# Optimal Invariant Bases for Atomistic Machine Learning

arXiv: https://arxiv.org/abs/2503.23515 · HTML: https://mlsanity.com/paper/2503.23515/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 81 references that are in the map.
- First posted on arXiv 30 Mar 2025.

## Builds on (most cited first, 22 of 22)

- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Tune: A Research Platform for Distributed Model Selection and Training](https://mlsanity.com/paper/1807.05118/) (2018)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Less is more: sampling chemical space with active learning](https://mlsanity.com/paper/1801.09319/) (2018)
- [TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials](https://mlsanity.com/paper/2306.06482/) (2023)
- [Hierarchical modeling of molecular energies using a deep neural network](https://mlsanity.com/paper/1710.00017/) (2017)
- [Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products](https://mlsanity.com/paper/2401.10216/) (2024)
- [Cartesian atomic cluster expansion for machine learning interatomic potentials](https://mlsanity.com/paper/2402.07472/) (2024)
- [Evaluation of the MACE Force Field Architecture: from Medicinal Chemistry to Materials Science](https://mlsanity.com/paper/2305.14247/) (2023)
- [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)
- [An introduction to graphical tensor notation for mechanistic interpretability](https://mlsanity.com/paper/2402.01790/) (2024)
- [Flexible Moment-Invariant Bases from Irreducible Tensors](https://mlsanity.com/paper/2503.21939/) (2025)

## Built on by (most cited first, 1 of 1)

- [Preserving Continuous Symmetry in Discrete Spaces: Geometric-Aware Quantization for SO(3)-Equivariant GNNs](https://mlsanity.com/paper/2603.05343/) (2026)

## Abstract (arXiv)

The representation of atomic configurations for machine learning models has led to the development of numerous descriptors, often to describe the local environment of atoms. However, many of these representations are incomplete and/or functionally dependent. Incomplete descriptor sets are unable to represent all meaningful changes in the atomic environment. Complete constructions of atomic environment descriptors, on the other hand, often suffer from a high degree of functional dependence, where some descriptors can be written as functions of the others. These redundant descriptors do not provide additional power to discriminate between different atomic environments and increase the computational burden. By employing techniques from the pattern recognition literature to existing atomistic representations, we remove descriptors that are functions of other descriptors to produce the smallest possible set that satisfies completeness. We apply this in two ways: first we refine an existing description, the Atomistic Cluster Expansion. We show that this yields a more efficient subset of descriptors. Second, we augment an incomplete construction based on a scalar neural network, yielding a new message-passing network architecture that can recognize up to 5-body patterns in each neuron by taking advantage of an optimal set of Cartesian tensor invariants. This architecture shows strong accuracy on state-of-the-art benchmarks while retaining low computational cost. Our results not only yield improved models, but point the way to classes of invariant bases that minimize cost while maximizing expressivity for a host of applications.