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Unified Representation of Molecules and Crystals for Machine Learning
Haoyan Huo, Matthias Rupp
TL;DR
First-principles atomistic simulations are expensive, creating a need for representations that support efficient kernel learning across arbitrary systems. The paper introduces MBTR, an invariant and differentiable many-body representation for molecules and crystals, and reports competitive energy and force predictions across molecular, crystalline, and dynamics tasks. It also demonstrates phase-diagram use while saving up to 48% of DFT calculations.
Problem
First-principles simulations are computationally costly, while kernel-based ML requires a common representation for arbitrary atomistic systems.
Method
MBTR represents element-stratified distributions of k-atom terms, with smoothing, weighting, and element correlation matrices, for finite and periodic systems.
Results
MBTR performs competitively across molecular and crystal prediction tasks and reduces sGDML errors by up to 50%–60%, especially with fewer than 100 training samples.
Takeaways & Limitations
MBTR provides a general representation for fast ML interpolation between quantum-mechanical calculations across molecules, crystals, and phase-diagram applications.
Takeaways & Limitations
With increasing system size and degrees of freedom, whole-system MBTR is likely to degrade, motivating local-environment representations and additive atomic energies.
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
The paper introduces MBTR to provide a general, invariant representation for kernel-based ML of finite and periodic atomistic systems. It demonstrates competitive interpolation across molecular structures, crystal chemistry, and dynamics, with applicability to alloy phase diagrams.
- Representation: The representation targets invariance, uniqueness, differentiability, coverage of molecules and crystals, fast computation, and data efficiency.These properties address requirements for kernel learning across arbitrary atomistic systems.
- Method: MBTR smooths element-stratified geometric distributions, applies distance-dependent weighting, and uses element correlation matrices to support alchemical learning.The method broadens geometric terms with a normal distribution and can downweight contributions from far-away atoms.
- Crystal chemistry: MBTR achieves 8.1 meV/atom RMSE and 4.7 meV/atom MAE for formation energies of elpasolite crystals using a training set of 9 k crystals.The evaluation covers 2,272 crystals containing 12 different elements.
- Molecular geometry: sGDML/MBTR matches or exceeds sGDML/CMmd for molecular energy and force prediction, with error reductions up to 50%–60%, especially below 100 training samples.The improvement is smaller but remains positive for more symmetric molecules.
- Representation: MBTR represents atomistic systems through distributions of k-atom terms stratified by chemical elements.Canonical terms include atom counts, inverse distances, angles, and dihedral angles.
Supplementary Material Unified Representation of Molecules and Crystals for Machine Learning
The supplementary material accompanies the paper by Haoyan Huo and Matthias Rupp.
- The authors listed are Haoyan Huo and Matthias Rupp.
S1 Parametrization of MBTR
The supplement lists the MBTR parametrizations used throughout the study and defines their common settings and notation.
- Table S1 lists all MBTR parametrizations used in the study.
- All experiments use a normal broadening distribution with standard deviation σ and identity element-correlation matrices.
- Dataset-dependent σ values marked σ∗ must be optimized.
- The notation defines Cartesian coordinates, coordinate differences, their norms, and element and atom indexing functions.
S2 Datasets
The supplement describes molecular and crystal datasets and reports evaluation procedures and prediction errors for energy, forces, and molecular dynamics.
- Molecular datasets: GDB7-13 (QM7b) contains 7,211 small organic molecules with up to seven atoms of C, N, O, S, and Cl, saturated with hydrogen.
- Molecular datasets: The MD17 dataset contains eight molecular-dynamics trajectory datasets with energy and force values for each configuration.
- Molecular dynamics: Energy-only models use 10 k training configurations and 2 k validation configurations for geometry changes in small organic molecules.
- Crystal datasets: For crystalline energy prediction, Table S4 covers 11 k elpasolites and 4.6 k ternary compounds, reporting RMSE and MAE in meV/atom.
- Crystal datasets: Figure S3 plots MBTR prediction errors against training-set size for ABC2D6 elpasolites and ABC2 ternary compounds.
S3 Implementation
The implementation linearizes MBTR element and atom tuples while removing redundancies from reversals and, for atoms, repeated indices.
- Tuple enumeration: MBTR element dimensions enumerate k-tuples with repetition, while atom interactions exclude repeated indices and reversed tuples.Repeated element indices capture same-element interactions; repeated atom indices would create undefined distances or angles.
- Tuple enumeration: Element-tuple dimensions use fixed lexicographic ordering after reversal invariance removes redundant tuples.Any fixed order that identifies tensor elements would work; lexicographic order is a convention.
- Number of interactions: The element-tuple count fe(n, k) allows repetition but excludes reversed sequences.
- Indexing functions: The indexing proof retains one tuple from each reversal pair by imposing constraints such as i1 ≤ i2 for k = 2 and i1 ≤ i3 for k = 3.
- Number of interactions: The atom-tuple count fa(n, k) is defined for tuples without repetitions and reversals, with explicit formulas given for k = 1 through 4.