Source-linked AI summary
Efficient and Accurate Machine-Learning Interpolation of Atomic Energies in Compositions with Many Species
Nongnuch Artrith, Alexander Urban, Gerbrand Ceder
TL;DR
Descriptors such as Behler–Parrinello symmetry functions scale quadratically with the number of chemical species. The paper assembles multicomponent Li–transition-metal oxide data sets and evaluates ANN potentials whose hidden-layer widths are held fixed across descriptor dimensions.
Problem
Existing local descriptors can require quadratically increasing complexity as the number of chemical species grows.
Method
The study constructs enumerated Li–transition-metal oxide configurations and uses feedforward ANN potentials with architecture specified by hidden layers and nodes per layer.
Results
16,047 atomic configurations comprise the complete reference data set for 11 chemical species.
Takeaways & Limitations
The data set and fixed-width ANN setup provide a basis for comparing structure and composition spaces across systems with different descriptor dimensions.
Abstract
from arXiv · showhide
Machine-learning potentials (MLPs) for atomistic simulations are a promising alternative to conventional classical potentials. Current approaches rely on descriptors of the local atomic environment with dimensions that increase quadratically with the number of chemical species. In this article, we demonstrate that such a scaling can be avoided in practice. We show that a mathematically simple and computationally efficient descriptor with constant complexity is sufficient to represent transition-metal oxide compositions and biomolecules containing 11 chemical species with a precision of around 3 meV/atom. This insight removes a perceived bound on the utility of MLPs and paves the way to investigate the physics of previously inaccessible materials with more than ten chemical species.
Appendix A: The LiMO2 data set
The LiMO2 data sets expand systematically from three to 11 chemical species, reaching 16,047 atomic configurations in the complete 11-species set.
- The three-species set contains 7,338 structures comprising Li, Ti, and O.These include TiO2 structures and systematically enumerated LiTiO2 configurations.
- The four-species set adds 1,343 configurations for LiNiO2 and Li2NiTiO4, reaching 8,681 configurations.
- The five-species set adds 1,494 configurations for LiMnO2 and Li2NiMnO4, reaching 10,175 configurations.
- 16,047 atomic configurations comprise the complete reference data set for 11 chemical species.
Appendix B: Artificial neural network potentials
The ANN potentials vary in complexity through descriptor dimension and network architecture, while training uses the aenet package with limited-memory BFGS.
- ANN model complexity is determined jointly by descriptor dimension and network architecture.For feedforward ANNs, architecture is specified by hidden-layer count and nodes per layer.
- The N-15-15-1 architecture uses two hidden layers with 15 nodes each, independent of descriptor dimension.
- The aenet software trains the ANN potentials using the limited-memory BFGS method.
Appendix C: Scaling behavior of existing local descriptors
Several existing multicomponent descriptors scale at least quadratically with species count, because they enumerate species-pair interactions or partial spectra; higher-order cluster expansions scale more steeply.
- The Behler–Parrinello descriptor and cluster-expansion basis scale at least quadratically with the number of chemical species.
- For N species, Behler–Parrinello angular symmetry functions contain N(N + 1)/2 interactions.The interactions enumerate all same- and cross-species pairs.
- After 3,000 training iterations, RMSE for 11-species systems has not converged and increases with descriptor size.
- Cluster-expansion basis functions scale quadratically for pair clusters and grow to cubic or higher orders for higher-body interactions.Triplets scale cubically, quadruplets to fourth order, and so on.
- The multicomponent SOAP approach also scales quadratically because it uses partial power spectra for each species pair.
Appendix D: Derivation of the expansion coefficients
The appendix derives radial and angular expansion coefficients for the descriptor from RDF and angular distributions using an orthogonal basis and its dual.
- The radial coefficient c(2)_α is obtained from the basis-set expansion of the radial distribution function.
- The orthogonal basis and dual basis satisfy the relation R ∫ ¯φβ(r)φα(r) dr = δβα.
- Because the RDF is nonzero only for 0 ≤ r ≤ Rc, its coefficient integral can be extended over the entire space.
- Substitution of the RDF expression produces the coefficient formula used in Eq. (6) of the main manuscript.
- The angular expansion coefficients c(3)_α follow by a completely analogous derivation.
Appendix E: Chebyshev polynomials of the first kind
The appendix defines Chebyshev polynomials through a recurrence and uses them to construct radial and angular basis functions on the relevant intervals.
- The Chebyshev basis begins with T0(x) = 1 and T1(x) = x, then uses Tn+1(x) = 2xTn(x) − Tn−1(x).
- The polynomials are orthogonal on the interval [−1, 1] with respect to a specified weight.
- The radial expansion uses basis functions and duals defined on 0 < r < Rc.
- For angular expansions, the radial interval is replaced by 0 ≤ θ < π, requiring Rc to be replaced by π in the corresponding equations.