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WACSF - Weighted Atom-Centered Symmetry Functions as Descriptors in Machine Learning Potentials

Michael Gastegger, Ludwig Schwiedrzik, Marius Bittermann, Florian Berzsenyi, Philipp Marquetand

arXiv:1712.05861v1physics.chem-phstat.ML

TL;DR

Conventional ACSFs become inefficient as the number of chemical elements increases because they require separate functions for element combinations. The paper develops weighted ACSFs, evaluates them in HDNNPs on QM9 enthalpies, and finds that shorter wACSF representations generalize better while genetic optimization mainly benefits smaller networks.

  • Problem

    Conventional ACSF descriptor size and computational cost grow with the number of chemical elements, while selecting their adjustable parameters is nontrivial.

  • Method

    The paper introduces wACSFs with element-dependent weighting functions, compares them with ACSFs in HDNNPs using 133855 QM9 molecules, and tests genetic optimization of descriptor parameters.

  • Results

    7.40 kcal/mol for ACSFs versus 1.83 kcal/mol for wACSFs was observed for comparable-length descriptor vectors, with wACSFs also showing better generalization.

  • Takeaways & Limitations

    A substantially smaller number of wACSFs can provide high-accuracy HDNNPs, while genetic optimization is most beneficial for relatively small neural-network potentials.

Abstract

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We introduce weighted atom-centered symmetry functions (wACSFs) as descriptors of a chemical system's geometry for use in the prediction of chemical properties such as enthalpies or potential energies via machine learning. The wACSFs are based on conventional atom-centered symmetry functions (ACSFs) but overcome the undesirable scaling of the latter with increasing number of different elements in a chemical system. The performance of these two descriptors is compared using them as inputs in high-dimensional neural network potentials (HDNNPs), employing the molecular structures and associated enthalpies of the 133855 molecules containing up to five different elements reported in the QM9 database as reference data. A substantially smaller number of wACSFs than ACSFs is needed to obtain a comparable spatial resolution of the molecular structures. At the same time, this smaller set of wACSFs leads to significantly better generalization performance in the machine learning potential than the large set of conventional ACSFs. Furthermore, we show that the intrinsic parameters of the descriptors can in principle be optimized with a genetic algorithm in a highly automated manner. For the wACSFs employed here, we find however that using a simple empirical parametrization scheme is sufficient in order to obtain HDNNPs with high accuracy.

I. INTRODUCTION

Machine-learning potentials use geometry descriptors to represent molecular environments while preserving invariances and enabling accurate property prediction. The paper introduces wACSFs to address the scaling limitations of conventional ACSFs and examines automated parameter optimization.

  • Motivation: Machine-learning potentials fit nonlinear expressions to electronic-structure reference data, targeting accurate predictions at much higher speed than ab initio methods.Applications include bond breaking, bond formation, atomization energies, dipole moments, material design, scattering simulations, and infrared spectra.
  • Descriptors: Descriptors transform Cartesian molecular geometries to encode roto-translational invariance, spatial cutoffs, and potentially linear computational scaling.Descriptor suitability depends strongly on the machine-learning model and remains an active research topic.
  • Research gap and contribution: Descriptor parameters must be predefined, making optimal parameter selection nontrivial and motivating approaches that integrate optimization into machine learning.The paper investigates a genetic algorithm after developing a descriptor variant with improved element-number scaling.
  • HDNNPs and ACSFs: HDNNPs use atom-centered descriptors and element-specific neural networks to predict molecular properties from local chemical environments.ACSFs are many-body descriptors based on radial and angular distribution functions.

B. Atom-centered Symmetry Functions

ACSFs describe local atomic environments through radial and angular distributions, using element-specific combinations and adjustable spatial and angular parameters. Their descriptor size and computational cost grow rapidly with the number of chemical elements.

  • Descriptor construction: ACSFs model each atom’s local chemical environment through radial and angular distributions of surrounding nuclei.These functions are primary descriptors used in HDNNPs.
  • Radial functions: Radial ACSFs use Gaussian functions whose width η and position µ control spatial sensitivity, while a cutoff function restricts the encoded region around the central atom.The cutoff radius rc specifies the surrounding region.
  • Angular functions: Angular ACSFs depend on θijk, λ, and ζ to represent angular distributions, with λ = ±1 shifting maxima between 0° and 180° and ζ controlling width.Terms involving rjk create asymmetric spatial extent for angles near 180°.
  • Element combinations: ACSFs use separate radial functions for element pairs and angular functions for element triples, with several parameter variants per combination.For H, C, and O, a hydrogen environment includes functions for H-H, H-C, and H-O combinations.
  • Scaling limitation: Nelem radial and Nelem(Nelem + 1) angular functions are needed to cover all combinations, so descriptor vectors and computational costs grow quickly with chemical diversity.Using five functions per pair and triple yields descriptor lengths of 40, 120, and 175 for systems containing two, four, and five elements, respectively.

