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Representations of molecules and materials for interpolation of quantum-mechanical simulations via machine learning

Marcel F. Langer, Alex Goeßmann, Matthias Rupp

arXiv:2003.12081v2physics.comp-phcond-mat.mtrl-scics.LGphysics.chem-ph

TL;DR

First-principles simulations are costly, while the space of possible molecules and materials is vast. This paper reviews atomistic representations in a unified mathematical framework and compares selected representations under controlled conditions; prediction errors improve with interaction order and for local over global representations.

  • Problem

    High ab initio simulation costs limit systematic study of molecular and materials spaces that grow combinatorially with atom count.

    Source: §1 · p. 2

  • Method

    The paper reviews atomistic representations in a unified framework based on k-atom terms, symmetrization, and tensor products, and compares selected representations under controlled conditions.

    Source: §9 · p. 9

  • Results

    Prediction errors improve with modeled interaction order and for local over global representations, with the magnitude of these effects varying across datasets.

    Source: Discussion · p. 9, §9 · p. 9

  • Takeaways & Limitations

    For large systems, the paper recommends using local representations.

    Source: §9 · p. 11

    The comparison includes only one global representation, so the locality aspect warrants further study.

    Source: Discussion · p. 9

Abstract

from arXiv · show

Computational study of molecules and materials from first principles is a cornerstone of physics, chemistry, and materials science, but limited by the cost of accurate and precise simulations. In settings involving many simulations, machine learning can reduce these costs, often by orders of magnitude, by interpolating between reference simulations. This requires representations that describe any molecule or material and support interpolation. We comprehensively review and discuss current representations and relations between them, using a unified mathematical framework based on many-body functions, group averaging, and tensor products. For selected state-of-the-art representations, we compare energy predictions for organic molecules, binary alloys, and Al-Ga-In sesquioxides in numerical experiments controlled for data distribution, regression method, and hyper-parameter optimization.

Glossary … 3 Requirements

Because ab initio simulations are costly and molecular and materials spaces grow combinatorially, machine learning can interpolate between reference simulations, making suitable representations central to QM/ML models. The paper reviews representation types and requirements and compares their performance under controlled conditions.

Glossary

  • The glossary defines abbreviations for representations, methodology, and supplementary information used throughout the paper.It includes terms such as SOAP, KRR, GPR, MAE, and RMSE.Source: Glossary · p. 1

1 Introduction

  • Machine learning exploits redundancy among related systems to interpolate between a small set of reference simulations, reducing the need for repeated ab initio calculations.The computational cost of simulations limits systematic exploration of molecular and materials spaces, which grow combinatorially with system size.Source: §1 · p. 2

Scope and structure

  • The review focuses on explicit Hilbert-space representations for finite and periodic systems and energy prediction, excluding incomplete features and learned or direct-kernel representations.It presents a unified framework, describes representations, compares them qualitatively, and benchmarks selected ones.Source: Scope and structure · p. 2

Related work

  • Reliable comparisons require controlling datasets, learning algorithms, hyperparameters, sampling, validation, and reported quantities; prior studies vary in automatic hyperparameter optimization.The paper distinguishes optimization of representation numerical and structural hyperparameters from regression-method hyperparameters.Source: Related work · p. 2

2 Role and types of representations

  • Representations map atomistic systems into spaces for regression; local descriptors cover atomic environments, while global ones describe finite or periodic systems.Local representations suit local properties and can model extensive global properties through additive approximations.Source: §2 · p. 3, Table 2 · p. 3

3 Requirements

  • Representations should be invariant to property-preserving transformations, unique across property-changing systems, continuous and ideally differentiable, efficient, regression-suitable, and general.Tensor predictions require rotational covariance; generality means encoding any atomistic system.Source: §3 · p. 4
  • Uniqueness prevents indistinguishable representations from forcing erroneous equal predictions, while continuity and differentiability support learning and force matching.Reference gradients can constrain interpolation and improve sample efficiency.Source: §3 · p. 4
  • Current representations do not satisfy every requirement exactly, and research remains limited for charged systems, excited states, continuous spins, isotopes, and external fields.Simple representations may trade uniqueness for coarse-graining or continuity for discontinuities; adequate interaction order depends on data type and amount.Source: §3 · p. 4

4 A unified framework

The framework builds representations from atoms to tuples, local environments, and whole systems, then uses symmetry handling and tensor operations to construct invariant features while retaining desired information.

