Accurate Interatomic Force Fields via Machine Learning with Covariant Kernels

Aldo Glielmo; Peter Sollich; Alessandro De Vita

Source-linked AI summary

Accurate Interatomic Force Fields via Machine Learning with Covariant Kernels

Aldo Glielmo, Peter Sollich, Alessandro De Vita

arXiv:1611.03877v2Nov 10, 2016physics.chem-phcond-mat.mtrl-sci

TL;DR

The paper addresses force learning as a vector-prediction problem requiring correct covariance under configuration transformations and invariance to database rotations. It develops vectorial Gaussian Processes with covariant kernels generated by group integration, and reports efficient learning and accurate force prediction across benchmark materials, with application-dependent trade-offs involving energy conservation.

Problem
Force-learning methods must predict vector forces with correct rotational behavior while remaining independent of arbitrary rotations applied to database configurations.
Method
The paper uses vectorial Gaussian Processes with matrix-valued covariant kernels constructed by integrating standard kernels over rotation and roto-reflection groups or summing over finite point groups.
Results
Covariant kernels improve learning efficiency and force prediction on nickel, silicon, and pure or defective iron systems, while quadratic kernels provide an effective covariant three-body field for covalent materials.
Takeaways & Limitations
Finite point-group covariance can provide the full error-convergence speed-up for systems with definite underlying symmetry, while full covariance is needed when no clear point symmetry exists.
Takeaways & Limitations
Kernels that do not conserve energy exactly should not replace conventional interatomic potentials for long NVE simulations because they may cause spurious energy drift.

Abstract

from arXiv · show

We present a novel scheme to accurately predict atomic forces as vector quantities, rather than sets of scalar components, by Gaussian Process (GP) Regression. This is based on matrix-valued kernel functions, on which we impose the requirements that the predicted force rotates with the target configuration and is independent of any rotations applied to the configuration database entries. We show that such covariant GP kernels can be obtained by integration over the elements of the rotation group SO(d) for the relevant dimensionality d. Remarkably, in specific cases the integration can be carried out analytically and yields a conservative force field that can be recast into a pair interaction form. Finally, we show that restricting the integration to a summation over the elements of a finite point group relevant to the target system is sufficient to recover an accurate GP. The accuracy of our kernels in predicting quantum-mechanical forces in real materials is investigated by tests on pure and defective Ni, Fe and Si crystalline systems.

I. INTRODUCTION

Growing computational resources and QM databases have made data-driven inference increasingly practical for atomistic modelling. The paper applies this opportunity to force learning using symmetry-aware vectorial Gaussian Processes.

Motivation: Large QM databases can support rapid prediction of properties for new configurations instead of repeating expensive QM calculations.The approach depends on databases being sufficiently large and representative, with controllable accuracy.
Prior work: Machine learning has been applied to atomisation energies, density functionals, Green’s functions, transport coefficients, potential energy surfaces, and free energy landscapes.
Prior work: Learning-on-the-fly methods combine inference with QM calculations when prediction is infeasible or insufficiently accurate.
Symmetry: Known transformation invariances can improve prediction, but exploiting non-invariant transformation properties has received less attention.
Contribution: The paper introduces vectorial Gaussian Processes with covariant matrix-valued kernels so predicted forces transform correctly under symmetries.The kernels exploit QM force databases together with prior knowledge of physical-system symmetries.

II. VECTORIAL GAUSSIAN PROCESS REGRESSION

The force is modeled as a vector-valued function of a local atomic environment using a Gaussian Process with a matrix-valued kernel. Bayesian conditioning on configuration–force data produces the posterior force predictor.

Force representation: The local environment ρ encodes neighboring atoms within a cutoff, with the target atom at the reference-frame origin.In the absence of long-range ionic interactions, nearsightedness supports this local mapping for finite-temperature systems.
Gaussian Process model: A vector-valued Gaussian Process assigns a multivariate Gaussian distribution to force values over finite sets of configurations.
Gaussian Process model: The matrix-valued kernel K(ρ, ρ′) represents correlations between vector forces at two configurations, while the mean function is usually set to zero.
Gaussian Process model: The kernel must be a positive semi-definite matrix function so the joint Gaussian model is valid.
Training and prediction: Bayesian conditioning on configuration–reference-force data yields a Gaussian-process posterior whose mean is the best estimate of the underlying force function.Gaussian observation noise with variance σ^2_n also regularizes the matrix inverse.
Symmetry requirements: Force learning must respect translation and same-species permutation invariance while treating rotations and reflections covariantly.