C. Weighted ACSFs

wACSFs modify ACSFs by weighting radial and angular contributions according to the participating elements rather than assigning separate functions to every element combination. This removes direct dependence on the number of elements and can yield shorter descriptors, with finer parameter variation compensating for denser information.

  • Construction: wACSFs introduce element-dependent weighting functions into radial and angular ACSFs to encode chemical composition implicitly.This replaces separate functions for different element combinations.
  • Weighting scheme: Setting g(Zj) = Zj and h(Zj, Zk) = ZjZk provides satisfying results without introducing additional parameters.The weighting functions modify each radial and angular contribution according to the atomic numbers involved.
  • Scaling: wACSF descriptor size no longer depends on the number of chemical elements, eliminating the combination-count limitation of standard ACSFs.The denser spatial and elemental information in each function may require finer variation of radial and angular parameters.
  • Descriptor length: Even after accounting for the need for finer parameter variation, wACSF descriptor vectors remain significantly shorter than ACSF counterparts.This supports using fewer functions while retaining a detailed environmental representation.

D. Parametrization of Symmetry Functions

The paper compares empirical parametrization schemes for wACSF and ACSF functions, focusing on Gaussian width and position parameters and angular parameters λ and ζ. A radial grid is used to cover the relevant spatial range, with centered and shifted schemes defined differently.

  • Descriptor parameters η, µ, λ, and ζ must be selected before HDNNP application, traditionally through trial and error requiring system knowledge and chemical intuition.
  • Angular functions generally use complementary λ = 1 and λ = −1 sets, while larger ζ emphasizes angles near 0° and 180° at the expense of angles near 90°.A few functions with higher ζ can help when many angular functions are needed relative to radial functions.
  • Radial Gaussian width η and position µ control spatial sensitivity and have the largest influence on descriptor performance.
  • The parametrization first selects N functions and constructs an equally spaced auxiliary grid spanning r0 to rN for balanced coverage until the cutoff radius rc.The grid contains N points, and ∆r is the spacing between neighboring points.
  • Centered functions set µi = 0 and derive widths from grid positions, whereas shifted functions place Gaussian centers at µi = ri and choose widths from adjacent centers.Both schemes aim to cover the relevant radial range; the shifted scheme maintains high radial resolution.

E. Optimization of wACSF Descriptors with a Genetic Algorithm

The paper uses a genetic algorithm to optimize wACSF descriptor vectors, encoding spatial and angular parameters in descriptor genomes. Fitness is evaluated with either small HDNNPs or faster linear ridge-regression models.

  • Automated descriptor construction builds on earlier heuristic and fitting-integrated approaches for selecting atomistic potential descriptors.
  • Genetic algorithms optimize descriptor vectors through population-based evolution using fitness scoring, crossover, mutation, and iterative generation updates.
  • A wACSF genome contains η and µ for radial functions and η, λ, and ζ for angular functions associated with each chemical element.
  • The genetic operators use real-parameter crossover and mutation for µ and η, while λ is sign-swapped and ζ is changed incrementally.
  • One fitness strategy scores genomes with small HDNNP models, keeping evaluation close to the final potential-fitting objective.
  • A second fitness strategy replaces neural networks with linear ridge regression, substantially reducing genome-evaluation cost but departing further from the final HDNNP objective.

III. COMPUTATIONAL DETAILS

The study evaluates wACSF and ACSF descriptors with HDNNPs on QM9 enthalpy data, comparing similarly sized and spatially matched descriptor sets. wACSFs achieve comparable or better predictive performance with far fewer functions and improved generalization.

  • QM9 provides 134 855 equilibrium structures and properties for small organic compounds containing H, C, N, O, and F, computed at the B3LYP/6-31G(2df,p) level.
  • HDNNPs use neural networks to model element-specific atomic contributions whose sum gives molecular properties such as energies or enthalpies.
  • wACSF models use 32 functions per atomic environment, while spatially matched ACSFs require 220 functions: 130 radial and 90 angular.
  • With similar descriptor sizes, the 32-function wACSF descriptor outperforms the 35-function minimal ACSF descriptor by more than a factor of three in MAE.
  • 220-function ACSFs reach performance comparable to wACSFs, but wACSF validation and test MAEs are more than 0.5 kcal/mol lower.
  • wACSFs share spatial functions across elements, allowing structurally similar motifs involving rare elements to contribute information across chemical species.The paper compares this behavior conceptually to weight sharing in neural networks.
  • Shorter wACSF descriptor vectors reduce computational cost, benefiting molecular dynamics simulations and high-throughput screening.

B. Comparison of Parametrization Strategies

Radial and angular wACSFs favor different Gaussian parametrization schemes. Shifted Gaussians improve radial resolution, whereas centered Gaussians preserve more useful overlap for angular descriptors.

  • Radial wACSFs: Shifted Gaussian functions outperform centered functions for radial wACSFs because they provide better radial resolution.
  • Radial wACSFs: Centered radial Gaussians overlap across broad regions, while shifted Gaussians localize signals to distinct shells around the central atom.
  • Angular wACSFs: Centered Gaussian functions offer a small but distinct advantage over shifted functions for angular wACSFs.
  • Angular wACSFs: For angular functions, products of highly localized shifted Gaussians can vanish easily, discarding spatial information; broader centered Gaussians retain more overlap.
  • General implication: The same centered-versus-shifted observations reported for wACSF parametrization also apply to ACSF-type functions.