Representing atoms, environments, systems

  • Atom information is encoded as kets, while k-atom relations use many-body functions and tensor products that attach each tuple’s properties.The relations may be geometric, chemical, or mixed in nature.Source: Representing atoms, environments,… · p. 4
  • Local environments keep a central atom fixed while relating it to neighbors; summing environments represents whole systems, but periodic sums require reciprocal-space treatment or strong weighting with one index in the unit cell.Weighting functions can reduce the influence of distant atoms.Source: Representing atoms, environments,… · pp. 4–5

Symmetries, tensor products, and projections

  • Representations enforce symmetry through invariant many-body functions or group averaging, while tensor contractions can yield rotationally invariant scalars.Group averaging integrates over the symmetry group using a right-invariant Haar measure.Source: Symmetries, tensor products, and… · p. 5
  • When group averaging removes desired information, tensoring a representation with itself can retain it by replacing H with H⊗ν.Source: Symmetries, tensor products, and… · p. 5
  • Projection onto orthogonal elements can retain only selected information; inner products and induced distances provide measures between representations.Source: Symmetries, tensor products, and… · p. 5

5 Selected representations

The section presents three openly implemented representations that meet the paper’s stated requirements and support finite and periodic systems; they are compared empirically later. Symmetry functions and SOAP describe local environments, while MBTR represents system-wide distributions of many-body terms.

5 Selected representations

  • The three selected representations have openly available implementations independent of regression algorithm and support finite and periodic systems.Their empirical comparison is deferred to Section 8.Source: §5 · p. 5

Symmetry functions

  • Symmetry functions encode local atomic environments using k-body relations, typically distances and angles, summed by chemical-element combinations and tapered to zero at a cutoff.Their numerical and structural hyperparameters control feature resolution and the choice of functions.Source: Symmetry functions · p. 5

Many-body tensor representation

  • MBTR represents a system with broadened distributions of k-body terms, summed and arranged by element combination.Weighting can reduce contributions from distant tuples; periodic systems can be handled with strong weighting and a unit-cell constraint, and central-atom tuples give a local variant.Source: Many-body tensor representation · pp. 5–6

Smooth overlap of atomic positions

  • SOAP expands a Gaussian-modeled local atomic density in radial and spherical-harmonic bases, then forms rotationally invariant features such as the power spectrum or bispectrum.The power spectrum captures up to three-body interactions; SNAP uses quadratic bispectrum terms.Source: Smooth overlap of atomic positions · p. 6

6 Other representations · 7 Analysis

The analysis relates diverse molecular and materials representations through shared mathematical structures, evaluates their trade-offs against representation requirements, and examines how locality and feature modeling affect their use. It also surveys other representations and their limitations.

6 Other representations

  • The Coulomb matrix4 (CM) encodes inverse interatomic distances globally and is fast and reconstructable, but its sorted and diagonalized forms respectively risk discontinuities and information loss.Directly encoding elements with atomic numbers is also problematic.Source: §6 · p. 6
  • Bag-of-bonds84 groups inverse-distance terms by element pair, while the BA-representation85 adds dressed atoms, distances, angles, and torsions; IDMBR106 and HDAD42 use element-grouped geometric terms.IDMBR employs higher powers of inverse distances; HDAD uses fewer bins than MBTR, without broadening or explicit weighting.Source: §6 · p. 6
  • FCHL104,105 represents atomic environments through scaled normal distributions over element identity, distances, and angles, while FCHL18 requires an integral kernel and FCHL19 supports standard vector kernels.Wavelet scattering transforms87–93 encode local-density variation across scales and orientations, with invariant features coupling multiple length scales.Source: §6 · p. 7

7 Analysis

  • The analysis examines representation relationships, fulfillment of requirements, local-versus-global trade-offs, and links to feature selection and other modeling techniques.Moment-tensor potentials95 use invariant polynomial contractions, while Gaussian moments118, NICE116, and ACE112–114 are related tensor-based approaches.Source: §7 · p. 7, §6 · p. 7

Relationships between representations

  • Representations handle invariances through invariant k-body functions or explicit symmetrization, and many share systematic connections through grids, tensor generation, and contraction.ACE, BS, GM, MTP, NICE, and SOAP generate tensors that are contracted into rotationally invariant features; SFs, MBTR, and HDAD are related through histogram-like constructions.Source: Relationships between representations · p. 7

