Hafnium and Hafnium Dioxide Machine Learned Interatomic Potentials with the Environment-Adaptive Proper Orthogonal Descriptors (EA-POD)

This repository contains the resources and some examples on how to train Environment-Adaptive MLIPs with the Proper Orthogonal Descriptors (POD). We provide the training data, input files and trained models for Hf and HfO2 for their most common polymorphs.

Data

This folder includes the training data grouped by solid phase. A mirror of the data set is also hosted on Zenodo.

EA-POD

This folder contains the trained models and can be used directly with LAMMPS.

    pair_style pod
	pair_coeff * * HfO2_param.pod HfO2_coefficients.pod Hf O

HfO2_param.pod contains the descriptor settings and parameters during training. HfO2_coefficients.pod contains the trained coefficients for the EA-POD potential. The element types must be provided in the order they appear in the training input file.

Note: The input files and training errors are also provided to reproduce the models, however hyperparameter optimization was not performed.

Training

This folder contains instructions on how to train custom EA-POD models.

Installation Requirements

POD MLIPs are trained directly in LAMMPS with the fitpod keyword. No external software is necessary. Please compile LAMMPS with the ML-POD package using cmake -D PKG_ML-POD=yes. Traditional make is not supported at this time.

For more information on the pair_style: https://docs.lammps.org/pair_pod.html

Supplementary Materials

This folder contains benchmarks of the model accuracy between EA-POD and other state-of-the-art MLIPs: vanilla POD, SNAP and ACE.

Citation and Publication Info

If you use this work or the Environment-Adaptive MLIP formulation in your research, please cite the following publications:

https://doi.org/10.1038/s41524-026-01984-4

Sema, D., Nguyen, N.C., Wyant, S. et al. Environment-adaptive machine-learned force fields for materials under extreme conditions: hafnium and hafnium dioxide polymorphs. npj Comput Mater (2026).

For the original Environment-Adaptive MLIP formulation:

@article{PhysRevB.110.064101,
  title = {Environment-adaptive machine learning potentials},
  author = {Nguyen, Ngoc Cuong and Sema, Dionysios},
  journal = {Phys. Rev. B},
  volume = {110},
  issue = {6},
  pages = {064101},
  numpages = {15},
  year = {2024},
  month = {Aug},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevB.110.064101},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.110.064101}
}

For the original POD formulation:

@article{PhysRevB.107.144103,
  title = {Fast proper orthogonal descriptors for many-body interatomic potentials},
  author = {Nguyen, Ngoc-Cuong},
  journal = {Phys. Rev. B},
  volume = {107},
  issue = {14},
  pages = {144103},
  numpages = {18},
  year = {2023},
  month = {Apr},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevB.107.144103},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.107.144103}
}