Source-linked AI summary
Accelerating crystal structure prediction by machine-learning interatomic potentials with active learning
Evgeny V. Podryabinkin, Evgeny V. Tikhonov, Alexander V. Shapeev, Artem R. Oganov
TL;DR
Crystal structure prediction is constrained by the computational cost of DFT and the limited transferability of conventional potentials. The paper combines USPEX with actively learned MTPs that replace DFT during search, then tests predicted structures with DFT. The methodology reproduced major carbon and sodium allotropes, treated boron structures exceeding 100 atoms, and discovered a 54-atom boron structure.
Problem
DFT-based crystal structure prediction is computationally expensive, while conventional potentials have limited transferability to unknown structures.
Method
The methodology combines the evolutionary algorithm USPEX with MTPs that detect extrapolative configurations and learn them on-the-fly using a D-optimality-based selection procedure.
Results
The method found all known high-pressure sodium phases, reproduced major carbon allotropes, treated boron structures exceeding 100 atoms, and discovered a new 54-atom boron structure.
Takeaways & Limitations
Active on-the-fly MTP training can replace DFT within USPEX while retaining DFT testing of predicted low-energy structures and achieving orders-of-magnitude higher computational efficiency.
Abstract
from arXiv · showhide
In this letter we propose a new methodology for crystal structure prediction, which is based on the evolutionary algorithm USPEX and the machine-learning interatomic potentials actively learning on-the-fly. Our methodology allows for an automated construction of an interatomic interaction model from scratch replacing the expensive DFT with a speedup of several orders of magnitude. Predicted low-energy structures are then tested on DFT, ensuring that our machine-learning model does not introduce any prediction error. We tested our methodology on a problem of prediction of carbon allotropes, dense sodium structures and boron allotropes including those which have more than 100 atoms in the primitive cell. All the the main allotropes have been reproduced and a new 54-atom structure of boron have been found at very modest computational efforts.
I. INTRODUCTION
Crystal structure prediction combines configuration-space sampling with relaxation, but its success depends on an accurate Hamiltonian. DFT is accurate yet computationally costly, motivating machine-learning potentials as a more flexible alternative.
- Crystal structure prediction searches for atomic structures with the lowest thermodynamic potential using sampling and local-minimum relaxation.
- DFT calculations scale cubically with atom number and typically consume more than 99.9% of prediction CPU time.This usually limits DFT-based prediction to systems with a few tens of atoms.
- Empirical potentials are efficient but are rarely transferable to new materials because their algebraic forms target a few known structures.
- Machine-learning interatomic potentials offer flexible functional forms that can systematically improve accuracy at the cost of computational efficiency.Existing approaches include neural-network, Gaussian-approximation, and linear-regression potentials.
- Machine-learning methods had already been applied to crystal-structure-related searches, including fitting during structure search and active learning for surface reconstructions.
II. MACHINE LEARNING INTERATOMIC POTENTIALS
The paper uses moment tensor potentials as a machine-learning interatomic interaction model, representing total energy through atom-centered contributions and fitted basis functions. The model balances physical symmetry, computational efficiency, and agreement with DFT training data.
- Moment tensor potentials partition the total energy into contributions from individual atoms and their local neighborhoods.Each atomic neighborhood contains relative positions within a cutoff sphere.
- Each atomic contribution is expressed as a linear combination of basis functions with fitted parameters.The basis functions depend on the atomic neighborhood, while the coefficients are the model parameters.
- The number of basis functions is chosen empirically to balance prediction accuracy and computational efficiency.The basis functions preserve rotation invariance and permutation invariance for atoms of the same type.
- The model parameters are fitted by matching predicted energies to DFT energies for a training set.This matching produces a system of linear algebraic equations, typically solved by least-squares minimization because it is overdetermined.
III. LEARNING ON THE FLY
The methodology trains an MTP during structure exploration by selecting configurations that extrapolate beyond its current training domain. The actively learned model then supplies energies, forces, and stresses for USPEX relaxation while DFT is reserved for selected configurations.
- Motivation: Offline-trained potentials can fail in crystal structure prediction because they may be unreliable outside their training domain.This transferability problem motivates learning configurations during exploration rather than relying on a fixed training set.
- Active selection: Active learning detects extrapolative configurations and adds them to the MTP training set.Configurations with sufficiently high extrapolation are evaluated with DFT and then used for learning.
- Active selection: A configuration is learned when its extrapolation grade γ(x*) = max_k(|c_k|) exceeds the threshold γ_tsh > 1.When the grade does not exceed the threshold, the model returns the energy, forces, and stresses without a DFT calculation.
- Active selection: The selection procedure uses maxvol and the D-optimality criterion to choose linearly independent configurations and increase |det(A)|.The stated determinant increase is linked to selecting configurations with extrapolation grade above the threshold.
- Integration with USPEX: The actively learning MTP replaces DFT inside USPEX, with relaxation adapted after retraining because the interaction model can change.The BFGS relaxation is forced to perform one gradient-descent iteration after a learning event.
- Computational efficiency: Pre-exploration samples 100 000 random structures, reducing initial DFT calculations by about a factor of 5.The initial training set contains m configurations, where m is the number of MTP parameters; learning from scratch otherwise concentrates most DFT calculations at the initial stage.