Requirements

  • Representations often satisfy requirements only under idealized limits, with fulfillment depending on hyperparameters and trade-offs among uniqueness, computational effort, and interaction order.Local representations with up to 4-body terms can have degenerate environments that impair learning efficiency and achievable errors; higher-order degeneracy remains unknown.Source: Requirements · p. 8

Global versus local representations

  • Local representations can model global energies through atomic contributions, which correspond to pairwise-sum kernels and scale favorably for large systems; global representations may need locality and directionality for local properties.A general way to make a global representation local is to require interactions to include the central atom, starting at k = 2.Source: Global versus local representations · p. 8

Relationships to other models and techniques

  • Feature selection and derived features change dimensionality and feature-space structure; their effectiveness depends on the target, data size, and distribution, while unrelated features or insufficient data can hurt performance.Kernels can be constructed directly between systems, and neural networks can learn implicit representations end to end; the same representation requirements apply.Source: Relationships to other models and… · p. 8

8 Empirical comparison

The comparison evaluates molecular and materials representations while controlling for data distribution, regression, and hyperparameter optimization. It finds that energy-prediction accuracy improves with modeled interaction order and for local over global representations, with effects that vary across datasets.

Data

  • The benchmarks span organic molecules, binary alloys, and ternary oxides, covering finite and periodic systems and ground-state and off-equilibrium structures.They use qm9, ba10, and relaxed nmd18r, respectively.Source: Data · p. 8

Method

  • Representations are compared using KRR with a Gaussian kernel, energy-only training, and a consistent automatic scheme for optimizing regression, numerical, and structural hyperparameters.The setup controls other modeling factors so the comparison focuses on representations.Source: Method · pp. 8–9, §8 · p. 8

Results

  • Learning curves estimate hold-out RMSE across training-set sizes using representative subsets, with predictions repeated 10 times to assess variability.Figure 5 compares SF, MBTR, and SOAP; hold-out sets contain 10 k qm9, 1 k ba10, and 0.6 k nmd18r samples.Source: Method · p. 8, Results · p. 9

Discussion

  • Energy-prediction accuracy improves with modeled interaction order and for local over global representations, though the magnitude of both effects varies across datasets.The locality advantage appears with sufficient training data on ba10; it is not observed on qm9, possibly because angular terms are immediately relevant to organic carbon scaffolds.Source: Discussion · p. 9
  • Representation computation costs tend to rise with predictive accuracy, so selection should reflect the target accuracy and available computing resources.The study also notes that its locality comparison includes only one global representation, while the impact of assuming additivity remains unclear.Source: Discussion · p. 9

9 Conclusions and outlook

The review places diverse atomistic representations in a unified framework based on k-atom terms, symmetrization, and tensor products. It also offers practical guidance and research directions for improving representation design and benchmarking.

Conclusions

  • With training data, regression, and hyper-parameter optimization controlled, SF, MBTR, and SOAP improve in prediction error and compute time with interaction order, while local representations improve over global ones.The comparison covers energy prediction for molecules and materials.Source: §9 · p. 9

Guidance

  • When prediction errors are sufficient, two-body SF and MBTR are recommended for speed; local representations are recommended for large systems.These recommendations address different conditions: error tolerance and system size.Source: §9 · p. 11
  • Under strong input-structure noise or bias, representation performance differences vanish, so cheaper features that do not meet the Section 3 descriptor requirements can suffice.The paper gives dataset nmd18u as an example.Source: §9 · p. 11

Research directions: representations

  • Future representation work should extend the shared framework to derive missing variants and cover systems such as charged, excited-state, spin, isotope, and externally field-driven systems.The goal includes principled reformulation of existing representations and potentially an efficient general implementation.Source: §9 · p. 11
  • Alchemical representations should be developed to encode similarities between chemical species and improve sample efficiency, while identifying which elemental features matter across charges, excitations, spins, and isotopes.Source: §9 · p. 11

