Source-linked AI summary
De novo exploration and self-guided learning of potential-energy surfaces
Noam Bernstein, Gábor Csányi, Volker L. Deringer
TL;DR
ML potentials require manually assembled reference databases and may not generalize beyond structures included in those databases. The paper presents a unified GAP-RSS protocol that explores and fits potential-energy surfaces de novo, selecting relevant configurations with a configuration-averaged kernel metric. It demonstrates self-guided discovery and accurate description across diverse structures and chemistries using relatively few single-point DFT calculations.
Problem
ML potentials require laborious, manually assembled DFT databases and may be unreasonable for crystal structures absent from those databases.
Method
The paper combines ML-driven de novo structural exploration with iterative fitting, using similarity-based selection of relevant configurations and single-point DFT evaluations.
Results
The protocol self-guidedly discovers α-B12 and covers diverse structures and chemistries without prior input about which structures are relevant.
Takeaways & Limitations
The approach supports more routine application of ML potentials in materials discovery by reducing required user input and computational effort.
Takeaways & Limitations
The demonstrated potentials are not expected to be reasonable for crystal structures absent from their databases, such as phases stable at very high pressures.
Abstract
from arXiv · showhide
Interatomic potential models based on machine learning (ML) are rapidly developing as tools for materials simulations. However, because of their flexibility, they require large fitting databases that are normally created with substantial manual selection and tuning of reference configurations. Here, we show that ML potentials can be built in a largely automated fashion, exploring and fitting potential-energy surfaces from the beginning (de novo) within one and the same protocol. The key enabling step is the use of a configuration-averaged kernel metric that allows one to select the few most relevant structures at each step. The resulting potentials are accurate and robust for the wide range of configurations that occur during structure searching, despite only requiring a relatively small number of single-point DFT calculations on small unit cells. We apply the method to materials with diverse chemical nature and coordination environments, marking a milestone toward the more routine application of ML potentials in physics, chemistry, and materials science.
INTRODUCTION
DFT provides confident atomistic descriptions but is computationally expensive, while ML potentials depend on manually assembled reference databases that may not generalize beyond included structures. This work introduces a unified, de novo approach that explores structural space and fits ML potentials without prior structural input.
- DFT-based materials simulations are highly reliable but severely limited by their computational cost.
- ML potentials accelerate simulations by approximating the potential-energy surface from DFT reference databases.
- Constructing ML fitting databases is usually a manual, laborious process guided by the physical problem under study.
- Existing potentials may be unreasonable for crystal structures absent from their training databases, including phases stable only at very high pressures.
- The proposed protocol explores and fits structural space de novo using ML-driven searching and similarity-based distance metrics without prior input about relevant structures.
- The approach is presented as an efficient and unified route for fitting ML potentials across a broad range of structures and chemistries.
RESULTS
The protocol combines automated structure generation, similarity-based selection, single-point DFT calculations, and iterative ML refinement. It learns diverse crystal structures, including boron, carbon, silicon, and titanium examples, with relatively few selected DFT evaluations.
- A unified framework for exploring and fitting structural space: The framework minimizes user effort by using single-point DFT calculations and general heuristics, with only characteristic distance and bonding type set manually.
- A unified framework for exploring and fitting structural space: The first iteration generates 10,000 randomized structures and selects the N most diverse candidates using leverage-score CUR with SOAP-based structural distances.
- A unified framework for exploring and fitting structural space: Each iteration relaxes randomized structures with the previous GAP, then selects relevant and diverse configurations from the resulting trajectories for database expansion.
- A unified framework for exploring and fitting structural space: The searches terminate after 2,500 DFT data points, which the authors find sufficient to discover and describe the structures studied.
- A unified framework for exploring and fitting structural space: The boron protocol discovers α-B12 self-guidedly, reducing the energy error below 10 meV/atom before 2,000 DFT evaluations and below 4 meV/atom after completion.
- A unified framework for exploring and fitting structural space: Adding N = 100 structures per step slightly outperforms N = 250, although both settings produce satisfactory results.
Learning diverse crystal structures
GAP-RSS learns diverse crystal structures across carbon, silicon, and titanium, reproducing key allotropes and their energy differences with DFT-level agreement. Energy–volume curves are generally well reproduced, though deviations remain for Ti at large volumes.
- Carbon graphite and diamond are both learned, with final errors below 1 meV/atom for sp3 allotropes and about 4 meV/atom for graphite.The diamond–lonsdaleite energy difference is also reproduced as 27 meV/atom versus 28 meV/atom with DFT.
- Silicon’s diamond-type ground state and high-pressure β-tin structure are discovered, while the unseen oS24 open framework receives a good description after a few iterations.The larger residual error for β-tin than diamond-type silicon is consistent with previous manually tuned potentials.
- Titanium’s hcp–ω energy difference is reproduced as 22 meV/atom with GAP-RSS versus 24 meV/atom with DFT.The zero-Kelvin ground state depends on the DFT method, with either hcp or ω obtained as the minimum.
- DFT energy–volume curves are generally well reproduced, but GAP-RSS deviates at large volumes for hcp and ω-type titanium.Those regions correspond to negative external pressure and are described as less relevant; pressure parameters could be modified for accurate elastic properties.
Entire potential-energy landscapes
The protocol explores low- and high-energy regions across carbon, silicon, and titanium, then maps their structural relationships using configuration-averaged SOAP distances. The resulting map organizes structures by coordination and highlights both chemically related families and unusual relaxation intermediates.
- The final minimization ensembles survey predicted-versus-DFT energies across three chemical systems and extend to 1 eV per atom.The energy scale remains continuous but switches from linear to logarithmic at 0.1 eV/at. to show both low- and higher-energy regions.
