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Unified Representation of Molecules and Crystals for Machine Learning
Haoyan Huo, Matthias Rupp
TL;DR
First-principles atomistic simulations are costly, and kernel learning needs representations that accommodate arbitrary atomistic systems. This paper introduces MBTR for molecules and crystals and reports competitive energy and force prediction errors across molecular and crystal evaluations.
Problem
Kernel learning for atomistic simulations needs representations spanning finite and periodic systems, while first-principles calculations remain costly for workloads requiring many evaluations.
Source: INTRODUCTION · p. 1Method
MBTR represents atomistic systems using distributions of k-atom terms stratified by chemical elements and describes both finite and periodic systems.
Source: INTRODUCTION · p. 1, DISCUSSION AND OUTLOOK · p. 6Results
Across molecular-structure, crystal-chemistry, and molecular-geometry evaluations, MBTR reports energy and force prediction performance, with sGDML/MBTR reducing errors over sGDML/CMmd by up to 50%–60%.
Source: Changes in molecular structure · p. 3, Changes in crystal chemistry · p. 3, Changes in molecular geometry · p. 4Takeaways & Limitations
The study demonstrates MBTR's applicability to Pt-group/transition-metal binary phase diagrams, where an energy threshold controls saved calculations and the probability of missing the correct convex hull.
Source: Phase diagrams · p. 5As atom count and degrees of freedom increase, the whole-system MBTR approach is likely to degrade, motivating representations of local chemical environments.
Source: DISCUSSION AND OUTLOOK · p. 6
Abstract
from arXiv · showhide
Accurate simulations of atomistic systems from first principles are limited by computational cost. In high-throughput settings, machine learning can reduce these costs significantly by accurately interpolating between reference calculations. For this, kernel learning approaches crucially require a representation that accommodates arbitrary atomistic systems. We introduce a many-body tensor representation that is invariant to translations, rotations, and nuclear permutations of same elements, unique, differentiable, can represent molecules and crystals, and is fast to compute. Empirical evidence for competitive energy and force prediction errors is presented for changes in molecular structure, crystal chemistry, and molecular dynamics using kernel regression and symmetric gradient-domain machine learning as models. Applicability is demonstrated for phase diagrams of Pt-group/transition-metal binary systems.
INTRODUCTION
Machine learning can lower the cost of repeated first-principles calculations by interpolating from reference data. The paper introduces MBTR, a representation for molecules and crystals, and evaluates it across molecular, crystalline, dynamical, and phase-diagram tasks.
INTRODUCTION
- MBTR supports interpolation between first-principles calculations, which can reduce costs in high-throughput searches, longer dynamics simulations, and studies of larger systems.It is related to Coulomb matrices, bag of bonds, histograms, symmetry functions, and cluster expansion.Source: INTRODUCTION · p. 1, DISCUSSION AND OUTLOOK · p. 6
METHOD
- MBTR maps k-atom terms stratified by chemical elements into distributions, and is designed to be invariant, unique, differentiable, fast, and applicable to finite and periodic systems.It broadens discrete distance terms into smooth distributions, can encode element similarities, and handles periodic sums using a unit-cell constraint and decaying weights.Source: INTRODUCTION · p. 1, METHOD · p. 2
Changes in molecular structure
- On 5 k training molecules, MBTR predicts atomization energies with MAE 0.6 kcal/mol, below chemical accuracy, and achieves similar performance with linear regression.The benchmark contains 7,211 small organic molecules; the reported properties were calculated with the hybrid PBE0 functional.Source: Changes in molecular structure · p. 3, Table I · p. 3
Changes in crystal chemistry
- With 9 k training crystals, MBTR predicts elpasolite formation energies with RMSE 8.1 meV/atom and MAE 4.7 meV/atom.For ABC2 ternary alloys spanning 22 elements, errors are larger: RMSE 31 meV/atom and MAE 23 meV/atom, versus 23 and 15 meV/atom for same-sized elpasolite training.Source: Changes in crystal chemistry · pp. 3–4, Figure 2 · p. 4
Changes in molecular geometry
- sGDML/MBTR matches or exceeds sGDML/CMmd for energy and force predictions, with error reductions up to 50%–60%, especially below 100 training samples.It performs better than PaiNN for energy predictions but worse for forces; its advantage is smaller for more symmetric molecules, though it remains an improvement.Source: Changes in molecular geometry · p. 4, Figure 3 · p. 5
Phase diagrams
- An active-learning scheme identified the correct convex hull while saving up to 48% of DFT calculations by evaluating only structures predicted to be low in energy.For AgPt, it used 357 DFT calculations for training and predicted 331 other structures with MAE 39 meV/atom; the energy threshold controls the savings-versus-failure trade-off.Source: Phase diagrams · pp. 4–5, Figure 4 · p. 6
Other uses
- Prior MBTR applications include predicting molecular and crystalline properties, visualizing conformational space, optimizing clusters, relaxing structures, and identifying defects.Examples include gas-particle partition coefficients [27], exchange spin couplings, and formation energies of Al-Ni and Cd-Te compounds.Source: Other uses · pp. 5–6
DISCUSSION AND OUTLOOK
- Whole-system MBTR may degrade as atom counts grow, motivating local-environment representations and additive atomic energy predictions.The paper also notes that model outliers can reveal errors in reference calculations, and that MBTR can support interpolation across growing collections of ab initio data.Source: DISCUSSION AND OUTLOOK · p. 6
S1 Parametrization of MBTR
The section catalogs the MBTR parametrizations used in the study, including shared distribution and element-correlation settings and dataset-dependent optimization of σ.