Research directions: benchmarking

  • Further work should analyze feature-space structure and data distributions, control representation complexity and cost for each dataset, and broaden benchmarks to more representations, datasets, forces, properties, and baselines.The proposed benchmark extensions retain control over regression method, data distribution, and hyper-parameter optimization; representation capacity and cost could be adapted automatically.Source: §9 · p. 11
  • Benchmarking should improve hyper-parameter optimization and optimize multiple properties and criteria, including predictive accuracy, computational cost, and uncertainty.The stochastic optimizer required multiple restarts to avoid sub-optimal results and reached limits in large search spaces; uncertainty also needs better analysis and local characterization.Source: §9 · p. 11

Related research directions

  • Research should address long-range interactions by coupling machine learning with physical models or teaching models to learn those interactions directly, and deepen formal links between quantum mechanics and kernel-based learning.Source: §9 · p. 11

Competing interests … 5 | Related work

The paper frames quantum-mechanical simulation as limited by computational cost and surveys how molecular and materials representations, data quality, and feature choices shape machine-learning interpolation. It situates its approach among prior studies spanning energy prediction, dynamics, sensitivity, and feature-space comparisons.

Competing interests; Data and code availability; Supplementary material for

1 | Cost of electronic structure calculations

  • Kohn–Sham density functional theory scales as O(N^3) for (semi)local and O(N^4) for hybrid functionals, so computational cost remains a major limit despite larger computers.Doubling system size increases compute time by roughly an order of magnitude, and large runtime prefactors also matter.Source: §1 · p. 21

2 | Size of molecular and materials spaces

  • The tolerance x sets k = 100/x; for n = 5 components and x = 5%, combinations containing four or fewer components are removed before choosing five elements from 30.Source: §2 · p. 21

3 | The role of data quality for QM/ML models

  • Errors in input data can cause outliers and qualitative problems that additional data or model fitting can address only to a limited extent.Conversely, fitting problems can indicate underlying data problems.Source: §3 · p. 21

4 | Explicit and implicit features

  • Representations may define features explicitly, whereas kernels on raw inputs and neural-network layers can define them implicitly; this work focuses on explicit Hilbert-space representations with Gaussian-kernel regression.The Gaussian kernel transforms representation components into regression features, while neural-network architectures can impose analogous requirements.Source: §4 · p. 22

5 | Related work

  • Prior comparisons include molecular and alloy energy studies, but report differing coverage of representation requirements and hyperparameter optimization [7, 8, 10, 11].Faber et al. compare representations and regression methods for qm9 atomization energies without optimizing hyperparameters; Himanen et al. optimize numerical, but not structural, hyperparameters. Schmitz et al. and Nyshadham et al. do not optimize representation hyperparameters.Source: §5 · p. 22
  • Other studies evaluate representations for dynamics and derived properties, molecular orbital energies, formaldehyde properties, and hydrogen adsorption [9, 12, 15, 16].Their settings include forces and stresses, multiple molecular datasets, and differing levels of numerical hyperparameter tuning.Source: §5 · p. 22
  • Separate work examines representation sensitivity, uniqueness, induced-distance correlations, and feature-space comparison, without consistently optimizing hyperparameters or evaluating prediction errors [13, 14, 17].Onat et al. do not investigate prediction errors; Goscinski et al. explore the impact of hyperparameter choices.Source: §5 · p. 22

2 Role and types of representations … 8 | Learning with atomic contributions

The section explains how representation-space structure affects regression, defines extensivity for finite and periodic systems, and presents atomic-contribution kernel learning as a route to scalable, size-extensive predictions.

7 | Extensive and intensive properties

  • An extensive property scales with system size, while an intensive property is size-independent; internal energy is extensive and band-gap energy intensive.Source: §7 · p. 22
  • For finite systems, extensivity means additivity for noninteracting systems, while for periodic systems it is defined using supercells of the same unit cell.In this minimal sense, total and atomization energies are extensive.Source: §7 · p. 22

6 | Structure and distribution of data

  • Figure S1 shows that linear regression suits the unrolled embedding of nearly one-dimensional data, but not its spiral embedding.The example illustrates how representation-space structure can determine suitability for regression.Source: Figure S1 · p. 22, §6 · p. 22

7 | Extensive and intensive properties

  • Because energies are not additive under general structural changes, energy models should be size-extensive; global representations can use normalization with a linear kernel, while local ones satisfy this automatically.Source: §7 · p. 23

8 | Learning with atomic contributions

  • Atomic-contribution learning predicts a system property as a sum of atomic contributions, assuming additivity and local atomic-environment representations, often enforced by a finite-radius cutoff.Source: §8 · p. 23
  • Atomic-kernel regression is equivalent to whole-system kernel regression when the system kernel sums atom-kernel values over every pair of atoms across two systems.The resulting system predictions are identical to those from atomic contributions.Source: §8 · p. 23
  • To reduce memory demands, the method computes atom-kernel matrix blocks on the fly and sums their entries; atomic-contribution learning is extensive.Source: §8 · p. 23

3 Requirements … 11 | k-body functions

The paper discusses how representations support regression and how their computational cost depends on system size. It frames representations through k-body functions, while noting that directly enumerating interactions of order four or higher is too costly for larger systems.

