# Descriptors for Machine Learning Model of Generalized Force Field in Condensed Matter Systems

arXiv: https://arxiv.org/abs/2201.00798 · HTML: https://mlsanity.com/paper/2201.00798/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 127 references that are in the map.
- First posted on arXiv 3 Jan 2022.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Machine learning for molecular simulation](https://mlsanity.com/paper/1911.02792/) (2019)
- [Continuously Differentiable Exponential Linear Units](https://mlsanity.com/paper/1704.07483/) (2017)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [DScribe: Library of Descriptors for Machine Learning in Materials Science](https://mlsanity.com/paper/1904.08875/) (2019)
- [Machine Learning for Condensed Matter Physics](https://mlsanity.com/paper/2005.14228/) (2020)
- [Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines](https://mlsanity.com/paper/1610.02746/) (2016)
- [Machine learning dynamics of phase separation in correlated electron magnets](https://mlsanity.com/paper/2006.04205/) (2020)
- [Machine learning electron correlation in a disordered medium](https://mlsanity.com/paper/1810.02323/) (2018)
- [Quantum deep field: data-driven wave function, electron density generation, and atomization energy prediction and extrapolation with machine learning](https://mlsanity.com/paper/2011.07923/) (2020)

## Built on by (most cited first, 16 of 16)

- [Machine learning for phase ordering dynamics of charge density waves](https://mlsanity.com/paper/2303.03493/) (2023)
- [Coarsening of chiral domains in itinerant electron magnets: A machine learning force field approach](https://mlsanity.com/paper/2403.11705/) (2024)
- [Kinetics of orbital ordering in cooperative Jahn-Teller models: Machine-learning enabled large-scale simulations](https://mlsanity.com/paper/2405.14776/) (2024)
- [Machine learning force-field model for kinetic Monte Carlo simulations of itinerant Ising magnets](https://mlsanity.com/paper/2411.19780/) (2024)
- [Machine-learning force-field models for dynamical simulations of metallic magnets](https://mlsanity.com/paper/2602.18213/) (2026)
- [Equivariant Neural Networks for Force-Field Models of Lattice Systems](https://mlsanity.com/paper/2601.04104/) (2026)
- [Graph Neural Networks in the Wilson Loop Representation of Abelian Lattice Gauge Theories](https://mlsanity.com/paper/2605.03901/) (2026)
- [Machine learning for structure-property relationships: Scalability and limitations](https://mlsanity.com/paper/2304.05502/) (2023)
- [Machine learning force-field models for metallic spin glass](https://mlsanity.com/paper/2311.16964/) (2023)
- [Echo State network for coarsening dynamics of charge density waves](https://mlsanity.com/paper/2412.11982/) (2024)
- [Machine learning nonequilibrium phase transitions in charge-density wave insulators](https://mlsanity.com/paper/2601.07583/) (2026)
- [Machine-learning modeling of magnetization dynamics in quasi-equilibrium and driven metallic spin systems](https://mlsanity.com/paper/2604.11513/) (2026)
- [Machine learning approach for vibronically renormalized electronic band structures](https://mlsanity.com/paper/2409.01523/) (2024)
- [Enhanced coarsening of charge density waves induced by electron correlation: Machine-learning enabled large-scale dynamical simulations](https://mlsanity.com/paper/2412.21072/) (2024)
- [Learning Degenerate Manifolds of Frustrated Magnets with Boltzmann Machines](https://mlsanity.com/paper/2511.19879/) (2025)
- [Graph Neural Network Force Fields for Spin Dynamics in Metallic Magnets](https://mlsanity.com/paper/2607.28537/) (2026)

## Abstract (arXiv)

We outline the general framework of machine learning (ML) methods for multi-scale dynamical modeling of condensed matter systems, and in particular of strongly correlated electron models. Complex spatial temporal behaviors in these systems often arise from the interplay between quasi-particles and the emergent dynamical classical degrees of freedom, such as local lattice distortions, spins, and order-parameters. Central to the proposed framework is the ML energy model that, by successfully emulating the time-consuming electronic structure calculation, can accurately predict a local energy based on the classical field in the intermediate neighborhood. In order to properly include the symmetry of the electron Hamiltonian, a crucial component of the ML energy model is the descriptor that transforms the neighborhood configuration into invariant feature variables, which are input to the learning model. A general theory of the descriptor for the classical fields is formulated, and two types of models are distinguished depending on the presence or absence of an internal symmetry for the classical field. Several specific approaches to the descriptor of the classical fields are presented. Our focus is on the group-theoretical method that offers a systematic and rigorous approach to compute invariants based on the bispectrum coefficients. We propose an efficient implementation of the bispectrum method based on the concept of reference irreducible representations. Finally, the implementations of the various descriptors are demonstrated on well-known electronic lattice models.