IV. CRYSTAL STRUCTURE PREDICTION WITH OUR METHODOLOGY
The methodology was tested for carbon allotropes, dense sodium structures under pressure, and boron allotropes without using prior low-energy structural information. The tests used an approximately 800-parameter MTP to balance accuracy and computational efficiency.
- The methodology was evaluated on carbon allotropes, sodium structures under pressure, and boron allotropes.The boron tests included structures with more than 100 atoms in the primitive cell.
- No a priori information about the low-energy structures was used in the three chemical-element tests.
- An MTP with about 800 parameters was used to balance accuracy and computational efficiency.Because the model only approximately reproduces DFT, structures predicted with the model and with DFT may not coincide.
A. Carbon structures
For carbon structures with 8 atoms per unit cell, the method recovered the principal allotropes while evaluating many more configurations with MTP than with DFT. The predicted-structure error remained below 40 meV/atom.
- The method correctly predicted graphite, diamond, and lonsdaleite among carbon structures with 8 atoms in the unit cell.
- More than 1.9·10^4 configurations were evaluated by MTP, compared with about 1300 DFT calculations.
- The actively selected training-set error was 86 meV/atom, while the error on predicted structures was less than 40 meV/atom.The training-set error overestimates the actual prediction error because active learning samples more extreme configurations.
B. Sodium under pressure
For sodium structures at 120–300 GPa with up to 20 atoms per unit cell, a single actively learned MTP recovered all known ground-state structures. The search used over 2·10^6 MTP evaluations and about 1500 DFT calculations.
- A single MTP correctly predicted the known ground-state sodium structures cI16, tI19, and hP4 across 120–300 GPa.
- The method evaluated more than 2·10^6 configurations while performing about 1500 DFT calculations.
- For this test, the method’s speedup was 100 times higher than in the previous test.The passage attributes this to higher efficiency when searching structures with more atoms.
- The MTP recognized and predicted hP4-Na, an electride phase, using only nuclear positions.The phase is described as Na+ cores with localized electron pairs acting as anions.
C. Boron allotropes
Boron allotropes provide a difficult test because of many local minima, small energy differences, and primitive cells exceeding 100 atoms. The two-MTP strategy focused accurate learning on low-energy configurations, reproduced established structures, and discovered a 54-atom boron structure.
- Boron is challenging for structure prediction because its energy surface has many local minima, small allotrope energy differences, and primitive cells exceeding 100 atoms.
- The two-stage scheme uses a reliable MTP to discard structures above −5.5 eV/atom, then applies an accurate MTP to lower-energy configurations.The accurate model is trained with a weight based on energy above the training-set minimum.
- 11 meV/atom RMS error was obtained for the 100 lowest-energy structures, using about 5000 DFT calculations while evaluating over 4 · 10^8 configurations.
- The search reproduced the lowest-energy Cccm structure and discovered a closely related 52-atom P¯42m structure with slightly lower energy.
- DFT predicts four related 26- and 52-atom structures to be dynamically stable within 8 meV/atom, but MTP captures only two of them.
- A new 54-atom metallic boron structure was found at 39 meV/atom above α-boron and 11 meV/atom above γ-boron.It has cubic Im-3 symmetry and can be viewed as the simplest periodic approximant of a strained aperiodic arrangement of complete icosahedra.
- The discovered 54-atom structure showed no transformation after 100 ps of NPT molecular dynamics at 1200 K, although its predicted superconducting transition temperature was below 1 K.
D. Prediction of β-boron structure
The methodology produced large boron structures without empirical structural constraints, including α-boron and β-boron-like approximants with 105–108 atoms per primitive cell. The predicted 106-atom structure closely matches a known β-boron approximant in energy.
- Fully theoretical prediction: The study performed a fully theoretical, unconstrained prediction of β-boron, using no empirical structural information beyond the approximate supercell size.Earlier theoretical attempts used experimental lattice parameters and most fully occupied atomic positions.
- Predicted structures: The method found the best known 108-atom boron structure, a supercell of α-boron, and β-boron-like approximants containing 105–108 atoms per primitive cell.The β-boron-like approximants were 14, 2, 8, and 8 meV/atom above α-boron for 105, 106, 107, and 108 atoms, respectively.
- Comparison with known structure: The predicted 106-atom structure had virtually the same energy as the known β-boron approximant, with DFT energies differing by less than 1 meV/atom.Both structures contain fused icosahedra with point defects and some non-icosahedral atoms.
V. CONCLUSION
The paper proposes an evolutionary crystal-structure-prediction methodology that actively trains a machine-learning interatomic model on-the-fly. Tests on sodium, carbon, and boron reproduced known phases and identified a new low-energy boron allotrope.
- Methodology: The methodology combines an actively trained machine-learning interatomic model with the evolutionary algorithm USPEX for crystal structure prediction.The model is trained automatically on-the-fly rather than from a manually assembled dataset.
- Efficiency: The methodology is orders of magnitude more computationally efficient than conventional DFT-based crystal-structure-prediction algorithms.It replaces DFT within the prediction workflow without significant changes to the crystal structure prediction algorithm.
- Validation: All tested high-pressure sodium phases, major carbon allotropes, and known pure boron allotropes were found, including disordered β-boron with 106 atoms per cell.The carbon structures included graphite, diamond, and lonsdaleite; sodium results included host-guest tI19 and electride hP4 phases.
- Boron results: For boron, the study demonstrated achievable accuracy of 11 meV/atom and predicted a new low-energy metallic allotrope.