# Gaussian Moments as Physically Inspired Molecular Descriptors for Accurate and Scalable Machine Learning Potentials

arXiv: https://arxiv.org/abs/2109.07421 · HTML: https://mlsanity.com/paper/2109.07421/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 51 references that are in the map.
- First posted on arXiv 15 Sep 2021.

## Builds on (most cited first, 5 of 5)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Hierarchical modeling of molecular energies using a deep neural network](https://mlsanity.com/paper/1710.00017/) (2017)

## Built on by (most cited first, 15 of 15)

- [Uncertainty-biased molecular dynamics for learning uniformly accurate interatomic potentials](https://mlsanity.com/paper/2312.01416/) (2023)
- [Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing](https://mlsanity.com/paper/2405.14253/) (2024)
- [Transfer learning for chemically accurate interatomic neural network potentials](https://mlsanity.com/paper/2212.03916/) (2022)
- [Tensor-reduced atomic density representations](https://mlsanity.com/paper/2210.01705/) (2022)
- [Fast and Sample-Efficient Interatomic Neural Network Potentials for Molecules and Materials Based on Gaussian Moments](https://mlsanity.com/paper/2109.09569/) (2021)
- [Wigner kernels: body-ordered equivariant machine learning without a basis](https://mlsanity.com/paper/2303.04124/) (2023)
- [FeNNol: an Efficient and Flexible Library for Building Force-field-enhanced Neural Network Potentials](https://mlsanity.com/paper/2405.01491/) (2024)
- [Physics-Informed Weakly Supervised Learning for Interatomic Potentials](https://mlsanity.com/paper/2408.05215/) (2024)
- [Evaluation of uncertainty estimations for Gaussian process regression based machine learning interatomic potentials](https://mlsanity.com/paper/2410.20398/) (2024)
- [Breaking scaling relations with inverse catalysts: a machine learning exploration of trends in CO2 hydrogenation energy barriers](https://mlsanity.com/paper/2504.16493/) (2025)
- [Efficient Parallelization of Message Passing Neural Network Potentials for Large-scale Molecular Dynamics](https://mlsanity.com/paper/2505.06711/) (2025)
- [Simplicial Message Passing for Chemical Property Prediction](https://mlsanity.com/paper/2307.05392/) (2023)
- [Predicting Properties of Periodic Systems from Cluster Data: A Case Study of Liquid Water](https://mlsanity.com/paper/2312.01414/) (2023)
- [Thermally Averaged Magnetic Anisotropy Tensors via Machine Learning Based on Gaussian Moments](https://mlsanity.com/paper/2312.01415/) (2023)
- [Performance of universal machine-learned potentials with explicit long-range interactions in biomolecular simulations](https://mlsanity.com/paper/2508.10841/) (2025)

## Abstract (arXiv)

Machine learning techniques allow a direct mapping of atomic positions and nuclear charges to the potential energy surface with almost ab-initio accuracy and the computational efficiency of empirical potentials. In this work we propose a machine learning method for constructing high-dimensional potential energy surfaces based on feed-forward neural networks. As input to the neural network we propose an extendable invariant local molecular descriptor constructed from geometric moments. Their formulation via pairwise distance vectors and tensor contractions allows a very efficient implementation on graphical processing units (GPUs). The atomic species is encoded in the molecular descriptor, which allows the restriction to one neural network for the training of all atomic species in the data set. We demonstrate that the accuracy of the developed approach in representing both chemical and configurational spaces is comparable to the one of several established machine learning models. Due to its high accuracy and efficiency, the proposed machine-learned potentials can be used for any further tasks, for example the optimization of molecular geometries, the calculation of rate constants or molecular dynamics.