- A configuration-averaged SOAP kernel defines distances between entire unit cells, which are reduced to a two-dimensional structural map.Cells are rescaled to a minimum bond length r0 = 1.0 Å before SOAP distances and principal component analysis are applied.
- The map progresses from threefold-coordinated graphite, through fourfold carbon and silicon networks, to high-pressure silicon and densely packed titanium structures.Species are encoded by symbols, while average coordination number is encoded by color; the plotted axis values are arbitrary.
- A high-energy, strongly disordered relaxation intermediate is structurally unlike the other entries and therefore appears at relatively large SOAP distances.It was added to the reference database even though it was not a local minimum.
DISCUSSION
The protocol automates structural-database generation and potential-energy-surface fitting with minimal computational and user effort. Its demonstrated scope is three-dimensional inorganic crystal structures, with possible conceptual extension to nanoparticles.
- DISCUSSION: Automated protocols generate structural databases and fit potential-energy surfaces in a self-guided way.The approach is designed to reduce both computational and user time.
- DISCUSSION: The study targets three-dimensional inorganic crystal structures, while conceptually similar approaches may apply to nanoparticles.Organic molecular materials and the use of RSS diversity for organic solids remain an open question.
Interatomic potential fitting
The fitting framework combines GAP with hierarchical two-, three-, and many-body descriptors, using heuristics to automate parameter choices where possible. Descriptor cutoffs and fitting weights are scaled according to characteristic radii and reference-energy differences.
- Interatomic potential fitting: GAP fitting uses a linear combination of two-, three-, and many-body descriptors, including SOAP for many-body environments.The framework develops heuristics to automate and generalize fitting-parameter choices where possible.
- Interatomic potential fitting: Descriptor cutoffs are ordered by range: two-body is longest, three-body is shortest for nearest neighbors, and SOAP is intermediate.The cutoffs are expressed in terms of each system’s characteristic radius.
- Interatomic potential fitting: Fitting weights for energies, forces, and stresses are set by diagonal Gaussian-process noise terms based on reference energy.Lower-energy structures receive greater accuracy near each volume, while higher-energy regions retain flexibility.
Comparing structures
The method compares entire unit cells using a configuration-averaged SOAP representation rather than averaging conventional atomic SOAP power spectra. Table I specifies the descriptor hyperparameters used in GAP fitting.
- Comparing structures: Configuration-averaged SOAP compares structural environments for selecting training data in the CUR step.The representation averages over all atoms in a cell.
- Comparing structures: Table I lists Gaussian widths, sparse-point counts, SOAP basis sizes, kernel exponent, and cutoffs expressed through characteristic radius.SOAP-specific parameters include nmax, lmax, and ζ.
- Comparing structures: Unit-cell similarity is constructed from averaged expansion coefficients and a rotationally invariant power spectrum for the entire cell.This representation is explicitly different from averaging ordinary atomic SOAP power spectra.
Iterative generation of reference data
Reference data are generated through randomized structures, pressure-biased enthalpy relaxation, and configuration selection for DFT evaluation. The procedure uses AIRSS-inspired volume heuristics and a Boltzmann-biased flat histogram with leverage-score CUR.
- Iterative generation of reference data: Randomized cells contain 6–24 atoms, use 1–8 symmetry operations, and impose a minimum separation of 1.8r.Initial volumes are centered on V0 = 14.5 r3 for covalent systems and V0 = 5.5 r3 for metallic systems, with half sampled from a ±25% wider range.
- Iterative generation of reference data: AIRSS seeds are biased toward distributions occurring in nature.This bias is described as central to the AIRSS formalism.
- Iterative generation of reference data: Candidate configurations are relaxed with preconditioned LBFGS until residual forces fall below 0.01 eV/Å while minimizing enthalpy.A random external pressure is applied during structural optimization.
- Iterative generation of reference data: The pressure protocol uses p0 = 1 GPa and β = 0.2, ensuring finite pressure and inclusion of smaller-volume structures.The same pressure range is used for all materials, although it can be adjusted for the target pressure region.
- Iterative generation of reference data: DFT-training configurations are selected at each iteration using a Boltzmann-biased flat histogram and leverage-score CUR.Selection probabilities are based on enthalpy distributions computed at the pressure used for each relaxation.
Computational details
Reference energies and forces were computed with DFT using PAW as implemented in VASP, with material-specific plane-wave cutoffs and exchange-correlation functionals.
- DFT reference energies and forces used PAW as implemented in VASP.
- Plane-wave cutoff energies were 500 eV for B, 800 eV for C, 400 eV for Si, and 285 eV for Ti.
- PBEsol described exchange and correlation for all materials except carbon, which used opt-B88-vdW to account for van der Waals interactions.
Supplementary Information for “De novo exploration and self-guided learning of
The supplementary information extends silicon energy–volume analysis to higher pressures and provides additional structural data for selected configurations and silicon database entries.
- Energy–volume curves for silicon allotropes extend to 50% of equilibrium volume, covering higher-pressure regions than the main-text analysis.
- The open-framework silicon structure oS24 collapses at approximately 16 Å3 in both DFT and GAP-RSS calculations.
- For diamond-type silicon, DFT and GAP-RSS agree over the full compressed-volume range, indicating learned repulsion at very small interatomic distances.
- Table S1 reports external pressure, energy relative to the ground state, and maximum DFT-computed force for selected structures shown in Fig. 5.
- Tables S2 and S3 provide lattice parameters and atomic coordinates for silicon structures (v) and (ix) added to the final GAP-RSS reference database.