9 | Computational cost

  • With bounded local environments and constant-size representations, local methods scale quadratically in system size, while global methods are dominated by the ck term; local kernels require more entries.The comparison assumes the average number of atoms per local environment is bounded; kernel-matrix overhead can dominate for small systems.Source: §9 · p. 23

10 | Role of representations

  • Representations map atomistic systems into spaces suited to regression; kernel regression operates in kernel feature space, and linear kernels leave the representation unchanged.Many nonlinear kernels, including the Gaussian kernel, act on representation space and rely on its structure and implied metric.Source: §10 · p. 24

11 | k-body functions

  • A k-body function maps information about k distinct atoms to an output, which may be a real number or a distribution.Atom information typically includes coordinates and proton number, but can also include properties such as neutron number.Source: §11 · p. 24
  • Examples include atomic-number counts (k=1), distances (k=2), angles (k=3), and less commonly, alchemical relationships based on periodic-table properties [27].Dihedral or torsional angles and volume-related terms are further examples.Source: §11 · p. 24
  • The study excludes directly enumerated interactions of order k=4 or higher because their combinatorial growth limits larger systems, including the nmd18 dataset.Source: §11 · p. 24
  • Glielmo et al. [28,29] define kernel order as the smallest k for which differentiation by k distinct atomic coordinates always vanishes; fixing a local atom lowers derivative order by one.Source: §11 · p. 24
  • Formal order need not mean full information: products of central-atom distances can be k=3 yet miss angles, and powers or products remain less expressive than full higher-body terms.The central-atom distance products lack the distance between the two other atoms, so they cannot resolve angular information.Source: §11 · p. 24

5 Representations … 18 | Sampling

The section defines local atomic neighborhoods and describes the datasets, sampling, and regression setup for comparing molecular and materials representations. It also characterizes learning curves as becoming linear on log-log plots after at most a few hundred samples.

12 | Local atomic neighbourhoods

  • Local atomic representations use atoms within a cutoff radius of a central atom, with a smooth cutoff function preventing discontinuities at the boundary.For periodic systems, unit cells are replicated to include interactions within the neighborhood.Source: §12 · p. 24

13 | qm9 dataset

  • The QM9 dataset contains 133,885 small H/C/N/O/F molecules with up to nine non-hydrogen atoms and B3LYP/6-31G(2df,p) properties [33,34].The study excludes structures flagged as difficult to converge or uncharacterized and converts total energies to atomization energies.Source: §13 · p. 24

14 | ba10 dataset

  • The ba10 dataset covers 10 binary alloys and 15,950 unrelaxed FCC, BCC, and HCP structures with formation enthalpies computed using PBE [11].The structures use Vegard’s-rule lattice parameters and unit cells spanning the dataset’s specified atom-count ranges.Source: §14 · p. 24

15 | nmd18 dataset

  • The nmd18 dataset contains 3,000 ternary Al-Ga-In oxides, with formation and band-gap energies predicted from relaxed or Vegard’s-rule geometries [41].The challenge predicts relaxed-structure properties from inexpensive Vegard’s-rule inputs, which introduce strong input noise or bias; reported errors are RMSE unless noted.Source: §15 · pp. 24–25

16 | Learning curves

  • Learning curves become linear after at most a few hundred samples, with log-log offset and slope used to characterize predictive performance.Asymptotic fits weight training-set sizes by the standard deviation across splits to reduce small-sample effects.Source: §16 · p. 25

17 | Subsets

  • Outer validation sets and multiple training-set sizes were sampled for each dataset, then each training set was split 80/20 into inner training and validation sets.The procedure was repeated 10 times, and structures with few atoms were excluded because too few were available for statistical learning.Source: §17 · p. 25, §18 · p. 25