C. Ratio of Radial to Angular Symmetry Functions

wACSF performance depends systematically on the balance between radial and angular functions. Radial functions provide strong spatial resolution but cannot distinguish angularly different environments, while too many angular functions reduce radial resolution.

  • 0:32 angular-only descriptors perform worst because centered Gaussian functions sacrifice radial resolution for angular-distribution information.
  • 32:0 radial wACSFs achieve relatively low MAEs because they provide refined spatial resolution for QM9 geometries.
  • Angular symmetry functions are crucial for distinguishing environments with identical central-neighbor distances but different relative angles.
  • The best descriptor ratio is 26:6, whereas performance decreases beyond 18:14 as lost radial resolution outweighs added angular information.
  • The optimal radial-to-angular ratio may differ outside QM9 because the database contains only equilibrium structures; molecular dynamics may benefit from more angular functions.

D. Genetic Algorithm Optimization of wACSF

The authors test whether a genetic algorithm can automate wACSF parameter selection for QM9 enthalpy HDNNPs. GA optimization substantially improves smaller networks when guided by HDNNP fitness, while larger networks gain less and LRR-based fitness is less reliable.

  • The genetic algorithm optimizes all wACSF parameters except the cutoff radius, starting from a 22:10 descriptor and using either LRR- or HDNNP-based fitness.
  • 3.11 kcal/mol to 2.19 kcal/mol: GA optimization lowers the NN10 test MAE when using an HDNNP-based fitness function.
  • 1.87 kcal/mol to 1.69 kcal/mol: NN10-50 models gain only 0.18 kcal/mol from optimizing the same descriptor.
  • GA-LRR improves its fitness measure from 5.67 kcal/mol to 5.18 kcal/mol but does not improve HDNNP accuracy and can worsen NN10 performance.
  • The GA makes only minor radial changes, including added resolution near C-H bond lengths around 1.09 Å, while angular parameters change more substantially.
  • The optimized angular functions retain a ratio close to 1:1, break the λ = ±1 parameter symmetry, and usually change ζ only from 1 to 2.

V. SUMMARY

wACSFs adapt ACSFs to avoid scaling problems with increasing chemical diversity while retaining accurate molecular-property predictions. On QM9, they achieved lower MAEs with far fewer symmetry functions, and genetic optimization helped mainly smaller neural networks.

  • wACSFs adapt conventional ACSFs to overcome shortcomings for systems containing moderate to large numbers of chemical elements.
  • 1.83 kcal/mol versus 7.40 kcal/mol was obtained for wACSFs and ACSFs, respectively, using descriptor vectors of similar length.The comparison used 133855 QM9 molecules and associated enthalpies.
  • 32 versus 220 symmetry functions were required in the chosen example, while wACSFs showed significantly better generalization at comparable spatial resolution.
  • Genetic optimization reduced the overall MAE from 3.11 to 2.19 kcal/mol for relatively small neural network potentials.
  • For a 10-50-NNP, the gain was negligible relative to the additional computational effort, changing the MAE from 1.87 to 1.69 kcal/mol.Larger networks can internally transform suboptimal descriptor vectors into more favorable representations.
  • Simple empirical parametrization schemes produced high-accuracy potentials without a tedious search procedure, although QM9 is an incomplete snapshot of chemical compound space.The authors suggest the observed trends may also hold for future studies of non-equilibrium structures and molecular forces.

SUPPORTING INFORMATION

Supporting analyses examine radial-to-angular function ratios and alternative genetic-algorithm descriptor optimization. Accuracy generally improves after adding some angular functions, while the benefit of optimization depends on model size and fitness function.

  • Adding a few angular functions generally improved accuracy, but performance deteriorated beyond a certain radial-to-angular ratio.The total number of symmetry functions remained fixed at 32, and angular-function counts changed in steps of two.
  • The alternative 18:14 descriptor omitted angular cross terms depending on r_jk and was compared with the original 22:10 wACSF descriptor.
  • The LRR-based fitness function produced no improvement for the small model, while the large model showed a gain similar to the HDNNP-based alternative.The authors conclude that LRR fitness should be used with care because its behavior depends on the descriptor being optimized.

Comparison of the 22:10 wACSFs pre and post GA optimization.

The supporting information compares the 22:10 wACSF descriptor before and after genetic-algorithm optimization across the H, N, O, and F atomic environments.

  • Figures S3 and S4 compare the radial and angular wACSF changes for H environments before and after optimization.
  • Figures S5 and S6 analyze the corresponding radial and angular descriptor changes for N environments.
  • Figures S7 and S8 compare the radial and angular descriptors for O environments before and after optimization.
  • Figures S9 and S10 analyze the optimized-versus-unoptimized wACSFs for F environments.
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