# Bayesian regression for

multi-level machine-learned potentials

### Subproject P03

The first-principles description of the properties of multi-component metal oxides is an exceedingly challenging problem. The reasons are that the configurational space grows exponentially with the number of species and standard Density Functional Theory (DFT) is often not accurate enough. The long-term objective of P03 is to accelerate first-principles calculations by developing machine-learning approaches for the description of the interatomic forces, Born effective charges, and other tensorial properties of multivalent oxides. The project will rely on kernel-based methods and Bayesian inference to implement fully automatic “on-the-fly” learning.

In the first project period, we will develop machine-learned force fields (MLFF) for DFT and DFT+U, whereby the number of components in the FF will be gradually increased. A concise framework for learning tensorial properties will be implemented. We will use this to simulate infrared spectra of oxide materials, which can be readily compared to the finite-temperature spectra measured by the experimental groups.

The difference between DFT and hybrid functionals will be machine-learned to go beyond semi-local functionals (Delta-learning). The long-term perspective is to extend this approach to highly accurate beyond-DFT methods, such as the random phase approximation and quantum chemistry (coupled cluster) methods. Although kernel-based methods are exceedingly accurate, they are often less efficient than NN. We will collaborate with other projects to recast the on-the-fly trained FF into NN potentials to address this issue.

## Expertise

The main research efforts of the group are directed towards the development of quantum-mechanical tools for atomic-scale simulations of properties and processes in materials and the application of these methodologies to key areas of condensed matter physics and materials research. An important pillar of the research is the Vienna Ab initio Simulation Package (VASP), a general-purpose *ab initio* code for solving the many-electron Schrödinger equation. The code is among the world leaders in its field, with more than 3500 licensees worldwide. We have expertise with simulations for a vast number of properties using many different techniques:

- Density functional theory (DFT), including spin and non-collinear DFT
- Linear response theory to calculate phonons and dielectric properties
- Hartree-Fock techniques and many flavors of hybrid functionals
- Many-body perturbation theory, including GW and Bethe-Salpeter
- Wavefunction-based correlated methods (Møller-Plesset perturbation theory)
- Surface science, including growth and oxide formation
- Simulation of nanostructures
- Semiconductor physics: charge trapping, polarons
- Electronic excitations
- Defect energies in extended systems

For TACO, we will adapt our machine-learning techniques to tensorial properties and correlated wavefunction techniques. These techniques are directly integrated into VASP and allow to accelerate finite-temperature simulations by many orders of magnitudes.

## Team

#### Former Members

## Publications

### 2022

Liu, Peitao; Verdi, Carla; Karsai, Ferenc; Kresse, Georg

Phase transitions of zirconia: Machine-learned force fields beyond density functional theory

Journal ArticleIn: Physical Review B, vol. 105, no. 6, pp. L060102, 2022.

Abstract | Links | BibTeX | Tags: P03

@article{Liu2022,

title = {Phase transitions of zirconia: Machine-learned force fields beyond density functional theory},

author = {Peitao Liu and Carla Verdi and Ferenc Karsai and Georg Kresse},

doi = {10.1103/physrevb.105.l060102},

year = {2022},

date = {2022-02-16},

journal = {Physical Review B},

volume = {105},

number = {6},

pages = {L060102},

publisher = {American Physical Society (APS)},

abstract = {Machine-learned force fields (MLFFs) are increasingly used to accelerate first-principles simulations of many materials properties. However, MLFFs are generally trained from density functional theory (DFT) data and thus suffer from the same limitations as DFT. To achieve more predictive accuracy, MLFFs based on higher levels of theory are required, but the training becomes exceptionally arduous. Here, we present an approach to generate MLFFs with beyond DFT accuracy which combines an efficient on-the-fly active learning method and Δ-machine learning. Using this approach, we generate an MLFF for zirconia based on the random phase approximation (RPA). Specifically, an MLFF trained on the fly during DFT-based molecular dynamics simulations is corrected by another MLFF that is trained on the differences between RPA and DFT calculated energies, forces, and stress tensors. We show that owing to the relatively smooth nature of these differences, the expensive RPA calculations can be performed only on a small number of representative structures of small unit cells selected by rank compression of the kernel matrix. This dramatically reduces the computational cost and allows one to generate an MLFF fully capable of reproducing high-level quantum-mechanical calculations beyond DFT. We carefully validate our approach and demonstrate its success in studying the phase transitions of zirconia. These results open the way to many-body calculations of finite-temperature properties of materials.},

keywords = {P03},

pubstate = {published},

tppubtype = {article}

}

### 2021

Verdi, Carla; Karsai, Ferenc; Liu, Peitao; Jinnouchi, Ryosuke; Kresse, Georg

Journal ArticleOpen AccessIn: npj Computational Materials, vol. 7, pp. 156, 2021.

Abstract | Links | BibTeX | Tags: P03

@article{Verdi2021,

title = {Thermal transport and phase transitions of zirconia by on-the-fly machine-learned interatomic potentials},

author = {Carla Verdi and Ferenc Karsai and Peitao Liu and Ryosuke Jinnouchi and Georg Kresse},

doi = {10.1038/s41524-021-00630-5},

year = {2021},

date = {2021-09-30},

urldate = {2021-09-30},

journal = {npj Computational Materials},

volume = {7},

pages = {156},

publisher = {Springer Science and Business Media LLC},

abstract = {Machine-learned interatomic potentials enable realistic finite temperature calculations of complex materials properties with first-principles accuracy. It is not yet clear, however, how accurately they describe anharmonic properties, which are crucial for predicting the lattice thermal conductivity and phase transitions in solids and, thus, shape their technological applications. Here we employ a recently developed on-the-fly learning technique based on molecular dynamics and Bayesian inference in order to generate an interatomic potential capable to describe the thermodynamic properties of zirconia, an important transition metal oxide. This machine-learned potential accurately captures the temperature-induced phase transitions below the melting point. We further showcase the predictive power of the potential by calculating the heat transport on the basis of Green–Kubo theory, which allows to account for anharmonic effects to all orders. This study indicates that machine-learned potentials trained on the fly offer a routine solution for accurate and efficient simulations of the thermodynamic properties of a vast class of anharmonic materials.},

keywords = {P03},

pubstate = {published},

tppubtype = {article}

}