18 | Sampling

  • Monte Carlo sampling kept subset statistics within predefined fractional thresholds of the parent dataset to reduce variance and bias.The matched statistics included composition, size, and binned energy, with specific counts and bins varying across QM9, ba10, and nmd18.Source: §18 · p. 25
  • Kernel ridge regression and Gaussian process regression yield equivalent predictions here, using a Gaussian kernel whose length scale and regularization strength are hyperparameters.Predictions are kernel expansions over training systems; the qmmlpack implementation was used.Source: §18 · pp. 25–26

20 | HP optimization · 21 | Kernel regression · 22 | Symmetry function HPs

The sections specify a consistent, automated hyperparameter-optimization procedure, a single-Gaussian-kernel KRR setup, and a constrained parameterization of symmetry functions. Together, they define shared search spaces and representation choices for comparing models.

20 | HP optimization

  • Hyperparameters were selected by minimizing inner-validation RMSE with tree-structured Parzen estimators and local grid search, using consistent ranges and grid spacings across representations.Uniform priors were used, with numerical hyperparameters discretized on linear or logarithmic grids; once defined, the search was fully automatic.Source: §20 · p. 26
  • Each training-set size used 2,000 optimization steps, repeated three times, while optimization ran on only one outer split to reduce computational cost.Representation hyperparameters and starting values for kernel length scale and regularization were drawn from priors, then refined by randomized local grid search.Source: §20 · p. 26

21 | Kernel regression

  • Kernel regression used KRR with a single Gaussian kernel, whose smoothness makes it better suited than less-smooth kernels under Requirement (iii.a).The two hyperparameters were kernel width σ and regularization strength λ; their search spaces were held constant across representations and learning curves.Source: §21 · p. 26
  • The kernel and labels were not centered, since centering is unnecessary for the Gaussian kernel; labels were normalized per atom or per system as needed.Both kernel hyperparameters were optimized on a base-2 logarithmic grid.Source: §21 · p. 26, Table S2 · p. 26

22 | Symmetry function HPs

  • Symmetry-function representations used RuNNer radial G2 and angular G4 functions, sharing the same functions across element combinations to limit the search space.Constant or low-variance features were considered unproblematic for Gaussian-kernel regression because they enter only through feature-vector difference norms.Source: §22 · p. 26
  • Radial symmetry functions used shifted and centered parameterization schemes, with the number of grid points varied from 2 to 10 in each scheme.The shifted scheme varies μ on a linear grid with fixed η; the centered scheme fixes μ = 0 and chooses η to match standard-deviation grid points.Source: §22 · pp. 26–27
  • Angular functions fixed λ = ±1 and ζ = 1, 2, or 4, while optimizing η on a log2 grid; the optimizer could toggle the three ζ = 2 or 4 functions.Radial functions and the two angular functions with ζ = 1 were always included, and cutoff radii varied in integer steps up to 9 Å.Source: §22 · p. 27
  • For KRR, RuNNer symmetry functions were organized into separate feature-vector blocks by central-atom type, and labels were normalized to per-system values.With the Gaussian kernel, representations belonging to different elements consequently had negligible kernel values.Source: §22 · p. 27

23 | Many-body tensor representation HPs … 26 | Error metrics

The section details hyperparameter choices for MBTR and SOAP and reports RMSE, MAE, and relative RMSE conventions for evaluating predictions. It also identifies the supplementary tables containing numerical learning-curve values and the dataset-wide standard deviations used for rRMSE.

23 | Many-body tensor representation HPs

  • MBTR was implemented in qmmlpack with optional ℓ1 or ℓ2 normalization, and its categorical hyperparameters determine which subsequent numerical hyperparameters apply.The selected weighting functions included conditional hyperparameters; for periodic nmd18, parameter ranges were manually restricted to avoid trials exceeding 30 seconds, with convergence threshold 0.001.Source: §23 · p. 27
  • MBTR used k-body functions for k=2 and 3, with 100-bin histograms and no one-body terms because atomization and formation energies already include linear element-count contributions.For k=2 and 3, the representations were concatenated; histogram ranges used the whole dataset, and broadening spanned at least one bin and at most a quarter of the function range.Source: §23 · p. 27
  • The full MBTR indexing scheme generates all element permutations; it was reported to improve consistency and accuracy for differently sized cells or supercells, at higher computational cost, while molecule accuracy was similar.Energies were per-system for qm9 and per-atom for ba10 and nmd18.Source: §23 · p. 27