III. COVARIANT KERNELS

Covariant kernels enforce the required transformation behavior of predicted forces and make predictions independent of arbitrary transformations applied to database configurations. They can be generated by integrating a base kernel over a symmetry group, or by summing over a finite point group.

Required properties: The two target properties are force transformation with the target configuration and prediction invariance under independent transformations of database entries.
Covariant kernels: A covariant kernel automatically guarantees both symmetry properties through its defining transformation law.
Database invariance: Covariance makes transformed-database factors cancel in the GP prediction, leaving the prediction unchanged.
Covariant integration: Covariant integration transforms a standard matrix kernel into a covariant kernel by integrating over group elements with normalized Haar measure.
Kernel construction: The construction preserves positive semi-definiteness, while a diagonal base kernel with scalar rotational invariance reduces the double integration to a single one.
Practical implementation: The resulting kernel can predict forces for arbitrary orientations, while discrete symmetry sums are sufficient when relevant configurations cluster around known point-group symmetries.At most 48 elements are required for the full O48 group.

IV. COVARIANT KERNELS FROM 1 TO 3 DIMENSIONS

The paper builds symmetry-aware kernels from Gaussian atomic representations and illustrates their behavior across one-, two-, and three-dimensional systems. The one-dimensional example shows how covariance recovers a reaction force absent from one-sided training data.

Configuration representation: Configurations are represented as normalized sums of Gaussian functions centered on atoms, enforcing translation and same-species permutation invariance.
Base kernel: A linear base kernel is defined by the overlap integral of two configuration functions and interpreted as a scalar product.
Base kernel: The induced permutation-invariant distance can be inserted into a squared-exponential kernel.
1D example: The one-dimensional Lennard-Jones dimer figure compares an unsymmetrised C1 kernel with a covariant D1 kernel trained using data from one atom only.C1 fails to learn the symmetric reaction force, whereas D1 does learn it.
Dimensional examples: The one-, two-, and three-dimensional cases provide conceptual illustrations followed by benchmarking on realistic systems simulated at DFT accuracy.

A. 1D systems

In 1D Lennard-Jones systems, covariant kernels exploit reflection symmetry to learn symmetry-related force configurations and reduce the data needed for accurate prediction.

Symmetry-based learning: Covariant kernels learn forces in symmetry-related configurations by appropriately reflecting information from configurations present in the database.The base kernel predicts its prior mean in the symmetry-related region when training data cover only the other side.
Learning efficiency: The covariant kernel performance is comparable to that of the base kernel with double the number of data points for training.Figure 2 reports average relative force error versus training-set size for a periodic 50-atom chain sampled from molecular dynamics.
Learning efficiency: Symmetrising over a relevant finite group gives an error drop approximately equivalent to a group-order-fold increase in training points.This effect is reported for one-, two-, and three-dimensional systems.
Computational implication: Because GP training has computational complexity O(N^3), the effective data reduction can produce significant computer time savings.The savings follow from achieving comparable performance with fewer explicit training configurations.

B. 2D systems

In 2D systems, covariant kernels can be constructed through orthogonal-group symmetry operations and can yield analytic, conservative pairwise force fields; relevant finite groups recover much of the benefit.

Kernel construction: The O(2) group includes all rotations and reflections, enabling roto-reflection covariant kernels for two-dimensional configurations.The construction combines continuous rotations with reflection operations represented by 2 × 2 matrices.
Kernel construction: For the linear base kernel, the covariant integration can be evaluated analytically, producing a rotation-covariant kernel.The analytic expression contains a modified Bessel function of the first kind.
Physical interpretation: The resulting predicted force is a sum of pairwise forces along neighbour directions, with distance-dependent magnitudes determined by the training database.The associated integral can be interpreted as a pairwise energy potential, making the force field conservative.
Finite-group performance: For a 2D triangular lattice, larger symmetry groups generally accelerate learning, but only when the group captures a true underlying lattice symmetry.The incorrect C4 group learns much more slowly than C3, whereas the full D6 point group performs almost as well as continuous-symmetry kernels.
Finite-group performance: The full D6 point group performs almost as well as continuous symmetry kernels, supporting finite-group summation when analytic covariant integration is unavailable.This provides a viable alternative for accurate force prediction in crystalline systems.

C. 3D systems

In three-dimensional systems, covariant kernels encode rotational force behavior and can be evaluated analytically in important cases, yielding conservative pairwise force fields. Tests on nickel, iron, and silicon show that symmetry-aware kernels substantially improve force prediction, while higher-order kernels better capture complex interactions.

