# A smooth basis for atomistic machine learning

arXiv: https://arxiv.org/abs/2209.01948 · HTML: https://mlsanity.com/paper/2209.01948/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 46 references that are in the map.
- First posted on arXiv 5 Sep 2022.

## Builds on (most cited first, 10 of 10)

- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [DScribe: Library of Descriptors for Machine Learning in Materials Science](https://mlsanity.com/paper/1904.08875/) (2019)
- [Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistry](https://mlsanity.com/paper/2105.14655/) (2021)
- [Unified theory of atom-centered representations and message-passing machine-learning schemes](https://mlsanity.com/paper/2202.01566/) (2022)
- [Improving Sample and Feature Selection with Principal Covariates Regression](https://mlsanity.com/paper/2012.12253/) (2020)
- [The role of feature space in atomistic learning](https://mlsanity.com/paper/2009.02741/) (2020)
- [Optimal radial basis for density-based atomic representations](https://mlsanity.com/paper/2105.08717/) (2021)
- [Feature Optimization for Atomistic Machine Learning Yields A Data-Driven Construction of the Periodic Table of the Elements](https://mlsanity.com/paper/1807.00236/) (2018)

## Built on by (most cited first, 4 of 4)

- [Smooth, exact rotational symmetrization for deep learning on point clouds](https://mlsanity.com/paper/2305.19302/) (2023)
- [Cartesian atomic cluster expansion for machine learning interatomic potentials](https://mlsanity.com/paper/2402.07472/) (2024)
- [Completeness of Atomic Structure Representations](https://mlsanity.com/paper/2302.14770/) (2023)
- [Wigner kernels: body-ordered equivariant machine learning without a basis](https://mlsanity.com/paper/2303.04124/) (2023)

## Abstract (arXiv)

Machine learning frameworks based on correlations of interatomic positions begin with a discretized description of the density of other atoms in the neighbourhood of each atom in the system. Symmetry considerations support the use of spherical harmonics to expand the angular dependence of this density, but there is as yet no clear rationale to choose one radial basis over another. Here we investigate the basis that results from the solution of the Laplacian eigenvalue problem within a sphere around the atom of interest. We show that this generates the smoothest possible basis of a given size within the sphere, and that a tensor product of Laplacian eigenstates also provides the smoothest possible basis for expanding any higher-order correlation of the atomic density within the appropriate hypersphere. We consider several unsupervised metrics of the quality of a basis for a given dataset, and show that the Laplacian eigenstate basis has a performance that is much better than some widely used basis sets and is competitive with data-driven bases that numerically optimize each metric. In supervised machine learning tests, we find that the optimal function smoothness of the Laplacian eigenstates leads to comparable or better performance than can be obtained from a data-driven basis of a similar size that has been optimized to describe the atom-density correlation for the specific dataset. We conclude that the smoothness of the basis functions is a key and hitherto largely overlooked aspect of successful atomic density representations.