24 | Smooth Overlap of Atomic Positions HPs

  • SOAP used DScribe with Gaussian-type orbitals, which provided more accurate predictions at lower computational cost than the quippy implementation.The reported GTO basis set had lower prediction errors and faster computation than the polynomial basis; labels were normalized per system, and lmax and nmax ranged from 2 to 8.Source: §24 · p. 27

25 | Prediction errors

  • Tables S3 and S4 provide numerical learning-curve values for RMSE and MAE; rRMSE uses dataset-wide standard deviations of 239.31 kcal mol−1, 178.86 meV, and 104.57 meV for qm9, ba10, and nmd18, respectively.These standard deviations differ from those computed over validation sets by around 1% or less in all cases.Source: §25 · p. 27

26 | Error metrics

  • RMSE and MAE are the absolute performance metrics, while rRMSE is included for scale-independent comparisons across datasets.RMSE is the canonical least-squares regression loss; MAE is also reported because it is common in the literature.Source: §26 · p. 27
  • rRMSE expresses RMSE relative to a baseline that always predicts the mean label; its interpretation assumes independent and identically distributed data.The passage notes that sample count matters more than whether the mean and standard deviation are computed from training or validation labels under this assumption.Source: §26 · p. 27

27 | Compute times … 31 | Comparison with DFT and experimental errors

The supplementary analyses report computational-cost measurements, examine error patterns across datasets and geometries, and compare prediction errors with literature and DFT benchmarks. They find that unrelaxed nmd18u geometries are harder to learn from than relaxed structures, while bulk-crystal prediction errors exceed DFT-intrinsic variations.

27 | Compute times

  • Compute-time benchmarking measured representation and kernel-matrix wall times on one Intel core, averaging three repetitions per system or kernel entry; Figure S3 plots MAE against representation time and Pareto frontiers.Table S7 summarizes compute times; the measurements used qm9, ba10, and nmd18 systems from the first outer validation set.Source: §27 · pp. 27, 31, Figure S3 · p. 30

28 | Analysis details

  • Predictive accuracy by rRMSE was worse for solid-state datasets than qm9, while MBTR particularly underperformed on nmd18r, where system size and computational cost hinder interpolation and hyperparameter optimization.The authors suggest periodic systems may pose harder learning tasks; MBTR lacks intrinsic scaling with atom count, impeding interpolation across unit-cell sizes.Source: §28 · p. 31
  • At 1,600 qm9 training samples, RMSE variability arose from high-error ring molecules absent from the hyperparameter-optimization training split, underscoring careful benchmark stratification.MAE showed no corresponding increase, and the outliers appeared in some, but not all, outer splits.Source: §28 · p. 31

29 | nmd18u dataset

  • For nmd18u approximate geometries, rRMSE increased 14–21%, representations converged to similar performance, and MAE–RMSE gaps indicated outliers; learning energies from unrelaxed structures was harder.These geometries can be obtained at almost no cost and may suit virtual screening; the findings suggest representation is not the limiting factor and other errors dominate.Source: §29 · p. 31

30 | Comparison with literature-reported errors

  • Comparisons with literature-reported performance remain qualitative because sampling, regression, and hyperparameter-optimization conditions differ, but this work’s errors appear compatible with reported estimates.Literature comparisons often report only MAE, which tends to be lower and de-emphasize outliers.Source: §30 · p. 31

31 | Comparison with DFT and experimental errors

  • Reported DFT MAEs against experiment were ≈2.6 kcal mol−1 for B3LYP molecular heats of formation [65] and 200 to 300 meV/atom for PBE cohesive energies [66–68].The reported errors depend on system, property, functional, and computational settings.Source: §31 · p. 31
  • 4.6 meV/atom and 3.3 meV/cation were the best bulk-crystal RMSEs on ba10 and nmd18, respectively, and prediction errors were at least ≈2–6 times larger than DFT-intrinsic variations [69].The PBE inter-code RMSE was 1.7 meV/atom (0.6 meV/atom for all-electron codes), while parametrization MAE was 2 meV/atom; those benchmarks span a ±6% volume interval, unlike predictions at energy minima [69].Source: §31 · p. 31

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