Covariant kernels: Analytical covariant integration over rotations produces a three-dimensional kernel whose predicted forces align with interatomic vectors, yielding a pairwise conservative force field.In three dimensions, rotations alone suffice because averaging over rotations around the connecting vector removes orthogonal force components.
Nickel: SO(3)-covariant nickel models reach force errors below 0.1 eV/Å with 10 training points at 500K and 80 at higher temperature.With N = 320, the errors are 0.0435(±0.0006) eV/Å and 0.095(±0.003) eV/Å, respectively.
Nickel: The nickel model achieves errors of 0.025 eV/Å and 0.052 eV/Å in the two tested cases, with the lower value comparing favorably to 0.09 eV/Å from an EAM potential.
Iron: For iron, squared exponential covariant kernels outperform linear kernels, especially near vacancies, reaching about 0.05 eV/Å in bulk and 0.075 eV/Å near the vacancy.The linear kernel is limited because it cannot reproduce all higher-order interactions in the reference quantum-mechanical force field.
Silicon: For crystalline silicon, the quadratic kernel eventually outperforms the linear kernel and the Stillinger–Weber potential at both 500K and 1000K.The errors are 0.05 eV/Å and 0.1 eV/Å, corresponding to approximately 4% and 6% of the mean force.

V. CONCLUSION

The paper presents covariant-kernel Gaussian Processes for learning quantum forces, showing that symmetry can improve force prediction while the best balance among accuracy, covariance, and energy conservation depends on the application.

Method: Covariant integration over the relevant roto-reflection group imposes force symmetry, and analytic O(d) kernels from a linear base kernel can generate conservative force fields.The method encodes prior symmetry knowledge in matrix-valued kernels and provides a general recipe for applying covariance to otherwise non-symmetric kernels.
Results: Symmetry incorporation consistently improved efficiency across tests in one, two, and three dimensions, including practical benchmarks on real systems.The three-dimensional tests used nickel, silicon, and iron systems, with pure and defective cases included among the materials tested.
Results: The recipes significantly improved force prediction over available classical potentials for nickel, silicon, and iron, while quadratic kernels produced effective O(3)-covariant three-body fields for covalent materials.The authors note that nonlinear kernels may be needed for complicated interactions such as plasticity, embrittlement, or fracture.
Accuracy and conservation: Force covariance can be achieved without exact energy conservation, and O48 covariance was effective for force prediction in crystalline iron with a squared-exponential kernel.The authors contrast this with the difficulty of analytically integrating the squared-exponential kernel over SO(3).
Scope boundary: Kernels that do not conserve energy exactly should not replace conventional interatomic potentials for long NVE simulations because they may produce spurious energy drift.The same kernels can remain suitable for NVT, LOTF, and dynamically updated force-model schemes.
Implementation: Exactly conservative covariant linear kernels support broad usage, while pairwise forms offer fast evaluation but may reduce accuracy for complex systems when transferability is required.The paper suggests locally optimized forces or potentials based on database subsets as one possible way to improve accuracy in different system regions.

Covariant integration

The covariant-integration derivation begins by separating radial and angular dependence because only the angular part changes under rotations.

Setup: The target integral is introduced before separating its radial and angular components.This separation exploits the fact that the radial component does not depend on rotations.
Setup: The radial part is rotation-independent, allowing the rotational calculation to focus on the angular contribution.

2D systems

The two-dimensional integration is simplified by aligning the relevant vectors and reducing the rotational average to integrals over angles whose symmetry eliminates most matrix elements.

Vector alignment: Rotation matrices are chosen to align the relevant vectors with the z-axis, simplifying the kernel integral through rotational invariance.The derivation inserts alignment rotations and changes variables so that the transformed vectors share a common axis.
One-angle reduction: In the alternative alignment step, the vectors are made parallel and the two-dimensional rotational average is parameterized by a single angle θ.
Analytic evaluation: The reduced angular integrals are evaluated analytically using the cosine and sine structure of the rotation matrix and the modified Bessel function I1.The odd sine contribution vanishes when integrated over an even domain.
Three-dimensional extension: In three dimensions, global rotational invariance of the base pairwise kernel is used to align one vector with the z-axis before integration.
Angular parametrization: Euler-angle parametrization shows that the exponential argument is invariant under rotations around the aligned z-axis.The integration uses the normalized Haar measure over the Euler angles.
Symmetry reduction: Most rotation-matrix elements vanish after angular integration because they contain sine or cosine factors integrated over a full period.The remaining contribution is associated with the matrix element aligned with the z-axis.