S1 Parametrization of MBTR
- Table S1 lists the MBTR parametrizations used across the study.Source: S1 Parametrization of MBTR · p. 11, Table S1 · p. 11
- Across experiments, D was a normal distribution with standard deviation σ, and element correlation matrices were identity matrices.Source: Table S1 · p. 11
- σ* denotes a dataset-dependent σ that must be optimized; the table defines its coordinate and indexing notation.It defines p, q, and r from Cartesian coordinates and notes that full and non-reversal indexing are described in Section S3.Source: Table S1 · p. 11
S2 Datasets
The section describes datasets and modeling setups for molecular-property prediction, molecular dynamics, and crystalline energy prediction. It reports energy-prediction results for ABC2D6 and outlines supplementary evaluations for MD17 and crystal learning curves.
S2 Datasets
- GDB7-13 (QM7b) contains 7,211 small organic molecules with up to seven C, N, O, S, or Cl atoms, and retains ground-state properties at the highest available theory level.It is a modified GDB7-12/QM7 dataset with relaxed geometries, 13 additional properties, and some Cl-containing molecules [1].Source: S2 Datasets · p. 11
distances:
- MD17 comprises eight molecular-dynamics trajectory datasets with energies and forces; energy-only models use an unsorted Coulomb-matrix variant encoding inverse interatomic distances.Because atom identities and order remain fixed through each simulation, sorting is unnecessary and element types are implicit in matrix positions.Source: S2 Datasets · p. 12, distances · p. 12
- For sGDML/MBTR on MD17, the study retains the Matérn kernel and symmetric atom permutations, treats each atom as its own element type, and reports smoother Gaussian-kernel performance than Laplacian for the modified Coulomb matrix.Table S3 covers energy-only models trained on 10k frames and evaluated on 2k other frames; the passage gives no error values.Source: distances · p. 12, Table S3 · p. 13
- At training size 3594, MBTR Gaussian achieved RMSE 31 and MAE 23, compared with MBTR Laplacian at 37 and 27 and MBTR Linear at 55 and 38.The crystalline energy prediction errors are in meV/atom.Source: distances · p. 14, Table S4 · p. 14
- Crystal MBTR learning curves cover 11k ABC2D6 elpasolites and 4.6k ABC2 ternary compounds, including training with chemical species removed; removing three species leaves too few ABC2D6 points for reliable estimation.The supplementary evaluation also reports MBTR performance for both crystal datasets.Source: distances · p. 12, Table S4 · p. 14, Figure S3 · p. 14
S3 Implementation
MBTR implementation enumerates element tuples and atom tuples differently: element tuples allow repeated element indices, while atom tuples exclude repeated atoms and reversals when the geometry function is reversal-invariant. Element tuples are mapped to tensor rows; atom-tuple enumeration needs no indexing.
S3 Implementation
- Element tuples allow repetition, omit reversed tuples for reversal-invariant geometry functions, and are enumerated lexicographically to support tensor-row indexing.Element-tuple indexing maps tuples to rows in the linearized tensor; the chosen fixed order is a convention.Source: S3 Implementation · p. 15
Enumeration of atom tuples
- Atom interactions use distinct indices and omit reversals when the geometry function is reversal-invariant, equivalent to enumerating k-permutations up to reversal.Atom-tuple enumeration order does not matter; repeated indices are excluded to avoid undefined quantities such as self-distances and angles with repeated atoms.Source: S3 Implementation · p. 16
Number of k-body interactions
- For k = 1 through 4, atom-interaction counts are n, n(n−1)/2, (n^3−3n^2+2n)/2, and (n^4−6n^3+11n^2−6n)/2, respectively.These counts exclude repeated indices and reversals.Source: S3 Implementation · p. 16