## Built on by (most cited first, 9 of 9)

- [SpookyNet: Learning Force Fields with Electronic Degrees of Freedom and Nonlocal Effects](https://mlsanity.com/paper/2105.00304/) (2021)
- [A practical guide to machine learning interatomic potentials -- Status and future](https://mlsanity.com/paper/2503.09814/) (2025)
- [Tensor-reduced atomic density representations](https://mlsanity.com/paper/2210.01705/) (2022)
- [Quantum neural networks force fields generation](https://mlsanity.com/paper/2203.04666/) (2022)
- [Lifelong Machine Learning Potentials](https://mlsanity.com/paper/2303.05911/) (2023)
- [LFaB: Low fidelity as Bias for Active Learning in the chemical configuration space](https://mlsanity.com/paper/2508.15577/) (2025)
- [Pre-training Graph Neural Networks with Structural Fingerprints for Materials Discovery](https://mlsanity.com/paper/2503.01227/) (2025)
- [Unsupervised Atomic Data Mining via Multi-Kernel Graph Autoencoders for Machine Learning Force Fields](https://mlsanity.com/paper/2509.12358/) (2025)
- [Lagrangian Density Space-Time Deep Neural Network Topology](https://mlsanity.com/paper/2207.12209/) (2022)

## Abstract (arXiv)

We introduce weighted atom-centered symmetry functions (wACSFs) as descriptors of a chemical system's geometry for use in the prediction of chemical properties such as enthalpies or potential energies via machine learning. The wACSFs are based on conventional atom-centered symmetry functions (ACSFs) but overcome the undesirable scaling of the latter with increasing number of different elements in a chemical system. The performance of these two descriptors is compared using them as inputs in high-dimensional neural network potentials (HDNNPs), employing the molecular structures and associated enthalpies of the 133855 molecules containing up to five different elements reported in the QM9 database as reference data. A substantially smaller number of wACSFs than ACSFs is needed to obtain a comparable spatial resolution of the molecular structures. At the same time, this smaller set of wACSFs leads to significantly better generalization performance in the machine learning potential than the large set of conventional ACSFs. Furthermore, we show that the intrinsic parameters of the descriptors can in principle be optimized with a genetic algorithm in a highly automated manner. For the wACSFs employed here, we find however that using a simple empirical parametrization scheme is sufficient in order to obtain HDNNPs with high accuracy.